A MODULATION SYSTEM AND METHOD, POLARIZATION CONTROL SYSTEM AND METHOD AND ISOLATOR DEVICE AND METHOD

Technical Field

[0001] The present invention relates generally to a modulation system and method, polarization control system and method, and isolator device and method.

Background

[0002] Existing beam steering systems using static or tunable Huygens Metasurfaces generally only function to control the carrier wave and do not involve sideband generation and manipulation.

[0003] In general, known dynamic beam control systems that generate sidebands do not utilize Huygens Metasurfaces. Further, existing dynamic beam control systems are currently unable to sufficiently control the direction, scattering and polarization of the generated electromagnetic sideband waves.

Summary

[0004] It is an object of the present invention to substantially overcome, or at least ameliorate, one or more disadvantages of existing arrangements.

[0005] Disclosed are arrangements which seek to address the above problems by controlling one or more electromagnetic sideband waves that are generated through the application of a periodic modulation signal to a modulator formed using a Dynamic Huygens Metasurface.

[0006] According to a first aspect of the present disclosure, there is provided modulation system for controlling at least one electromagnetic sideband wave generated from an incident wave directed at a modulator, the system comprising: a modulator comprising a metasurface, the metasurface comprising at least one metasurface unit having a controlled response to both electric and magnetic fields, wherein the at least one metasurface unit produces scattered electromagnetic fields in a forward and/or backward direction in a ratio determined by, at least, the structure of the metasurface unit, and has a tunable element formed therein; and a control system comprising a modulation signal generator arranged to generate at least one periodic modulation signal, wherein the control system is arranged to apply the generated periodic modulation signal to the modulator to control at least one electromagnetic sideband wave that is generated when an incident wave impacts the modulator.

[0007] According to a second aspect of the present disclosure, there is provided a polarization control system for controlling the polarization of at least one electromagnetic sideband wave generated from an incident wave directed at a polarization converter, the system comprising: a polarization converter comprising a metasurface, the metasurface comprising at least a first and second metasurface unit, each metasurface unit having a controlled response to both electric and magnetic fields, wherein each of the first and second metasurface units produce scattered electromagnetic fields in a forward and/or backward direction in a ratio determined by, at least, the structure of the metasurface unit, and have a tunable element formed therein, wherein the first metasurface unit is arranged in a first orientation and the second metasurface unit is arranged in a second orientation that is different from the first orientation, and a control system comprising a modulation signal generator that is arranged to generate at least one periodic modulation signal, and the control system is arranged to apply the generated periodic modulation signal to the first and second metasurface units to control the polarization of at least one electromagnetic sideband wave that is generated when an incident wave impacts the polarization converter.

[0008] According to a third aspect of the present disclosure, there is provided a modulation method for controlling at least one electromagnetic sideband wave generated from an incident wave directed at a modulator, the method comprising the steps of: generating at least one periodic modulation signal, and applying the generated periodic modulation signal to a modulator to control at least one electromagnetic sideband wave that is generated when an incident wave impacts the modulator, wherein the modulator comprises a metasurface, and the metasurface comprises at least one metasurface unit having a controlled response to both electric and magnetic fields, wherein the at least one metasurface unit produces scattered electromagnetic fields in a forward and/or backward direction in a ratio determined by, at least, the structure of the metasurface unit, and has a tunable element formed therein.

[0009] According to a fourth aspect of the present disclosure, there is provided a polarization control method for controlling the polarization of at least one electromagnetic sideband wave generated from an incident wave directed at a polarization converter, wherein the polarization converter comprises a metasurface, and the metasurface comprises at least a first and a second metasurface unit, each metasurface unit having a controlled response to both electric and magnetic fields, wherein each of the first and second metasurface units produce scattered electromagnetic fields in a forward and/or backward direction in a ratio determined by, at least, the structure of the metasurface unit, and have a tunable element formed therein, wherein the first metasurface unit is arranged in a first orientation and the second metasurface unit is arranged in a second orientation that is different from the first orientation, the method comprising the steps of: generating at least one periodic modulation signal, and applying the generated periodic modulation signal to the first and second metasurface units to control polarization of at least one electromagnetic sideband wave that is generated when an incident wave impacts the polarization converter.

[0010] Other aspects are also disclosed.

Brief Description of the Drawings

[001 1] At least one embodiment of the present invention will now be described with reference to the drawings and appendices, in which:

[0012] Fig. 1 shows an example of a Dynamic Huygens Metasurface with an electric and magnetic element forming a metasurface unit according to the present disclosure;

[0013] Figs. 2A - 2C show how electromagnetic sideband waves may be controlled according to the present disclosure;

[0014] Fig. 3 shows a modulation system according to the present disclosure;

[0015] Fig. 4 shows a further modulation system according to the present disclosure;

[0016] Figs. 5A - 5F show measured sideband spectra generated using a modulation system and the applied modulation waveforms acting on the electric and magnetic elements according to the present disclosure;

[0017] Figs. 6A - 6B show examples of an isolator according to the present disclosure;

[0018] Figs. 7A - 7D show various configurations of metasurfaces according to the present disclosure; [0019] Fig. 8 shows a radar sensing system using a modulation system according to the present disclosure;

[0020] Fig. 9 shows an imaging system using a modulation system according to the present disclosure;

[0021] Fig. 10 shows a circular-waveguide-based polarization converter with a metasurface unit according to the present disclosure;

[0022] Fig. 1 1 of the Appendix shows a schematic of time-varying Huygens' meta-devices for parametric waves according to the present disclosure;

[0023] Fig. 12 of the Appendix shows a design of a Huygens' unit element according to the present disclosure;

[0024] Fig. 13 of the Appendix shows simulated normalized sideband spectra for forward and backward scattering according to the present disclosure;

[0025] Fig. 14 of the Appendix shows a schematic of the microwave experimental setup according to the present disclosure;

[0026] Fig. 15 of the Appendix shows Measured normalized sideband spectra of a pair of electric and magnetic meta-atoms in a rectangular waveguide according to the present disclosure;

[0027] Fig. 16 of the Appendix shows a Schematic representation of the dynamic control of sideband scattering in the rectangular waveguide according to the present disclosure;

[0028] Fig. 17 of the Appendix shows Photographs of the fabricated time-varying Huygens' metadevice according to the present disclosure;

[0029] Fig. 18 of the Appendix shows a normalized time-varying signals for three types of ideal directive sideband scatteringaccording to the present disclosure;

[0030] Fig. 19 of the Appendix shows a schematic of the series RLC equivalent circuit for a time-varying meta-atom according to the present disclosure; [0031] Fig. 20 of the Appendix shows a relative error of the sideband power calculated using the impedance Model according to the present disclosure;

[0032] Fig. 21 of the Appendix shows design of the Huygens' unit of the array structure according to the present disclosure;

[0033] Fig. 22 of the Appendix shows a schematic of the experimental setup for the time- varying Huygens' array according to the present disclosure;

[0034] Fig. 23 of the Appendix shows a photograph of the experimental platform of time- varying Huygens' array according to the present disclosure;

[0035] Fig. 24 of the Appendix shows a flow chart of genetic algorithm according to the present disclosure.

Detailed Description including Best Mode

[0036] Fig. 1 shows an example of a Dynamic Huygens Metasurface with a single magnetic element 101 and an electric element 103 forming a metasurface unit 105. It will be understood that the surface may include a plurality of metasurface units.

[0037] The electric and magnetic elements are also known as meta-atoms and are controlled, in one example, with independent modulation signals. Alternatively, the same modulation signal may be applied to each of the electric and magnetic elements. The electric and magnetic elements may be controlled separately or controlled together.

[0038] In one example, the metasurface unit may have magnetic and electric elements that are combined or interleaved together. In a further example, the metasurface unit may have magnetic and electric elements that are arranged parallel to each other or orthogonal to each other. In yet a further example, the metasurface unit may have magnetic and electric elements that are arranged both orthogonal to each other and are interleaved.

[0039] A plurality of metasurface units may be arranged to form a modulator in a modulation system for controlling one or more electromagnetic sideband waves that are generated when an incident wave impacts (or is directed towards) the modulator. That is, each metasurface unit has a controlled response to both electric and magnetic fields. Each metasurface unit produces scattered electromagnetic fields in a forward and/or backward direction in a ratio determined by, at least, the structure of the metasurface unit. A scattered field would be understood to mean the electromagnetic field that is scattered from the electric and magnetic elements back to the environment during their interaction with an incident wave that is directed towards the metasurface.

[0040] As shown in Fig. 1 , each of the magnetic element 101 and electric element 103 has an associated tunable element (107, 109) formed therein. In this example, the tunable elements (107, 109) are diodes. The tunable element may also be a tunable material.

[0041] The tunable element may be one or more of a diode, a transistor, an amplifier, a tunable attenuator, a tunable resistor, a tunable capacitor, a tunable inductor, a MEMS (micro- electro-mechanical system), an optomechanical system, a transducer, a microfluidic system, a liquid crystal, a magnetoelastic material, a ferromagnetic material, a ferrimagnetic material, a ferroelectric material, a ferrielectric material, a piezoelectric material, a piezomagnetic material, a phase-change material, a superconductor, a quantum well, a quantum dot, a quantum wire, a topological insulator, a 2D material, a semiconductor, a nonlinear material, and a photosensitive material.

[0042] Figs. 2A - 2C show how electromagnetic sideband waves may be controlled using a periodic modulation signal that is applied to each of the metasurface units in the metasurface while an incident wave is directed towards and impacts the modulator.

[0043] Fig. 2A shows multiple metasurface units (105A - 105F) number as n, n-1 , n+1 etc. forming a metasurface having a linear array 201 of metasurface units. An incident wave impacts the surface while a periodic modulation signal (not shown) is applied to the metasurface.

[0044] The incident wave carrier frequency ωο is converted to sidebands ωη=ωο+ηΩ, the directionality (i.e. forwards or backwards) of which can be controlled by the relative modulation phase of the electric and magnetic elements (meta-atoms), while the steering angle (any angle in a forwards or backwards direction) can be controlled by the phase gradient along the metasurface of the linear array. By applying appropriate periodic modulation waveforms to the metasurface units, the carrier frequency can be converted to the desirable sidebands with high efficiency as well as obtaining control of the directive sideband scattering. It will further be understood that the array may have a non-uniform spacing of metasurface units. [0045] In Fig. 2A, double-sideband bi-directional scattering is obtained. The modulation of electric and magnetic elements should have a phase difference φ _Ε - φ _Μ ~ ±90° . The

modulation signal will have a sinusoidal like symmetric waveform so that the energy is converted to the sidebands of orders ±1 evenly.

[0046] In Fig. 2B, double-sideband unidirectional (forward or backward) scattering is obtained. The modulation of electric and magnetic elements should be in phase (φ _Ε - φ _Μ « 0 ) to achieve directive forward scattering, and be out of phase (φ _Ε - φ _Μ ∞λ 80°) to achieve directive backward scattering. The modulation signal will have a sinusoidal like symmetric waveform so that the energy is converted to the sidebands of orders ±1 evenly.

[0047] In Fig. 2C single sideband unidirectional scattering is obtained. The modulation of electric and magnetic elements should be in phase (φ _Ε - φ _Μ ∞0 ) \ο achieve directive forward scattering, and be out of phase (φ _Ε - φ _Μ ^ λ 80°) to achieve directive backward scattering. The modulation signal will have a sawtooth like asymmetric waveform so that the energy is converted to the sidebands of orders ±1 unevenly, and preferably to only one of the sidebands.

[0048] Fig. 3 shows a modulation system 300. A modulator is provided that has a

metasurface made up of multiple metasurface units 301. Each metasurface unit 301 has a magnetic element 101 and an electric element 103 arranged on a substrate. The substrate may be any suitable substrate for supporting the magnetic element and electric element. In one example, the substrate used was Rogers4003C substrate based on thermoset plastics and ceramic glass. According to other examples, the substrate may be one or a composite of the following materials: Polytetrafluoroethylene (PTFE), glass, epoxy, polyolefin, thermoplastic, ceramic, Alumina, Aluminium Nitride, Beryllium Oxide, Gallium Arsenide, Gallium Nitride, Indium Phosphide, Porcelain, Quartz, Sapphire, Silicon, Silicon Carbide, and other suitable metals. The metasurface units 301 are arranged on surface 303. The magnetic element and electric element in each metasurface unit is controlled via a control system 307 that has a modulation signal generator 309 and control circuitry including connections 305, a processor 311 , a memory 313, a power supply 315 and an interface 317. The memory 313 holds instructions for the processor 31 1 to carry out a pre-programmed algorithm to enable the modulation signal generator 309 to generate the required periodic modulation signals that are applied to the metasurface units via the connections 305. It will be understood that the processor may be any suitable device that can operate under a set of defined instructions such as, for example, a microcontroller, microprocessor or field-programmable gate array (FPGA).

