Average Phase Read-out in Photodetection Chains

• Average phase read-out is a technique that estimates the mean phase by correcting systematic delay errors via pilot-tone methods.
• It employs a coherent analog-to-digital chain with IQ demodulation, low-pass filtering, and calibration to reconstruct accurate phase signals.
• The method suppresses low-frequency phase drift by averaging multiple corrected channels, yielding up to a 100× improvement in stability.

Average phase read-out denotes a class of phase-estimation procedures in which the measured phase is averaged only after systematic readout errors have been suppressed or modeled, so that averaging improves precision rather than accumulating low-frequency drift. In photodetection chains, the term has a particularly concrete meaning: it is the estimation of the mean phase of a detected RF or heterodyne beat reconstructed from photodiode current after transimpedance amplification, low-pass filtering, analog-to-digital conversion, and typically IQ demodulation, with explicit removal of delay-induced phase errors that would otherwise appear as phase walk. In that setting, average phase read-out is closely associated with optical pilot-tone correction of the entire photodetection chain (Schultze et al., 2023).

1. Definition and domain-specific scope

In photodetection systems, “phase readout” is the phase of a detected RF or heterodyne beat reconstructed from the photodiode current, while “average phase readout” refers to estimating the mean phase over time and/or across multiple detectors or channels after removing systematic phase errors. The purpose is to prevent low-frequency drift from biasing the average, especially in the region below $10~\mathrm{mHz}$, where phase walk is prominent (Schultze et al., 2023).

The phrase is not terminologically uniform across the literature. This suggests that “average phase read-out” is best understood as a methodological label whose precise meaning depends on the measurement architecture.

Context	Meaning of phase read-out	Representative source
Photodetection chains	Mean phase after correcting readout-chain delay errors	(Schultze et al., 2023)
Dispersive Floquet spectroscopy	Extraction of cycle-averaged dynamical and geometric phases from transmission peaks	(Kohler, 2017)
Phonon-mediated KIDs	Averaged phase pulse template used in matched filtering	(Bellini et al., 2016)
Tone-tracking microwave resonators	Stabilized phase-sensitive I/Q axis via delay monitoring	(Silva-Feaver et al., 2022)
Oscillator-network reservoirs	Low-dimensional read-out built from average phases of oscillator populations	(Jong, 31 Aug 2025)

2. Photodetection-chain formulation

In the photodetection-chain setting, average phase read-out is motivated by the fact that detector delay is not constant. Intensity, reverse bias, and temperature all modify the delay of the photodiode and its electronics, and therefore shift the measured phase. The relevant slow effects include carrier-transport changes in the photodiode, temperature-dependent RC phase shifts in the analog front-end, op-amp slew-rate limitations, and ADC input behavior. Phase walk is described as a slow, often $1/f$-like phase drift, dominated in the cited work by intensity-to-phase conversion in the photodiode, reverse-bias dependence, temperature effects in the sensor and analog front-end, and ADC behavior (Schultze et al., 2023).

The central relationship is the conversion of delay variation into phase error at signal frequency $f_s$: $\Delta\phi(t)=2\pi f_s\,\Delta\tau(I,V,T,t).$ The measured signal phase and pilot phase are modeled as

$$\phi_s(t)=\phi_{\mathrm{true}}(t)+\Delta\phi_{\mathrm{readout}}(t),$$

$\phi_p(t)=\phi_{p0}+2\pi f_p\,\Delta\tau(I,V,T,t).$

Estimating the delay variation from the pilot gives

$$\hat{\Delta\tau}(t)=\frac{\phi_p(t)-\phi_{p0}}{2\pi f_p},$$

and the corrected signal phase becomes

$$\phi_{\mathrm{corr}}(t)=\phi_s(t)-2\pi f_s\,\hat{\Delta\tau}(t) =\phi_s(t)-\frac{f_s}{f_p}\big(\phi_p(t)-\phi_{p0}\big).$$

Phase noise and timing jitter are related through

$S_{\phi}(f)=\big(2\pi f_s\big)^2 S_{\tau}(f).$

The physical sources of delay variation are heterogeneous. In photodiodes, the carrier drift velocity in the substrate depends on the internal field, which is set by the reverse bias and influenced by photocurrent; changing intensity or bias changes transit time, and temperature shifts quantum efficiency and dark current. In the analog front-end, first-order RC filters contribute

$\phi=-\tan^{-1}(2\pi RC f),$

with temperature coefficients of several ppm/K even for NP0/C0G components. Additional phase errors arise if $1/f$0 approaches the op-amp slew-rate limit, and ADC mean-signal-to-phase coupling was measured at $1/f$1 over $1/f$2 full-scale in one device, with hysteresis when the mean-signal gradient changed (Schultze et al., 2023).

