Description

Title of Invention

LIGHT STEERING AND FOCUSING BY DIELECTROPHORESIS Cross Reference to Related Applications

[001] This invention claims the benefit of U.S. patent applications: [1] U.S. 62/456,614 filed by patent attorney Christopher Peil for the inventor Leo D. DiDo- menico on 2017 February 08 and entitled, "Active Microfluidic Beam Steering," U.S. 62/456,614 is hereby incorporated in its entirety; [2] U.S. 62/460,080 filed by patent attorney Christopher Peil for the inventor inventor Leo D. DiDomenico on 2017 February 16 and entitled, "Active Microfluidic Beam Steering," U.S. 62/460,080 is hereby incorporated in its entirety.

Technical Field

[002] This invention relates to using electronically controlled opto-microfluid- ics using a dielectrophoresis process on nanoparticles in a liquid to redirect and generally control the propagation of light electronically and non- mechanically for beam-steering and focusing applications.

Background Art— Fluidic Light Control

[003] There are several fluidic methods discussed in the prior art for electronically redirecting light. Broadly speaking micro-optofluidic light guides and lenses are utilized for redirecting light and they are enabled by (1) changing the properties throughout a volume of a liquid (pressure, electric energy, orientation of suspended liquid crystals etc..) or (2) by changing the shape of a liquid's opti- cal surface by using surface effects. Somewhat more specifically, electrostatic and quasi electrostatic fields in a liquid can provide a means for these changes and can provide: a net force on the bulk volume of a liquid and/or change the contact angle of a liquid's surface with a contacting solid by electronically changing the surface energy of liquids and solids.

[004] In the case of substantially conductive liquids the surface shape is often changed electrically by means of an electrowetting precess, which requires that an insulator covers electrodes so that the electrodes are not shorted to ground through the liquid. A voltage across an electrode pair induces polarization charges in the insulators near the electrodes. These polarization charges interact with a polar liquid to change the liquid's surface energy and the wettability of the insulation contacting the liquid. This results in forces on a liquid in contact with 5 the insulator that can move the liquid. It can also change the shape of the surface of a drop of a liquid as a function of applied electrode voltage.

[005] However, in the case of substantially non-conductive liquids the surface shape is often changed electrically by means of dielectrophoresis, wherein the electrodes can now be in direct contact with a liquid that is surrounded by a gas

10 or vacuum. A non-uniform electric field can then induce the liquid to move or to change shape by means of Kelvin forces.

[006] The use of liquid drops in electrowetting and dielectrophoresis based light steering and focusing has disadvantages, including the need for: (1) small optics with limited light gathering ability, (2) multiple lenses to correct aberrations,

15 (3) a limited range of refractive index due to the relatively low refractive index of most liquids so that the refractive power is small and light steering not appreciable.

[007] There are numerous variations in the prior art for using the electrowetting approach including for example U.S. patent 7,898,740 entitled "Tunable Optical Array Device Comprising Liquid Cells" , issued to Jason Heikenfeld et. al.

20 on 2011 Mar. 1. This patent shows an array of liquid-filled cells comprising at least two transparent and non-miscible fluids, each having a different refractive index. Voltages applied to the cells induce polarization charges that change the wetting angle of the boundary between the two non-miscible fluids, by a process called electrowetting, and this causes the optical boundary to change orientation so that

25 light is refracted into different directions.

[008] There are also numerous variations of the dielectrophoresis based approach of light control, including that discussed in a 2013 review article in the Journal of Physics D: Applied Physics, by Su Xu, et. al, entitled "Dielectrophoretically tunable optofluidic devices" and having digital object identifier doi: 10.1088/0022-

30 3727/46/48/483001 J. Phys. D: Appl. Phys. 46 (2013) 483001 (14pp). The article describes using dielectrophoresis for manipulating a quantity of liquid so that the shape of the liquid changes to provide lensing or guiding of light.

[009] One particularly important point about the prior arts' use of dielectrophoresis for light control is that it is often the case that the focus of attention is a tiny liquid particle (e.g. a liquid drop), which is used in the dielectrophoresis process. The forces of the dielectrophoresis are directed on the liquid drops, which is immersed in a gas, another immiscible liquid or a vacuum medium and moved about by nonuniform electric fields.

[010] I ⁿ contradistinction, in this disclosure typically solid nanopar- ticles are immersed in a liquid medium as described in detail later.

[Oil] Alternately, in the prior-art the shape of liquid surfaces may be modified indirectly using soft-matter transparent elastomeric membranes, such as poly- dimethylsiloxane, in combination with actuators like piezoelectrics. Changing the fluid pressure causes liquid that is behind a membrane to push the membrane into a curved surface that mimics a lens shape that can subsequently redirect light.

[012] The use of membrane-based light steering and focusing has disadvantages, including: (1) slow response time, (2) limited controllability of the membrane shape and (3) the need for actuators that may be complex and take up additional volume.

[013] There are also numerous variations of a fluidic and membrane based approach of light control, including those discussed in the book "Introduction to Adaptive Lenses," by Hongwen Ren and Shin-Tson Wu, printed in 2012 by John Wiley & Sons, Inc., ISBN 978-1-118-01899-6, which provides a review of light control techniques and documents optical surfaces by using elastomeric membranes, piezoelectric actuators, electrowetting, dielectrophoretic optics comprising one more liquids, liquid crystals, polymer dispersed liquid crystal lenses.

[014] In yet another approach of the prior art two or more flowing liquids are used in guiding and/or focusing light. In particular, different flowing liquids that are moving relative to each other provide a refractive index gradient that is formed and controlled by the local geometry controlling the flow and speed of contacting liquids as a diffusion process works to mix some of one liquid having a low refractive index into another liquid having a higher refractive index. The gradient in the refractive index then provides an optical medium to modify the direction of light propagation.

[015] The use of flowing liquids has disadvantages, including the need for: (1) an expendable feedstock of each liquid, (2) a pump to move each liquid in a flow, (3) the need for significant pump power to overcome friction and momentum and (4) the slow response time of flowing liquid system. [016] There are also numerous variations of the liquid flow and induced refractive index gradient based approach of light control, including that discussed in an article published by the Optical Society of America by authors Richard S. Conroy, et. al., entitled "Optical waveguiding in suspensions of dielectric particles," in Vol. 44, No. 36 of Applies Optics, 2005 December 20. The article focuses on refractive index gradients formed by flowing liquids of differing refractive index.

[017] Another example of controlling light by flowing fluids can be found in the journal review article "Micro-optofluidic Lenses: A review," by Nam-Trung Nguyen and published by the American Institute of Physics in BioMicroFluidics 4, 031501 2010 and having digital object identifier doi: 10.1063/1.3460392. The article notes various techniques of how flowing fluids, including those that have nanoparticle suspensions can redirect light by forming a refractive index gradient.

[018] In yet another approach of the prior art liquid crystals, including polymer dispersed liquid crystals, focus and redirect light not necessarily based on the shape of an optical surface, but rather on the orientation of liquid crystals within an optical volume to provide the needed phase shifts to allow the focusing and steering of light.

[019] There are numerous variations on using liquid crystals for beam steering and focusing including: U.S. Patent 6,958,868 entitled "Motion-Free Tracking Solar Concentrator" , issued to John George Pender on 2005 Oct. 25, which shows liquid crystal filled prism arrays that can steer light. Another example is U.S. patent 8,311,372 entitled "Liquid Crystal Waveguide Having Refractive Shapes For Dynamically Controlling Light" , issued to Michael H. Anderson et. al. on 2012 Nov. 13. This patent shows how to use evanescent field coupling to liquid crystals to steer light. Yet another example is U.S. patent application 2012/0188467 entitled "Beam steering devices including stacked liquid crystal polarization gratings and related methods of operation" , issued to Michael J. Escuti et. al. on 2012 July 26. This patent shows stacks of polarization holograms (instead of the more common phase and amplitude holograms) and is used to steer light. The polariza- tion holograms are formed from electronically controlled liquid crystals with cyclic variations in the liquid crystal orientations.

[020] The use of liquid crystals has disadvantages, including the need for: (1) restrictions in the polarization of the incident and transmitted light and (2) the small difference in refractive index between the extreme values provided by the liquid crystal orientation, which limits the ability to redirect light over large angles.

[021] Therefore, it is clear that there are many methods for steering light and that these methods have an assortment of significant shortcomings. Consequently, there is a clear need for a method of controlling light that can overcome the shortcomings of the prior-art.

Summary of the Invention

Technical Problem

[022] The technical problem addressed in this patent disclosure is to answer the question of how to configure a nanoparticle colloid to electronically control light. This disclosure then provides derivative methods and device embodiments that allow electronic control of the propagation direction of light for beam steering, focusing, concentration and other optical operations.

The devices that are described herein typically have most of the following properties: low loss, broad spectral range, large angular steering range, high angu- lar steering precision, high angular steering accuracy, polarization independence, voltage controllability, low power consumption, intensity-independent operation, fast responding, thin profile, configurable to steer, focus and/or concentrate light. Light redirection of up to 2 radians in two dimensions and up to 4 steradians in three dimensions is possible. Moreover, with proper selection of solid materials, liquids and control algorithms the devices are capable of working over temperature ranges that exceed the typical Mil-spec temperature range of -55°C to +125°C.

In Fig. 1 several applications of the this disclosure are shown as part of a discussion on integration of the optical devices to the overall functional end-use platform and the need for compact and conformal optical devices.

In particular, Fig. 1A shows a smartphone camera lens in the upper left corner that focuses light electronically over a large range of environmental conditions while providing corrections to image aberrations that have historically taken large complex compound lens configurations but are now contained in a single lens element having graded refraction index optics to reduce the size, complexity and cost. The device is ideally conformal to the smartphone, compact and electronically controlled.

In Fig. IB a LiDAR system that is nearly conformal to a car's roof provides digital and non-mechanical beam steering of laser light over extreme environmental conditions. The beam steering is over an azimuth angle of 2 radians and has a modest amount of elevation scan angle to allow the car's LiDAR system to paint a full scene of the road and adjacent areas with pulsed laser energy to recreate an accurate representation of the space around the vehicle. Whether the conformal surface is the car's hull or the windshields there is a way to integrate the beam control systems discussed in this disclosure therein. Both single beam LiDAR and imaging LiDAR are compatible with the beam steering technology discussed herein.

In Fig. 1C a LiDAR system that is nearly conformal to a drone aircraft is shown as different angular regions of a landing area are searched for hazards. As can be seen, instead of the LiDAR beam steering constrained predominantly to a planar circle (as for the car) the beam steering is now over a hemisphere of 2 steradians beneath the drone. Systems with spherical coverage of 4 steradians are also possible. Here again the large beam steering angle is shown in the context of conformal or nearly conformal beam directors into the hull of the craft.

In Fig. ID a several light beams are shown for LiDAR and directed energy applications, which are directly integrated into the hull of a military stealth drone aircraft to allow stealth and aerodynamic performance to be maintained. The aircraft is shown in a head-on view from which the need to maintain the craft's smooth aerodynamic surfaces are clearly seen. Additionally, note that different portions of the hull of the aircraft are clearly being used to send out (and potentially receive) the beams of energy.

In Fig. IE shows a robot vision application that provides a combination of imaging and LiDAR to sense the environment for a robot.

These and many more applications often have the need for conformal optical elements that are capable of fully electronic beam steering and focusing of light.

Solution of the Problem

[023] Dielectrophoresis (DEP) is one of several phenomenon that may be used to steer light electronically. In particular, DEP provides pondermotive forces on particles in a liquid using particle interactions with a non-uniform electric field. This type of force does not require the particle to be charged or to even be a dielec- trie. In fact, all particles exhibit dielectrophoresis in the presence of nonuniform electric fields. However, a special case of particular interest is DEP utilizing a transparent dielectric driven by a harmonic non-uniform electric field, which can provide a (non-harmonic) average pondermotive force in one direction. This unidirectional pondermotive force is a result of forces on a particle not being symmetric during each full oscillation of the electric field. The resulting average force can produce a constant drift velocity because the friction of the viscous liquid on the DEP particle resists the unbounded acceleration of the particle and the velocity may quickly saturate to a constant. Also, the harmonic form of DEP avoids many parasitic phenomena, such as electrophorsis, electro-osmosis and electrolysis, which may make light control more difficult in practice.

[024] When we apply this pondermotive force to large numbers of nanoparticles in a colloid we can control the distribution of those nanoparticles and the resulting refractive index as a function of space and time due to impressed electrode voltages. Different size distributions of nanoparticles and different materials will separate differently in space and this allows for an optical element having many degrees of freedom so that various lens or beam steering aberrations are reduced or eliminated in one compact device instead of needing to use multiple lenses or other optical devices. For example, by having some nanoparticles with positive material dispersion and other nanoparticles with negative material dispersion we can better manage chromatic aberrations in a single lens element.

[025] A colloid is a homogeneous, noncrystalline substance comprising nanoparticles of one or more substances mixed and dispersed through a second substance. The nanoparticles do not settle or precipitate from a colloid and cannot be separated out by mechanical means like filtering or centrifuging the mixture. This is unlike nanoparticles in a suspension where such mechanical actions can separate the nanoparticles from the liquid. Additionally, a colloid is often made stable by means of adding surfactants and other chemicals into the liquid or onto the nanoparticles. In this document the nanoparticles of interest typically have diameters of 1-lOOnm, however this size range is not meant to be limiting to the use of other particle sizes in this disclosure.

[026] Consequently, as the size of the nanoparticles can be much smaller than the wavelength of light being steered, e.g. visible light is from about 390- 700nm, we can approximately average the colloidal refractive index by using the volume fraction of the nanoparticles in the liquid. The direction of light passing through the colloid of nanoparticles is then controlled by using strategically located electrode voltages at to induce nonuniform electric fields and change the nanopar- ticle spatial distribution within the liquid guides to redirect light. When used to steer or focus light this is called Dielectrophoresis Beam Steering (DBS). [027] However, nanoparticles that are substantially smaller than the wavelength of light cost more than larger nanoparticles and also have less surface area so that larger voltages are required to separate the nanoparticles and form a refractive index gradient. Therefore, the following disclosure also shows how to use larger particles by using a refractive index averaging process that includes diffraction effects of the light around the nanoparticles. This is accomplished by leveraging the theory of anomalous diffraction as applied to a colloid to properly describe the effective refractive index in a DEP driven colloid. It will also be seen that different probability distributions of nanoparticle size and material composition will interact in a complex way to further change the refractive index averaging process and that this can be leveraged to provide many additional degrees of freedom to control the redirection, concentration and focusing of light.

[028] In this disclosure a distribution of dielectric nanoparticles in a liquid forms a colloid that is called an Index Gradient Liquid (IGL). This is different than an Index Matching Liquid (IML), also synonymously called an Index Matching Fluid (IMF), which has been used by this author in other disclosures for soft-matter beam steering. Thus, the refractive index of an IGL is electronically controlled to achieve a refractive index gradient for steering and focusing light, typically within a micro-Fluidic Control Channel (//FCC), gap or non-solid volume in a hard and transparent host material. In contradistinction, an IML requires advection of the optical liquid to achieve beam steering. IMLs and IGL can have different operating parameters making one more or less desirable than the other for specific applications.

Brief Description of Drawings

[029] The foregoing discussion is only an introduction and other objects, features, aspects, and advantages will become apparent from the following detailed description and drawings of physical principles given by way of illustration. Note that figures are often drawn for improved clarity of the underlying physical principles and are not necessarily to scale and have certain idealizations introduced to show the essence of the method and embodiments and to make descriptions clear. FIG. 1A shows a smartphone using dielectrophoresis-based focusing for high performance imaging.

FIG. IB shows an automotive LiDAR with a dielectrophoresis-based beam steering system for a LiDAR system that is conformally mounted to a car.

FIG. 1C shows a drone LiDAR with a dielectrophoresis-based beam steering system for a collision avoidance system that is conformally mounted to a drone.

FIG. ID shows LiDAR and directed energy beam steering on a high-performance military drone.

FIG. IE shows a hybrid imaging LiDAR system conformally integrated into a robot for machine vision systems.

FIG. 2 shows a cross section of a fluid control channel having free and bound charges associated with a nano-scale dielectric sphere embedded in a dielectric liquid and energized by electrodes on the control channel.

FIG. 3A shows the real am imaginary parts of the Clausius-Mossotti factor for K ₀ = 1 and K _∞ =—0.5.

FIG. 3B shows the real am imaginary parts of the Clausius-Mossotti factor for K ₀ =— 1 and K _∞ = 0.5.

FIG. 3C shows the real am imaginary parts of the Clausius-Mossotti factor for K ₀ = 1 and K _∞ = 0.5.

FIG. 3D shows the real am imaginary parts of the Clausius-Mossotti factor for K ₀ =— 1 and K _∞ =—0.5.

FIG. 3E shows the real am imaginary parts of the Clausius-Mossotti factor for

FIG. 3F shows the real am imaginary parts of the Clausius-Mossotti factor for

FIG. 4 shows a modified Smith Chart that relates the complex dielectric constants of the colloidal liquid and nanoparticles to the complex Clausius-Mossotti factor parameterized by the frequency of harmonic excitation.

FIG. 5 An input plane wave is shown interacting with a single dielectric nanosphere to produce both a diffractive scattered wave and a diffractive transmissive diffractive that is phase shifted through the sphere. The annulus of area dA is in the plane of the screen. The line of length r is not to scale in this image and should be thought of as nearly parallel to the z axis so that the far-field point C is far away from the nanosphere's center. Line AB has length 2b sin r, which is associated with the phase lag of the input wave at angle r.

FIG. 6 A shows a prior art embodiment for redirecting light that is based on ad- vection of a refractive index matching liquid within a microfluidic control channel.

FIG. 6B shows the on- and off-state refractive indices as a function of radius for the device in Fig. 6A.

FIG. 7A shows one embodiment of dielectrophoresis based beam steering to form a graded refractive index sufficient to curve the path of the light until it is released for free-space propagation.

FIG. 7B shows the refractive index as a function of radius for the device in Fig. 7A.

FIG. 8A shows in cross section electrodes on the surfaces of a wedge shaped con- tainment vessel, where an Index Gradient Liquid is subject to a nonuniform electric field. The wedge is oriented for a positive radius sign convention.

FIG. 8B shows electrodes on the surfaces of a wedge shaped containment vessel, where an Index Gradient Liquid IGL is subject to a nonuniform electric field. The wedge is oriented for a negative radius sign convention.

FIG. 9 shows the cross section of electrodes and refractive index gradient liquid at steady-state conditions when the real part of the complex Clausius-Mossotti factor is greater than zero for a light beam propagating into the page of the figure.

