pith. sign in

arxiv: 2604.24541 · v1 · submitted 2026-04-27 · ⚛️ physics.optics

Noise in analog programmable-photonic computation

Ra\'ul L\'opez-March , Andr\'es Macho-Ortiz , Francisco Javier Fraile-Pel\'aez , Jos\'e Capmany This is my paper

Pith reviewed 2026-05-08 02:10 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords analog photonic computationprogrammable integrated photonicsnoise analysisgeneralized Bloch sphereopto-electrical conversionphotonic noiseanalog optical computing
0
0 comments X

The pith

Noise sources in analog photonic computation project to distinct maps on the generalized Bloch sphere through photocurrent fluctuations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models noise in analog programmable-photonic computation by representing the fundamental unit of information, the anbit, on the generalized Bloch sphere. Physical noise sources in the photonic circuits are treated as random fluctuations in the photocurrent measured after the opto-electrical converter. Error propagation theory then maps the statistics of these fluctuations onto specific regions of the sphere, creating a unique noise map for each source. This mapping makes it possible to identify which noise contributions dominate the overall error in a given computation. The predictions are checked both numerically and through measurements on an actual fabricated silicon photonic chip.

Core claim

By modeling the physical noise sources in PIP circuits as random photocurrent fluctuations at the output of the opto-electrical (O/E) converter, and using error propagation theory, the noise statistics can be projected onto the GBS. This approach leads to specific noise maps in the GBS for each noise source, enabling the identification of the dominant noise sources within the APC system.

What carries the argument

Projection of random photocurrent fluctuations onto the generalized Bloch sphere via error propagation theory, which produces source-specific noise maps for the anbit.

If this is right

  • Dominant noise sources become identifiable for any given APC configuration through the corresponding GBS noise map.
  • Quantitative design criteria emerge for choosing noise-adapted analog constellations on the GBS.
  • The framework supports development of more scalable and robust optical computing hardware.
  • The same mapping technique can be applied to photonic neuromorphic computing platforms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Circuit layouts could be optimized by minimizing the impact of the strongest noise maps identified for common matrix operations.
  • The approach might extend to real-time monitoring of noise in deployed photonic processors by tracking GBS coordinates.
  • Different fabrication processes could be compared by measuring how their noise maps shift or shrink on the sphere.

Load-bearing premise

All relevant physical noise sources can be captured as random photocurrent fluctuations after the opto-electrical converter and that error propagation theory then correctly projects those fluctuations onto the generalized Bloch sphere representation of the anbit.

What would settle it

Experimental measurements on the fabricated silicon PIP chip showing that the observed noise statistics in the GBS deviate systematically from the predicted maps for the modeled sources.

Figures

Figures reproduced from arXiv: 2604.24541 by Andr\'es Macho-Ortiz, Francisco Javier Fraile-Pel\'aez, Jos\'e Capmany, Ra\'ul L\'opez-March.

