U.S. patent application number 16/524136 was filed with the patent office on 2019-11-14 for universal linear components.
The applicant listed for this patent is David A.B. Miller. Invention is credited to David A.B. Miller.
Application Number | 20190346685 16/524136 |
Document ID | / |
Family ID | 51653802 |
Filed Date | 2019-11-14 |
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United States Patent
Application |
20190346685 |
Kind Code |
A1 |
Miller; David A.B. |
November 14, 2019 |
Universal Linear Components
Abstract
Universal linear components are provided. In general, a P input
and Q output wave combiner is connected to a Q input and R output
wave mode synthesizer via Q amplitude and/or phase modulators. The
wave combiner and wave mode synthesizer are both linear, reciprocal
and lossless. The wave combiner and wave mode synthesizer can be
implemented using waveguide technology. This device can provide any
desired linear transformation of spatial modes between its inputs
and its outputs. This capability can be generalized to any linear
transformation by using representation converters to convert other
quantities to spatial mode patterns. The wave combiner and wave
mode synthesizer are also useful separately, and can enable
applications including self-adjusting mode coupling, optimal
multi-mode communication, and add-drop capability in a multi-mode
system. Control of the wave combiner and wave mode synthesizer can
be implemented with single-variable optimizations.
Inventors: |
Miller; David A.B.;
(Stanford, CA) |
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Applicant: |
Name |
City |
State |
Country |
Type |
Miller; David A.B. |
Stanford |
CA |
US |
|
|
Family ID: |
51653802 |
Appl. No.: |
16/524136 |
Filed: |
July 28, 2019 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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16298531 |
Mar 11, 2019 |
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16524136 |
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14092565 |
Nov 27, 2013 |
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16298531 |
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61730448 |
Nov 27, 2012 |
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61846043 |
Jul 14, 2013 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G02B 27/145 20130101;
G02F 2001/212 20130101; G02F 1/0136 20130101; G02F 1/31
20130101 |
International
Class: |
G02B 27/14 20060101
G02B027/14; G02F 1/31 20060101 G02F001/31; G02F 1/01 20060101
G02F001/01 |
Goverment Interests
GOVERNMENT SPONSORSHIP
[0005] This invention was made with Government support under
contracts FA9550-09-1-0704 and FA9550-10-1-0264 awarded by the Air
Force Office of Scientific Research, and under contract
W911NF-10-1-0395 awarded by the Department of the Army. The
Government has certain rights in the invention.
Claims
1. A component comprising: a wave combiner having P inputs and Q
outputs configured such that a contribution of each of the P inputs
to each of the Q outputs of the wave combiner is adjustable in at
least one of amplitude and relative phase; a wave mode synthesizer
having R inputs and S outputs configured such that the contribution
of each of the R inputs to the S outputs of the wave mode
synthesizer is adjustable in at least one of amplitude and phase;
and T amplitude and/or phase modulators coupled between outputs of
the wave combiner and inputs of the wave mode synthesizer; wherein
a configuration of at least one of the wave combiner, wave mode
synthesizer and amplitude and/or phase modulators is determined
using a singular value decomposition.
2. The component of claim 1, wherein the T modulators include at
least one of Mach-Zehnder modulators and phase modulators.
3. The component of claim 2, wherein the T modulators include phase
shifters to configure phase.
4. The component of claim 1, wherein signals propagate sequentially
from the wave combiner to the modulators, and then from the
modulators to the wave mode synthesizer.
5. The component of claim 1, wherein signals propagate sequentially
from the wave mode synthesizer to the modulators, and then from the
modulators to the wave combiner.
6. The component of claim 1, wherein signals propagate
bidirectionally between the wave combiner and the modulators and
between the modulators and the wave mode synthesizer.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application is a divisional of U.S. patent application
Ser. No. 16/298,531, filed on Mar. 11, 2019, and hereby
incorporated by reference in its entirety
[0002] Application Ser. No. 16/298,531 is a continuation of U.S.
patent application Ser. No. 14/092,565, filed on Nov. 27, 2013, and
hereby incorporated by reference in its entirety.
[0003] Application Ser. No. 14/092,565 claims the benefit of U.S.
provisional patent application 61/730,448, filed on Nov. 27, 2012,
and hereby incorporated by reference in its entirety.
[0004] Application Ser. No. 14/092,565 claims the benefit of U.S.
provisional patent application 61/846,043, filed on Jul. 14, 2013,
and hereby incorporated by reference in its entirety.
FIELD OF THE INVENTION
[0006] This invention relates to mode coupling and mode
transformation in wave propagation.
BACKGROUND
[0007] Optical systems can be usefully classified as being
single-mode or multi-mode. In a single mode system, there is only
one possible spatial pattern (i.e. the "mode") of optical amplitude
and phase. In a multi-mode system, there are two or more such
possible spatial patterns of optical amplitude and phase. A free
space optical system can be regarded as having an infinite number
of modes, although in practice there are effectively a finite
number of relevant modes. For example, the number of resolvable
spots in an imaging system would be on the same order as the number
of relevant modes in that system.
[0008] In principle, each mode can be accessed independently of the
other modes. For example, a multi-mode fiber telecommunication
system using fiber that supports 100 modes would in principle have
100 independent communication channels on that single fiber (all at
the same optical wavelength).
[0009] However, the difficulties in actually providing independent
access to these 100 different modes are formidable, especially
because small perturbations to the fiber (which can vary in time)
will cause the relative phases (incurred in transmission) of the
fiber's modes to change. Thus, any approach for accessing the 100
modes in the fiber of this example would have to adapt in real time
to account for these changing relative phases, which can completely
alter the received intensity pattern from the fiber by constructive
and destructive interference.
[0010] In fact, long haul telecommunications uses single-mode fiber
in nearly all cases, in large part to avoid complexities such as
those described above. Accordingly, it would be an advance in the
art to provide improved handling of multi-mode wave
propagation.
SUMMARY
[0011] Universal linear components are provided. In general, a P
input and Q output wave combiner is connected to a Q input and R
output wave mode synthesizer via Q amplitude modulators. The wave
combiner and wave mode synthesizer are both linear, reciprocal and
lossless. For the combiner, the contribution of each of the P
inputs to each of the Q outputs is adjustable in both amplitude and
relative phase. For the wave mode synthesizer the contribution of
each of the Q inputs to each of the R outputs is adjustable in both
amplitude and relative phase. Preferably, the wave combiner and
wave mode synthesizer are implemented using waveguide Mach-Zehnder
modulator technology. Such a device can provide any desired linear
transformation of spatial modes between its inputs and its outputs.
By using representation transformers between other quantities
(e.g., polarization, frequency, etc.) and spatial mode pattern,
this capability can be generalized to any linear transformation at
all.
[0012] The wave combiner and wave mode synthesizer are also useful
separately, and can enable applications such as self-adjusting mode
coupling, optimal multi-mode communication, and add-drop capability
in a multi-mode system. Control of the wave combiner can be
facilitated using detectors at its control ports, or detectors at
its outputs. Control of the wave mode synthesizer can be
facilitated using detectors at its control ports, or detectors at
its inputs. Control of the wave combiner and wave mode synthesizer
can be implemented with simple single-variable optimizations.
BRIEF DESCRIPTION OF THE DRAWINGS
[0013] FIGS. 1A-B show exemplary embodiments of the invention (wave
combiners) having detectors at the control ports.
[0014] FIGS. 2A-C show exemplary embodiments of the invention
implemented with Mach-Zehnder modulators (MZMs).
[0015] FIG. 3 shows use of grating couplers to sample an incident
optical beam into several waveguides.
[0016] FIGS. 4A-B show exemplary embodiments of the invention (wave
combiners) having detectors at the outputs.
[0017] FIG. 5 shows a binary tree arrangement of Mach-Zehnder
modulators.
[0018] FIGS. 6A-B show exemplary embodiments of the invention (wave
mode synthesizers) having detectors at the control ports.
[0019] FIG. 6C shows an MZM implementation of the example of FIG.
6B.
[0020] FIGS. 7A-B show exemplary embodiments of the invention (wave
mode synthesizers) having detectors at the inputs.
[0021] FIGS. 8A-C show universal components having a wave combiner
connected to a wave mode synthesizer via modulators.
[0022] FIG. 9 shows an MZM implementation of the example of FIG.
8C.
[0023] FIG. 10A is a block diagram of a universal component.
[0024] FIG. 10B shows an application of the example of FIG. 10A to
polarization mode conversion.
[0025] FIG. 11 shows an implementation of polarization mode
conversion.
[0026] FIGS. 12A-B show exemplary time delays for splitting signals
into different time windows.
[0027] FIGS. 13A-B show combination of circulators with a universal
reciprocal component to provide a universal non-reciprocal
component.
[0028] FIG. 14 sets forth notation used for analysis of a wave
combiner.
[0029] FIG. 15 sets forth notation used for analysis of a beam
splitter.
[0030] FIG. 16 set forth notation used for analysis of a
Mach-Zehnder modulator.
[0031] FIG. 17 shows communication between a waveguide and a free
space source using a self-configuring mode coupler.
[0032] FIG. 18A shows communication between two single-channel
self-configuring mode couplers in free space.
[0033] FIG. 18B shows communication between two single-channel
self-configuring mode couplers with intervening scattering.
[0034] FIG. 19 shows communication between two multi-channel
self-configuring mode couplers with intervening scattering.
[0035] FIG. 20A shows use of a wave combiner and a wave mode
synthesizer to provide add-drop capability in a multi-mode
system.
[0036] FIG. 20B shows use of grating couplers to provide beam
coupling for the example of FIG. 20A.
[0037] FIG. 21 shows addition of dummy devices to the example of
FIG. 20A to equalize path lengths/loss.
[0038] FIG. 22 shows an exemplary implementation of bypass
capability.
[0039] FIG. 23 shows a simplified implementation of the add-drop
device.
[0040] FIG. 24 shows a multi-channel add-drop device.
DETAILED DESCRIPTION
[0041] In section A, self-configuring universal mode couplers are
described. These devices can be regarded as being wave combiners (#
of outputs.ltoreq.# of inputs) or wave mode synthesizers (# of
inputs.ltoreq.# of outputs). Section B relates to universal linear
components that can be constructed with the wave combiners and wave
mode synthesizers. Section C describes an application of this
technology to optimal multi-mode free space communication. Section
D describes an application of this technology to providing add-drop
capability in a multi-mode system.
A) Wave Combiner
A1) Multiple Detector Alignment
[0042] Coupling to waveguides remains challenging in optics,
especially if alignment or precise focusing cannot be guaranteed,
or when coupling higher-order (e.g., multimode fiber) or
complicated (e.g., angular momentum) modes. Simultaneous coupling
of multiple overlapping input modes without splitting loss has had
few known solutions. Here we provide a device, using standard
integrated optical components, detectors and simple local feedback
loops, and without moving parts, that automatically optimally
couples itself to arbitrary optical input beams. The approach could
be applied to other waves, such as radio waves, microwaves or
acoustics. It works when beams are misaligned, defocused, or even
moving, and it can separate multiple arbitrary overlapping
orthogonal inputs without fundamental splitting loss.
[0043] FIGS. 1A-B show a conceptual schematic of the approach.
Diagonal rectangles represent controllable partial reflectors.
Vertical rectangles represent controllable phase shifters. FIG. 1A
shows a coupler for a single input beam with four beam splitter
blocks (numbered 1-4), phase shifters P1-P4 and controllable
partial reflectors R1-R3 (R4 is a normal high reflectance mirror
that need not be controllable). Detectors D1-D3 provide signals
that go to feedback electronics 102. FIG. 1B shows a coupler for
two simultaneous orthogonal input beams (connections from detectors
to feedback electronics omitted for clarity).
[0044] For illustration we divide the arbitrary input beam into 4
pieces, each incident on a different one of the 4 beam splitter
blocks. Each block includes a variable reflector (except number 4,
which is 100% reflecting) and a phase shifter. (The phase shifter
P1 is optional, allowing the overall output phase of the beam to be
controlled.) For simplicity, we consider a beam varying only in the
lateral direction. We presume loss-less devices whose reflectivity
and phase shift can be set independently, for example, by applied
voltages for electrooptic or thermal control. For the moment, we
neglect diffraction inside the optics and presume that the phase
shifters, reflectors, and detectors operate equally on the whole
beam going through one beamsplitter.
[0045] We shine the input beam onto the beamsplitter blocks as
shown. To start, the phase shifter and reflectivity settings can be
arbitrary as long as the reflectivities are non-zero so that we
start with non-zero powers on the detectors. First, we adjust the
phase shifter P4 to minimize the power on detector D3. Doing so
ensures that the wave reflected downwards from beamsplitter 3 is in
antiphase with any wave transmitted from the top through
beamsplitter 3. Then we adjust the reflectivity R3 to minimize the
power in detector D3 again, now completely cancelling the
transmitted and reflected beams coming out of the bottom of
beamsplitter 3. (If there are small phase changes associated with
adjusting reflectivity, then we can iterate this process, adjusting
the phase shifter again, then the reflectivity, and so on, to
minimize the D3 signal.)
