Phase Loss: Theory and Applications

• Phase loss is a class of phenomena where phase shifts are intrinsically linked with amplitude attenuation due to mechanisms like free-carrier absorption, thermal diffusion, and non-unitary processes.
• In integrated photonics and RIS systems, phase-dependent loss affects device performance, as demonstrated by trade-offs between phase modulation efficiency and insertion loss in modern silicon photonics and quantum communication devices.
• Phase loss also serves as an optimization objective in inverse problems, where loss functions enforce phase consistency in spectrogram reconstruction and speech enhancement algorithms.

Searching arXiv for recent and foundational papers relevant to “phase loss” across optics, communications, quantum metrology, and signal processing. “Phase loss” does not denote a single universal construct. In the technical literature it names a family of coupled phase–attenuation phenomena and objective functions: insertion loss incurred while imposing a phase shift in integrated photonics; phase-dependent loss (PDL), where amplitude depends on the selected phase state in RISs or silicon-photonic quantum transmitters; degradation of phase sensitivity by optical loss in interferometry; and loss functions that constrain phase or phase consistency in reconstruction and speech enhancement (Harris et al., 2014, Singh et al., 2022, Crowley et al., 2012, Ku et al., 2024). Across these settings, the common structure is that phase is not an isolated degree of freedom: it is constrained by absorption, carrier-induced attenuation, thermal diffusion, measurement incompatibility, or consistency conditions in the underlying representation.

1. Terminological scope and recurring mathematical forms

The literature uses “phase loss” in at least four technically distinct senses. In photonic devices, phase shift and loss are linked through the complex refractive index, so any imposed phase can carry a nonzero attenuation penalty. In wireless RIS models and silicon-photonic QKD, the loss is explicitly phase dependent, so the amplitude response is a function of the programmed phase. In quantum metrology, loss is a nuisance parameter or a dissipative channel that limits attainable phase precision. In signal processing and inverse problems, “phase loss” can mean a training objective over phase variables or a consistency-preserving constraint on a complex spectrogram (Li et al., 2018, Kolodynski et al., 2010, McRae, 24 Nov 2025, Ku et al., 2024).

Domain	Meaning of “phase loss”	Representative formulation
Integrated photonics	Insertion loss accompanying a phase shift	$\Delta\phi=\frac{2\pi}{\lambda}\Delta n_{\rm eff}L$
RIS / QKD	Amplitude depends on programmed phase	$L(\phi)$ or $T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}]$
Quantum interferometry	Photon loss degrades phase estimation	$\Delta^2\phi\ge 1/F_{\phi\phi}$ under lossy dynamics
Reconstruction / enhancement	Optimization loss acting on phase or phase consistency	$\mathcal L_{EC}(H)=\sum_{m,n}|\Delta_{m,n}(H)|^2$

A recurrent formal pattern is the decomposition of the optical response into a phase-bearing real part and a loss-bearing imaginary part. For guided-wave phase shifters, the phase shift follows from the effective-index perturbation,

$$\Delta\phi \;=\;\frac{2\pi}{\lambda}\;\Delta n_{\rm eff}\;L,$$

while the effective-index change itself is an overlap of the thermo-optic response and the modal intensity profile,

$$\Delta n_{\rm eff} = \frac{\iint \bigl(\tfrac{dn}{dT}\bigr)\; \Delta T(x,y)\; |E(x,y)|^2 \, dx \, dy} {\iint |E(x,y)|^2 \, dx \, dy}.$$

In PDL settings, the phase-bearing map becomes explicitly non-unitary; for example, RIS elements are modeled with a phase-dependent amplitude $L(\phi)$, and silicon-photonic QKD transmitters use the intensity transmission

$$T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}] \equiv L(\phi)$$

with $\mathrm{PDL}[{\rm dB}]=-10\log_{10}L$ (Harris et al., 2014, Singh et al., 2022, Li et al., 2018).

2. Integrated phase shifters: insertion loss as a design variable

In silicon thermo-optic phase shifters, minimizing “phase loss” means maximizing thermal overlap with the optical mode while suppressing free-carrier absorption. A representative SOI device uses a $L(\phi)$0-wide silicon ridge waveguide $L(\phi)$1 high), two narrow $L(\phi)$2-wide) thermal channels in a $L(\phi)$3 slab, heavily boron-doped $L(\phi)$4 routing arms, and moderately doped $L(\phi)$5 silicon only under the ridge. This confines most of the Joule heat within $L(\phi)$6 of the optical mode while keeping free-carrier absorption low. COMSOL simulations show that $L(\phi)$7 of $L(\phi)$8 lies within the optical mode area. A $L(\phi)$9-long phase shifter fabricated in a CMOS process with oxide cladding and two metal layers achieved $T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}]$0, $T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}]$1, $T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}]$2, a $T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}]$3 modulation bandwidth of $T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}]$4, and insertion loss $T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}]$5 for $T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}]$6 devices across an $T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}]$7-inch wafer. To keep thermally induced crosstalk below $T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}]$8 for $T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}]$9, a lateral separation of $\Delta^2\phi\ge 1/F_{\phi\phi}$0 in $\Delta^2\phi\ge 1/F_{\phi\phi}$1 SOI is sufficient (Harris et al., 2014).