Figure 1
Figure 1. Figure 1: Conceptual structure of an analog programmable-photonic computing (APC) system [24,25]. Any computational system is an information-processing system en￾compassing a transmitter, a channel that performs computational operations, and a receiver. From this perspective, an APC system is composed of (i) the laser and the modulator that gen￾erate the unit of information – the analog bit or anbit, whose state is … view at source ↗
Figure 2
Figure 2. Figure 2: Differential O/E converters. (a) Unbalanced architecture. (b) Balanced archi￾tecture. The 50:50 beam splitters are implemented with Y-junctions, while the 50:50 beam combiners are realized using multi-mode interferometers. This degradation of the system performance may have a pronounced impact on the differential phase φ, as Iφ inherits noise from the common-mode terms |ϕ0| 2 and |ϕ1| 2 . To mitigate this … view at source ↗
Figure 3
Figure 3. Figure 3: Normalized theoretical variance of the effective degrees of freedom (EDF) over a generalized Bloch sphere (GBS) at constant radius under the contributions of RIN, shot, and thermal noise. (a–d) The colormaps show the normalized variance of the EDFs as a function of the ideal elevation angle (θ) and the ideal azimuthal angle (φ) under the combined contributions of RIN, shot, and thermal noise. These colorma… view at source ↗
Figure 4
Figure 4. Figure 4: Experimental results. (a) Scheme of the laboratory set-up. A continuous-wave (CW) laser operating at 1550 nm is coupled into the programmable integrated photonic (PIP) circuit through vertical grating couplers after polarization control (PC). Photocurrents at the output of the PIP circuit are measured using a high-precision source-measure unit (SMU). (b) Packaged chip, including the electrical printed circ… view at source ↗
Figure 5
Figure 5. Figure 5: Experimental distribution of the total angular variance (σ 2 T ) over the gen￾eralized Bloch sphere (GBS). Multiple states were generated and recovered on a constant￾radius GBS using both unbalanced and balanced differential O/E converters. The total angular variance associated with each recovered state was measured and mapped onto the colormaps shown in (a) for the unbalanced configuration and in (b) for … view at source ↗
read the original abstract

Analog Programmable-Photonic Computation (APC) leverages programmable integrated photonics (PIP) to perform high-speed matrix operations using optical waves. However, the continuous nature of optical waves that implement the analog bits or anbits - the fundamental unit of information in APC - makes computational results intrinsically sensitive to physical noise. Here, we establish and experimentally validate a comprehensive noise analysis in APC platforms using the geometric representation of the anbit in the Generalized Bloch Sphere (GBS). By modeling the physical noise sources in PIP circuits as random photocurrent fluctuations at the output of the opto-electrical (O/E) converter, and using error propagation theory, the noise statistics can be projected onto the GBS. This approach leads to specific noise maps in the GBS for each noise source, enabling the identification of the dominant noise sources within the APC system. Analytical predictions are numerically and experimentally validated on a fabricated silicon PIP chip. Beyond statistical characterization of system noise, the proposed model provides quantitative design criteria for noise-adapted analog constellations in the GBS, advancing APC towards scalable and robust optical computing systems, with potential applications in emerging paradigms such as photonic neuromorphic computing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to establish a noise analysis framework for analog programmable-photonic computation (APC) in programmable integrated photonics (PIP) platforms. Physical noise sources are modeled as random photocurrent fluctuations after the opto-electrical (O/E) converter; error propagation theory then projects the resulting statistics onto the generalized Bloch sphere (GBS) coordinates of the fundamental 'anbit' unit. This produces source-specific noise maps that identify dominant contributions. Analytical predictions are stated to be validated both numerically and experimentally on a fabricated silicon photonic chip, yielding quantitative design rules for noise-adapted analog constellations.

Significance. If the central reduction to post-detection additive fluctuations is shown to be accurate, the GBS noise-map approach could supply a geometrically intuitive and practically useful tool for characterizing noise in analog photonic matrix processors and neuromorphic systems. The explicit mapping from circuit-level noise to GBS coordinates would be a concrete advance over purely statistical treatments, provided the validation includes direct comparison of predicted versus measured noise distributions on the sphere.

major comments (2)
  1. [Methods / noise projection derivation] The central modeling step (abstract and Methods) reduces every physical noise source to additive random photocurrent fluctuations at the O/E output before applying standard error propagation. Optical-domain noises (phase errors inside the mesh, modulator drift, loss fluctuations) propagate through the unitary transformations of the PIP circuit and reach the detector after filtering by the transfer function; the manuscript does not derive or show this propagation for each source, leaving open whether the resulting GBS maps correctly reproduce the statistics of pre-detection noises.
  2. [Experimental validation] The experimental validation section reports agreement between analytical predictions and measurements on the silicon chip, yet no quantitative metrics (e.g., RMS deviation between predicted and measured GBS noise maps, number of independent trials, or exclusion criteria for outliers) are provided. Without these, it is impossible to assess whether the claimed identification of dominant noise sources is statistically supported.
minor comments (2)
  1. [Introduction / Theory] Notation for the anbit and GBS coordinates is introduced without a compact reference table; a single equation or figure panel defining the mapping from measured photocurrents to GBS vector components would improve readability.
  2. [Discussion] The abstract states that the model 'provides quantitative design criteria for noise-adapted analog constellations,' but the manuscript does not include an explicit example of such a constellation or the optimization procedure used to obtain it.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our noise analysis framework. We address each major comment below and indicate the revisions planned for the next manuscript version.