[0049] The interface 317 may be connected to a keyboard 319 and display unit 321 to enable a user to control the system as well as to update the stored algorithms that are used to generate new periodic modulation signals as required.

[0050] When the generated periodic modulation signals are applied to the modulator, one or more electromagnetic sideband waves may be controlled upon generation when an incident wave 325 generated by the incident wave generator 323 is directed towards and impacts the modulator.

[0051] The modulation signal generator may be arranged to generate the periodic modulation signal based on a fundamental frequency signal and a series of harmonic frequency signals. Further, the amplitudes or phases of the fundamental frequency signal and the series of harmonic frequency signals are variable when the amplitude or phase of the generated periodic modulation signal is changed.

[0052] The control system is therefore arranged to control the directionality of the

electromagnetic sideband wave by changing the amplitude and/or the phase of the periodic modulation signal acting on the metasurface unit. Further, the control system is arranged to control a steering angle of the electromagnetic sideband wave by changing the modulation amplitude and/or the gradient of the modulation phase of the periodic modulation signal along the metasurface.

[0053] The gradient of the modulation phase is defined by the relative modulation phase of the adjacent metasurface units over the distance between the two units. The relative modulation phase can be defined by the time delay of the modulation signals.

[0054] The relative phase may be calculated by choosing the same point on the modulation waveform (e.g. the lowest point), and estimating the time difference t _n - t _n- i of this point for the modulation waveform signals that are acting on the different metasurface units. The relative modulation phase is: φη-φη-1= Qx(tn-tn-l) [0055] where φ„ is the modulation phase for the modulation waveform acting on the n metasurface unit and φ _η -ι is the modulation phase for the modulation waveform acting on the n-1 metasurface unit.

[0056] The gradient of the modulation phase is:

[0057] where A is the distance between metasurface units n and n-1. Therefore, the phase gradient along the metasurface (i.e. multiple metasurface units) is effectively the cumulative time delay of the modulation waveform as it is applied to each of the metasurface units. In other words, each of the metasurface units may be stimulated at a different phase of the modulation waveform. Some metasurface units may be stimulated at the same phase. It will be understood that for a zero degree steering angle, the modulation phase will be the same along the metasurface.

[0058] It will be understood that the metasurface units 301 may be arranged in an array or a linear array configuration. For an array of metasurface units, the metasurface units are arranged in an p x q array, where row (p) >=1 and column (q) > 1. A linear array includes a single row (p) of metasurface units where p = 1. That is 1 x q metasurface units are arranged to form a linear array, where q > 1.

[0059] Fig. 4 shows a further modulation system that is controlled using the same control system as described with reference to Fig. 3. In this modulation system combined

electric/magnetic elements (327A and 327B) are used instead of separate magnetic and electric elements. Each of the combined magnetic/electric elements are positioned opposite another combined magnetic/electric element on either side of a surface 303.

[0060] Figs. 5A - 5C show measured sideband spectra generated using a modulation system. In Fig. 5A, measured sideband spectra are shown for single-sideband forward directional scattering. An associated applied modulation waveform is shown in Fig. 5D. In Fig. 5B, measured sideband spectra are shown for double-sideband forward scattering. An associated applied modulation waveform is shown in Fig. 5E. In Fig. 5C, measured sideband spectra are shown for double-sideband bidirectional scattering. The spectra are normalized to the total scattered power. An associated applied modulation waveform is shown in Fig. 5F. ISOLATOR

[0061] In accordance with a further aspect, an isolator is now described. The isolator uses various components of the modulation system as hereinbefore described in addition with one or more band pass filters. The metasurface units are arranged and controlled such that a generated electromagnetic sideband wave's frequency and direction is controlled in accordance with the band pass frequency of the band pass filters to enable a wave having the same frequency as an incident wave (that is initially used to generate the sideband wave) to effectively pass through the isolator in a first direction, but block any reflected wave from passing back through the isolator in a direction opposite the first direction.

[0062] Fig. 6A shows an example isolator 601 with a first modulator 603 and second modulator 605. It will be understood that each modulator at a minimum may include a single metasurface unit as herein before described. Alternatively, each modulator may include more than one metasurface unit as hereinbefore described.

[0063] The first modulator 603 is controlled by a control system in the same form as the control system described with reference to Fig. 3. A first periodic modulation signal applied to the first modulator 603 causes the modulator to produce a single sideband frequency output ω+ι when an incident wave ωο is directed towards and impacts the first modulator 603. The single sideband frequency output ω+ι is controlled by the adjustment of the periodic modulation signal to be at a higher frequency than the incident wave. That is, the first modulator is configured (by way of the first periodic modulation signal) to act as a frequency up converter.

[0064] A narrow band pass filter 607 is chosen to have a band pass frequency that is the same as this higher frequency and as such allows the single sideband frequency output ω+i to pass through the band pass filter. A second periodic modulation signal applied to the second modulator 605 causes the modulator to produce a single sideband frequency output ooowhen the single sideband frequency output ω+ι is directed towards and impacts the second modulator 605. This single sideband frequency output ωο is the same frequency as the initial incident wave. The second modulator is therefore configured (by way of the second periodic modulation signal) to act as a frequency down converter.

[0065] When the single sideband frequency output ωο is reflected back to the second modulator 605, the second modulator again down converts this signal (based on the second periodic modulation signal) producing an output sideband wave ω ι which is then blocked by the band pass filter 607 as it is not the correct frequency.

[0066] Therefore the isolator includes the modulation system as described herein where at least two metasurface units are provided along with at least one band pass filter. The modulation signal generator generates a first periodic modulation signal, which is then applied to a first metasurface unit to generate a first electromagnetic sideband wave that has a higher frequency than the first incident wave that was applied to the input of the first metasurface unit. A second periodic modulation signal is generated by the modulation signal generator, which is then applied to a second metasurface unit to generate a second electromagnetic sideband wave having a lower frequency than the second incident wave (i.e. the first electromagnetic sideband wave) that was applied to the input of the second metasurface unit. The band pass filter is then arranged to allow the first electromagnetic wave to pass and further arranged to block the second electromagnetic wave.

[0067] As an alternative to the isolator described with reference to Fig. 6A, the first modulator 603 may be replaced with a down converter and the second modulator 605 may be replaced with an up converter. In this example, an alternative isolator 608 is shown in Fig. 6B, the first modulator 609 may be controlled by a control system in the same form as the control system described with reference to Fig. 3. A first periodic modulation signal applied to the first modulator 609 causes the modulator to produce a single sideband frequency output ω-ι when an incident wave ωο is directed towards and impacts the first modulator. The single sideband frequency output ω-ι is controlled by the adjustment of the periodic modulation signal to be at a lower frequency than the incident wave. That is, the first modulator is configured (by way of the first periodic modulation signal) to act as a frequency down converter.

[0068] A narrow band pass filter 61 1 is chosen to have a band pass frequency that is the same as this lower frequency and as such allows the single sideband frequency output ω-i to pass through the band pass filter. A second periodic modulation signal applied to the second modulator 613 causes the modulator to produce a single sideband frequency output ooowhen the single sideband frequency output ω-ι is directed towards and impacts the second modulator. This single sideband frequency output ωο is the same frequency as the initial incident wave. The second modulator 613 is therefore configured (by way of the second periodic modulation signal) to act as a frequency up converter. [0069] When the single sideband frequency output ωο is reflected back to the second modulator 613, the second modulator again up converts this signal (based on the second periodic modulation signal) producing an output sideband wave ω+ι which is then blocked by the band pass filter 61 1 as it is not the correct frequency.

[0070] Therefore the isolator 608 includes the modulation system as described herein where at least two metasurface units are provided along with at least one band pass filter. The modulation signal generator generates a first periodic modulation signal, which is then applied to a first metasurface unit to generate a first electromagnetic sideband wave that has a lower frequency than the first incident wave that was applied to the input of the first metasurface unit. A second periodic modulation signal is generated by the modulation signal generator, which is then applied to a second metasurface unit to generate a second electromagnetic sideband wave having a higher frequency than the second incident wave {i.e. the first electromagnetic sideband wave) that was applied to the input of the second metasurface unit. The band pass filter is then arranged to allow the first electromagnetic wave to pass and further arranged to block the second electromagnetic wave.

[0071] Further examples of isolators are also envisaged where a single metasurface unit with multiple band pass filters provide the same isolator functionality.

[0072] For example, an isolator is also provided that has a modulation system as described herein and at least two band pass filters. The periodic modulation signal is generated by the modulation signal generator and is applied to the metasurface unit of the modulation system to generate the at least one electromagnetic sideband wave having a higher or lower frequency than the incident wave. A first of the at least two band pass filters is then arranged to allow a first frequency wave to pass and a second of the at least two band pass filters is arranged to allow a second frequency wave to pass, where the first and second frequencies are different.

[0073] For example, a first ωο band pass filter may allow a frequency ωο ίο pass thus allowing that signal to be applied as an incident wave to a modulator system arranged to function as an up-converter. This causes the modulator system to produce a sideband wave having a frequency ω+ι, which is then allowed to pass through the second ω+ι band pass filter. When the sideband wave ω+ι is reflected back to the isolator, the second ω+ι band pass filter allows the reflected signal to pass through and is applied to the modulator system acting as the up converter. This causes the modulator system to produce an output signal having a frequency second ω+ ₂ , which is not allowed to pass through the first ωο band pass filter.

[0074] In a similar example to the above example, the first and second pass filters may be switched so that the first band pass filter is a ω+ι band pass filter and the second band pass filter is a ooo band pass filter. These used in conjunction with a modulation system acting as a down converter in between the band pass filters operates in a similar manner when an initial incident wave of frequency ω+ι is applied to the modulation system.

POLARIZATION CONTROL SYSTEM

[0075] In accordance with a further aspect, a polarization control system is now described. The polarization control system uses various components of the modulation system as hereinbefore described. Figs. 7A - 7D show various configurations of metasurfaces where Figs. 7B and 7D are used to control the polarization of one or more generated electromagnetic sideband waves. Figs. 7A and 7C depict an array of metasurface units of a modulation system as described above with reference to Figs. 1 -5.

[0076] As explained above, the metasurface unit may have magnetic and electric elements that are arranged orthogonal to each other. Further the metasurface unit may have magnetic and electric elements that are arranged both orthogonal to each other and interleaved with each other.

[0077] In this example, a polarization converter is formed using the metasurface, where the metasurface has at least a first and second metasurface unit. Each of the metasurface units have a controlled response to both electric and magnetic fields and each of the first and second metasurface units produce scattered electromagnetic fields in a forward and/or backward direction in a ratio determined by, at least, the structure of the metasurface unit. Further, each have a tunable element formed therein. In this example, the first metasurface unit is arranged in a first orientation and the second metasurface unit is arranged in a second orientation that is different to the first orientation.

[0078] A polarization control system is provided that includes the modulation system (i.e. the modulator and control system) as described above in relation to Figs. 1 -5. In particular, the system comprises the same components as those described with reference to Fig. 3 except that the arrangement of the metasurface units is different.

[0079] In this system, the metasurface has at least a first and second metasurface unit where the first metasurface unit is arranged in a first orientation and the second metasurface unit is arranged in a second orientation that is different to the first orientation. For example, the first orientation may be orthogonal to the second orientation. As a further example, the first orientation may be at 45 degrees to the second orientation. It will be understood that the amount of polarization conversion will change based on the variation between the first and second orientations from 0 degrees to 90 degrees.

[0080] The control system of this polarization control system is arranged to independently control each of the electromagnetic sideband waves generated by each of the first and second metasurface units in order to control the polarization of the sideband waves. That is, each metasurface unit comprises one electric and one magnetic element that are tunable

independently by the modulation control system.

[0081] Referring to Fig. 7B, each block 701 depicts a metasurface unit that is arranged in a first orientation, and each block 703 depicts a metasurface unit that is arranged in a second orientation that is orthogonal to the first orientation. The array of metasurface units has a response in both x and y directions.

[0082] In Fig. 7A, the scattered electric field at the sideband frequency can only have a linear polarization in x direction, while in Fig. 7B the scattered field can have a desired polarization state as long as the incident polarization can excite the units in both x and y directions where the polarization of the scattered sideband wave is controlled by the modulation amplitude and phase of the meta-atoms (elements) in x and y directions.