FIG. 10 shows the cross section of electrodes and refractive index gradient liquid at steady-state conditions when the real part of the complex Clausius-Mossotti factor is greater than zero for a light beam propagating into the page of the figure.

FIG. 11 shows a cut-away perspective of an example dielectrophoresis beam steering device based on a wedge-shaped microfluidics channel and electrodes to induce a gradient refractive index over a portion of the circular device. A typical light beam trajectory is shown, including its release into free-space. FIG. 12 shows in perspective view a segmented set of electrodes to be used to control light steering in Fig. 11 , where the electrodes are arranged about the refractive index gradient liquid.

FIG. 13 shows a plot of the potential solution functions for steady state diffusion of nanoparticles under the influence of a nonuniform electric field from non- parallel flat electrodes forming the boundary of a wedge shaped containment region for an index gradient fluid. The solutions correspond to Fig. 9-12.

FIG. 14 shows a plot of the normalized nanoparticle refractive index vs the normalized radius for different volume fractions of the solid nanoparticles v _$ ranging from 10%-90% by 20% steps assuming that normalized radius of TZ = 50 and the nanosphere radius is b = lOOnm at a wavelength of λο = 532nm.

FIG. 15 shows an example of additional electrodes used to add elevation control the already existing azimuth control of beam steering form Fig. 9.

FIG. 16A shows circular beam steering based on dielectrophoresis to form a gradient refractive index medium to confine and redirect light.

FIG. 16B shows the gradient refractive index distribution for Fig. 16A as well as the uniform refractive index state.

FIG. 17 shows in cross section a distribution of nanoparticles in a micro-fluidic control channel that can provide graded refractive index confinement of a light beam similar to that provided by graded refractive index fiber optics, except that the refractive index is electronically controllable by a dielectrophoresis process.

FIG. 18A shows a cross section of a dielectrophoresis-based electronically controllable lens.

FIG. 18B shows the refractive index as a function of lens radius for a converging lens when the real part of the complex valued Clausius-Mossotti factor is greater than zero.

FIG. 18C shows the refractive index as a function of lens radius for a diverging lens when the real part of the complex valued Clausius-Mossotti factor is greater than zero.

FIG. 19 shows a perspective view of annular electrodes that are used to provide nonuniform electric fields for a dielectrophoresis process to variably focus light in an electronic lens.

FIG. 20 shows a perspective cut-away view of the integrated annular electrodes and containment vessel used in the formation of a dielectrophoresis based lens utilizing concentrations of nanoparticles.

FIG. 21A shows a polyphase voltage excitation that effectively provides an in- phase and quadrature electric field excitation and a traveling voltage wave.

FIG. 21B shows a polyphase and amplitude voltage excitation with arbitrary in- phase and quadrature electric field excitation.

FIG. 22A shows the real part of a traveling wave electric field above electrodes when there is an in-phase and quadrature filed components.

FIG. 22B shows the imaginary part of a traveling wave electric field above electrodes when there is an in-phase and quadrature filed components.

FIG. 23 shows an example amplitude variation and its derivative relative to elec- trodes.

FIG. 24A shows the real part of a traveling wave electric field above electrodes when there are in-phase and quadrature field components and an amplitude reversal along the electrodes.

FIG. 24B shows the imaginary part of a traveling wave electric field above elec- trodes when there are in-phase and quadrature field components and an amplitude reversal along the electrodes.

FIG. 25 shows in perspective view a light beam steering system that can redirect light into 2 radians of azimuth and over ± /4 radian of elevation by the use of a discrete electrode array.

FIG. 26 shows a perspective and cut-away view of a dielectrophoresis-based lens that utilizes traveling voltage waves on a circular and concentric array of electrodes.

FIG. 27A shows an example of a basic unstructured angular mode in the form of a spherical harmonic function.

FIG. 27B shows an example of a basic unstructured radial mode in the form of an Airy function, where the mode is only shown in the microfluidic control volume.

FIG. 28 shows a basic unstructured mode propagating around the symmetry axis on non-geodesic trajectories that are formed just above an inner solid dielectric sphere in the region of a microfluidic control volume.

FIG. 29 A shows an example of a structured angular mode in the form of a superposition of gaussian weighted spherical harmonics at φ = 0.

FIG. 29B shows an structured finite power radial mode in the form of an Airy function, where the mode is only contained in the microfluidic control volume.

FIG. 30 shows an example structured angular trajectory and intensity of a light beam's main lobe on a non-geodesic trajectory that is confined to constant latitudes.

FIG. 31 shows an example structured angular trajectory and intensity of a light beam's main lobe on a non-geodesic trajectory that is not confined to a simple constant latitude. Theory and Description of Embodiments

[030] The following pages provide substantial detail of the underlying theory and embodiments of DBS for both opto-electronic light steering and light focusing. The theory is provided so that there is no question as to exactly what it is that is being discussed in a complex technical area. The detailed technical discussion also allows insights into the limits of prior-art and ultimately allows those less familiar with the material to have easier access to the present material using one consistent and unified notation. This material is inherently multi-discipline in nature and draws from a number of diverse fields including, but not limited to: fluid flows, heat transfer, continuum mechanics, electromagentics, electronics, chemistry, material science, optics, thermodynamics and mathematics.

Dyadic Calculus for DEP Processes

[031] Consider 3-dimensional vector functions of space and time A and B, which are arbitrary and not necessarily field quantities from Maxwell's equations. The dyadic product of these three dimensional vectors is represented as AB and has nine components that may be written as a matrix having elements <¾ = AiB _j and is given formally using the Einstein summation convention as

AB = AiXi BiXj = ΑγΒγΧγΧγ + Λ V A3B3X3X3 , (1) where double subscripts indicate summation over the three spatial dimensions and in general AB φ BA. No distinction is made here between covariant and con- travariant vectors. In the following analysis all derivations are provided using cartesian dyadics, however the resulting equations are correct across any self-consistent system of coordinates.

[032] We shall use the Levi-Civita tensor to make curl and other operations easy to compute. In particular, the Levi-Civita tensor takes the values

+ 1 if (*,j,£;) is (l,2,3), (2,3,1), or (3,l,2)

e _ilk = { -\ if (i _{7 j7} k) is (3,2,1), (1,3,2), or (2,1,3) (2) 0 if i = j, j = k, or k = i and we can show by direct computation that

^ijk^klm ^kij^klm <¾¾m &im jl (3) where Sij is a Kronecker Delta function. The Levi-Civita tensor is used to write the cross product of A and B as

A x B = XkekijAiBj (4) and the curl of A as

V x A = XkekijdiAj , (5) where as usual we have applied the Einstein summation convention.

[033] The first result of importance is the divergence of a dyadic product. It may be expanded as follows

V · (AB) = (x _k dk) ^■ (xiAiXjBj)

= d _k (AiBj) S _ik Xj

= [BjidiAij + Xj (6) = (xjBj)(diXi) ^■ (AiXi) + (AiXi) ^■ (diXi)(xjBj) (7) = B(V -A) + A-(VB) . (8)

V ^■ (AB) = B(V ^■ A) + A-VB (9) where in the future we may not always write obvious unit factors like (¾ · ¾) in intermediate results. Also note in the above equation we have used the fact that

(A-V)B = A - (VB) = A-VB . (10)

[034] Next, the cross product of the curl of a vector is developed. It may be expanded as x (VxB) = Xi i _jk A _j diB _rn

= Xi(A _j diB _j — B _j d _j Ai) . (11) Similarly, its compliment is

B x (V x A) = XiiB _j diA _j - A _j d _j Bi) . (12)

Moreover,

(A-V)-B = A _j d _j BiXi (13) (B ^■ V)A = B _j d _j AiXi (14)

(15) so that

A x (V x B) + B x (V x A) + (A- V)B + (B ^■ V)A = + B _j diA _j )

= XidiiA _j B _j )

= V(A-B) (16)

A x (V x B) + B x (V x A) + (A ^■ V)B + (B ^■ V)A = V(A · B) (17)

A special case of the above equation that is useful for analysis of Quasi Electro Static (QES) systems and DEP in particular specifies irrotational fields so that VxA = Vx-B = 0. Therefore, if we set A = B in Eq.17 and assume irrotational fields then

A. A = ( ) =V. ( /) , (18) where the last equality in the above equation was obtained by observing that for an arbitrary scalar U = A ² /2 we can write the equation using the 3x3 identity matrix I. In particular,

V-t/ = V-(UI)

= x _k d _k ^■ (UxiX _j Si _j )

= d _k S _ki Uxi

= diUxi

= U . (19) so that

V · U = V · (UI) = VU when U is a scalar. (20)

[035] Next, by using Eq.9 and its compliment, which is formed by swapping B for A and vice versa, we can solve separately for (B ^■ V)A and (B ^■ V)

(A-V)B = V · (AB)— B(V ^■ A) (21) (B ^■ V)A = V · (BA) - A(V · B) (22) and now substitute these equations back into Eq.17 so that

V(A-B) = Ax (V x B) + B x (V x A)

+ {V-(AB) - (V-A)B}

+ {V-(B4)-(V-B)A} . (23)

A special case for QES systems is where A and B are irrotational so that VxA = V x B = 0, then Eq.23 can be written as

V · (AB + BA) = V(A · B) + A(V · B) + B(V ^■ A) (24)

By setting A = B in the above equation we obtain

V · (AA) = A(V · A) + V ( j (25) where A ² = A ^■ A. Now using Eq.18 in Eq.25 we obtain

V · (AA) = V (^j + V ( ) = V(A ² ) (26) so that

V · (AA) = V (A ² ) = V · {A ² 1) when A is only irrotational. (27) Additionally, from Eq.25 and 27 we also have that

A(V-A) =v(^ ) . (28) We summarize Eqs.18 and 28 as

A(V · A) = A - VA = V ^ ^" TT^ ⁼ V ' ( ^~ 2 ^~~ ^) ^wnen ^ ^{lii on} ^J irrotational.

(29)

Notice that if A is solenoidal in Eq.25 so that V · A = 0 then we have from Eq.25 that

V · (AA) = V (— J when A is irrotational and solenoidal. (30) Again, notice how different Eq.30 is from Eq.27.

[036] The next relation that is needed is derived by using Eq.2 and 3 so that the divergence of the ordered difference is

V · (AB - BA) = diiAiB _j - B^x _j

⁼ (fiilfijm ^~ 8i _m Sji)di(AiB _m )Xj

= -Xjtjikdi [A x B] _k

= — V x (Ax B)

= V x (B xA) . (31)

V · (AB - BA) = V x (B x A) (32) This result can be combined with Eq.9 so that

V x (B x A) = V · (AB) - V · (BA)

= {(V-A)B + (A-V)B)}

-{(V-B)A + (B-V)A)} , (33) which in the special case when A and B are solenoidal, i.e. with V· A = V B = 0, then Eq.33 reduces the above equation to

V x (B x A) = A ^■ VB - B ^■ VA when A & B are solenoidal. (34) Maxwell's Equations & Notation

[037] We start with the space-time version of Maxwell's equations for macroscopic systems in SI units and the constitutive relations for a homogeneous, isotropic and linear material. In particular, Maxwell's equations in SI units are

V x 5(r, _t ) = (35)

d >(r t)

V x¾(r,t) = +— J ^. + J _f (r,t) (36)

V-2?(r,t) = p _f (r,t) (37)

V-B(r,t) = 0 (38) and the constitutive relations

T>(r,t) = e(r,t)£(r,t) (39) B(r,t) = μ(ν,ήΗ(ν,ί) (40) J _f (r,t) = a(r,t)£(r,t) . (41)

The electromagnetic quantities above include the electric field intensity £, the electric field density X>, the magnetic field intensity Ή, the magnetic field density B, the free electric current density ^ f, the free electric charge density pf, the permittivity e, the permeability μ and the conductivity σ. If we assume that quantities are harmonic then we can decompose the space and time functions of fields and charges into functional products in phasor space and material properties into functions of space and frequency, which are to be calculated or assumed as the situation requires, so that in the frequency domain we have

V(r,t) → D(r)e ^iwt (42)

£(r,t) → E{r) ^lwt (43)

B(r,t) → B(r)e ^lwt (44)

U(r,t) → H(r)e ^{t lt} (45) p _f (r,t) → p _f (r)e^ (46) a(r,t) →· σ(ν,ω) (47) e(r,t) → e(r, ) (48)

/i(r,t) → ^■ x(r,w) , (49) where quantities like -E(r) are in general complex valued quantities unless noted otherwise and % = Alternately, we can write that £{r,t) = Re [E{r)e ^lujt ]. Either way we can rewrite Maxwell's equations as

V x E{r) = -ίωμ{τ,ω)Η{τ) (50)

V H(r) = +iue(r,u)E(r) (51)

V· {i(r,j)E(r)} = 0 (52)

V · {fr(r, )H(r)} = 0 (53) where the third and fourth equations above follows from taking the divergence of the first and second equations and using the vector identity V · (V x A) = 0, where A is an arbitrary vector over space coordinates. Furthermore, we find that on using the constitutive relation ^~ f = OfS that e(r, ) = e(r, ) -i ^A (54) = {e _R (r, ) -i _I {r, )}-i ^a ^ ^r ^ ⁾ (55)

= e _R (r, ) -i e _I (r, ) + ^af{ ^ ^{r )} ^ (56)

μ(ν,ω) = μ(ν,ω) (58)

The above equations are provided to establish notation and sign conventions used in the DEP analysis that follows.

Time Averaged Maxwell's Stress Tensor

[038] In addition to Maxwell's equations Eqs. 35-38 and the constitutive equations Eqs.39-41 there is the Lorentz force equation, which for a single charge is given by

F = q£ + qv x B (59) where q is the charge and v is the velocity vector of the charge in an inertial frame of reference. If we multiply both sides by N, which is the concentration of charged particles in units of number of charges per unit volume. Note that N is different than the concentration of nanoparticles per unite volume u. Therefore, then we can identify = Nq as the free charge density, = Nqv as the free vector current density and the force per unit volume as J-y = NJ-. Therefore, the force per unit volume, i.e. the force density, is v = p _f £ + J x B . (60) Also, because the speed of light squared is c ² = l/(ex) the Poynting vector divided by the speed of light squared in media is

S £ x U £ x B

- _j = 5— = = T> x B . (61) c ^z c ^z μ

Therefore,

Combining results by inserting Maxwell's equations Eq. 35-38 into the volume forces Eq.61 and further exploiting Eq.62 we obtain

T _Y = p _f £ + x B

= (V · V) S + (V x U) x B - x B

= (ν-τ>)£ + (νχη)χΒ-

= ( -V)£ + ( x U) x B- ^- ( ] -V x ( x £) (63)

ot ^c /

and on rearranging terms and strategically adding a zero term using V · B = 0 then v = -^- ( ] +£ (V · V) - T> x (V x £)+Ui · B) - B x (V U) (.64) ot \c ^z J ·> _v ' _v '

Term-1 (Ti) for Electric Field Term-2 (T ₂ ) for Magnetic Field

Therefore, focusing initially on term-1 we find

^j^i^i (¾i¾m im jl) ^j^l^m

= SjdiVi— VjdiSj + VjdjEi

= e (—£jdi£j + £jdi£i + SjdjSj) Reordered and constant e = e —£,<¾£, ^■ + £jdi£i + ¾<¾£,) With i ¾ j in 3rd term

--E _k E _k bij + Because <¾(<¾<¾) = 2<¾<¾<¾ e<¾ (——£ ² 6ij + ¾£ _j ] By Einstein Summation Convention

2

— )· V · C |£ \ ^' I + e££ ) Where i ^" is the identity matrix. (65) e

-V (-|£| ² j +V- ( _e ££) (66) Where the first term in the last line is minus the gradient of the electric field energy density and the second term is the divergence of the dyadic product of field quantities.

[039] Similarly, for T ₂ we can let >— B, £— Ή and e— y l/μ so that

T ₂ = ui ■ B) - & x (v x u)

→ V· ( -— \B\ ² I + -BB ) (67)

2μ μ J

Therefore, the Maxwell stress tensor, comprising field-induced mechanical stress matrix elements, is

^T ={-\^ ^{I + t££} ) ⁺ {- ^ ^{I + BB} ) ^■ < ⁶⁸ >

The force per unit volume is given as

^- ⁼ -!(!)+ ^ν'τ · ⁽⁶⁹⁾ which has a first term due to radiation pressure and a second term due to field induced stresses.