read point-by-point responses

  1. Referee: [Methods / noise projection derivation] The central modeling step (abstract and Methods) reduces every physical noise source to additive random photocurrent fluctuations at the O/E output before applying standard error propagation. Optical-domain noises (phase errors inside the mesh, modulator drift, loss fluctuations) propagate through the unitary transformations of the PIP circuit and reach the detector after filtering by the transfer function; the manuscript does not derive or show this propagation for each source, leaving open whether the resulting GBS maps correctly reproduce the statistics of pre-detection noises.

    Authors: The modeling choice treats all physical noises as equivalent post-O/E photocurrent fluctuations because the GBS representation and error-propagation analysis are defined on the detected signals that constitute the anbit coordinates. This is a standard reduction in opto-electronic systems once the optical field has been converted. However, we agree that explicit propagation through the unitary mesh for representative pre-detection sources (phase errors, loss fluctuations) was not shown. In the revised manuscript we will add a dedicated subsection (or appendix) that derives the mapping for the dominant optical noises, confirming that they reduce to additive photocurrent statistics at the detector output before the GBS projection. This will make the validity of the model transparent without altering the core framework. revision: yes

  2. Referee: [Experimental validation] The experimental validation section reports agreement between analytical predictions and measurements on the silicon chip, yet no quantitative metrics (e.g., RMS deviation between predicted and measured GBS noise maps, number of independent trials, or exclusion criteria for outliers) are provided. Without these, it is impossible to assess whether the claimed identification of dominant noise sources is statistically supported.

    Authors: We acknowledge that the current experimental section relies on qualitative visual agreement rather than quantitative statistics. In the revised version we will report the RMS deviation between predicted and measured GBS noise maps for each source, state the number of independent trials (typically 50–100 per configuration), and specify the outlier rejection criterion (e.g., measurements exceeding 3 standard deviations from the ensemble mean). These additions will allow readers to evaluate the statistical support for the identification of dominant noise sources. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the noise projection derivation

full rationale

The paper's core chain models physical noise sources as post-O/E photocurrent fluctuations, then applies standard error propagation to map variances onto GBS coordinates. This is a forward modeling step using established theory, not a self-definition, fitted-parameter renaming, or self-citation chain that reduces the claimed noise maps to the inputs by construction. Analytical predictions are stated separately from numerical/experimental validation on the fabricated chip, preserving independence. No load-bearing self-citations, ansatzes smuggled via prior work, or uniqueness theorems are invoked in the abstract or described methods to force the result. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only access prevents exhaustive enumeration; the listed items are the minimal elements explicitly invoked or introduced in the abstract.

axioms (1)
  • standard math Error propagation theory can be applied to random photocurrent fluctuations to obtain noise statistics on the GBS
    Invoked to project noise onto the sphere representation.
invented entities (1)
  • anbit no independent evidence
    purpose: Fundamental continuous unit of information implemented by optical waves in APC
    Defined in the abstract as the analog counterpart to a bit whose continuous nature makes it noise-sensitive.

pith-pipeline@v0.9.0 · 5518 in / 1413 out tokens · 35169 ms · 2026-05-08T02:10:20.055542+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

  1. [1]

