Part I · Today's module · Chapter 6 of 18

Skew, drift and calibration

Lane skew against eye opening, heater drift as a random walk, the calibration state machine and what it costs in availability.

Skew against eye opening: one lane's drive, folded into an eyeillustrative

Speed

Lane 1 skew0.14 UIChannel bandwidth:

Step 18 symbols of lane 1 through a 0.75 × baud channel, delayed 0.14 UI and sampled where an unskewed lane peaks: lowest '1' 0.979, highest '0' 0.0209, so the drive eye is 0.958 of a full swing.

Lane 1’s NRZ drive through a band-limited channel, overlaid symbol by symbol over two unit intervals. All lanes are sampled where an unskewed lane peaks, so delaying lane 1 moves its samples off its own peak and lets the neighbouring bits leak in. Computed with the port of the simulator’s ISI model.

\hl{drive}{v_k} = \sum_{m=-6}^{6} p\big(t_0 - \tau + m\big)\, b_{k-m}

p(t) = \Phi\!\Big(\frac{t}{\sigma}\Big) - \Phi\!\Big(\frac{t-1}{\sigma}\Big),\quad \sigma = \frac{\sqrt{\ln 2}}{2\pi B}

What this stage does

Two things go wrong slowly or quietly. The four electrical lanes do not arrive perfectly aligned, and the module samples them all at once, so a skewed lane is read off its peak. And the interferometer is tuned by thermo-optic heaters (17 patent) whose phases drift with temperature and age. A microcontroller with monitor photodiodes calibrates the heaters at power-up, then tracks them with small pilot perturbations while data flows; when tracking fails, the stream pauses for a recalibration.

How it’s modelled

Skew. Each lane’s NRZ drive passes a Gaussian channel of bandwidth 0.75 × baud illustrative (swept down to 0.5 × baud); one lane is delayed; every lane is sampled at the unskewed optimum; the sequence is periodic, truncated at 6 symbols illustrative each side.

Drift and tracking. Each heater’s phase is a random walk (0.01 rad/√s illustrative, swept) plus a common temperature walk. A first-order tracker (gain 0.1 illustrative) is updated once per pilot period (1,000 frames patent) and evaluated in closed form per monitoring period (0.5 s patent). Each period sends a marker frame of 256 words illustrative; a failure is re-applied once (true patent) and a second failure pauses the stream for 1 s illustrative.

Calibration. The simulator’s calibration runs ordered coarse and fine scans per heater in mesh order, a joint refinement on the truth-table residuals, DAC quantisation (12 bits illustrative, phase growing as V²) and a pass check at max |deviation| below 0.5 level steps patent. Its code is not part of this site’s vendored copy, so the animation below runs a simplified stand-in written for this site: the same steps on the same ported analogue chain, with 64 illustrative coarse points, 3 illustrative sweeps and a damped least-squares refinement. The simulator’s own refinement already stands in for the patent’s procedure, whose details are not given.

What we found

Skew

The channel bandwidth matters more than the skew. At 0.75 × baud the eye opening (1 = ideal, over 2,048 random symbols) is 0.9575 with no skew and 0.9155 with 10 ps (0.14 UI) on one lane; the error-rate target holds up to 0.4023 UI. At 0.5 × baud the eye starts at 0.464, falls to 0.3928 at 10 ps, and the target holds only to 0.2449 UI.