[0046] We then repeat this procedure for the next beamsplitter
block, adjusting first phase shifter P3 to minimize the power in
detector D2, and then reflectivity R2 to minimize the D2 signal
again. We repeat this procedure along the line of phase shifters,
beamsplitters and detectors. Finally, all the power in the incident
beam emerges from the output port on the right. This approach could
also be used to coherently combine multiple beams of the same
frequency but of unknown relative phases, as in fiber laser
systems, with each beam incident on a separate beamsplitter
block.
[0047] Unlike typical adaptive optical schemes, this method is
progressive rather than iterative--the process is complete once we
have stepped once through setting the elements one by one--and only
requires local feedback for minimization on one variable at a
time--no global calculation of a merit function or simultaneous
multiparameter optimization is required. Simple low-speed
electronics could implement the feedback.
[0048] To optimize this beam coupling continually, we can leave
this feedback system running as we use the device, stepping through
the minimizations as discussed. This would allow real-time tracking
and adjustments for misalignments or to retain coupling to moving
sources. For static sources, we could use an alternate algorithm
based only on maximizing output beam power (see section A2
below).
[0049] FIGS. 2A-C show waveguide versions based on Mach-Zehnder
interferometers (MZIs) as the adjustable "reflectors" and phase
shifters. Device numberings correspond to those of FIGS. 1A-B. FIG.
2A shows a coupler for a single input beam divided into 4 pieces.
FIG. 2B shows a coupler as in FIG. 2A with dummy devices (in dashed
line boxes) added to ensure equal path lengths and background
losses. FIG. 2C shows a coupler for two simultaneous modes. The
lower portions in the bottom row of Mach-Zehnder devices are
optional arms for symmetry only; simple controllable phase shifters
could be substituted for these Mach-Zehnder devices.
[0050] A MZI gives variable overall phase shift of both outputs
based on the common mode drive of the controllable phase elements
in each arm and variable "reflectivity" (i.e., splitting between
the output ports) based on the differential arm drive. Such a
waveguide approach avoids diffractions inside the apparatus and
allows equal path lengths for all the beam segments. Equal path
lengths are important for operation over a broad wavelength range
or bandwidth; otherwise the relative propagation phase changes with
wavelength in the different waveguide paths.
[0051] For further equality of beam paths and losses, we could add
dummy MZIs in paths 1, 2, and 3, respectively, as shown in FIG. 2B,
to give the same number of MZIs in every beam path through the
device; the dummy devices would be set so as not to couple between
the adjacent waveguides (i.e., the "bar" rather than the "cross"
state), and to give a standard phase shift. Note that as long as no
power is lost from the system out of the "open" arms--here, the top
right ports of the top two dummy devices in FIG. 2B--the settings
of these dummy devices are not critical; the subsequent setting of
MZIs 1-4 can compensate for any such loss-less modification of the
input waves. We could add further detectors at those top right
ports, adjusting the dummy MZI reflectivities to minimize the
signals in such detectors, ensuring loss-less operation. We note
that systems with large numbers (e.g., 2048) of MZIs have been
demonstrated experimentally, with low overall loss.
[0052] An alternative scheme to that of FIGS. 2A-B using a binary
tree of devices for coupling to a single input beam is presented in
section A3.
[0053] To use the waveguide scheme with a spatially continuous
input beam, we need to put the different portions of the beam into
the different waveguides. We could use one grating coupler per
waveguide as explored for angular momentum beams or phased-array
antennas. For full 2D arrays, we need space to pass the waveguides
between the grating couplers.
[0054] FIG. 3 shows a top-view schematic of an array of grating
couplers 302 with a set of Mach-Zehnder devices 304 and detectors
306 to produce one beam at the output waveguide 308, analogous to
FIG. 2A. For graphic simplicity, we omit here any additional
lengths of waveguide and possible dummy Mach-Zehnder devices to
equalize path lengths and losses. A lenslet array can be added to
this example to improve the fill factor. The input beam is shone
onto the grating coupler surface or onto the lenslet array.
[0055] We could either allow an imperfect fill factor, shining the
whole beam onto the top of the grating coupler array, or we could
use an array of lenslets focusing the beam portions onto the
grating couplers to improve the fraction of the beam that lands on
the grating couplers. Grating coupler approaches are also known
that can separate polarizations to two separate channels, allowing
the input mode of interest to have arbitrary polarization content
at the necessary expense of twice as many channels in the device
overall.
[0056] We can extend this concept to detecting multiple orthogonal
modes simultaneously. In this case, we would use detectors that
were mostly transparent, such as silicon defect-enhanced
photodetectors in telecommunications wavelength ranges, sampling
only a small amount of the power and transmitting the rest. In this
case, (FIGS. 1B and 2C), we first set the "top" row of phase shifts
and reflections (devices 11-14) as before while shining the first
beam on the device, which gives output beam 1. Then, if we shine a
second, orthogonal beam on the device, it will transmit completely
through the "top" row of beamsplitters and photodetectors, becoming
an input beam for the second row. We then use the same alignment
process as before, now with the second row of phase shifters and
reflectors (devices 21-23), and the detectors D21-D22, leading to
output beam 2. We can repeat this process for further orthogonal
beams using further rows, up to the point where the number of rows
of beamsplitters (and the number of output beams) equals the number
of beamsplitter blocks in the first row. (Generally, we can leave
all the preceding beams on, if we wish, as we adjust for successive
added orthogonal beams.) We could analogously apply the same
approach to the structure of FIG. 3, adding further "rows" as in
FIG. 2C to allow simultaneous detection of multiple orthogonal 2D
beams shone on the grating couplers or lenslets.
[0057] We could also add some identifying coding to each orthogonal
input beam, such as a small amplitude modulation at a different
frequency for each beam; then, we can have all beams on at once,
with each detector row set to look only for the specific frequency
of one beam. Such an approach, combined with continuous cycling
through the different rows as above, allows continuous tracking and
alignment adjustment of all the beams.
[0058] The number of portions or subdivisions we need to use for a
given beam depends on how complex a mode we want to select or how
complicated a correction we want to apply.
[0059] If we want to be able to select one specific input mode form
out of M.sub.I orthogonal possibilities, we need at least M.sub.I
beamsplitter blocks in the (first) row. Subsequent rows to select
other specific modes from this set need, progressively, one fewer
beamsplitter block.
[0060] At radio or microwave frequencies, we could use antennas
instead of grating couplers. Various microwave splitters and phase
shifters are routinely possible. Use of nanometallic or plasmonic
antennas, waveguides, modulators and detectors is also conceivable
for subwavelength circuitry in optics, allowing possibly very small
and highly functional mode separation and detection schemes.
[0061] In conclusion, we have shown a general method for coupling
an arbitrary input beam to one specific output beam, such as a
waveguide mode, with an automatic method for setting the necessary
coefficients in the array of adjustable reflectors and phase
shifters based on signals from photodetectors, and with extensions
to allow multiple orthogonal input beams to be separated without
fundamental splitting loss. This should open a broad range of
flexible and adaptable optical functions and components, with
analogous possibilities for other forms of waves such as microwaves
and acoustics.
A2) Single Detector Alignment
[0062] As an alternative to the use of multiple detectors when
aligning a single beam with the device, we could use only a
detector in the output beam, with a different algorithm. FIGS. 4A-B
show single output and multiple output versions of this,
respectively. Detectors Do1 and Do2 are placed in the path of the
output beams. Preferably these detectors are nearly transparent as
indicated above. We first set all the reflectors in the beam
splitter blocks in FIG. 4A to be 100% transmitting, except the last
one--beamsplitter block 4, which is set permanently to 100%
reflection--and the second last one (block 3), which we set to some
intermediate value of reflectivity. Then, monitoring detector Do1
in the output beam, we adjust the phase shifter P4 on the right of
beamsplitter block 4 to maximize the output power. We then adjust
the reflectivity R3 in beamsplitter block 3 to maximize the output
power again (these two steps in sequence arrange that there is no
power emerging from the bottom of block 3). We then proceed along
the beamsplitter blocks in a similar fashion, setting the next
beamsplitter reflectivity to some initial intermediate value,
adjusting the phase shifter just to its left to maximize the
output, then adjusting the reflectivity in this block to maximize
output again, and so on along the beamsplitter blocks. (In this
case, we would not be able to do continuous feedback on the
settings while the system was running because we need to set some
of the reflectors temporarily to 100% transmission during the
optimization steps.)
A3) Alternative Configuration
[0063] For coupling a single beam, an alternate configuration of
phase shifters, MZ interferometers and detectors is shown in FIG.
5. In this approach, phase shifter P1 is adjusted to minimize the
signal in detector DA1, and then the split ratio ("reflectivity")
of MZI MA1 is adjusted through differential drive of the arms to
minimize the DA1 signal again. Similar processes can be used
simultaneously with P2, DA2 and MA2, with P3, DA3 and MA3, and with
P4, DA4 and MA4. Next, the overall phase is adjusted in MA1 to
minimize signal in DB1, and then the split ratio ("reflectivity")
of MB1 is adjusted to minimize DB1 signal again. A similar process
can be run simultaneously with MA3, DB2 and MB2. Finally, in this
example, the phase in MB1 is adjusted to minimize DC1 signal, and
then the split ratio ("reflectivity") of MC1 is adjusted to
minimize DC1 signal again. Dummy phase shifters (dashed lines) can
be incorporated in the input paths for beams 2, 4, 6, and 8, as
shown to help ensure equality of path lengths in the system
overall.
[0064] This approach has the advantages of requiring no dummy
interferometers and allowing simultaneous feedback loop
adjustments, first in the DA column of detectors, then in the DB
column, and finally in the DC column. In contrast to the approach
of FIG. 2A, the MZI devices are arranged in a binary tree rather
than a linear sequence, and so the device is shorter and a given
beam travels through fewer MZI devices, possibly reducing loss. It
would be possible to extend this approach also for coupling
multiple orthogonal beams (e.g., by using beams transmitted through
mostly transparent versions of the detectors and into analogous
trees of devices); but, in contrast to the approach of FIG. 2C, we
would require crossing waveguides and/or multiple stacked planar
circuits if we used a planar optical approach.
A4) N to 1 and 1 to N Operation
[0065] The device of FIG. 1A can be regarded as an example of a
coherent N to one wave combiner having N.gtoreq.2 inputs and a
single output. The combiner includes a coherent wave superposition
network that is substantially linear, reciprocal and lossless, and
which includes one or more series-connected 2.times.2 wave
splitters configured such that the contribution of each of the N
inputs to the output is adjustable in both amplitude and relative
phase. The coherent wave superposition network provides N-1 control
ports at which waves not coupled to the output are emitted. N-1
detectors are disposed at the control ports. Amplitude splits and
phase shifts of the coherent wave superposition network are
determined in operation by adjusting them in sequence (e.g., P4,
then R3, then P3, then R2 etc.) to sequentially null signals (e.g.,
from D3, then from D2, then from D1) from the N-1 detectors. The
2.times.2 wave splitters can be implemented as waveguide
Mach-Zehnder interferometers, e.g., as on FIGS. 2A-B.
[0066] The extension to the multiple output case can be regarded as
providing one N to 1 wave combiner as described above for each
output. More specifically, a coherent P input to Q output wave
combiner having P.gtoreq.2 and Q.ltoreq.P would include Q single
output combiners as described above. These combiners can indexed by
an integer i (1.ltoreq.i.ltoreq.Q), where combiner i for
1.ltoreq.i.ltoreq.min(Q, P-1) is a P+1-i to one combiner as above.
Inputs of combiner i for i.gtoreq.2 are provided by the control
ports of combiner i-1. The detectors of combiner i for i<Q are
tap detectors that absorb less than 50% (preferably 10% absorption
or less) of the incident light and transmit the remainder. FIGS. 1B
and 2C show examples of the P=4, Q=2 case.
[0067] Similarly, the device of FIG. 4A can also be regarded as an
example of a coherent N to one wave combiner having N.gtoreq.2
inputs and a single output. The combiner includes a coherent wave
superposition network that is substantially linear, reciprocal and
lossless, and which includes one or more series-connected 2.times.2
wave splitters configured such that the contribution of each of the
N inputs to the output is adjustable in both amplitude and relative
phase. The coherent wave superposition network provides N-1 control
ports (e.g., 402, 404, and 406 on FIG. 4A) at which waves not
coupled to the output are emitted. An output detector is disposed
at the output. Amplitude splits and phase shifts of the coherent
wave superposition network are determined in operation by adjusting
them in sequence (e.g., P4, then R3, then P3, then R2 etc.) to
maximize the signal from the output detector. The 2.times.2 wave
splitters can be implemented as waveguide Mach-Zehnder
interferometers. The output detector is preferably a tap detector
that absorbs less than 50% (more preferably 10% absorption or less)
of the incident light and transmits the remainder.
[0068] FIG. 4B shows a multi-output example of the approach of FIG.