[040] For qusi electrostatic DEP systems the Maxwell stress tensor needs to be evaluated for the time average volume force, accordingly if we assume harmonic electric and magnetic fields with phase shifts then we may separate space and time. For example, for electric fields we can write

£(r,t) = Re[E(r)e ^ijt ] = E(r) cos(wt) (70) V(r,t) = Re[D(r)e ^ijt ] = D(r)cos(jt- φ _ε ) (71) where φ _Ε = φ _ε (ν), E(r) and D(r) are real valued functions of position r. We can then write the time average Maxwell stress tensor as

= 0 + V- [{T _E ) + {TB)] ■ (72)

Let us assume that electrodes are introduced such that the electric field intensity has an in-phase and a quadrature component so that by design

£ = E _R cos( t) + Eis ( t) (73) > = D _R cos(wt - φ _ε ) + Όι sin(wt - φ _ε ) (74) where E _R is not parallel to E and the phase shift φ _Ε is due to the material properties. Then we consider the dyadic product £ = D _R E _R cos{ t— φ _ε ) cos(wt) + DREI cos{ t— φ _ε ) s ( t)

+DIER s (ut— φ _ε ) cos(wt) + DIEI sin(wt— φ _ε ) sin(wt) .(75)

The time average can be obtained by first using the trigonometric identities for arbitrary angles a and b

2 cos a cos b = cos(a + b) + cos(a— b) (76)

2 cos a sin b = sin(a + b)— sin(a— b) (77)

2 sin a cos b = sin(a + b) + sin(a— b) (78)

2 sin a sin b = — cos(a + b) + cos(a— b) (79) so that the time average of >S over an integer number of cycles is formed. For example,

1 cos φ _ε cos( t— φ _ε ) cos(wt) =— [cos(2wt— φ _ε ) + cos φ _ε ] ^— -> —— - (80) and applying this method to each term in Eq.75 we obtain the time average dyadic product as

1

CDS) = - [D _R E _R cos φ _€ + ORE sin φ _€ — O E _R sin _e + OJEJ cos _e ]

1

= - Re [(cos + % sin _e ) (£> _Λ + iD _r ) (E _R - iE^}

1

= - Re [|e| (cos φ _ε + i sin ,)(£¾ + iEj)(E _R - iE _r )]

= ^ Re [(¾ + ¾)(¾ + ¾£,)(£¾ -¾£,)]

= -Re\Ei ^* E ^* }

2 ^{[ J}

= - ^l e[ED ^* ] (81) where the asterisk indicates complex conjugate and Re[] is the real part operator. Also note that the complex permittivity is

^~ = _R -i _! . (82)

Additionally, we also find using the same procedure that for an inner product the time average is

CD ^■ £) = i Re [E ^■ D ^* ] . (83) Therefore, the time average Maxwell stress tensor is

1

(TE) (τ> ε)ΐ + τ>ε

1

Re E■ E *)I + e *EE *

2

E _R + iE _T )■ (E _R - iE _T ))I + (½ + iei) (E _R + iEi) {E _R

-CRE ² ! + CR (ERER + EIEI) + i (EREI - EIER) (84)

However, the electric component of the average pondermotive force is

(FE) = V · (T _E ) (85) and we immediately see a need to reduce the expressions for V · (EREI— EIER) and V · {ERER + EJEJ) . We shall assume that we are in the quasi electro static regime where both the real and imaginary parts of the electric field intensity are irrotational so from Eq. 32 we find that

V · [E _R Ej - E _T E _R ] = V x (E _T X E R (86) and from Eq. 27 with irrotational fields V x ER = V x Ej = 0 for quasi electrostatics we have

V · [E _R E _R + EJEJ] = V {E _R ² + E ² ) = V · {E ² l) (87) so, the quasi-electrostatic force, i.e. the Kelvin force density, is obtained from Eq. 84 as

^' DEP ) (ER ^■ ER + E · E ) V x (e _T Ei x E R (88)

This is the Kelvin force density applied to a spherical nanoparticle of radius b and volume

V = -7Γ& ³ . (89) 3 ^v

[041] The value of e = £R— i i associated with Eq. 88 is now derived heuris- tically by considering Fig. 2, where a schematic of a spherical dielectric nanoparticle within a liquid medium is shown between electrodes. In particular, regions Ωο and Ω ₃ comprise metallic electrodes 2a and 2b respectively. A dielectric liquid 2c fills region Ωχ and a spherical dielectric nanoparticle 2d is in region Ω2. The boundary 2e between regions Ω ₀ and Ω _χ is ω _χ . The spherical boundary 2f between Ω _χ and Ω2 is u)2 - The boundary 2g between regions Ωχ and Ω3 is ω . The materials are to be replaced by bound and free charges in the regions and on the boundaries between the regions. The angle Θ, which is used to describe the charge distributions, is also shown in Fig. 2. Finally, Fig. 2 also shows the volume polarization of atoms with ovals containing positive and negative charges; the figure also shows the surface charges at each boundary. The surface charges include free charges from conducting electrodes and bound charges on dielectric boundaries.

[042] We seek to find an expression for the local field in region Ω2 so that we can find an equation for the polarizability «5 that relates the electric field intensity in the liquid EL to the polarization of the dielectric sphere. We start by considering the total local electric field in region Ω ₂ as the sum of the electric field intensity in the liquid plus the field due to the bound volume polarization and the field from the bound surface polarization on surface ω ₂ . This results in a Lorentz relation between macroscopic effective fields {EL and Ei _oc ) and polarization fields (E ₂ and Ε _Ω2 ) , so that by inspection of field directions in Fig. 2

where EL depends on the voltages applied to the electrodes, EQ ₂ also depends on the specific spatial configuration (i.e. the lattice structure) of the atoms and molecules making up the spherical nanoparticle and Ε _ω2 also depends on the net charge at the spherical boundary. Once found Ei _oc allows us a way to find the polarizability «5 of the nanoparticle so that we can find the effective (average) polarization of the nanoparticle by

OisEi _oc (91)

This can be related to the potential of an average dipole moment. As we shall see this then provides the effective dielectric constant, which is the objective of our calculation.

[043] To find the three field components of the Lorentz relation we need to find the various electric charges in the system first. In general, because the divergence of the macroscopic polarization is the volume charge density V · P = Pb we have from Gauss's law V · E = (pf + Pb) that

P = D - e ₀ E (92) and the bound-surface-charge density per unit area is (93) Also, by integrating V · D = pf over volume and using Gauss's integral theorem we obtain the free-surface-charge boundary condition a _f = (D ₂ - Dj - n (94) where n is the normal unit vector pointing out of medium- 1 into medium-2. At each of the boundaries we can write expressions for the bound and free charge densities.

[044] For boundary i the free, bound and total charge densities are a _fl = (D ₁ - D ₀ ) ^■ £ = e ₀ i _RL E _L (95) a _bl = P _L · £ = -e ₀ (e _rL - l)E _L (96) ti = Vfi + σ ₆ ι = e ₀ E _L . (97) where the complex relative permittivity was used to account for losses and phase shifts between the application of a harmonic electric field intensity and an induced polarization. Similarly, for the boundary ω we have σ _/3 = (D ₃ - D ₂ ) ^■ z = -e ₀ i _RL E _L (98) a _b3 = P _L ^■ £ = e ₀ (e _rL - 1)E _L (99) σ _ί3 = σ _/3 + a _b3 = -e ₀ E _L , (100) which shows that the total surface charge densities on the electrodes (i.e. a _tl and σ _ί3 ) are of equal and opposite sign. It also shows that one component of the total electric field intensity within the dielectric is E^ . However, this is not the only component because of the bound charges induced on the sphere. Therefore, the total local field must be EL plus a electric field component due to the sphere.

[045] In particular, for the spherical boundary ω ₂ the charge densities are e ₀ e _r E

-eo (erL - 1) -EL cos

+eo (erS - l)-¾ cos

σ/2 + ° ^" 62L + ^<J b2S

0 - e ₀ (e _rL - l)E cos ( + e ₀ (e _r5 - l)/¾ cos #

-e ₀ (e _r L - l)E _L cos Θ + e ₀ (e _r5 - 1) cos O [Using Eq. 101]

(<¾? - <¾)<¾-¾ cos fl

(105) We can use the net charge distribution a _t 2 on the surface ω<ι to obtain the total electric field at the origin of the nanosphere by integration from the source on the spherical surface to the observation point at the center of the sphere, i.e. in the (— r) direction so that

j _? _ f °¾2 j-f) dA

(<¾ - £ _T L)E _L Z [ ^Π 2 .

cos Θ sm θ αθ

=0

Because we have taken the sum to ensure that we have the total electric field at the center of the sphere of radius b, we can identify that , _{2 (Sphere) =} d¾^£ ₌ ¾ (w _{( 107)} so that

¾ ₂ (Liquid) = ^ ( ∞uum) = ^ ¾ ₂ (Sphere) (108)

So that in the liquid we can finally write that relative to the local background medium of a liquid that

¾ ₂ (Liquid) = — . (109)

6 _r L

[046] Next, we consider the bound polarization charges within the nanopar- ticle. We will assume that the atoms are on a cubic grid— this is only a convenient assumption that can in general be relaxed. That is, the result is approximately correct for other geometries, including amorphous media like glass, but the calculation can become very complex in these other cases. In particular, the total electric field intensity is given by summing over source charges ¾· at positions Vj so that

¹ - Qj (r - j

Ώ ₂

(110) where p = qd and d is the displacement of the charges in the direction of the z-axis.

Setting p = pz and r _m = x _m x + y _m y + z _m z we obtain

But for a cubic lattice we are summing an antisymmetric function over a symmetric domain for the x and y components, which sums to zero for these two components. For the z-com onent we recognize that we can write = x _m ² + y^ + and

therefore for a cubic lattice and as a good approximation for many other practical materials we have that

E _Q2 = 0 . (113)

[047] From the above calculations we find that the Lorentz relation can be evaluated for the local electric field intensity with the spherical nanoparticle removed and the fields referenced to the liquid. The result is

^{E~R5 + 2E L} E _L z . (114)

3e _rL

Therefore, with a concentration of u particles per unit volume and a polarizability of ,s then using Eqs. 109 and 114

(e _r s - <¾) £ _r s + 2e _rL \

P = eo = ua _s —— E _L (115) so that

~ - (116) There is one particle with a volume of V = § & ³ , where b is the particle radius, so that the sphere's polarizability is as = nb ^s ₀ ( " ^~rS ) . (117)

The effective polarization density of a collection of nanoparticles with concentration per unit volume u is then ps = ua _s Re[i _rL ]E _L = 3e ₀ Re[e _rL ] ( " ^{rS ~} ' ^rL £ _L = 3Re[e _L ] ^{¾ ~ ¾} ) E _L

(118) nd

(119)

Note the real part of the is used to ensure consistency with the classical expression for electric dipole potential as a function of radius and angle— e.g. "Classical Electrodynamics 2nd Ed." by J. D. Jackson.

[048] Note that the above derivation is only exemplary and additional layers may be added to the dielectric nanoparticle to provide other functions for the complex dielectric constant that are consistent with multilayer spherical geometry.

[049] For example, to ensure a stable colloid a surfactant may be added as a coating to the nanoparticle's outer layer. This would require that another layer of total bound charges at the boundary between the inner dielectric and the outer coating layer needing to be included into the analysis and a new expression for e is developed.

[050] Additionally, note that from Eq. 56 and assuming for convenience the imaginary part of the dielectric constant equal to zero we have

£5 es - i— (120) ω

e _L - i— (121) ω

so that

¾ - ¾

K (122) ¾ + 2e _L

K _R + iKi (123) which is the complex Clausius Mossotti (CM) factor for the solid sphere embedded in a dielectric liquid. Additionally, after some algebra we can isolate the real and imaginary parts of the CM factor so that

where

(126) 2a _L

es - £L

(127)

£5 - f 2e _L

es ^■ + 2e _L

T ^~ MW (128)

^ 2a _L Note that KR and K can take on both positive and negative values. As we shall see this allows forces and drift velocities on nanoparticles to take on both positive or negative values depending on the frequency ω of the harmonic excitation of the voltages on electrodes. In the case of e _$ > ¾ we have that K _∞ > 0. Therefore, to make KQ < 0 we would choose the liquid, which may be a mixture of several chemicals, to have > σ _$ . This will provide a means to focus, defocus and collimate light using DEPS optics as needed for a particular application simply by changing the frequency and voltage amplitude on the electrodes, as is demonstrated later in specific embodiments. Figures 3A-F show log-linear plots of the real and imaginary parts of the CM factor over frequency for specific values of KQ and K _∞ .

[051] Equation 122 is a modified bilinear transformation on the complex plane. Taking z = ¾/¾ we can rewrite Eq. 122 as

1 + 2K

(129)

K

On setting z = ZR + izj and K = KR + iKj we can with some algebra obtain the from the real and imaginary parts of the resulting equation a set of parametric equations

2

2z _R

K R Kf = , , . _, ⁹ _ολ? (131)

2(2 + ¾) J ¹ 4(¾ + 2) ²

which are orthonormal circles forming a modified Smith Chart that relates the dielectric constants of the liquid and the nanosphere to the CM factor. As it turns out the force and drift velocity of a nanoparticle is proportional to the CM factor as we shall see. Therefore, the modified Smith chart, which is shown in Fig. 4, helps to convert between force (or drift velocity) and the material parameters.

[052] The Smith chart in Fig. 4 shows a number of trajectories parameterized by the harmonic excitation frequency ω of the driving electrodes. The trajectories correspond to Figs. 3A-F. For example, the trajectory formed by the semi-circle from Smith Chart point 4a to Smith Chart point 4b corresponds to Fig. 3D. The trajectory formed by the semi-circle from Smith Chart point 4a to Smith Chart point 4d corresponds to Fig. 3B and displays both positive and negative values for the real part of the complex CM factor. Each of the Smith Chart points 4a, 4b, 4d and 4e correspond to either KQ or K _∞ . When the Smith Chart trajectory includes the origin Smith Chart point 4c then there will be both positive and negative values to the real part of the CM equation, which provides both attractive and repulsive forces on nanoparticles by switching the excitation frequency on electrodes. Additionally, at the top of each semi-circular trajectory there is a zero force point, for example at zero-force point 4f, where there are no DEP forces on a nanoparticle.

Time Averaged DEP Forces

[053] A nanoparticle that is subject to a DEP based pondermotive force will also feel the effects of its viscous fluid environment. In particular, Newton's second law at steady-state conditions becomes

d ² r

m ^{s ~} d^ ⁼ ^ ^DEP ^ ^r ^ ^{~ F∑)ra} 9

≡ 0 (132) where, the well known Stokes Law for spherical particle drag in a fluid was used to develop the second term and η is the dynamic viscosity of the fluid surrounding the nanoparticle. Clearly, the ratio of the average force to the steady-state drift velocity is given by the intrinsic mobility factor

The DEP drift velocity of nanoparticles through a fluid is obtained from Eq. 88 by

x ((3Kj Re [¾] )_£/ _/ x J¾

= 7 { V (^ ^ - + V x ((¾r¾Re» x E _R ) }

= 7 {ν (7ι£ · £) + V x (7 ₂ # _/ x ER) } . (134)

Therefore, in summary v _DE p{r) = 7 {V ( _ι Ε - E) + V x (7 ₂ ^ _/ x E _R ) } (135) where we define the relative mobility factors as

₇₁ = nb ³ K _R Re[i _L ] (136)

7 ₂ = 2nb ³ K _I Re[i _L ] . (137) Clearly the average force can be written in terms of a conservative potential energy — U, i.e. a stress tensor, so that the net flux of particles through the liquid under equilibrium conditions (i.e. not a function of time) is given by the drift current j _dr if _t ir) = u(r)v _drift (r) = -u(r) \7 ^■ U(r) ( 138) where u(r) is the particle concentration per unit volume [number particles per unit volume] and the diffusion of the particles due to thermal agitation Brownian motion is given by

jdiffusion(r) = -D(r)V ^■ u(r) , ( 139) where u(r) = Iu(r) and u is the particle concentration, i.e. the number particles per unit volume, and D is the to-be-determined diffusion coefficient of our analysis. Under equilibrium conditions the net vector flux of particle is zero so that j drift + J 'diffusion = 0. Therefore, j drift + jdif fusion = -u(r)^V ^■ U(r) - D(r)V ^■ u(r) = 0 . ( 140)

[054] Observe that by the definition of the gradient if W is the energy then VW ^■ dl = dW and VM · dl = du, where dl is the length element in the direction of the shortest distance between iso- value surfaces so that the gradient is a maximum. Therefore,

dW du

( 141)

VW l Vu - l

and

and

(V · (ul)) ^■ I = (V · iWI)) ^■ I ( 143) or on introducing additional terms that have an inner product with I that is zero we find that

/,-, \ i du

V · W + V · (V x F) +V · (GH - HG) ^■ I ( 144)

≡o v

where the second term on the right sets the energy level to an arbitrary value and the third term has the property that the commutator of the outer products (GH— HG) is always a skew-symmetric matrix wherein V · (GH— HG) = V x (G x H). For this to occur we can imagine that the complex electric field intensity E is broken into an in-phase and a quadrature component that are directed along the x- and y-axis respectively and have no z-directed variations. In this way

[V · {GH - HG)] l = [V · {Ε _Η Ε _τ - EJER)] ^■ I

= [V x (E _R x E _! )] - l

= [x{d _y Q _z - d _z Qy) - y{d _x Q _z - d _z Q _x ) + z(d _x Q _y - d _y Q _x )} ^■ I

= [x(d _y (0) - (0)Q _y ) - y(d _x (0) - (0)Q _X ) + z(d _x Q _y - d _y Q _x )} · I

= [z(d _x Q _y - d _y Q _x )} ^■ I

o z■ I

≡ 0 (145) where for example we might take

E _R + iE _r

g(x, y)x + ih(x, y)y (146) where functions g and h are real valued functions of the coordinates x and y only. Also, note that ER and E are really not restricted to just the xy-plane. For example in cylindrical coordinates we might take a tangent plane to a unit cylinder at each polar azimuth angle as the plane wherein the in-phase and quadrature components of the electric field intensity are defined. This will have important consequences for beam steering from arbitrary formed hulls, e.g. the hull an aircraft having complex form for proper aerodynamic properties.

[055] Consequently, we can write the equations

du

V · u V · [W + V] (147) dW

u

and

du

V · u V - U (148) dW

However, in statistical mechanics the Maxwell Boltzmann statistics are used to describe the average distribution of non-interacting material particles over allowed energy states in thermal equilibrium. This distribution is applicable when the temperature is high enough or the particle density is low enough to render quantum effects negligible. Using this observation we write u(r) = Ae (149) where A is a constant related to the total number of particles in a unit volume and /¾ is Boltzmann's constant. Consequently,

du u

(150) dW knT

Therefore,

Jdrift J dif fusion 0

du

wy + D V - U 0 (151) dW

and

hence

D = ₇ £¾T (153) which is an extension of the well known Einstein Diffusion Coefficient for the case where the Maxwell stress tensor has non-zero off-diagonal elements.

[056] The quasi-electrostatic approximation for the time averaged force associated with DEP assumes that the amplitude of the electric field intensity E is invariant in space. This can be relaxed if time variations are slow compared to the period of the harmonic excitation r = 2π/ω. Therefore, if t ^> τ then approximately E = E(r, t) = E^(r, t) + iEj(r, t) so that the DEP particle flux is jde _P (r, t) = u(r, t)v _drift (r, t)

= u(r, t) {V (71 -E ^■ E) + X7 x ( 2-E/ x -¾)} (154) By way of comparison, for the diffusion process the particle flux is jdif (r, t) = -DVu{r, t) , (155) for advection

jadv(r, t) = u(r)v(r) (156) and for electrophoresis (EP) for positive/negative charged particles jep(r, t) = ±η(τ, ί)μ(τ, ϊ)Ε . (157) The total particle flux is then

M

j ( ^r , *) =∑ Jm(r, t) = jdep(r, t) + j _di f (r, t) + · · · (158) m=l

and by continuity we can write

du(r, t)

-V - j (r, t) + Q(r, t) (159) dt

where Q(r, t) is a volume source in units of particle per unit volume.

