A Scanning Mirror Apparatus and Methods of Designing and Fabrication

FIELD OF THE INVENTION

The disclosed technique relates to optical devices in general, and to a micro electro mechanical system and method to provide non-sinusoidal oscillatory motion to a scanner.

BACKGROUND OF THE INVENTION

Micro Electro Mechanical System (MEMS) scanning devices and their applications, for example as optical scanners are known in the art. One limitation of MEMS devices is the limited force which may be generated by force producing MEMS elements. Thus, it is advantageous to use resonating systems with mechanical gain to be able to achieve large motion span under limited forces. Oscillating mirrors are employed to scan objects and raster-scan displays. Such a mirror is generally connected to two vibrating flexural beams, thereby forming a single degree-of-freedom (SDOF) structure, wherein the structure has a single torsional natural frequency at which the device can resonate. Such scanners oscillate according to a sinusoidal waveform. The high gain (i.e. , large compliance) which is exhibited by a second order system at its natural frequency (especially when damping is low), gives rise to a significant angular deflection under a small or moderate sinusoidal torque.

Sinusoidal motion of the mirror reflects the light beam in a nonuniform manner, thereby yielding non-uniform intensity and pixel spacing and hence low optical display quality. It is possible to improve the display quality, if the mirror oscillates according at a uniform rate i.e. according to a triangular waveform.

However, the level of the torque, which is to be applied to the mirror in order to provide oscillatory motion having the triangular waveform, is approximately two orders of magnitude greater than in the case of sinusoidal motion. In large scale applications, where large torques can be produced, it is possible to produce this additional torque. However in small scale applications, such as micro-electromechanical systems (MOEMS), due to the inherently small dimensions and the limitation of the commonly used electrostatic excitation, it is much more difficult to provide the needed torque.

WO04095111 entitled "scanning mirror", to Velger, Bucher and

Zimmerman; filed 2004-04-20 discloses: Periodic-waveform oscillator for processing light, the geometric waveform oscillator including a plurality of masses, at least one force producing element, and a plurality of elastic elements, each of the force producing elements being coupled with a respective one of the masses, at least one of the masses including a light processing module, each of the force producing elements applying a force to the masses, the elastic elements coupling the masses together, the elastic elements coupling the masses with a respective support, wherein the mass values of the masses, the force values of the forces, and the stiffness coefficients of the elastic elements, are selected such that the light processing module oscillates according to the periodic-waveform.

The disclosed device as described in the WO04095111 has a symmetrical construction. Specifically, WO04095111 discloses a five masses and six springs configuration. The disclosed device as described in the WO04095111 is based on narrow resonance peaks resulting from the interaction among masses and springs composing the disclosed device. Said resonance peaks must have predetermined ratio of frequencies in order to produce the desired periodic- waveform oscillation. WO04095111 The disclosed device as described in the WO04095111 uses five masses in symmetric construction wherein the scanning mirror is in the device's center to create a scanning mirror device for non sinusoidal

motion. Figure 3 in WO04095111 depicts the device of Velger, Bucher and Zimmerman.

As can easily be seen, the device used in the art comprises five masses (202,204,210,206 and 208) and six elastic elements in a symmetric configuration.

Generally, for N Degrees Of Freedom (DOF), the configuration used in the art requires 2*N-1 masses and 2*N elastic elements. The large number of elements may adversely affect the total size of the device and its complexity; hence it may adversely affect the manufacturing requirements, manufacturing yield and fabrication and packaging costs.

References

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[3] Chiao M and Lin L 2003 Post-packaging tuning of microresonators by pulsed laser deposition 12th International Conference on TRANSDUCERS, Solid-State Sensors, Actuators and Microsystems 2 1820-3

[4] Syms R.R.A. 1998 Electrothermal frequency tuning of folded and coupled vibrating micromechanical resonators Journal of Microelectromechanical Systems 7 Issue: 2 pp 164-171

[5] Madou Mark J 2001 Fundamentals of MicroFabrication: the Science of Miniaturization 2'nd edition (NY: CRC Press)

[6] Daphne Joachim and Liwei Lin 1999 Localized Deposition of Poly silicon for MEMS post-fabrication processing MEMS-VoI. 1 Microeletromechancial systems (MEMS) - 1999 ASME 1999

[7] K. Tarlaku, YMochida, S.Sugimolo K.Moriya, T. Husegawa, K.Atsuchi, and K. Ohwuda 1995 A MICROMACHINED VIBRATING GYROSCOPE

[8] Ki Bang Lee Lin, L. Young-Ho Cho 2004 A frequency-tunable microactuator with a varied comb-width profile Micro Electro Mechanical Systems, 2004. 17th IEEE International Conference on. (MEMS) On page(s): 257- 260

[9] Ki Bang Lee and Young-Ho Cho 1997 Frequency Tuning of a Laterally Driven Microresonator Using an Electrostatic Comb Array of Linearly Varied Length Transducers 1997 International Conference on Solid-state Sensors and Actuators

[10] Yao J J and MacDonald N C 1996 A micromachined single crystal silicon tunable resonator J. Micromech. Microeng. 6 257-264

[11] Kun Wang, Ark-Chew Wong, Wan-Thai Hsu, and Clark T.-C. Nguyen 1997 Frequency Trimming and Q-Factor Enhancement of Micromechanical Resonators Via Localized Filament Annealing Transducers '97 international Conference on Solid-state Sensors and Actuators .Chicago, USA.

[12] Fan Z, Wang J and Goodman E 2005 An Evolutionary Approach For Robust Layout Synthesis of MEMS Proceedings of the 2005 IEEE/ASME International Conference on Advanced Intelligent Mechatronics Monterey, California, USA.

[13] Mawardi A and Pitchumani R 2005 Design of Microresonators under Uncertainty J. of Microθlectromechanical Sys. 14 no. 1

[14] Liu R, Paden B and Turner K 2002 MEMS Resonators That Are Robust to Process-Induced Feature Width Variations J. of Microelectromechanical Sys. 11 no. 5

[15] Naftali M Supervisor Assoc. Prof. David E 2004 Towards a Linear Response of Vertical Comb-Drive Actuators M. Sc Thesis Faculty of Mechanical Engineering Technion Israel

[16] Minhang Bao 2005 Analysis and design principles of MEMS devices 1'st edition (Amsterdam, Netherlands: ELSEVIER B.V.)

[17] Minikes A, Bucher I and Avivi G 2005 Damping of a micro-resonator torsion mirror in rarefied gas ambient J. Micromech. Microeng 15 1762-9

[18] Meriam J L and Kraige L G 2003 Engineering Mechanics VoH Statics 5'th edition SI version (USA: John Wiley & Sons, Inc.) pg 485

SUMMARY OF THE INVENTION

According to an exemplary embodiment of the current invention a periodic-waveform oscillator is provided comprising: at least one support; a plurality of masses; at least one force producing element, said at least one force producing element applying force to at least one of said masses; and a plurality of elastic elements, said elastic elements coupling said masses together, said elastic elements coupling at least one of said masses with a respective to at least one support, wherein the mass values of said masses, the force value of said at least one force, and the stiffness coefficients of said elastic elements, are selected such that at least one mass oscillates according to a periodic-waveform, and wherein said configuration of said masses and said elastic elements is asymmetric.

In some embodiments the geometric waveform is non-sinusoidal.

In some embodiments the non-sinusoidal geometric waveform is triangular.

In some embodiments the oscillator is substantially made of semiconductor.

According to an exemplary embodiment of the current invention a periodic-waveform oscillator is provided comprising: A first and second support; a first, second and third mass; a first elastic element connecting said first support to said first mass; a second elastic element connecting said first mass to said second mass; a third elastic element connecting said second mass to said third mass; a force producing element connected to said third mass; and a forth elastic element connecting said third mass to said second support.

In some embodiments the third mass comprises an optical mirror.. In some embodiments the oscillator is substantially made of semiconductor.

According to an exemplary embodiment of the current invention a periodic-waveform oscillator produced by lithographically producing trenches in a substrate, having reduced sensitivity to manufacturing inaccuracies is provided comprising: at least one support; at least a first a second and a third masses; at least one force producing element , said at least one force producing element applying force to at least one of said masses; and at least a first, a second and a third elastic elements, wherein said first elastic element connects said support to said first mass, said second elastic element connects said first mass to said second mass, and said thirds elastic element connects said second mass to said third mass; wherein the ratio of the change of inertial properties of first mass to the change in trench width due to change in trench width caused by lithographic inaccuracies is substantially the same as the ratio of the change of inertial properties of second mass to said change in trench width due to said change in trench width.

In some embodiments the ratio of the change of stiffness of said first elastic element due to said change in trench width to said change in trench

width is substantially the same as the ratio of the change of stiffness of said second elastic element due to said change in trench to said change in trench width

In some embodiments the third mass comprises an optical mirror. In some embodiments the substrate is substantially made of semiconductor.