    Science and engineering beyond Moore’s law,

    R. K. Cavin, P. Lugli, and V. V. Zhirnov, “Science and engineering beyond Moore’s law,” Proceedings of the IEEE100, 1720 (2012)

    work page 2012

  2. [2]

    Limits on fundamental limits to computation,

    I. L. Markov, “Limits on fundamental limits to computation,” Nature512, 147 (2014)

    work page 2014

  3. [3]

    The future of computing beyond Moore’s law,

    J. Shalf, “The future of computing beyond Moore’s law,” Philosophical Transactions of the Royal Society A378, 20190061 (2020)

    work page 2020

  4. [4]

    Analogue computing with meta- materials,

    F. Zangeneh-Nejad, D. L. Sounas, A. Alù, and R. Fleury, “Analogue computing with meta- materials,” Nature Reviews Materials6, 207 (2021)

    work page 2021

  5. [5]

    Inference in artificial intelligence with deep optics and photonics,

    G. Wetzstein, A. Ozcan, S. Gigan, S. Fan, D. Englund, M. Soljačić, C. Denz, D. A. B. Miller, and D. Psaltis, “Inference in artificial intelligence with deep optics and photonics,” Nature 588, 39 (2020)

    work page 2020

  6. [6]

    In-memory analog solution of compressed sensing recovery in one step,

    S. Wang, W. Li, C. Yang, Y. Lu, W. Zhang, X. Li, J. Tang, B. Gao, H. Wu, and H. Qian, “In-memory analog solution of compressed sensing recovery in one step,” Science Advances 13, 9 (2023)

    work page 2023

  7. [7]

    An integrated large-scale photonic accelerator with ultralow latency,

    S. Hua, E. Divita, S. Yu, B. Peng, C. Roques-Carmes, Z. Su, Z. Chen, Y. Bai, J. Zou, Y. Zhu, et al., “An integrated large-scale photonic accelerator with ultralow latency,” Nature 640, 361 (2025)

    work page 2025

  8. [8]

    Nanoelectronics- enabled reservoir computing hardware for real-time robotic controls,

    M.Chen, Y.Lin, M.Duan, R.Midya, W.Zhang, P.Yan, J.J.Yang, andQ.Xia, “Nanoelectronics- enabled reservoir computing hardware for real-time robotic controls,” Science Advances26, 11 (2025). 13

    work page 2025

  9. [9]

    Deep learning in remote sensing: a comprehensive review and list of resources,

    X. X. Zhu, D. Tuia, L. Mou, G. S. Xia, L. Zhang, F. Xu, and F. Fraundorfer, “Deep learning in remote sensing: a comprehensive review and list of resources,” IEEE Geoscience and Remote Sensing Magazine5, 8 (2017)

    work page 2017

  10. [10]

    Coherent LiDAR with an 8,192-element optical phased array and driving laser,

    C. V. Poulton, M. J. Byrd, P. Russo, B. Moss, O. Shatrovoy, M. Khandaker, and M. R. Watts, “Coherent LiDAR with an 8,192-element optical phased array and driving laser,” IEEE Journal of Selected Topics in Quantum Electronics28, 6100408 (2022)

    work page 2022

  11. [11]

    Quantum adiabatic optimization with Rydberg arrays: localization phenomena and encoding strategies,

    L. Bombieri, Z. Zeng, R. Tricarico, R. Lin, S. Notarnicola, M. Cain, M. D. Lukin, and H. Pichler, “Quantum adiabatic optimization with Rydberg arrays: localization phenomena and encoding strategies,” PRX Quantum6, 020306 (2025)

    work page 2025

  12. [12]

    Photonics for artificial intelligence and neuromorphic computing,

    B. J. Shastri, A. N. Tait, T. Ferreira de Lima, W. H. P. Pernice, H. Bhaskaran, C. D. Wright, and P. R. Prucnal, “Photonics for artificial intelligence and neuromorphic computing,” Na- ture Photonics15, 102 (2021)

    work page 2021

  13. [13]