[057] For the special case where only DEP and diffusion are the dominant phenomena, i.e. gravity is ignored when surfactant coatings are used on nanopar- ticles to maintain a stable and separated colloid in a gravity field. We then obtain a total particle flux of j {r, t) = _M (r, i) ₇ {V ( i£ · E) + V x ( ₂ -Ej x E _R ) } - D u{r, t) (160)

Therefore, when the volume source is zero Q(r, t) = 0 then

= -V · [ ¾(r , t) V (7 ₁ E - E)} - V · [η ^η {τ, t) V x {η ₂ Ε _Ι x E _R )] + D V ² u(r , t) dt

(161) and on assuming for the moment that 7 _{1 ?} ηι and 7 are constant, which is an assumption that we can relax later as needed, and not functions of space. Therefore, writing u = u(r, t) and using the definition of the diffusion coefficient we obtain the master equation for the nanoparticle spatial distribution

— = - _77l V · - 77 ₂ V · [uV x (Ej x E _R )} +≠ _B T ² u . (162)

Summary of DEP-Based Nanoparticle Concentration

[058] The following equation assumes that only DEP and diffusion process are at play where a harmonic voltage excitation is provided to electrodes that are in the vicinity of a colloidal suspension of solid nano-particles in a liquid, both of which are optically transparent but typically not of the same refractive index as a function of wavelength. In particular, [059] - 772V · [MV X (Ei x E _R )] + D V ² u (163)

u(r, t) = Nano-particle concentration in liquid colloidal suspension L ^~3 ]

E(r, t) = Amplitude of the complex Electric Field Intensity [LM ^-3 / ^-1 ]

ER _, (r, t) = Real part of the amplitude of complex Electric Field Intensity [LM ^-3 / ^-1 ]

Ei(r, t) = Imaginary part of the amplitude of complex Electric Field Intensity [LM ^-3 / ^-1 ] r = Position vector [L]

t = Time when t 3> (2π/ω) to allow for quasi-electrostatic approximation [T] ω = Radian frequency of the harmonic excitation of electrodes [ ^-1 ]

7 = Intrinsic Mobility [M ^~l T]

1 η = Dynamic viscosity [L ^{- 1} MT ^{- 1} ]

b = Spherical nanoparticle radius [L]

V = Nanoparticle volume [L ³ ]

-

3 > TT& ³

71

72 = Relative mobility in rotating fields [M ^-1 T ⁴ / ² ]

= 2.6 ³ _J fi _/ e _L

KR = Real part of the Clausius-Mossotti equation [Unitless]

K + [ ^O - ^oo 1

°° [ l + (UJTMW) ² _

Kj = Imaginary part of the Clausius-Mossotti equation [Unitless]

(K _∞ - K ₀ )

1 + {UJTMW ) ²

£5 - ^£ L

T ^~ MW = Maxwell- Wagner time constant [T]

¾ + 2e _L ,s = Permittivity of solid nanoparticles [L ^~ M ^-1 T ⁴ / ² ] [060] ei, = Permittivity of liquid medium [L M ¹ T ⁴ I ² ]

= e ₀ e;

e _s = Relative Permittivity of solid nanoparticle [Unitless] i = Relative Permittivity of liquid medium [Unitless] e ₀ = Permittivity of Free Space [L ^~ ^ Μ ^{~ λ} Τ I ² }

= 8.85418782 x 1(Γ ¹² m ^"3 kg ^_1 s ⁴ A ²

a = Conductivity of solid nanoparticle [M ^{_ 1} L ^_ T ³ / ² ] a _L = Conductivity of liquid medium [M ^_1 L ^_3 T ³ / ² ]

D = Einstein diffusion coefficient [L ² T ^_1 ]

= lk _B T

T = Absolute temperature [Θ]

k _B = Boltzmann constant [L ² ΜΤ^Θ ^{' 1} }

= 1.38064852 x 10 ^"23 m ² kg s ^"2 K ^{" 1}

Volume Averaged Refractive Index

[061] The objective of this section is to show that DEP can be used to create refractive index gradients that allow a light beam to be steered. In particular, a colloid comprising transparent sub-wavelength-scale particles in a transparent liquid host medium provides a volume averaged refractive index. By changing the local volume fraction of nanoparticles within a colloid the point-to-point refractive index may be changed dynamically in time.

[062] If we assume that scattering processes have no impact on the average refractive index then the volume fractions for the liquid L and solid z/ _$ components of a colloid must sum to unity so that

V + vs = 1, (164) and we can derive a relation for the average refractive index. This is based on our intuition of how the refractive averaging process works and provides an average real refractive index of n _A = n _L u _L + n _s u _s

¾ (1 - u _s ) + n _s u _s

n _L + (n _s - n _L )u _s (165)

The volume fraction of the solid nanoparticles i/g has no units and is a pure real number between zero and one such that i/g is obtained by multiplying the particle concentration by the volume per nanoparticle so that u _s ≡ uV (166) whereby it is possible to provide a direct linear relation between the nanoparticle concentration u and the average refractive index as

This is a direct linear relation between nanoparticle concentration we found in the previous sections and the average refractive index of the colloidal suspension.

[063] Strictly speaking Eq. 167 is only correct when the radius of the particle approaches zero, i.e. when 6 —^ 0 and scattering processes are essentially zero. As the particle size approaches zero the required voltage for changing the spatial distribution of nanoparticles goes to infinity. So we are motivated to obtain a more realistic averaging equation between TIL and ng so that voltages are in a practical range for engineering devices.

[064] Additionally, note that when the particle size is much smaller than the free-space wavelength of light, typically b < λο/20, then we have Rayleigh scattering as a light beam passes through the colloid comprising the dispersed nanoparticles. As the particle size increases then typically we would use the full Mie scattering theory for nanoparticles of comparable size to the wavelength of light to describe the scattering process.

[065] For the purposes of this discussion the scattering can be broken down into roughly two components. The first component is scattering far from the direction of wave propagation. For example, perpendicular to the direction of propagation of the incident light. This type of scattering is highly dependent on the size of the particle relative to the wavelength of the light. The second component is light scattering that is roughly in the direction of the incident light, which tends to be approximately independent of particle diameter. See for example "Light Scattering by Small Particles," by H. C. Van de Hulst having ISBN 0-486-64228-3 and "Refraction by spherical particles in the intermediate scattering region," by G. H. 5 Meeten, 1996, Optics Communications 134 (1997) 233-240.

[066] When the nanoparticle is made of a material that is transparent, but of a refractive index η$ that is different than the surrounding liquid medium TIL then it can be shown that particle size has a very weak dependence on the scattering process and we can simultaneously account for both diffraction and particle induced

10 phase shifts. This means that even for particles that are large compared to the wavelength that an expression for the scattering cross-section close to the direction of propagation is roughly independent of the type of scattering theory that is used to derive it. That said, we are going to use the theory of Anomalous Diffraction (AD) to obtain analytic expressions for the scattering process and then use that

15 information to develop an expression for the average refractive index that is similar to Eq. 165, but modified to account for scattering processes of practical larger nanoparticles.

[067] For DEP-based beam steering the colloid or suspension comprises a liquid and nanoparticles having two operational frequencies. The first frequency

20 is that associated with DEP control of the nanoparticle distribution in the liquid.

This frequency is roughly on the order of 10 ¹ to 10 ⁶ Hz and at this frequency there are nonzero loss tangents so that there is a real and imaginary component to the dielectric constants.

[068] In contradistinction, the optical frequency of light is approximately

25 10 ¹⁴ -10 ¹⁵ Hz, where by convention we tend to specify free-space wavelength rather than frequency (i.e. roughly 1000-100 nm respectively). We will assume that the materials are lossless at this higher optical frequency. This is not a requirement, but it makes the calculations simpler and the interpretation clear.

[069] So, for the sake of simplifying the description to its important points,

30 in this document we assume a nonzero loss tangent only for the low frequency effects associated with DEP. In actual devices there will always be some losses at all frequencies.

[070] As we shall come to see, the scattering of light by many lossless colloidal nanoparticles removes light from the direction of propagation. Thus, we will be able to introduce an effective complex refractive index to model the loss process. However, it must be stressed that the loss is a result of scattering by diffraction and not ohmic heating of the liquid or nanoparticles.

[071] Consider an incident light field propagating in the z-direction given by its electric field phasor

where we have used a scalar to represent the incident electric field strength EQ instead of a vector quantity because spherical nanoparticles do not have a preferred orientation and we can therefore ignore the light's polarization in this special case. In the far field the scattered electric field intensity will take the form

ikr

where S(0) is the normalized amplitude function of the scattering particle, r is the distance from the center of the nanoparticle to the far-field point (x, y, z) and Θ is the angular direction of scattering relative to the direction of incidence, i.e. the z-direction.

[072] The total field E of the incident and scattered field is then the super- position

^—ik(r—z)

E = E ₀ e -ikz

S(0) - (170) ikr

but, in the far- field where z is large relative to x and y, we can use a Taylor expansion approximation so that

(a? + + f) ¹ ' ² z + 7Γ ( ^χ2 + y ² ) (171) and in the direction of incidence Θ = 0 therefore

E = E- 1 + S(0) (172) ikr where = E§e ^~%kz . This process can be extended for many identical spherical nanoparticles having a concentration of u [number of particles per m ³ ] in a thin infinite planar slab of thickness z = I according to

g

E = E- ∑ S(0) (173) ikr, [073] However, z ^> Λ Χ ² + y ² = ζ and the distance to the observation point is r _a « r so that for large numbers of particles observed from the far-field, after the light passes through the layer of nano particles, we have in the limit that

E = E- e ^{2r <} * u dz p άζ άφ (174)

Moreover, the integral in ζ is eas to evaluate by observing that

where a = ikj (2r). So on substituting this result we find the expression reduces to

E = E- ulS(0) - 7T «iS(0) (176) k ²

where in general we have S(0) is a complex quantity.

[074] Let n = n'— in" represent the effective complex refractive index of the colloidal suspension of nanoparticles that includes diffractive scattering losses from a large number of otherwise lossless dielectric nanaospheres at the optical free-space wavelength λο· That is we say that in the direction of Θ = 0 we have losses and a non-zero imaginary component of the effective refractive index. Then the complex phase shift is

The effective complex refractive index of the colloid is then defined by writing so that

uS(0) = i ( (179)

where S'(0) = e[S(0)] and S"(0) = l [S(0)] so that on equating real and imaginary components on the left and right hand sides of the above equation we find a relation between the effective refractive index and the scattering amplitude as

n" = n _L ^uS'(0) . (182) Moreover, we can use the fact that the volume fraction of the nanoparticles is s = uV and the individual nanoparticle volume is V = § & ³ so that

and therefore

3 u _{s c} „ _n

nL + n _L -—S (0)

2 Y ⁶

3 us

n n _L -~^S (0) (185) where the nanoparticle size parameter as χ = kb was utilized. Then by using Eq. 182 the magnitude of the Poynting vector is

\ E \ ²

2%

^i

e-°^ ^ul , (186) 2% ^{K }} where C _ex t is defined to be the extinction cross-sectional area and

4

- ^

= ^S'(0) , (187) X ²

where G = irb ² is the geometric cross sectional area of a spherical nanoparticle. We further define the real part of the extinction efficiency in the direction of the incident light as

and the imaginary part of the extinction efficiency is

C, ext

Qeext -S"

G

Therefore, using Eqs. 184-185 n' = n _L + n _L ^S"(0)

2 X ^s

^ 3 u _s AS"(0)

= n _L + n _L - —

8 X X ²

= n _L + n _L u _s -^ (190)

8 X and similarly

So that we have developed the components of the complex effective refractive index in terms of the extinction scattering efficiency, the liquid's refractive index and the volume fraction of the spherical nanoparticles.

[076] Next, consider a plane wave incident on a spherical particle as shown in Fig. 5, where we see a single dielectric nanosphere 5a having incident light field 5b. The light field is assumed to be a plane wave propagating in the positive z- direction. The input plane wave interacts with a single dielectric nanosphere and produces both a diffractive scattered wave and a transmissive wave with a phase shift through the sphere. The differential annulus 5c of area dA is in the plane of the screen. The distance from the annulus to a far-field point 5d (also identified as point C in this figure) is r, note the image is not to scale so that this line should be nearly parallel to the z axis. Line AB has length 2b sin r, which is associated with the phase lag of the input wave at angle r.

[077] The diffraction process is accounted for by using Huygen's principle, which states that each point of a wavefront can be considered to be the origin and source of wavelets that radiate spherically. For example a plane wave

E ₁ = E ₀ e- ^ikz (192) can have an array of area elements, each having area dA across its surface, which serve as the sources for Huygen's wavelets. In this case we expect that in the far-field that

g— ikr

dE ₂ = q E ₀ dA (193) r

where the proportionality constant (not charge) q is a to-be-determined and

1 C ²

r « z +— (x ² + y ² ) = z + (194) so that

where the last integral can be evaluated using Eq. 175 and the fact that r f¾ z in the far-field. On setting the result equal to the original plane wave, as it must be in free space propagation, we have

_El = ^ ^E ^- ^lkZ _{≡ Ei e} -^ . _{( 196)}

i

Therefore, the proportionality constant is q = i/X and dE = E ₀ - ^t ^—dA . (197)

rX

[078] This is the disturbance caused by a wavefront having area dA and amplitude EQ at a point having a distance r from the area element, where r is in a direction that is approximately in the direction of propagation of the incident wavefront. It is clear that when dA = rX that there is no focusing or defocusing of the wave. Therefore, the width of the source must be approximately w = yfrX. The implication is that it is impossible to have rays traced from a particle having a width (diameter) of λ or smaller.

[079] By way of example: the light scattering that gives rise to the apparent white color of a white paint is produced by large (1-10 micron sized) spherical micro-spheres of transparent titanium dioxide that are included in the paint by the manufacturer to scatter the light and provide the appearance of a white paint. However, the white color will typically not be present for small diameter nanospheres (e.g. <50nm) , which are sized significantly less than the wavelength of visible light. This idea may be generalized to any wavelength of interest and it is very important because we do not want the addition of nanoparticles to cloud and substantially scatter light from the IGL beam steering liquid.

[080] With this analysis and example in mind we now return to Fig. 5, wherein a dielectric nanosphere whose bulk material is substantially transparent is scattering a plane wave traveling in the z-direction. We can now apply Babinet's principle, which is a theorem from optics that states that the diffraction pattern from an opaque body is identical to that from a hole, i.e. formed by the spherical nanosphere, in an opaque screen 5e. The hole is of the same size and shape as the particle cross section. Moreover, for complementary screens comprising the hole and the infinite area around the hole, the sum of the wave diffracted around a finite opaque screen plus the wave diffracted through the complementary hole, is the same as if no screen were present. Therefore, Babinet's principle is

E p. article E hole E, incident (198)

[081] This theorem can be modified slightly to account for a transparent dielectric particle that introduces optical phase shifts into the incident field. In particular, if from each area element of the hole we subtract off the complex ampli- tude of the incident wave and then add back in the complex phase-modified field we will have accounted for the phase shifts of the transparent bulk medium of the nanoparticle. In this way we can approximately account for the phase lag induced by the nanosphere of a refractive index different than its surrounding medium. Therefore, Babinet's principle in the far-field becomes

[082]

ie -kir

E ₀ -ίψ

'p„article l) dA = E ₀ e —ikz

(199)

Hole ^Τλ

where from Fig. 5 the phase lag is

(2b sin r) (η _$ n _L )

p sin r (200) and

Clearly, p is the phase lag directly through the central diameter of the nanosphere. Moreover, from Eq. 199 we have

'particle Ene- EQ l dA

where the identification of S(0) was obtained by comparison to Eq. 172. Moreover, from Fig. 5 we can see that the annular area element is a function of r so that the scattering amplitude in the direction of the incident wave is

k ² Γ

S(0) = — (1 - _e - ^ipsinr ) dA

;i-e- ^¾psin ) (27r6cosr)(6sin (203)

Let s = sinr then the complex scatterin amplitude is

S(0)

-ip _e -

X (204) ip (ip) ²

From Eqs. 188-189 we get the complex extinction efficiency as

-ip _p -vp

ip

Next, plug in the expression for Q" _xt into Eq. 190 and simplifying using Eq. 201 we find

smp cosp

n _L + _s (n _s -n _L )3 (206) p- ^J p*

Thus, on setting TIA = n' for the average real refractive index we get

^' sinp cosp"

n _A =n _L + v _s (n _s -n _L )3 p3 p2 (207) where the phase lag through the diameter of a nanosphere is

and where λο is the free-space wavelength of the light. These equations should be compared with the intuitively derived expression of Eq.165, where we note that in the limit of small particles that the correction factor

sin p cos p\

lim 3 —- ^y = 1 209 so that Eq. 165 and Eq. 207 are identical. Notice also that no attempt to further developed the expression for n" has been undertaken because the scattering losses are more strongly regulated by scattering processes roughly perpendicular to the direction of the incident light and are strongly size dependent. This requires the use of an appropriately chosen scattering formalism (e.g. Mie scattering) to properly account for the loss processes. Finally, the equation for the refractive index Eq. 207 can be connected to Eq. 163 by noting that s(r, t) = u(r, t)V (210) where V = § & ³ and we consider the volume fraction a function of space and time under the control of voltages from electrodes. Then

f ^sm P cos p\

n _A (r, t) = n _L + v _s {r, t) {n _s - n _L ) 3 I— —j . (211)

[083] By way of a numeric example, if b = 25nm, (η _$ — UL) = 0.5 and λο = 500nm then p = /ΙΟ and the correction factor would be about 0.9902, so Eq. 165 is off by about 1%. However, the needed voltages for a 25nm radius nanoparticle are high and will require high DEP voltages to control and potentially longer time periods to stabilize at the steady state. Therefore, if b = lOOnm we find that the correction factor is 0.8507 and represents nearly a 15% error that does need to be taken into account.

[084] In summary, light passing through a colloid or suspension will encounter liquid and nanospheres that are made of a material that in bulk is trans- parent. If at the optical frequencies the bulk materials are lossless then there will still be an extinction process that is governed by scattering of light from the nanoparticles. While scatting in directions substantially different than the direction of incidence is a strong function of nanoparticle size, this is not the case for light scattered into a narrow range of angles about the incident direction where the dependence on nanoparticle size is much weaker. In this case we can avoid using the exact Mie scattering theory or specific scattering approximations that are optimized for particle size relative the wavelength, e.g. the Rayleigh scattering approximation for particles small relative to wavelength, and use a scattering approximation called the anomalous diffraction approximation, which provides a compact analytic expression for the average refractive index of the mixture of liquid and nanoparticles.

[085] The Anomalous diffraction approximation, was originally developed by Dutch astronomer van de Hulst describing light scattering for optically soft spheres. It also goes by the name of the van de Hulst approximation, eikonal approximation, high energy approximation or soft particle approximation and it allows us to replace the lossless refractive indices of the liquid and nanoparticles with a complex, and therefore lossy dielectric constant n = n' '—in" of a homogenous material. The essence of the approximation was that n could be related to the scattering amplitude in the forward direction S(0) as the loss mechanism instead of a ohmic loss. The value of S(0) being derived from using a modified version of Babinet's principle to account for both diffraction and phase lags introduced in the incident wave from the transparent sphere. The technique can be generalized to account for particles that are not spherical and for intrinsically lossy liquids and nanoparticles, though the mathematics is more involved.