According to an exemplary embodiment of the current invention a method for designing a periodic-waveform oscillator comprising plurality of masses and plurality of elastic elements, produced by lithography, and having reduced sensitivity to manufacturing inaccuracies is provided comprising the steps of: selecting geometrical parameters of at least first, second and third mass such that the ratio of the change of inertial properties of first mass due to small change in lithography conditions to inertial properties of said second mass due to said small change in lithography conditions is substantially the same as the ratio of the change of inertial properties of first mass due to said small change in lithography conditions to inertial properties of said third mass due to said small change in lithography conditions.

In some embodiments the method further comprising the steps of: selecting geometrical parameters of at least first, second third elastic elements such that the ratio of the change of elastic properties of first elastic element due to small change in lithography conditions to elastic properties of said second elastic element due to said small change in lithography conditions is substantially the same as the ratio of the change of elastic properties of first elastic element due to said small change in lithography conditions to elastic properties of said third elastic element due to said small change in lithography conditions.

In some embodiments the change in lithography conditions comprises change in width of lithographically produced trenches.

According to an exemplary embodiment of the current invention a method for the evaluation and correction of natural frequency accuracy in a multi-degree of freedom system having at least three resonance frequencies is provided comprising the steps of: measuring frequencies f-t, f ₂ and f ₃ of first,

second and third resonance respectively; calculating deviation parameters C ₂ and C ₃ wherein C ₂ equal to (f ₂ - fι ^* n ₂ )/fi and C ₃ equal to (f- fi*n ₃ )/f1 wherein n ₂ and n ₃ are the desired frequency multiplications of f ₂ and f ₃ respectively; determine from a model of said multi-degree of freedom system the dependencies of said C ₂ and C ₃ on physical parameters of said system; and affecting at least one of said physical parameters of said multi-degree of freedom system to correct the natural frequency multiplication of said multi- degree of freedom system, wherein said affecting at least one of said physical parameters is based on calculated C ₂ and C ₃ . In some embodiments the affecting at least one of said physical parameters of said multi-degree of freedom system comprises changing the mass of at least one mass in said system.

In some embodiments the affecting at least one of said physical parameters of said multi-degree of freedom system comprises changing the stiffness of at least one elastic element in said system.

In some embodiments the changing the stiffness of at least one elastic element in said system is done electrically.

In some embodiments the changing the stiffness of said at least one elastic element in said system comprises etching said at least one elastic element.

In some embodiments the changing the mass of said at least one mass in said system comprises etching said at least one mass.

Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. Although methods and materials similar or equivalent to those described herein can be used in the practice or testing of the present invention, suitable methods and materials are described below. In case of conflict, the patent specification, including definitions, will control. In addition, the materials, methods, and examples are illustrative only and not intended to be limiting.

The foregoing summary, as well as the following detailed description of certain embodiments of the present invention, will be better understood when

read in conjunction with the appended drawings. The figures illustrate diagrams of the functional blocks of various embodiments. The functional blocks are not necessarily indicative of the division between hardware circuitry. Thus, for example, one or more of the functional blocks (e.g., processors or memories) may be implemented in a single piece of hardware (e.g., a general purpose signal processor or a block or random access memory, hard disk, or the like). Similarly, the programs may be stand alone programs, may be incorporated as subroutines in an operating system, may be functions in an installed imaging software package, and the like. It should be understood that the various embodiments are not limited to the arrangements and instrumentality shown in the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

Some embodiments of the invention are herein described, by way of example only, with reference to the accompanying drawings. With specific reference now to the drawings in detail, it is stressed that the particulars shown are by way of example and for purposes of illustrative discussion of the preferred embodiments of the present invention only, and are presented in the cause of providing what is believed to be the most useful and readily understood description of the principles and conceptual aspects of the invention. In this regard, no attempt is made to show structural details of the invention in more detail than is necessary for a fundamental understanding of the invention, the description taken with the drawings making apparent to those skilled in the art how the several forms of the invention may be embodied in practice In the drawings:

Figure 1 schematically depicts a top view of the moving part of a scanning mirror device according to an exemplary embodiment of the current invention.

Figure 2a schematically depicts a top view of the moving part of a scanning mirror device having more than three masses according to another exemplary embodiment of the current invention.

Figure 2b schematically depicts a top view of the moving part of a scanning mirror device according to an exemplary embodiment of the current invention.

Figure 3 schematically depicts the mathematical representation of a 3 DOF system 309 according to the current invention:

Figure 4 depicts the measured frequency response of the motion of m ₃ to a sinusoidal force exerted on rri ₃ in rotational system such as depicted in figure 1.

Figure 5 schematically depicts a semiconducting wafer 500 and fabricated MEMS devices.

Figure 6 schematically depicts an exemplary topological design of an asymmetric 3 DOF system according to an embodiment of the current invention.

Figure 7 schematically depicts the variations in feature dimensions caused by variation during fabrication process

Figure 8 shows a plot showing the matching of ^ to ^ as a function of the dimension w of the mass, as calculated according to an exemplary method of the current invention.

Figure 9 shows a plot of & for i=1 ,2,3 as a function of spring width w _s as calculated according to an exemplary embodiment of the current invention.

Figure 10 shows the impact of trench tolerance TWT on c ₃ c ₅ plane accuracy for the method according to the current invention.

Figure 11 shows the impact of trench tolerance TWT on fi accuracy for the method according to the current invention.

Figures 12a, 12b and 12c show a summarizing graph for systems having a total length of 2500 μm designed according to an exemplary embodiment of the current invention.

Figures 13a and 13b show a summarizing graph for systems having same width spring as calculated according to embodiments of the current invention.

Figure 14 depicts designs of five distinct different devices designed to be located in the origin and four quarters of the C ₃ C ₅ according to the current invention.

Figure 15 shows a photograph of a fabricated 3 DOF device according to the current invention showing the VCD structure.

Figure 16 shows experimental results of deviation from design goals, measured from fabricated 3 DOF devices designed according to the current invention.

Figure 17 illustrates the measured deflection angle as a function of time of a mirror with accuracies of (c ₃ ,c ₅ )=(-0.15%, 0.20%) taken at

atmospheric pressure and a DC and AC maximum voltage of 50 V operated at a basic frequency of 13.396 KHz.

Figure 18 shows a graph an exemplary measurement results of frequency response for a specific manufactured device.

Figure 19 shows depicts the measured deviation of the above example plotted in the C ₃ C ₅ plane according to an embodiment of the invention.

Figure 20 depicts the change of C ₃ C ₅ for change in the mass of the scanning mirror m ₃ according to an embodiment of the invention.

Figure 21 depicts the change of C ₃ C ₅ for change in the masses m-i, m ₂ , m ₃ and the springs k-i, k ₂ and k ₃ according to an embodiment of the invention.

Figure 22 depicts the changes in C ₃ C ₅ for increase in the masses m-i, m ₂ , m ₃ according to an embodiment of the invention.

DESCRIPTION OF THE EMBODIMENTS

The invention is herein described, by way of example only, with reference to the accompanying drawings. With specific reference now to the drawings in detail, it is stressed that the particulars shown are by way of example and for purposes of illustrative discussion of the preferred embodiments of the present invention only, and are presented in the cause of providing what is believed to be the most useful and readily understood description of the principles and conceptual aspects of the invention. In this regard, no attempt is made to show structural details of the invention in more detail than is necessary for a fundamental understanding of the invention, the

description taken with the drawings making apparent to those skilled in the art how the several forms of the invention may be embodied in practice.

Before explaining at least one embodiment of the invention in detail, it is to be understood that the invention is not limited in its application to the details of construction and the arrangement of the components set forth in the following description or illustrated in the drawings. The invention is capable of other embodiments or of being practiced or carried out in various ways. Also, it is to be understood that the phraseology and terminology employed herein is for the purpose of description and should not be regarded as limiting. In discussion of the various figures described herein below, like numbers refer to like parts.

The drawings are generally not to scale. Some optional parts were drawn using dashed lines.

For clarity, non-essential elements were omitted from some of the drawings.

Asymmetric construction of a scanning mirror device.

One aspect of the invention is to. provide a scanning mirror device having asymmetric configuration.

Asymmetric device according to first aspect of the invention

In contrast to the device of the art, the device according to the current invention uses a non-symmetric construction, wherein the scanning mirror is at one end of a chain of masses connected one to another, and to at least one support at the other end of the chain with elastic elements. Preferably, the chain of masses is connected to a second support at the scanning mass end as well. In one embodiment of the invention, the device is constructed of three masses wherein the scanning mirror is the last mass in the chain. However in

other embodiments of the current invention might have a larger number of masses in the chain of masses.

At least one force generating element exerts force on at least one of the masses. Preferably, a force generating element is connected directly to the scanning mirror.

Alternatively or additionally, force generating element is connected to other masses in the chain.