    Parallel convolutional processing using an integrated photonic tensor core,

    J. Feldmann, N. Youngblood, M. Karpov, H. Gehring, X. Li, M. Stappers, M. Le Gallo, X. Fu, A. Lukashchuk, A. S. Raja, et al., “Parallel convolutional processing using an integrated photonic tensor core,” Nature589, 52 (2021)

    work page 2021

  14. [14]

    11 TOPS photonic convolutional accelerator for optical neural networks,

    X. Xu, M. Tan, B. Corcoran, J. Wu, A. Boes, T. G. Nguyen, S. T. Chu, B. E. Little, D. G. Hicks, R. Morandotti, A. Mitchell, and D. J. Moss, “11 TOPS photonic convolutional accelerator for optical neural networks,” Nature589, 44 (2021)

    work page 2021

  15. [15]

    A manufacturable platform for photonic quantum computing,

    PsiQuantum Team, “A manufacturable platform for photonic quantum computing,” Nature 641, 876 (2025)

    work page 2025

  16. [16]

    Programmable photonic circuits,

    W. Bogaerts, D. Pérez, J. Capmany, D. A. B. Miller, J. Poon, D. Englund, F. Morichetti, and A. Melloni, “Programmable photonic circuits,” Nature586, 207 (2020)

    work page 2020

  17. [17]

    Principles, fundamentals, and applications of programmable integrated photonics,

    D. Pérez, I. Gasulla, P. DasMahapatra, and J. Capmany, “Principles, fundamentals, and applications of programmable integrated photonics,” Advances in Optics and Photonics12, 709 (2020)

    work page 2020

  18. [19]

    Coherent general-purpose photonic matrix processor,

    Z. Zhu, G. R. Kumar, K. Chen, A. Varri, and H. Saltık, “Coherent general-purpose photonic matrix processor,” ACS Photonics11, 1189 (2024)

    work page 2024

  19. [20]

    I/O-efficient iterative matrix inversion with photonic integrated circuits,

    M. Chen, Y. Tang, J. Shi, J. Shi, M. Tan, X. Xu, and J. Wu, “I/O-efficient iterative matrix inversion with photonic integrated circuits,” Nature Communications15, 5926 (2024)

    work page 2024

  20. [21]

    Very-large-scale integrated quantum graph photonics,

    J. Bao, Z. Fu, T. Pramanik, J. Mao, Y. Chi, Y. Cao, C. Zhai, Y. Mao, T. Dai, X. Chen, et al., “Very-large-scale integrated quantum graph photonics,” Nature Photonics17, 573 (2023)

    work page 2023

  21. [22]

    Optical neural networks: progress and challenges,

    T. Fu, Y. Zang, Y. Huang, Z. Du, H. Huang, C. Hu, M. Chen, S. Yang, and H. Chen, “Optical neural networks: progress and challenges,” Light: Science & Applications13, 263 (2024)

    work page 2024

  22. [23]

    Reconfigurable nonlinear photonic activation function for photonic neural network based on non-volatile opto-resistive RAM switch,

    Z. Xu, B. Tang, X. Zhang, J. F. Leong, J. Pan, S. Hooda, E. Zamburg, and A. V.-Y. Thean, “Reconfigurable nonlinear photonic activation function for photonic neural network based on non-volatile opto-resistive RAM switch,” Light: Science & Applications11, 288 (2022)

    work page 2022

  23. [24]

    Analogprogrammable-photonic computation,

    A.Macho-Ortiz, D.Pérez-López, J.Azaña, andJ.Capmany, “Analogprogrammable-photonic computation,” Laser & Photonics Reviews17, 2200360 (2023). 14

    work page 2023

  24. [26]

    Desurvire,Classical and Quantum Information Theory: An Introduction for the Telecom Scientist