Eye opening vs skew of lane 1illustrative

Data table

Eye opening vs skew of lane 1
skew (UI)	0.5×baud	0.75×baud
0	0.464	0.9575
0.02	0.4762	0.9593
0.04	0.4868	0.9576
0.06	0.4916	0.953
0.08	0.4709	0.9469
0.1	0.4476	0.9389
0.12	0.4217	0.9286
0.14	0.3928	0.9155
0.16	0.361	0.8991
0.18	0.3262	0.8787
0.2	0.2882	0.8537
0.22	0.2471	0.8235
0.24	0.203	0.7876
0.26	0.1558	0.7454
0.28	0.1057	0.6966
0.3	0.05288	0.6408
0.32	−0.002429	0.578
0.34	−0.05998	0.5084
0.36	−0.1195	0.4322
0.38	−0.1805	0.3502
0.4	−0.2428	0.2631
0.42	−0.3058	0.1721
0.44	−0.3692	0.0784
0.46	−0.4326	−0.01662
0.48	−0.4955	−0.1115
0.5	−0.5575	−0.2049

Word error rate vs skewillustrative

Data table

Word error rate vs skew
skew (UI)	0.5×baud	0.75×baud
0	1.743 × 10⁻¹⁸	1 × 10⁻²⁰
0.02	3.38 × 10⁻¹⁹	1 × 10⁻²⁰
0.04	8.822 × 10⁻²⁰	1 × 10⁻²⁰
0.06	1.452 × 10⁻¹⁹	1 × 10⁻²⁰
0.08	2.102 × 10⁻¹⁸	1 × 10⁻²⁰
0.1	4.429 × 10⁻¹⁷	1 × 10⁻²⁰
0.12	1.123 × 10⁻¹⁵	1 × 10⁻²⁰
0.14	3.268 × 10⁻¹⁴	1 × 10⁻²⁰
0.16	1.032 × 10⁻¹²	1 × 10⁻²⁰
0.18	3.304 × 10⁻¹¹	1 × 10⁻²⁰
0.2	9.887 × 10⁻¹⁰	1 × 10⁻²⁰
0.22	2.528 × 10⁻⁸	1 × 10⁻²⁰
0.24	5.024 × 10⁻⁷	1 × 10⁻²⁰
0.26	7.112 × 10⁻⁶	1 × 10⁻²⁰
0.28	6.782 × 10⁻⁵	1 × 10⁻²⁰
0.3	0.0004399	1 × 10⁻²⁰
0.32	0.002093	1 × 10⁻²⁰
0.34	0.007757	1.758 × 10⁻¹⁹
0.36	0.02256	6.995 × 10⁻¹⁵
0.38	0.05187	1.094 × 10⁻¹⁰
0.4	0.1003	4.298 × 10⁻⁷
0.42	0.1706	0.0002878
0.44	0.2287	0.02446
0.46	0.2449	0.1889
0.48	0.2514	0.2437
0.5	0.2732	0.2453

Drift as a random walk

One heater drifting, and what the tracker leavesillustrativepatent

A random walk at one of the drift study's rates, tracked at two of its pilot periods. Normals come from the same counter-based stream as the simulator's Monte Carlo.

\hl{phase}{e_{\text{after}}} = (1-g)^{n}\,\big(e_{\text{before}} + \sigma\sqrt{T}\,\xi\big)

With a pilot every 10 frames the tracker gets about 10,050 updates per heater per monitoring period, and availability stays above the study’s threshold at every swept drift rate, up to 1 rad/√s. With a pilot every 100,000 frames it gets about 1.005, and availability holds only up to 0.01 rad/√s: at 0.1 rad/√s it is 0.6708, and at the top of the sweep 0.3333, almost all of it spent recalibrating. Even a pilot every 1,000 frames loses ground at 1 rad/√s (availability 0.9408, 71 recalibrations in 1,200 s, one every 16.9 s).