The dominant optical penalty in that device is free-carrier absorption,

$\Delta^2\phi\ge 1/F_{\phi\phi}$2

with $\Delta^2\phi\ge 1/F_{\phi\phi}$3 and $\Delta^2\phi\ge 1/F_{\phi\phi}$4 at $\Delta^2\phi\ge 1/F_{\phi\phi}$5. The connection to insertion loss is written as

$\Delta^2\phi\ge 1/F_{\phi\phi}$6

This formulation makes the design problem explicit: phase efficiency is governed by the overlap integral $\Delta^2\phi\ge 1/F_{\phi\phi}$7, while loss is governed by carrier-induced $\Delta^2\phi\ge 1/F_{\phi\phi}$8 or $\Delta^2\phi\ge 1/F_{\phi\phi}$9 in regions sampled by the mode (Harris et al., 2014).

Comparable tradeoffs appear in non-volatile PCM phase shifters. A GeSe-based SOI device with a uniform TiN heater $\mathcal L_{EC}(H)=\sum_{m,n}|\Delta_{m,n}(H)|^2$0 achieves a $\mathcal L_{EC}(H)=\sum_{m,n}|\Delta_{m,n}(H)|^2$1 phase shift for a $\mathcal L_{EC}(H)=\sum_{m,n}|\Delta_{m,n}(H)|^2$2 pulse, but its phase levels are highly non-linear. A segmented-heater design with $\mathcal L_{EC}(H)=\sum_{m,n}|\Delta_{m,n}(H)|^2$3 contiguous TiN segments, each length $\mathcal L_{EC}(H)=\sum_{m,n}|\Delta_{m,n}(H)|^2$4 and widths increasing from $\mathcal L_{EC}(H)=\sum_{m,n}|\Delta_{m,n}(H)|^2$5 to $\mathcal L_{EC}(H)=\sum_{m,n}|\Delta_{m,n}(H)|^2$6 in $\mathcal L_{EC}(H)=\sum_{m,n}|\Delta_{m,n}(H)|^2$7 steps, produces a graded temperature profile and near-linear phase programming. Under PWM, the uniform heater yields $\mathcal L_{EC}(H)=\sum_{m,n}|\Delta_{m,n}(H)|^2$8 distinct levels with $\mathcal L_{EC}(H)=\sum_{m,n}|\Delta_{m,n}(H)|^2$9 up to $$\Delta\phi \;=\;\frac{2\pi}{\lambda}\;\Delta n_{\rm eff}\;L,$$0 for $$\Delta\phi \;=\;\frac{2\pi}{\lambda}\;\Delta n_{\rm eff}\;L,$$1 steps, whereas the segmented heater yields $$\Delta\phi \;=\;\frac{2\pi}{\lambda}\;\Delta n_{\rm eff}\;L,$$2 levels with $$\Delta\phi \;=\;\frac{2\pi}{\lambda}\;\Delta n_{\rm eff}\;L,$$3; insertion loss varies from $$\Delta\phi \;=\;\frac{2\pi}{\lambda}\;\Delta n_{\rm eff}\;L,$$4 $$\Delta\phi \;=\;\frac{2\pi}{\lambda}\;\Delta n_{\rm eff}\;L,$$5 to $$\Delta\phi \;=\;\frac{2\pi}{\lambda}\;\Delta n_{\rm eff}\;L,$$6 $$\Delta\phi \;=\;\frac{2\pi}{\lambda}\;\Delta n_{\rm eff}\;L,$$7. Under PAM at $$\Delta\phi \;=\;\frac{2\pi}{\lambda}\;\Delta n_{\rm eff}\;L,$$8, the segmented heater again delivers $$\Delta\phi \;=\;\frac{2\pi}{\lambda}\;\Delta n_{\rm eff}\;L,$$9 well-spaced levels with $$\Delta n_{\rm eff} = \frac{\iint \bigl(\tfrac{dn}{dT}\bigr)\; \Delta T(x,y)\; |E(x,y)|^2 \, dx \, dy} {\iint |E(x,y)|^2 \, dx \, dy}.$$0 (Dwivedi et al., 21 Dec 2025).

