Phase Cancellation: Principles and Applications

• Phase cancellation is a method that suppresses unwanted phase fluctuations by exploiting a predictable relation between a disturbed signal and a phase reference, applicable in optics, control, and quantum systems.
• Techniques such as delay-line interferometry, pilot-assisted estimation, and feedforward correction are employed to mitigate phase noise across different frequency ranges and applications.
• Practical implementations demonstrate significant noise suppression—up to 38 dB in lasers and stabilization in fiber links—enhancing the performance of communication systems and sensitive quantum measurements.

Phase cancellation denotes a family of techniques that suppress unwanted phase fluctuations or adverse phase effects by exploiting a predictable relation between a disturbed signal and an available phase reference. In optics and photonics, it commonly refers to the cancellation of laser or fiber-induced phase noise through round-trip comparison, delayed self-interference, pilot-assisted estimation, or feedforward correction; in wireless systems, it includes receiver-side removal of phase-noise-induced rotation and intercarrier interference; in classical control, it also denotes partial cancellation of the adverse phase effect of a non-minimum-phase zero by fractional-order pre-compensation; and in quantum control and quantum algorithms, it includes cancellation of phase transients and finite-runtime geometric-phase errors by virtual frame changes or symmetry-based protocol combinations (Hua et al., 13 Jan 2025, Quadri et al., 2019, Merrikh-Bayat et al., 2016, Hu et al., 2020, Stasiuk et al., 2023, Kiumi, 22 Apr 2026).

1. Conceptual scope and governing mechanisms

Across these settings, phase cancellation is not a single device architecture but a structural principle. One class of methods subtracts a delayed or otherwise correlated copy of the same phase process, as in delay-line interferometry, where the phase discriminator is

$G(f)=1-e^{-2\pi i \tau f},$

so the delayed and undelayed phase fluctuations subtract in a frequency-dependent way (Parniak et al., 2020). A second class estimates the phase disturbance from an embedded reference, such as a pilot tone or pilot subcarrier, and removes the inferred phase rotation digitally, as in OFDM-based communication receivers (Quadri et al., 2019). A third class uses round-trip propagation to convert distributed phase perturbations into a measurable correction variable that is then fed back with an acousto-optic modulator; this is the basis of active phase-noise cancellation in optical fiber dissemination and related sensing systems (Rauf et al., 2018, Noe et al., 2023).

A distinct but closely related class uses feedforward rather than feedback. In frequency-doubled lasers, the relevant observable is the pump phase noise, because the second-harmonic phase tracks the pump phase almost exactly but doubled, subject to a first-order low-pass effect from the SHG enhancement cavity (Hua et al., 13 Jan 2025). In injection-locked optical amplification, the residual locked-state phase error is measured by heterodyne detection and copied forward to a second actuator so that it subtracts from the amplified output field (Simmons et al., 20 Apr 2026). Passive or open-loop optical schemes similarly embed the phase perturbation into an RF signal and reapply it as an optical frequency shift, thereby avoiding conventional phase discrimination and dynamic phase tracking (Hu et al., 2020).

In control theory, the phrase is used in a different technical sense. There, phase cancellation refers not to stochastic phase-noise removal but to the weakening of the adverse phase contribution of a right-half-plane zero by fractional-order pre-compensation. The central transformation is

$$1-\frac{s}{z}\quad \rightarrow \quad 1-\left(\frac{s}{z}\right)^\alpha,\qquad 0<\alpha<1,$$

which does not remove the non-minimum-phase zero exactly, but attenuates its effect without creating the internal instability associated with exact unstable pole-zero cancellation (Merrikh-Bayat et al., 2016, Merrikh-Bayat, 2012).

