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Self-Configuring Universal Linear Optical Component

David A. B. Miller

arXiv:1303.4602v2physics.optics

TL;DR

The paper addresses how arbitrary linear optical functions can be physically realized and maintained without difficult global design and calibration. It presents a progressive, self-configuring architecture trained on orthogonal input-output pairs with local feedback, and shows constructive universality in principle alongside practical spatial and polarization implementations.

  • Problem

    Physical methods for implementing arbitrary linear transformations across spatial, polarization, frequency, time, and non-reciprocal operations have not been generally clear.

  • Method

    The device is configured progressively by training desired orthogonal input-output pairs and using local feedback loops to set individual optical elements.

  • Results

    The paper provides a constructive method for arbitrary spatial, polarization, and spectral linear mappings, with self-configuration and practical implementations for some spatial and polarization uses.

  • Takeaways & Limitations

    The approach extends beyond optics in principle to radio-frequency electromagnetics, acoustics, and quantum mechanical waves and superpositions.

  • Takeaways & Limitations

    Non-reciprocal functions require additional non-reciprocal elements, and the idealized beam-splitter implementation is sensitive to wavelength because of unequal path lengths.

Abstract

from arXiv · show

We show how to design an optical device that can perform any linear function or coupling between inputs and outputs. This design method is progressive, requiring no global optimization. We also show how the device can configure itself progressively, avoiding design calculations and allowing the device to stabilize itself against drifts in component properties and to continually adjust itself to changing conditions. This self-configuration operates by training with the desired pairs of orthogonal input and output functions, using sets of detectors and local feedback loops to set individual optical elements within the device, with no global feedback or multiparameter optimization required. Simple mappings, such as spatial mode conversions and polarization control, can be implemented using standard planar integrated optics. In the spirit of a universal machine, we show that other linear operations, including frequency and time mappings, as well as non-reciprocal operation, are possible in principle, even if very challenging in practice, thus proving there is at least one constructive design for any conceivable linear optical component; such a universal device can also be self-configuring. This approach is general for linear waves, and could be applied to microwaves, acoustics and quantum mechanical superpositions.

1. INTRODUCTION

The paper presents a constructive, progressive method for designing arbitrary linear optical devices and making them self-configuring. The approach addresses spatial, polarization, temporal, frequency, and potentially non-reciprocal transformations without global optimization.

  • Arbitrary linear optical operations are mathematically simple to define, but their physical implementation has not been generally clear.
  • The proposed design method sets required optical components sequentially after specifying the desired device function.
  • Spatial-mode devices can use standard optics and integrated optical approaches, while polarization can be handled by mapping polarization states into separate spatial modes.
  • Linear temporal devices can in principle map an input spectrum to an output spectrum through prescribed time variation of the refractive index.
  • Self-configuration uses desired input-output training pairs, local single-parameter feedback loops, and progressive adjustment without calculations or global optimization.

2. Device concept for spatial beams

The spatial device uses self-aligning input and output couplers connected through amplitude-and-phase modulators, trained progressively on orthogonal input-output pairs. The same architecture supports practical polarization stabilization and conversion, while idealized beam-splitter implementations have wavelength sensitivity.

  • The device connects self-aligning input and output universal beam couplers through modulators that control amplitude and phase.
  • 2.1 Single beam case: Training a single mapping uses the desired input beam at the input coupler and a reversed, phase-conjugated desired output beam at the output coupler.
  • 2.1 Single beam case: After training, the desired input mode produces the desired output mode, while the modulator sets the overall emerging amplitude and phase.
  • Additional orthogonal input-output pairs can be trained sequentially, adding rows until the number of specified couplings equals the number of device blocks.
  • 2.3 Implementation with Mach-Zehnder interferometers: The idealized beam-splitter implementation assumes negligible internal diffraction and uniform beam segments, and unequal path lengths make it sensitive to wavelength changes.
  • 2.4 Extension to polarization: Continuous feedback can stabilize a polarization output against input polarization drift without global feedback, simultaneous multiparameter optimization, or calculations in the loop.

3. Mathematical discussion

The device is modeled by a linear operator that maps input functions to output functions and can be factorized into orthogonal input and output channels with associated singular values. The physical architecture and parameter count correspond directly to this factorization, while finite coupling limits make the relevant spaces effectively finite-dimensional.

  • A linear device operator D maps an input wave to an output wave through φ_O = Dφ_I.
  • Singular-value decomposition factorizes D into unitary input and output transformations and a diagonal matrix of singular values.The columns of U and V are orthonormal input and output mode sets, respectively.
  • Each corresponding pair of singular vectors defines an orthogonal channel through the device, with coupling strength given by its singular value.
  • In the spatial implementation, the input coupler realizes U†, modulators realize the diagonal singular values, and the output coupler realizes V.
  • The required adjustable parameters match the device complexity: for example, the first input mode requires 2M_I − 2 real numbers, and a maximally functional device requires N_D = M_C(M_I + M_O − M_C).For M_I = 4, the first row has six adjustable reflectors and phase shifters.
  • The phase-shifter and reflector settings can be calculated sequentially, although the paper emphasizes self-configuration as an alternative.
  • Only finitely many channels need consideration because coupling between transmitting, input, and output spaces is strongly significant for a finite number of modes.A diffraction-limited example is the practically finite number of distinct spots formed on one surface from sources on another.