Outside silicon PICs, analogous device-level “phase loss” figures appear in other platforms. A mechanically reconfigurable WR-15 gap-waveguide phase shifter provides a maximum phase shift of $$\Delta n_{\rm eff} = \frac{\iint \bigl(\tfrac{dn}{dT}\bigr)\; \Delta T(x,y)\; |E(x,y)|^2 \, dx \, dy} {\iint |E(x,y)|^2 \, dx \, dy}.$$1 with mean insertion loss $$\Delta n_{\rm eff} = \frac{\iint \bigl(\tfrac{dn}{dT}\bigr)\; \Delta T(x,y)\; |E(x,y)|^2 \, dx \, dy} {\iint |E(x,y)|^2 \, dx \, dy}.$$2 and maximum insertion loss $$\Delta n_{\rm eff} = \frac{\iint \bigl(\tfrac{dn}{dT}\bigr)\; \Delta T(x,y)\; |E(x,y)|^2 \, dx \, dy} {\iint |E(x,y)|^2 \, dx \, dy}.$$3 over $$\Delta n_{\rm eff} = \frac{\iint \bigl(\tfrac{dn}{dT}\bigr)\; \Delta T(x,y)\; |E(x,y)|^2 \, dx \, dy} {\iint |E(x,y)|^2 \, dx \, dy}.$$4 to $$\Delta n_{\rm eff} = \frac{\iint \bigl(\tfrac{dn}{dT}\bigr)\; \Delta T(x,y)\; |E(x,y)|^2 \, dx \, dy} {\iint |E(x,y)|^2 \, dx \, dy}.$$5; a warm $$\Delta n_{\rm eff} = \frac{\iint \bigl(\tfrac{dn}{dT}\bigr)\; \Delta T(x,y)\; |E(x,y)|^2 \, dx \, dy} {\iint |E(x,y)|^2 \, dx \, dy}.$$6 all-optical modulator demonstrates $$\Delta n_{\rm eff} = \frac{\iint \bigl(\tfrac{dn}{dT}\bigr)\; \Delta T(x,y)\; |E(x,y)|^2 \, dx \, dy} {\iint |E(x,y)|^2 \, dx \, dy}.$$7 with $$\Delta n_{\rm eff} = \frac{\iint \bigl(\tfrac{dn}{dT}\bigr)\; \Delta T(x,y)\; |E(x,y)|^2 \, dx \, dy} {\iint |E(x,y)|^2 \, dx \, dy}.$$8 transmission and $$\Delta n_{\rm eff} = \frac{\iint \bigl(\tfrac{dn}{dT}\bigr)\; \Delta T(x,y)\; |E(x,y)|^2 \, dx \, dy} {\iint |E(x,y)|^2 \, dx \, dy}.$$9 bandwidth (Palomares-Caballero et al., 2020, Ormerod et al., 2023). These results do not collapse to a single metric, but they all instantiate the same design question: how much loss is incurred per useful phase excursion.

3. Phase-dependent loss and non-Hermitian phase control

Phase-dependent loss is explicit in RIS channel models. For an $L(\phi)$0-element RIS assisting a SIMO uplink, the overall channel is

$L(\phi)$1

and the per-element amplitude loss is modeled as

$L(\phi)$2

Using the lossless-optimal RIS phases, the mean SNR can be written in closed form in terms of $L(\phi)$3, $L(\phi)$4, and a correlation term $L(\phi)$5. The coherent RIS gain is reduced by $L(\phi)$6, the diffuse RIS power by $L(\phi)$7, and the $L(\phi)$8 scaling is retained but with a reduced constant factor. Numerical results show perfect match between Monte-Carlo simulation and the closed-form $L(\phi)$9; the mean SNR can fall by $$T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}] \equiv L(\phi)$$0–$$T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}] \equiv L(\phi)$$1 relative to the ideal lossless case for typical semiconductor-derived parameters and $$T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}] \equiv L(\phi)$$2; and varying $$T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}] \equiv L(\phi)$$3 has zero effect on the averaged SNR because all phases are uniform in $$T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}] \equiv L(\phi)$$4 (Singh et al., 2022).

