Arbitrary and reconfigurable optics - new opportunities for integrated photonics

Arbitrary and reconfigurable optics - new opportunities for integrated photonics David Miller, Stanford University For a copy of these slides, please e-mail dabm@ee.stanford.edu

How to design any linear optical component David Miller, Stanford University For a copy of these slides, please e-mail dabm@ee.stanford.edu

How to design any linear optical component and how to avoid it! David Miller, Stanford University For a copy of these slides, please e-mail dabm@ee.stanford.edu

How to design any linear optical component Caveat Not really going to show how to make any very complicated optical component This approach has obvious practical limits But we can prove that any linear optical component that does not otherwise violate laws of physics is possible in principle And it could be practical and very interesting in some cases especially exploiting silicon photonics in new ways And we really can do this without any calculations! real-time adaptive, self-designing, self-stabilizing optics

Applications demultiplexing of signals in multimode fibers and waveguides losslessly separating overlapping light beams of different shapes sensor preprocessing, arbitrary linear transforms, neural networks, linear optical quantum circuits self-aligning beam couplers and beam combiners tracking a source in three dimensions, undoing optical scattering A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. B. Miller, A. Melloni, and F. Morichetti, Unscrambling light automatically undoing strong mixing between modes, Light Science & Applications 6, e17110 (2017)

+ = An example problem Separating overlapping beams

Example - Separating overlapping beams Suppose we have two different (orthogonal) beams e.g., from an optical fiber such as a single bump beam intensity field and a two bump beam

Example - Separating overlapping beams If both of these beams emerge simultaneously from the fiber how can we separate them for example to different fibers without loss? loss-less mode separator

Example - Separating overlapping beams If both of these beams emerge simultaneously from the fiber how can we separate them for example to different fibers without loss? loss-less mode separator

Example - Separating overlapping beams If both of these beams emerge simultaneously from the fiber how can we separate them for example to different fibers without loss? + = loss-less mode separator

Example - Separating overlapping beams In situations with fixed highly symmetric beams good specific low-loss separation solutions are known + = loss-less mode separator But for general cases of lower symmetry and/or higher complexity or where the beams change in time general solutions have not been known

Dividing the beam into patches We can approach the beam-separation problem by presuming it will be good enough to imagine that we can divide the beam up into a finite number of patches

Dividing the beam into patches We can approach the beam-separation problem by presuming it will be good enough to imagine that we can divide the beam up into a finite number of patches We treat each of these patches as if it was approximately uniform in intensity and in phase At least with a sufficiently large number of patches this could be a good enough approximation and sampling loss may be small

Dividing the beam into patches Even relatively small numbers of patches are sufficient to distinguish beams of low or moderate complexity

Dividing the beam into patches Even relatively small numbers of patches are sufficient to distinguish beams of low or moderate complexity

Dividing the beam into patches Even relatively small numbers of patches are sufficient to distinguish beams of low or moderate complexity

Input beam + + + Output beam 0 1 0 1 0 1 The first step A self-aligning universal beam coupler

Coupling an arbitrary input beam Input beam Self-aligning universal beam coupler, Opt. Express 21, 6360-6370 (2013) For illustration suppose, for simplicity, that an arbitrary input beam can be adequately described by splitting it into 4 sections

Coupling an arbitrary input beam Input beam Self-aligning universal beam coupler, Opt. Express 21, 6360-6370 (2013) Phase fronts For illustration suppose, for simplicity, that an arbitrary input beam can be adequately described by splitting it into 4 sections each approximately uniform in intensity and flat in phase For simplicity, neglect diffraction for the moment assuming each of these sections will propagate as a square section of the beam

Self-aligning beam coupler Phase shifters Beamsplitter blocks Controllable reflectors Detectors 0 1 0 1 0 1

Self-aligning beam coupler Input beam Output beam Reflected wave Transmitted wave 0 1 0 1 0 1 Adjust phase shifter in first block to minimize power in first detector Adjust reflectivity in first block to minimize power again in first detector Repeat for each block Leaves no power in detectors, all input power in output beam