4. Universal linear device

The universal-device concept extends the linear mapping framework beyond spatial and polarization modes to additional attributes such as frequency and time. Representation converters can encode these multimodal basis functions into a common spatial form, apply the desired mapping, and convert the result back.

  • The underlying mathematics can represent linear mappings involving frequency, time, and other attributes in addition to spatial and polarization modes.
  • A universal device can be conceived by forming direct-product Hilbert spaces from basis functions for the different attributes.
  • Representation converters can transform each multimodal basis function into a monochromatic spatial mode for processing inside the spatial device.
  • The desired operator D remains mathematically unchanged while the physical representation of modes is changed inside the device and restored at its output.

Example temporal device interfering two colors

The red-blue interference device converts two frequency components into a common representation, enabling trained mappings between orthogonal spectral input and output combinations. Relative phase distinguishes inputs with identical power spectra, allowing them to serve as separate channels.

  • Device concept: A dichroic beamsplitter separates red and blue components, and a frequency shifter converts red to blue in a separate waveguide.This places both components in a common frequency representation for interference.
  • Device concept: The device can be trained to detect any selected red-blue input combination and produce any selected red-blue output combination.Its operation is analogous to polarization control, but acts on two colors.
  • Example operation: A 180-degree delay applied to the blue input redirects the interferometer output, producing a red output after frequency shifting.The alternative path sends the signal through the frequency shifter before output.
  • Example operation: The device maps orthogonal spectral inputs to orthogonal spectral outputs while remaining linear in the signal field.The red + blue and red - blue combinations form a two-dimensional orthogonal spectral basis.
  • Spectral encoding: Red + blue and red - blue have identical power spectra but are orthogonal through their relative phase, enabling separate communication channels.This differs from ordinary power-spectrum orthogonality, which requires non-overlapping positive spectra.
  • Spectral encoding: With N wavelengths, multiple equal-power orthogonal spectra could in principle support simultaneous spread-spectrum communication channels.A binary phase-inversion approach could provide log2N different orthogonal spectra.

Universal device

The universal apparatus converts spatial, polarization, and frequency content into common orthogonal channels, applies independent modulation, and converts the channels back to arbitrary output fields. In principle, this supports arbitrary linear mappings beyond permutations, including broadcast, multicast, and cross-representation transformations.

  • Universal apparatus: The apparatus converts continuous input fields into waveguides, separates polarization and wavelength components, and recombines them after processing.Representation converters provide the interfaces between fields and the central spatial mode converter.
  • Universal apparatus: For four spatial modes, two polarizations, and three frequencies, the apparatus forms 24 orthogonal channels that can be modulated independently.The channel count is 4 × 2 × 3 = 24.
  • Implementation: Known components can implement the representation converters, including grating couplers, fiber arrays, polarization splitters, and wavelength splitters.These components are established at least in principle, though the overall implementation may be challenging.
  • Temporal extension: Fourier decomposition and frequency shifters can, in principle, extend the architecture from spatial and spectral mappings to finite-time signals.Time-multiplexing is offered as an alternative to frequency shifting.
  • Capabilities: The approach maps arbitrary linear combinations of inputs to arbitrary linear combinations of outputs, encompassing permutations, broadcast, multicast, and cross-representation mappings.Examples include converting spatial modes at one frequency into distinct spectra in one spatial mode.

Devices with forward and backward waves

The described device handles forward propagation directly and reciprocal backward propagation by running beams through it in reverse. Non-reciprocal functions require additional non-reciprocal elements.

  • Directional operation: The device as described initially operates with inputs entering one side or port and outputs leaving the other.Bidirectional operation is therefore an additional universality requirement.
  • Directional operation: Reciprocal functions can operate backward by sending beams through the device in reverse.This follows from optical reciprocity.
  • Directional operation: Non-reciprocal functions, such as Faraday isolation, cannot be provided by the basic device and require added non-reciprocal elements.The paper discusses such additions separately.

Cloaking

The universal architecture can emulate propagation through a cloaked volume by routing connecting waveguides around it and implementing the required field mapping. Perfect cloaking is limited by additional propagation delay.