3. Optical pilot-tone correction and averaging strategy

The optical pilot-tone method superimposes an amplitude-modulated optical tone on the detected optical signal. The pilot has known phase and frequency inside the photodiode bandwidth and is deliberately placed close to the signal frequency so that pilot and signal experience nearly identical delays. In the reported experiments, amplitude modulation produced pilot tones at $1/f$3 and $1/f$4, both derived from a $1/f$5 master clock to maintain coherence across generators and the FPGA (Schultze et al., 2023).

Because the pilot shares the optical path and the full photodetection and electronic chain, its measured phase encodes instantaneous delay variations of the readout. Subtracting the pilot-derived error corrects the signal phase either in real time or in post-processing. In heterodyne configurations with two optical frequencies $1/f$6 and $1/f$7, each can carry its own pilot tone, allowing separation of detector-chain phase errors from optical-source or time-base phase errors via

$1/f$8

Average phase read-out after pilot correction proceeds in three ways. First, time averaging computes

$1/f$9

with $f_s$0 chosen to suppress residual white phase noise without reintroducing bias from low-frequency drift. Second, multi-channel averaging combines corrected phases with inverse-variance weights,

$f_s$1

Third, channel differencing removes common pilot residuals tied to source or time-base noise. The cited experiments emphasize combinations such as

$f_s$2

which improve low-frequency stability and thereby enable bias-free averaging (Schultze et al., 2023).

Implementation was fully coherent. The demodulation chain used transimpedance amplifiers, $f_s$3-bit SAR ADCs at $f_s$4, and digital IQ demodulation with a fourth-order low-pass filter. FPGA processing multiplied by reference $f_s$5 signals synchronized to the $f_s$6 clock. A central practical point is that the corrected phase is derived from pilot phase, not instantaneous amplitude, so residual intensity fluctuations do not bias the corrected phase. This makes averaging robust in the presence of large RIN or varying mean intensity (Schultze et al., 2023).

4. Experimental demonstrations and quantitative performance

Two experimental configurations were used: photodiode characterization and a heterodyne Mach–Zehnder interferometer at $f_s$7. In the photodiode characterization setup, an LED was driven by two AC generators producing sinusoidal amplitude modulation at $f_s$8 and $f_s$9; the modulated light was split onto two photodiodes, a second LED adjusted the mean intensity, and the photocurrents were amplified, digitized, and IQ-demodulated in an FPGA (Schultze et al., 2023).

The measured coupling factors showed strong dependence of phase on intensity, reverse bias, and temperature:

Photodiode	Measured coupling factors	Notes
Hamamatsu G12180 (InGaAs PIN)	intensity $\Delta\phi(t)=2\pi f_s\,\Delta\tau(I,V,T,t).$0, bias $\Delta\phi(t)=2\pi f_s\,\Delta\tau(I,V,T,t).$1	measured at $\Delta\phi(t)=2\pi f_s\,\Delta\tau(I,V,T,t).$2 bias for intensity
Hamamatsu S3071 (Si PIN)	intensity $\Delta\phi(t)=2\pi f_s\,\Delta\tau(I,V,T,t).$3, bias $\Delta\phi(t)=2\pi f_s\,\Delta\tau(I,V,T,t).$4, temperature $\Delta\phi(t)=2\pi f_s\,\Delta\tau(I,V,T,t).$5	strong bias and temperature sensitivity
Centronic OSD15-5T (Si)	intensity $\Delta\phi(t)=2\pi f_s\,\Delta\tau(I,V,T,t).$6, bias $\Delta\phi(t)=2\pi f_s\,\Delta\tau(I,V,T,t).$7	weaker bias dependence

These measurements confirmed significant phase shifts at low incident light and strong dependence on both intensity and reverse bias. One photodiode exhibited a working point with negligible phase gradient under specific frequencies and intensities. Differential pilot-tone referencing suppressed intensity-induced phase conversion by $\Delta\phi(t)=2\pi f_s\,\Delta\tau(I,V,T,t).$8 and temperature-induced phase shifts by a similar factor, with measured temperature coupling $\Delta\phi(t)=2\pi f_s\,\Delta\tau(I,V,T,t).$9 after correction (Schultze et al., 2023).