Mach-Zehnder interferometer as controllable reflector and phase shifter Top Top av /2 Right Left Right Phase shifters Left av /2 Bottom Bottom 50% splitters Beam splitter Waveguide Mach- Zehnder interferometer A Mach-Zehnder interferometer functions both as a controllable reflector Using differential drive of the two phase shifter arms And as a controllable phase shifter Using common mode drive av of the two phase shifter arms To control amplitude and phase of the outputs

Mach-Zehnder self-aligning implementations Input beam (sampled into waveguides) 1 2 3 4 4 3 2 D3 1 D2 D1 Output beam Self-aligning universal beam coupler, Opt. Express 21, 6360-6370 (2013) Now use Mach-Zehnder waveguide interferometers instead of beamsplitters spatially sampling the input beam into the waveguides or combining power from different laser beams (Note there are is no diffraction inside the system in this approach)

Mach-Zehnder self-aligning implementations Input beam (sampled into waveguides) 1 2 3 4 4 3 2 D3 1 D2 D1 Output beam Self-aligning universal beam coupler, Opt. Express 21, 6360-6370 (2013) Adjust phase in device 4 to minimize power on D3 Adjust split ratio in device 3 to minimize power on D3 Adjust phase in device 3 to minimize power on D2 Adjust split ratio in device 2 to minimize power on D2 Adjust phase in device 2 to minimize power on D1 Adjust split ratio in device 1 to minimize power on D1 All power from the input waveguides now in output beam

Mach-Zehnder self-aligning implementations Grating couplers Output waveguide Top view Perspective view Photodetectors Optional lenslet array Grating couplers would allow us to couple a free-space beam to a Mach-Zehnder implementation of the device

Input beams Output beam 1 Output beam 2 Output beam 3 Output beam 4 The second step Separating multiple orthogonal beams

Separating multiple orthogonal beams Input beams Self-aligning universal beam coupler, Opt. Express 21, 6360-6370 (2013) 14 13 Once we have aligned beam 1 using detectors D11 D13 an orthogonal input beam 2 passes through the nearly transparent detectors to the second row where we can self-align it using detectors D21 D22 separating two overlapping orthogonal beams to separate outputs 12 11 Output beam 1 Detectors D13 D12 D11 (nearly transparent) 23 22 21 Output beam 2 Detectors D22 D21

2 beam Mach-Zehnder implementation Input beam(s) (sampled into waveguides) 1 2 3 4 12 13 14 D13 23 11 D12 22 D11 21 D22 D21 1 2 Output beams D11, D12, D13 are mostly-transparent detectors Since alignment and re-alignment need not be performed at data-rate speeds only need small signals from the detectors Many ways of making such mostlytransparent detectors

Separating multiple orthogonal beams 14 Input beams 13 12 D13 D12 23 22 11 D11 21 Output beam 1 Output beam 2 Adding more rows and selfalignments separates a number of orthogonal beams equal to the number of beam segments here, 4 D22 D21 32 31 Output beam 3 D31 41 Output beam 4

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D. A. B. Miller, A. Melloni, and F. Morichetti, Unscrambling light automatically undoing strong mixing between modes, Light Science & Applications 6, e17110 (2017)

Waveguide Layout A. Ribeiro, A. Ruocco, L. Vanacker, and W. Bogaerts, "Demonstration of a 4 4-port universal linear circuit," Optica 3, 1348-1357 (2016) output grating couplers ABCD input grating couplers 1234 Outputs of the in-circuit tap couplers

Input beams Output beams The third step A self-configuring universal spatial device

Arbitrary mode converter Input beam Output beam Beamsplitter blocks 4 P4 3 R3 P3 2 R2 P2 1 R1 P1 SD1 P1 1 2 3 4 P2 P3 P4 R1 R2 R3 Detectors D3 D2 D1 Modulator D1 D2 D3 Self-aligning input coupler Self-aligning output coupler Exploit two back-to-back self-aligning universal beam couplers Self-align input coupler by shining in beam of interest and adjusting using local feedback loops as before "Self-configuring universal linear optical component," Photon. Res. 1, 1-15 (2013)

Arbitrary mode converter Reversed output beam Beamsplitter blocks 4 P4 3 R3 P3 2 R2 P2 1 R1 P1 SD1 P1 1 2 3 4 P2 P3 P4 R1 R2 R3 Detectors D3 D2 D1 Modulator D1 D2 D3 Self-aligning input coupler Self-aligning output coupler Self-align the output coupler shining desired output beam backwards into the output coupler technically, the phase conjugate of the desired output beam And adjusting using local feedback loops as before But now in the output coupler