2. Optical source-level phase-noise cancellation

A representative feedforward realization is the cancellation of high-frequency phase noise in frequency-doubled light. For a pump field

$$E_p \propto \exp\!\left[i\left(\omega_p t+\phi_p(t)\right)\right],$$

second-harmonic generation yields a generated term oscillating as

$$\exp\!\left[i\left(2\omega_p t + 2\phi_p(t)\right)\right],$$

so for ideal phase matching the SHG phase obeys

$\phi_{\mathrm{SHG}}(t)=2\phi_p(t).$

This is the basis for the statement that the phase-noise PSD of the second harmonic is 6 dB higher than that of the pump. In the demonstrated 420-nm system, the practical relation is modified by the SHG enhancement cavity, whose response is fit by

$\frac{A}{1+f^2/f_{\mathrm{SHG}}^2},$

with fitted values $A \approx 4.47$ and $f_{\mathrm{SHG}} \approx 2.09~\text{MHz}$, so the doubled light inherits a low-pass-filtered version of the pump phase noise. Using the pump’s phase noise as an error signal and applying a compensating phase shift to the 420-nm output with an electro-optic modulator, the experiment realized a 25-dB suppression of the servo noise bump near 1 MHz on the 420-nm light, and an average suppression of 30 dB for strong injected noise ranging from 100 kHz to 20 MHz (Hua et al., 13 Jan 2025).

A different source-level scheme uses an unbalanced Mach–Zehnder interferometer as a phase discriminator for MHz-band laser noise. With a 50 m fiber delay line, corresponding to $\tau \approx 250$ ns, the transfer function

$|G(f)|^2 = 4\sin^2(\pi f\tau)$

gives a comb-like sensitivity pattern with maxima near $$1-\frac{s}{z}\quad \rightarrow \quad 1-\left(\frac{s}{z}\right)^\alpha,\qquad 0<\alpha<1,$$0 and zeros at $$1-\frac{s}{z}\quad \rightarrow \quad 1-\left(\frac{s}{z}\right)^\alpha,\qquad 0<\alpha<1,$$1. In the demonstrated Ti:sapphire-laser system, the controller was centered at $$1-\frac{s}{z}\quad \rightarrow \quad 1-\left(\frac{s}{z}\right)^\alpha,\qquad 0<\alpha<1,$$2 MHz, yielding noise reduction in bands at least 300 kHz wide, more than 10 dB reduction in the out-of-loop noise, and a best phase-noise value around $$1-\frac{s}{z}\quad \rightarrow \quad 1-\left(\frac{s}{z}\right)^\alpha,\qquad 0<\alpha<1,$$3 at 1.5 MHz offset. The same work emphasized that in-loop noise squashing is not a reliable measure of true suppression; the out-of-loop detector is the relevant indicator (Parniak et al., 2020).

Residual phase-noise cancellation has also been applied to optical resonant amplification by injection locking. In that setting, injection locking transfers the reference phase only imperfectly, with residual noise following

$$1-\frac{s}{z}\quad \rightarrow \quad 1-\left(\frac{s}{z}\right)^\alpha,\qquad 0<\alpha<1,$$4

The residual error is measured by optical heterodyne detection, and a second acousto-optic modulator applies the negative of that error to the amplified branch. The reported result was up to 38 dB phase-noise reduction at Fourier frequencies above 1 kHz for injection ratios down to -57 dB, with delay compensation identified as critical for high-frequency suppression (Simmons et al., 20 Apr 2026).

3. Fiber links, dissemination, and distributed optical phase stabilization

In optical frequency transfer, active phase-noise cancellation is typically implemented by comparing a local oscillator with a round-trip returned signal and driving an acousto-optic modulator so that the transmitted phase remains fixed relative to the source. A passive variant embeds the optical phase information into an RF signal and shifts the optical frequency by the amount of phase noise introduced by the link, without phase discrimination or active phase tracking. For this passive scheme, the residual remote-end phase noise is delay-limited:

$$1-\frac{s}{z}\quad \rightarrow \quad 1-\left(\frac{s}{z}\right)^\alpha,\qquad 0<\alpha<1,$$5

