Part I · Today's module · Chapter 4 of 18

Homodyne detection

Where the down-conversion happens: square-law photodiodes, a balanced pair and a local oscillator split from the same laser.

Where the down-conversion happenstextbookillustrative

LO phase ψ 0°input x = [1,0,1,1]

Speed

Step 1Output 1's optical field oscillates at the optical carrier (drawn enormously slowed down): amplitude 10.37 √µW, far too fast for any electronics to follow.

One output of the worked example, x = [1,0,1,1], meeting the local oscillator. The waveforms are drawn at a hugely slowed optical frequency; the averages, the balanced current and the four outputs are the model’s numbers. Pick an output and turn the LO phase to see the receiver read a different quadrature.

i_\pm = R\,\Big(\frac{\hl{field}{e_s} \pm \hl{lo}{e_{\mathrm{LO}}}}{\sqrt2}\Big)^2

\langle i_\pm \rangle = \tfrac{R}{2}\big(|Y|^2 + E_{\mathrm{LO}}^2\big) \pm \hl{product}{R\,\mathrm{Re}(Y E_{\mathrm{LO}}^{*})}

\hl{current}{I} = i_+ - i_- = 2R\,E_{\mathrm{LO}}\,\mathrm{Re}\big(Y e^{-j\psi}\big)

What this stage does

The OFT’s outputs are optical fields oscillating at about two hundred terahertz. No electronics can follow that, and a plain photodiode would only report each output’s power, losing the sign and the phase that the DFT needs. The module therefore reads each output coherently: it mixes it with a strong local oscillator (LO) split from the same laser, a self-homodyne receiver.

The down-conversion happens inside the photodiode. Its current follows the square of the total field, so the sum of signal and LO produces each one’s own power, their product, and terms at twice the optical frequency. The photodiode is far too slow for the double-frequency terms, so they average away, and the product is left as a current proportional to the signal’s projection onto the LO. A balanced pair (one photodiode at each output of a 3 dB coupler) subtracts the two currents: the self-terms, and the LO’s intensity noise with them, cancel, and the product doubles.

The LO’s phase ψ picks the quadrature. Outputs 1, 2 and 3 read the real part of their bin and output 4 the imaginary part, which is all the 4-point DFT of a real input needs.

How it’s modelled

The simulator’s receiver is exactly the last equation, $I_k = 2R\,E_{\mathrm{LO}}\,\{\mathrm{Re},\mathrm{Im}\}\,Y_k e^{-j\psi}$ , with responsivity 0.65 A/W illustrative and LO power −5 dBm illustrative per balanced pair, giving a conversion gain of 23.1 µA/√µW and an LO field of 17.8 √µW. Noise is added at the TIA input: shot noise of the LO and of the signal power on each pair, TIA noise, and the LO’s relative intensity noise reduced by the pair’s common-mode rejection (20 dB illustrative).

Because the LO comes from the same laser, the laser’s phase noise mostly cancels. What is left depends on the path mismatch between signal and LO (1 mm illustrative) and the linewidth (1 MHz illustrative): about 0.52° rms, which the Monte Carlo includes.

What we found

For the worked example the four receivers carry 239.7 µA, 0 µA, 36.56 µA and −36.56 µA. The large current on output 1 is mostly the carrier pedestal (130 µA) from the modulators.

The LO phase must be held, but not tightly. A static LO-phase error leaks the other quadrature into each output; the word error stays below 1 × 10⁻⁶ up to 28.31° on output 1, 18.59° on output 2, 35.67° on output 3 and 18.39° with the same error on all outputs. Output 3 tolerates most because its bin (X2) is always real, so a rotation only shrinks it; outputs 2 and 4 suffer most because their bins have both parts. At 30° on all outputs the worst level error is 0.634 steps, past the half-step margin.

What would need measured data

The receiver numbers are a mix of datasheet values (the TIA’s noise) and illustrative ones (responsivity, LO power, RIN, CMRR, path mismatch, linewidth). A measured model would need each pair’s responsivity balance and common-mode rejection across frequency, the real LO power reaching each pair after the split, the noise spectrum at the TIA output, the laser’s linewidth and RIN, and the signal/LO path lengths on the chip.