4A. More specifically, a coherent P input to Q output wave combiner
having P.gtoreq.2 and Q.ltoreq.P would include Q single output
combiners as described above in connection with FIG. 4A. These
combiners can indexed by an integer i (1.ltoreq.i.ltoreq.Q), where
combiner i for 1.ltoreq.i.ltoreq.min(Q, P-1) is a P+1-i to one
combiner as above. Inputs of combiner i for i.gtoreq.2 are provided
by the control ports of combiner i-1.
[0069] The preceding examples all relate to wave combiners having a
number of outputs that is less than or equal to the number of
inputs. It is also possible to implement structures having a number
of outputs that is greater than or equal to the number of inputs.
It is convenient to refer to such structures as wave mode
synthesizers. The reason for these names can be most clearly seen
in the N to 1 and 1 to N cases. An N to 1 device is clearly a
combiner, while a 1 to N device synthesizes a mode pattern of its N
outputs based on its input.
[0070] Operation of wave mode synthesizers will be described in
detail below, so here it will be convenient to illustrate some of
the possible configurations for wave mode synthesizers. FIG. 6A
shows a wave mode synthesizer having 1 input and 4 outputs. It is
analogous to the wave combiner of FIG. 1A. FIG. 6B shows a
4.times.4 wave mode synthesizer. FIG. 6C shows an MZM
implementation of the example of FIG. 6B.
[0071] The device of FIG. 6A can be regarded as an example of a
coherent one to N wave mode synthesizer having one input and
N.gtoreq.2 outputs. The wave mode synthesizer includes a coherent
wave superposition network that is substantially linear, reciprocal
and lossless, and which includes one or more series-connected
2.times.2 wave splitters configured such that the contribution of
the inputs to each of the N outputs is adjustable in both amplitude
and relative phase. The coherent wave superposition network
provides N-1 control ports at which waves incident on the outputs
can be emitted. N-1 detectors are disposed at the control ports.
Amplitude splits and phase shifts of the coherent wave
superposition network are determined in operation by adjusting them
in sequence (e.g., P4, then R3, then P3, then R2 etc.) to
sequentially null signals (e.g., from D3, then from D2, then from
D1) from the N-1 detectors when the N outputs are illuminated with
a phase-conjugated version of a desired output mode, as described
in greater detail below. The 2.times.2 wave splitters can be
implemented as waveguide Mach-Zehnder interferometers.
[0072] The extension to the multiple input case can be regarded as
providing one 1 to N wave mode synthesizer as described above for
each input. More specifically, a coherent Q input to R output wave
mode synthesizer having Q.gtoreq.2 and Q.ltoreq.R would include Q
wave mode synthesizers. These mode synthesizers can be indexed by
an integer i (1.ltoreq.i.ltoreq.Q), where each mode synthesizer i
for 1.ltoreq.i.ltoreq.min(Q, R-1) is a one to R+1-i mode
synthesizer as above. Outputs of mode synthesizer i for i.gtoreq.2
are provided to the control ports of mode synthesizer i-1 as
inputs. The detectors of mode synthesizer i for i<R are tap
detectors that absorb less than 50% (preferably absorption is 10%
or less) of the incident light and transmit the remainder. FIG. 6B
shows the Q=R=4 case, and from here it is apparent why the Q=R=4
case only has three mode synthesizers as in FIG. 6A--the last row
on FIG. 6B (i.e., block 41) is trivial, having no adjustable
reflector. More generally, the last row of any Q=R wave mode
synthesizer, or of any P=Q wave combiner is similarly trivial.
[0073] The wave mode synthesizers described thus far have detectors
at the control ports. It is also possible to have the detectors at
the inputs instead, analogous to the case of wave combiners with
detectors at the outputs. FIG. 7A shows a wave mode synthesizer
having an input detector Di1 (analogous to the combiner of FIG. 4A
with its output detector). FIG. 7B shows a 4.times.4 wave mode
synthesizer having input detectors Di1, Di2, Di3, and Di4
(analogous to the combiner of FIG. 4B with its output
detectors).
[0074] The device of FIG. 7A can be regarded as an example of a
coherent one to N wave mode synthesizer having one input and
N.gtoreq.2 outputs. The mode synthesizer includes a coherent wave
superposition network that is substantially linear, reciprocal and
lossless, and which includes one or more series-connected 2.times.2
wave splitters configured such that the contribution of the input
to each of the N outputs is adjustable in both amplitude and
relative phase. The coherent wave superposition network provides
N-1 control ports at which waves incident on the outputs can be
emitted. An input detector is disposed at the input and is capable
of detecting radiation incident on the outputs that is coupled to
the input. Amplitude splits and phase shifts of the coherent wave
superposition network are determined in operation by adjusting them
in sequence to maximize a signal from the input detector when the N
outputs are illuminated with a phase-conjugated version of a
desired output mode. The input detector is preferably a tap
detector that absorbs less than 50% (more preferably less than 10%)
of the incident light and transmits the remainder.
[0075] The extension of this to the multiple input case can be
regarded as providing one 1 to N wave mode synthesizer as above for
each of the inputs. More specifically, a coherent Q input to R
output wave mode synthesizer having Q.gtoreq.2 and Q R includes Q
mode synthesizers indexed by an integer i (1.ltoreq.i.ltoreq.Q).
Each mode synthesizer i for 1.ltoreq.i.ltoreq.min(Q, R-1) is a one
to R+1-i mode synthesizer as above. Outputs of mode synthesizer i
for i.gtoreq.2 are provided to the control ports of mode
synthesizer i-1 as inputs.
B) Universal Linear Component
[0076] In this section, we show how to construct an optical device
that can configure itself to perform any linear function or
coupling, of arbitrary strength, between inputs and outputs. The
device is configured by training it 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, 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.
B1) Introduction
[0077] There has been growing recent interest in optical devices
that can perform functions such as converting spatial modes from
one form to another, offering new kinds of optical frequency
filtering, providing optical delays, or enabling invisibility
cloaking. All these operations are linear. Many other linear
transformations on waves are mathematically conceivable, involving
spatial form, polarization, frequency or time, and non-reciprocal
operations. Despite the mathematical simplicity of defining such
linear operations, it has apparently not generally been understood
how to execute an arbitrary linear operation on waves physically,
even in principle. The usual linear optical components, such as
lenses, gratings and mirrors, only implement a subset of all the
possible linear relations between inputs and outputs. Other
components such as volume holograms or matrix-vector multipliers
can implement more complex relations; it is difficult, however, to
make such approaches efficient--for example, avoiding a loss factor
of 1/M when working with M different beams. Interactions between
designs for different inputs leave it unclear how, or even if, we
could design and/or fabricate an efficient arbitrary design
constrained only by general physical laws. Indeed, some designs
resort to blind optimization based in part on random or exhaustive
searches among designs with no guarantee of the existence of any
solution. Even with some design approach for an arbitrary desired
linear operation, the resulting device could be quite complicated.
Furthermore, operations on waves can require interferometric
precision, and configuring many analog elements precisely to
construct such a design could be very challenging.
[0078] In section A above, we showed how to make a self-aligning
optical beam coupler that can configure itself to couple arbitrary
input spatial beams to simple beam outputs (e.g., single-mode
waveguides). Here, first, we extend that work, using the
mathematical understanding of arbitrary linear optical components,
to show how to make an optical device that can perform an arbitrary
spatial mapping between inputs and outputs. This device shares with
the self-aligning beam coupler the feature that all the necessary
analog settings of individual optical components to define the
necessary mode mappings can be set based on local feedback loops,
each adjusting only a single measurable quantity. Though we show we
can calculate the design values externally, the use of these
feedback loops avoids the calibration of multiple analog components
so they can be set precisely to calculated design values. Such
spatial devices could be implemented with current integrated optics
technologies.
[0079] We then extend the device concept to show in principle how
arbitrary linear optical devices can be constructed, including
polarization, frequency, temporal, and non-reciprocal linear
functions. In the spirit of a universal machine, such as the Turing
machine in computing, this approach therefore proves that there is
at least one constructive approach to an optical device that can
perform any linear operation on waves. This general approach shares
the ability of the simpler spatial devices that the device can
configure itself to perform such arbitrary linear operations, again
based only on simple local feedback loops.
[0080] We discuss the basic device concept for spatial beams in
Section B2. The mathematics is developed further in Section B3. We
generalize to a universal linear optical machine in Section B4.
Section B5 relates to an approach for calculating the
reflectivities and phase shifts a priori. Further details relating
to the Mach-Zehnder implementations are given in Section B6, and we
draw conclusions in Section B7.
B2) Spatial Beams
[0081] The concept of the approach is shown in FIGS. 8A-C,
illustrated here first for a spatial mode example with the inputs
and outputs sampled to four channels. It includes two self-aligning
universal wave couplers, one, CI, at the input, and another, CO, at
the output. These are connected back-to-back through modulators
(SD1, SD2, SD3, and SD4) that can set amplitude and phase; these
modulators could also incorporate gain elements. The self-aligning
couplers require controllable reflectors and phase shifters
together with photodetectors that are connected in selectable
feedback loops to control the reflectors and phase shifters. Dashed
rectangle phase shifters are not required, but may be present
depending on the way the devices are implemented, and might be
desirable for symmetry and equality of path lengths.
[0082] We presume that, for our optical device, we know what set of
orthogonal inputs we want to connect, one by one, to what set of
orthogonal outputs. If we know what we want the component to do,
any linear component can be completely described this way. The
simplest case is that we want the device to convert from one
spatial input mode to one spatial output mode (FIG. 8A).
B2.1) Single-Beam Case
[0083] To train the device as in FIG. 8A, we first shine the input
mode or beam onto the top of the input self-aligning coupler CI.
Then we proceed to set the phases and reflectivities in the beam
splitter blocks in CI as described above. Briefly, this involves
first setting phase shifter P4 to minimize the power in detector
D3; this aligns the relative phases of the transmitted and
reflected beams from the bottom of beamsplitter 3 so that they are
opposite, therefore giving maximum destructive interference. Then
we set the reflectivity R3 to minimize the D3 signal again;
presuming that the change of reflectivity makes no change in phase,
the D3 signal will now be zero because of complete cancellation of
the reflected and transmitted light. Next, we set phase shifter P3
to minimize the D2 signal, then adjust R2 to minimize the D2 signal
again. Proceeding along all the beamsplitter blocks in this way
will lead to all the power in the input mode emerging in the single
output beam on the right.
[0084] The second part of the training is to shine a reversed
(technically, phase-conjugated) version of the desired output mode
onto the output coupler CO; that is, if we want some specific mode
to emerge from the device (i.e., out of the top of CO), then we
should at this point shine that mode back into this "output". We
set the values of the phase shifters and reflectivities in coupler
CO by a similar process to that used for coupler CI, which will
lead to this "reversed" beam emerging from the left of the row of
beamsplitter blocks, for the moment going backwards into modulator
SD1 from the right.
[0085] Now that we have set the required reflectivity and phase
values in coupler CO, we imagine that we turn off the training beam
that was shining backwards onto the top of coupler CO and shine a
beam instead from the output of modulator SD1 into CO. It is
obvious that will lead to all the power coming out of the top of
coupler CO and that the resulting amplitudes (and powers) of beams
emitted from the tops of the beamsplitters will be the same as the
ones incident during the training. To understand why the phases are
set using a phase conjugate beam during training, we can formally
derive the mathematics of the design, as discussed below and in
section B5; we can also understand this intuitively. Note, for
example, that if, during training, the (backward) beam incident on
the top of beamsplitter block 4 (of CO) had a slight relative phase
lead compared to that incident on the top of beamsplitter block 3
(as would be the case if it was a plane wave incident from the top
right), then we would have added a phase delay in phase shifter P4
to achieve constructive interference along the line of beam
splitters. Running instead in the "forward" mode of operation,
then, the beam that emerges vertically from beamsplitter block 4
will now have a phase delay compared to that emerging from block 3
(as would be the case if it was a plane wave heading out to the top
right). The resulting phase front emerging from the top of coupler
CO is therefore of the same shape (at least in this sampled
version) as the backward (phase conjugated) beam we used in
training, but propagating in the opposite direction as desired.
[0086] So, with the device trained in this way, shining the desired
input mode onto CI will lead to the desired output mode emerging
from CO. Finally, we set modulator SD1 to get the desired overall
amplitude and phase in the emerging beam; choosing these is the
only part of this process that does not set itself during the
training. Modulator SD1 could also be used to impose a modulation
on the output beam, and an amplifier could also be incorporated
here if desired for larger output power.
B2.2) Multiple Beam Case
[0087] The process can be extended to more than one orthogonal
beam. In FIG. 8B, having trained the device for the desired "first"
input and output beams, we can now train it similarly with a
"second" pair of input and output beams that are orthogonal to the
"first" beams. Since the device is now set so that all of the
"first" beam shone onto the top of CI will emerge into modulator
SD1, then any "second" beam that is orthogonal to the "first" beam
will instead pass entirely into the photodetectors D11-D13 (or,
actually, through them, since now we make them mostly transparent,
as discussed above). Though this second beam is changed by passing
through the top (first) row of beamsplitters, it is entirely
transmitted through them to the second row of beamsplitter blocks.