with a characteristic bandwidth of $$1-\frac{s}{z}\quad \rightarrow \quad 1-\left(\frac{s}{z}\right)^\alpha,\qquad 0<\alpha<1,$$6. On a 145 km fiber spool, the stabilized transfer achieved an Allan deviation of $$1-\frac{s}{z}\quad \rightarrow \quad 1-\left(\frac{s}{z}\right)^\alpha,\qquad 0<\alpha<1,$$7 at $$1-\frac{s}{z}\quad \rightarrow \quad 1-\left(\frac{s}{z}\right)^\alpha,\qquad 0<\alpha<1,$$8, improving to $$1-\frac{s}{z}\quad \rightarrow \quad 1-\left(\frac{s}{z}\right)^\alpha,\qquad 0<\alpha<1,$$9 at $$E_p \propto \exp\!\left[i\left(\omega_p t+\phi_p(t)\right)\right],$$0, and the work explicitly noted the absence of strong servo bumps typical of conventional active cancellation (Hu et al., 2020).

An active 578 nm implementation using all polarisation-maintaining fibers employed imbalance interferometers, retro-reflected round-trip phase sensing, tracking voltage-controlled oscillators implemented, and PI-controlled AOM feedback. The beatnote was first tracked by a VCO with 3 MHz bandwidth to prevent cycle slips under low optical power. The reported performance included no cycle slips in 20 hours, $$E_p \propto \exp\!\left[i\left(\omega_p t+\phi_p(t)\right)\right],$$1 at $$E_p \propto \exp\!\left[i\left(\omega_p t+\phi_p(t)\right)\right],$$2, uncertainty $$E_p \propto \exp\!\left[i\left(\omega_p t+\phi_p(t)\right)\right],$$3, and stability and accuracy below $$E_p \propto \exp\!\left[i\left(\omega_p t+\phi_p(t)\right)\right],$$4 after 10,000 s of averaging (Rauf et al., 2018).

The same round-trip compensation principle has been repurposed as a sensor. In long-range fiber-optic earthquake sensing by active phase noise cancellation, the compensation frequency is the observable:

$$E_p \propto \exp\!\left[i\left(\omega_p t+\phi_p(t)\right)\right],$$5

On the 123 km Bern–Basel link, recorded at 500 Hz, the compensation frequency revealed the 10 September 2022 M3.9 Mulhouse earthquake after low-pass filtering at 5 Hz. Comparison with spectral-element simulations gave $$E_p \propto \exp\!\left[i\left(\omega_p t+\phi_p(t)\right)\right],$$6 for 7–25 s and $$E_p \propto \exp\!\left[i\left(\omega_p t+\phi_p(t)\right)\right],$$7 for 3–10 s, indicating strong strain transfer into the cable. The compatibility of the method with bidirectional amplification was identified as the reason it can scale to links beyond 1000 km (Noe et al., 2023).

In opto-terahertz dissemination, ordinary common-mode stabilization is insufficient because chromatic dispersion makes the two optical wavelengths accumulate different phase noise. The dual-channel round-trip architecture measures the noise on each wavelength independently and forms

$$E_p \propto \exp\!\left[i\left(\omega_p t+\phi_p(t)\right)\right],$$8

which directly contains the differential link noise corrupting the THz beat note. Closed-loop suppression is described by

$$E_p \propto \exp\!\left[i\left(\omega_p t+\phi_p(t)\right)\right],$$9

Over 38 km of standard single-mode fiber, the system delivered opto-THz carriers at 150, 300, and 600 GHz with fractional frequency instabilities below $$\exp\!\left[i\left(2\omega_p t + 2\phi_p(t)\right)\right],$$0 at 10,000 seconds of averaging (Heffernan et al., 17 Apr 2026).