Self-configuring linear spatial device Input beam Output beam Beamsplitter blocks 4 P4 3 R3 P3 2 R2 P2 1 R1 P1 SD1 P1 1 2 3 4 P2 P3 P4 R1 R2 R3 Detectors D3 D2 D1 Modulator D1 D2 D3 Self-aligning input coupler Self-aligning output coupler Now any single desired input beam is converted to any single desired output beam with no calculations by training the device with the desired beams Can also adjust modulator to give desired coupling strength

General multiple mode converter Input beams Output beams 14 13 12 11 SD1 11 12 13 14 D13 D12 D11 D11 D12 D13 Can extend to all the possible orthogonal beams the device can support 23 D22 22 32 21 D21 31 D31 SD2 SD3 21 22 23 D21 D22 31 32 D31 41 SD4 41

Mach-Zehnder implementation Input waveguides Modulators Self-aligning output coupler WI1 WI2 WI3 WI4 M14 M13 M11 M12 D12 D13 M23 D11 M22 SD1 D31 M32 D22 M23 D13 M21 SD2 M31 D21 M22 D12 D21 M31 SD3 M21 D11 M12 D22 M32 D31 SD4 M11 M13 M14 WO4 WO3 WO2 WO1 Self-aligning input coupler Output waveguides Same concept can be implemented in a planar Mach-Zehnder form No crossing waveguides required Example here has the output order flipped for compactness "Self-configuring universal linear optical component," Photon. Res. 1, 1-15 (2013)

Describing an arbitrary linear optical component Any linear optical component can be described by a linear device or scattering operator D and we can perform the singular value decomposition (SVD) D=VD diag U U and V are unitary operators and D diag is a diagonal operator If we can emulate any SVD for given input and output spaces then we can make any linear optical device for those spaces input or source space device or scatterer output or receiving space "All linear optical devices are mode converters," Opt. Express 20, 23985 (2012)

Describing an arbitrary linear optical component Because of the SVD D=VDdiagU all linear optical devices are mode converters Any linear optical device can be completely described as a mapping from the mode-converter input functions the columns of U with scattering strengths or couplings the elements of D diag to the mode-converter output functions the columns of V input or source space device or scatterer output or receiving space "All linear optical devices are mode converters," Opt. Express 20, 23985 (2012)

Formal mathematics singular value decomposition These configurations implement the singular value decomposition with full generality so we can design and make any linear optical component for these input and output spaces Proves any linear optical component is possible D diag U V D diag V Photon. Res. 1, 1-15 (2013) U

Radiation laws Updating and extending Kirchhoff s law

Radiation laws Kirchhoff s radiation law the absorptivity of a surface the fraction of the incident light power it absorbs at a given wavelength must equal the emissivity of a surface the amount of thermal radiation power it emits at a given wavelength relative to the amount a black body would emit Required so that two bodies at the same temperature cannot exchange thermal radiation in such a way that one of them heats up the other one which would violate the second law of thermodynamics

A directional Kirchhoff radiation law This law is commonly extended also to state that the absorptivity of a wave in a given direction incident on a surface is equal to the emissivity back into the same direction giving a directional radiation law Object

Problems with Kirchhoff radiation laws This directional derivation was done neglecting diffraction so it is not clear it applies to small objects such as wavelength-scale or smaller light beams especially focused ones without a unique direction explicitly presuming reciprocity so this directional law does not work for non-reciprocal objects which could, e.g., absorb nothing for light in one direction while being able to emit in that direction Object

First new Radiation Law The absorptivity of any mode-converter input mode of an object is equal to the emissivity into the corresponding modeconverter output mode Note this is not the emissivity back into the reversed input mode it is the emissivity into the corresponding scattered mode this is a new radiation law and it applies to reciprocal and non-reciprocal objects this law lets us prove three other laws DM, L. Zhu, and S. Fan, PNAS 114, 4336 (2017)