Optionally, driving forces to different force generating elements is different in at least one of: amplitude, spectral content and phase.

Optionally, other force generating means may be used, for example: parallel plate electrostatic actuator, for example the type used in the device of the art; magnetic actuator, thermo-elastic actuator, or piezo-electric actuator.

The actuator may be connected to a mass using an elastic element. The asymmetric construction according to the embodiment enables separation of the force generating element, preferably connected to the scanning mass, and the dynamic chain of masses.

In the preferred embodiment, the force generating element is a linear actuation device in the form of a of Vertical Comb-Drive actuator (VCD). Optionally, VCD is situated between the scanning mass and the elastic element connecting to the second support. However, the VCD may be connected to the scanning mass and to an elastic element connected to the dynamic chain.

Figure 1 schematically depicts a top view of the moving part of a scanning mirror device according to an exemplary embodiment of the current invention.

Note that the stator of the VCD is not shown in this drawing. In this embodiment, the device is fixed at both ends.

Moving part 100 of the scanning mirror device according to an exemplary embodiment of the current invention is attached to the stator of the device (not seen in this drawing) at first and second supports 110a and 110b respectively. Moving part 100 further comprises masses 130a, 130b and

130c. Mass 130c comprises moving mirror 132 and force generating structure 134. In this embodiment, mirror 132 is circular. In the depicted embodiment force generating structure 134 is a VCDA structure.

Elastic elements 120a, 120b, 120c and 12Od in the form of long narrow bars connect the first support, the first second and third masses and second support respectively.

It should be noted that the second support 110b and forth elastic element 12Od are optional and are used for stabilizing the moving part 100. It also should be noted that order of moving mirror 132 and force generating structure 134 may be exchanged. In this exemplary embodiment, moving mirror 132 and force generating structure 134 are rigidly connected to each other.

Figure 2a schematically depicts a top view of the moving part of a scanning mirror device having more than three masses according to another exemplary embodiment of the current invention.

Note that the stator of the VCD is not shown in this drawing. In this embodiment, the device is fixed at both ends.

Moving part 200 of the scanning mirror device according to an exemplary embodiment of the current invention is attached to the stator of the device (not seen in this drawing) at first and second supports 210a and 210b respectively. Moving part 200 further comprises n masses 130(1), 130(2) ...

130(n) and mass 230c. Mass 230c comprises moving mirror 232 and force generating structure 234. In this embodiment, mirror 232 is circular. In the depicted embodiment force generating structure 234 is a VCDA structure. Elastic elements 220(1), 220(2),... 220(n), 220(n+1) and 220d in the form of long narrow bars connect the masses to each other and to the supports.

It should be noted that the second support 210b and forth elastic element 22Od are optional and are used for stabilizing the moving part 200. It also should be noted that order of moving mirror 232 and force generating structure 234 may be exchanged. In this exemplary embodiment, moving

mirror 232 and force generating structure 234 are rigidly connected to each other.

The drawing beiow schematically depicts a scanning mirror device according to another embodiment of the invention having one support and a limited span force generating element.

Figure 2b schematically depicts a top view of the moving part of a scanning mirror device according to an exemplary embodiment of the current invention.

Note that the stator of the VCD is not shown in this drawing. In this embodiment, the device is fixed at both ends.

Moving part 150 of the scanning mirror device according to an exemplary embodiment of the current invention is attached to the stator of the device (not seen in this drawing) at support 160a. Moving part 150 further comprises masses 180a, 180b and 182c. Mass 182 is a moving mirror 182. In this embodiment, mirror 182 is circular. In the depicted embodiment force generating structure 184 is a VCDA structure.

Elastic elements 170a, 170b, 170c and 17Od in the form of long narrow bars connect the support, the masses and force generating structure.

In this exemplary embodiment, moving mirror 182 and force generating structure 184 are connected to each other via elastic element 17Od.

As can clearly seen, for the asymmetric configuration with N DOF, according to the current invention, the number of masses is N; and the number of elastic elements is N+1 (N for single support system; N+2 for limited span force generating element and dual support). The asymmetric configuration uses fewer elements. Thus the configuration according to the embodiments of the invention has reduced total size and reduced complexity. The reduced size, number of elements and complexity may: ease production, increase robustness and accuracy, increase production yield and decrease fabrication and packaging costs. Below is theoretical mathematical support showing the ability to design an asymmetric scanning device.

Theory

A periodic triangular response signal (motion of 01 ₃ ) may be decomposed to its Fourier components according to:

θ ₍ λ = ^cos ^ ^{h ω} o '^ _] ∞ ^β β' ^β fr 'O , cos(5 ^• <■% ^{• .} -) _| cos(7-a> ₀ -Q

I ² 3 ² 5 ² 7 ²

A limited accuracy reproduction of the signal may be constructed using the first few terms in the Fourier infinite sum. For example taking the first three terms of frequencies ωo, 3ω ₀ and 5ω ₀ .

Figure 3 schematically depicts the mathematical representation of a 3 DOF system 309 according to the current invention:

In this representation, q _\ is the motion of mass (i); and the force is assumed to be applied to the last mass (1TI ₃ , far from anchor 303) in this case). A 3 DOF system designed to have resonance components compatible with the Fourier representation approximating a triangular wave is mathematically depicted by it ratios among value of components:

Masses, where the values of the mass obey:

And springs, where the value of the springs obey:

₇ ⁵⁰ ₂ 90 ₂ , _{1 C 2}

A ₁ = CO _Q - Tn ₁ K ₂ = ω _a - m ₃ £ ₃ = 15 ^• ω _ϋ ^■ m ₃

These desired ratios among values of components is obtained by solving the Newtonian equation of motion for the system comprising masses and spring and requiring that the desired values for resonance frequencies match the values as depicted by the Fourier transform of the desired approximated waveform.

It should be noted that more complex systems and waveforms may similarly analyzed.

To translate this method of mathematical analysis to the torsion and rotation system such as depicted in figures 1 , 2a and 2b, the mass value of the masses are replaced with its moment of inertia, the spring linear values of each elastic bar with the tortional elasticity; the velocities with angular velocities and forces with torsions. The equations and their solutions are generally unchanged. Real structure may be treated more accurately by adding second order effects such as energy dissipation, for example due to air friction and energy dissipation in twisted bar; flexure of the masses; moment of inertia of the elastic bars, etc. Such complex systems may produce equation requiring numerical solution or iterative solutions. However, methods for solving the behavior of these systems are known in the art of mechanical engineering.

Figure 4 depicts the measured frequency response of the motion of m ₃ to a sinusoidal force exerted on m ₃ in rotational system such as depicted in figure 1.

It should be noted that the system according to the invention has high mechanical gain at the desired frequencies (3ωo, 5ωo, etc).

Generally, the gain amplitude and spectral width depends on damping of the structure caused by friction and inelastic components. Physical driving force generators are sometimes non linear. Driving such non-linear driving force generators at frequency ωo, unavoidably generates force having frequency components at 2ωo, 4ω ₀ , etc.

It can be seen that the system according to the current invention has very low response for the even harmonics (2ω ₀ , 4ω ₀ ) of the driving force. It also should be noted that the anti-resonance frequencies (also known as zeros) are almost exactly symmetrically interlaced between the natural frequencies.

The new and original ideas characterizing the depicted system according to the current invention are: 1. symmetric configuration.

2. The system has exactly the number of degrees of freedom (masses) that corresponds to the number of required natural frequencies.

3. Location of zeros (anti resonance frequencies) is not controlled but they are inherently located between the peak resonances (poles).

Geometrical variance matching for the elimination/reduction of inaccuracy in the natural frequency multiplications in a multi DOF scanning mirror

According to a second aspect of the invention, a method for designing a multi DOF system is provided. The method according to the invention ensures that the fabricated device would have reduced sensitivity to manufacturing imperfections.

The inventive method may reduce or eliminate all together the need to have any type of post fabrication corrective process in order to enhance the compliance of the device to triangular wave form.

Such a method translates to higher accuracy and yield, thus may reduce fabrication cost.

It should be noted that some exemplary embodiments are given for a 3-mass asymmetric construction, but the method may be extended to other forms, such as symmetric configuration or to larger number of masses.

The invention is suitable for any arbitrary, periodic waveform generator. By Fourier decomposition of the desired wave form, the number and strength of the resonances may be determined and a suitable system may than be designed and fabricated. It should also be noted that that some exemplary embodiments are given for a device used as a scanning mirror, and specifically for a scanning mirror with scanning waveform approximating a triangular motion , but the method may be extended to other waveforms and other applications.