4. Communication-system phase cancellation

In mmWave OFDM, phase noise cancellation is often a receiver-side DSP procedure. A real-time 60 GHz testbed implemented a practical pilot-aided phase noise cancellation scheme in which the DC subcarrier (index 0) is used as the pilot, $$\exp\!\left[i\left(2\omega_p t + 2\phi_p(t)\right)\right],$$1 neighboring subcarriers on each side are nulled as guards, and experiments show $$\exp\!\left[i\left(2\omega_p t + 2\phi_p(t)\right)\right],$$2 is sufficient. After FFT, the receiver zeros out payload-bearing subcarriers, performs an IFFT to reconstruct the phase-noise waveform, and removes the estimated angle:

$$\exp\!\left[i\left(2\omega_p t + 2\phi_p(t)\right)\right],$$3

The reported measurements showed phase-noise standard deviation dropping from about 0.26 rad to about 0.09 rad, EVM improving from about $$\exp\!\left[i\left(2\omega_p t + 2\phi_p(t)\right)\right],$$4 dB to about $$\exp\!\left[i\left(2\omega_p t + 2\phi_p(t)\right)\right],$$5 dB on average, and real-time video streaming with average EVM about $$\exp\!\left[i\left(2\omega_p t + 2\phi_p(t)\right)\right],$$6 dB (Quadri et al., 2019).

In optical OFDM employing an RF pilot tone, the pilot acts as a common phase reference, but dispersion prevents complete cancellation because different subcarriers experience different delays. The key delay parameter is

$$\exp\!\left[i\left(2\omega_p t + 2\phi_p(t)\right)\right],$$7

and the residual phase-noise penalty is decomposed into common phase error and inter-carrier interference. The analysis explicitly included the fact that the correlation signal detection filters the phase noise, reducing the phase-difference variance by a factor of $$\exp\!\left[i\left(2\omega_p t + 2\phi_p(t)\right)\right],$$8. Under the numerical setup of 200 OFDM channels, 1 GS/s, and 4 MHz linewidths, the main practical conclusion was that CO-OFDM with 4PSK supports about 225 km over G.652 fiber, while DD-OFDM with 4PSK supports about 40 km, and 16PSK systems only about 10–20 km (Jacobsen et al., 2016).

In full-duplex OFDM radios, phase cancellation is tied to self-interference suppression. One digital-domain method first estimates the common phase error term $$\exp\!\left[i\left(2\omega_p t + 2\phi_p(t)\right)\right],$$9 by least squares on pilots, then uses a linear MMSE estimator for the remaining phase-noise coefficients, reconstructs the resulting intercarrier interference,

$\phi_{\mathrm{SHG}}(t)=2\phi_p(t).$0

and subtracts it. The reported gain was up to 9 dB more self-interference cancellation than existing digital-domain cancellation schemes that ignore the intercarrier interference suppression (Ahmed et al., 2013).

Analytical studies of full-duplex transceivers identified oscillator phase noise as a severe limit on self-interference cancellation in both independent-oscillator and common-oscillator architectures, while also showing that the common oscillator yields clearly lower residual self-interference levels because of partial phase-noise self-cancellation. A major conclusion was that, in practical scenarios, the subcarrier-wise phase-noise spread of the multipath components of the self-interference channel causes most of the residual phase-noise effect when high amounts of self-interference cancellation is desired (Syrjala et al., 2014). A related architecture study concluded that phase noise in the transmit and receive local oscillators is the dominant bottleneck in current full-duplex systems and that digital cancellation cannot substantially reduce residuals that are already phase-noise-dominated and therefore poorly correlated with the original self-interference waveform (Sahai et al., 2012).

5. Control-theoretic phase cancellation of non-minimum-phase zeros

In classical feedback control, phase cancellation refers to the partial cancellation of a non-minimum-phase zero rather than exact noise suppression. Exact cancellation of a right-half-plane zero by a controller pole is classically forbidden because it leads to internal instability. The fractional-order alternative is to insert a pre-compensator in series with the plant so that

$\phi_{\mathrm{SHG}}(t)=2\phi_p(t).$1

is transformed into

$\phi_{\mathrm{SHG}}(t)=2\phi_p(t).$2

This is the core mechanism described as cancellation on the Riemann surface: the zero is not removed outright, but weakened into a fractional-order zero (Merrikh-Bayat et al., 2016).