[086] In practice, obtaining spherical nanoparticles of only one size b is often difficult or expensive. Typically there is a distribution of sizes that are available. With that in mind we can generalize Eq. 207 to

« = ¾ + (% - ¾) 3 ' ( ²¹² )

where the total volume fraction of the solid nanospheres is z/$ , which is divided into M distinct different sphere radii b _m . The different radii then provide different phase diameter phase lags

Pm {n _s - n _L ) = b _m , (213)

where a is a convenient constant to be used shortly. Moreover, the sum over the individual number of nanoparticles in each size is

N _T = N ₁ + N ₂ + N ₃ + - - - + N _M (214) where N _T is the total number of nanoparticles. Therefore,

(215) which can be rewritten in terms of the probability mass function f _m

Therefore, the volume fraction of the ^th sphere size is the ratio given by the volume of the nanospheres to the total volume as

where Vo is the maximum unit cell volume that allows one nanoparticle to fit therein and VoNr is the total volume of a colloid filled with nanoparticles. By using Eq. 217 and taking the limit of many nanoparticles we can move to a continuum of nanoparticle sizes so that n _A = n _L + {n _s - n _L ) — ⁴⁷ / f°° f _b ( ,b) 1.3

b ⁶ ( ^sm' P ^cos P

Vo J Jo V P ³ P

where f _b (b) is the probability density function (PDF) of the random variable b. If we take

p = ab (219) where

4

(220) n _A = n _L + (n _s - n _L ) (^) ft ρ» ( - ) dp . (221)

Thus with different particle size distributions we will get different correction factors.

[087] For example, if we assume that nanospheres are fabricated with a uniform distribution of sizes, perhaps by using a nano-sieve during fabrication then

then Eq. 221 can be integrated directly and provides that

, 4 b ^: _nv ( 2 2 cos p sin rA

n _A = n _L + (,,, -„ _L ) { -^ ) - - -J! - - L j , (223) where for uniform probability distributions

P = ^^ (n _s - n _L ) . (224) This can be written in several ways, however let's observe that the average particle volume is

then for uniform distributions of nanospheres having radii ranging from 0 to b _max we find

. . . ( 2 2 cos p sin p

n _A = n _L + (n _s - n _L ) ( _s ) 12 ( - -— \ . (226) where the average volume fraction of nanospheres in the colloid or suspension is

= - (227)

Also, note that in the limit of small particles that

2 cos p sinp

lim 12 — (22 p→0 p ⁴

so that we can see that very small particles with a range of sizes do not have a great impact on the overall performance if the particles are small. The impact of using such small particles is the need for high DEP voltages and perhaps high nanoparticle cost, which may not be desired.

[088] Conversely, by moving to other probability distribution functions for particle sizes, and potentially a range of refractive indices, we can gain added degrees of freedom for forming optical elements having different layers and mitigating optical aberrations. The resulting average refractive index will take the form n _A = n _L + δ ^η (ZAJI ) F _t (p) + δ ^η ₂ (i/ _S2 ) F ₂ {p) H h δ ^η _Μ (v _SM ) F _M {p) (229) where Sri _j = ris _j — nL , the refractive index of the ^th material, the correction factors are F _j (p) and (vs _j ) ^are t ^ne average volume fractions for the ^th material used such that the total volume fraction is vs = ( si) + {vs2) + ^{■ ■ ■} + {VSM ) ^■ (230)

These different forms of averaging the refractive index provide the basis for correcting chromatic, spherical and other forms of aberrations in an optical system.

DEP-Based Light Steering Using In-Phase Excitation

[089] Having built up the needed physics in the previous sections we will next consider beam steering along a planar circle using an in-phase harmonic excitation comprising only cos(wt) harmonic excitation. This is to be contrasted with the development in a subsequent section where we will consider beam steering using both in-phase and quadrature harmonic excitation, i.e. where cos(wt) and sin(wt) electric field components are both present in non-collinear directions.

[090] Additionally, the focus on circular beam steering for achieving large angular extents is not to be construed to be restrictive to other beam steering embodiments that are not only along a planar circle. This is because a general arbitrary path in three dimensional space may be described by radius of curvature of an osculating circle in an osculating plane, which is tangent to each point along the arbitrary path in three dimensional space. Said another way, if beam steering oc- curs along a circle then it is also possible for beam steering along an arbitrary path without loss of generality as long as an appropriate /FCC is provided. Nonetheless, beam steering using circular beam steering devices has practical embodiments and exact theoretical formulations and it can be used directly for many diverse applications. In short, the circular beam steering technique can be applied to more complex situations if desired.

[091] As an introduction to DEP-based beam steering it will prove instructive to first consider the prior art of the current author as provided in Fig. 6A, wherein a circular /FCC 6a is cut into a transparent dielectric 6b block. The channel is filled with a vacuum except for a small region that contains a small quantity of IML 6c, which is positioned at the location shown by means of advec- tion induced by electronics— e.g. electrowetting or dielectrophoresis of fluid drops. The electronics to accomplish advection is not shown to keep the figure uncluttered and general, but in general requires electrodes along the /FCC. Two rays are input 6d (source not shown) and are directed in a counterclockwise propagation direction 6e around the inside boundary of the /FCC guide and the resulting output rays 6f are shown to diverge on as external output rays 6g outside the transparent dielectric.

[092] Two possible states of the refractive index as "seen" by the laser beam are shown in Fig. 6B, which represents the refractive index towards the direction arrows 6h and 6i. The inside radius of the circular /FCC is (ξο— δξ) , the outside radius is (£o + 5£) and the center is at radius ξ ₀ . As can be seen the refractive index in the /FCC varies between the vacuum refractive index 6j and UQ , which is the solid medium refractive index 6k.

[093] The extent of the total angular divergence of the output rays in Fig. 6 may be shown by analytic ra tracing to be

when ξ ₀ ^ δζ where ξ ₀ is the radius of curvature and d is the diameter of the collimated input light source . When ξο is large or d is small the divergence can be made to approach zero. Another approach is to approximate the circle by straight/FCC segments (that have ξο =∞) so that θτ = 0.

[094] To overcome the problem of light divergence and the need for liquid advection we consider Fig. 7A, where a continuous (non-segmented) /FCC 7a is located. Input rays 7b are launched within the /FCC in a counterclockwise propagation direction 7c— source not shown.

[095] An IGL 7d completely fills the /FCC and it is controlled electronically to form a refractive index gradient on demand. The idea is to form a gradient to steer the light without divergence of the rays until just the correct position is encountered by the rays, at which point we electronically spoil the Graded Refractive INdex (GRIN) in the gradient spoiled region 7e and remove the gradient and launch the beam out of the /FCC into output rays 7f . In this case the channel is filled completely with liquid that never flows, but responds to electric fields passing through it to form a GRIN channel. The electrodes and composition of the liquids used to form the electric fields are not shown in Fig. 7A, but are described in more detail later in this disclosure. Ultimately the output rays leave the solid transpar- ent material 7g as external output rays 7h. The shape of the solid transparent material may be in any convenient shape, but it is shown as a square (a cube in three dimensions) in Fig. 7A.

[096] Next, we develop the required refractive index gradient profile as seen looking towards the direction arrows 7i and 7j so that circular propagation is maintained. The required linearized refractive index 7k gradient is then shown in context in Fig. 7B.

[097] In particular, from Fig. 7 at a circular radius ξ the refractive index is n and at a radius ξ + δξ the refractive index is n + δη and at radius ξ— δξ the refractive index is n— δη. Then the optical path length at the smaller radius is OPLi = θξη, where Θ is the angular arc that one side of the laser beam traverses in Fig. 7A. The other side of the laser beam must then accrue an optical path length of OPL ₂ = θ(ξ + δξ) (η + δη) . For non-diverging trajectories we would need OPLi = OPL ₂ so after rejecting second-order differentials and taking the limit to infinitesimals we find that

(233) ξ

where n _C i _r is the refractive index needed for a circular trajectory of the beam and the constant of integration C was determined by assuming that at ξ = ξ ₀ the refractive index is n = UQ . Next we take the Taylor expansion of the refractive index to first order to linearize the equation about ξ = ξ ₀ , the result is η _ατ ~ η ₀ - (ξ - ξ ₀ ) . (234) ςο

A /FCC having a width of (2δξ) symmetrically about ₀ at the center of the /FCC would then provide a refractive index at the inner and outer channel boundary of η _σιτ .(ξ ₀ ± δξ) ¾ n ₀ =F ¾— = η ₀ δη , (235) ςο

where δη = ο(δξ/ξο), as shown in Fig. 7B. For example, if we wanted to make the optics "big" then we can choose ξ ₀ = 25 mm, δξ = 0.25 mm (i.e. a channel width of 0.50 mm) and UQ = 1.50. Then we find that δη = 0.0150 and this is possible for certain liquid mixtures that provide more than 2δη = 0.030 of refractive index change.

[098] One of the key points to be made in the above example is that the ratio of δξ/ξο sets the amount of change in the refractive index needed. Moreover, as the /FCC are small in width 2δξ the ratio δξ/ξο can be made very small, thereby keeping the needed refractive index change small even with circular beam steering of 2 radians and so practical colloids are available for the IGL.

[099] In Fig. 7B, the required linearized refractive index 7k is spoiled by changing the amplitude or frequency of a DEP based IGL. The gradient spoiled region 7e of the device corresponds to uniform refractive index region 7m region in Fig. 7B. The uniform refractive index region 7m may be set equal to the refractive index of the solid transparent material 7g as shown by the solid refractive index regions 7n and 7o. Refractive index regions 7n and 7o may also be higher or lower than n ₀ to facilitate containment of the beam in the presence of imperfections in the manufactured DBS device. Note that the gradient spoiled region 7e obtains its uniform refractive index by changing the voltage level or the frequency on electrodes that are energizing the gradient spoiled region 7e. The electrodes for this operation are not shown in Fig. 7 to avoid cluttering the figure.

[100] The IGL comprises a mixture of one or more liquids with one or more tiny particle types therein (with potentially many millions of individual particles) so that the bulk refractive index is a spatial average over the properties of the liquids and solid particles. In this way controlling the spatial distribution of solid particles in liquids also controls the volume average refractive index. This can be accomplished at fast electronic speeds, without the need for liquid advection.

[101] Next, we will consider Fig. 8 and assume that we have an expression v(r) for the steady state concentration of nanoparticles in a colloidal suspension. Then for the geometry that we have shown in Fig. 7 we have a one dimensional problem and r — > r . Additionally, we can write the steady-state nanoparticle concentration as v(r) = u(r, oo) so that Eq. 167 becomes Near the radius of r = TQ we can use a Taylor approximation to rewrite this as

Moreover, we can now equate Eq. 237 to Eq. 234 after we first relate the coordinate system in Fig. 7 to the coordinate system in Fig. 8A.

[102] Note that in Fig. 8A the apex of the DEPS electrodes is pointing towards the symmetry axis of rotation of the electrodes and this forms the circular channel in Fig. 7. Said another way, Fig. 7 provides a top view and Fig. 8 provides a cross sectional view that includes the center of the device and electrode orientation is as shown. The beam steering device 8a has a circular /FCC having electrodes 8b and 8c. Internal to the beam steering device is a wedge shaped /FCC that contains the refractive IGL 8d. A collimated light beam 8e is propagating into the page and out of the page on the other symmetric side. Figure 8B is identical except that the orientation of the wedge is reversed and this will require KR < 0 compared to Fig. 8A, which requires KR > 0. Note that— r ₀ is used as the coordinate convention as shown in Fig. 8B. [103] From Fig.8A we can see that ξ = R + r (238) ξο = R + r ₀ (239) where R is the distance from the axis of rotation to the point where the electrodes would intersect and r ₀ is the position of the portion of the /FCC used to control the optical radiation. We can rewrite Eq. 234 as ricir ~ n ₀ - ^U ° ((R + r) - (R + r ₀ ))

R + r ₀

= ο - -^— (Γ - Γ ₀ ) . (240)

R + r ₀

Therefore, on setting the required n _circ equal the DEPS based n we obtain

/ \ / \ ^^(^o) \ \ \ / \ n _L + v(r ₀ ) (n _s - n _L )V -\ — (r - r ₀ ) (n _s - n _L )V = n ₀ -— (r - r ₀ ) , (241)

(IT IX -\- TQ

which we can see by matching coefficients of (r— r ₀ ) that n ₀ = n _L + v(r ₀ ) (n _s - n _L )V (242) d ^v{ro) -(n _s - n _L )V , (243)

R + r ₀ dr

which can be used to eliminate UQ SO that

^ v(r ₀ ) + (R + r ₀ ) ^ ± . (244)

V n _L - n _s J ^{7 7} dr

This is the high-level master equation that must be satisfied for a light beam to move in a circular trajectory at DEP steady-state. To go any further we must solve the quasi-electrostatic boundary value problem for the electric fields generated by the electrodes and then solve the DEP master equation Eq. 163 under steady-state conditions. Finally, note that the case of Fig. 8B is obtained by setting 7¾ — > — 7¾ in the above equation.

[104] With that objective in mind, consider an example arrangement of two electrodes as shown in cross-section in Fig. 9 where an angle of 2Θ ₀ is formed between two flat electrodes 9a and 9b. The electrodes form a v-shape or wedge shape and follow a circular arc that follows the electric field 9c in the plane of the page and can also form a circular /FCC in the direction into the plane of the page. The total voltage amplitude between the electrodes is 2VQ , i.e. from +VQ to— VQ, and there are dashed constant potential lines such as 9d. The volume between the electrodes is filled with a nanoparticle colloid that forms a IGL 9e. The gradient of the IGL is represented in the figure by the density of dots shown and the density of nanoparticles is greatest near the apex of the electrode wedge. In this region is a light beam 9f that propagates into the plane of the page and is redirected into a circular trajectory as the light propagates into the page along the circular /FCC.

[105] Note that Fig. 9 is consistent with KR > 0 and is the same as Fig. 10 except that Fig. 10 is consistent with KR < 0. This can be seen by the IGL 9e gradient, i.e. the density of dots, in the nanoparticle colloid compared to the IGL 10a gradient of nanoparticles. The position of the light beam 10b that is being steered in a circle has shifted relative to the light beam 9f. When KR > 0 the process is a positive DEP (pDEP) and when KR < 0 the process is a negative DEP (nDEP) . If K ₀ and K _∞ in Eq. 124 are of opposite sign then we can switch between pDEP and nDEP operation by changing the frequency ω. Additionally, the pDEP configuration of Fig. 9 is used in Fig. 8A, while the nDEP configuration is used in Fig. 8B.

[106] Figure 11 shows a cut-away perspective of an example DBS device based on forming a circular path of the wedge-shaped microfiuidics channel and electrodes about a common symmetry discussed above for Figs. 8-9. A typical curved light trajectory 11a is shown, including its release into a free-space light beam lib.

[107] A disk shaped and transparent containment vessel 11c is shown in a cutaway perspective view. The reader is to imagine that the disk also fills the region along the cut-away region lid. The containment vessel contains a wedge- shaped profile that is revolved around the symmetry axis to form a /FCC lie that contains within its volume an IGL based on a colloid comprising optically transparent nanoparticles in an optically transparent liquid, where each component of the colloid has a different bulk refractive index. Solid nanoparticles have a refractive index that is typically (but not necessarily) higher than the refractive index of the liquid and transparent containment vessel 11c. This allows the refractive index of the nanoparticles to be averaged with the liquid as a colloid so that the colloid's effective (volume averaged) refractive index is close to that of the solid containment vessel when there are no DEP based forces on nanoparticles, i.e. as shown in Fig. 7 at the uniform refractive index region 7m. This occurs when the voltage difference across the electrodes is zero or when the harmonic electrode ex- citation radian frequency times the Maxwell- Wagner time constant JTMW is such that the real part of the Clausius-Mossotti factor is zero, e.g. this is seen in Fig. 3A when the solid curve crosses the K = 0 line and it also occurs at the top of the semi-circular trajectory in Fig. 4, such as at zero-force point 4f.

[108] On the top and bottom of the /FCC lie are electrodes llf and llg.

The electrodes may each comprise a number of sub-electrodes that are separate and identical in form, but which may be independently energized so that separate angular regions in the /FCC may be energized for light steering, as described below for Fig. 12. A first region is where there is a refractive index gradient is formed in the IGL and a second region is where there is no gradient of the refractive index formed in the IGL. In Fig. 11 the first and second regions have a electrode boundary llh. The curved light trajectory 11a exists on one side of the boundary and straight propagation in a volume exists on the other side of the boundary.

[109] In Fig. 12 a set of segmented electrodes 12a that are used in Fig. 11 are shown here in detail to avoid the clutter it would have generated in Fig. 11. In particular, about a symmetry axis 12b are arranged segmented upper electrodes and lower electrodes. An example of an upper electrode 12c and its associated lower electrode 12d are provided. The upper electrodes are typically almost all at the same phasor voltage level +V _Q . The lower electrodes are typically almost all at the same phasor voltage level — VQ . The exception to these voltage levels being where the DBS is to be deactivated to allow the light beam to escape into rectilinear propagation. The deactivated electrodes will have the voltage set to zero or the harmonic excitation frequency set to that value that sets the real part of the complex Clausius-Mossotti factor to zero. Additionally, in actual devices there may be an ability to provide electrode-to-electrode variations in the electrode voltages to compensate for non-ideal effects like temperature effects, beam spread, imperfect input light direction and a distributions of nanoparticle diameters and refractive indices that may not have been well characterized. Electrodes in the off-state angular range 12e are deactivated to allow the light beam to escape for the example beam depicted in Fig. 11. It is important to note that there may be many electrode segments to allow for high angular resolution. Additionally, they can be set into an off-state over multiple segments so that multiple pulsed beams can be launched almost simultaneously. This is important in applications like LiDAR that have a need for high sample rates using pulsed laser light and pulse-position-encoded laser light.

