Inside a Photonic Link Module
Part I · Today's module · Chapter 3 of 18

The optical Fourier transform

A passive mesh of couplers and phase shifters computes a 4-point DFT of four optical fields in a single pass of light.

The 4-point DFT as a mesh of couplerspaperillustrative
Input word x =
3 dB coupler3 dB coupler3 dB coupler3 dB coupler−jE0 5.97E2 5.97E1 2.81E3 5.97a0 8.45a1 0b0 6.21b1 −2.24a0 8.45b0 6.21a1 0−j·b1 2.24jY0 10.4Y2 1.58Y1 1.58jY3 −1.58jY0 10.4Y1 1.58jY2 1.58Y3 −1.58jinputs E (√µW)outputs Y = ½·DFT(E)
3 dB coupler3 dB coupler3 dB coupler3 dB coupler−j5.97E05.97E22.81E15.97E38.4506.21−2.248.456.2102.24j10.41.581.58j−1.58j10.4Y01.58jY11.58Y2−1.58jY3
Stage 1x = [1,0,1,1] enters as four fields E = [5.97, 2.81, 5.97, 5.97] õW; a 0 is the carrier, not darkness. Rows are drawn x0, x2, x1, x3.
Stage 1: x = [1,0,1,1] enters as four fields E = [5.97, 2.81, 5.97, 5.97] õW; a 0 is the carrier, not darkness. Rows are drawn x0, x2, x1, x3.
Two stages of 3 dB couplers with a −j phase shift between them: the two-point butterflies of a 4-point FFT, built in waveguides. Path width follows the field amplitude on each waveguide; the numbers are the complex fields (√µW) the model computes at each point. Toggle the input bits to send another word.
Yk=12∑n=03En (−j)nk\hl{oft}{Y_k} = \tfrac12 \sum_{n=0}^{3} \hl{field}{E_n}\,(-j)^{nk}
(Y0Y1Y2Y3)=12(11111−j−1j1−11−11j−1−j)(E0E1E2E3)\begin{pmatrix}Y_0\\Y_1\\Y_2\\Y_3\end{pmatrix} = \tfrac12\begin{pmatrix}1&1&1&1\\1&\hl{phase}{-j}&-1&j\\1&-1&1&-1\\1&j&-1&-j\end{pmatrix}\begin{pmatrix}E_0\\E_1\\E_2\\E_3\end{pmatrix}

What this stage does

The optical Fourier transform (OFT) is a passive interferometer: four waveguides in, four out, and in between only couplers and phase shifters. Light entering it is mixed so that each output carries one bin of the 4-point discrete Fourier transform of the four input fields. There is no clock and no arithmetic: the transform happens in the time light takes to cross a few millimetres of chip.

The factor ½ keeps the mesh lossless (it is unitary: the power in equals the power out). The −j between the stages is the transform’s one non-trivial twiddle factor, and it is set by a heater.

How it’s modelled

The ideal OFT is the matrix ½·DFT. For calibration studies the simulator builds it as a two-stage butterfly mesh of tunable Mach–Zehnder couplers and phase shifters, each with a heater error; at zero error the mesh is exactly ½·DFT, and it stays unitary for any heater setting (both are tested). The real layout is not published, so this topology is an illustrative stand-in. Random fabrication error is modelled as U = ½·DFT·exp(j·s·H) with H a random Hermitian matrix and scale s (default 0 illustrative).

Below, pick one of the mesh’s heaters and give it an error. The model recomputes all 16 inputs’ outputs and, after re-measuring the carrier pedestal as calibration would, the worst level error over every input and output.

All 16 inputs through the mesh, ideal and with a heater errorillustrative
Constellation explorer (loads when scrolled into view)
Each panel is one output in the complex plane. The receivers of outputs 1, 2 and 3 read the real part; output 4 reads the imaginary part.

What we found

A level error of half a step is where decisions start to fail. With an input-phase error on one lane the worst level error is 0.1736 steps at 10° and 0.5 steps at 30°; at the default LO power the word error stays below 1 × 10⁻⁶ up to 21.24°. That is a comfortable tolerance: a heater only has to hold its phase to within some tens of degrees.

Uncorrected fabrication error matters more. At scale s = 0.01 the worst level error over eight random draws is 0.07178 steps; at s = 0.05 it reaches 0.367 steps and the worst draw’s word error 2.558 × 10⁻⁷; at s = 0.1 even the median draw fails, at 0.05914. The mesh has to be calibrated, which is what its heaters are for (chapter 6).