Earlier formulations expressed the same idea with

$\phi_{\mathrm{SHG}}(t)=2\phi_p(t).$3

so that the fractionally cancelled plant becomes

$\phi_{\mathrm{SHG}}(t)=2\phi_p(t).$4

The reported consequences were slower phase decay, faster magnitude decay at high frequencies, increased phase margin, increased gain margin, and reduced undershoot and overshoot, while avoiding the internal instability of exact unstable pole-zero cancellation. The analysis invoked Matignon’s fractional-order stability criterion to show that the fractional-order cancellation does not change the internal stability status of the feedback system (Merrikh-Bayat, 2012).

The method was further developed for undershoot-less control of flexible-link robots. In that formulation, a pre-compensator $\phi_{\mathrm{SHG}}(t)=2\phi_p(t).$5 partially cancels each non-minimum-phase zero to an arbitrary degree, after which a classical controller $\phi_{\mathrm{SHG}}(t)=2\phi_p(t).$6 is designed for the modified plant. For a one-link flexible robot arm with a non-minimum phase zero at $\phi_{\mathrm{SHG}}(t)=2\phi_p(t).$7, using the pre-compensator with $\phi_{\mathrm{SHG}}(t)=2\phi_p(t).$8 and a PD controller $\phi_{\mathrm{SHG}}(t)=2\phi_p(t).$9 yielded step responses with very small or no sensible initial undershoot. The work emphasized the tradeoff governed by $\frac{A}{1+f^2/f_{\mathrm{SHG}}^2},$0: larger $\frac{A}{1+f^2/f_{\mathrm{SHG}}^2},$1 gives closer approximation to full cancellation and less undershoot, but lower open-loop bandwidth, more control effort, and more complex realization (Merrikh-Bayat et al., 2013).

6. Quantum-control and geometric-phase error cancellation

Phase cancellation in quantum control includes compensation of deterministic pulse imperfections. In solid-state NMR, short, high-power nominal $\frac{A}{1+f^2/f_{\mathrm{SHG}}^2},$2 pulses acquire a phase-transient control error that is approximately quadrature-shifted by $\frac{A}{1+f^2/f_{\mathrm{SHG}}^2},$3 relative to the intended pulse axis. The experimental pulse is modeled as

$\frac{A}{1+f^2/f_{\mathrm{SHG}}^2},$4

with a tune-up-dependent transient phase angle $\frac{A}{1+f^2/f_{\mathrm{SHG}}^2},$5 measured by a nulling experiment; in the example shown, $\frac{A}{1+f^2/f_{\mathrm{SHG}}^2},$6. The exact correction is implemented by virtual $\frac{A}{1+f^2/f_{\mathrm{SHG}}^2},$7 rotations through a frame change,

$\frac{A}{1+f^2/f_{\mathrm{SHG}}^2},$8

so the phase of pulse $\frac{A}{1+f^2/f_{\mathrm{SHG}}^2},$9 is shifted by $A \approx 4.47$0. The reported improvements included slower decay and longer signal lifetimes in time-suspension and dynamical-decoupling sequences, more consistent quadrature correlations in MREV-8, and significantly slower decay of echo amplitude in Loschmidt-echo measurements (Stasiuk et al., 2023).

A more abstract use appears in Berry phase estimation. There, finite-time adiabatic evolution produces a phase error

$A \approx 4.47$1

so the basic Berry-phase bias is $A \approx 4.47$2. The key cancellation mechanism is to combine finite-runtime evolutions under $A \approx 4.47$3 and $A \approx 4.47$4. Because the leading $A \approx 4.47$5 term changes sign while the Berry phase does not, forward + reverse adiabatic evolution cancels the leading $A \approx 4.47$6 phase error exactly, leaving an $A \approx 4.47$7 residual (Kiumi, 22 Apr 2026).