[110] Next, the nanoparticle concentration in the /FCC lie is developed for a the specific example case of the wedge shaped electrodes having harmonic voltage excitation. The voltage phasors across the top and bottom electrodes are

V(Top, t) = +V ₀ e ^{i Jt} (245) F (Bottom, *) = -V ₀ e ^{i jt} . (246) In phasor form the voltage in the volume between the electrodes is the solution of

V ² \ (r, 0) = O (247) subject to the boundary conditions of

V(n < r < r ₂ , +0 _o /2) = +V ₀ (248) V{n < r < r ₂ , -θ ₀ /2) = -V ₀ . (249)

Thus, we can see that we can approximate the voltage between the electrodes at -0o < Θ < +θ ₀ as

(¾ ( ²⁵ °) Therefore, the electric field intensity in cylindrical coordinates is

W(r, 0)

,dV l dV dV

Or r dO dz

so that

and v^ ² - — I— ^dE2 )

0 ' dz

Therefore,

Also, we can identify the real and imaginary parts of the electric field intensity

(0, 0, 0)

ER 0 . (255)

Using the above analysis for V · (uVE ² ) and E x ER the master DEP equation Eq. 163 can be written as

du D du , d ² u

277i u D (256) dr r dr dr ² therefore

1 du d ² u β/D du 2β/Ό

u (257) D ^~ dt dr ² dr

Next, we will normalize this equation and make it unitless. This will simplify the equation somewhat and provide some insight into to the final functional form of the solution u. First note that the units of β are obtained by first expanding, whereby

Therefore, the units of are

L'T- ¹ . (259)

Given the differential equation Eq. 257 we seek a product solution of the form u ~ r ^a t ^b D ^{c d} u ₀ ^e (260) so that so that all of the quantities in Eq. 257 are included as well as the initial spatial condition u(r, 0) = u ₀ is a constant at concentration at time zero. In terms of the units

-3) = L ^a T ⁶ (L ² T- ¹ ) ^c (L ⁴ T- ¹ ) ^d (# L -3\ e (261) so that a + 2c + Ad - 3e (262) 6— c— d 0 (263)

1 (264)

Therefore,

d = b— c (265) and

c = - + 2b (266) 2

Therefore,

or more generally on summing over different proportionality constants A _a ^ and exponents we can provide a series solution of the form

or equivalently we can write

u = u ₀ g (p, T) (269) where g is a function that needs to be determined and where we have introduced the dimensionless similarity variables

T = t- (271) β

Consequently,

and at steady-state v(r) = u(r, oo) = u ₀ g (273)

where the functions g and G are yet to be found by rewriting

du f D ² \ dg

(274)

Έ - ^Uo T

d ² u D \ d ² g

UQ (276) dr ^{2 υ} β J

resulting in a normalized differential equation

An initial condition and boundary condition are taken respectively as g(p, o) = i (278) g(oo, oo) = 0 . (279)

At steady state when t— > 00 let us define G{p) = g(p, 00) where

The infinitely large wedge region is an approximation that allows to avoidance of calculating fringing fields and is reasonable given that most of the DEP forces are near the wedge apex anyway. The first condition above says that the electric field has no impact on the IGL far away from the wedge's apex. The second condition bounds the solution, however for the point particles being used in this model the concentration may go to infinity and we will have to manually restrict the concentration. This will be explained in detail later in the analysis.

[Ill] Before proceeding note that there are other possible ways to model the situation. For example, by assuming that the number of nanoparticles, in a fixed region between r _x < r < r ₂ , is invariant due to the walls of a finite containment vessel, then

u(r, t)r dr d8 = ζ = const. , (281)

whereby another similarity analysis yields solutions when β φ 0) of the form

and the resulting differential equation Eq. 277 does not change, even though the expression for the solution is different. Thus, Eq. 272 transforms into Eq. 282 so that UQ — ζΌ/β would be needed in subsequent equations. For the sake of retaining a simple model with features that are useful for this disclosure we now proceed using Eq. 272.

[112] Therefore, at stead-state we ex ect that

The general solution to the steady-state equation may be found in several ways, however for the sake of expedience note that the solution is most likely an exponential. So by trial and error we can try trial-solutions of the form Exp [ap ^~2 ], Exp[ap ^_1 ] , Expfap ¹ ] and Exp[ap ² ] . From this we find that the exponential Exp[ p ^~2 ] solves the equation and that a second independent solution may be found for the second order differential e uation by the method of reduction-of-order so that

where Ei[^] is the exponential inte ral function

The first term G to the steady-state solution is a pure exponential and is always greater than zero and decreasing as p increases. The second term G ₂ is not always positive and increasing without bound as p increases as shown in Fig. 13. The G curve is therefore not a physically realizable solution. Notice that the Gi solutions are also labeled as positive dielectrophoresis pDEP 13a and negative dielectrophoresis nDEP 13b where KR > 0 and KR < 0 respectively.

[113] In general we have the steady-state concentration v(p) = u(p, 00) = UQG(P) where = 1 as required for our normalized differential equation, so that

1

v{p) = u ₀ exp (286)

2 ²

From Eqs. 89, 124, 133, 153, 258 and 270 we can write Eq. 286 as and we can clearly see that KR can take on both positive and negative values depending on the values of K ₀ , K _∞ , TMW ^an d the frequency ω. Additionally, we can identify Re[e] from Eq. 119, the arc length we identify as s = r(2#o) and the total voltage drop across the electrodes as 2VQ SO we can further identify the electric field E and electric field energy within a nanoparticle W as

Ref (288)

Therefore, the steady-state nanoparticle concentration is a function of the ratio of (1) the energy in the volume of the nanoparticle and (2) the thermal energy. In particular

[ 1 W 1

v = M ₀ exp -—— . (291)

2 k _B T

[114] The factor of 1/2 in Eq. 291 is specific to the form of the impressed electric field used and may change for other electrode geometries. Clearly, the simple form of the above equation shows that an alternate variable to use instead of p is the ratio of the energies. Note that we can switch between the two forms of Gi shown in Fig. 13 simply by changing the frequency ω of the voltage impressed on the drive electrodes to change the sign of KR , i.e. when one of KQ and K _∞ takes on a negative value and the other takes on a positive value, whereby the energy W can be either positive or negative, which creates the physically realizable positive dielectrophoresis pDEP 13a and negative dielectrophoresis nDEP 13b curves that are explicitly shown in Fig. 13 for the wedge electrode example. [115] From Eqs. 153, 258 and 270 we can write

r

P (292)

P

0

Po 293)

where we call p the normalizing radius and r ₀ is the distance from the electrode apex to where the light is confined for steering as shown in Figs. 8-9. Consequently, we can rewrite Eq. 244 as

1 dv(po)

v(po) + {TZ + po) (295) V n _L - n _{S /} dp

where, similar to the definition of p in Eq. 270

n R

(296) P

Therefore,

dv(p)

-u ₀ — exp (297) dp 2p ²

and

where the volume fraction of the solid spheres is z/ _$ = UQV, occurs on completely mixing the IGL's liquid and nanoparticle components in a beaker outside if an energized circuit. Now solving for the ratio of refractive indices we obtain for the configuration of Fig. 8A the equation

! _{= 1 +} rf≤£B*i _{. (299)} n _L i/ _s {R ^, + Po - Po)

Notice, that as the normalized radius of the pFCC increases towards infinity, TZ— > oo that the ratio of refractive indices approaches unity because the trajectory of the light is a straight line and a GRIN profile is not needed. Additionally, for the case shown in Fig. 8B we make the substitution o - ~Po ^s ° that

5 ^£ _{= 1} _ ^ ^H "°t (30°)

Finally, for the sake of completeness, when extending Eq. 299 from Rayleigh to Mie scattering with larger nanoparticle sizes, i.e. b > λο/20, we have where p is given by Eq. 208 and o ^can take on both positive and negative values to account for the different configurations in Figs. 8A-B respectively and the refractive indices are a function of free space wavelength λο and temperature T. Equation 301 is plotted in Fig. 14 for polydimethylsiloxane, which is a low vis- cosity, chemically stable and large-temperature-range transparent liquid that is representative of dielectric liquids intended for the applications in this disclosure.

[116] Note that some care has to be taken in using the above equations because Eq. 163 essentially assumes that the nanoparticle spheres are points. This means that the points can get arbitrarily close to each other so that the concen- tration can tend to infinity as r— y 0. This cannot happen in practice as the solid spheres would touch each other and stop any further concentration from occurring. In particular, for uniformly sized spheres in three dimensions the densest packing is approximately 74%, i.e. the fraction of the volume, for the face centered cubic (FCC) packing and the hexagonal close packing (HCP) . In contradistinction, a random packing of equal spheres generally has a density around 64%. Although the analysis in this document has focused on spherical nanoparticles, it is clear that other geometries for the nanoparticles can be utilized.

[117] The maximum uniformly-sized spherical packing density provides a volume fraction of

7Γ

z s (Regular Packing) =—= (302)

3v 2

and the random packing provides

2

1/5 (Irregular Packing) f¾ - . (303)

3

As a practical matter for spherical nanoparticles

0 < vs ^< ^ ■ (304) and because v$ = u V

- 1/2

In (305)

[118] Thus far in the analysis of beam steering the electrodes llf and llg in Fig. 11 have been configured to provide light steering in a plane. However, this is not a requirement and it is possible to add additional steering electrodes so that the light beam can be steered in elevation as well as azimuth. [119] In Fig. 15 additional electrodes are added to gain modest control over the elevation angle. Some applications, like LiDAR for automotive applications, need perhaps just as few as 5-10 degrees of elevation beam control to be able to cover the anticipated field of view. The elevation control electrodes 15a and 15b provide this additional degree of freedom that shifts the beam to a new elevation beam position 15c. Any electrode configurations that can change the distribution of nanoparticles may be used for both the azimuth and elevation control not just the wedge shaped electrodes shown. The electrodes in Fig. 15 are only one possible example.

[120] The DBS device just described is based on a wedge shaped /FCC that has electrodes with opposite polarity placed on the upper and lower surfaces of the/FCC to form a wedge geometry. An alternative method is shown in Figure 16 where a different electrode symmetry is used and different principles for guiding the light are utilized.

[121] In particular, it is now desired to create a GRIN liquid that confines and guides light much as a fiber optic confines and guides light: by having a core region with higher refractive index than the cladding region. Most fiber optics have a step change in the refractive index between the core and the cladding. Some fiber optics uses a GRIN refractive index profile so that a gradual change occurs between the core and the cladding refractive index. That said, fiber optics have static refractive index profiles while the DBS systems that are described below have dynamic GRIN profiles so that the guided light beam can be steered to a desired azimuth (or elevation) direction and then released, i.e. by removing the GRIN profile, so that the light is launched into straight-line propagation and out into free-space.

[122] In Fig. 16A input light 16a is injected from a light source (not shown) into a GRIN light guide that is based on the dielectrophoresis of nanoparticles in a colloidal IGL, which is located within an /FCC 16b. The light has a general propagation direction 16c. However, due to the GRIN profile we may consider that rays undulate around the guide as shown in the figure and this can lead to some light divergence at the output. Similar to fiberscope, borescope or endoscope the ray divergence can be controlled by choosing the cladding and core diameters strategically. Along the /FCC there is a plurality of discrete electrodes along the angular direction of propagation (not shown in this figure) that are used to form a GRIN medium to confine the light until a DEP-disrupted region 16d of the /FCC is reached. In the disrupted region the electrodes are intentionally configured so that either the harmonic voltage amplitude is set to zero or the frequency ω of harmonic excitation is set so that the real part of the complex Clausius-Mossotti factor of Eq. 122 is zero to eliminate pondermotive forces on the nanoparticles and eliminate the GRIN effect. Consequently, after the DEP process is disrupted the light will have rectilinear propagation 16e in the medium of the solid transparent containment vessel 16f and will refract out of the containment vessel into free-space light propagation 16g. The angular divergence of the beam will be determined by the effective numerical aperture of the /FCC based GRIN light guide. The shape of the containment vessel is arbitrary, but has been shown as square in cross section for convenience in Fig. 16A.

[123] In Fig. 16B the refractive index profile as a function of radius ξ is provided looking towards direction arrows 16h and 16i is shown. The containment vessel refractive index 16j and 16k is UQ and at the center of the /FCC has a refractive index of n _{1 ?} which is achieved via a GRIN profile 16m. When the DEP process is disrupted the GRIN reverts to the uniform refractive index 16n.

[124] In Fig. 17 the configuration of electrodes that are used to support the GRIN of Fig. 16 is provided. It consists of two separate and substantially sepa- rated electrodes 17a and 17b, which support harmonic voltage amplitudes ±¼. Additional electrodes may also be used (but are not shown here) , for example the well-known quadruple configuration, so that greater control over the distribution of nanoparticles within the refractive IGL 17c can be obtained. The symmetry of the configuration is provided by the dashed equipotential curves 17d and the electric field curves 17e. Note that the light that is directed around the circular/FCC in Fig. 16A would now be centered between the electrodes and directed into the page of the figure in Fig. 17.

DEP-Based Light Focusing Using In-Phase Excitation

[125] The same principles, materials and structures used in beam steering can also be used in focusing a lens. An example of this is shown in Fig. 18A where a cross sectional image of a lens is provided. Input light 18a enters the lens and is focused (or defocused if desired) on passing through the lens.

[126] The lens has a transparent containment vessel to hold an IGL, which comprises a first containment structure 18b and a second containment structure 18c. These containment structures provide a volume within which both the refractive IGL 18d and the electric fields used in DEP are located.

[127] Circular annular electrodes comprising a first annular electrode 18e and a second annular electrode 18f are separated by an electrical insulator 18g. The first and second annular electrodes support phasor voltages +VQ and— VQ and produce electric fields 18h, which are indicated in Fig. 18A by dashed lines. Note that the equations already developed for nanoparticle concentration u for wedge- shaped /FCC beam steering still apply here, but now the usable portion of the electric fields fill the outside of the wedge.

[128] Input light 18a that enters the DEP optoelectronic lens along the symmetry axis 18i is focused as output light 18j. The solutions shown in Fig. 13 still apply approximately for nDEP and pDEP and are plotted in the context of the lens in Fig. 18 in Figs. 18B-C. If an nDEP process is utilized then a converging lens is formed as shown in Fig. 18B by the converging lens refractive index 18k, which corresponds to a conventional convex lens. If an pDEP process is utilized then a diverging lens is formed as shown in Fig. 18C by the diverging lens refractive index 18m, which corresponds to a conventional concave lens.

[129] Note that a conventional lens uses a constant refractive index and a variable thickness as a function of lens radius to form the lens. In contradistinction the lens in Fig. 18A has a variable refractive index as a function of lens radius and a constant thickness. Additionally, it is also possible for the lens in Fig. 18A to also include a variable thickness. This might be done to eliminate lens aberrations. Alternately, additional electrodes (not shown) placed on the containment vessel surfaces can provide additional degrees of freedom to shape the distribution of nanoparticles and better control focusing and aberrations.

[130] A perspective view of the annular electrodes is shown in Fig. 19 and comprises a first annular electrode 19a and a second annular electrode 19b. The electrodes are separated by an electrical insulator 19c. The first and second annular electrodes support phasor voltages +VQ and— VQ and produce nonuniform electric fields by means of the tapered electrode 19d section of the electrodes that has variable thickness function of radius.

[131] A perspective and cut-away view of the annular electrodes and containment vessel of a electrophoresis-based lens are shown in Fig. 20. This figure is consistent with Figs. 18-19. Input light 20a is focused to output light 20b via an optoelectronic dielectrophoresis lens 20c that is typically (but necessarily) symmetric about the optical axis 20d. A first annular electrode 20e and a second annular electrode 20f are separated by an electrical insulator 20g. The electrodes provide the nonuniform electric field needed for changing the distribution of nanoparticles and the optical refractive index to electronically focus light passing through the lens. The light is focused by a colloid based GRIN profile formed in the containment vessel, which is formed by the electrodes, insulator, first containment structure 20h and second containment structure 20i.

Light Steering Using In-Phase &ε Quadrature Excitation

Up to this point in the disclosure the focus has been on the use of only an in-phase harmonic excitation of the electric field. In this case the master equation Eq. 163 shows that pondermotive forces on nanoparticles can be achieved from only the first term. However, for in-phase & quadrature excitation the first two terms of Eq. 163 are required. This can be seen when the master equation Eq. 163 is rewritten as

— = - _77l V · (uV [\E _R \ ² + |£ ₇ | ² ] ) -77 ₂ V · [uV x (E _r x E _R )} +D ² u (306) at v _v -· _v '

Non-Uniform Field Circulating Field where i¾ is the in-phase electric field and Ej is the quadrature electric field, which is not in the same direction as En- One way to induce the in-phase and quadrature excitations in the above equation is to use a poly-phase harmonic traveling-voltage-wave on the electrode array in Fig. 21A. Under suitable conditions this has the potential to provide forces on nanoparticles in both the x- and y-directions as indicated in the figure. In the example shown in Fig. 21 A only three phases are shown

¼ = V ₀ e ^t0 (307) V ₂ = be^ (308) v ₃ = V ₀ e^ , (309) but more or less phases are also possible with equal phase separation between adjacent electrodes. The voltage sources are connected to electrodes, such as electrode 21a. The electrodes may be placed on or imbedded in a solid dielectric material 21b that is typically transparent at the optical wavelengths of interest. In contact with the solid dielectric material is a colloidal nanoparticle fluid 21c. As discussed before the nanoparticles are dispersed in the liquid and the particular spatial distribution of the nanoparticles provides an optical gradient to control light. Although it is not shown in Fig. 21A the electrode array is formed on at least one surface of a //FCC, which has been discussed throughout this disclosure. The linear electrode array may be extended into two dimensional surface, i.e. a manifold, which is embedded in a three dimensional space— see for example Fig. 25 for a cylindrical manifold. Additionally, the array may have counterpart electrodes that are formed on other surfaces of the /FCC.

Added degrees of freedom to the nanoparticle distribution are applied to the electrodes by energizing them with different voltage amplitudes, timing, phases and frequencies in order to better control the forces on transparent dielectric nanopar- tides and the resulting distribution of the refractive index as a function of space and time. This is shown in Fig. 21B, where forces on colloidal nanoparticles are in one or both of the x- and y-directions. As in the previous paragraph Fig. 21B has electrodes such as 21d, which is on or in a solid dielectric material 21e. In contact with the solid dielectric material is a colloidal nanoparticle fluid 21f.

The electric field may be represented by a harmonic wave traveling from left to right. A convenient model for the phasor form for the electric field to first order is given by

E = E ₀ ( _i x + y)f(x)e- ^tKX - ^K (310) where f (x) is a real valued function that models the driving electronic's ability to change the amplitude of the harmonic voltage excitation as a function of electrode position, κ is the spatial radian frequency of the fields and x and y are unit vectors in the x- and y-directions. Note that when the amplitude has a change from VQ to — V _Q this represents a radian phase shift. Also, there is an exponential decay of the electric field strength as the point of observation moves away from the plane of the electrodes in the y-direction.