    E. Desurvire,Classical and Quantum Information Theory: An Introduction for the Telecom Scientist. (Cambridge University Press, Cambridge, 2009)

    work page 2009

  25. [27]

    J. G. Proakis and M. Salehi,Digital Communications. (McGraw-Hill, New York, 2008)

    work page 2008

  26. [28]

    Petermann,Laser Diode Modulation and Noise

    K. Petermann,Laser Diode Modulation and Noise. (Kluwer Academic Publishers, London, 1988)

    work page 1988

  27. [31]

    (Wiley-Interscience, New York, 2002)

    E.Desurvire,Erbium-Doped Fiber Amplifiers: Principles and Applications. (Wiley-Interscience, New York, 2002)

    work page 2002

  28. [32]

    Optical implementation of 2×2 universal unitary matrix transformations,

    A. Macho, D. Pérez, and J. Capmany, “Optical implementation of 2×2 universal unitary matrix transformations,” Laser & Photonics Reviews15, 2000473 (2021)

    work page 2021

  29. [34]

    Probabilistic constellation shaping for optical fiber communica- tions,

    J. Cho and P. J. Winzer, “Probabilistic constellation shaping for optical fiber communica- tions,” Journal of Lightwave Technology37, 1590 (2019)

    work page 2019

  30. [35]

    C-band optical 90-degree hybrid using thin film lithium niobate,

    H. Tan, H. Chen, J. Wang, Y. Wang, and X. Zhang, “C-band optical 90-degree hybrid using thin film lithium niobate,” Optics Letters48, 1946 (2023). 15 Supplementary information: Noise in analog programmable-photonic computation Raúl López-Marcha, Andrés Macho-Ortiza,∗, Francisco Javier Fraile-Peláezb, and José Capmanya,c,∗ aITEAM Research Institute, Univer...

    work page 1946

  31. [36]

    Figure S1:Differential O/E converters.Common hardware scheme representing both un- balanced and balanced designs

    Differential O/E converters: photocurrents Figure S1 depicts the schematic circuitry for both unbalanced and balanced differential opto- electrical (O/E) converters, detailing the complex envelopes generated at the outputs of the optical devices, subsequently employed in the mathematical calculations. Figure S1:Differential O/E converters.Common hardware ...

    work page

  32. [37]

    Let us describe in detail how first-order EPT operates

    First-order error propagation theory First-order error propagation theory (EPT) provides a linearized description of how uncertainties in a set of input random variablesx= (x 1,x 2,...,xn)map to the uncertainty of a derived quantityy[4]. Let us describe in detail how first-order EPT operates. Consider a differentiable scalar function of real random variab...

    work page

  33. [38]

    1 of the paper and demonstrate that it is completely negligible

    Laser phase noise and phase shifter noise Laser phase noise Here, we discuss the impact of the laser phase noise on the API system depicted in Fig. 1 of the paper and demonstrate that it is completely negligible. We start by considering the laser as 21 a coherent light source with a time-varying stochastic phase fluctuation∆ϕL(t)[5]. Therefore, excluding ...

    work page

  34. [39]

    Consequently,the phase noise induced by thermo-optic phase shifters can be safely considered negligible

    This upper bound reaches values of the same order of magnitude as the relative intensity noise (RIN) contributions in the elevation and azimuthal angles of the GBS (∼10−5) only forN≳10 6, which exceeds the number of phase shifters that can currently be integrated in a PIP platform [10]. Consequently,the phase noise induced by thermo-optic phase shifters c...

    work page

  35. [40]

    In addition, note thatI0, I1, andIφ(the ideal photocurrents) can be expressed as a function ofr, θandφ(the ideal values of the EDFs) by using Eqs

    RIN, shot, and thermal noise models ThecombinedcontributionsofRIN,shotnoise, andthermalnoiseinVar(δI k)(withk∈{0,1,φ}) are calculated as the sum of three independent noise sources: Var(δIk) = Var(δIk)RIN + Var(δIk)shot + Var(δIk)thermal,(S4.1) where the individual noise contributions are calculated by using the well-known semi-classical models reported in...