[0083] To generate a particular polarization state at the sideband frequency, the incident polarization at the carrier frequency should be arranged in such a way that it can excite the metasurface units in both orthogonal directions as shown in Fig. 7B. The amplitude and phase of the sidebands generated by the two orthogonal metasurface units can be independently tuned by the modulation control system (by adjusting the periodic modulation signal), to enable desired polarization states to occur. [0084] Further, the polarization control may be combined with the beam steering technique as described herein when a modulation phase gradient is introduced along the array of

metasurface units.

[0085] Although a minimum of two orthogonal metasurface units are required to control polarization of the generated sidebands, it will be understood that more than two metasurface units arranged in an array (such as that shown in Fig. 7D) may be used, where a first set of metasurface units are arranged in a first orientation and a second set of metasurface units are arranged in a second orientation that is different to the first orientation.

FURTHER EXAMPLES

[0086] Fig. 8 shows a radar sensing system 800 using a modulation system as herein before described. In this example, the signals that are output from the modulation system are all generated at the same time by multiple metasurface units making up the metasurface 801 by way of applying one or more periodic modulation signals to the separate metasurface units to produce multiple sideband signals (803 A-C) at different frequencies. These generated signals are then used by the radar sensing system to detect objects. In existing radar sensing systems, the radar system is required to generate a single output and then sweep that output over time to detect objects in the path of the waveforms. The herein described system allows sending signals of different frequencies to different directions, and is therefore capable of detecting objects in multiple directions simultaneously.

[0087] Fig. 9 shows an imaging system 900 using a modulation system 901 as herein before described as part of an emitter 903. In this example, the metasurface 905 produces a series of complex waves 907 having defined patterns that are known to the system based on the periodic modulation signals and incident wave that are being used. The complex waves 907 are directed towards an object 909 that is to be measured and reflected signals 91 1 are picked up by a receiver 913. The receiver determines a difference between the patterns of the emitted complex waves and the received reflected signals, and uses this difference to calculate the shape of the object 909.

[0088] Fig. 10 shows a circular waveguide-based polarization converter 1001 with a metasurface unit. [0089] Due to the circular symmetry of the waveguide, the circular waveguide can support an arbitrary polarization state in the same way as an arbitrary polarization state can be supported in free-space. By controlling the modulation amplitudes and/or phases acting on the two orthogonal metasurface units, the incident carrier wave, which usually has a fixed polarization, can be converted to one or more sidebands with arbitrary polarization states. The polarization states can be tuned dynamically by changing the modulation signals. This may be useful for polarization division multiplexing or mode division multiplexing communication, where signals are encoded in different polarizations/modes and transmitted through waveguides or fibres.

[0090] Therefore, the modulation system as described herein is capable of generating a periodic modulation signal that is in a form that can control one or more of i) a direction of at least one electromagnetic sideband wave, ii) a steering angle of at least one electromagnetic sideband wave, iii) scattering of at least one electromagnetic sideband wave, iv) a frequency of at least one electromagnetic sideband wave, and v) polarisation of at least one electromagnetic sideband wave.

Industrial Applicability

[0091] The arrangements described are applicable to beam modulation systems.

[0092] The foregoing describes only some embodiments of the present invention, and modifications and/or changes can be made thereto without departing from the scope and spirit of the invention, the embodiments being illustrative and not restrictive.

[0093] In the context of this specification, the word "comprising" means "including principally but not necessarily solely" or "having" or "including", and not "consisting only of". Variations of the word "comprising", such as "comprise" and "comprises" have correspondingly varied meanings.

[0094] The following Appendix provides further information as follows: Appendix

Huygens' metasurfaces have demonstrated almost arbitrary control over the shape of a scattered beam, however its spatial profile is typically fixed at fabrication time. Dynamic reconfiguration of this beam profile with tunable elements remains challenging, due to the need to maintain the Huygens' condition across the tuning range. In this work, we experimentally demonstrate that a time- varying meta-device which performs frequency conversion, can steer transmitted or reflected beams in an almost arbitrary manner, with fully dynamic control. Our time- varying Huygens' meta-device is made of both electric and magnetic meta-atoms with independently controlled modulation, and the phase of this modulation is imprinted on the scattered parametric waves, controlling their shapes and directions. We develop a theory which shows how the scattering directionality, phase and conversion efficiency of sidebands can be manipulated almost arbitrarily. We demonstrate novel effects including all-angle beam steering and frequency-multiplexed functionalities at microwave frequencies around 4 GHz, using varactor diodes as tunable elements. We believe that the concept can be extended to other frequency bands, enabling metasurfaces with arbitrary phase pattern that can be dynamically tuned over the complete 2π range.

I. INTRODUCTION

Recent advances in Huygens' metasurfaces and meta-devices provide new insight into highly efficient wavefront shaping and scattering manipulation by incorporating electric and magnetic responses [1-13], extending earlier studies of blazed gratings [14]. While a large amount of work has been done on static Huygens' metasurfaces and meta-devices, advanced applications require that the performance of meta-devices can be tuned to adapt to varying operating conditions or requirements. Ideally, this would mean that the amplitude and phase response of a Huygens' meta- device can be tuned dynamically in an arbitrary fashion. Here, "arbitrary" means the amplitude and phase response can be independently tuned to all the possible states; for example, to tune the phase response over a complete 2π range without changing the amplitude response, or to change the amplitude from zero to maximum while keeping the phase unchanged. While there have been some recent attempts to achieve independent tuning of Huygens' metasurfaces [15], fully arbitrary control remains very challenging to realize in practice, since truly arbitrary tuning requires independent control of not only the electric and magnetic responses, but also of the gain and loss.

To solve this problem, we propose parametric meta-devices, where some parameters of the structure can be dynamically modulated. When an electromagnetic wave with central frequency Q¾ interacts with a system having properties modulated with frequency Ω < coo, new sideband frequencies G¾ = (flo + «Ω (n e Έ) are generated. The manifestation of this process can be found in systems of different scales, ranging from the energy level splitting of trapped cold atoms [16, 17], Raman scattering of vibrational molecules [18-20], Brillouin scattering due to opto-acoustic coupling [21-23], to the amplitude and frequency modulation of radio signals [24], and the micro- Doppler effect widely employed in radar sensing [25, 26]. The manipulation and detection of these sidebands are of great importance from both fundamental and application points of view. A variety of novel effects explored recently in optical systems with dynamic modulation also rely on sideband control, such as sideband cooling [16, 27-29], magnet-free optical isolation [30-33], and optomechanical interaction in the resolved sideband regime [34, 35].

Understanding and harnessing the effects of dynamic modulation in artificial subwavelegnth electromagnetic systems such as resonant particles and metasurfaces, could lead to various ultra- compact tunable devices and novel functionalities via the introduction of an additional degree of freedom - time varying properties [36-44]. One important feature of sideband generation in subwavelength systems is that the scattered phases of the sidebands have a gauge freedom and can be controlled by the dynamic modulation process. However, due to the symmetry of dipole scattering, a thin-layer of modulated elements scatters sidebands in both forward and backward directions. In addition, the limited modulation strength available in practice further suppresses the efficiency of energy conversion from the carrier wave to sidebands at the subwavelength scale.

FIG. 11. Schematic of time- varying Huygens' meta-devices for parametric waves. By independently modulating the electric and magnetic polarizations P and M, the sideband scattering can be manipulated almost arbitrarily, and multiple functionalities can be achieved with the same Huygens' meta-device. (Top) Single sideband directive scattering. By changing the relative modulation phases <j¾ and (pM, the directionality can be tuned between forward (top right) and backward (top left) mode. (Middle) Double sideband directive scattering, including double-sideband bidirectional scattering (middle left) and unidirectional scattering (middle right), where the two sidebands can be steered towards different directions via a linear phase gradient along the units of the meta-device. When a more complicated phase gradient is introduced, a time-varying Huygens' meta-device can function as different devices simultaneously for different sidebands. For example, under double-sideband bidirectional scattering mode, it can function as a concave mirror for one sideband and a convex lens for the other (bottom left); while under double- sideband forward scattering mode, it can function as lenses of opposite focusing properties for different sidebands (bottom right).

Here, we develop and experimentally verify a theory of time- varying Huygens' meta-devices for parametric waves, which consist of electric and magnetic meta-atoms that can be modulated independently via external stimuli. Compared to conventional static Huygens' meta-devices, these time-varying Huygens' meta-devices do not aim to manipulate the carrier wave at frequency a¾, but instead efficiently convert the carrier wave into sideband frequencies (On via dynamic modulation (see Fig. 11). Unlike static Huygens' meta-devices, where the amplitude and phase response to the carrier wave are determined by the intrinsic resonances of the meta-atoms, the sidebands scattered from a time-varying meta-device are fully controlled by the amplitude and phase of modulation.

Since the parities of electric and magnetic dipole radiation are preserved under modulation, the directivity of sideband scattering can be manipulated by the relative modulation phase of electric and magnetic meta-atoms, and high conversion efficiency from the carrier frequency to sidebands can be achieved within a layer of subwavelength thickness. By introducing a gradient of modu- lation phase along the unit cells of a metasurface, the sidebands can be steered in the transverse direction. We note that some recent studies already provided vivid numerical demonstrations of sideband generation and steering in time-varying Huy gens' metasurfaces [45^17]; in this study we reveal the physical mechanism and highlight the key components for achieving high conversion efficiency and full spatial control of parametric waves. We demonstrate this concept experimentally in the microwave frequency range, where we measure the sideband scattering of independently modulated electric and magnetic meta-atoms. Using optimized modulation signals, we achieve a high conversion efficiency of over 75% from the carrier wave to the target sidebands and successfully demonstrate various sideband scattering regimes with controlled directionality, including single-sideband unidirectional scattering [Fig. 11 (top)], double-sideband unidirectional scattering [Fig. 11 (middle right)], and double-sideband bi-directional scattering [Fig. 11 (middle left)]. Using a finite time-varying Huygens' meta-device, we demonstrate novel effects including all-angle (360-degree) beam steering and frequency-multiplexed functionalities [Fig. 11 (bottom)]. These results set the foundation of more advanced Huygens' meta-devices for parametric waves.

ANALYTICAL MODEL A. General description based on the effective impedance model

The physics of time- varying Huygens' meta-devices can be described by the boundary conditions of a thin sheet with space-time-dependent electric and magnetic polarizations P(r,i) and M(r,r), dubbed the generalized sheet transition conditions (GSTCs) [48]:

where n is the normal vector of the surface; the superscripts 'i', 'f and 'b' represent the incident, forward scattered and backward scattered fields, respectively; '||' denotes the components parallel to the surface. The more general form of GSTCs also includes the normal components of polarizations Ρ and M (see Sec. A), but in this work, we limit our discussion to a meta-device with polarizations only in the lateral directions (Ρ = Μχ = 0).

When a slow periodic modulation with a fundamental frequency of Ω is introduced in the polarizations, i.e. P| = Ρ|| (Γ, Ωί) and My = M | (r, Ωί ), all the time-dependent components of Eqs. (1) and (2) can be decomposed in the frequency domain in the form of F(r, t) = ^_ _∞ F _N {T)e ^~l0}nt , where ft¾ = (OQ + ηΩ. is the sideband frequencies [24], with 0Q the carrier frequency of the incident field. The electric and magnetic polarizations can be achieved using time-varying meta- atoms with effective Fourier components of electric and magnetic dipole moments p„ = P„A, and m„ = Μ„Α/μ, respectively, where A is the area of the unit-cell, and μ is the permeability of the medium surrounding the meta-device. The boundary equations (1) and (2) for sideband ft¾ are then given by - ¾ ₀ Hi _{; ||} ) = -i<¾p _w ,||/A, (3)

δηθ is the Kronecker delta, which is 1 for n = 0, and 0 otherwise. It is clear that the space-time dependency of the scattered fields on the two sides of a meta-device are determined by the space- time variation of the effective dipole moments; only in the case of static meta-devices (Ω = 0) is the frequency of the scattered fields identical to the incident one.