In one specific embodiment, the method comprises of the following steps:

1. A theoretical lumped model of the system is created. Preferably, in his model, the elements (masses and elastic elements) are defined by their lumped mechanical properties such as elastic compliance and the rotational and or linear moments of inertia. 2. According to the example herein, the third mass is used as a scanning mirror for an optical system. The requirements for the scanning mirror are usually size and flatness. During motion, the mass withstands large accelerations. The requirement of minimal dynamic flatness combines with other parameters such as the shape; the scanning frequency; angular motion span; and material parameters such as elasticity and density mandates a minimal thickness to the mirror. Preferably, a minimal thickness is chosen for the design. Preferably, the device is made from a layer of material, preferably Silicon having uniform thickness. Knowing the shape, the thickness and the density, the mass (and moment of inertia) of the third mass is determined. Optionally, the force creating element is integrated into the third mass and thus its inertia is added.

3. Using the lump model of the system, inertial values for the masses and compliance values for the elastic elements are calculated. Preferably, requiring resonance frequencies of three times and five times the scanning frequency as mandated by the desire to produce triangular scanning waveform.

4. Using the shape of the third mass, the mass variation parameter ψ(3) for the third mass can be calculated.

5. Requiring that all mass variation parameters would be equal to ψ(3) yields finite number of solutions to all other masses. Preferably, the solutions obtained for the shapes of each of the other masses are rectangles oriented along or perpendicular to the system's axis. Preferably, the perpendicular solution is chosen as it yields a shorter overall system length.

6. Using the compliance values of the elastic elements while requiring that the elastic variation parameter φ of all the elastic elements would be equal, yields limitations on the shape of the elastic elements. Preferably, elastic elements shaped as straight bar are chosen. Optionally,

the total length of the system is in trade off to the value of the elastic variation parameter.

7. Once the initial shapes of all masses and elastic elements were selected using the above steps, a finite element model of the system is created using these initial shapes.

8. Finite element model is used for optimizing the suggested shape of the system requiring at least that the resonance frequencies of the system would be as desired, and optionally that the variation parameters of all masses would be the same and optionally that the variation parameter of all the elastic elements would be the same.

9. The optimization is preferably done in iterative way. Stopping the optimization is preferably determined by a convergence requirement or optionally by maximum number of iterations. The results are used for designing the masks used for producing the device. It should be noted that optimization criteria may comprise one or few of: requirements on the base operation frequency and the response of the system to the base frequency and some of its harmonics; tolerance to manufacturing inaccuracy including but not limited to trench width variations. 10. Optimally, plurality such devices are manufactured from a single wafer, for example by using step and repeat lithography method.

It should be noted that the order of steps could be modified and some of the steps may be eliminated of modified within the scope of the current invention.

More details, explanations and embodiments for the method according to the current invention are given below.

1. Introduction

We have established that in order to benefit from the high compliance of the multi Degrees Of Freedom (DOF) system, only the multiplications of the natural frequencies should be accurate.

Herein we will analyze the causes for natural frequencies multiplications inaccuracy. We will then introduce an innovative method that enables to reduce or even eliminate inaccuracy due to the contributing factors, which are: • Material properties; and • Dimensional accuracy.

We will demonstrate the method as an example on a lumped 3 degrees of freedom model made from the same isotropic material.

The specific example herein is a lumped model of a system mounted on one side only made of isotropic material.

2. Lumped Model Analysis

2.1 General Consideration: Scaling of the generalized eigenvalue problem

The natural frequencies of an oscillatory system are related to the mass and stiffness matrices

Where both M and K can relate either to a translational or to rotational momentum and stiffness multi DOF system.

The natural frequencies correspond to the eigenvectors

(M ^~ι K}-v = λ-v (5)

We will now point out an important observation that will be used later, that if M ^-1 ZiS multiplied by a scalar αthen

It is evident that a-λ meets the condition to be an eigenvalue of a-M ^~l K while the eigenvector is unchanged, so

Similarly we can show that if a-λ is an eigenvalue of a- M ^~l K then λ is an eigenvalue of M ^~l K , so if λ is an eigenvalue of M ^-1 Z then a-λ is an eigenvalue of a -W ¹ K and vise versa.

Thus the accuracy of natural frequencies multiplications in matrix

M ^-1 ZWiII not change in the scaled matrix a-M ^~l K .

This simple yet important observation brings us to the core idea behind the proposed method that if changes in the original matrix AT ¹ Z , due to any contributing factors, will result only in the scaling of M ^~l K then the intended natural frequency multiplication accuracy will be kept invariant.

Before continuing we will present a new multi DOF structure on which the method will be demonstrated.

2.2 An improved 3 DOF Structure for uniform rate scanning

The symmetric 5 DOF Micro Mirror design presented in [1] enabled the selection of all 5 natural frequencies, 3 of them are the resonance frequencies and 2 of them coincide with anti resonance frequencies.

The minimal order of a multi DOF system for the creation of a 3 term triangular is only 3. But the theoretical development was of a 5 DOF system as implemented in [1] in order to be able to predetermine the anti resonance frequencies. A close examination of the theoretical basis of the concept as discussed in [1] reveals that the asymmetric part having free-clamped boundary conditions has exactly the same odd-numbered natural frequencies as the full system. A simple analysis shows that anti-resonance frequencies in

such a case are located between the resonances owing to the well know interlacing property. Although the location of those anti resonances cannot be pre-determined by altering the mass and stiffness values in the same manner as before, its location is close to were we would have wanted them to be in order to suppress the even harmonics 2ω ₀ , 4ω ₀ .

The proposed new structure is depicted schematically in figure 3. Actuation, denoted as force F, is exerted on mass πi ₃ which is the mirror.

The values and mass and stiffness can be easily found using the algorithm described in [1] and are given in equations (8) through (12).

OT ₁ = OT ₃ (8)

7

_ 15

/7Z ₂ OT ₃ (9)

^~ 7

90 h - ^■ ω ² • m-. (11)

^" 7

k ₃ - = 15 - ω ² - OT ₃ (12)

The main and most important benefit from utilizing the 3 DOF asymmetric design compared to the 5 DOF design, is the geometrical separation of the system into two distinct regions: the dynamic side (right side in figure 3), responsible for the high compliance to the triangular wave and the actuation side (left side in figure 3). Any actuation method can be used with the proposed design. This separation enables great flexibility in the design and implementation of an actual MEMS device.

Other benefits, compared to the 5 DOF symmetric systems are: length reduction by a factor of 2 which will result in reduced die size and thus significantly reduce fabrication and packaging costs and higher stiffness of the springs, and, as will become apparent later, reduces the system sensitivity to fabrication inaccuracy.

2.3 General Consideration: elements form in the matrix M ¹ K

Let us now continue the discussion and consider the 3 DOF systems shown in figure 3 as a typical case, the general approach is also valid for similar systems.

The matrix M ¹ K takes the form

So all non-zero elements in matrix (13) have the same form, soon to be found of great value:

(14) m _n

Where m,n = 1,2,3

2.4 Material properties effect on multiplications of the natural frequencies

Most of the resonating MEMS devices are implemented using Silicon

On Insulator (SOI) technology using a Deep Reactive Ion Etching (DRIE)

process. Thus all elements in the system are "carved out" from the device layer which is made from Single Crystal Silicone (SCS).

Material properties of SCS are reported to have different values [5] pp 198 and fabrication process is also known or assumed to have an effect on the Material properties, this usually introduces a difficulty if absolute value of natural frequencies is required.

Figure 5 schematically depicts a semiconducting wafer 500 and fabricated MEMS devices.

Since the device layer is made from a Single Crystal, its uniformity is good over the entire layer and the amount of defects, if any, are assumed to be very low, thus we can assume that mechanical material properties are uniform within a unique MEMS system region, which is contained within a small region of the device layer, usually only few thousands of microns in length as depicted in figure 5. It should be emphasized that material properties might change from the area of one system on the wafer to another, as depicted as areas a,b and c in figure 5 showing different elasticity modules and density at different regions.

We can also reasonably assume that process variations effect on material properties are territory based and are continuous in nature so we can assume that our assumption regarding the uniformity of material properties within a unique system still holds.

We will continue our discussion and for simplicity limit our proof to isotropic material. Generally, mass (or inertia) and stiffness depend on material properties and geometry alone, furthermore they are usually linear functions of material properties so matrix elements m _n , k _m can be represented as K -K -K (16)

Where m _n ,k _m are functions of geometry only and are independent of the material properties: density and elasticity, respectively. We will also note at this point that E represents a generalized elasticity module and can represent shear modules G as well.

We have already established that within a unique single system, mass and spring elements have exactly the same material properties. Using that assumption and (14) matrix IW ¹ K can be represented as:

M ^~1 K = —M ^~1 K (17) P

Where M, K depend on the geometry only and are thus independent of the material properties.

If a system, represented by matrix l\ZT ¹ K,was designed to have perfect natural frequency multiplications, based on original material properties p,E and in reality other material properties are present p _a ,E _a but the intended geometry is maintained then the matrix representing the actual system becomes

Meaning that the modified material properties matrix, left hand side of

(18), is a scaling of the original matrix M ¹ K so we can deduce that under the specified assumptions, natural frequency multiplications accuracy is invariant to material properties. This is an extremely important conclusion since it also states that inaccuracy is exclusively attributed to dimensional inaccuracy.