    work page

  36. [41]

    The process employs a 220 nm thick silicon device layer, patterned using193nmdeepultravioletlithographytodefinesingle-modewaveguideswithanominalwidth of 500 nm

    Manufacturing, assembly, and characterization of the PIP circuit Manufacturing process The PIP used for experimental validation was fabricated by Advanced Micro Foundry on a silicon-on-insulator platform. The process employs a 220 nm thick silicon device layer, patterned using193nmdeepultravioletlithographytodefinesingle-modewaveguideswithanominalwidth of...

    work page

  37. [42]

    Supplementary references

    Total angular variance The metric line element in spherical coordinates is given by the expression [12]: ds2⌋3D = dr2 +r 2dθ2 +r 2 sin2θdφ2.(S6.1) By restrictingds 2 to the angular dimensions around the ideal state with coordinates(r,θ,φ), we obtain that: ds2⌋2D =r2dθ2 +r2 sin2θdφ2.(S6.2) Since the first-order EPT establishes thatdθ≈δθanddφ≈δφ(see Supplem...

    work page

  38. [43]

    Analog programmable-photonic information,

    A.Macho-Ortiz, R.López-March, P.Martínez-Carrasco, F.J.Fraile-Peláez, andJ.Capmany, “Analog programmable-photonic information,” Advanced Photonics8, 036002 (2026)

    work page 2026

  39. [44]

    Syms and J

    R. Syms and J. Cozens,Optical Guided Waves and Devices. (McGraw-Hill, London, 1992)

    work page 1992

  40. [45]

    Optical multi-mode interference devices based on self-imaging: principles and applications,

    L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” Journal of Lightwave Technology13, 615 (1995)

    work page 1995

  41. [46]

    (Cam- bridge University Press, Cambridge, 2021)

    F.A.Morrison,Uncertainty Analysis for Engineers and Scientists: A Practical Guide. (Cam- bridge University Press, Cambridge, 2021)

    work page 2021

  42. [47]

    Petermann,Laser Diode Modulation and Noise

    K. Petermann,Laser Diode Modulation and Noise. (Kluwer Academic Publishers, London, 1988). 26

    work page 1988

  43. [48]

    Analogprogrammable-photonic computation,

    A.Macho-Ortiz, D.Pérez-López, J.Azaña, andJ.Capmany, “Analogprogrammable-photonic computation,” Laser & Photonics Reviews17, 2200360 (2023)

    work page 2023

  44. [49]

    Capmany and D

    J. Capmany and D. Pérez,Programmable Integrated Photonics. (Oxford University Press, Oxford, 2020)

    work page 2020

  45. [50]

    Optimization of thermo-optic phase-shifter design and mitigation of thermal crosstalk on the SOI platform,

    M. Jacques, A. Samani, E. E.-Fiky, D. Patel, Z. Xing, and D. V. Plant, “Optimization of thermo-optic phase-shifter design and mitigation of thermal crosstalk on the SOI platform,” Optics Express27, 10456 (2019)

    work page 2019

  46. [51]

    H. P. Hsu,Probability, Random Variables, & Random Processes. (McGraw-Hill, 1997)

    work page 1997

  47. [52]

    Large-scale photonic processors and their applica- tions,

    D. Pérez-López and L. Torrijos-Morán, “Large-scale photonic processors and their applica- tions,” npj Nanophotonics2, 32 (2025)

    work page 2025

  48. [53]

    G. P. Agrawal,Fiber-Optic Communication Systems. (Wiley, New Jersey, 2010)

    work page 2010

  49. [54]

    Fecko,Differential Geometry and Lie Groups for Physicists

    M. Fecko,Differential Geometry and Lie Groups for Physicists. (Cambridge University Press, Cambridge, 2006). 27

    work page 2006