However, the Fourier components of the effective dipole moments are not independent but interact with each others via parametric modulation. In order to describe this parametric process in the meta-device, we extend our previous models for static meta-devices [49, 50]. Generally, the interaction between the electric (magnetic) meta-atom at position r and the incident wave can be expressed in a compact form: Z _E ( _M )/ _E ( _M ) = V _E (M) _> where Z _E ( _M ) is the effective impedance of the meta-atom in the array with all the mutual interaction taken into account (see Sec. C for details); / _E ( _M ) is the mode amplitude (current amplitude); V _E (M) is the effective electromotive force evaluated by the overlap of the incident field and the mode profile [49, 50]. When a periodic modulation /(Ωί— φ) is introduced to the meta-atoms, Z, / and V become time-dependent and they can be expanded in the frequency domain by a Fourier series. Representing each meta-atom as a series RLC circuit, the input electromotive force is equal to the total voltage at any point in time, which requires that the coefficients of the Fourier decomposition of Z(t), /(f) and V(f) satisfy the following equation (see Sec. C for details)

Here, I = n— m. The subscripts denote the order of the Fourier coefficients of the time-varying Z(t), I{t) and V(f), while the superscript in denotes that the Fourier coefficient is calculated at frequency (¾,. φ is the relative phase introduced from the modulation, which is effectively a time-delayed replica of the modulation waveform. The dynamics of the modulated meta-atom at position r can then be expressed in a compact matrix form:

¾I = V, (6) where

The zeroth order impedance describes the linear response of the meta-atom at frequency C0m, where the effect of mutual interaction with other meta-atoms has been taken into account; the

(n)

(m— order component Z^_ _n characterizes the frequency conversion from (dm to (On, which is originated from the dynamic modulation of the self-impedance. For more details regarding the construction of Eq. (7), see Sec. C.

Under normally incident plane wave excitation E ¹ , the Fourier component of the effective electromotive force acting on the electric and magnetic meta-atoms can be expressed as

VM,/I = J M,n ^■ E ^x d ³ r = i&oE'wM.n, (11)

"E,n and "M,n = "* _n /½,n are the Fourier components of the normalized effective electric and magnetic dipole moments, which can be defined by the Fourier component of the normalized current distribution j _n [49, 50] (see Sec. C for more details). Solving Eq. (6) independently for both electric and magnetic meta-atoms, we can find the current amplitudes ½(M),n' ^men the dipole moments at frequency (On can be readily found as

m _n (r) = /M _I „(r) _MM ^ ^<PM(r) . (13) Here, ¾(M) * ^{s me} relative phase introduced from the modulation of the electric (magnetic) meta atom.

Once we substitute Eqs. (12) and (13) into Eqs. (3) and (4), it becomes clear that an additional phase of ηφ is introduced to the sideband fields (E _n ,H _n ) via modulation. This feature of time- varying Huygens' meta-devices offers a unique opportunity in controlling wave scattering - since the modulation phase φ can be chosen arbitrarily for different unit cells, one can generate any desired spatial phase pattern along the meta-device simply by modifying the local modulation phase <p(r). As a result, the phase of each unit cell can be dynamically tuned over a complete 2π range, giving our structure an advantage over existing static Huygens' meta-devices that rely on the deliberate overlap and balance of electric and magnetic resonances. Moreover, unlike conventional sideband scattering processes, the functionality of time-varying Huygens' meta-devices can be actively tuned by changing the modulation phase difference between the electric and magnetic meta-atoms.

B. Modulation with uniform amplitude and a linear phase gradient

To give an illustrative example, we consider the simple case of a homogeneous metasurface consisting of identical units, where the response of the meta-atoms is polarization-independent. The incident carrier wave propagates in the normal direction, and the dynamic modulation of the meta-atoms (of the same type) has identical amplitude but a periodic linear phase gradient: φθ = θ + Gy, with G being the spatial frequency of modulation and being the modulation phase at y— 0. As a result, the solution of the system is a series of Floquet modes characterized

ΐϋ (14)

where each sideband frequency <¾ is scattered with a transverse wave vector β _η = j¾ + nG.

When the modulation frequency Ω is much smaller than the linewidth of the resonance, the difference between the impedance evaluated at ft¾ and (% is negligible, and the effective impedance matrix in Eq. (7) can be simplified as (for details, see Sec.

Ζβ _n is the zeroth order effective impedance of Floquet mode ((ύη, β _η ). For a normally incident carrier wave (j¾ = 0), the sidebands c¾ and ω_„ have opposite transverse wave vectors: β _η = —β-η, i.e. they are deflected towards opposite transverse directions, and their impedances can be approximated as identical: Ζβ _η — Ζβ _ _η . In the special situation of uniform modulation without any phase gradient (G = 0), all the sidebands are scattered in the normal direction (j3„ = 0), and the effective impedance can be considered the same for all sidebands: Ζβ _n = Ζβ _Μ , in this special situation, Eq. (C33) becomes a Toeplitz matrix (see Sec. C).

We further assume that the change of the mode profile is negligible during modulation (i.e. j„ = 0 for n 0), and only the zeroth order terms remain in the effective electromotive force V, as well as in the normalized dipole moments WE and UM of Eq. (10) and (11), then

V = ν_ _η , · · · ,ν ₀ ,· · · ,ν„ (16) with V _n — V5 _n Q. These approximations work well in the adiabatic limit of modulation, and the sideband spectra calculated based on the impedance model converge and agree well with the full- wave simulation when the order of truncation increases. See Sec. E for details.

In order to give a simple explicit expression, we truncate both the sideband and the modulation to the first order; for purely reactive modulation, we define Z\ = Z_\ =—ϊξ, where ξ £ R (see Sec. E for more discussion). The effective impedance and the electromotive force at position y are simplified as

Zo = Ζβ o and Ζβ = Ζβ _±1 are the effective impedances for the carrier wave and the first order sidebands. The mode amplitudes can be expressed in a compact form:

where σ = ΖοΖ _β + 2ξ ² . Equation (19) shows that while the zeroth order response IQ is only affected by the modulation amplitude ξ , both the amplitude ξ and the modulation phase φ play a key role at the sideband frequencies.

For a particular Floquet mode, the effective impedance of the meta-atom can be characterized with Z = -iX + R^ + fl(° ^hm ), where X, fl( ^ohm ) and correspond to frequency-dependent reactance, ohmic loss and radiative loss, respectively. From the passivity condition, the following relations for electric and magnetic meta-atoms hold for the TE polarized Floquet mode (ί¾, /3„) (see Sec. D for details):

Tj is the wave impedance of the surrounding space; k _n — £¾/c is the total wave vector and K _n — ^ ^— n is the longitudinal component. Applying Eq. (19) for both electric and magnetic meta- atoms via Eqs. (12) and (13), and using the relations in Eqs. (10), (11) and (20), we can derive the dipole moments p _n and m _n , as well as the forward and backward fields and via Eqs. (3) and (4) [for details see Eqs. (A5) and (A6)]. The generalized scattering parameters for TE polarization can be defined as r _n = originates from the local power conservation [51] (see Sec. D for more details). The final expressions are given by (23)

(24)

To have frequency conversion with maximal efficiency (limited only by the fraction of energy dissipated in the meta-atoms), no energy may be transmitted or reflected at the carrier frequency, i.e. to = ro = 0. Applying this condition to Eqs. (21) and (22), we get the requirement for the modulation term: ξ ² — ¾), which should be satisfied independently for electric and magnetic meta-atoms. It is clear from Eqs. (21) and (22) that due to the symmetry of the dipole scattered field, perfect conversion is impossible if only electric or magnetic dipole response is employed. For a lossless metasurface, R( ^ohm ) = 0, the condition for complete frequency conversion can be simplified as \ξ | ² = ^Z _Q Z^, which can be strictly satisfied when ZQ and Ζβ have the same phase angle. From the relations in Eq. (20), this implies that Z _E β = ^Ζ^β,Ζ _Μ β = ^ΖΜ,Ο; ^{as a} result, the required impedance modulation | ¾ | ² = I ¾ ,o 1 ² _> I M | ² = I ¾f ,o 1 ² · The corresponding sideband scattering parameters in Eqs. (23) and (24) are then simplified as

Γ±ι = ^^(ΨΕ±ΦΕ) [! _ ₆ ψ±Αφ)} _j (25)

2v 2

V¾(M) ⁼ Arg(l¾(M),ol/¾(M),o) * ^{s me} intrinsic linear phase response of the electric (magnetic) meta-atoms and Αψ— ψΜ— ΨΕ is the intrinsic phase difference; Δφ = — (j¾ is the phase difference between the modulation signals applied to the electric and magnetic meta-atoms.

Equations (25) and (26) highlight the capability of time-varying Huygens' meta-devices to control the directionality of sidebands by changing the modulation phase difference between electric and magnetic meta-atoms. For example, for overlapping electric and magnetic resonances (Αψ = 0), |r±i | = 0 and |t±i | = 1/V2 when the relative modulation phase Δφ = 0; while |r±i | = 1/ 2 and |t±i | = 0 when Δφ = 180°. This indicates that a single time- varying Huy gens' meta- surface can function as either a transmissive (see middle-right panel of Fig. 11) or reflective device simply by changing the relative phase Δφ. Another example is when Αψ = Αφ = ±90°, the +l _si and — 1s _t orders can be well separated in opposite directions: |r_i | = |t ₊₁ 1 = 0, |t_i | = |r ₊₁ 1 = 1 /y/2 (see middle-left panel of Fig. 11). These effects will be demonstrated experimentally in Sec. IV and highlighted in Figs. 16 and 17.

In contrast to static metasurfaces, overlapped and balanced electric and magnetic resonances at the carrier frequency is not always required in time- varying Huy gens' meta-devices, for two reasons: 1) different types of directive sideband scattering require different detuning between the electric and magnetic resonances; 2) the unbalanced linear response of electric and magnetic meta-atoms can be compensated in the parametric process by adjusting the amplitudes, phases and waveforms of the dynamic modulation of electric and magnetic meta-atoms.

We emphasize that while the above discussion provides some useful insight, the condition for perfect conversion shown above is based on the approximation where both the impedance modulation and the sidebands can be truncated to the first order. In practice however, this approximation is difficult to achieve since sideband generation is a cascaded process. Moreover, impedance modulation of resonators generally changes both the amplitude and the phase of the carrier wave, and the phase changes nonlinearly with modulation signal, which inevitably introduces high order sidebands. This effect becomes more pronounced for strong modulation. In order to maximize the energy conversion to the first order sidebands, we need to introduce higher order correction terms in the modulation waveform to suppress the undesirable high order sidebands, as will be shown below.

III. DESIGN OF HUYGENS' UNITS FOR PARAMETRIC WAVES

To validate the concept, we design electric and magnetic meta-atoms working in the microwave regime using full-wave simulation (CST Microwave studio). As a first step, we study a Huygens' unit (a pair of electric and magnetic meta-atoms) in a metallic rectangular waveguide, which is easier to characterize experimentally. We emphasize that while the theoretical discussion above focused on a meta-device excited by a normally -incident plane wave, the impedance model is very general and can be easily extended to other situations. Here, the pair of closely spaced resonators are positioned at the center of the waveguide and are excited by the fundamental waveguiding mode. Although the effective impedance Z of the meta-atom in a rectangular waveguide is different from the one in a periodic array, the whole system is still closely related to the situation discussed in Sec. II B, since in both cases the scattered fields from the electric and magnetic meta- atoms have opposite parities, and the waveguide system can be considered as a special situation discussed in Sec. II B, in which the carrier wave and the sidebands share the same scattering channels (forward and backward in normal direction). Therefore, we expect that the features discussed in Sec. II B can also be observed in this system. Indeed, we find that even a basic Huygens' unit made of a pair of electric and magnetic meta-atoms is sufficient to demonstrate the effect of directional sideband scattering in conjunction with a high conversion efficiency.

FIG. 12. (a) Design of a Huygens' unit element (sizes in mm): XM— 7.87, = 12, XE— 14.25, = 8, = 3.89, d = 6.5. The track width and gap size are both 1 mm. Measured at 3.7 GHz (λ f¾ 81 mm), the overall size of the Huygens' unit is subwavelength: ¾ ~ 0.18A , 0.15λ,ά ¾ 0.08A. The thickness of the substrates is 0.4 mm. (b) The simulated transmission and reflection coefficients at =3.7 GHz as a function of the DC bias voltages (£%, UM), in the absence of dynamic modulation.

The Huygens' unit consists of an electric and a magnetic split-ring resonators, which are printed individually on a Rogers RO4003 substrate (e _r = 3.5, loss tangent 0.0027, substrate thickness 0.4 mm), and the whole unit is positioned at the center of a WR229 rectangular waveguide that supports a TEio mode [see geometry and parameters of meta-atoms in the caption of Fig. 12 (a)]. We use varactor diodes (SMV1405) as the voltage-tunable capacitors. Under appropriate DC bias voltages, the electric and magnetic resonances overlap. Figure 12 (b) shows the simulated linear response of the unit at ωο/(2π) =3.7 GHz as a function of the DC bias voltages (L¾, UM)- There exists a parameter regime around UE/UM∞ 2, where the reflection approaches zero and the transmission phase changes by around 2π, which is a typical manifestation of the Huygens' condition.