In the second row of beamsplitter blocks, we can run an exactly
similar alignment procedure, now using detectors D21-D22 to
minimize the signal based on adjustments of the phase shifters and
reflectivities. We can proceed similarly by shining the reversed
(phase conjugated) version of the desired second (orthogonal)
output beam into the top of coupler CO. Then, shining the second
input beam into CI will lead to the desired second output beam
emerging form CO. If our device requires us to specify more than
two mode couplings, we can continue this process, adding more rows
until the number of rows equals the number of blocks (here 4) on
the first row. FIG. 8C illustrates a device for 4 beams. Note that
once we have set the device for the first 3 desired orthogonal
pairs, then the final (here, fourth) orthogonal pair is
automatically defined for us, as required by orthogonality.
Formally, the number of rows we require in our device here is equal
to the mode coupling number, M.sub.C.
B2.3) Implementation with Mach Zehnder Interferometers Similarly to
such bulk beamsplitter versions discussed above, the configurations
in FIGS. 8A-C are idealized. We are neglecting any diffraction
inside the apparatus, we are presuming that our reflectors and
phase shifters are operating equally on the entire beam segment
incident on their surfaces, and we are presuming that each such
beam segment is approximately uniform over the beamsplitter width.
The path lengths through the structure are also not equal for all
the different beam paths, which would make this device very
sensitive to wavelength; different wavelengths would have different
phase delays through the apparatus, so the phase shifters would
have to be reset even for small changes in wavelength. An
alternative and more practical solution is to use Mach-Zehnder
interferometers (MZIs) in a waveguide configuration; diffraction
inside the apparatus is then avoided, and equalizing waveguide
lengths can eliminate the excessive sensitivity to wavelength. FIG.
9 illustrates such a planar optics configuration. Here wave
combiner 902 is analogous to CI of FIG. 8C, and wave synthesizer
906 is analogous to CO of FIG. 8C. Not shown are devices such as
grating couplers that would couple different segments of the input
and output beams into and out of the waveguides WI1-WI4 and
WO1-WO4, respectively.
[0088] Common mode (i.e., equal) drive of the two phase shifting
arms of such an MZI changes the phase of the output; differential
(i.e., opposite) drive of the arms changes the "reflectivity"
(i.e., the split ratio between the outputs) (see section B6 for a
detailed discussion of the properties of the MZIs as phase shifters
and variable reflectors).
[0089] The use of sets of grating couplers connected to the input
waveguides WI1-WI4 and to the output waveguides WO1-WO4 is one way
in which this device could be connected to the input and output
beams, as discussed above. In this case, though the wave is still
sampled at only a finite number of points, we can at least obtain
true cancellation of the fields in the single mode guides even if
the field on the grating couplers is not actually uniform. The
geometry of FIG. 9 also shows that we can make a device that has
substantially equal time delays between all inputs and outputs
because all the waveguide paths are essentially the same length. As
discussed above, such equality is important if the device is to
operate over a broad bandwidth.
[0090] The example so far has considered a beam varying only in the
horizontal direction, and using only four segments to represent the
beam. Of course, the number of segments we need to use depends on
the complexity of the linear device we want to make, and the number
could well be much larger than 4; we will discuss such complexities
in Section B3. Additionally, we would likely want to be able to
work with two-dimensional beams, in which case we could imagine
two-dimensional arrays of grating couplers coupling into the
one-dimensional arrays of waveguides of FIG. 9, as discussed
above.
[0091] The configuration in FIG. 9 formally differs mathematically
from that in FIG. 8C in that we have reflected the output
self-aligning coupler CO about a horizontal axis to achieve a more
compact device. This reflection makes no difference to the
operation of the device; since the device can couple arbitrary
beams, the labeling or ordering of the waveguides is of no
importance. (This reflection would be equivalent to similarly
reflecting the self-aligning output coupler CO in FIG. 8C about a
horizontal axis, which would lead to the output beam coming out of
the bottom, rather than the top, of the device.) Schemes are also
discussed above for ensuring equal numbers of MZIs in all optical
paths for greater path length and loss equality by the insertion of
dummy devices, and such schemes could be implemented here also.
B3) Mathematical Discussion Quite generally, any linear optical
device can be described mathematically in terms of a linear
"device" operator D that relates an input wave, |.PHI..sub.I, to an
output wave |.PHI..sub.O through
|.PHI..sub.O=D|.PHI..sub.I (1)
It can be shown that essentially any such linear operator D
corresponding to a linear physical wave interaction can be
factorized using the singular value decomposition (SVD) to yield an
expression
D = m s Dm .phi. DOm .phi. DIm ( 2 ) ##EQU00001## or,
equivalently,
D=VD.sub.diagU.sup..dagger. (3)
where U (V) is a unitary operator that in matrix form has the
vectors |.PHI..sub.DIm (|.PHI..sub.DOm) as its column vectors and
D.sub.diag, is a diagonal matrix with complex elements (the
singular values) s.sub.Dm. The sets of vectors |.PHI..sub.DIm and
|.PHI..sub.DOm form complete orthonormal sets for describing the
input and output mathematical spaces H.sub.I and H.sub.O
respectively. The resulting singular values are uniquely specified,
and the unitary operators U and V (and hence the sets
|.PHI..sub.DIm and |.PHI..sub.DOm) are also unique (at least within
phase factors and orthogonal linear combinations of functions
corresponding to the same magnitude of singular value, as is usual
in degenerate eigenvalue problems). An input |.PHI..sub.DIm leads
to an output s.sub.Dm|.PHI..sub.DOm so these pairs of vectors
define the orthogonal (mode-converter) "channels" through the
device.
[0092] In a practical device, we may have a physical input space
that we would describe with M.sub.I modes or basis functions and
similarly an output space that we would describe using M.sub.O
modes or basis functions. For example, the input mathematical space
might consist of a set of M.sub.I Gauss-Laguerre angular momentum
beams, and the output space might be a set of M.sub.O waveguide
modes or M.sub.O different single-mode waveguides, with M.sub.I and
M.sub.O not necessarily the same number. Alternatively, we might be
describing the input space with a set of M.sub.I waveguide modes,
and the output space might be described with a plane-wave or
Fourier basis of M.sub.O functions, as appropriate for free-space
propagation. In any of these cases, the actual number of orthogonal
channels, Me, going through the device might be smaller than either
M.sub.I or M.sub.O (or both); for example, we could have large
plane wave basis sets for describing the input and output fields of
a 3-moded waveguide; no matter how big these input and output sets
are, however, there will only practically be M.sub.C=3 orthogonal
channels through the device.
[0093] In the example devices of FIGS. 8A-C, the most obvious
choices for the input and output basis function sets are the
"rectangular" functions that correspond to uniform waves that fill
exactly the (top) surface of each single beamsplitter block; in
this example, we have chosen equal numbers (M.sub.I and M.sub.O
each equal to 4) of such blocks on both the input and the output,
though there is no general requirement to do that, and the number
M.sub.C of channels through the device is the number of rows of
beamsplitter blocks (1 in FIG. 8A, 2 in FIG. 8B, and 4 in FIG. 8C).
In those devices also, the (complex) transmissions of the
modulators SD1-SD4 correspond mathematically to the singular values
s.sub.Dm.
[0094] In these cases of possibly different values for each of
M.sub.I, M.sub.O, and M.sub.C it is more useful and meaningful to
define the matrix U as an M.sub.I.times.M.sub.C matrix (so
U.sup..dagger. is a M.sub.C.times.M.sub.I matrix) and the matrix V
as an M.sub.O.times.M.sub.C matrix. With these choices, the matrix
D.sub.diag becomes the M.sub.C.times.M.sub.C square diagonal matrix
with the (generally non-zero) singular values s.sub.Dm as its
elements. If there are only M.sub.C possible orthogonal channels
through the device, then there are only M.sub.C singular values
that are possibly non-zero also. Using these possibly rectangular
(rather than square) forms for U and/or V means we are only working
with the channels that could potentially have non-zero couplings
(of strengths given by the singular values) between inputs and
outputs. In the device of FIGS. 8A-C, the input coupler CI
corresponds to the matrix U.sup..dagger., the vertical line of
modulators corresponds to the diagonal line of possibly non-zero
diagonal elements in D.sub.diag, and the output coupler CO
corresponds to the matrix V. In the cases of FIGS. 8A-B, the
matrices U and V are not square. Because they are not square, in
this amended way of writing the mathematics, they are not therefore
unitary, but we eliminated elements in our mathematics that serve
no purpose; we have essentially avoided having our mathematics
describe rows of beam splitters and modulators that do not exist
physically. Despite that fact that U and V are no longer
necessarily unitary, the forms of Eqs. (1)-(3) remain valid. The
sets of functions |.PHI..sub.DIm and |.PHI..sub.DOm are complete
for representing input and output functions corresponding to
non-zero couplings (i.e., non-zero singular values) and are still
the columns of the matrices U and V, respectively. (The settings of
the phase shifters and reflectors in the full unitary forms of
couplers CI and CO as shown in FIG. 8C would each correspond to a
Gaussian-elimination-like factorization of a unitary matrix; other
forms, such as the multilayer binary tree form considered above,
would correspond to other possible factorizations of such unitary
matrices.)
[0095] Though here we will emphasize the self-configuring approach,
the specific settings of the phase shifters and reflectors can
instead be calculated straightforwardly given the desired function
of the device. See section B5 for an explicit sequential row-by-row
and block-by-block physical design process for the partial
reflector and phase shifter parameters. Section B6 gives the formal
analysis for the MZI implementation of variable reflectors and
phase shifters.
[0096] One final formal issue for an arbitrary device is that the
input and output Hilbert function spaces, H.sub.I and H.sub.O
respectively, in which |.PHI..sub.I and |.PHI..sub.O exist
mathematically, may well each have infinite numbers of dimensions,
whereas our device has finite dimensionality. To resolve this, note
first that the input waves |.PHI..sub.I come from some a wave
source in another volume (generally, a "transmitting" Hilbert space
H.sub.T), through some coupling operator G.sub.TI. Because of a sum
rule, there is only a finite number of channels between H.sub.T and
H.sub.I that are strongly enough coupled to be of interest. A
familiar example is the finite number of distinct "spots" that can
be formed on one surface from sources on another, consistent with
diffraction. A similar argument holds at the output with output
waves |.PHI..sub.O leading to resulting waves in some "receiving"
space H.sub.R. Hence, we can practically presume that D can be
written as a matrix with finite dimensions to any degree of
approximation we wish.
B4) Universal Linear Device
[0097] So far, we have only considered spatial input and output
modes for the device concept, though the underlying mathematical
discussion above can consider any additional linear attributes
also, such as polarization (or, more generally, quantum mechanical
spin), frequency or time. We can at least conceive of a universal
machine that would attempt to perform any linear mapping between
inputs and outputs. Mathematically, it is straightforward to
construct the necessary Hilbert spaces, which would be formed by
direct products of the different basis functions corresponding to
each attribute separately.
[0098] One example of such a universal component includes a wave
combiner and wave synthesizer as described above (with or without
the detectors) and amplitude and/or phase modulators connected
between outputs of the wave combiner and inputs of the wave mode
synthesizer. More specifically, the combiner can be a linear,
reciprocal and lossless wave combiner having P inputs and Q outputs
with 2.ltoreq.Q.ltoreq.P configured such that the contribution of
each of the P inputs to each of the Q outputs of the wave combiner
is adjustable in both amplitude and relative phase. Similarly, the
synthesizer can be a linear, reciprocal and lossless wave mode
synthesizer having Q inputs and R outputs with Q.ltoreq.R
configured such that the contribution of each of the Q inputs to
each of the R outputs of the wave mode synthesizer is adjustable in
both amplitude and relative phase. FIG. 10A shows an example, where
wave combiner 1002 and wave mode synthesizer 1004 are connected to
each other via modulators M-1, M-2, M-3, . . . , M-Q. Combiner 1002
has P inputs I-1, I-2, I-3, . . . , I-P. Synthesizer 1004 has R
outputs O-1, O-2, O-3, . . . , O-R. The modulators can provide
amplitude and/or phase modulation.
[0099] Such a device provides, in principle, arbitrary
transformations of spatial modes. To provide other kinds of linear
operations on waves, representation transformers can be placed at
the inputs and/or outputs of the device to convert between other
wave properties and spatial modes. FIG. 10B schematically shows an
example where input transformer 1006 transforms input polarization
modes into spatial modes I1 and I2 provided to combiner 1002, and
output transformer 1008 transforms the output modes O-1 and O-2 of
synthesizer 1004 to polarization modes. In this manner universal
polarization transformation becomes possible. As described in
greater detail below, any linear property of a wave can be
transformed to a spatial mode pattern, so a universal wave spatial
mode transformer combined with representation transformers (e.g.,
1006 and 1008 on FIG. 10B) can provide truly universal
functionality.