By expanding the last equation using the well-known Euler's formula e ^%e = cos Θ + i sin Θ we obtain E = ER + iEj such that

E _R = E ₀ e ^~Ky f(x) [x s (Kx) + y cos(Kx)] (311) Ei = E ₀ e ^~Ky f(x) [x cos(Kx) - y sm(Kx)] . (312) from which we find

ER - ER = E ₀ e- ^2K *f ² (x) (313)

EJ - EJ = Eoe- ² "vf (x) (314)

E _R - E _r = 0 (315)

E ² = 2E ₀ e- ^2Ky f ² (x) (316)

EJ X ER = E ₀ f ² (x)e- ² "*z (317) where E ² = \ER \ ² + |-E _/ | ² . The forces on the nanoparticles therefore have contributions that are proportional to - ^L VE ² = +2Ele- ² y [f {x)f{x)x - Kf ² {x)y] (318) V X (E! X ER) = -2E ² e- ² "y [f (x)f'(x)y + Kf ² (x)x] , (319) where the primes indicate derivatives with respect to x. If f(x) = 1 then jE ² = -2E ² Ke- ^2Ky y (320) V x (Ej x E _R ) = -2E ² e- ^2Ky x (321) and steady-state nanoparticle flows induced by induced forces varying in the y- direction. Therefore, the steady-state concentration is u(r) f¾ u(y) and the concentration contributions are

V · [u(y)VE ² ] = -2E ² K d _y [u(y) _e - ^2Ky ] (322) V · [u(y)V x (Ej x E _R )] = 0 . (323)

When we f (x) is not a constant then both of the above terms are in general not zero and there are more degrees of freedom for controlling the GRIN profile and light beam steering.

Thus, approximately only y-directed variations in the nanoparticle concentration u are anticipated when f(x) = 1. The real and imaginary electric fields when f(x) = 1 are shown in Figs. 22A-B. The fields are shown relative to electrodes like 22a and 22b. The y-directed variations can be generalized and used to control a beam of light.

However, additional degrees of freedom for controlling light can be obtained by providing electronics that provide electrode-to-electrode variations in the amplitude so that f (x) φ constant. For example, if the amplitude changes as shown in Fig. 23 relative to the electrodes 23a shown then we obtain control over nanoparticle flows in both the x- and y-directions of Fig. 2 IB. The variations in amplitude 23b and the derivative of the amplitude 23c in Fig. 23 are shown as fields in Figs. 24A-B. Notice that the field structure in Fig. 22 is different that of Fig. 24.

A DBS device with 2 radians of azimuth steering and over ± /4 radians of elevation steering is shown in Fig. 25. The DBS device comprises a solid outer transparent radome 25a, which has a radome inner surface 25b forming one surface of a //FCC. Additionally, an optional inner solid 25c, which may be transparent, provides an outer surface of the inner solid 25d. The gap between the radome inner surface 25b and the outer surface of the inner solid 25d is the //FCC 25e.

Additionally, a nanoparticle colloid 25f located in a //FCC formed either between the outer transparent radome 25a and the optional inner solid 25c; or between the outer transparent radome 25a and the spatial extent of spatially varying electric fields used to contain the light by a GRIN profile.

An input light beam 25g (source not shown) is injected into the //FCC and it is reflected by means of an angle selective electronic mirror 25h into an upward spiraling light trajectory 25i. The upward spiraling light trajectory may optionally be converted into a downward spiraling light trajectory 25j by a reflection device 25k, which is typically just total internal reflection. The injection of the light into the //FCC may be accomplished by any means so that the angle selective electronic mirror 25h is only provided as an example.

A means to control the nanoparticle colloid concentration u(r, t) is also provided and described later. When the means to control the nanoparticle colloid is activated the light trajectory remains spiraling in the upward or downward direc- tions of Fig. 25. When the means to control the nanoparticle colloid is deactivated the light is launched, for example at point 25m, and then has an internal rectilinear trajectory 25n and the light beam is refracted into a free-space beam at point 25o on an external rectilinear trajectory 25p.

There are three different modes for controlling the trajectory of the light beams. The first mode uses in-phase and quadrature electrode excitation from a traveling voltage wave on electrodes to induce a radial gradient in the refractive index that allows circular light propagation. This first mode (mode-1) has the same requirements as set forth in Eq. 235, sets f(x) = 1 and has the concentration u exponentially decaying away from the two dimensional electrode array into the IGL in the /FCC. In Fig. 25 we can see the first magnification inset 25q, which shows an array of electrodes on the outer surface of the inner solid 25d in a small region of the upward spiraling light trajectory 25i. The electrode array may also be placed on the radome inner surface 25b, but may require setting ω so that the opposite sign of the Clausius-Mossotti factor is used. The radius of curvature of the osculating circle tangent to the spiral trajectory is known so that voltage and harmonic frequency ω can be chosen to ensure that a decaying concentration exists in the radial direction that ensures a spiraling trajectory of the light. The electrodes may be energized as pixels, as shown in Fig. 25, or long strip electrodes (not shown) around the optional inner solid 25c cylinder.

The second mode (mode-2) sets f(x) to a non-constant function of x, such that two separate traveling voltage waves are generated: either both propagating away from or towards each side of the the upward spiraling light trajectory 25i. This allows nanoparticles to accumulate around the light's trajectory to induce a GRIN waveguide similar to a GRIN fiber optic. Therefore, while mode-1 creates a gradient in the radial direction, mode-2 creates a gradient predominately in the angular and axial directions, i.e. in a tangent plane to the outer surface of the inner solid 25d such that the gradient vector is always towards or away from the light's spiraling trajectory. This induces a greater refractive index about the desired trajectory and provides a light guide for the light. For the avoidance the doubt Fig. 25 shows an examples: a mode- 1 radial nanoparticle gradient vector 25r and tangent mode-2 nanoparticle gradient vectors 25s and 25t. A mode-3 can also be imagined that is combination of mode-1 and mode-2.

A second magnification inset 25u shows a detail of transparent transistors 25v and transparent electrodes 25w that are printed on at least one of the surfaces of the/FCC as an array that can have each of the electrode pixels selected and energized to hold a charge by capacitive action between refresh signals to the transistors shown. A fast microprocessor could exploit a GRIN strategy like mode-1 or mode- 2 and energize the electrode array so that a light beam would progress along a spiral until the light is launched out of the /FCC at (for example) point 25m.

While the above description was made for a cylindrical DBS device it is equally valid for other structural forms, for example a sphere or hemisphere. In such a case the /FCC would be formed between an outer spherical radome and an inner spherical material. Many variations of electrode geometry are also possible. Light Focusing Using In-Phase &ε Quadrature Excitation

In the previous section the electrodes have a geometry that allows their coverage to extend to full or nearly full coverage of the supporting manifold. For example in a flat plane electrodes can extend in one or two dimensions. One possible tessellation is a rectangular grid of square electrodes on a plane, another is a linear grid of electrode strips on a plane and other tessellations are possible. The planar grid many be changed to a cylinder, sphere or other two-dimensional surface.

In an alternative approach, the geometry of the electrodes may be chosen to allow a simple traveling wave to still accumulate nanoparticle in a local re- gion. Consider the one dimensional electrode array in Fig. 21A, which is excited by harmonic polyphase voltages to provide a traveling electromagnetic wave and providing a pondermotive force on a nanoparticle in the x- and y-directions. At steady-state there will be a non-zero gradient component in the y-direction and (approximately) a zero gradient component in the x-direction. However, if a wall was inserted at an x-location then the boundary would allow an accumulation of nanoparticles and an optical GRIN profile.

By using a cylindrical geometry the same thing can be accomplished to form a lens. Therefore, consider the perspective and cutaway view of an electronically controllable lens in Fig. 26, where a set of concentric electrodes 26a is positioned inside a colloid containment volume 26b, which holds the IGL . The electrodes are formed from a transparent conductor, such as but not limited to: indium tin oxide, conductive polymers, silver nanoparticle ink and graphene. The electrode array shown may be the only one used or another electrode array may be formed on the upper surface of the containment volume 26b.

The colloid containment volume is a /FCC that contains the IGL and electrodes used to control GRIN profile in the liquid. A traveling wave on the concentric electrodes forces nanoparticles towards or away from the optical axis 26c so that at steady-state the constant thickness containment volume has a GRIN profile in the radial direction. A conventional circular lens, i.e. either convex and converging or concave and diverging, has an optical path length that varies radially from the input surface to the output surface of the lens. This same radial function is made using this DBS device, but now the shaped input and output lenses surfaces are replaced with a GRIN volume. The function of a converging or diverging lens is now obtained by the sign of the real part of the complex Clausius-Mossotti factor, which is controlled by the harmonic frequency of excitation of the electrodes. In this way input light 26d becomes focused output light 26f after it passes through the colloid containment volume 26b and its corresponding IGL containment structure 26e. It also allows for a dynamic zoom-in and zoom-out function as the magnification is changed by changing the harmonic electrode excitation frequency ω, the voltage amplitude or both. In practice the perspective cutaway view in Fig. 26 is made whole by symmetrically completing the structure along cutaway arc 26g.

It should also be noted that if the transparent nanoparticles are replaced by opaque nanoparticles it is possible to have an electronic shutter function or neutral density filter implemented. By this example we can also see that while the main focus of this disclosure has been for transparent nanoparticle there are also obvious extensions to nanoparticles that are opaque or even absorbing in certain optical bands to make a spectral lens filter.

Accelerating Waves and Non-Geodesic Beam Trajectories In this section we consider an alternative method and embodiment to using the DBS devices and techniques already described, in particular the range of light sources is now extended beyond light without specific modal structure, e.g. ray- optics types of sources, and instead special modal solutions to the wave equation are developed that allow greater control of the light steering process. By carefully choosing modes to excite the /FCC control volume the light's energy can be made to move along a new class of "accelerating" trajectories within the curved control volume forming the /FCC. This has substantial practical applications for building compact beam directors for applications like Light Detection and Ranging (LiDAR) where potentially millions of laser pulses per second need to be steered into many distinct angular directions. In what follows a quick review of select historical results is followed by an extension of the prior art theory to the new situation of spatially varying refractive index in a colloidal IGL. The resulting modes are use to redirect light into specific angular directions over large solid angles.

Over thirty yers ago, in the paper "Nonspreading wave packet" by M. V. Berry [Am. J. Phys. 47(3) , Mar. 1979] it was observed that a probability wave taking the form of an Airy function can propagate in free space without distortion and with constant acceleration without violating the Ehrenfest's theorem on non- accelerating wave packets. In particular, one possible solution to the Shcrodinger equation in free space

is an Airy wave given by

which can be demonstrated by means of direct substitution and the Airy differential equation and use of the Airy differential equation

It can be shown that, like plane waves, this Airy wave packet can propagate without changing form and remain diffraction free. No other functions than the plane wave and the Airy wave have this property in a linear medium. Thus, if one could find the electromagnetic counterpart to the above quantum wave functions then the unusual feature of the Airy packets to remain diffraction-free over long distances by free acceleration during propagation might be realized. Again, it is worth emphasizing that this diffraction-free property is realized without the need for a special nonlinear medium and the property is to be maintained at least within the DBS device.

The staring point is the time-domain version of the wave equation, in particular from Maxwell's equations it is easy to show for free propagation that

1 d ² £

V ² S (327) c ² dt ²

If we substitute the following assumed field configuration (which temporarily breaks with the time sign convention in this disclosure so as to be consistent with the cited literature)

£(r, t) = E{r)e ^{l kz} - ^Wt) (32 into Eq. 327 then we obtain

which has historically been difficult to solve unless a paraxial approximation made. This requires that the field varies gradually along the z-axis so that

dE{r) d ² E{r)

2k (330) dz > dz ² whereby Eq. 329 becomes

d ² d ² d_

2ik E{r) = 0. (331) dx ² dy ² dz

This can be written generically as a kind of Shcrodinger equation similar of Eq. 324. In particular, so that

■ dE(x, Q

(332) dx ²

where c (333) x

x (334) x ₀

2πη

k (335) λ o

and χ is the normalized transverse direction, IS 8b convenient transverse metric like the beam width, ζ is the normalized propagation distance and k is the wavenumber in the medium with refractive index n. E uation 332 has a solution

which clearly shows that the intensity profile of the wave is invariant in its propagation. This result does not require a nonlinear medium. Moreover, the excitation of the Airy beam can be accomplished through a truncation via an exponential aperture function at the light beam source so that

E(x, 0) = E ₀ Ai lx] e (337) where 0 < α < 1, which ensures that for negative χ-coordinate values that the exponential goes to zero and effectively truncates the Airy beam. The resulting extended beam is found by integrating Eq. 332 so that a practical beam takes the form

(338)

The Fourier transform of which is proportional to exp[a£; ² ] exp[i£; ³ /3] so that one way the Airy beam is generated is from a Gaussian beam by means of an optical Fourier transformation after a cubic phase is impressed on the Gaussian. The details of this are further discussed by G. A. Siviloglou et. al. in the paper entitled "Observation of Accelerating Airy Beams," in Physical Review Letters [PRL 99, 213901 (2001)] .

Unfortunately, the Airy beams formed in the paraxial approximation are limited to small angles. However, more recent solutions to the full Maxwell's equations have been found that bear a striking resemblance to the paraxial Airy beam solutions and are capable of large angular deviations like those that are found in DBS. For example, the journal paper entitled "Nondiffracting Accelerating Wave Packets of Maxwell's Equations," by Ido Kaminer et. al. in Physical Review Letters [PRL 108, 163901 (2012)] demonstrates accelerating solutions to Maxwell's equations for circular propagating non-diffracting waves. Also, in the full three dimensions accelerating waves on spherical manifolds were also found and experimentally observed in the journal paper entitled "Observation of Accelerating Waves Packets in Curved Space," by Anatoly Patsyk et. al. in [Physical Review X 8, 011001 (2018), DOI: 10.1103/PhysRevX.8.011001]. This last journal paper is of particular interest in this disclosure.

The main feature of accelerating electromagnetic light beams is that their center of momentum (similar to the center of mass) follows a geodesic formed by the/FCC, but the radiation's main lobes follows a curved path that is shape-preserving (non diffracting) over large angular extents while the main radiation lobes do not propagate on a geodesic. In contradistinction to the accelerating beams in flat space, accelerating beams in curved space change their acceleration trajectory due to an interaction that exists between the space curvature and interference effects so that the light's main lobe may follow non-geodesic trajectories while the beam focuses and defocuses periodically due to the spatial curvature of the medium that they traverse.

For the avoidance of doubt, a geodesic is the shortest possible curve between two points on a curved surface or manifold. In the case of a sphere in three spatial dimensions a geodesic is a circle that cuts the sphere such that the center of the cutting circle coincides with the center of the sphere. Examples of a geodesies on a sphere are circles of longitude, while examples of non-geodesics on a sphere are circles of latitude— except for the equator circle. Light that is launched just within and tangent to the glass sphere's surface, which is in contact with an external medium of a lower refractive index, will propagate along a geodesic trajectory due to total internal reflection. Similar considerations are possible for other geometric shapes, like cylinders and ellipsoids. For example, a glass cylinder provides a geodesic trajectory called a geodesic helix. The fact that light obeys this principle can be derived in several ways, e.g. for ray optics it is possible to start with Fermat's principle and use the variational calculus. The idea that there is a "loop hole" of sorts in the physics that allows a portion of a light field to propagate on non-geodesic and "accelerating" trajectories is exploited for dielectrophoresis-based beam steering in this disclosure. The loop-hole is simply that the peak intensity radiation lobe of the light field can follow a non-geodesic if other parts of the light field compensate so that on average the light's average momentum is on a geodesic. Thus if one is willing to expend some extra optical energy it is possible to obtain new ways to control the main lobe of the optical radiation.

The modal solutions that are developed herein allow a control volume com- prising a thin spherical shell //FCC, which is simple and cost effective to manufacture, to support many separate laser pulses that are directed along different non-geodesic trajectories, e.g. latitude circles instead of being restricted to just path-length-minimizing geodesies, where the IGL is maintained with a beam confining gradient. The accelerating laser pulses are steered within the /FCC to the correct angular location and the IGL is relaxed from a gradient state into a uniform refractive index that is matched to the surrounding medium so as to release the light beam into a straight line trajectory that ultimately becomes a free space light in rectilinear motion. The light may be continuous or pulsed.

Some of the main conceptual points being developed herein include: 1. Thin shells comprising an IGL have a non-uniform refractive index.

2. The curved shell can take any shape, but we will analyze a sphere herein.

3. The curved shell is a transparent liquid.

4. A shape-preserving intensity profile develops within the curved shell.

5. Shape preservation ( "acceleration" ) implies diffraction-free propagation. 6. Only a linear medium is required to support accelerating beams.

7. Acceleration of wave allows the main diffraction lobe to not follow a geodesic. 8. Controlling the modal structure within the IGL allows control of output beam direction.

9. New accelerating radial modes found include Airy and parabolic- cylindrical- D functions. 10. Radial Airy modes are truncated by light scattering perpendicular to propagation within the IGL.

11. Accelerating light on a curved manifold focuses and defocuses periodically.

12. This focusing and defocusing can occur on trajectories that are complex and not just circles of latitude. The development of DBS actually uses a different assumption from the prior art by assuming that a DBS device needs to have a liquid and nanoparticle based colloidal dielectric shell that has a graded refractive index formed therein. In what follows a new analysis with new solutions compared to the prior-art is provided wherein a refractive index that varies radially within a thin spherical dielectric shell is considered. Furthermore, the thin spherical dielectric shell is not a solid but a colloidal IGL.

In particular, consider a thin spherical shell that has a graded refractive index that is only a function of the radius. Moreover, assume that refractive index monotonically decreases as the radius increases, which is a convenient but not a necessary assumption. An example, of this is obtained by generalizing Eq. 234 for the colloid's refractive index as n _L (r) = n ₀ - ( ) ^ (r - i?) , (339) where R is the radius to the center of the thin dielectric shell that forms the //FCC, ri _Q is the refractive index at the center of the /FCC and a is a real refractive index slope adjustment that is obtained in practice by electronically setting harmonic excitation voltages, electrode-to-electrode phase shift and frequencies that influence the quasi-static pondermotive forces on the nanoparticles in the IGL. Of course other linearized functions may be used, but Eq. 339 provides a reasonable example for this disclosure.

Specifically, the linear relationship in Eq. 339 is formed from the electric field decay as the point of observation moves away from the /FCC surface where, for example, in-phase and quadrature electrodes provide a harmonic traveling wave excitation— i.e. as was introduced in Eq. 310. Therefore, for a spherical dielectric shell the quasi-electrostatic field is

E = Ε ₀ {ίφ + ϊ)ί(φ)ε- ^'ικφ - ^κτ . (340)

There is clearly an exponentially decaying field as the point of observation moves from the inner surface at r _\ = R—Sr towards the outer surface at Τ2 = R+Sr, where 0 < Sr < R. This decaying field provides pondermotive forces on nanoparticles via a gradient formed by a decaying electric field as r increases. Additionally, and for the sake of clarity, it should be noted that electrodes can also be placed on or about the outer surface of the thin spherical surface so as well, however the sign of the real part of the complex Clausius-Mossotti factor would have to be reversed by choosing that quasi-static excitation frequency appropriately as can be seen in Fig. 3. Thus both nDEP and pDEP processes are possible within the /FCC.