We are interested in the behavior of the unit around the overlapped resonance when dynamic modulation is introduced. We choose the carrier frequency to be <Μο/(2π) =3.7 GHz, and assume that the modulation frequency Ω is sufficiently low such that the sidebands produced by the dynamic modulation can be calculated in the adiabatic limit (see Sec. F for more discussion), where we approximate the time-varying transmission and reflection signals with voltage-dependent stationary linear responses (see Sec. F for more discussion). In this paper, we only study the behavior in the adiabatic limit of modulation, while a detailed theoretical discussion of modulation beyond the adiabatic limit can be found in Ref. [52].

The modulation voltage is defined as t/ _E (M)( = ¾M),am _P ( + ¾M),off _s> with t/ _E(M))amp , and C E(M),offs being the modulation amplitude and the DC offset; f(t) =∑¥ ₌ ι α^ _Μ ^08[/(Ωί— Ε(Μ) )] + ^E(M) ^sm [^(^t— Ε(Μ))] i ^{s me} normalized waveform constructed using a Fourier series. As has been noted in Sec. Π Β, we introduced high order terms in the modulation waveform to suppress the undesirable higher order sidebands, and we truncate the highest order to N = 8 since higher order terms (N > 8) only bring in negligible improvement. For each set of modulation signals [¾(t), £/M( ] _> ^{we can} obtain the modulated scattering parameters at each time step via a one-to-one mapping of the voltage-dependent stationary linear response from Fig. 12 (b), and the corresponding sidebands can be calculated via Fourier transformation (see Sec. F for more details).

FIG. 13. Simulated normalized sideband spectra for forward and backward scattering under the optimized modulation waveforms for (a) single-sideband directive forward scattering, (b) double- sideband directive forward scattering, and (c) double sideband bidirectional scattering. The spectra are normalized to the total scattered power of all sidebands up to the tenth order, including the carrier frequency; ζ„ is the normalized scattered power for order n in the forward/backward direction. For clarity, only sidebands up-to the third orders, with a normalized power larger than 10 ^~4 are shown. Stars indicate the dominant sidebands. The corresponding modulation voltage signals, as well as the resulting modulation of transmission and reflection are plotted in (d), (e) and (f).

We employ a genetic algorithm to optimize the waveform in order to maximize both the directivity and the power of the chosen sidebands (see Sec. Sec. H for details). As an example, Figs. 13 (a) to (c) depict the normalized spectra for three different types of directive sideband scattering. The spectra are normalized to the total scattered power of all sidebands including the carrier frequency, and more than 80% of the scattered power is converted to the targeted sidebands. The corresponding modulation waveforms and the resulting modulation in transmission and reflection are plotted in Figure 13 (d) to (f). To simplify the optimization procedure, we use the same normalized waveform for L¾ and i½- This restriction is not essential, however our empirical tests indicated no improvement in efficiency when allowing these waveforms to differ. The optimized coefficients of the modulating signals are detailed in the Table I of Sec. H.

The results of the optimization process reveal that different types of sideband generation require very different modulation waveforms. To achieve high conversion efficiency for single-sideband forward scattering [Fig. 13 (a)], the transmission coefficient is ideally modulated linearly over 360° (see Sec. B for more discussion); the optimized result indeed shows that the phase of transmission changes almost linearly over 360°, and the required modulation waveform has a highly asymmetric sawtooth-like shape [Fig. 13 (d)]. In contrast, for double-sideband unidirectional forward scattering [Fig. 13 (b)], the transmission is ideally amplitude modulated by a purely sinusoidal waveform (see Sec. B), thus the optimized L¾ and UM have a sinusoidal-like waveform and are modulated in phase (Δφ ¾ 0) [Fig. 13 (e)]. Note that the reflection remains low during the modulation cycle and thus the transmission amplitude modulation is achieved via the absorption modulation of the meta-device. The most demanding situation is double-sideband bidirectional scattering [Fig. 13 (c)], which ideally requires that both the transmission and reflection phases are modulated linearly over 360°, but with opposite signs of the slopes (see Sec. B). This challenging requirement can be maximally satisfied using electric and magnetic meta-atoms simultaneously since we can find a parameter regime from Fig. 12 (b) where the transmission and reflection phases change dramatically. The final optimized result from Fig. 13 (f) confirms that the transmission phase is indeed modulated over 360° with a positive slope while the reflection phase is modulated with a negative slope; L¾ and UM are modulated with a phase-lag Δφ fa—90°, indicating there is an intrinsic phase difference of Αψ fa—90° between the electric and magnetic response.

These results are consistent with our previous discussion based on Eqs. (25) and (26), which shows the Δφ required for different types of sideband control. In fact, the optimized sideband spectra can also be reproduced using the impedance model introduced in Sec. II B by taking into account the higher order terms in the impedance matrix shown in Eq. (C33). To evaluate the accuracy of the impedance model, we calculate the Fourier coefficients Z _E ( _M ) _W using the time- varying scattering coefficients shown in Figs. 13 (d) to (f), and apply these Fourier coefficients in the impedance model to calculate the mode amplitudes -¾(M),W ^{as we} H ^{as me} sideband spectra (see Sec. E for details). The results clearly show that as the order of truncation increases, the spectra calculated with the impedance model converge and agree well with the full-wave simulation. In fact, the maximum relative error of the dominant sidebands is already below 5% when we truncate the harmonics to the first order sidebands (see Fig. 20 in Sec. E). Note that different from the simplified case discussed in Eq. (17), the second order terms Z± ₂ are also preserved in Eq. (C33), which allows us to calculate the case where the modulation waveform is asymmetric, such as single-sideband conversion. IV. EXPERIMENTAL REALIZATION

A. A Huygens' unit in a rectangular waveguide

FIG. 14. Schematic of the microwave experimental setup. RF Signal Generator: HP-8673B; Vector Network Analyzer: Rohde & Schwarz ZVB-20; Spectrum Analyzer: Rohde & Schwarz FSV-30; Arbitrary Function Generator: Tektronix AFG 3022B; Switch: Agilent 8763B coaxial switch. For clarity, the substrates of the meta-atoms are not shown.

To demonstrate the dynamic control of sideband scattering, we fabricated the electric and magnetic meta-atoms separately on a Rogers RO4003 printed circuit board; varactor diodes were soldered into the additional gaps in the meta-atoms as voltage tunable elements. The schematic of the microwave setup is shown in Fig. 14. The pair of meta-atoms were fixed on a polystyrene foam holder (with a relative permittivity close to 1) forming a Huygens' unit, and positioned at the center of a rectangular waveguide (see Fig. 14). We first employed the network analyzer to measure the resonance spectrum of the sample, and set the DC bias of the arbitrary function generator to tune the two meta-atoms such that their resonances overlap. The resonant frequencies of the fabricated meta-atoms red- shift compared to the simulation, due to the slight difference in substrate permittivity and the connection to the bias network; nevertheless, the resonant frequencies are still in the designed range and can be easily tuned to overlap. When the electric and magnetic resonances are separated, two reflection peaks were observed; as the DC bias voltages (¾offs _> t½,offs) ~ (— 2.0V,—2.5V), the resonances of the pair overlap, with a reflection dip observed at around 3.64 GHz and the measured transmission phase varies over 2π, which is a clear evidence for the Huygens condition. We set this point as the initial point for optimization and introduced a microwave CW carrier signal with too/ (2ττ) = 3.64 GHz from the RF signal generator. Periodic modulation signals with a base frequency Ω/(2π) = 2 MHz were generated by the arbitrary function generator to modulate the meta-atoms; the interaction between the carrier wave and the modulated meta-atoms generate sidebands on both sides of the carrier frequency, which were measured by the spectrum analyzer.

FIG. 15. Measured normalized sideband spectra of a pair of electric and magnetic meta-atoms in a rectangular waveguide. The spectra are normalized to the total scattered power of all sidebands up to the tenth order, (a) single-sideband forward scattering, (b) double-sideband forward scattering, and (c) double-sideband bi-directional scattering. Stars indicate the dominant side- bands, (d) to (f) are the corresponding modulation waveforms and the time-varying transmission and reflection signals.

The modulation voltage signals were constructed using the same definition of Fourier series shown in Sec. ΠΙ. The offset voltage £ E(M),offs _> ^me amplitude ¾(M),amp _> the phase (¾(M) as well as the coefficients of waveform were optimized using genetic algorithm to maximize the power conversion to the desirable sidebands (see Sec. H for details). Figure 15 (a), (b) and (c) show the normalized scattering spectra for three different types of directive sideband scattering; the normalized power of the dominant sidebands is around 75% for single- sideband unidirectional scattering and double-sideband bidirectional scattering, and around 84% for double-sideband unidirectional scattering. The corresponding modulation waveforms and the time-varying transmission and reflection signals (measured with I/Q mode of the spectrum analyzer) are shown in Fig. 15 (d), (e) and (f). All the important features of time- varying amplitude and phase match well the theoretical predictions shown in the Fig. 13 (d) to (f). The optimized coefficients of the modulation signals are listed from Table II to Table IV in Sec. H. These results confirm that, with appropriate modulation waveforms, a Huygens' design can achieve various directional sideband scattering with high conversion efficiency.

FIG. 16. (top) Schematic representation of the dynamic control of sideband scattering in the rectangular waveguide, where the directionality of sidebands can be tuned by the modulation phase difference φ _\ ι— <j¾. (a) to (c) Measured normalized scattered power of the first order sidebands as a function of the modulation phase difference for three different types of directional sideband scattering, (a) Single-sideband unidirectional scattering; (b) double- sideband unidirectional scattering; (c) double- sideband bidirectional scattering. The sideband powers are normalized to the total scattered power of all sidebands (including the carrier frequency). For clarity, the curves are scaled by a factor of 2 in figures (b) and (c).

Further, to demonstrate the capability to dynamically tune the directionality, we kept the optimized waveforms unchanged and only tuned the relative modulation phase of electric and magnetic meta-atoms. Remarkably, for all three types of directive sideband scattering, the directionality of the sideband scattering can be dynamically tuned in between forward and backward, without sacrificing the total conversion efficiency [Fig. 16 (a), (b) and (c)]. This property is particularly attractive since a time-varying Huygens' meta-device can function as a lens (transrnissive mode), a mirror (reflective mode), or something in between, without requiring additional optimization of the waveform but a simple change of the relative phase of modulation. The sinusoidal-like change of the power with respect to Δφ = — <j¾ is also consistent with the prediction from Eqs. (25) and (26): |r±i | ² °= 1 -∞5(Αψ± Αφ) and |t _±1 | ² « 1 + cos(Ai^± A<]()).

B. A finite Huygens' meta-device in a parallel-plate waveguide

FIG. 17. (a) Photographs of the fabricated time-varying Huygens' metadevice. The electric meta-atoms (top) and magnetic meta-atoms (bottom) are fabricated on the opposite sides of the substrate. For detailed geometries, see Fig. 21 of Sec. F. (b) and (c) Measured scattered field intensity at sideband frequencies ω±ι under the modulation phase pattern G3 [see (f)], showing simultaneous beam-steering of the two side-bands in different directions, (d) and (e) Measured scattered field intensity under phase pattern G5, showing simultaneous focusing and defocusing of the two sidebands, (f) Measured angular distributions of the scattered power at sideband (0-\ under different modulation phase patterns. The plots are normalized to the peak power of plot "Gi , <PE = ΨΜ ί·6·, under uniform modulation phase and forward scattering mode, (g) Polar plots of the angular distribution. For clarity, each plot has been normalized to its peak power, (h) Schematic of the modulation phase patterns.

One advantage of time-varying Huygens' meta-devices is that they allow dynamic complex wavefront control in ways impossible using conventional linear tunable metasurfaces, such as all- angle beam steering and frequency-multiplexed functionalities. To demonstrate these effects, we designed and fabricated an array of Huygens' units working inside a two-dimensional field scanner based on a parallel-plate waveguide, as shown in Fig. 17 (a) (for the detailed designed geometries, see Fig. 21 of Sec. F). The electric and magnetic meta-atoms are printed on the opposite sides of a single substrate to enhance the mechanical stability.

The array is composed of 16 identical units, each consisting of an electric and a magnetic meta- atom that can be modulated independently. To mimic normally incident plane wave excitation, we generated a collimated beam using a parabolic mirror to direct the signals from a monopole source antenna. We employed four dual channel arbitrary function generators to introduce 8 channels of independent modulation, where four of them were applied to the electric meta-atoms, and the other four used to modulate the magnetic meta-atoms. For the details of experimental setup, see Fig. 22 and 23 and the corresponding discussion in Sec. G.