B4.1) Universal Device with Representation Converters
[0100] One general approach that would work in principle for a
universal device is to physically convert each direct product basis
function (e.g., one with specific spatial, temporal and
polarization characteristics) to a monochromatic spatial mode with
a specific polarization, a mode we can then feed through a version
of the spatial device we discussed above. In other words, we can
convert the representation to a simple monochromatic spatial one
(e.g., in fiber or waveguide modes), perform the desired
mathematical device operation (i.e., the mathematical operator D),
using our spatial approach discussed above, and then convert the
representation back to its full spatial, temporal and polarization
form. That is, we make "representation converters" to convert into
and back out of the single-frequency, single-polarization,
waveguide mode representation we use in our universal spatial
device, or general spatial mode converter, as discussed above. The
mathematical operator D that describes that mapping from input
modes to output modes is not changed, but the physical
representation of those modes is changed inside the device, and is
changed back before we leave the device.
Polarization Controller Example
[0101] As a very simple example of a device that operates based on
such representation conversion, consider a simple polarization
converter as in FIG. 11. Light incident on the grating coupler 1106
in self-aligning coupler 1102 is split by its incident polarization
into the two waveguides, and similarly light from the waveguides
going into the grating coupler 1108 in self-aligning coupler 1104
appears on the two different polarizations on the output light
beam. PI and PO are phase shifters; the similar but unlabeled boxes
are optional dummy phase shifters. Optionally, a phase shifter and
its dummy partner could instead be driven in push-pull to double
the available relative phase shift. MZI and MZO are Mach-Zehnder
interferometers, and DI and DO are detectors.
[0102] In this example, an incident beam is split into two
orthogonal polarizations, for example, using a polarization
demultiplexing grating coupler 1106. The polarization demultiplexer
here is converting the physical representation from a polarization
basis on a single spatial mode to a representation in two spatial
modes (the waveguide modes) on a single polarization. Then the
simple two-channel self-aligning coupler 1102 combines the fields
and powers from the two polarizations loss-lessly into one
single-mode waveguide beam. Here, as before, we adjust phase
shifter PI to minimize the power in detector D1, and then adjust
the "reflectivity" of the Mach-Zehnder interferometer MZI (by
differential drive of the phase shifters in the two arms) to
minimize the power in detector D1. All the power from both incident
polarizations is now in one beam in one polarization in waveguide
WIO. In many situations, this may be the desired output, and we
could take this output from waveguide WIO at the point of the
dashed line in FIG. 11. If we wish, instead, we can change the wave
from the output grating coupler 1108 into any desired polarization
using the second, output self-aligned coupler 1104; we can program
this desired output polarization by training with the desired
polarization state running backwards into that output grating
coupler and running the feedback loops with PO, MZO and DO in the
same way as we did for the input. With this device operating with
circular polarizations, if we train with a right circular
polarization going "in" to the output coupler from the outside, for
example, the beam emerging from the output coupler under actual
operation will also be right circularly polarized. Note that, in
contrast to other polarization state controllers, this device
requires no global feedback loop and no simultaneous multiple
parameter optimization. It also requires no calculation in the
feedback loop.
Universal Device
[0103] More generally, we can expand the idea shown in the simple
polarization controller above with other representation converters.
For example, we could first convert from a continuous input field
to waveguides using the spatial single mode converters. Then, in
this example, we split the polarizations, converting to (twice as
many) waveguide modes all in the same polarization. Next we split
each such waveguide mode into separate wavelength components.
Finally, we use wavelength converters (frequency shifters) to
change each of those components to being at the same wavelength
(frequency). Now the input field that was originally a continuous
beam with possibly spatially varying polarization content and with
multiple frequency components or time-dependence (possibly
different for each spatial and polarization component) has been
converted into a representation in a set of spatial modes all at
the same frequency and polarization. This set of modes is then fed
into our device as described above, with the U.sup..dagger. and V
blocks representing the self-aligning couplers CI and CO
respectively (e.g., in the planar configuration of FIG. 9) and
D.sub.diag representing the vertical line of modulators SD1, SD2, .
. . , etc. On the right side of the device, we perform the inverse
set of representation conversions to that on the left to obtain the
final output field.
[0104] Methods for making each of the "representation converter"
devices considered above are known, at least in principle. Various
approaches exist to convert from one spatial mode form to another,
including the grating coupler approach. If we started with a
two-dimensional (2D) spatial input field, we could sample it with a
2D array of spatial single mode converters into optical fibers, and
then rearrange the outputs of those fibers into a 1D line of
inputs. Polarization splitters are standard components that can
exist in many different forms. Wavelength splitters, such as
gratings, separate different frequencies to different spatial
channels.
[0105] For a finite input time range or repetition time, we know we
can always Fourier-decompose a signal into a set of amplitudes of
each of an equally spaced comb of frequencies. We can then, at
least in principle (though with greater practical difficulty),
convert each frequency component to a standard frequency using
frequency shifters; electro-optic frequency shifters could in
principle be driven from the beating of the different comb
elements, thus retaining well defined phase relative to the input
field. In this way, at least in principle, we can convert an
arbitrary Fourier decomposition in different frequency modes
emerging from the wavelength splitters into different spatial modes
all at the same frequency. Note, incidentally, that such frequency
shifters are linear optical components in that they are linear in
the optical field being frequency-shifted; in the case of
modulator-based frequency shifters, it is largely a matter of taste
whether we regard them as being non-linear optical devices. Such
devices can all, at least in principle, be run backwards at the
output.
[0106] The spatial modes, now all in the same polarization and at
the same frequency, pass through the general spatial mode converter
(e.g., like FIG. 9). Finally, we pass back through another
representation converter to create the output field. In this way,
we can in principle perform any linear transformation of the input
field, including its spatial, spectral, and polarization forms.
[0107] This general approach is reminiscent of switching fabrics in
optical telecommunications, and this approach can certainly
implement the permutations required in such fabrics. The present
approach, however, goes well beyond permutations, allowing
arbitrary linear combinations of inputs to be mapped to arbitrary
linear combinations of outputs, including as other special cases
all broadcast and multicast functionalities.
[0108] As an alternative to the frequency splitting and frequency
conversion considered above, in principle we could split an input
pulse into different time windows, then pass each of those through
the general spatial mode converter. Idealized time delay units for
implementing a time (rather than frequency) version of the approach
are shown in FIGS. 12A-B. Here the switches rotate through
positions 1, 2, and 3, with a dwell time of .DELTA.t at each
position, taking a total time of 3.DELTA.t to cycle through all 3
positions before returning to position 1. FIG. 12A shows the switch
used at the input side. FIG. 12B shows the switch used at the
output side. At the input side, the paths connected to points 2 and
1 have additional propagation delays compared to the path connected
to point 3 of .DELTA.t and 2.DELTA.t, respectively. Thus the
signals from three successive time windows of duration .DELTA.t
appear simultaneously at the three outputs on the right, allowing
them then to be fed into the general spatial mode converter (or
into the next stage of the preparatory representation conversion
stages). A similar apparatus can be used at the output, but
operated with the delays reversed to reconstruct a signal segment
of duration 3.DELTA.t at the final output, with each .DELTA.t time
slot in that signal being an arbitrary linear combination of 3
incident .DELTA.t time slots.
[0109] Devices with forward and backward waves So far, we have only
considered devices that operate with input waves coming from one
side or port and output waves leaving from the other. If the device
is to be truly universal, it would have to handle waves going in
the other directions also. Furthermore, the device shown so far is
reciprocal, and cannot therefore emulate a non-reciprocal device (a
Faraday isolator being a simple example).
[0110] To handle non-reciprocal optical elements in this approach,
or any element where we want forward and backward waves in the
ports of the device (as in cloaking), we can in principle add
forward/backward splitters to the left and right sides of the
apparatus of FIG. 10A, e.g. as shown in FIGS. 13A-B. Here FIG. 13A
shows a schematic of a 3-port optical circulator 1302. The dashed
lines show the effective paths of waves in different directions
between the three ports. FIG. 13B shows a universal 4-port
"two-way", potentially non-reciprocal device, with input and output
beams in each of two paths at both the left (1310) and right (1312)
of the device. The central "U.sup..dagger.", "D.sub.diag", and "V"
units (converter 1304, modulators 1308 and synthesizer 1306,
respectively) form a general spatial mode converter as above. This
can be regarded as an example of starting with a universal
combiner--modulators--synthesizer device and adding one or more
three port optical circulators connected to an input of the
combiner and to an output of the synthesizer. This can provide a
universal non-reciprocal linear component.
[0111] This example approach is based on the use of 3-port optical
circulators to separate forward and backward waves. Backward waves
coming into the right of the structure are separated from the
forward waves and fed as additional inputs into the left of the
general spatial mode converter in the middle. Two of the four
outputs from the general spatial mode converter are fed to the
optical circulators on the left to give the backward propagating
output beams on the left.
[0112] The addition of such circulator devices, which are
non-reciprocal by definition, allows the whole optical arrangement
to be non-reciprocal if required, while leaving the core general
spatial mode converter itself as a reciprocal device that always
runs only from front to back (left to right).
Cloaking
[0113] To implement "cloaking" in principle, we flow the fibers
connected to the left or right ports in FIG. 13B, round the volume
to be "cloaked" and use the general spatial mode converter to
implement the required mapping between input and output fields to
emulate free-space propagation through the cloaked volume. Note
that, as with all "transmission" cloaks, we generally have
additional propagation delay that prevents truly perfect cloaking.
The overall additional time delay in our universal device is the
one sense in which it cannot be made perfect.
B4.2) Self-Configuring Operation
[0114] So far, for this universal device, we have shown that in
principle any such linear transforming device can be made, though
we have not explicitly discussed the self-configuration in this
general context. There are two sophistications we have to consider
compared to the simple spatial case, the first related to the time
behavior and the second to the non-reciprocal behavior.
Temporal Self-Configuring
[0115] Suppose first that we are operating with the
wavelength-splitting version of the universal device. We presume
that we work with frequency converters that, when run with waves
propagating in the opposite direction, perform the opposite
frequency conversion; that is, if when run with a "forward" wave a
converter changes the wave frequency from .omega. to
.omega.+.delta..omega., then with a wave propagating backwards into
it, it will convert from .omega.+.delta..omega. to .omega.. With
such a frequency converter, the mapping from spatial to frequency
modes and the mapping from frequency to spatial modes are just
inverses of one another. Electro-optic frequency converters can
operate in this way, for example.
[0116] Suppose, then, that we want to train the device to output a
pulse f(t) in a particular spatial mode in response to some
specific input. Then, in training, we send the same pulse f(t)
propagating backwards, i.e., in the phase-conjugated version of the
spatial mode. Phase conjugation changes the spatial direction of
propagation by changing the sign of the spatial variation of the
phase, but it does not time-reverse the pulse envelope (despite the
occasional, and somewhat misleading, description of phase
conjugation as time-reversal); the different frequency components
in this phase-conjugated pulse have the same relative complex
amplitudes at any point in space in both the "forward" and
phase-conjugated versions, consistent with the time behavior of the
pulse being of the same form. Hence, we need make no change to the
apparatus described above, with the frequency splitting and
conversion, to allow it to be self-configuring, as long as the
frequency converters operate as discussed here when run
backwards.
[0117] If we are operating using the time-domain rather than
frequency-domain devices, i.e., using units as in FIGS. 12A-B
rather than wavelength splitters and converters, and we want to
train the device to output a pulse of temporal form f(t) for a
given input, then, at least if using the time-delay units of FIGS.
12A-B, we would need to train with a time-reversed pulse, i.e., of
form f(-t) running in each spatial mode back into the device;
otherwise we do not get the desired relative delays of each segment
of the pulse so that they are all lined up in time within the
central general spatial mode converter.
Non-Reciprocal Self-Configuring
[0118] If we are using a non-reciprocal device configuration (e.g.,
as on FIG. 13B), then during training we need to reverse the sense
of the circulators; i.e., the rotation arrows should be flipped
form clockwise to anticlockwise at the input and from anticlockwise
to clockwise at the output. Such a change might be achieved by
changing the direction of the static magnetic fields in circulators
based on Faraday isolation.
B5) Progressive Calculation Method
[0119] Though the device can operate in a self-configuring mode, we
can also formally calculate what the reflectivities and phases need
to be in all of the beamsplitter blocks. FIG. 14 shows one unitary
transformer (here for U.sup..dagger.) with the reflectivities and
phase shifts labeled, analogous to coupler CI in FIG. 8C.
(Detectors are omitted here.) Here the reflectivities and phase
shifts are labeled for each beamsplitter block. The diagonal mirror
has 100% reflectivity.
[0120] The reflectors and phase shifters in FIG. 14 (and in FIGS.