We should note at this point that although natural frequency multiplication accuracy is invariant to material properties, the actual natural

frequencies themselves will definitely change from one system to another according to the change in material properties.

The geometrical mass and geometrical stiffness for the 3 DOF system, using (8) - (12) and (15)-(16), will now become

15 „ m2 = ~ ^m 3 (20)

r ⁵⁰ 2 P ~ (21) 1 7 G ³

^ ₃ =15- ffl ² -^- w ₃ (23)

G

The values of the geometrical mass and geometrical stiffness are in 1/m

3. Elimination of dimensional In-plane inaccuracy on natural frequency multiplications accuracy

If we can find a set of dimensions such that the effect of dimensional inaccuracy will result only in the scaling of the geometry matrix M ^~x k then natural frequency multiplications accuracy will be invariant to dimensional inaccuracy.

A simple way to achieve such a goal is to recall that all the elements in the matrix M ^~l k have the same form of (14), and that all the elements are functions of geometry only.

Assuming small geometrical variations, small variations of stiffness and mass are introduced thus their new value will be

K, = k _m ± dk _m (24)

m _n ' = tn _n + dfh _n (25)

If we can find a set of dimensions such that relative change of stiffness will be kept to one constant value and relative change of mass will be kept to another constant value

dL ^ ' - tconst (26)

dm _n _

^~ - ψ Const (27) tn,.

Then using equations (24) and (25) k _m ' =k _m +dk _m =k _m {\ + φ _Const ) (28) m _n ' = Th _n + dm _n = ih _n (1 + ψ _Comt ) (29)

And the corresponding matrix elements (14) will take the form

∑k _m ' _ ₊ β + φco _ns t)∑K, (30) m _n ' ^~ {\ + ψCo _n st )™n

So the new geometrical matrix, corresponding to geometrical variations will take the form

M^K'= ^ ^φconst \ -M- ^ι k (31)

(1 + ψ 'Const)

We see that the geometry modified matrix M' ^"l K' is again a scaling of the original geometry matrix M ^~l K so that the natural frequency multiplications accuracy is invariant to dimensional inaccuracy thus simply achieving our invariant goal. We will now consider how equations (26) and (27) can be enforced and although general in nature we will introduce the following details regarding the application of a scanning mirror

In the application of a scanning mirror the matching of mass variation in eq (27) is different then matching stiffness variation since in the first case ψconst is usually predetermined since mirror dimensions, both in-plane and thickness t , are usually dictated by optical considerations and are considered to be known, so dm _{3 /ooN}

ψ Const = ^~ (32) m3

Now, let us start by considering the requirements on/% , a similar analysis can, of course, be performed regarding m ₂ . Mass m _x needs to satisfy both equation (19) and (27)

Since m _x is a function of geometry only, and in our specific case, of arbitrary in-plane dimensions that will be denoted Xi,X2, ...x _q and the known thickness t it takes the general form of

Since m _x needs to meet both (19) and (27) we can deduce that it should be a function of least two independent in-plane dimensions. We will refer to the simplest case were m _{ is a function of only 2 in-plane dimensions

/W ₁ = m _x {x^X _j j) (34)

Simple examples of in-plane geometries that are driven by two independent dimensions are rectangle and hollow circle.

4. Topological Design of the 3 POF system

At this point and for the sake of demonstrating the proposed method we will present the topological structure of the asymmetric 3 DOF system shown in a 2D plane view as depicted in figure 6. The mirror has a circular shape, while springs and masses have a rectangular shape. All elements have nominal thickness t We emphasize that since the device is intended for scanning proposes all the relevant DOF of the system are rotational around the depicted center line.

Figure 6 schematically depicts an exemplary topological design of an asymmetric 3 DOF system according to an embodiment of the current invention.

For simplicity, the second support, the elastic member connecting to the second support and the force generating structure were eliminated.

Let us continue and consider the evaluation of (27). Using (27) and (32) we get

(35) m. m-x TW ₃ Js also a function of arbitrary in-plane dimensions yi,yz -.-y _s and thickness t so m ₃ =m ₃ (y _l ,y ₂ ,...y _t ,t) (36)

So we can write that a mass differential of the general geometrical mass function of dimensions Zi,z ₂ , ...z _r ,t m(z _ι ,z ₂ ,...z _r ,ή as

,_ _o-7 , dm = 37)

For the time being we will limit our discussion to the case where dt = 0 (38) It is easy to see that without knowledge about dimension differentials we cannot equate both sides of (35). But if we can find a way to equalize the absolute value of dimensions differentials denoted as dz so \ = \dz ₂ \ = = dz _const (39)

And using (38), we can now rule out dz from both sides of equation (35) so a new value can be introduced instead of (27) which is the sum of normalized mass gradients. We will be denote this value as ψ _comt

ψccoonnsstt (40)

Intuitively φ _const can be understood as the amount of relative variation of the geometrical mass due to DRIE in plane dimensions. Where the ± sign in (40) is determined according to the specific sign of the dimension differential i.e. sign(dz _/ ) or \dz\/dz

Similar consideration can be made regarding (26) but pointing out that the matching of stiffness elements is not constrained to any predetermined value as in (32)

At this point we will discuss the effect of DRIE trench width variation on in plane dimensions and see how we can achieve the important condition stated in (39)

5. DRIE process and feature dimensions tolerance

In the DRIE process features are created by removal of material in the shape of trenches so feature dimensions are directly related to trench edges. Let us consider the effect of trench width variation on feature dimension.

We have concluded that within a single system material properties are the same and we are also aware that DRIE and related fabrication processes are usually location variant we can thus assume that process parameter are constant within a single system so we can conclude that trenches that have the same nominal width will change exactly the same as depicted in figure 7.

Figure 7 schematically depicts the variations in feature dimensions caused by variation during fabrication process according to an exemplary embodiment of the current invention.

We will also assume that the trench center location does not change. In figure 7 bold lines show the trench edges in case of a nominal sized trench while the dotted lines show the case of an over etched trench. Under etched trenches are also accounted for by the following equations.

Were NTW - Nominal Trench Width, ATW - Actual Trench Width, TWT- Trench Width Tolerance, Axe - Actual x close edges, Nxc - Nominal x close edges, Axf- Actual x far edges, Nxf- Nominal x far edges

First lest us define the Trench Width Tolerance as

TWT = ATW-NTW (42)

It is obvious that dimensions that are related to close trench edges like Nxc will decrease as Trench Width increases while dimensions that are related to far trench edges like λ/xfwill increase.

According to our assumptions • regarding to close edges features dimensions and regarding the trench center lines we can write

Nxc + 2 --NTW = Axc + 2 --ATW (43)

2 2 ^V '

So Axe -Nxc = NTW -ATW = -TWT (44)

Meaning that xc decreases in the amount of the Trench Tolerance. Similarly, regarding to far edges features dimensions we can show that

Ax/ -Nxf = TWT (45)

We have established that if the features are bounded by trenches of the same nominal width then all features dimensions tolerances will have the same absolute value and that this value is exactly the tolerance of the trench width variation TWT.

We have now finished laying down all the needed theoretical background of the proposed method which can be now described in a single sentence:

In order to implement the method we need to satisfy (40) and (41) while surrounding all features with trenches of the same nominal width so (39) will be met.

6. Numerical example of the proposed method

We will now present a numerical example and will consider the case of the system as depicted in figure 6.

The normalized moment of inertia of the circular mirror is, see [18] m ₃ = —tπD ² (3D ² + 4t ² ) (46)

For W ₁ Of a rectangular shape, see [18]

J_ (w ² ₊ t ² ) (47)

¹ 1 129.

Since all dimensions are bounded by close trench edges, and if we make sure that those trenches will be of the same nominal width as explained earlier then da = dw = dD = -TWT (48)

And equating (40) for 7% and /« ₃ we get

As mentioned earlier, the diameter and thickness of the mirror are usually predetermined by optical considerations so (49) combined with equation (19) is a polynomial of order 6 that can be solved numerically for w and a. As a numerical example we take a circular mirror having the dimensions of D=500μm t=50μm and we obtain 2 real solutions for w, a ; wherein n/ and a are the mass dimensions as depicted in figure 6.

Solutioni = {a1 = 1563.99 μm, w1 = 406.43 μm } (50)

Solution2 = {a1 = 253.10 μm, w1 = 748.45 μm } (51)

Figure 8 shows a plot showing the matching of ψ _x to φ ₃ as a function of the dimension w of the mass, as calculated according to an exemplary method of the current invention.