To demonstrate dynamic metasurface functionality, we first tuned the bias voltages such that the two resonances overlap around 4 GHz. Then we introduced a CW carrier signal at ωο/(2π) =4 GHz and modulated the array with 4 synchronized arbitrary function generators. The modulation waveforms were first optimized to achieve double-sideband forward scattering for sidebands (0± _\ under a uniform modulation phase pattern Gl [see Fig. 17 (h)]; the optimized coefficients of the modulation signals are listed in Table V in Sec. H. For simplicity, we only introduced different modulation phases at different units while keeping the modulation waveforms of all the electric or magnetic meta-atoms identical (see Fig. 17 (h) for the phase patterns). Due to the limited number of independent modulation channels, we only generated phase patterns with 4 discrete phase levels.

Since different sidebands experience different imprinted phases, the time-varying meta-device allows frequency-multiplexed functionalities. When the first order term is dominant in the modulation waveforms, the phase imprinted on sidebands ω± can be considered as conjugated. We confirmed this unique feature by mapping the two-dimensional distribution of the field intensity under the modulation phase patterns G3 and G5, as shown from Fig. 17 (b) to (e). The phase pattern G3 allows large angle beam steering (>60 degrees) of sidebands (ύ and (0 _\ towards different directions, which was further confirmed from the scattering pattern in Fig. 17 (f). A more interesting effect occurred when the phase pattern G5 was applied - the time- varying meta-device functioned as a convex lens for ω ₁ and a concave lens for ω-ι at the same time. By properly engineering the modulation waveforms, additional functionalities can be multiplexed into higher order sidebands.

We measured the angular distribution of sideband power via a circular scan around the center of the sample. By introducing different modulation phase patterns, we achieved directional beam steering of the sidebands over the whole 360°. Figure 17 (f) depicts the measured angle-dependent power for sideband (ύ- _\ . For clarity, we normalize each measurement and plot them in Fig. 17 (g). As has been shown in Fig. 16, adding an additional π difference between the modulation phases <j¾ and <PM allows the beam to be steered from forward (marked by circles) to backward (marked by squares) direction. Such an all-angle directional beam steering highlights the unique capability of time-varying meta-devices as it is a highly nontrivial task for conventional tunable metasurfaces or beam deflectors.

The asymmetry of the measured scattered power in Fig. 17 (f) under conjugated phases originates from the inhomogeneous response of the Huygens' units. This was mainly introduced during the hand-soldering of the varactor diodes and resistors, and it can be overcame by employing a more sophisticated fabrication process. We also note that the peak power reduces at large beam- steering angle, which can be attributed to two main reasons. The first one is due to the fact that we kept the modulation waveforms unchanged during beam steering. In fact, due to the mutual interaction among the units, the effective impedance of the Floquet mode will have a noticeable change at large steering angles compared to the scattering at normal direction. We expect that additional improvement of the conversion efficiency can be obtained if the modulation waveforms are optimized for each steering angle. The second reason is the reduced directivity due to the decreased effective aperture at large steering angle. In fact, this is a common problem in all array antenna systems [53], and the directivity can be further improved by increasing the sample size (currently ~ 2λ) and adapting more sophisticated designs of the meta- atoms.

V. DISCUSSION AND CONCLUSION

The concept of time- varying Huygens' meta-devices studied here could provide new tools and possibilities for research and applications that generate and manipulate sidebands. The ability to achieve high conversion efficiency of sidebands is particularly attractive since it could enable various ultra-compact devices. One potential example is compact isolators. A naive design could consist of a bandpass filter that only transmits <¾¾ at the input side, followed by a time- varying Huygens' meta-device that achieves single sideband up-conversion (ωο— > (0\ ), and a bandpass filter that only transmits <0\ at the output side. Since the process of sideband generation is nonreciprocal, a wave propagating in the reverse direction with frequency ω ₁ would be further up-converted to ft¾ after passing the time- varying Huygens' meta-device, and be blocked by the filter at the input side.

Another potential application based on the effects shown in Fig. 17 is a compact beam deflector that could steer one or more sidebands at different directions and scan them over the full 4π solid angle. This functionality could benefit the development of new compact radar systems for full- angle and multi-target detection. For example, existing frequency-modulated continuous wave (FMCW) radars widely employed in vehicles for obstacle detection can only aAIJseeaAI things within a certain angle range, and thus multiple radars are required to cover 360 degrees of view. This solution might become too costly and bulky when it is applied to light-weight platforms such as drones and small robots. A naive vision is to utilise a time-varying metasurface as the transmitter of the radar, from which different sidebands can be steered to different directions. The angular directions of the obstacles all around can be detected simultaneously with a single monopole receiver by identifying the frequencies of the signals bounced back.

The proof-of-concept design presented in this paper shows a promising start, yet further optimization and even new designs are required to facilitate implementation in more practical areas and in a more scalable fashion. We note that while the relatively high quality factor of the meta- atoms employed in the current design allows large modulation with a small voltage (< 10 V), the peak absorption of the system can reach 65% when the electric and magnetic resonances overlap. From simulation we notice that the percentages of the energy lost in the varactor diodes, metal tracks and substrates are around 66%, 20% and 14%, respectively. To reduce the absorption loss, one can design meta-atoms with a lower Q factor by increasing the scattering losses, and avoiding the self-resonance of the diode. In addition, one can employ tuning mechanisms that have lower loss, such as MEMS NEMS [54-57], tunable capacitors based on ferroelectric thin films [58, 59], or transistors.

While the demonstrated dynamic sideband control is limited to the simple case of one- dimensional array with only 8 independent channels, this is not a fundamental limitation. Using an FPGA or micro-controller to achieve independent modulation over a large two-dimensional array is technically possible, as has been shown recently in the static tuning of a Huygens' meta-device [60]. The extension of the idea to terahertz and even optical frequencies is feasible by using other high-speed modulation mechanisms, such as electro-optical [61-63], optomechanical/acousto- optical [64—66] and nonlinear optical effects [67-70].

To conclude, we introduced the concept of time- varying Huygens' meta-devices and studied the sideband generation and manipulation in this type of system. We showed both theoretically and numerically that dynamic modulation of Huygens' meta-devices provides unique opportunities in manipulating parametric wave scattering. Importantly, we designed and fabricated prototype meta- devices working at microwave frequencies, and successfully demonstrated controlled generation and directive scattering in the experiment, with a high conversion efficiency (> 75%) from the carrier wave to the target sidebands. We also demonstrated novel effects that are difficult to achieve with conventional tunable linear meta-devices, including all-angle beam steering and frequency- multiplexed functionalities. Our study provides new insights for realizing highly efficient and ultra-compact devices based on dynamic modulation, allowing dynamic tuning of electromagnetic waves in an almost arbitrary way, which should find potential applications in many areas including radar and compressive sensing. TECHNICAL DETAILS

Section A: Boundary conditions

The physics behind the concept of time-varying Huy gens' metasurfaces can be described by the generalized sheet transition conditions (GSTCs) [48]:

n x ( - H ^~ ) =— Pi, - -n x VM _± (Al)

V II II / dt ¹¹ μ

n x (E+ - E ^" ) = -^M|| - ^n x VP (A2) where n is the normal vector of the surface; '+' and '— ' represent field components on the two sides of the surface; and ' || ' denote the components being normal and parallel to the surface. In the paper, we limit our discussion to a flat meta-device with polarizabilities only in the transverse direction (P^ = Mj_ = 0), which is described by Eqs. (1) and (2). Without loss of generality, we assume that the meta-device is on the plane z = 0, with P = P _x ^■ x + P _y ^■ y and M = M _x - x + M _y - y, and the wave is propagating in the y— z plane.

When a slow periodic modulation is introduced, the time-dependent components of Eqs. (1) and (2) can be decomposed in the frequency domain in the form of F(r, t)— ^^_ _∞ F _n (r)e ^→(0nt , with (On = α¾ + ηΩ, with £0Q and Ω being the carrier frequency of the incident field and the modulation base frequency of the polarizations, respectively. The boundary conditions Eqs. (1) and (2) for TE (transverse electric) polarization at frequency £¾ are then given by

Hy, _n +H _n - H _yfl 5 _n0 = ιωηΡχ,η (A3) ,η - Εχ,η - <(Ao = i<¾M _y , _n (A4)

The subscripts 'i', 'b' and denote the incident, back-scattered and forward-scattered field components, respectively; 5„Q is the Kronecker delta function. The conditions for TM (transverse magnetic) polarization can be obtained from duality (E _x — > H _x ,H _y →—E _y ,P _x → M _x ,M _y — »—Py).

In the simple case discussed in Sec. II B, the response of the meta-device is isotropic (i.e. polarization-independent in x and y directions); the carrier wave propagates in the normal direction while the scattered sidebands are Floquet mode with transverse wave vectors β _η .

For TE polarized propagating waves, the ratio of the transverse electric and magnetic fields is given by E _n /H _n = 7]k _n /K _n . η = is the wave impedance of the surroundings, k _n = (On/c is the wave vector and κ _η = — is the longitudinal component. By substituting the magnetic field H _n with the electric field E _n in the boundary equations, we can get the transverse field components in the forward and backward directions as n ^{= ~} 2A ^~ ^ _n ^{Pn ~ ~} c ^{mn) ' ( }}

Pn ⁼ Pn-A, m _n = Μ„Α/μ are the Fourier components of the effective electric and magnetic dipole moments, respectively, where A is the area of the unit-cell and μ is the permeability of the surroundings. Equations (A5) and (A6) are employed to calculate the generalized scattering parameters: r„ = as shown in Sec. II B.

Section B: The ideal condition for different types of sideband control

Equations (A5) to (A6) directly link the field scattered from a Huy gens' meta-device to the Fourier components of the modulated electric and magnetic dipoles. By assigning the desired scattered fields E _n in the forward and backward directions, we can retrieve the required Fourier components of the modulated dipole moments p _n and m _n based on Eq. (A5) and (A6).

FIG. 18. The normalized time-varying signals t = E ^f /E ^l and r = E ^b /E ¹ , for the three types of ideal directive sideband scattering: (a) single-sideband forward scattering, (b) double-sideband forward scattering, and (c) double-sideband bi-directional scattering.

As an example, we examine the required dipole moments for the three types of perfect directive sideband scattering that are discussed in Sec. Ill, where all the sidebands are scattering in the normal direction (κ = k), as indicated in Fig. 18. In the ideal situation where all the energy from the carrier wave is converted to the desired sidebands: E _Q — E _Q — 0 |E' | ² _> it requires that po = mo/c = ϊΕ ¹ Α/(ωοη).

We can get the time varying electric fields E ^f ( ^b ) (i) = nE^e-^, as well as the required dipole moments p _n and m _n for the three different types of directive sideband scattering:

(a) single-sideband forward scattering

E ^f (t) = E_ ^f _{i e} ^iilt = £ν ^Ωοί , (B l) E ^b (t) = 0, (B2) which requires p-i = m-i/c, (B3) where

\ρ- ₁ \ = Ε ^ί Α/(ω- ₁ η). (B4)

(b) double-sideband forward scattering

E ^b (i) = 0, (B6) which requires where

(c) double-sideband bi-directional scattering which requires where

The resulting normalized time- varying signals t(f) = E ^f (t)/E ^l and r(i) = E ^h (t)/E ¹ are given in Fig. 18. In the case where only one sideband exists on one side (front or back side) of the meta-device, it requires a pure linear phase modulation of t or/and r with a phase variation of 360° [see Figs. 18 (a) and (c)]. When both sidebands exist on the same side of the meta-device [see Fig. 18 (b)], a pure sinusoidal amplitude modulation is required. It should be noted that there exist instances where the amplitude of the normalized signal goes above one for double-sideband forward scattering. Such an oscillation is due to the temporal interference of the two sideband waves, but the total energy remains conserved after time averaging over one modulation cycle. However, we do notice that in the strict adiabatic limit of modulation (Ω/ωο—· 0), the signals t and r at each time-instance should correspond to the stationary transmission and reflection coefficients of the carrier wave, which should not exceed one at any instance for a purely passive meta-device. Therefore, the required signal in Fig. 18 (b) does imply that at least for adiabatic modulation, balanced gain and loss are required during each modulation cycle in order to achieve a 100% lossless conversion to double sidebands. This is related to the recent findings in static Huygens meta-devices, which showed that balanced spatial-dependent gain and loss are required for perfect beam- steering [13] in the absence of bianisotropy or spatial dispersion [10, 11].

Section C: The interaction impedance matrix

In general, mutual interaction exists when the meta-atom is positioned in an array or in certain environments such as a waveguide or close to a ground-plane. The linear response of the meta- atom at position r ₇ can be described by the coupled equation:

¾ _elf (r;)/(r;) + £z _mut (r , r _i )/(r _i ) = V^) .