Proof of the First Law Single mode Black body B 1 Because we know we can construct any linear optical machine we know we can make one that maps the output from a singlemode black body to a given mode-converter input mode and maps the resulting scattered output which is all into the corresponding mode-converter output mode back into the single-mode black body Input ports PNAS 114, 4336 (2017) 1 2 N Optical machine M p Object D Free space Circulator C 1 p Circulator C N 1 2 N Single mode Black body B N Circulator C 2 Single mode Black body B 2 Output ports

Proof of the First Law Single mode Black body B 1 This lets us set up a thermal equilibrium argument between the black body and the light absorbed, scattered and emitted in this specific mode-converter function pair and this Law follows simply as a result Input ports 1 2 N Optical machine M p Object D Free space Circulator C 1 p Circulator C N 1 2 N Single mode Black body B N Output ports Circulator C 2 PNAS 114, 4336 (2017) Single mode Black body B 2

Proof of the First Law Single mode Black body B 1 Some power from the black body B 1 is absorbed by the object the rest is scattered back to the black body B 1 and no other power is scattered to B 1 For equilibrium, the object must emit as much power as it absorbs so the absorptivity of the input mode-converter beam equals the emissivity into the corresponding output modeconverter beam Input ports PNAS 114, 4336 (2017) 1 2 N Optical machine M p Object D Free space Circulator C 1 p Circulator C N 1 2 N Single mode Black body B N Circulator C 2 Single mode Black body B 2 Output ports

Another law Modal law for reciprocal objects For any given input beam at a given frequency incident on a reciprocal object, the absorptivity is equal to the emissivity into the phase-conjugated version of that beam Notes: For reciprocal objects this is what we actually wanted the directional version of Kirchhoff s radiation law to be it gives equality of the absorptivity of any input beam and the emissivity into its reversed version as long as the object is reciprocal so it gives a simple radiation law valid for any scale of object and any beam (for reciprocal objects) PNAS 114, 4336 (2017)

Conclusions

Conclusions New algorithms and design approaches offer wide range of opportunities for nano and silicon photonics allowing us to exploit the available complexity Supported by deep mathematics Singular value decomposition All linear optical devices are mode converters Applications in communications, sensing signal processing and computing linear optical quantum circuits New classes of self-configuring and self-optimizing optics adaptable, manufacturable, complex optics Field-programmable linear arrays Setting up arbitrary forward-only optics and we sorted out and extended Kirchhoff s radiation laws! For a copy of these slides, please e-mail dabm@ee.stanford.edu

Funding DARPA InPho program AFOSR MURIs Integrated Hybrid Nanophotonic Circuits FA9550-10-1-0264 Robust and Complex On-Chip Nanophotonics FA9550-09- 0704 For a copy of these slides, please e-mail dabm@ee.stanford.edu

SVD and self-configuring optics references DM, L. Zhu, and S. Fan, Universal modal radiation laws for all thermal emitters, PNAS 114, no. 17, 4336-4341 (2017) Annoni et al., Unscrambling light automatically undoing strong mixing between modes, Light Science & Applications 6, e17110 (2017) Wilkes et al., "60 db high-extinction auto-configured Mach Zehnder interferometer," Opt. Lett. 41, 5318-5321 (2016) Perfect optics from imperfect components, Optica 2, 747-750 (2015) Sorting out light, Science 347, 1423-1424 (2015) Designing Linear Optical Components, Optics in 2013 Special Issue, Optics and Photonics News, December 2013, p. 38 Establishing optimal wave communication channels automatically, J. Lightwave Technol. 31, 3987 3994 (2013) Reconfigurable add-drop multiplexer for spatial modes, Opt. Express 21, 20220-20229 (2013) "Self-configuring universal linear optical component," Photon. Res. 1, 1-15 (2013) Self-aligning universal beam coupler, Opt. Express 21, 6360-6370 (2013) "How complicated must an optical component be?" J. Opt. Soc. Am. A 30, 238-251 (2013) "All linear optical devices are mode converters," Opt. Express 20, 23985-23993 (2012) Communicating with Waves Between Volumes Evaluating Orthogonal Spatial Channels and Limits on Coupling Strengths, Appl. Opt. 39, 1681 1699 (2000) For an overview, including all these links, see http://www-ee.stanford.edu/~dabm/selfalign.html For a copy of these slides, please e-mail dabm@ee.stanford.edu