Figure 8 shows a plot of ψ _γ (805), which is the left hand side of equation (49) ,Vs w/ for a mass that meets (19) and plot of ψ ₃ (815) which is constant due to optical requirements. The solutions are the intersection points 810 and 820 of the two. The left solution 810 (w ~400 μm) corresponds to a wide rectangle and the right solution 820 (w ~750 μm) to a narrow one. If the total length of the system is of importance then the narrow rectangle solution might be preferred.

We will now refer to matching of spring variation and consider the case of a rectangular cross section torsion bar see figure 6. This geometry has two independent in-plane dimensions so it is suitable for our purpose.

Normalized spring stiffness is, stemming from [16] pp 59-62 is ( ₅₂₎

Where spring dimensions are L length, w _s width and t thickness where w _s <=t

And β is a function of w _s ,t

(53)

Since w _s is bounded by close edge trenches dw _s = -TWT (54)

And since L is the length of the trench dL _^ TWT (55)

We get from (41) and (54)-(55)

Since there is no value of φ to match we can theoretically set it to any arbitrary value, but it is very important and will become more so later to try to minimize the value of φ as possible.

In order to better understand φ we will plot the value of φ vs. the spring width w _s we again assume that mirror dimension are known.

By equating (52) to any of (21)-(23) we can extract L and substitute its value to (56) to get 4 =4 (w _v ') ( ⁵⁷ )

Figure 9 shows a plot of φ _t for 1-1,2,3 as a function of spring width w _s as calculated according to an exemplary embodiment of the current invention. The plot of φ _t vs. w _s for i=1, 2,3 is shown in figure 9 for values of t=50μm, D=500μm, G=79e9 pa , ω = 2π-15OOθi, p = 2330^- s m We can understand that reducing φ will result in large width and length of each spring as well as the total length of the system. We will soon see why it is very worth while to explore such a design

7. Numerical Investigation of the proposed method

We will now investigate several cases in order to evaluate the actual benefit of the approach we have introduced.

The first case is the straight forward implementation of the new approach where

ψ\ =ψi =ψτ ( ⁵⁸ )

And <k =h =h ( ⁵⁹⁾

In order for a unique design to be calculated another constrain must be implemented. In our case the total length of the system is a natural constrain. In this example the total length was limited to 2500 μm and with a mirror of D=500 μm and t=50 μm We can iteratively compute φ _const as a function of the total length of the system until a total length of 2500 μm is achieved, this yield in i φ _const ≤ -0.09731 — and from (40) or right hand side of (49) we get μm

Vco _n st = -T^- = -0-007947^- (60)

19000 μm μm

And the calculated corresponding dimensions are listed in table 1 referring again to figure 6.

Table 1. Dimensions of a 3 DOF system with a total length of 2500///« implementing the new approach nension Value (μm)

W _S 1 26.31

U 699.15

A ₁ 253.10

Wi 748.45

Ws2 26.55

L ₂ 397.46

A ₂ 306.70

W ₂ 637.48

W _s3 26.64

L ₃ 343.56

D ₃ 500

We can see that the difference in w _s is in the order of 0.1 μm . In order to implement such a difference an accurate lithography mask must be implemented so all dimensional variations will be only process related and thus (39) will be met. Such lithography masks are available although higher in cost.

Figure 10 shows the impact of trench tolerance TWT on C ₃ C5 plane accuracy for the method according to the current invention.

We define q as equal to (ή- fι*ni)/fi wherein rij is the desired integer frequency multiplications of fi to get ή . So for example: measuring frequencies fi, f ₂ and f ₃ of first, second and third natural frequency, respectively; we can calculate C ₂ and C ₃ wherein C ₂ equal to (f ₂ - f-ι*n ₂ )/fι and C ₃ equal to (f ₃ - fι ^* n ₃ )/fi wherein n ₂ and n ₃ are the desired integer frequency multiplications of f ₂ and f ₃ respectively. We can indicate the departure of a fabricated system from its design goals by calculation the deviation parameters C ₂ ; C ₃ ; C ₄ ; C ₅ ; etc. We can look at Cj as a vector that its length (distance from the origin) and direction in the Ci multidimensional space indicates the deviation from the design goals. Specifically, we can plot the projection of the deviation vector in {c _k ; c _m} 2D plane, for example the {c ₃ ; C ₅ } plane. In order to see the effect of Trench Width Tolerance (TWT) on C ₃ C ₅ plane we will plot the distance the origin in C ₃ C ₅ plane as a function of TWT as TWT changes from 0 to 1 μm as can be seen in figure 10. This case will be denoted as case 1.

We see that implementing the proposed method results in a reduced sensitivity to dimensional inaccuracy that for practical purposes can be considered zero.

Another important aspect for the analysis is the variation of the first natural frequency as a function of TWT as can be seen in figure 11.

Figure 11 shows the impact of trench tolerance TWT on fή accuracy for the method according to the current invention.

In this figure, f _{ή nomma\} is the intended designed value, in our case 15 KHz. We see that the first natural frequency is reduced up to about 5% as the trench tolerance is increased up to 1 μm, this is very interesting to observe knowing that the implementation of the proposed method resulted in c ₃ c ₅ changes that are 3 orders of magnitude smaller and have the maximum value of about 0.003% as can be seen in figure 10.

We will also point that the relative change in ψ is much smaller then the relative change in φ this means that the sensitivity to geometry variation on the springs is much higher , this is due to the small relative width size of the springs relative to any of the mass dimensions. This observation suggest to choose wider springs when ever possible in general.

We will now consider two cases in which φ, and ψ _t will be intentionally different while the total length of the system is kept at 2500 μm.

The specific dimensions of the corresponding elements can be calculated using (19)-(23) and (40)-(41) with the values of φ, and ψ, listed in table 2. The case denoted as case 1 is the straight forward implementation of the proposed method and serves as a base for comparison. In case 2 φ, is kept constant to the value of the proposed method, while ψ, is intentionally different and in case 3 ψ _x is kept constant while fa is intentionally different Table 2. Values of fa and ψ _t for systems having the same total length of 2500 μm

0.0973 0.09731 0.0973 0.007947 0.00794 1 1 7

-0.015 -0.007947

0.0973 0.09731 0.0973 0.007029 1 1 8

-0.2 -0.3 -0.007947 0.0689 0.007947 0.00794 3 7

Graphs for systems having the same total length of 2500 μm are shown in figure 12(a), 12(b) and 12(c).

Figures 12a, 12b and 12c show a summarizing graph for systems having a total length of 2500 μm designed according to an exemplary embodiment of the current invention.

It is evident to see by looking at figure 12(a) that mismatching of springs , case 3, will cause an very large impact up to about 12% in harmonies accuracy while mass mismatch, case 2, will have a much smaller impact with max of 0.14% Vs 0.003% as can bee seen in figure 12(b). This again shows the importance of spring variance matching. This phenomenon cannot be predicted by looking at the change of the first natural frequency alone as can be seen in figure 12(c) in which all cases resemble each other.

We will now consider other three cases of interest in which spring widths are kept the same for all springs, see table 3. This case is of interest since, if there is no specific reason, it is very reasonable to keep spring width the same, We will note that system's length is not fixed but ψ _t is kept constant at the nominal value of (60). Results are shown in figure 13(a) and 13(b). Table 3. Values for systems same width springs

Case w _s1 w _s2 w _s3 φl φl Total Lengtfo

(μm) μm) (μm) (μm) a 26 26 26 - - - 2438.5

0.0 0.0999 .100

987 5 4

7 b 10 10 10 - -.3215 - 1161.5

.30 .327

55 5 25

C 15 15 15 - -.1959 - 1378.4

.19 .197

08 9

(b)

Figures 13a and 13b show a summarizing graph for systems having same width spring as calculated according to embodiments of the current invention.

We now see that wider springs are better both for first resonance robustness and also for harmonics accuracy. We can also note that although wide springs, case a, that are close in values to the best design, case 1 , exhibit small sensitivity to trench variations and have a maximum of about 0.08% they are still much higher then when the proposed approach according to the invention is implemented with a maximum of only 0.003%

8. Let us now also consider the case where dt ≠ O

For simplicity we will refer to the geometrical mass differential in equation (37), similar consideration can be made regarding the differential of geometrical spring stiffness. Equation (37) can be written as the sum of two parts

If we have maintained the condition in (40) then So the first part of (62) is the same for all the mass elements and the only part to consider is the second part.

If m is a linear function of t like in (53) we can generally write m = γt (63)

Then (62) becomes dm „ _τ I dt ~^ = ψ comf ^dz \ + —

Since device layer thickness is attributed to chemical-mechanical of polishing-grinding process we can again assume that both thickness t and thickness variation dt are the same for all elements within a unique system thus in the case of a linear dependency of thickness there is no impact on

harmonies accuracy even in case of thickness variation. This is in our favor since we have already seen that spring stiffness variation mismatch has the largest impact on harmonies accuracy.