8A-C) are shown as rectangles only in the middle of the
beamsplitter blocks, but it is understood that they act on the
entire beam passing through each block. A completely arbitrary
unitary transformer would require the phase shifters at the right
in the dashed rectangles so as to set the overall phases of the
outputs on the right, and we will use these in our algebra here,
though we do not need these in the architecture of FIGS. 8A-C
because the singular value modulators SD1-SD4 can set any specific
phase required between the beamsplitter blocks for U.sup..dagger.
and V.
[0121] To discuss the phases involved in the beamsplitter, we need
some formal definitions. FIG. 15 shows a (lossless) beamsplitter
(without any additional phase shifter) with definitions of field
reflection and transmission factors and nominal labels of the
beamsplitter ports as top, bottom, left and right. We can define
complex field transmission factors t.sup.(TB) from top to bottom
and t.sup.(LR) from left to right, and similarly define field
reflection factors r.sup.(TR) and r.sup.(LB). These factors include
the phase shifts between the respective inputs and outputs as their
arguments; for example, the phase delay between top and bottom is
.theta..sup.(TB) in the expression
t.sup.(TB)=|t.sup.(TB)|exp(i.theta..sup.(TB)) (4)
and similarly for the other transmission and reflections. Because
the beamsplitter is lossless
|t.sup.(TB)|.sup.2=1-|r.sup.(TR)|.sup.2=|t.sup.(LR)|.sup.2=1-r.sup.(LB)|-
.sup.2 (5)
and, obviously from Eq. (5), |r.sup.(TR)|.sup.2=|t.sup.(LB)|.sup.2.
Also,
.theta..sup.(TR)+.theta..sup.(LB)-.theta..sup.(TB)-.theta..sup.(LR)=.+-.-
.pi. (6)
(at least within some additive phases in units of 2r, which we
neglect for simplicity in the algebra).
[0122] We will formally write any of our input basis functions
|.PHI..sub.DIm as a linear combination of the "modes" (rectangular
functions) corresponding to the inputs to the individual
columns
.phi. DIm = n = 1 M a mn .phi. 1 n ( 7 ) ##EQU00002##
where by |.PHI..sub.1n we mean the (input) mode (rectangular
function) incident on the top row in the nth column. The idea of
this unitary transformer is that, if we illuminate from the top
with the function |.PHI..sub.DI1, all the power will come out of
port 1 at the right. Similarly, illuminating with function
|.PHI..sub.DI2 will lead to all the power coming out of port 2 at
the right, and so on. To understand how to set the reflectivities r
and phase shifts .theta. in the top row mathematically, we imagine
for the moment that we are running the device backwards, shining a
beam into port 1 on the right and looking at the beams coming out
of the ports at the top. We presume that we are dealing only with
reciprocal optics in our beam splitters and phase shifters so that
the phase delays and the magnitudes of the reflectivities are the
same forwards and backwards. The output amplitudes that we want our
device to generate at the top in this backwards case should
therefore be the complex conjugates a*.sub.1n of the amplitudes in
Eq. (7); if we generate some phase delays in running the device
backwards, then we should have corresponding phase leads in the
input beams when running the device forwards so all the beams add
up with the correct phase at output 1 on the right.
[0123] Hence, for the top right block in FIG. 14, we should
choose
r.sub.11.sup.(TR) exp(i.theta..sub.11)=a*.sub.11 (8)
In operation, when we choose the magnitude of a given r.sup.(TB),
for example by setting phase delay in a Mach-Zehnder interferometer
implementation of a variable beam splitter, the phase
.theta..sup.(TR) associated with r.sup.(TR) will also be set as a
result and we will know what it is. (Note in our mathematics here
we are allowing for possible changes in phase associated with
changes in reflectivity, though in the self-configuring versions of
the device discussed in the main text, we prefer to work with
components that do not change phase as they change reflectivity
because it makes the feedback loops simpler.) We will then choose
the phase shifter phase delay (e.g., the .theta..sub.11, in Eq.
(8)) so as to satisfy the necessary overall design requirement on
phase, as in Eq. (8) here.
[0124] Now knowing r.sub.11.sup.(TR) (and hence, from Eq. (5) also
t.sub.11.sup.(LR)) and .theta..sub.1, we can proceed to the next
block in this first row. The field that will emerge from top in the
second column is
t.sub.11.sup.(LR)r.sub.12.sup.(TR)
exp[i(.theta..sub.11+.theta..sub.12)]=a*.sub.12 (9)
so we should choose
r.sub.12.sup.(TR) exp(i.theta..sub.12)=a*.sub.12
exp(-i.theta..sub.11)/t.sub.11.sup.(LR) (10)
We can continue progressively along the top row, with the
reflectivity and phase in the nth column being chosen to
satisfy
r 1 n ( TR ) exp ( i .theta. 1 n ) = a 1 n * exp ( - i p = 1 n - 1
.theta. 1 p ) / q = 1 n - 1 t 1 q ( LR ) ( 11 ) ##EQU00003##
where we understand that when n=1 the summation term will be 0 and
the product term will be 1. (Note that the magnitude of the last
reflectivity, |r.sub.1M.sup.(TR)|, will always be 1, which is
ultimately guaranteed by the lossless nature of this set of
beamsplitters and the consequent unitarity of the operators.)
[0125] Now we consider what happens when we shine the second basis
function |.PHI..sub.DI2 into the top of the set of beamsplitters.
First we need to set up some notation. For a field arriving at the
top of the uth row of beamsplitter blocks, we can choose to
write
.phi. ( u ) = j = 1 M - u + 1 a j ( u ) .phi. uj ( 12 )
##EQU00004##
where, in an extension from the kind of notation used in Eq. (7),
by |.PHI..sub.uj we mean the (input) rectangular "mode" incident on
the uth row in the jth column. Given that we know all the
reflectivities (and hence transmissivities) and phases of the first
row of beamsplitter blocks, given some field |.PHI..sup.(1)
incident on the top row, we can deduce what field |.PHI..sup.(2)
will arrive at the top of the second row. We can formally write
this linear relation in terms of a matrix C( )
|.PHI..sup.(2)=C.sup.(1)|.PHI..sup.(1) (13)
where C.sup.(1) is the first of a family of (M-u).times.(M-u+1)
matrices
C ( u ) = [ t u 1 ( TB ) c 12 ( u ) c 13 ( u ) c 1 ( M - u ) ( u )
c 1 ( M - u + 1 ) ( u ) 0 t u 2 ( TB ) c 23 ( u ) 0 t u 3 ( TB ) 0
0 0 t u ( M - u ) ( TB ) c ( M - u ) ( M - u + 1 ) ( u ) ] ( 14 )
##EQU00005##
where c.sub.sj.sup.(u) is the "complex fraction" (i.e., the
multiplier) of the field incident on column j of row u that
contributes to the field incident on the top of column s of row
u+1. For the diagonal elements,
c.sub.ss.sup.(u)=t.sub.us.sup.(TB) (15)
For the elements to the right of the diagonal,
c sj ( u ) = r u j ( TR ) r u s ( LB ) [ p = s + 1 j - 1 t up ( LR
) ] exp [ i p = s + 1 j .theta. up ] ( 16 ) ##EQU00006##
[0126] This element is the product of (i) the field reflectivity
r.sub.uj.sup.(TR) of the "sideways" reflecting beamsplitter in
block uj that reflects into row u, (ii) the field reflectivity
r.sub.us.sup.(LB) in the "downwards reflecting" beamsplitter in
block us that reflects down into row u+1, (iii) the product of all
the "sideways" transmissions in all the intervening blocks, and
(iv) the phase factors from all of the phase shifters encountered
on this path.
[0127] So, given that we have calculated all the reflectivities and
phases for the first row, we can now calculate C.sup.(1), and hence
when we shine the second basis function |.PHI..sub.DI2 onto the top
of the whole device, we will obtain a field
.phi. DI 2 ( 2 ) .ident. j = 1 M - 1 a 2 j ( 2 ) .phi. 2 j = C ( 1
) .phi. DI 2 ( 17 ) ##EQU00007##
at the top of the second row.
[0128] Now to calculate the settings of the reflection and phase
factors for the second row, we proceed in a similar fashion to that
used for the first row, but with input amplitudes on the top of the
nth column of the second row of a.sub.2n.sup.(2) instead of the
amplitudes a.sub.1n we used in calculating the first row reflection
and phase factors.
[0129] For the third row, having calculated all the reflections and
phases in the second row, we can calculate the matrix C.sup.(2) and
hence calculate amplitudes a.sub.3n.sup.(3) that will appear at the
top of the third row when we illuminated the top of the device with
the third basis function |.PHI..sub.DI3
.phi. DI 3 ( 3 ) .ident. j = 1 M - 2 a 3 j ( 3 ) .phi. 3 j = C ( 2
) C ( 1 ) .phi. DI 3 ( 18 ) ##EQU00008##
We proceed similarly to calculate progressively all subsequent
rows, thereby completing the design mathematically.
[0130] Note that shining the second basis input |.PHI..sub.DI2 on
the top of the structure produces no output from port 1 on the
right. The unitarity of the overall operation means that orthogonal
inputs always give orthogonal outputs (unitarity preserves all
inner products). Because |.PHI..sub.DI2 is orthogonal to
|.PHI..sub.DI1, then their outputs must also be orthogonal. Since
the output with |.PHI..sub.DI1 is solely from the top port,
|.PHI..sub.DI2 can therefore have no component emerging from the
top port. Similar behavior follows for all subsequent orthogonal
inputs, each of which leads only to output from one (different)
port at the right of the structure.
[0131] To calculate the reflections and phases in the device
implementing the unitary transformation V, for which we want output
functions
.phi. DO m = n = 1 M b mn .beta. 1 n ( 19 ) ##EQU00009##
where by |.beta..sub.uj we mean the (output) mode leaving the top
of the uth row in the jth column, we can proceed similarly. Here,
when we shine light into a port on the left of the output coupler
structure (as in CO in FIG. 8C), we want to create the actual
output fields for a given output basis function, so we do not take
the complex conjugates of the amplitudes b.sub.mn for our
calculations. That is, where we have a*.sub.mn, in Eqs. (8)-(11),
we will use b.sub.mn in the analogous equations for V.
B6) Mach-Zehnder Reflectivity and Phase Shift
[0132] The Mach-Zehnder waveguide modulator configuration as in
FIG. 9 implements the necessary control of reflectivity and phase
using two phase shifters within the modulator. FIG. 16 shows the
modulator configuration in detail. More specifically, this is a
Mach-Zehnder waveguide modulator configuration with 50% ("3 dB")
splitters notionally implemented here with coupled waveguides and
two arms each with a phase shifting element. The grey rectangles
represent the phase shifting control elements (e.g., electrodes).
The labeling of the ports corresponds with the notation used in
FIG. 15.
[0133] The phase shifting could be accomplished with electrooptic
materials with voltages applied through electrodes or with thermal
devices, which here for simplicity of description we take to have
phase shift also set by some voltage. (For such thermal phase
shifters, negative voltages would not, however, give negative phase
shifts, so in that case, we can imagine the voltages we discuss
here to be in addition to some positive bias so that all actual
voltages are positive in the thermal case). Nominally defining the
phase delays in the phase shifters as being between points A and C
(B and D) for the upper (lower) phase shifter, the average voltage
controls the common-mode phase shift .theta..sub.av and the
difference between the voltages controls the differential phase
shift .DELTA..theta.. The device is presumed perfectly symmetric;
in a real device we might add one or more control phase shifting
electrodes inside the beamsplitter sections to achieve symmetric
behavior in practice. Here we formally analyze the Mach-Zehnder
interferometers, showing how to relate their behavior and settings
to those of the "conventional" beam splitters and phase shifters of
FIGS. 8A-C and the discussion of section B5 on the required values
in an actual design.
[0134] In a symmetric Mach-Zehnder device as in FIG. 16, the 50%
splitters are each identical symmetrical loss-less beam splitters.
Reflection within these 50% splitters corresponds to the paths
"Top"--C, "Left--D", F--"Right", and G--"Bottom". The phase delays
associated with these reflections, .theta..sub.TC. .theta..sub.LD,
.theta..sub.FR, and .theta..sub.GB, respectively are all equal,
i.e.,
.theta..sub.refl=.theta..sub.TC=.theta..sub.RD=.theta..sub.FL=.theta..su-
b.GB (20)
Similarly for the transmission phases, with obvious notation,
.theta..sub.trans=.theta..sub.TD=.theta..sub.LC=.theta..sub.FB=.theta..s-
ub.GR (21)
Similarly, the magnitudes of the various transmissions and
reflections through these 50% splitters are all equal at a value 1/
{square root over (2)} (which leads to the 50% power splitting).
There may be an additional fixed phase delay .theta..sub.ex
associated with any other waveguide propagations not accounted for
in phase delays in the 50% splitters and the phase shifters.