¥J

(CI)

Z _sel f(r ) is the self-impedance of the meta-atom at position r , and ¾n _U t( ^r 7, ^r i) is the mutual- impedance that describe the interaction between meta-atoms at r _m and r„, and Vj (r^ ) is the effective electromotive force (input voltage). When dynamic modulation is introduced, Z _se if(r _m ) can be expanded into a matrix to describe the nonlinear parametric process that occurs locally within the meta-atom, while ^(Γ-,, ΓΒ) can be expanded into a matrix to describe the linear mutual interaction at each individual sideband frequency (as will be shown below).

The effect of mutual interaction can be incorporated with the self-impedance as an effective impedance Z _e ff, and Eq. (CI) can be simplified as

Z _eff (ry)/(r ) = VHry) . (C2)

For simplicity, we have written the effective impedance of the electric (magnetic) meta-atom as ¾(M) ^{m me} main text, and its interaction with the incident wave can be expressed as: Z _E ( _M )/ _E ( _M ) = ^Ε(Μ) · Note that if we retrieve the effective impedance of the meta-atom in a periodic array or waveguide via the scattering parameters from full-wave simulations, then the interaction effects are automatically taken into account. The form of Vj depends on the incident polarization and the mode profile of the meta-atom. For simplicity, we assume that the incident wave propagates in the y— z plane, the response of the meta-atoms is isotropic in the x— y plane, and the meta-atoms are electrically thin in the z direction. The effective electromotive force acting on the electric and magnetic meta-atoms under a plane wave excitation E ^+^ can be expressed explicitly. For TE wave excitation,

= JJM - Ε ³ Γ = iK& jP ^y . (C4)

For TM wave, we have

^M ^M) = / J ^' M · E ³ r = ik& jP ^y ., (C6) where K, β and k are the longitudinal, transverse and total wave vectors, respectively. jE(r) and jNi(r) are the normalized electric current distributions of the modes on the electric and magnetic meta-atoms. At normal incidence, the expressions are simplified as Eqs. (10) and (1 1), respectively, ME and "M are the normalized effective electric and magnetic dipole moments of the meta- atoms, which can be defined from the normalized electric current distribution j (r)

^{UM =} J ^{R X} J (r)d ³ r, (C8) where the integral is performed over the volume of the meta-atom. Note that since the normalized current j (r) has a unit of m ^-2 , «E has a unit of m, while «M has a unit of m ² .

The effective impedance can also be defined rigorously based on the mode of the meta-atoms, where the details can be found in our previous studies [49, 50].

FIG. 19. Schematic of the series RLC equivalent circuit for a time- varying meta-atom.

Below, we derive Eq. (5) in the main text. When dynamic modulation is introduced to the meta- atom at position Xj, the effective impedance Z _e fr, mode amplitude / and effective electromotive force V become time-dependent. Based on Kirchoff 's voltage law, the input voltage should equal the total voltage across the series RLC equivalent circuit (see Fig. 19):

V _i (f) = Vt( + V ^r c( + V _R ( (C9) where

_{T7 /} . dLI _T . . dL _T , . dl

= Q(t)L(t) + L(t)Q(t), (CIO)

Vc(t) ( S(t)Q(t) , (Cl l)

C(t)

V _K (t) = R(t)Q(t) (C12)

Here S = 1/C is the elastance, and we use charge Q since it yields purely differential equations rather than integro-differential equations. Vi is the input voltage generated by a particular set of polarization and mode, represented by one of Eqs. (C3) to (C6). For illustration, we use V{(t)— E ^x u _E {t) .

When the temporal modulation of the impedance is periodic in time with a frequency « fi), each time varying effective circuit parameter in the above equations can be expanded in a Fourier series of the form: F{t) and the charge can also be expanded as Q(t) = _e -i<»t∑+∞_ _∞ Q _ie -il(Cit- ⁽ p) φ is the modulation phase corresponding to a constant time-offset for the modulation waveform. Then we have

¾( = £ Rie ^~l'l£u e ^ll( P -i £ e> _m Q _m e-^e→ (C15)

For electric meta-atoms excited by a normally incident field

V _i (t) = E ¹ £ u^e-^e^. (C16)

Satisfying Eq. (C9) at every instant in time requires that the Fourier coefficients satisfy following equation

l=—∞m=—∞ Orthogonality of the exponential functions implies that equivalent terms on each side of Eq. (CI 7) must be equal. This yields the following equation for each order n:

+∞

∑ - (o^ + ©2 ) _L ( ₊ s( _ _ia ?( _Q (m) _e -i(i+m)(at- ₉ ) _{= E} i _{UE ne} -in(at-cp) _{^ (C 18)} where I = n— m. We further define I _m -i((Om + l .)Li + iSi/cOm + Ri, and V _n — E ^l UE, _n , then Eq. (C18) becomes

+∞

(C19) which is Eq. (5) in the main text. Note that for brevity, we have omitted the notation ry in the above equations.

Now we bring the position notation ry back and write the coefficients of Eq. (CI 9) in a matrix form:

¾r )I(r ) = V(r, (C20) where

(C22)

V(r ) (C23) q> _j is the modulation phase at ry. It is important to note that while we have incorporated the effect of mutual interaction in the effective impedance _¾ _e ff, we have assumed that only the modulation of self-impedance contributes to the higher order terms 0), i.e.

¾ry)I(ry) = ¾ry)I(ry) +∑Qry, r _i -)I(r _i )

(C24) where

¾ei _f ( ^r y) * ^s ^ ^{ε zerom} order self -impedance of the meta-atom at position r , evaluated at frequency (Dm. The mutual-impedance matrix describes the linear interaction at each sideband frequency and thus it is a diagonal matrix with only the zeroth order terms

Therefore, the value of the zeroth order effective impedance Z™ in Eq. (C21) is determined by not only the zeroth order self-impedance, but also the mutual impedances and the mode amplitudes of all other interacting meta-atoms at sideband frequency c¾„. In general, Z _Q ^ Z _Q ^ ύ ^' ηι φ η, except for some special cases.

When the modulation frequency Ω is much smaller than the linewidth of resonance of the meta- atoms, the difference between the impedance elements of the same order evaluated at different sideband frequencies G¾ and (½ becomes negligible. We can simplified the notation as Z _se ] = and Zm _Ut = Z _MU ( = Z _MUT , and the impedance matrices Eq. (C25) and (C26) can be simplified as Toeplitz matrices:

(C27)

(C28)

For the special example discussed in Sec. II B, where the dynamic modulation of meta-atoms has identical modulation amplitude but a periodic linear phase gradient along y direction: (p(y) = φο + Gy, the solution of the system Eq. (C22) should be a series of Floquet modes:

Ito) = I(y.) G to ¾) (C29) where

(C31)

G is the spatial frequency of modulation, β _η = /¾ + nG is the transverse wave vector of the Floquet mode, and φο is the modulation phase at y = 0. Applying Eqs. (C27) to (C31) in Eq. (C24), it becomes clear that the effective impedance matrix of the series of Floquet modes is given by

where the higher order components Z _n are generated directly from the modulation of self- impedance, while Zp _m {yj) = Z _se \f(yj) +∑ _i Z _mut (yj,yi)e ^lmG ( ^y j ^~yi " ¹ is the zeroth order effective impedance of Floquet mode (<¾¾„, j3 _m ). With a normally incident carrier wave ( ¾ = 0), we have Ζβ _m = Zp _ _m , and the effective impedance matrix is given by Eq. (15). Only in the special case where the sidebands are also scattered in the normal direction (G = 0), the effective impedance matrix becomes a Toeplitz matrix:

(C33)

Generally, the Fourier coefficient Z _n and Z_ are complex but not conjugated. However, in the special situation where only reactive or resistive modulation exists, Z\ = Z* _j =— ϊξ , and the phase of the complex value ξ is determined by the time delay of the modulation waveform. Since we already introduce the phase parameter φ to describe the time delay of the modulation signal, it is convenient to set the phase of ξ to zero in Eq. (17) such that ξ G R.

Section D: The relation between radiative loss and normalized dipole moments

Below we derive the relation between the radiative loss term /?( ^rad ) and the normalized dipole moments for the Floquet mode (t¾, j8„) under TE polarization, as shown in Eq. (20) of the main text:

n(rad) _ Π "Ε½ p(rad) _ ^M^ ¹ ^ Τ»Ή E> ^W - 2AfC„ ' - 2A · ^Di Note that the above relations are only valid for Floquet mode (ω _η , β _η ) under TE polarization. These intrinsic relations are determined by the current distributions of the modes, and thus should hold regardless of the modulation condition. Therefore, it is convenient to derive the relations in an unmodulated meta-device consisting of identical meta-atoms in each unit, where both the excitation and the scattering are propagating plane waves with a transverse wave vector β _η . For a lossless meta-device (/?( ^ohm ) — 0), the fields in the forward and backward directions are given by

E ^f = E ⁱ +E ^p + E ^m , (D2) E ^b = E ^v - E ^m . (D3) where E and E ^m are the scattered plane waves generated by the array of electric and magnetic meta-atoms, respectively. From the passivity condition, the power should be conserved:

From Eqs. (D2), (D3) and (D4), we have the following relation:

EP + (EyE ^m ] + |E ^P | ² + |E ^m | ² = 0. (D5)

This relation should hold regardless of the ratio of E ^p and E ^m , i.e. it should also be satisfied even for an array of pure electric or pure magnetic meta-atoms, and thus requires the electric fields generated by electric and magnetic meta-atoms satisfy the following relations individually:

ΈΙε[(Ε ^ί )*Εν] + \Ε*\ ² = 0, (D6) Re[(E ⁱ YE ^m ] + |E ^m | ² - 0, (D7)

Substitute with the following relations:

(D9) and use the identity for lossless meta-atoms Re(l/Z) = Re(Z*) /|Z| ² = 7? ^(rad V|Z| ² , we get Eq. (20). Section E: Evaluating the accuracy of the impedance model

Below, we evaluate the accuracy of the impedance model based on the Toeplitz matrix shown by Eq. (C33), which is a special case of Eq. (15). One way to calculate the impedance matrix Eq. (C33) is to retrieve the time-varying impedance Z(t) from the time-varying transmission and reflection coefficients, and calculate the Fourier coefficients Z _n via a Fourier transformation of

Z{t).

As can be inferred from Eqs. (21) and (22), the stationary transmission and reflection coefficients of a static meta-device excited by a normally incident plane wave can be expressed as

t = l - ^ ^_ (E2) from which we can retrieve the effective impedance ZE and ZM as

Z _E = 2/ ^rad) /(l - t - r) , (E3) Z _M = 2 ^ad) /(l - t + r) (E4)

The time-varying impedance Ζβ(ί) and ZM( under a slow modulation (L¾(t) and £/Μ( ) can then be approximated by calculating the stationary response at each time-step with Eqs. (E3) and (E4), and the Fourier coefficients Z _E ( _M )„ can be obtained via Z _E ( _M ) _N — ^

where T = 2π/Ω is the modulation period.

Once we get the impedance matrix Z _E (M) _> ^{we can} calculate the mode amplitude I _E ( ) using Eq. (6), and the corresponding dipole moments p _n and m _n from Eqs. (12) and (13), as well as the scattered fields from Eqs. (A5) and (A6). At normal incidence the generalized scattering parameters can be defined as r„ = E^/E ¹ and t _n = E^/E ¹ . The final expression of the generalized scattering parameters can be expressed in a com act form:

where 3 < = [¾-jv, · ^■ · , ¾o _> · ^■ · , SON] ⁷ , with ¾v being the Kronecker delta function; r=[r_/v, ^■■■ , ro and · ^■ · ,to, · · · , tw] ^r - N is the order of truncation. FIG. 20. (a) to (c) The relative error of the sideband power calculated using the impedance model when comparing to the ones shown in Fig. 13 (a) to (c).

To show the accuracy of the impedance model as a function of the order of truncation N, we utilize the impedance model to calculate the sideband spectra under different optimized modulation waveforms shown in Figs. 13 (d) to (f), and evaluate the accuracy by comparing the full- wave results depicted in Figs. 13 (a) to (c). Figures 20 (a) to (c) depict the relative calculation error of the impedance model when comparing to the sideband spectra calculated with full-wave simulation: |^ _fu u— Cimp l /Cfuii; f° ^r clarity, we only show the relative error of the dominant sidebands in each type of modulation. It is clear that as the order of truncation increases in the impedance model, the result converges and approach to the full- wave calculation. Surprisingly, the maximum relative error in the three cases studied is already below 5% even when we truncate to the first order sideband, as shown in Fig. 20 (b) for the case of double-sideband forward scattering. Note that different from the simplified case discussed in Eq. (17), the second order terms Z± ₂ are also preserved in the impedance matrix Eq. (C33) when we truncate the sidebands to N = 1. The introduction of higher order impedance terms > 2) allows us to calculate the case where the modulation waveform is asymmetric, as is required for the case of single-sideband conversion.