In the heterodyne interferometer, optical pilot tones at $$\phi_s(t)=\phi_{\mathrm{true}}(t)+\Delta\phi_{\mathrm{readout}}(t),$$0 and $$\phi_s(t)=\phi_{\mathrm{true}}(t)+\Delta\phi_{\mathrm{readout}}(t),$$1 were superimposed on the measurement and local-oscillator beams and recovered on two photodiodes. The single-channel pilot phase $$\phi_s(t)=\phi_{\mathrm{true}}(t)+\Delta\phi_{\mathrm{readout}}(t),$$2 correlated strongly with total intensity. Subtracting $$\phi_s(t)=\phi_{\mathrm{true}}(t)+\Delta\phi_{\mathrm{readout}}(t),$$3 yielded $$\phi_s(t)=\phi_{\mathrm{true}}(t)+\Delta\phi_{\mathrm{readout}}(t),$$4 with dramatically reduced detector-induced noise. A further subtraction using the second detector,

$$\phi_s(t)=\phi_{\mathrm{true}}(t)+\Delta\phi_{\mathrm{readout}}(t),$$5

reached $$\phi_s(t)=\phi_{\mathrm{true}}(t)+\Delta\phi_{\mathrm{readout}}(t),$$6 stability at low frequencies, corresponding to a $$\phi_s(t)=\phi_{\mathrm{true}}(t)+\Delta\phi_{\mathrm{readout}}(t),$$7 improvement over the uncorrected readout. Phase walk below $$\phi_s(t)=\phi_{\mathrm{true}}(t)+\Delta\phi_{\mathrm{readout}}(t),$$8 was reduced to negligible levels under ambient, unstabilized conditions (Schultze et al., 2023).

The dynamic-range penalty of sharing ADC range between tones was also explicit. With two tones on one ADC, each beat used about half the range, slightly raising quantization noise; the cited mitigations were oversampling, higher ADC resolution, and driving the input signals to full scale (Schultze et al., 2023).

5. Practical limits and comparison with alternatives

Average phase read-out by pilot correction is limited by pilot SNR, signal–pilot frequency proximity, time-base coherence, analog linearity, crosstalk, calibration, and frequency-dependent delay mismatch. The pilot amplitude must dominate detector and electronics noise in the demodulated band. In ADC-offset tests, pilot amplitudes of $$\phi_s(t)=\phi_{\mathrm{true}}(t)+\Delta\phi_{\mathrm{readout}}(t),$$9 and $\phi_p(t)=\phi_{p0}+2\pi f_p\,\Delta\tau(I,V,T,t).$0 suppressed ADC dc-to-phase coupling by $\phi_p(t)=\phi_{p0}+2\pi f_p\,\Delta\tau(I,V,T,t).$1. Signal and pilot frequencies should be close enough to see the same dispersive delay, and all sources and demodulation references should share a common master clock. The reported work noted that two channels of a single generator may not share an exact time base, whereas separate generators locked to the same master performed better (Schultze et al., 2023).

Linearity constraints remain important. The superposed pilot increases total modulation depth, so the transimpedance and ADC stages must avoid saturation, and op-amp slew-rate limits must satisfy

$\phi_p(t)=\phi_{p0}+2\pi f_p\,\Delta\tau(I,V,T,t).$2

Residuals can also arise if the chain introduces appreciable dispersion across the signal–pilot separation. In that case, placing $\phi_p(t)=\phi_{p0}+2\pi f_p\,\Delta\tau(I,V,T,t).$3 near $\phi_p(t)=\phi_{p0}+2\pi f_p\,\Delta\tau(I,V,T,t).$4 mitigates the mismatch. Calibration of $\phi_p(t)=\phi_{p0}+2\pi f_p\,\Delta\tau(I,V,T,t).$5 is obtained under stable conditions, after which drift is tracked continuously. Correction bandwidth is set by the demodulation low-pass filters and pilot-tracking rate, and must cover the relevant low-frequency drift region, including below $\phi_p(t)=\phi_{p0}+2\pi f_p\,\Delta\tau(I,V,T,t).$6, without injecting high-frequency noise (Schultze et al., 2023).

The method is distinct from several alternatives. Electrical pilot tones injected at the ADC front-end can correct ADC timing jitter and downstream electronics, as in the LISA phasemeter, but they do not sense phase errors originating in the photodiode or transimpedance amplifier. Temperature stabilization and bias optimization reduce drift but require tight control and do not suppress ADC dc-to-phase coupling or slew-rate effects. PLL-based tracking follows phase variations but cannot distinguish true signal phase from physics-based delay changes without a reference. Balanced detection reduces intensity noise but does not inherently correct intensity-induced phase delay in each diode’s junction unless both arms are perfectly matched (Schultze et al., 2023).

6. Other technical uses of the expression

In dispersive readout of adiabatic phases, average phase read-out has a different meaning. There, cavity transmission peaks are measured as a function of drive frequency, and the extracted quantities are the dynamical and geometric phases averaged over one Floquet period. The central resonance relation is

$\phi_p(t)=\phi_{p0}+2\pi f_p\,\Delta\tau(I,V,T,t).$7

so the slope yields the dynamical contribution and the offset yields the geometric phase difference $\phi_p(t)=\phi_{p0}+2\pi f_p\,\Delta\tau(I,V,T,t).$8 (Kohler, 2017).