Richardson extrapolation then removes the non-oscillatory $A \approx 4.47$8 contribution, leaving an oscillatory endpoint-controlled term with coefficient

$A \approx 4.47$9

Runtime randomization suppresses that oscillatory contribution further; for suitable smooth runtime distributions, the remaining oscillatory bias can be reduced to $f_{\mathrm{SHG}} \approx 2.09~\text{MHz}$0 for any fixed $f_{\mathrm{SHG}} \approx 2.09~\text{MHz}$1. The resulting randomized Hadamard-test procedure was presented as a Berry phase estimation algorithm over the full range $f_{\mathrm{SHG}} \approx 2.09~\text{MHz}$2 (Kiumi, 22 Apr 2026).

7. Limits, trade-offs, and applications

A recurring limitation is imperfect matching of the reference and correction paths. In SHG feedforward cancellation, suppression above 40 dB up to 20 MHz requires mismatch less than about 1% in gain and 0.08 ns in group delay (Hua et al., 13 Jan 2025). In passive optical frequency transfer, the residual suppression remains fundamentally delay-limited by the one-way propagation time $f_{\mathrm{SHG}} \approx 2.09~\text{MHz}$3, even though the open-loop design avoids phase discrimination and dynamic phase tracking (Hu et al., 2020). In dual-wavelength THz links, the same delay physics limits the effective common-mode correction bandwidth to about $f_{\mathrm{SHG}} \approx 2.09~\text{MHz}$4 for a 38 km round trip (Heffernan et al., 17 Apr 2026).

Dispersion, multipath, and bandwidth selectivity are equally fundamental. In optical OFDM with RF pilot cancellation, dispersion-induced delay differences make the pilot only a partial phase reference, leaving residual common phase error and intercarrier interference that worsen with distance and subcarrier index (Jacobsen et al., 2016). In full-duplex OFDM radios, multipath components experience different effective phase-noise terms, so phase-noise-induced intercarrier interference becomes the dominant residual after strong analog and digital cancellation (Syrjala et al., 2014). Delay-line interferometers likewise provide only frequency-selective transduction: they offer strong suppression near the targeted band, but do not act as broadband phase erasers (Parniak et al., 2020).

A common misconception is that phase cancellation is necessarily a feedback operation or that exact cancellation is always desirable. The surveyed literature includes open-loop feedforward correction in frequency-doubled lasers and injection-locked amplifiers, passive/open-loop optical frequency transfer, and receiver-side digital cancellation in OFDM (Hua et al., 13 Jan 2025, Simmons et al., 20 Apr 2026, Hu et al., 2020, Quadri et al., 2019). In control theory, exact cancellation of a non-minimum-phase zero is specifically disallowed because it causes internal instability; only partial, fractional-order cancellation is admissible in the proposed framework (Merrikh-Bayat et al., 2013).

The applications are correspondingly broad. In optics and AMO physics, low-phase-noise blue or ultraviolet light is important for precision control of atoms and molecules, improved Rydberg excitation, and adiabatic creation of ultracold molecules (Hua et al., 13 Jan 2025). In optomechanics, MHz-band laser-noise suppression supports deep ground-state cooling of mechanical motion (Parniak et al., 2020). In optical metrology and time transfer, phase cancellation underpins optical frequency standard comparisons, clockworks for future optical atomic clocks, and ultra-stable fiber dissemination (Hu et al., 2020, Rauf et al., 2018). In sensing, compensation-frequency logging turns stabilized fiber links into environmental deformation sensors and potential contributors to earthquake detection and early warning in the oceans (Noe et al., 2023). In communications, pilot-aided phase cancellation enables stable mmWave OFDM and real-time video streaming, while phase-noise-aware self-interference cancellation is necessary for practical full-duplex radios (Quadri et al., 2019, Ahmed et al., 2013). In robotics and control, fractional-order cancellation improves transient response and robustness for flexible-link robots (Merrikh-Bayat et al., 2013). In quantum control and algorithms, virtual frame changes and adiabatic symmetry constructions cancel coherent phase errors without introducing substantial additional hardware overhead (Stasiuk et al., 2023, Kiumi, 22 Apr 2026).