Regarding functions that are not linear to t ,for example (47), equation (62) will become . ,|* , I| _{+ 7} dt ₊ f j It ¹ \ J ₇ dt ( _/n 64 _λ ,)

Obviously the first two terms in (66) are again constant but the last term is dependent of w. Substituting typical values of the system from table 1 and assuming thickness variation of dt=1 μm reveals that the relative change is minor and produces a maximal change of ^c ₃ ² + c ₅ ² = 0.005% this is also due to the already seen phenomena that in our case mass variation mismatch has a minor impact on harmonies accuracy.

Preferring similar w values for the different masses and a thicker mirror might also be considered in order to enhance robustness in the where dt ≠ 0 and as long these preferences so not contradict any other design considerations.

9. MEMS design of the 3 PQF Asymmetric Mirror

For the MEMS implementation of the proposed design an actuation method was determined and designed. A single layer Vertical comb drive (VCD) actuation, (see [15]), has been found to be optimal for the demonstration of the proposed device and method since it is easy to manufacture, produces relatively large and linear torque and enables large deflection angles. Other methods can be implemented as well.

In order to reduce the risk of side pull of the VCD a counter spring was used, this changes the lumped model that has been described in figure 1 but a solution can still be found to satisfy the resonance frequencies of ωθ,3ωθ,5ωθ although anti resonance frequencies shift slightly from the known location of 2ωO, 4ωO.

As mentioned earlier, the details of the Finite Element (FE) model including assumptions, convergence and final dimensions were dealt within the course of the research and published elsewhere. We will point out that the diameter of the mirror was 500 μm and the total length of the system, from the end of the first spring to the end of the opposing spring, was 3688.5 μm.

Figure 14 depicts designs of five distinct different devices designed to be located in the origin and four quarters the of the C ₃ C5 according to the current invention.

Another aspect of the MEMS design included the design of five distinct different devices designed to be located in the origin and four quarters the of the C ₃ C ₅ plane as depicted in Figure 14.

Beside the device group that is naturally located at the origin four other groups were designed at locations were |c ₃ | = 0.75% and |c ₅ | = 0.75% . Some variance of the groups was assumed, shown as hatched squares in figure 15, since we are aware that our previous theoretical assumptions regarding material and trench uniformity cannot be perfectly met in practice. The different groups were located far enough from each other so such that they will be statistically distinct.

The rational behind the design of those groups was to see if we can find any systematic behavior of those groups relative to each other and absolutely i.e. regarding the origin, if so we will be able to predict next batch performance. The MEMS implementation of the different group of devices was by designing different spring length for each type of device while other dimensions like VCD features, mirror diameter and other masses dimensions was kept exactly the same. The shift of a system in the C ₃ C ₅ plane as a function of change in springs and masses is a very important and interesting issue but it is also far beyond the scope of this paper.

10. Fabrication

A typical fabricated device is shown in figure 16 taken under a microscope. It can be classified as comb driven single layer SOI device. 118 dies sized 6mm x 6mm each containing one system was designed per 4" wafer and few wafers were fabricated during the research period.

The fabrication process was designed to utilize well known and stable processes in order to enable prompt progress in the research. The terraced comb drive structure is achieved by fixing the stator with a UV curable adhesive dispensed on the flexures after it has been lowered 25 microns using a manual micro manipulator. To enable parallel lowering of the stator it was designed to have very low stiffness in the out of plane direction and high stiffness in other directions utilizing a serpentine shaped springs. The point of force excretion for parallel lowering of the stator was marked by the circular gold marks. The micro manipulator exerted the lowering force on the center of the circular marks.

Figure 15 shows a photograph of a fabricated 3 DOF device according to the current invention showing the VCD structure.

Sacrificial areas were designed in order to keep trench width the same around the structure so equation (39) will be met while enabling large openings around vibrating elements in order to keep air damping, both squeeze and drag, to a minimum. Since squeeze damping is highly dependent on proximity to objects parallel to the vibrating masses, see [17] a 400 μm handle layer was selected for fabrication purposes. The fabrication of the device started with a SOI wafer consisting of a

50 μm device layer 400 μm handle layer and a 2 μm BOX layer and consisted of the following 6 primary steps, only a textual description is brought since these steps are common and well known.

Step 1 : Etch Back of METAL layer deposited by sputtering Seed layer of Cr 900 A and an Au Layer 5000 A to create pads, circular marks and labeling

Step 2; Double side oxidation by PECVD Front Side 5000 A, Back Side 4.5 μm as a preparation for DRIE steps

Step 3: DRIE process on the handle side is used in order to free the 3 mass mirror system from the handle layer. Step 4: DRIE process on the device side in order to free the 3 mass mirror system from the device layer.

Step 5: Dicing while the structure is still held together by the BOX layer

Step 6: Etching of the Oxide layer is used to free the structure from the Oxide layer thus enabling it to vibrate freely.

11. Experimental Results

The experimental set-up consisted of a laser vibration and displacement sensor, a multi-channel, 16-bit data-acquisition system sampling at a rate of 200 kHz for the Frequency Response Function (FRF) tests and a 300-MHz oscilloscope sampling at a rate of 12 MHz for the triangular wave tests. Using an optical microscope the measuring laser beam is being reduced to a diameter of about 5 μm. The laser beam was located near the edge of the mirror rotor far from the rotation axis, thus attaining maximum sensitivity in the rotational mode of motion. On each of the two stators an equal DC voltage with opposite potentials was applied. The rotor was subjected to an AC voltage at the excitation frequency; this is a common actuation method see [16] pp 206.

FRF results were as expected and resemble the ones showing the FRF measurement of the 5 mass system as depicted in [1] so they are not shown again.

11.1 Accuracy in caCs plane

Figure 16 shows experimental results of deviation from design goals, measured from fabricated 3 DOF devices designed according to the current invention.

Accuracy in the c ₃ c ₅ plane of a total of 128 3 DOF devices utilizing the proposed method is shown in figure 16. The different groups are marked Q1 ,

Q2 etc to indicate their pre-designed location in the different quarters of the

C ₃ C ₅ plane. A + (plus) mark is located at the average location of the Origin group and other four similar signs are located with a difference of |δc ₃ | = 0.75% and |δc ₅ j = 0.75% relative to the origin group average. We notice that different groups are both distinctly different and that they also maintained their relative position. The scattering of each group is in the order of 0.1% which is considered to be a very satisfactory result. The fact that we can pre design the relative location of the devices in the C ₃ C ₅ plane and that the scattering is extremely low suggests that we can iteratively adjust the mask dimensions so most of the population of the origin group will have an accuracy of only a few 0.1 %.

In contrast, measurements from 50 systems of 5 DOF, which were fabricated in different batches without the implementation of the proposed method of the current invention exhibited large deviation, as large as 6% and

8% in C ₃ and C ₅ respectively.

It is easy to see that the new devices have very low scattering relative to the prior design. Furthermore, we demonstrate for the first time a pre- designed control of the location of the groups in the c ₃ c ₅ plane thus establishing such analytical and practical ability.

11.2 Triangular wave experiments

In the second type of experiments, an arbitrary waveform generator was used in order to generate an excitation wave form described by:

V ₁ (t) = B _x cos {cot + β _[ ) + B ₃ cos (3ωt + P ₃ ) + B ₅ cos (5ωt + β ₅ ) (70)

Were V ₁ (Q is the voltage applied to the VCD, see [16] pp 206 and a tuning process was conducted by iteratively changing B _u β _u B ₃ ,β ₃ ,B ₅ ,β ₅ until the best approximation of a triangular waveform was obtained.

Figure 17 illustrates the measured deflection angle as a function of time of a mirror with accuracies of (c ₃ ,c ₅ )=(-0.15%, 0.20%) taken at atmospheric pressure and a DC and AC maximum voltage of 50 V operated at a basic frequency of 13.396 KHz. The measured deflection angle 171 is very close to the perfect Three-

Term triangular wave, thus the plotted difference between measured and the Three-Term triangular wave 172 has maximal value of 0.0617° compare to the total deflection of approximately +/-5 degrees.

The maximal angular displacement was limited only due to electrical breakdown of the BOX layer ('Box Isolation Technique' is a known in the art of 'Shallow trench isolation' (STI) fabrication) which is expected to be simply solved by employing thicker layer Silicon On Insulator (SOI) wafer. Extrapolation predicts that a voltage level of 67 V should be sufficient to achieve angular amplitude of 10° which was the goal of research.

Evaluation and correction

According to the third aspect of the invention, a method for detecting, evaluating and correcting inaccuracies caused during manufacturing of the device is provided.

We introduce a new way to measure the natural frequency multiplication accuracy using the previously defined q deviation parameters.

The above definitions represent a new domain which is the natural frequency multiplication domain. For example for a 3 resonance system the domain is a two-dimensional ({c ₃ ; c ₅ } 2-D) plane.

Having defined the components of the system and using the proposed new domain we can analyze an existing system and also predict how the system will perform after introducing a change in mass and/or stiffness elements. The diagram indicates which elements need to be modified in order to eliminate the deviations completely.