With that in mind we consider an optical transverse magnetic (TM) electric field given by E = (E _r , Eg, Εψ) = (0, Eg, 0) , where the conventional meaning of the spherical coordinates include: r is the radius, Θ as the polar angle and φ the azimuth angle. The wave equation Eq. 327 is then excited by a field £ = E(r, θ, φ)β ^ιωί which gives a Helmholtz wave equation ν ² Ε _θ (τ; θ, φ) + }ζ ² (τ-)Εβ (τ; θ, φ) = 0 . (341)

The V ² i¾ term has a sub-terms with r factors, however we can make the approximation that for factors that do not include derivatives with respect to the radius that r = R ± Sr is really just r f¾ R so that Eq. 341 becomes

d ² E _e 1 (d ² Ee 1 δΕ _θ 1 δ ² Ε _θ \ _{; 2} . . ^ _n , . dr ² R ² 0Θ ² tan 6> δθ sin θ δφ ² J

This may be solved by the method of separation of variables by letting EQ (T, θ, φ) = Χ{τ)φ(θ, φ) so that = ⁰ · ⁽³⁴³⁾

Where we proceed in the usual way and assign constants +β and—β to the terms in X and φ separately. Then the equations for X(r) and φ(θ, φ) become ^ _. + ( _fc 2 _(r) _ _β ) _{X {r) = 0 (344)} -^- ^{+ +} sin ² δφ ² ) ^{+ βΦ θ} ^ ^{Φ) =} ° - ⁽³⁴⁵⁾

The equation in φ can also be separated by setting φ(θ, φ) = Υ(θ)Ζ(φ) so that Eq. 345 becomes

1 d ² Z\ ( ₉ 1 d ² Y _n A dY „ \ . .

sin ² + sin 0 cos Θ-— + /3i? ² sin ² 0 = 0 , (346)

Z άφ ² J Y άθ ² Y άθ which has terms in Θ and φ only that are set equal to opposite signed constants ±m ² . The resulting set of separated equations becomes

d ² X(r)

+ (k ² (r)

dr ² - 0) X(r) = 0 (347) sin ² #^¾^ + sin Θ cos θ^^- + (/3i? ² sin ² Θ - ηι ² )Υ(θ) = 0 (348) d ² Z ) „ , ,

^^ + m ² ( ) = O . (349)

Note that by making the change of variables x = cos Θ we find that Eq. 348 can be written as

₀ , d ² Y(x) dY(x) m ² \ , _N , . which is the well-known standard form for the associated Legendre equation, which is solved for using a series solution as described by the well-known method of Frobenius— see for example "Special Functions for Scientists and Engineers," by W. W. Bell, 1996, ISBN 0-486-43521-0. The resulting recursion relations from the Frobenius method require that the value of βΒ ² and \m\ are constrained as βΒ ² = 1 1 + 1) V / = 0, 1 , 2, 3, . . . (351) To solve the system of equations Eqs. 347-349 we need to make the observation that the graded refractive index within the relatively very thin /FCC is nearly linear so we can expand k ² (r) about the center of the thin spherical /FCC region at r = R via a Taylor expansion and then retain the zeroth, first, second, etc.. order terms as needed to accurately describe the situation. The expansion of k ² (r) to first order is k ² (r) « k ² (R) - 2k(R) \ k' (R) \ (r - R) (353) where by assumption k'(r) < 0 for R— Sr < r < R + SR in the /FCC is assumed in the analysis to draw out the negative sign in front of the second term so as to obtain a real valued solution to the radial modes. The k'(r) < 0 assumption is not a requirement, but is done for convenience in the example being developed here. For example, in the case of the refractive index of Eq. 339 we find that k(r) = k ₀ n(r) so that

Using Eq. 353 in Eq. 347 provides a general solution in terms of the Airy functions of the first kind Ai) and the Airy of the second kind (Bi) so that

Alternately, the general expansion of k ² (r) to second order is k ² {r) « k{R) ² + 2k{R)k'{R) {r - R) + [k(R)k"(R) + k'{R) ² ] (r - i?) ² (356)

Using Eq. 356 in Eq. 347 provides a general solution in terms of the Parabolic Cylindrical D-function D _v (£ where

which is further given in terms of the confluent hypergeometric function— see for example Abramowitz and Stegun (1965) . In particular, the solution takes the form

X(r) = C _X D _VX + C ₂ D _V2 [A ₂ r + B ₂ ] (358) where {Α , A ₂ , Β , B ₂ , i , v ₂ } are substantially complicated functions of the parameters {k(R) , k'(R) , k"(R) , β} and are not provided here to save space. A detailed analysis of this radial mode as well as higher order radial modes in (r— R for j > 2 are not provided here, but are to be included as part of the disclosure for a non-uniform refractive index in the /FCC. The solution to Eq. 348 is given in terms of the associated Legendre polynomials of the first kind (P polynomials) and the associated Legendre polynomials of the second kind (Q polynomials) so that y(0) = C ₃ P™ (cos Θ) + C ₄ Q (cos Θ) (359)

The solution to Eq. 349 is given in terms of the complex exponential functions

Ζ(φ) = C ₅ e+ ^'→ + C ₆ e- ^→ , (360) where the requirement that Ζ(φ) = Ζ(φ + 2 ) requires that e ^±i27rm = 1 and that m = 0, ±1 , ±2, . . . and therefore in general m is an integer. Note that the superposition of Eq. 360 represents propagation into ± directions.

It is clear that we can choose to have the electric field propagate in only one direction of φ so we choose a direction and set C = 0. It is also clear that associated Legendre polynomials of the second kind are not bounded and are unphysical for the device configuration being analyzed, therefore C ₄ = 0. Finally, the Airy function of the second kind is unbounded and also unphysical for the device configuration being analyzed, therefore = 0. Additionally, the function φ(θ, φ) = Υ(θ)Ζ(φ) is a spherical harmonic and written formally as the normalized eigenmodes

Φι Θ, Φ) = ¾n P, ^H (cos 0)e ^im * (361)

/ = 0, 1 , 2, 3, . . . (363) m = 0, ±1 , ±2, ±3, . . . , ±/ (364) where N^ _m is the normalization amplitude of the mode that forms an orthonormal set of spherical harmonics— note the slightly different definitions given by "Physical Chemistry, A Molecular Approach," by Donald A McQuarrie, ISBN 0-935702-99-7 and "Mathematical Methods for Physicists, 3rd ed," by Arfken, ISBN 0-12-059820- 5.

However, on using Eq. 351 in Eq. 355 with = 0 for just a single mode having modal numbers / and m we obtain the simplest allowed modal structures, which in this disclosure are called the unstructured angular modes, which are

) - k ² (R) + 2(r - R)k(R) \ k'(R) \

Ε _θ (τ; θ, φ 1, ητ) = E ₀ Ai _ΐ ,τη(θ, φ)

{2k(R) \ k'(R) \ } ^2/3

(365) and in the specific case of the graded refractive index via Eq. 354 becomes

Ε _θ (Γ, θ, φ; 1, ηι) = E ₀ Ai

where a = k^n^R ² .

The unstructured angular modes in 366 are configured with an unstructured radial mode, which in this case is an Airy beam having an electric field amplitude of EQ . Examples of the unstructured modes are plotted in Fig. 27A-27B. Note that in Fig. 27B the radial region between R— Sr < r < R + Sr is the region of the/FCC where the colloidal IGL is located. A plurality of DEP electrodes operating at harmonic frequency ω are located on or about the radial surfaces r _x = R— Sr and Γ2 = R + Sr. These electrodes are not shown in the figure. Additionally, in Fig. 27B only the Airy modal structure within the /FCC region is shown in preparation for a discussion of the structured radial modes.

A different way of showing an example of an unstructured mode is provided in Fig. 28, where the radiation lobes, which are shown as bands, are substantially following non-geodesic trajectories around a spherical dielectric manifold 28a. For the sake of clarity, in Fig. 28 geodesies are those circles on a sphere where the center of the sphere is the same as the center of the circle, an example is a meridian geodesic 28b. Non-geodesic trajectories include a main lobe 28c and a side lobe 28d. Note that the thickness of the trajectory curves is representative of the radiation's intensity. When the circulating wave energy is released, i.e. tangent to the accelerating trajectory, it then becomes linearly propagating light 28e and it follows a straight trajectory through a transparent optical radome (not shown) and refracts into the external environment. The radome may have an outer spherical surface or it may take on other forms.

Having just described the unstructured angular and transverse modal struc- tures, we now look at the structured angular and radial modes, which are formed by the superposition of the basic unstructured modes. The structured modes provide additional useful features for end-use applications like LiDAR and are now described.

Structured radiation modes within the DBS device are formed as a superpo- sition of basic unstructured eigenmodes. For example, we can write in general that a structured angular angular mode ψι (θ, φ, Ιο, πΐο) is given by

m

which can impact the trajectory of the light in the 2-dimensional (θ, φ) space. Different complex apodization weighting coefficients A^ _m will impact the manifold trajectory differently. There are essentially an infinite number of potential candi- dates for A^ _m however, as a specific example we consider the structured angular mode formed from a gaussian weighting, which is chosen mealy as an example that connects back with discussions of the prior art, in particular φ^θ, φ, o, m ₀ ) =∑ e ^o-o ⁾ ψ _ι>ηι (θ, φ) (368) m

where the gaussian weighting parameter is (/, m) = αο,ο + ^a o,i ^m + ^a i,o I + o ^, _{1 1} lm + a m ² + <¾o I ² + l ^N m ^N . (369)

In general the trajectories of the radiation in the /FCC vary as a function of how the input light is modally structured.

If M(100, 90) = αο,ο = 1 then an example of a structured mode is shown in Fig. 30 and there are a series of bands in the Θ direction that represent a main and side lobes. As the wave progresses around the spherical shell of the /FCC in the φ direction the radiation focuses periodically as shown. The main lobe of the radiation can be launched out of the /FCC at the time the radiation is focused and maximized. In practice the values of and niQ may be quite large as integers with values in the thousands to hundreds of thousands. The values of / ₀ = 100 and niQ = 90 are only taken to provide a manageable example. Additionally, note that the angular distance of a main lobe from the equator (Θ = /2) is proportional to mo) in this example.

However, as another example, if (/ ₀ , m ₀ ) = m ₀ + l then (100, 90) = 11 and a different type of structured mode is shown in Fig. 31 and there is an oscillating trajectory and an angular locations (θ, φ) for a focus peak. Thus, a means exists for the peak intensity lobe reaching any point (θ, φ) on a trajectory Θ = θ(φ, mo) and we can also control the slope δψθ(φ, 1 ₀ , ηι ₀ ) at this focus.

This allows light to be launched into any direction by controlling the modal structure of the radiation in the /FCC and the harmonic excitation voltages, frequency and phase on the electrodes controlling nanoparticle pondermotive forces within the control volume /FCC— i.e. launching a beam of light by locally shutting off the DEP radiation containment fields in the /FCC.

In general, as Eq. 351 shows β is a function of the mode number / so that X(r) becomes Xi (r) and mathematically "infinite power" and shape-preserving radial mode functions, like an Airy mode, can be made to have finite power by a suitable weighted superposition. Multiple main lobes may evolve by higher order superpositions

Ε _θ ( _Γ , θ, φ) = Ε ₀ ∑ ΧΜ (θ, φ, 1 ₀ , πι ₀ ) . (370)

l ₀ ,m ₀

where in general Xi is a structured finite power mode (unlike a pure Airy function) that is induced by the scattering losses induced by the effective refractive index of the IGL. As noted previously in this disclosure the IGL has a refractive index that is complex due for example to the non-ohmic scattering loss in the directions roughly perpendicular to the direction of propagation. Moreover, the induced gradient in the distribution of nanoparticles in the IGL in the /FCC places a higher particle density at the inner radius r _\ = R— Sr and therefore provides stronger scattering at the inner surface, which is consistent with maintaining a truncated Airy beam with its peak intensity at the outer surface Τ2 = R + Sr.

Given a /FCC with a colloidal IGL dispersed therein the geometry of electrodes, the voltage amplitude on the electrodes, the phase difference between electrodes, the harmonic excitation frequencies on electrodes, the size distribution of the nanoparticles and the electrical properties (dielectric constant & conductivity) of both liquid and nanoparticles will come together to provide electronic control over the wavenumber slope of k'(R) , which directly impacts the propagation of the light in the /xFCC.

In summary, an "accelerating beam" that maintains its main lobe on a trajec- tory within a DBS device and this trajectory does not have to be a geodesic. This flexibility provides significant opportunities for laser directors used in applications like LiDAR, where there is a need to have large numbers of laser pulses sent into potentially millions of directions per second for ranging.

Scope of Invention

[132] While the above descriptions in each of the sections provided contains many specific details for dielectrophoresis based light control, these details should not be construed as limiting the scope of the invention, but merely providing illustrations of some of the possible methods and physical embodiments. The present invention is thus not limited to the above theoretical modeling and physical embodiments, but can be changed or modified in various ways on the basis of the general principles of the invention.

Note that the theoretical discussion provided in this disclosure reuses some mathematical symbols to mean different things in different locations of the text for historical and pragmatic reasons. The meaning is readily discernible by those skilled in the art.

Thus the scope of the invention should be determined by the appended claims and their legal equivalent and not only by the examples, embodiments and physical theory provided in this disclosure.

Industrial Applicability

[133] This invention has applicability for controlling light. The control includes such optical operations as beam steering and focusing as well as general wavefront modification over large angles and solid angles without significant restrictions due to polarization, direction of input light and direction output light. Specific applications include, but are not limited to: LiDAR, electronically focused lenses for cameras and cell phones, electronically steered automotive headlights, projector display systems, light-art, free-space photonic network configurations for comput- ing, directed-energy laser steering for welding, topographic mapping, automated inspection, remote sensing, analysis of biological samples in a microfluidic lab on a chip and point-to-point communications

Reference Signs List

[134]

2a Electrode 4b Smith Chart Point

2b Electrode 35 4c Smith Chart Point

2c Dielectric Liquid 4d Smith Chart Point

2d Spherical Dielectric Nanoparticle 4e Smith Chart Point

2e Boundary 4f Zero-Force Point

2f Boundary 5a Dielectric Nano- Sphere

2g Boundary 40 5b Incident Light Field

4a Smith Chart Point 5c Differential Annulus 5d Far- Field- Point 35 9c Electric Field

5e Opaque Screen 9d Constant Potential Line

6a Micro Fluidic Control Channel 9e Index Gradient Liquid

6b Transparent Dielectric 9f Light Beam

6c Index Matching Liquid 10a Index Gradient Liquid

6d Input Rays 40 10b Light Beam

6e Propagation Direction 11a Curved Light Trajectory

6f Input Rays lib Free-Space Light Beam

6g External Output Rays 11c Containment Vessel

6h Direction Arrow lid Cut- Away Region

6i Direction Arrow 45 lie Micro-Fluidic Control Channel

6j Vacuum Refractive Index llf Electrodes

6k solid medium refractive index llg Electrodes

7a Microfluidic Control Channel llh Electrode Boundary

7b Input Rays 12a Segmented Electrodes

7c Propagation Direction 50 12b Symmetry Axis

7d Index Gradient Liquid (IGL) 12c Upper Electrode

7e Gradient Spoiled Region 12d Lower Electrode

7f Output Rays 12e Off-State Angular Range 7g Solid Transparent Material 13a Positive Dielectrophoresis

7h External Output Rays 55 13b Negative Dielectrophoresis

7i Direction Arrow 15a Elevation Control Electrodes

7j Direction Arrow 15b Elevation Control Electrodes

7k Linearized Refractive Index 15c New Elevation Beam Position 7m Uniform Refractive Index 16a Input Light

7n Solid Refractive Index Region 60 16b Microfluidic Control Channel

7o Solid Refractive Index Region 16c Light Propagation Direction

8a Beam Steering Device 16d Dielectrophoresis Disrupted Reg

8b Electrode 16e Rectilinear Propagation

8c Electrode 16f Transparent Containment Vessel

8d Index Gradient Liquid 65 16g Free-Space Light Propagation

8e Light Beam 16h Direction Arrow

9a Electrode 16i Direction Arrow

9b Electrode 16j Containment Vessel Refractive dex 35 20i Second Containment Structure 16k Containment Vessel Refractive In21a Electrode

dex 21b Solid Dielectric Material

16m GRIN Profile 21c Colloidal Nanoparticle Fluid 5 16n Uniform Refractive Index 21d Electrode

17a Electrode 40 21e Solid Dielectric Material

17b Electrode 21f Colloidal Nanoparticle Fluid

17c Refractive Index Gradient Liquid 22a Electrode

17d Equipotential Curves 22b Electrode

10 17e Electric Field Curves 23a Electrode

18a Input Light 45 23b Amplitude Variations

18b First Containment Structure 23c Derivative of Amplitude Variations

18c Second Containment Structure 25a Transparent Radome

18d Index Gradient Liquid 25b Radome Inner Surface

15 18e First Annular Electrode 25c Optional Inner Solid

18f Second Annular Electrode 50 25d Outer Surface of the Inner Solid

18g Electrical Insulator 25e Microfluidic Control Channel

18h Electric Fields 25f Nanoparticle Colloid

18i Symmetry Axis 25g Input Light Beam

20 18j Output Light 25h Angle Selective Mirror

18k Converging Lens Refractive Indexi5 25i Upward Spiraling Light Trajectory

18m Diverging Lens Refractive Index 25j Downward Spiraling Light Trajec¬

19a First Annular Electrode tory

19b Second Annular Electrode 25k Reflection Device

25 19c Electrical Insulator 25m Point

19d Tapered Electrode 60 25n Internal Rectilinear Trajectory

20a Input Light 25o Point

20b Output Light 25p Internal Rectilinear Trajectory

20c Dielectrophoresis Lens 25q First Magnification Inset

30 20d Optical Axis 25r Nanoparticle Gradient Vector

20e First Annular Electrode 65 25s Nanoparticle Gradient Vector

20f Second Annular Electrode 25t Nanoparticle Gradient Vector

20g Electrical Insulator 25u Second Magnification Inset

20h First Containment Structure 25 v Transparent Transistors 25w Transparent Electrodes 26g Cutaway Arc

26a Concentric Electrodes 28a Dielectric Spherical Manifold 26b Colloid Containment Volume 28b Meridian Geodesic

26c Optical Axis 28c Main Lobe

26d Input Light 28d Side Lobe

26e Containment Structure 28e Linearly Propagating Light 26f Focused Output Light