[0135] Adding the fields on the two "transmission" paths through
the different 50% splitters and phase shifters, the overall complex
field transmissions t.sup.(TB) and t.sup.(LR) are both therefore
given by
t.sup.(TB)=t.sup.(LR)=exp(i.theta..sub.S)exp(i.theta..sub.av)
(22)
where
t=cos(.DELTA..theta./2) (23)
and the background "static" phase .theta..sub.S is the sum
.theta..sub.S=.theta..sub.ex+.theta..sub.trans+.theta..sub.refl
(24)
Before adding up the phases for the reflection paths, we note from
Eq. (6) above, with Eqs. (20) and (21) that we can write
.theta. trans = .theta. refl .+-. .pi. 2 ( 25 ) ##EQU00010##
Whether we use the "+" or the "-" here depends on the detailed
design of the 50% splitters. (It is also possible in principle that
there are additional amounts of phase in units of K that could be
added to the right of Eq. (25), but we neglect those for
simplicity.) Adding the fields on the two "reflection" paths, we
obtain
r.sup.(TR)=-r.sup.(LB).-+.r exp(i.theta..sub.S)exp(i.theta..sub.av)
(26)
where
r=sin(.DELTA..theta./2) (27)
In formally designing using this kind of dual phase-shifter
Mach-Zehnder device, we can drop the additional phase factors of
the form exp(i.theta..sub.up) as in Eqs. (8)-(11) and (16), because
all the necessary phase factors are included in the field
reflection and transmission coefficients r.sub.(TR), r.sub.(LB),
t.sup.(TB) and t.sup.(LR). We use the choice of .DELTA..theta. to
set the magnitude of r.sup.(TR) and the choice of .theta..sub.av
sets its phase, with the magnitudes and phases of r.sub.(LB),
t.sup.(TB) and t.sup.(LR) being therefore set also.
[0136] When used as an amplitude modulator as part of implementing
the singular values s.sub.Dm in an architecture such as that of
FIG. 9, the power out of the "bottom" port will be dumped.
B7) Conclusions
[0137] In conclusion, we have shown that there is at least one
constructive method to design an arbitrary linear optical component
capable in principle of any spatial, polarization, and spectral
linear mapping. This method can also be self-configuring. Only
local feedback loops, optimizing one parameter at a time, are
required. This feedback-based operation avoids the necessity of
setting calculated analog values with interferometric precision in
collections of optical components. The method can be extended to
other linear wave problems generally. In particular, such an
approach can allow simultaneous and separately modulated
conversions from multiple orthogonal inputs to corresponding
orthogonal outputs. Versions for certain specific uses, such as
arbitrary polarization and spatial mode conversions and
modulations, appear practical with current planar optical
technology.
C) Application to Establishing Optimal Wave Communication
Channels
[0138] We show how optimal orthogonal channels for communicating
with waves between two objects can be established automatically
using controllable beamsplitters, detectors and simple local
feedback loops, without moving parts. Applications include multiple
simultaneous orthogonal spatial channels in multimode optical
fibers without fundamental splitting loss, automatically focused
power delivery with waves, communication through scattering or
lossy media, and real-time-optimized focused channels to and from
moving objects. The approach could be exploited in optics,
acoustics, and radio-frequency (e.g., microwave) electromagnetics.
It corresponds mathematically to automatic singular value
decomposition of the wave coupling between the objects, and is
equivalent in its effect to the beam forming in a laser resonator
with phase-conjugate mirrors.
C1) Introduction
[0139] Establishing optimal communication channels with waves is of
obvious importance in many applications in electromagnetics and
acoustics, including remote or biological sensing, and wireless or
optical communications. For example, present increasing demands for
telecommunications capacity are forcing the consideration of
spatial degrees of freedom in multimode optical fibers so as to
provide more orthogonal communications channels. One difficulty in
fully exploiting multiple communications channels simultaneously is
that there may be scattering between simple channels during
propagation--for example, between the different spatial modes in an
optical fiber or between simple beams when propagating through a
scattering environment. The effects of such scattering could be
relatively simply deconvolved spatially if the loss in different
modes is equal because the mathematical transform between inputs
and outputs remains unitary (within an equal multiplying factor
across all modes), and unitary operations preserve
orthogonality--in other words, in that equal loss case, if the
channels are spatially orthogonal at the input, they are spatially
orthogonal at the output. Based on our explicit design for a device
that can separate arbitrary orthogonal waves to simple (e.g.,
single-mode waveguide) channels as described above, any desired
spatially orthogonal inputs in such a unitary case can be separated
automatically as simple orthogonal channels at the output. But, if
there is different loss for different modes (i.e., mode-dependent
loss or different amounts of scattering for different beams), then
such simple separation is no longer possible (though optimizing a
single channel is possible using overall optimization algorithms).
Here, however, we describe an automatic approach to establishing
the independent physical communications channels or modes even for
channels with unequal transmission and for arbitrary scattering of
waves.
[0140] When different modes have different losses or coupling
strengths, the mathematical operator describing the mapping from
inputs to outputs is no longer unitary (even within an overall
multiplying constant), and non-unitary operations do not in general
retain orthogonality--in other words, just choosing orthogonal
input channels in this non-equal-loss case does not guarantee
output channels without cross-talk. Fortunately, it is true that
for essentially any linear operator (technically any compact linear
operator), there is some set of orthogonal inputs that lead to some
set of orthogonal outputs, with these sets being called the
mode-converter basis sets. Mathematically, if we know the operator,
we can always find these sets by singular value decomposition
(SVD).
[0141] On the face of it, though, we might therefore need to go
through the process of measuring that operator explicitly--for
example, by measuring the phase and amplitude of each possible
coupling from every input mode to every output mode; then in a
separate SVD computational operation, we could find the necessary
mode-converter basis sets to allow us to run orthogonal (i.e.,
cross-talk free) channels (or "communications modes") through the
system. Note that when the "component" on which we are performing
the SVD corresponds to the entire set of optics and propagation
medium between outputs at one end of a communication system and the
resulting inputs at the other end, the mathematical idea of
establishing communications modes through this system is identical
to finding the mode-converter basis sets of this whole system
considered as a mode-converting "component". In this section,
however, we show how this operation of finding the orthogonal
channels can be completed automatically, based only on simple local
feedback loops driving variable reflectors and phase shifters. The
net result is to physically establish the actual orthogonal
channels or communications modes through the system, even in the
presence of arbitrary scattering and/or loss in the medium.
[0142] The approach here is quite general for any two objects
connected linearly by waves of any kind (e.g., optical, acoustic,
radio-frequency, quantum mechanical), with arbitrary linear
scattering media between the objects. For each kind of wave, this
approach requires only (i) simple wave sources, (ii) sets of wave
couplers (e.g., grating couplers or antennas), (iii) intensity,
power or, in general, squared-modulus detectors, (iv) controllable
reflectors and phase shifters, and (v) simple local feed-back loops
to minimize power in each detector by adjusting a local phase
shifter and reflector.
[0143] In Section C2 we discuss the simplest case of finding the
optimum single channel from a simple source. In Section C3, we
extend this to optimizing both ends (transmitter and receiver) for
a single channel. Optimization of multiple channels simultaneously
is discussed in Section C4, and we draw conclusions on Section
C5.
C2) Single-Ended Single Channel Optimization
[0144] To describe how this approach works, consider first the
simple free-space situation in FIG. 17, shown for a one-dimensional
array of grating coupler elements for graphical simplicity.
(Two-dimensional versions of the self-aligning universal wave
coupler 1701 in FIG. 17 are discussed above, and those could also
be employed here.)
[0145] A beam 1704 from an external "inward" source 1702 shines
onto the grating couplers (G1-G4) and is coupled into waveguides
W1-W4. The couplers are the inputs to a self-aligning universal
wave coupler as described above, which adjusts the Mach-Zehnder
(MZ) couplers MZ1-MZ4, based on signals from the photodetectors
D1-D3, so as to couple essentially all the light from the grating
couplers into one single-mode input/output waveguide. That light
emerges as indicated by arrow 1708 from the single-mode
input/output waveguide at the bottom right of FIG. 17.
[0146] The operation of the self-aligning coupler is as described
above. Briefly, in the present example, light coupled in through
grating coupler G4 passes through MZ4, which is operated only as a
phase shifter; phase shift in a MZ can be set by the common mode
drive of the phase shifters in its two arms. (A simple phase shift
element could also be substituted for MZ4). That phase shifter is
adjusted to minimize the detected power (coming from both G4 and
G3) in detector D3. Then we adjust what we can call the
"reflectivity" of MZ3 (through the differential drive of the phase
shifters in the MZ arms) to minimize the power in D3 again (ideally
now to zero power). (By "reflectivity" here we mean effectively how
the incident power in one input arm of the MZ interferometer is
split between the two output arms, in the spirit of operating the
MZ device as a variable beam splitter; we do not mean
back-reflection from the interferometer, which we presume here to
be effectively zero in all cases.) Next, the power in D2 is
minimized similarly, first through the common mode drive of MZ3 and
then through the differential drive of MZ2, and so on for any
successive detectors and MZ couplers. The net result of this whole
process is to end up with settings of the MZ interferometers such
that there is negligible or zero power into any of the detectors,
and so all the "inward wave" power now emerges from the single-mode
input/output waveguide.
[0147] So far, we have merely summarized the operation of the
self-aligning universal mode coupler. Now, however, we shine a
"backward wave" into the single-mode input/output waveguide, as
shown by the arrow 1710 at the bottom of FIG. 17. This wave will
experience the same phase delays and amplitude splitting as the
inward wave did as it propagates back to the grating couplers. We
can think of the set of (complex) amplitudes in the single modes in
the waveguides W1-W4 connected to the grating couplers as being a
supermode of those single-mode guides. The net result of this
process is that the backward wave going into the grating couplers
in W1-W4 represents the phase conjugate of the inward wave in W1-W4
coming from the grating couplers.
[0148] To understand this backwards behavior more intuitively, note
first the self-aligning universal mode coupler will adjust its
internal phase delays and reflectivities so that all the "inward"
waves add in phase by the time they get to the single-mode
input/output guide. Since, in FIG. 17, an inward phase front will
arrive last at grating coupler G1, then this path will be set
during the self-alignment to have the shortest phase delay. When
running the device backwards now, the phase front will therefore
get first to G1, so it will launch first, as required so as to get
the backward phase fronts 1706 curved as shown. We can argue
similarly for the phase delays to the other grating couplers. The
net result is to generate phase fronts 1706 going back towards the
original inward source 1702 (at least approximately so).
[0149] Of course, this device is only sampling the phase fronts at
discrete points or regions (in FIG. 17, four averaged points
corresponding to the four grating couplers). But, in the spirit of
Huygens' principle, the wavelets (sketched as dashed lines) from
these sources construct their best representation of the
backward-propagating phase fronts 1706. We also understand that
there are various other ways of implementing the self-aligning
universal mode coupler in FIG. 17, as discussed above. These
include the addition of lenslets in the optical case and, in a
microwave case, the use of antennas instead of the grating
couplers.
[0150] The device of FIG. 17, as it stands, is already useful for
establishing a backwards channel to this inwards source, a channel
that could be used for communicating efficiently back to that
source in a bidirectional link, for example. Such a channel could
also be used for remote power delivery from the backward wave
source to some appropriate power detector at the inward wave
source. Alternatively, by partially back-reflecting the inward
power leaving the single-mode guide so that this reflected power
gives a backward wave (i.e., the beam represented by arrow 1710
could be formed from some reflected power from the beam represented
by arrow 1708), the optimized bidirectional channel could be
created without the need for another source for the backward wave.
Conceivably, in that case detected power in the self-aligned wave
coupler could be used to power that entire device and its feedback
loop electronics, allowing a remote element with no local power
source for, e.g., environmental sensing or biological applications.
Note, too, that an approach like this will also work even if there
is a scattering object in the way between the inward source and the
self-aligning universal mode coupler, as is well known for phase
conjugate optics.
[0151] We could also run the device of FIG. 17 starting with light
into the single-mode input/output waveguide as the optical source
(i.e., arrow 1710 represents the source light), leading to a beam
propagating out towards the position of source 1702; the scattering
off source 1702 could constitute the backwards wave 1704 and 1708
that we would then use to optimize wave 1706 so that it focused
more effectively onto source 1702. Such an operation mode could
also be useful for addressing remote sensors, including power
delivery to them.
[0152] Since the feedback sequence in this self-aligning mode
coupler can be left running continuously, this approach could also
be used even if source 1702 and the self-aligned universal mode
coupler are moving relative to one another, thus allowing this
bidirectional channel to "track" with relative movement.
[0153] The inward wave 1704 and backward wave 1706 could be at the
same frequency here, or they could be at different frequencies or
with finite bandwidths of sources provided only that the relative
phase delays for the different frequencies or bandwidth range in
the different optical paths from grating couplers to the
single-mode input/output waveguide are all substantially similar
within the frequency or wavelength range of interest. Keeping all
the optical path lengths from the different grating couplers to the
single-mode input/output waveguide approximately equal is desirable
for keeping such approximate frequency-independence of the relative
phase delays in the different paths.