Word error rate vs phase errorillustrative
05101520253010⁻²⁰10⁻¹⁶10⁻¹²10⁻⁸10⁻⁴1phase error (deg)word error ratetarget
Analytic word error at the default LO power, with one phase error at a time (values below the plotting floor are drawn at the floor).
Data table
Word error rate vs phase error
phase error (deg)input lane 1input lane 2LO output 1LO output 2LO output 3LO all outputs
01 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰
11 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰
21 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰
31 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰
41 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰
51 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰
61 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰
71 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰
81 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰
91 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰
101 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰
111 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰
121 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰1.997 × 10⁻²⁰
131 × 10⁻²⁰1 × 10⁻²⁰1 × 10⁻²⁰4.577 × 10⁻¹⁸1 × 10⁻²⁰9.154 × 10⁻¹⁸
141.388 × 10⁻¹⁸1.388 × 10⁻¹⁸1 × 10⁻²⁰1.38 × 10⁻¹⁵1 × 10⁻²⁰2.76 × 10⁻¹⁵
151.681 × 10⁻¹⁶1.681 × 10⁻¹⁶1 × 10⁻²⁰2.694 × 10⁻¹³1 × 10⁻²⁰5.387 × 10⁻¹³
161.455 × 10⁻¹⁴1.455 × 10⁻¹⁴1 × 10⁻²⁰3.35 × 10⁻¹¹1 × 10⁻²⁰6.701 × 10⁻¹¹
179.032 × 10⁻¹³9.032 × 10⁻¹³1 × 10⁻²⁰2.618 × 10⁻⁹1 × 10⁻²⁰5.236 × 10⁻⁹
184.039 × 10⁻¹¹4.039 × 10⁻¹¹1 × 10⁻²⁰1.268 × 10⁻⁷1 × 10⁻²⁰2.537 × 10⁻⁷
191.307 × 10⁻⁹1.307 × 10⁻⁹1 × 10⁻²⁰3.768 × 10⁻⁶1 × 10⁻²⁰7.535 × 10⁻⁶
203.08 × 10⁻⁸3.08 × 10⁻⁸1 × 10⁻²⁰6.803 × 10⁻⁵1 × 10⁻²⁰0.000136
215.31 × 10⁻⁷5.31 × 10⁻⁷1 × 10⁻²⁰0.00074321 × 10⁻²⁰0.001482
226.748 × 10⁻⁶6.748 × 10⁻⁶6.949 × 10⁻²⁰0.004921 × 10⁻²⁰0.009647
236.366 × 10⁻⁵6.366 × 10⁻⁵1.716 × 10⁻¹⁷0.019991 × 10⁻²⁰0.0368
240.00044960.00044963.504 × 10⁻¹⁵0.051461 × 10⁻²⁰0.08192
250.00240.00245.618 × 10⁻¹³0.089611 × 10⁻²⁰0.1167
260.0097880.0097886.703 × 10⁻¹¹0.11831 × 10⁻²⁰0.1337
270.030910.030915.639 × 10⁻⁹0.13821 × 10⁻²⁰0.1558
280.076740.076743.169 × 10⁻⁷0.16312.346 × 10⁻²⁰0.2016
290.15280.15281.128 × 10⁻⁵0.20132.949 × 10⁻¹⁸0.2777
300.250.250.00024160.253.144 × 10⁻¹⁶0.3751
Worst level deviation vs phase errorillustrative
05101520253000.20.40.6phase error (deg)max deviation (steps)half step
The worst noiseless level error over the 16 inputs. Half a step is the decision margin.
Data table
Worst level deviation vs phase error
phase error (deg)input lane 1input lane 2LO output 1LO output 2LO output 3LO all outputs
08.882 × 10⁻¹⁶8.882 × 10⁻¹⁶8.882 × 10⁻¹⁶8.882 × 10⁻¹⁶8.882 × 10⁻¹⁶8.882 × 10⁻¹⁶
10.017450.017450.00060920.01760.00030460.0176
20.03490.03490.0024370.035510.0012180.03551
30.052340.052340.0054820.053710.0027410.05371
40.069760.069760.0097440.072190.0048720.07219
50.087160.087160.015220.090960.0076110.09096
60.10450.10450.021910.110.010960.11
70.12190.12190.029820.12930.014910.1293
80.13920.13920.038930.14890.019460.1489
90.15640.15640.049250.16870.024620.1687
100.17360.17360.060770.18880.030380.1888
110.19080.19080.073490.20920.036750.2092
120.20790.20790.087410.22980.04370.2298
130.2250.2250.10250.25060.051260.2506
140.24190.24190.11880.27160.059410.2716
150.25880.25880.13630.29290.068150.2929
160.27560.27560.1550.31440.077480.3144
170.29240.29240.17480.33610.087390.3361
180.3090.3090.19580.3580.097890.358
190.32560.32560.21790.380.1090.38
200.3420.3420.24120.40230.12060.4023
210.35840.35840.26570.42480.13280.4248
220.37460.37460.29130.44740.14560.4474
230.39070.39070.3180.47020.1590.4702
240.40670.40670.34580.49320.17290.4932
250.42260.42260.37480.51630.18740.5163
260.43840.43840.40480.53960.20240.5396
270.4540.4540.4360.5630.2180.563
280.46950.46950.46820.58650.23410.5865
290.48480.48480.50150.61020.25080.6102
300.50.50.53590.6340.26790.634

What would need measured data

The model knows the OFT only as a matrix. A measured version would need the chip’s actual 4×4 transfer matrix (amplitudes and phases, for instance from a swept-input characterisation), how each heater moves it including thermal cross-talk between neighbouring heaters, the insertion loss and its wavelength dependence, and how the matrix drifts with temperature. Any of these can replace the ideal unitary in the simulator without touching the layers above it.