Silicon-photonic QKD treats the same phenomenon as a security problem. Plasma-dispersion modulators induce both $$T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}] \equiv L(\phi)$$5 and $$T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}] \equiv L(\phi)$$6, so the transmitted intensity becomes encoding dependent. The model

$$T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}] \equiv L(\phi)$$7

is paired with $$T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}] \equiv L(\phi)$$8. The security restoration strategy is to regard only state-independent single photons as “untagged” and secure, then use post-selection on the stronger polarization so that the preparation density matrix becomes maximally mixed. In the asymptotic regime, the secure key rate is bounded by

$$T(\phi)=\exp[-\Delta\alpha(\phi)L_{\rm mod}] \equiv L(\phi)$$9

The optimal post-selection probability in the low-background, low-$\mathrm{PDL}[{\rm dB}]=-10\log_{10}L$0 regime is

$\mathrm{PDL}[{\rm dB}]=-10\log_{10}L$1

For PDL up to $\mathrm{PDL}[{\rm dB}]=-10\log_{10}L$2, introducing post-selection improves the final key rate by $\mathrm{PDL}[{\rm dB}]=-10\log_{10}L$3 at $\mathrm{PDL}[{\rm dB}]=-10\log_{10}L$4, $\mathrm{PDL}[{\rm dB}]=-10\log_{10}L$5 at $\mathrm{PDL}[{\rm dB}]=-10\log_{10}L$6, and $\mathrm{PDL}[{\rm dB}]=-10\log_{10}L$7 at $\mathrm{PDL}[{\rm dB}]=-10\log_{10}L$8, and allows any positive rate even at $\mathrm{PDL}[{\rm dB}]=-10\log_{10}L$9 (Li et al., 2018).

A conceptually different development reverses the role of loss and treats it as the phase-control mechanism itself. A non-Hermitian synthetic phase shifter uses two independently controlled loss-modulation stages with parameters $L(\phi)$00 and a three-port waveguide-array scattering matrix

$L(\phi)$01

with transfer function

$L(\phi)$02

Closed contours in the $L(\phi)$03 plane can be chosen so that $L(\phi)$04 while the phase winds by $L(\phi)$05: $L(\phi)$06 Measured constant-$L(\phi)$07 contours from $L(\phi)$08 to $L(\phi)$09 encircle isolated zeros in the $L(\phi)$10–$L(\phi)$11 plane; the measured phase spans the full $L(\phi)$12 around each ellipse; and the measured phase jitter is $L(\phi)$13 (Zelaya et al., 27 Apr 2026). This suggests a precise distinction within the literature: PDL is usually treated as an impairment, whereas loss-induced synthetic phase control treats non-unitarity as the resource.

4. Quantum metrology: loss-limited phase information

In quantum interferometry, phase and loss are jointly estimable but not jointly saturable in the single-measurement sense. For a fixed-photon-number probe

$L(\phi)$14

with phase $L(\phi)$15 and transmissivity $L(\phi)$16 applied in one arm, the QFI matrix for $L(\phi)$17 is diagonal,

$L(\phi)$18

with

$L(\phi)$19

However, the symmetric logarithmic derivatives do not commute on average: $L(\phi)$20 Accordingly, no single measurement can saturate both $L(\phi)$21 and $L(\phi)$22 simultaneously. If the eigenbasis of $L(\phi)$23 is used, phase precision reaches $L(\phi)$24 but the classical Fisher information for loss becomes

$L(\phi)$25

This is the explicit trade-off between Hamiltonian and dissipative parameter estimation in the same interferometer (Crowley et al., 2012).

A separate global-estimation result shows that loss removes Heisenberg scaling when no prior phase knowledge is available. For an arbitrary pure $L(\phi)$26-photon state in a lossy two-mode interferometer, the minimal average cost obeys

$L(\phi)$27

and asymptotically

$L(\phi)$28

The corresponding statement is that nonzero loss confines the quantum enhancement to at most a constant factor over classical strategies, rather than restoring $L(\phi)$29 scaling (Kolodynski et al., 2010).

Loss also dictates optimal operating points in classical-light interferometry. In a lossy Mach–Zehnder interferometer with coherent-state input and internal loss $L(\phi)$30 in one arm, the generalized bound called the standard interferometric limit is

$L(\phi)$31

For difference-intensity detection, $L(\phi)$32 and $L(\phi)$33 achieve the SIL; at $L(\phi)$34, optimizing $L(\phi)$35 produces a $L(\phi)$36 sensitivity improvement, equivalent to a $L(\phi)$37 sensitivity improvement in single-intensity detection (Huang et al., 2023).

For SU(1,1) interferometers, parity detection provides slightly better optimal phase sensitivity in the absence of loss but becomes more fragile under internal photon loss. The phase sensitivity is

$L(\phi)$38

and the loss threshold for beating the shot-noise limit is numerically about $L(\phi)$39 for vacuum input and $L(\phi)$40, about $L(\phi)$41 for one-coherent input with $L(\phi)$42, and about $L(\phi)$43 for coherent $L(\phi)$44 squeezed-vacuum input with $L(\phi)$45 and $L(\phi)$46 (Li et al., 2018).