In phonon-mediated kinetic inductance detectors, the expression refers to building a stable phase template by averaging many phase pulses aligned in time, then combining amplitude and phase with a covariance-aware matched filter. In that context, the cited prototype reached a baseline resolution of $\phi_p(t)=\phi_{p0}+2\pi f_p\,\Delta\tau(I,V,T,t).$9 with a single Aluminum KID on a $$\hat{\Delta\tau}(t)=\frac{\phi_p(t)-\phi_{p0}}{2\pi f_p},$$0 silicon substrate, improving on an earlier $$\hat{\Delta\tau}(t)=\frac{\phi_p(t)-\phi_{p0}}{2\pi f_p},$$1 result obtained with four Al KIDs (Bellini et al., 2016).

In tone-tracking readout for superconducting microwave resonators, average phase read-out denotes choosing and maintaining a phase-sensitive measurement axis by monitoring system delay with pilot tones distributed across the band. The phase drift is modeled as

$$\hat{\Delta\tau}(t)=\frac{\phi_p(t)-\phi_{p0}}{2\pi f_p},$$2

and the paper estimated $$\hat{\Delta\tau}(t)=\frac{\phi_p(t)-\phi_{p0}}{2\pi f_p},$$3 degrees per $$\hat{\Delta\tau}(t)=\frac{\phi_p(t)-\phi_{p0}}{2\pi f_p},$$4 phase drift at $$\hat{\Delta\tau}(t)=\frac{\phi_p(t)-\phi_{p0}}{2\pi f_p},$$5 for a $$\hat{\Delta\tau}(t)=\frac{\phi_p(t)-\phi_{p0}}{2\pi f_p},$$6 round trip of solid-PTFE coax, with pilot-tone-based correction expected to reduce angle drift by $$\hat{\Delta\tau}(t)=\frac{\phi_p(t)-\phi_{p0}}{2\pi f_p},$$7 across channels (Silva-Feaver et al., 2022).

In oscillator-network reservoir computing, the read-out is explicitly “a function on the average phases with respect to each oscillator population.” There the low-dimensional read-out is built from the population-average phases $$\hat{\Delta\tau}(t)=\frac{\phi_p(t)-\phi_{p0}}{2\pi f_p},$$8, and numerical evidence indicated that at least $$\hat{\Delta\tau}(t)=\frac{\phi_p(t)-\phi_{p0}}{2\pi f_p},$$9 oscillator populations were necessary to learn chaotic target dynamics (Jong, 31 Aug 2025).

This suggests that the phrase marks a common strategy—deriving robust information from phase averages—rather than a single universally fixed estimator.

7. Applications and implications

In optical metrology, pilot-tone-aided average phase read-out directly benefits applications that require accurate timing or signal phase determination with photodiodes. The cited work identifies precision interferometry, heterodyne displacement and angle metrology, differential wavefront sensing, RF photonics and timing distribution, optical telemeters and ranging, and optical clocks and frequency distribution as direct beneficiaries (Schultze et al., 2023).

The reported residuals imply sub-picometer displacement and $$\phi_{\mathrm{corr}}(t)=\phi_s(t)-2\pi f_s\,\hat{\Delta\tau}(t) =\phi_s(t)-\frac{f_s}{f_p}\big(\phi_p(t)-\phi_{p0}\big).$$0 angular errors in QPD differential wavefront sensing under realistic conditions and without active stabilization. More generally, the demonstrated suppression of detector-induced phase noise by $$\phi_{\mathrm{corr}}(t)=\phi_s(t)-2\pi f_s\,\hat{\Delta\tau}(t) =\phi_s(t)-\frac{f_s}{f_p}\big(\phi_p(t)-\phi_{p0}\big).$$1 for intensity and temperature stimuli, together with $$\phi_{\mathrm{corr}}(t)=\phi_s(t)-2\pi f_s\,\hat{\Delta\tau}(t) =\phi_s(t)-\frac{f_s}{f_p}\big(\phi_p(t)-\phi_{p0}\big).$$2 overall low-frequency phase-stability improvement using multi-pilot and multi-detector referencing, shows that averaging can become precision-enhancing rather than drift-limited (Schultze et al., 2023).

A plausible implication is that average phase read-out is most useful when the dominant obstacle is not white measurement noise but the integration of slow, systematic phase errors. In photodetection chains, optical pilot tones provide a direct way to measure those errors at the detector input. In other architectures, analogous roles are played by cycle averaging, phase-template averaging, delay tracking, or population-level phase reduction. Across these domains, the unifying objective is the same: to make the averaged phase represent the underlying physical quantity rather than the readout apparatus.