C3) Double-Ended Single Channel Optimization
[0154] The approach as shown in FIG. 17 does not itself incorporate
any optimization of the inward source form. Such optimization could
result in better coupling efficiencies overall in such a
bidirectional system. FIGS. 18A-B show versions in which
self-aligning mode couplers are used at both ends of the link. Mode
coupler 1802 is on the left and mode coupler 1804 is on the right.
FIG. 18A shows the free space case, while FIG. 18B shows the case
of an intervening scattering object 1806.
[0155] An interesting question now is to understand the nature of
the wave that will form in the case of FIGS. 18A-B if two
self-aligning universal mode couplers are running one into the
other. To understand this mathematically, we can use the result
that, for essentially any linear operator relating waves or sources
in one ("left") space to waves in another ("right") space, we can
perform the singular value decomposition (SVD) of the resulting
coupling operator to define two complete orthonormal
("mode-converter") basis sets, one set |.PHI..sub.Lmfor the "left"
space and a corresponding set |.PHI..sub.Rm for the "right" space.
Here, for example, we could think of the "left" ("right") space
being the waves on the surface of the grating couplers on the left
(right). It may be mathematically cleaner and simpler, however, to
consider functions in the "left" mathematical space as being the
set of amplitudes of the single modes in the waveguides going into
the grating couplers (i.e., a supermodes of those waveguides) in
the "left" side coupler 1802, and the "right" space as being
correspondingly the set of amplitudes of the single modes (or
supermodes) in the waveguides coming out of the grating couplers in
the "right" side coupler 1804.
[0156] For the example structures in FIGS. 18A-B, any function in
either of these mathematical spaces is therefore representable as a
four (complex) element vector (at least if we consider
monochromatic fields for the moment); in general, with similar
systems each with M grating couplers and waveguides on each side,
the corresponding basis (supermode) functions would be M
(complex)-element vectors. With the mathematical spaces chosen this
way, the coupling "device" operator D between the "left" space and
the "right" space is an M.times.M matrix. The fact that we can
essentially always perform the SVD of D means not only that we
could obtain the two sets |.PHI..sub.Lmand |.PHI..sub.Rm, each
comprising M different vectors, but also that these vectors are
connected one by one with specific (field) coupling strengths given
by the singular values s.sub.m. That is, for each of the M
different orthogonal "left" functions (or supermodes)
|.PHI..sub.Lm
D|.PHI..sub.Lm=s.sub.m|.PHI..sub.Rm (28)
Hence, at least mathematically, we can have M orthogonal channels
(or, in a generalized sense, communications modes), with the
orthogonal "left" output supermodes |.PHI..sub.Lm giving orthogonal
"right" input supermodes |.PHI..sub.Rm, each with corresponding
amplitude s.sub.m. Note that these sets of functions |.PHI..sub.Lm
and |.PHI..sub.Rm are each mathematically complete for their
respective spaces.
[0157] Suppose, then, that we start out with some arbitrary setting
of the "left" set of MZ interferometers, and shine a left input
beam LI1 into the single-mode input/output waveguide on the left.
We will therefore have some output supermode going to the "left"
grating couplers in the "left" waveguides, which we can expand in
the complete set |.PHI..sub.Lm, i.e.,
.phi. L ( 1 ) = A L ( 1 ) m = 1 M a m .phi. Lm ( 29 )
##EQU00011##
where A.sub.L.sup.(1) is a factor related to the power in the LI1
beam, and the coefficients a.sub.m result from the arbitrary
initial setting of the MZ interferometers. Now we send this out
through the "left" grating couplers and the intervening space
(which might include some scattering object 1806) to the "right"
grating couplers. The resulting supermode set of amplitudes in the
waveguides coming out of the "right" grating couplers is, when
expanded on the set |.PHI..sub.Rm,
.phi. R ( 1 ) = D .phi. L ( 1 ) = A L ( 1 ) m = 1 M s m a m .phi.
Rm ( 30 ) ##EQU00012##
Now, we run the self-alignment process in the "right" self-aligning
mode coupler, fixing the resulting "right" MZ settings and
effectively maximizing RO1, the received light from the LI1 source.
Next, we turn on the backwards light source RI1, generating a
backwards propagating wave. We presume the optical system through
the grating couplers and the intervening space or scatterer is made
from materials with symmetric permittivity and permeability
tensors, as is nearly always the case in optical materials in the
absence of static magnetic fields, and that this system is not
time-varying, or that any time variation is negligible over the
time scales of interest. As a result, this optical system shows
reciprocity. Because of that reciprocity, we know that if a given
supermode from the "left" waveguides propagated forward through the
system to one particular "right" supermode, without scattering to
any of the other "right" modes, and with a complex transmission
factor s.sub.m, then the same supermode going backwards (i.e., in a
phase-conjugated fashion) will also proceed without scattering to
other backwards modes, and will have the same complex transmission
factor s.sub.m (i.e., the same magnitude of field transmission and
the same phase pick-up) going backwards. Hence each supermode will
return to the original "left" device, in phase-conjugated form,
with an amplitude
s.sub.m.sup.2a.sub.mA.sub.R.sup.(1)A.sub.L.sup.(1) where now
A.sub.R.sup.(1) is a factor related to the power in the RI1
beam.
[0158] Next, we run the self-alignment process in the "left"
self-aligning mode coupler. The LI1 beam will now lead to some
output
.phi. L ( 2 ) = A L ( 2 ) m = 1 M s m 2 a m .phi. Lm ( 31 )
##EQU00013##
where A.sub.L.sup.(2) is some factor related to the LI1 beam power.
Going round this loop again--i.e., running the self-alignment
process in the "right" self-aligning mode coupler, using the RI1
light source to send the beam back again, and running the "left"
self-alignment process once more--will lead to some LI1 output
.phi. L ( 2 ) = A L ( 3 ) m = 1 M s m 4 a m .phi. Lm ( 32 )
##EQU00014##
[0159] We can see now that what is happening is that the
communications mode with the largest singular value (i.e., field
coupling strength) magnitude will progressively dominate over the
other modes as these singular values s.sub.m are raised to
progressively higher powers every time we go round this loop.
[0160] Since the numbering of the communications modes here is up
to us, we can always choose to number them by decreasing magnitude
of s.sub.m. So, regardless of the initial (presumably non-zero)
strength (i.e., its coefficient a.sub.1) of the mode with the
largest singular value, eventually, after sufficiently many
iterations of running the self-alignment processes on the "right"
and "left" self-aligning mode couplers, the output will tend to the
most strongly coupled mode.
[0161] Thus, this system in FIGS. 18A-B, when run with only local
optimizations in the self-aligning mode couplers at each side, will
find the most strongly coupled communications mode between the
waveguides in these two devices, regardless of the form of the
scattering between the devices (and regardless of the form of the
grating couplers or other wave coupling devices in and out of the
waveguides). This whole process is, of course, strongly analogous
to the build up of the mode with the highest gain in a laser
resonator; the analogy between communications modes and the laser
modes of a resonator with phase conjugate mirrors at both ends has
been pointed out in the literature for the simple free-space case,
and we have generalized this result here.
C4) Double-Ended Multiple Channel Optimization
[0162] The devices in FIGS. 17 and 18A-B only show a single spatial
channel associated with a given self-aligning mode coupler.
Following the previous discussions of related devices, we can also
make versions of such devices that can handle multiple orthogonal
modes simultaneously.
[0163] FIG. 19 shows a version of the device of FIGS. 18A-B, but
configured with additional MZ interferometers and detectors in the
left side 1902 and the right side 1904. The various detectors in
FIG. 19 are presumed to be mostly transparent, sampling only enough
of the beam power passing through them to give enough signal to run
the feedback loops. In general with such devices, with M input
couplers, we can make a device that can handle M orthogonal modes
simultaneously. The device in FIG. 19 can handle 4 orthogonal
modes, and beams and waves are sketched explicitly for two
orthogonal modes. Here the dashed diagonal lines indicate different
"rows" of Mach-Zehnder interferometers and detectors used to set
the different orthogonal modes. Thus left side 1902 has "rows"
1902a, 1902b, and 1902c. Similarly, right side 1904 has "rows"
1904a, 1904b, and 1904c.
[0164] To set up two orthogonal channels in the device of FIG. 19,
we could first proceed as for the device of FIGS. 18A-B, turning on
the LI1 and RI1 beams input into waveguides LW1 on the left and RW1
on the right, respectively, and adjusting the MZ settings using the
signals from their corresponding detectors (now the interferometers
and detectors in the diagonal rows 1902a and 1904a of FIG. 19) as
described above to establish the first optimized channel, with
corresponding supermodes |.PHI..sub.L1 and |.PHI..sub.R1 in the
waveguides leading to the grating couplers on the left and on the
right, respectively.
[0165] A key point to note now is that, regardless of the settings
of the MZ interferometers in rows 1902b and 1904b, shining light
into waveguide 2 (e.g., light LI2 into waveguide LW2 on the left or
shining light RI2 into waveguide RW2 on the right) leads to LI2 and
RI2 supermodes in the waveguides connected to the grating couplers
that are orthogonal to the supermodes |.PHI..sub.L1 and
|.PHI..sub.R1, respectively. None of the power shining into
waveguide 2 (i.e., LW2 or RW2) on either side gets to waveguide 1
(i.e., LW1 or RW1) on the other side. So, now, leaving the devices
in rows 1902a and 1904a fixed at the settings we just established
for the optimized LI1-RI1 channel 1, we can run the same kind of
optimization procedure for the devices in rows 1902b and 1904b with
the LI2 and RI2 beams that we originally ran for the devices in
rows 1902a and 1904a with the LI1 and RI1 beams. This process leads
to supermodes |.PHI..sub.L2 and |.PHI..sub.R2 that are orthogonal,
respectively, to the supermodes |.PHI..sub.L1 and
|.PHI..sub.R1.
[0166] Now we have established two orthogonal channels. Light
shining into waveguide LW1 on the left appears only at waveguide
RW1 on the right. Similarly, light shining into waveguide LW2 on
the left appears only at waveguide RW2 on the right. We can repeat
a similar procedure for any remaining rows of devices. In our
example here, we have only one remaining row, which we can set to
establish a third orthogonal channel, now between waveguides LW3
and RW3. Automatically, we also establish the final orthogonal
channel--between waveguides LW4 and RW4. In general, to establish M
orthogonal channels, we need to run our optimization process on M-1
rows of interferometers and detectors.
[0167] Hence, we have now established all the independent
orthogonal communications mode channels between one side and the
other, regardless of the scattering medium between the two sides
(as long as it is reciprocal). These channels are in order of
decreasing coupling strength.
[0168] If we have no other way of distinguishing between the
various beams in the detectors, the scheme described here does
require that, in setting the device, we first work with the LI1 and
RI1 beams, with the other beams turned off, and only progressively
turn on the light in the other waveguides, holding fixed the
interferometer settings in preceding rows. This particular scheme
cannot be left running continuously to optimize all the modes as
the system changes (e.g., for relative movement of the two object
or for changes in the scattering); to re-optimize the LI1-RI1
channel 1, we would need to turn off the other beams, for example.
There are, however, many ways we could effectively make the beams
distinguishable in the signals they give in the detectors. For
example, we could impose different low frequency small modulations
on each of the beam sources. The feedback electronics connected to
each row of detectors could then be programmed to look only for the
corresponding modulation frequency (for example, using lock-in
detection). Alternatively some other code could similarly be
imposed on each of the beams, with corresponding decoders to pick
out such coding. With such schemes, the different modes could all
be optimized simultaneously with all the beams running, for
example, by stepping through the optimizations of the modes one by
one, in sequence from mode 1 upwards, and continuing to cycle
through all the modes in such a manner.
[0169] The use of heterodyne or homodyne detection approaches in
the detectors could lead to very specific identification of
channels. For example, if we use back reflected LI1 light to
provide the RI1 input in these methods, then, by mixing with
portions of the original LI1 beam, we have the option of homodyne
detection in the detectors on the left, which could allow very
specific discrimination against any other signals of even slightly
different frequencies incident on the grating couplers on the left,
e.g., from the environment. Imposing a modulation, e.g., from a
modulator inserted in the input/output waveguide just before the
back-reflection on the right, would allow heterodyne detection on
the left; looking in the detector outputs on the left for the
desired frequency sideband imposed by the modulation on the right
would also allow discrimination against any direct back-scatter
from the scatterer or from optical components in the path, of the
original LI1 beam from the left back into the left self-aligned
mode-coupler on the left; only the actual beam from the device on
the right would have this side-band present.
C5) Conclusions
[0170] We have presented an approach that allows two objects to
establish optimal channels for communication in the presence of
arbitrary scattering or loss in different physical modes. The
approach is based only on local feedback loops minimizing detector
signals as parameters of beamsplitters, intereferometers or phase
shifters are adjusted progressively. Importantly, multiple
optimized orthogonal spatial channels can be set up and maintained
simultaneously, without any fundamental splitting loss. The
different modes can be optimized sequentially by cycling through
the optimizations of them. The mathematics of the resulting
optimized communications modes is analogous to the formation of
laser modes in a cavity with two phase-conjugate mirrors, though n