Section F: Numerical simulation

To validate the practicality of the concept, we design electric and magnetic meta-atoms working in the microwave regime using full wave simulation (CST Microwave studio).

For a single Huy gens' unit in a rectangular waveguide, we employ electric boundary condition on the waveguide sidewalls and open boundary condition otherwise. The meta-atoms are positioned in the middle of the waveguide and excited with the fundamental mode of the rectangular waveguide.

FIG. 21. (a) Design of the Huygens' unit of the array structure. The electric and magnetic meta- atoms are on the opposite sides of the substrate (0.8 mm thick Rogers4003C). xs— 13.52, y — 13.20, wi = 1.00, w ₂ = 0.25, w ₃ = 0.50,x _M = 6.64,y _M = 6A6,x _E = 9.52,)¾ = 6.80,g _M = l-00,g _E = 0.20,As=10, (mm). Operating at 4 GHz (λ ¾ 75mm), the overall size of the unit (x _$ x Js ^x As) is 0.19A x 0.18A x 0.13A. The gap between the parallel plates is around 15.5 mm (~ 0.21A). (b) The simulated transmission and reflection coefficients of the periodic array at 4 GHz as a function of the DC bias voltages, in the absence of modulation, (c) Simulated scattering patterns of a finite array of 16 Huygens' units.

For the Huygens' array, we re-optimize the designs in the parallel-plate waveguide. We simulate the structure under periodic boundary conditions along the direction of the array and perfect electric boundaries on the top and bottom waveguide walls. The unit is excited with the fundamental TEM mode. The detailed geometries can be found in Fig. 21. To enhance the mechanical stability, the electric and magnetic meta-atoms are designed on the opposite sides of a 0.8 mm thick substrate (Rogers RO4003), and the center-to-center distance between neighboring units is 10 mm. High-resistance (10 ka>) resistors are employed for the bias-lines to avoid the out-coupling of microwave signals to the external modulators.

To calculate the sidebands in the adiabatic limit, we first run a full-wave simulation of the system by replacing the diodes with discrete ports; the full impedance matrix of the system can be extracted and used to perform a circuit simulation, where we define the property of the varactor diode based on its SPICE model. The transmission and reflection coefficients under different bias voltage l¾ and Uyi are calculated by a parameter scan in the circuit simulation of CST. The simulated response of the array [Fig. 21 (b)] has the same feature as the one of a single unit in the waveguide [Fig. 12 (b)]. The time-dependent transmission and reflection under periodic modulation of and are found via a one-to-one voltage-scattering parameter mapping; finally a Fourier transformation of the time-dependent signals gives the information of sidebands in both forward and backward directions.

This adiabatic approximation is valid when the modulation frequency Ω, the range of resonant frequency modulation Δ and the linewidth of the resonance y satisfy ΔΩ < γ ² [52]. In our studied system, Ω/ ¾ 0.009, Α/γ < 1.7, therefore the adiabatic approximation is valid. This approximation could provide more physical insight and higher efficiency during optimization compared to first-principle calculation methods such as FDTD, which become very inefficient when critical time scales vary by several orders of magnitude.

The SPICE model of the SMV1405 varactor diodes is obtained from the datasheet [71], and has the following parameters: is=le-014, rs=0.8, n=l, eg=l.ll, xti=3, ibv=le-03, cjo=2.37e-12, cp=0.29e-12, ls=0.7e-9, vj=0.77, m=0.5, fc=0.5, tt=0, kf=0, af=l.

For the scattering of a finite array under different modulation phase patterns, due to the limited numerical simulation capacity, we can only perform a linear simulation to give a reference, as we assume that the mode profiles of the meta-atoms will not have a substantial difference at the sidebands and the carrier frequency when the modulation frequency satisfies the adiabatic approx- imation. To do so, we perform a linear simulation of a periodic array and calculate the far-field scattering of a single unit-cell at the frequency with minimal reflection, which gives directional scattering. Then we assign different phase patterns and simulate the scattering power pattern of an array of 16 units to mimic the situation shown in Fig. 17. The results have a reasonably good agreement with the measured ones in Fig. 17 (f), even though in the simulation we assume that the scattering amplitude from each unit is identical and does not change under different modulation phase patterns [see Fig. 21 (c)].

Section G: Experimental measurement

FIG. 22 Schematic of the experimental setup for the time-varying Huygens' array.

FIG. 23 Photograph of the experimental platform of time- varying Huygens' array.

To confirm the phase control of sidebands can also be employed in dynamic beam steering, we fabricated an array composed of 16 Huygens' units and positioned the array inside a two dimensional field scanner based on parallel plate waveguide. Each unit has 2 ports (4 patches) connected to the external modulation, as can be seen from Fig. 21 - two patches connect to grounds, one patch for the electric meta-atom and one patch for the magnetic meta-atom, and all 64 patches of the array were independently connected with thin enamel wires that connect to the arbitrary function generators. We identified the resonances of the meta-atoms by measuring the scattering in the forward direction with the network analyzer, which shows two dips in the spectrum. We tuned the bias voltages such that the two resonances overlap around 4 GHz when the bias voltages are UE =— 2V, UM =— 2V, which were chosen as the starting point of optimization. The array was then excited with a CW signal at 4 GHz (λ 75mm), which was a collimated beam generated by a parabolic mirror with a monopole source antenna positioned around 315 mm (~ 4.2λ) away from the array center (see Figs. 22 and 23). We employed four double-channel arbitrary function generators to provide eight channels of modulation waveforms, four of them were applied on the electric meta-atoms and the other four were used to modulate the magnetic meta-atoms. Due to the limited number of modulation channels available, we only generated modulation phase patterns with 4 discrete phase levels. The patterns were reconfigured by rewiring the connections on a breadboard circuit. The optimization was perform using the scattered fields measured in the forward (0 degree) and backward (180 degree) directions. After optimization, a 2D line scan over a circle centered around the Huygens array was performed to get the scattering pattern. Due to the limited scanning range, the radius of the circle is limited to 2λ.

Section H: Genetic algorithm

The optimization of the modulation waveforms in this paper was performed via a Genetic Algorithm (GA) [72-75]. GA is a bio-inspired method that has been widely applied in optimization and search problems. The general procedure of GAs includes several operations that are inspired by evolution and natural selection, including mutation, crossover and selection. Below we will explain the operations in the chart flow of Fig. 24 (a) in the context of waveform optimization used in this paper.

FIG. 24. (a) Flow chart of genetic algorithm, (b) Schematic of the arithmetic-crossover operation. Two selected parental data sets (chromosomes) from generation N are hybridized to generate a new data set for generation N+l via convex combination, g £ (0, 1) is a random number.

As has been introduced in Sec. Ill, the modulation voltage signal is defined as ¾(Μ) ( — ¾M),am _P /( + ¾M),offs _> with t/ _E(M);amp , and t/ _E(M ) _;0ffs being the modulation amplitude and the DC offset; f{t) =∑ _=l cos[l(Qj - (fe _(M) )] + ^E(M) ^sm [^(^i— <¾(M) )] is the normalized waveform constructed using a Fourier series, where we take N=8 as the highest order term. For brevity, we will neglect the subscript of "E(M)" below and write the sets of coefficients a and as {a} and {b}. The problem we need to solve is to find the optimal set of coefficients, i.e. ({a}, {b}, C/amp, U ₀ f _s , φ), in order to achieve the desired types of sideband scattering.

The flow of optimization, as depicted in Fig. 24 (a), includes many rounds (generations) of operations. In each round a total number of N sets of coefficients (genes) are generated, and they are used to construct N different modulation signals, which gives N different values of the objective function (O.F.). Here we define the objective function as O.F. = Ργ Χ Bp; Ργ is the total scattered power of the desired sidebands, and Bp— Ώήα(ζ _η /ζ _ηι , ζ _ηι /ζ _η ) is a function that gives maximum value when the power of the two desired sidebands are equal. This objective function is employed to maximize the conversion efficiency while maintaining a balanced power when two sidebands are involved.

The best M genes that give the highest values of O.F. are selected as parents to produce Cff = M(M— l)/2 new genes (offspring) for the next generation via crossovers and mutation. There are many different approaches for crossover operations [73], such as one-point-crossovers, multi-point-crossovers, uniform-crossovers. Here we employ arithmetic-crossovers that produce offspring via a convex combination of the genes from two parents, as depicted in Fig. 24 (b). After the crossover, mutation is performed by introducing small random values into both the new genes and the old genes to ensure the diversity; then, genes with the lowest values of O.F. are replaced by the best parents (without mutation) and the best "ancestors" from previous generations in the new generation. This iterative process continues until the stopping criteria (e.g. the desired value of O.F.) are met. In both the simulation and the experiment, we employed 40 genes in each generation, and the results converge well after around 30 to 40 generations.

The optimized coefficients for the three types of sideband scattering shown in Fig. 13 (d) to (f) are summarized in Table I, where we used the same normalized waveform for electric and magnetic meta-atoms. For double-sideband scattering, we employed symmetric waveforms such that = 0.

In the experiment, we set all coefficients to free parameters, and they are summarized in Table II, III , IV and V, respectively. We note that φ is not an independent parameter since the effect of time-delay can also be accomplished by changing all a and b ; however, it is much more convenient to use φ, in particular when we change the relative time-delay of the two modulation signals while maintaining the waveforms unchanged, as has been shown in Fig. 16 and 17.

TABLE I. Optimized waveform coefficients for the three types of sideband scattering shown in Fig. 13 (d) to (f)

Single- Double- Double- sideband sideband sideband

forward forward bidirectional aE — ^a M 0.357 -1.121 -1.186

"E — "M 0.166 0.055 0.002 aE — ^a M -0.124 0.215 0.297

a E ^W - ^— a M ^{4) -0.159 -0.037 0.010

a E ⁽⁵⁾ - ^— « M ⁽⁵⁾ -0.033 -0.068 -0.131

a E ⁽⁶⁾ - ^— « M ⁽⁶⁾ 0.073 0.004 -0.004

a "E ⁽⁷⁾ - ^— « M ⁽⁷⁾ 0.063 0.040 0.070

fl "E ⁽⁸⁾ - ^— « M ⁽⁸⁾ 0.005 0.003 0.002

°E ^— °M -0.119 0 0

J2) _ . (2)

°E ^— °M 0.239 0 0

J3) _ 3)

°E ^— °M 0.173 0 0

°E —°M -0.037 0 0

J5) (5)

°E ^— °M -0.122 0 0

J6) _ 6)

°E ^— °M -0.068 0 0

°E —°M 0.021 0 0

°E — ^D M 0.004 0 0

t _E) amp(V) -2.994 -0.739 -0.774

i _E) off _S (V) -2.028 -2.164 -1.597

t¾amp(V) -2.268 -0.623 -0.687

i½,offs(V) -1.113 -1.197 -1.518

<PM - PE(°) 0 0 -87.6 TABLE Π. Optimized waveform coefficients for single-sideband forward scattering shown in Fig. 15 (d)

TABLE III. Optimized waveform coefficients for double-sideband forward scattering shown in Fig. 15 (e)

I fl "E ^(,) _b (

°Έ a ^{l) _b {i)

1 -0.745 0.521 -0.772 0.481

2 0.021 -0.038 0.025 -0.036

3 -0.009 -0.067 0.001 -0.068

4 -0.021 -0.008 -0.019 -0.012

5 0.083 0.019 0.075 0.040

6 -0.023 0.009 -0.024 0.001

7 -0.051 0.022 -0.055 0.002

8 0.024 -0.042 0.039 -0.028

ΨΜ - ΨΕΠ i E,amp(V) %,offs(V) t¾amp(V) f¼offs(V)

0 -1.142 -2.424 -1.686 -3.128 TABLE IV. Optimized waveform coefficients for double-sideband bidirectional scattering shown in Fig. 15 (f)

TABLE V. Optimized waveform coefficients for double-sideband umdirectional forward scattering for the array shown in Fig. 17 (f) and (g), for plots <PE = Ψ _Μ ·

I _b (

"E ^D E a ^{l) _b {i)

1 0.956 0.988 1.058 0.984

2 -0.021 -0.003 0.140 0.048

3 -0.085 0.106 -0.126 0.1 17

4 0.049 -0.011 -0.065 0.149

5 -0.013 0.003 -0.026 -0.052

6 0.076 -0.068 -0.019 0.023

7 0.009 0.025 -0.010 0.030

8 -0.027 0.038 -0.025 -0.063

<PM - <PE(°) %,amp(V) C _E) o£fs(V) t¾amp(V) t¼offs(V)

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