There are other methods for calibrating the natural frequencies of an existing system but the uniqueness of the proposed method is that it directly evaluates and tackles the compliance to an N term wave form and disregards the values of specific frequencies it therefore simplifies the analysis considerably. Furthermore, with this approach, it is evident how to choose the minimal correction that would render the system exact.

It should be noted that although some of the description may be given for a specific exemplary embodiment, for example the three mass asymmetric configurations, the method according to the current invention may be extended systems with larger DOF or to symmetrical configurations.

The method according to the invention comprises the following steps:

1. Measuring the frequency response of the device.

2. Evaluation of the measured response: a. If the measured response is within the desired limits - no action is needed. b. If the response deviates from the desired limits - calculate the needed correction or corrections.

3. Performing the correction by modifying at least one of the elements of the device. 4. Optionally verify correct operation of the device.

5. Optionally iteratively performing adjustment of the device by repeating step 1 , 2 and 3.

1. Measuring the frequency response of the system after production

Measuring the frequency response of the device may be done by exciting the device using the force generating element or elements.

Preferably this is best done while the device is electrically connected. Preferably an electrical signal is connected to the force generating element. Preferably, interrogation of the response is done optically, preferably using deflection of light from the scanning mirror.

Alternatively external source of excitation is used, for example application of force by an external actuator or vibration applied to the entire device. Application of external force is advantageous before electrical leads are connected to the device. Optionally, evaluation and optionally correction is done before the individual devices are separated from the wafer.

Optionally, measuring the frequency response comprises applying a frequency varying signal, for example a chirp sweeping of the frequency range of interest. Optionally measurements are done only at the vicinity of the base frequency and its first relevant odd harmonics. Optionally, measurements are done at the vicinity of the first relevant even harmonics. Optionally both amplitude and phase of the frequency response are measured.

Optionally, measuring the frequency response comprises applying a time varying signal, for example "white noise". Optionally frequency response is determined by the device's response to a step function or an impulse function.

Optionally, measuring the frequency response comprises using spectrum analyzer as known in the art.

Figure 18 shows a graph an exemplary measurement results of frequency response for a specific manufactured device.

The nominal base frequency for this device (as designed) was 15 kHz, and its two first odd harmonics were 45 and 75 kHz as marked by the green markings 1801 , 1803 and 1805 respectively.

As can be seen, from the graph, the actual measured frequencies of the resonances (marked as 1811 , 1813 and 1815 respectively) were: -12.50 kHz, ~37.39 kHz and 63.59 kHz for the base frequency, the first and second odd harmonics respectively.

It should be noted that for proper operation of the device, that is the generation of waveform close to the desired triangular shape, it is the ratio of the frequencies that has to be close to the 1 :3:5 ratios.

In this exemplary graph it is easy to see that while the measured frequency 1813 of first odd harmonics is close to three times the frequency

1823 of the measured base frequency 1811 , the measured frequency of second harmonics 1815 deviates more from the value of five times the base frequency 1825.

2. Evaluation of measured response.

Normalized deviation parameters

We define the following normalized deviation parameters:

And

^C 5 = -

/ ₅

Figure 19 shows depicts the measured deviation of the above example plotted in the C ₃ C ₅ plane.

It should be noted that according to the general aspect of the current invention, the dimensionality of the deviation plot above is N-1 wherein N is the number of DOF.

Sensitivity of the Normalized deviation parameters to deviation of values of physical elements. We can calculate and plot on the deviation plot, the sensitivity of the normalized deviation parameters to changing values of the device elements in a parametric way.

Calculation of the sensitivities may be done using one of: • Lump theoretical model of the device may be used. • Alternatively, finite elements model may be used.

• Alternatively, actual measurements may be performed on system built with elements deviated from the nominal values or modified system with known modification.

For simplicity, we define the relative change of element parameter as R _x wherein x is the parameter. For example, a change of a mass of a specific mass "i" relative to the nominal value mi_ _nO minai is defined as: m ₃ = Rm, -m ₃ _ _norϊύλal

Figure 20 depicts the change of C ₃ C ₅ for change in the mass of the scanning mirror m ₃ m ₃ = Rm ₃ - m ₃ _,, _ominal over the relative change: 1.2 > Rm ₃ > 0.8

Note that 20% change in the mass creates only 2% to 3% change in C ₅ and 0.3% change in C ₃ .

Similar plots may be created for other element parameters, for example elastic parameters of the elastic elements defined as Rkj = kj / ki_nominai wherein I is the spring number and k, (kj_nominai ) are the actual (nominal) elasticity of spring i.

Figure 21 depicts the change of C ₃ C ₅ for change in the masses m-i, ιτ) ₂ , 1TI3 and the springs ki, k ₂ and k ₃ . Note that some element-pairs have similar effects. For example: m ₃ and k ₃ . and m ₂ , and k ₂ .

Any point in the C ₃ C ₅ plane may be decomposed as a vector sum of deviations caused by plurality of deviations of the element values.

Since generally the manufacturing inaccuracies are small, the assumption of linearity of the graphs near the origin holds and the decomposition to vector components yields accurate results.

Since it is important only to bring the frequency multiplication to the correct ratio, it is not necessary to bring all the values of the elements to the nominal values. In fact, it is possible to compensate a change of a value of an element by modifying the value of one or plurality of other elements.

Similarly it is possible to compensate a change of values of plurality of element by modifying the value of one or plurality of other elements.

As will be apparent from the graph below, it is possible to compensate for any combination of any changes in element parameters by changing values of two elements.

It should be noted that compensation option may be limited by the technology employed for modifying the elements' values.

For example, if the technology used for adjusting the device is mass addition by gluing small masses, than only Rm's may be changed and in the positive direction only.

Similarly, if the technology is mass removal by abrasion or laser trimming of the masses, than only Rm's may be changed and in the negative direction only.

Similarly, if the technology is spring thinning by abrasion, etching or laser trimming of the springs, than only Rk's may be changed, etc.

Figure 22 depicts the changes in C ₃ C ₅ for increase in the masses m-i, m ₂ , m ₃

As can be seen, any point on the C ₃ C ₅ plane may be reached by combination of at most two Rm's.

It was found that any single technology, for example mass reduction (increasing), spring weakening or spring strengthening have the same property that any adjustment may be done by modifying at most two elements using said technology.

It should be noted that for devices with larger DOF, all these conclusions hold, but the C ₃ C ₅ plane is replaced with a higher dimensionality space having N-1 dimensions. For large deviation, wherein the assumption of linearity may not hold, a numerical computation may replace the vector decomposition.

3. Correction of device performance o Mass may be: added for example by deposition, evaporation or gluing of small mass reduced for example by abrasion, etching, focused ion beam and laser trimming of the masses

o Stiffness of elastic members may be reduced for example by: abrasion, etching, focused ion beam, laser trimming and voltage controlled electrostatic force .

It should be noted that if the adjustment technology allows increasing and decreasing the value of the elements' parameter, any adjustment may be performed by adjusting chosen two of the same type of elements. For example mi and m ₂ .

On the other hand, if the adjustment technology allows only increasing or decreasing the value of the elements' parameter, any adjustment may be performed by adjusting two the three of the same type of elements. For example mi and m ₂ ; mi and m ₃ or m ₂ and nri ₃ .

Similar conditions apply to adjustments using modification of spring's stiffness.

Base frequency may also be corrected; however doing this may require modifying values of three elements.

Observing the graphs indicates that some elements may, for example k ₃ and m ₃ may have the same effect on the device and thus their combination may not be enough for correcting some imperfections.

Some adjustment technologies may be used under closed loop while system response is measured. For example, laser trimming may be applied to a mass while the system response is monitored using any of the abovementioned techniques. Using close loop, the trimming may be terminated when the desired performance were achieved.

Alternatively, the correction and evaluation steps may be iteratively repeated.

More details, explanations and embodiments for the devices systems and methods according to the current invention are given in the attached paper entitled "A method for eliminating the inaccuracy of natural frequency multiplications in a multi DOF micro scanning mirror" to GaI Avivi and Izhak Bucher; to be published.

It is appreciated that certain features of the invention, which are, for clarity, described in the context of separate embodiments, may also be provided in combination in a single embodiment. Conversely, various features of the invention, which are, for brevity, described in the context of a single embodiment, may also be provided separately or in any suitable sub combination.

Although the invention has been described in conjunction with specific embodiments thereof, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art. Accordingly, it is intended to embrace all such alternatives, modifications and variations that fall within the spirit and broad scope of the appended claims. All publications, patents and patent applications mentioned in this specification are herein incorporated in their entirety by reference into the specification, to the same extent as if each individual publication, patent or patent application was specifically and individually indicated to be incorporated herein by reference. In addition, citation or identification of any reference in this application shall not be construed as an admission that such reference is available as prior art to the present invention.