Availability vs drift rateillustrativepatent

Data table

Availability vs drift rate
drift (rad/√s)	pilot 10 frames	pilot 1000 frames	pilot 10000 frames	pilot 100000 frames
0.003	1	1	1	1
0.01	1	1	1	1
0.03	1	1	1	0.9983
0.1	1	1	0.9992	0.6708
0.3	1	1	0.41	0.3347
1	1	0.9408	0.3333	0.3333

Mean word error while trackingillustrative

Data table

Mean word error while tracking
drift (rad/√s)	pilot 10 frames	pilot 1000 frames	pilot 10000 frames	pilot 100000 frames
0.003	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
0.01	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
0.03	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1.482 × 10⁻⁵
0.1	1 × 10⁻²⁰	1 × 10⁻²⁰	3.841 × 10⁻⁵	0.03418
0.3	1 × 10⁻²⁰	1 × 10⁻²⁰	0.2119	0.6896
1	1 × 10⁻²⁰	0.003199	0.8718	0.9326

The calibration state machine

Calibrate, track, fail, retry, recalibratepatentillustrative

Every frame is a real evaluation of the ported analogue chain over the 16-input truth table. The calibration is this site's simplified stand-in for the simulator's (see how it's modelled); tracking uses one of the drift study's faster drift rates and its slowest pilot, where failures are common.

How good does calibration have to be?

Phase errors that calibration leaves behind are forgiving. Word error stays below one in a million for an input-phase error up to 21.24° on one lane, and for LO-phase errors up to 28.31° on output 1, 18.59° on output 2 and 18.39° on all outputs at once. Output 3 tolerates 35.67°.

Word error rate vs phase errorillustrative

Data table

Word error rate vs phase error
phase error (deg)	input lane 1	input lane 2	LO output 1	LO output 2	LO output 3	LO all outputs
0	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
1	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
2	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
3	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
4	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
5	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
6	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
7	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
8	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
9	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
10	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
11	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰
12	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	1.997 × 10⁻²⁰
13	1 × 10⁻²⁰	1 × 10⁻²⁰	1 × 10⁻²⁰	4.577 × 10⁻¹⁸	1 × 10⁻²⁰	9.154 × 10⁻¹⁸
14	1.388 × 10⁻¹⁸	1.388 × 10⁻¹⁸	1 × 10⁻²⁰	1.38 × 10⁻¹⁵	1 × 10⁻²⁰	2.76 × 10⁻¹⁵
15	1.681 × 10⁻¹⁶	1.681 × 10⁻¹⁶	1 × 10⁻²⁰	2.694 × 10⁻¹³	1 × 10⁻²⁰	5.387 × 10⁻¹³
16	1.455 × 10⁻¹⁴	1.455 × 10⁻¹⁴	1 × 10⁻²⁰	3.35 × 10⁻¹¹	1 × 10⁻²⁰	6.701 × 10⁻¹¹
17	9.032 × 10⁻¹³	9.032 × 10⁻¹³	1 × 10⁻²⁰	2.618 × 10⁻⁹	1 × 10⁻²⁰	5.236 × 10⁻⁹
18	4.039 × 10⁻¹¹	4.039 × 10⁻¹¹	1 × 10⁻²⁰	1.268 × 10⁻⁷	1 × 10⁻²⁰	2.537 × 10⁻⁷
19	1.307 × 10⁻⁹	1.307 × 10⁻⁹	1 × 10⁻²⁰	3.768 × 10⁻⁶	1 × 10⁻²⁰	7.535 × 10⁻⁶
20	3.08 × 10⁻⁸	3.08 × 10⁻⁸	1 × 10⁻²⁰	6.803 × 10⁻⁵	1 × 10⁻²⁰	0.000136
21	5.31 × 10⁻⁷	5.31 × 10⁻⁷	1 × 10⁻²⁰	0.0007432	1 × 10⁻²⁰	0.001482
22	6.748 × 10⁻⁶	6.748 × 10⁻⁶	6.949 × 10⁻²⁰	0.00492	1 × 10⁻²⁰	0.009647
23	6.366 × 10⁻⁵	6.366 × 10⁻⁵	1.716 × 10⁻¹⁷	0.01999	1 × 10⁻²⁰	0.0368
24	0.0004496	0.0004496	3.504 × 10⁻¹⁵	0.05146	1 × 10⁻²⁰	0.08192
25	0.0024	0.0024	5.618 × 10⁻¹³	0.08961	1 × 10⁻²⁰	0.1167
26	0.009788	0.009788	6.703 × 10⁻¹¹	0.1183	1 × 10⁻²⁰	0.1337
27	0.03091	0.03091	5.639 × 10⁻⁹	0.1382	1 × 10⁻²⁰	0.1558
28	0.07674	0.07674	3.169 × 10⁻⁷	0.1631	2.346 × 10⁻²⁰	0.2016
29	0.1528	0.1528	1.128 × 10⁻⁵	0.2013	2.949 × 10⁻¹⁸	0.2777
30	0.25	0.25	0.0002416	0.25	3.144 × 10⁻¹⁶	0.3751