5. Fundamental phase–loss bounds and material limits

The most general optical statement is that phase modulation in a linear dielectric is constrained by the same complex material response that produces loss. If a section of length $L(\phi)$47 has refractive-index change $L(\phi)$48, then

$L(\phi)$49

and

$L(\phi)$50

A material figure of merit

$L(\phi)$51

leads to the general phase–loss relation

$L(\phi)$52

and the corresponding lower bound on insertion loss

$L(\phi)$53

The same framework states that filtering, resonance and critical coupling could be of advantage in approaching the limit (Zanotto et al., 2018).

Material choice then determines how closely a practical phase shifter can approach that bound. A recent topological photonic-crystal implementation replaces GST with Sb$L(\phi)$54Se$L(\phi)$55, for which spectroscopic ellipsometry at $L(\phi)$56 gives $L(\phi)$57, $L(\phi)$58 and $L(\phi)$59, $L(\phi)$60. The index contrast $L(\phi)$61 enables topological inversion while $L(\phi)$62 keeps added absorption negligible. With $L(\phi)$63, the bulk absorption coefficient

$L(\phi)$64

is $L(\phi)$65, corresponding to an absorption length $L(\phi)$66. Simulations predict $L(\phi)$67 in both amorphous and crystalline states, and experimentally the total $L(\phi)$68 stays on the order of $L(\phi)$69 in both states, nearly an order of magnitude higher in the crystalline phase than for GST-based reference data (Uemura et al., 29 Dec 2025). This suggests that the practical meaning of “phase loss” is often material-limited before it is geometry-limited.

A related loss–phase tradeoff appears in squeezed-state readout with phase-sensitive amplification. For an input squeezed state of parameter $L(\phi)$70, OPA gain $L(\phi)$71, detection efficiency $L(\phi)$72, and Gaussian phase jitter variance $L(\phi)$73, the measured variance is

$L(\phi)$74

The effective measurable squeezing is $L(\phi)$75, and the effective detection efficiency is

$L(\phi)$76

In that analysis, phase noise acts as a “phase-loss” channel because it mixes antisqueezed noise into the measured quadrature (Kwan et al., 2024).

6. Phase loss as an optimization objective

In reconstruction and enhancement, “phase loss” often names an objective rather than a physical attenuation. A recent example is the explicit consistency-preserving loss for STFT phase reconstruction and speech enhancement. For a complex spectrogram $L(\phi)$77, consistency is enforced through

$L(\phi)$78

and the loss is

$L(\phi)$79

For a constructed spectrogram $L(\phi)$80, the objective becomes

$L(\phi)$81

The key point is that the loss never references the “true” phase; it only requires that $L(\phi)$82 and $L(\phi)$83 form a consistent STFT of some real signal. The loss is invariant to global phase shifts $L(\phi)$84, which explains why it tolerates sign or phase-bias indeterminacy. On VoiceBank-DEMAND phase reconstruction, EC achieves PESQ $L(\phi)$85, ESTOI $L(\phi)$86, CSIG $L(\phi)$87, and COVL $L(\phi)$88; on speech enhancement, EC with MetricGAN reaches PESQ $L(\phi)$89 on VB-DMD and $L(\phi)$90 on WSJ0-CHiME3, comparing favourably to cosine-distance and anti-wrapping losses (Ku et al., 2024).

A more classical inverse-problem usage is the nonsmooth amplitude least-squares loss for phase retrieval,

$L(\phi)$91

and its overparametrized generalization

$L(\phi)$92

The PhaseCut-style smooth reformulation introduces

$L(\phi)$93

The deterministic landscape results state that, in the noiseless case, any second-order critical point achieves exactly zero loss and therefore recovers $L(\phi)$94; under isotropic sub-Gaussian measurements, if $L(\phi)$95 and $L(\phi)$96 for absolute constants, every second-order critical point satisfies statistically optimal recovery guarantees with only a constant amount of overparametrization (McRae, 24 Nov 2025). In this usage, “phase loss” is not attenuation but the variational landscape through which phase information is recovered.

The coexistence of these meanings is sometimes a source of confusion. In photonics and wireless communications, phase loss is usually a non-unitary transfer effect; in quantum metrology, it is the precision penalty induced by a dissipative channel; in learning-based reconstruction, it is a training objective over phase or phase consistency. The literature nevertheless converges on a common lesson: phase cannot be treated independently of the mechanism that realizes, perturbs, or estimates it.