  • Cloaking mechanism: Cloaking can be implemented in principle by routing waveguides around the volume and mapping input fields to outputs that emulate free-space propagation through it.The general spatial mode converter performs the required input-output mapping.
  • Cloaking limitation: The universal device cannot make transmission cloaking truly perfect because the routed paths generally introduce additional propagation delay.The paper identifies this delay as the device’s relevant imperfection for cloaking.
  • Self-configuration: Self-configuration of the universal device requires care in the time domain when training the output side.The paper then distinguishes wavelength-splitting, time-multiplexed, and non-reciprocal training procedures.

5. Conclusions

The paper presents a constructive method for arbitrary linear optical components that can also self-configure using local feedback. The approach supports combined spatial, polarization, and spectral mappings and extends to other linear-wave systems.

  • The method can constructively implement arbitrary spatial, polarization, and spectral linear mappings in any combination.
  • Self-configuration requires only local feedback loops that optimize one parameter at a time.This avoids setting calculated analog values with interferometric precision.
  • The architecture can simultaneously and separately modulate conversions from multiple orthogonal inputs to corresponding orthogonal outputs.
  • The method extends beyond optics to radio-frequency electromagnetics, acoustics, and quantum mechanical waves and superpositions.
  • Certain arbitrary polarization and spatial-mode conversions and modulations appear practical with current planar optical technology.

Appendix A. Progressive calculation of reflectivities and phase shifts

The appendix derives a unitary mode transformer by progressively calculating beamsplitter reflectivities and phases. Orthogonality and losslessness then ensure distinct basis inputs emerge from distinct output ports.

  • The reflectivities and phase shifts for a unitary transformer can be formally calculated for every beamsplitter block.The corresponding M = 4 transformer is illustrated with labeled settings.
  • For each block, field reflection and transmission factors encode both magnitude and phase between named beamsplitter ports.
  • The calculation begins with the first row and proceeds sequentially through later rows using the fields produced by previously configured rows.Matrices C^(u) propagate the relevant field amplitudes between rows.
  • The last reflectivity is always 1 because the lossless beamsplitter network implements a unitary transformation.
  • Orthogonal input basis functions produce orthogonal outputs, so each subsequent basis input has no component at the output port used by the preceding input.Thus each orthogonal input leads only to a different output port.

Appendix B. Implementation with Mach-Zehnder interferometers

A symmetric Mach-Zehnder interferometer provides the variable beamsplitter and phase controls needed for the optical architecture. Common-mode drive controls phase, while differential drive controls splitting ratio.

  • Common-mode phase shifting changes the output phase, whereas differential arm drive changes the effective reflectivity or output split ratio.
  • The implementation uses symmetric Mach-Zehnder interferometers with two phase-shifting arms and nominally 50% splitters.The splitters can be implemented with coupled waveguides, while grey control elements represent electrodes or analogous actuators.
  • For the symmetric device, all 50% splitter transmission and reflection magnitudes equal 1/2, producing 50% power splitting.
  • The formal design uses the differential phase to set reflectivity magnitude and the average phase to set its phase.
  • The modulator can implement singular-value amplitudes by dumping power from the bottom port when used as an amplitude modulator.

Appendix C. Non-Reciprocal devices

Non-reciprocal operation can be added by placing optical circulators around the reciprocal spatial mode converter. The circulators separate forward and backward waves and support a four-port two-way device.

  • Forward/backward splitters based on 3-port optical circulators can surround the spatial mode converter to create a general four-port device.
  • Backward waves entering from the right are separated and routed as additional inputs into the converter, while selected outputs return as backward-propagating beams on the left.
  • Adding non-reciprocal circulators allows the overall arrangement to be non-reciprocal while the central spatial mode converter remains reciprocal and operates left to right.
  • During self-configuration, the circulator directions must be reversed when training the output coupler with reversed desired output beams.The paper suggests changing static magnetic-field directions in Faraday-isolator circulators as one possible implementation.

Appendix D. Time-Multiplexing Representation converters

The appendix describes a time-domain representation converter that multiplexes successive temporal windows into spatial channels, applies spatial-mode processing, and recombines them. Training such a device requires time-reversed input pulses to obtain the desired temporal output.

  • Time-domain conversion: Input pulses can be split into successive time windows and passed through a general spatial mode converter as an alternative to frequency-based conversion.The time-domain approach replaces wavelength splitters and frequency converters with temporal multiplexing and time-delay units.
  • Time-delay units: The idealized delay unit cycles through three switch positions, dwelling for t_Δ at each and requiring 3 t_Δ to complete one cycle.The figure presents versions for both input-side and output-side switching.
  • Time-domain representation: The delays make three successive windows of duration t_Δ appear simultaneously at three outputs, enabling processing by the spatial converter or further preparation stages.The temporal sequence is thereby represented across spatial channels.
  • Output reconstruction: At the output, reversed delays reconstruct a 3 t_Δ signal segment whose individual t_Δ slots can have arbitrary temporal forms.For training toward an output pulse f(t), the device must receive a time-reversed pulse f(-t) in each spatial mode.
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