Worst level deviation vs phase errorillustrative

Data table

Worst level deviation vs phase error
phase error (deg)	input lane 1	input lane 2	LO output 1	LO output 2	LO output 3	LO all outputs
0	8.882 × 10⁻¹⁶	8.882 × 10⁻¹⁶	8.882 × 10⁻¹⁶	8.882 × 10⁻¹⁶	8.882 × 10⁻¹⁶	8.882 × 10⁻¹⁶
1	0.01745	0.01745	0.0006092	0.0176	0.0003046	0.0176
2	0.0349	0.0349	0.002437	0.03551	0.001218	0.03551
3	0.05234	0.05234	0.005482	0.05371	0.002741	0.05371
4	0.06976	0.06976	0.009744	0.07219	0.004872	0.07219
5	0.08716	0.08716	0.01522	0.09096	0.007611	0.09096
6	0.1045	0.1045	0.02191	0.11	0.01096	0.11
7	0.1219	0.1219	0.02982	0.1293	0.01491	0.1293
8	0.1392	0.1392	0.03893	0.1489	0.01946	0.1489
9	0.1564	0.1564	0.04925	0.1687	0.02462	0.1687
10	0.1736	0.1736	0.06077	0.1888	0.03038	0.1888
11	0.1908	0.1908	0.07349	0.2092	0.03675	0.2092
12	0.2079	0.2079	0.08741	0.2298	0.0437	0.2298
13	0.225	0.225	0.1025	0.2506	0.05126	0.2506
14	0.2419	0.2419	0.1188	0.2716	0.05941	0.2716
15	0.2588	0.2588	0.1363	0.2929	0.06815	0.2929
16	0.2756	0.2756	0.155	0.3144	0.07748	0.3144
17	0.2924	0.2924	0.1748	0.3361	0.08739	0.3361
18	0.309	0.309	0.1958	0.358	0.09789	0.358
19	0.3256	0.3256	0.2179	0.38	0.109	0.38
20	0.342	0.342	0.2412	0.4023	0.1206	0.4023
21	0.3584	0.3584	0.2657	0.4248	0.1328	0.4248
22	0.3746	0.3746	0.2913	0.4474	0.1456	0.4474
23	0.3907	0.3907	0.318	0.4702	0.159	0.4702
24	0.4067	0.4067	0.3458	0.4932	0.1729	0.4932
25	0.4226	0.4226	0.3748	0.5163	0.1874	0.5163
26	0.4384	0.4384	0.4048	0.5396	0.2024	0.5396
27	0.454	0.454	0.436	0.563	0.218	0.563
28	0.4695	0.4695	0.4682	0.5865	0.2341	0.5865
29	0.4848	0.4848	0.5015	0.6102	0.2508	0.6102
30	0.5	0.5	0.5359	0.634	0.2679	0.634

What would need measured data

Every drift rate, tracker gain and recalibration time here is illustrative; so are the channel bandwidths and the heater mesh layout (inferred, since the real layout is not given). Measured lane skew and channel responses, the OFT’s actual heater cross-talk, drift statistics against temperature and the real calibration sequence and timings would replace them. The pilot period, monitoring period and retry-once rule come from the patent.