pith. machine review for the scientific record.
sign in

arxiv: 2605.03910 · v1 · submitted 2026-05-05 · ⚛️ physics.optics · quant-ph

Inverse-designed release-free optomechanical crystal with high photon-phonon coupling

David Hambraeus , Paul Burger , Johan Kolvik , Philippe Tassin , Rapha\"el Van Laer This is my paper

Pith reviewed 2026-05-07 14:13 UTC · model grok-4.3

classification ⚛️ physics.optics quant-ph
keywords optomechanicsinverse designsilicon photonicsoptomechanical crystalsphoton-phonon couplingrelease-free devicesvacuum coupling rate
0
0 comments X

The pith

A release-free silicon optomechanical crystal reaches a vacuum coupling rate of 800 kHz through inverse design.

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

The paper establishes that a release-free optomechanical crystal can match the strong photon-phonon coupling of suspended devices while remaining anchored to the substrate for better heat dissipation. This matters for applications that need both fast light-sound interactions and low noise from laser heating in quantum technologies and signal processing. The authors reach this performance by combining guided intuition with a new multiphysics inverse-design algorithm that simultaneously tunes optical and mechanical resonances. The fabricated device shows a vacuum optomechanical coupling rate of roughly 800 kHz and an optomechanical scattering rate of 1.1 kHz, nearly double prior release-free results. If the approach holds, it removes the previous trade-off between coupling strength and thermal robustness.

Core claim

The authors introduce a multiphysics inverse-design algorithm for resonant optomechanical structures and apply it to produce a release-free silicon optomechanical crystal. The resulting device exhibits a vacuum optomechanical coupling rate g_OM/(2π) ≈ 800 kHz, comparable to suspended state-of-the-art devices, together with an optomechanical scattering rate Γ_OM/(2π) = 1.1 kHz that is nearly twice as large as earlier release-free implementations. The design improves thermal anchoring without sacrificing coupling strength, thereby strengthening release-free optomechanical crystals as a platform for fast, low-noise classical and quantum optomechanics.

What carries the argument

A multiphysics inverse-design algorithm that co-optimizes optical and mechanical resonances and eigenmodes.

If this is right

  • Stronger photon-phonon interactions become possible in thermally robust, release-free structures.
  • The inverse-design framework extends to broader co-optimization of optical and mechanical properties.
  • Applications in classical signal processing and quantum technologies gain reduced parasitic laser heating.
  • Performance comparable to suspended devices is achieved without the fabrication challenges of full suspension.

Where Pith is reading between the lines

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

  • The same algorithm could support integrated circuits where multiple optomechanical elements share a common substrate and thermal environment.
  • Testing the device at single-photon or quantum-limited regimes would reveal whether the improved thermal anchoring translates into lower added noise.
  • Extending the method to other high-index materials could further increase coupling rates while retaining the release-free advantage.

Load-bearing premise

The fabricated device matches the designed geometry closely enough that its measured coupling rate equals the simulated vacuum rate without major fabrication deviations or measurement artifacts.

What would settle it

Fabricating the device and measuring a vacuum optomechanical coupling rate significantly below 800 kHz would show that the design did not translate accurately into the physical structure.

Figures

Figures reproduced from arXiv: 2605.03910 by David Hambraeus, Johan Kolvik, Paul Burger, Philippe Tassin, Rapha\"el Van Laer.

Figure 1
Figure 1. Figure 1: Release-free optomechanical crystal cavity with improved photon-phonon overlap. (a) The release-free OMC with X-HOPE design pattern showing the transverse component of the normalized electric field Ex of the optical mode (top) and the longitudinal normalized displacement uy of the mechanical mode (bottom). (b) Diagram of unit cell geometry. The basic cell is an elliptical hole with major and minor radii rx… view at source ↗
Figure 2
Figure 2. Figure 2: Inverse-design of the optomechanical cavity enhances the optomechanical scattering rate. (a) A diagram depicting the inverse design algorithm. (i) The geometry is imported into the simulation software. (ii) An eigenmode simulation is performed, yielding the eigenmode fields and frequencies. (iii) The gradient of the figure-of-merit w.r.t. the fields is computed. (iv) An adjoint simulation is performed, yie… view at source ↗
Figure 3
Figure 3. Figure 3: Measurement of inverse-designed X-HOPE cavity. (a) Spectral sweep of optical resonance. Lorentzian fit yields internal quality factor of 1.2 · 105 , with a total linewidth of κ = 2.37 GHz (b) Scanning electron micrograph of the device and measurement setup featuring a variable optical attenuator, electro-optic modulator, photodetector, spectrum analyzer, and vector network analyzer. (c) Mechanical spectrum… view at source ↗
Figure 4
Figure 4. Figure 4: Thermo-optic robustness of the X￾HOPE cavity. Thermo-optic shift as function of number of intracavity photons nc. The X-HOPE￾design from this paper is compared with the conven￾tionally designed release-free implementation and a suspended OMC [20, 21]. the device optically and probe the mechanical re￾sponse with thermal sideband spectroscopy (Fig￾ure 3b) under ambient conditions (see methods section). The o… view at source ↗
read the original abstract

Interactions between light and mechanics provide a powerful interface between optical and microwave-frequency signals, with applications spanning classical signal processing and quantum technologies. High-performance optomechanical devices require both strong photon-phonon coupling and tolerance to parasitic laser heating. Release-free optomechanical crystals provide improved thermal anchoring compared to suspended nanobeams, but have so far exhibited weaker vacuum optomechanical coupling rates, leaving a trade-off between coupling strength and thermal robustness. Here, we largely close this gap: we design and experimentally demonstrate a release-free silicon optomechanical crystal with a record vacuum optomechanical coupling rate of about $g_\text{OM} / (2 \pi) = 800$ kHz, comparable to suspended state-of-the-art devices. The resulting optomechanical scattering rate $\Gamma_\text{OM}/(2 \pi)= 1.1$ kHz is nearly twice that of previous release-free implementations. This performance is achieved by combining physics-guided human intuition with a multiphysics inverse-design algorithm introduced here for resonant optomechanical structures. Beyond the specific device demonstrated, the inverse-design framework is applicable to co-optimizing optical and mechanical resonances and eigenmodes more broadly. These results strengthen release-free optomechanical crystals as a platform for fast, low-noise classical and quantum optomechanics.

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

0 major / 3 minor

Summary. The paper claims to design and experimentally demonstrate a release-free silicon optomechanical crystal using a multiphysics inverse-design algorithm (combined with human intuition) that achieves a record vacuum optomechanical coupling rate g_OM/(2π) ≈ 800 kHz—comparable to suspended state-of-the-art devices—along with an optomechanical scattering rate Γ_OM/(2π) = 1.1 kHz that is nearly twice prior release-free implementations, while providing improved thermal anchoring.

Significance. If the experimental results hold, this work meaningfully advances release-free optomechanical crystals as a platform by largely eliminating the prior trade-off between strong photon-phonon coupling and thermal robustness. The multiphysics inverse-design framework for co-optimizing optical and mechanical resonances/eigenmodes is a broadly applicable methodological contribution. The manuscript supplies concrete numerical performance metrics and an experimental validation that strengthens applications in classical signal processing and quantum optomechanics.

minor comments (3)
  1. The abstract states the coupling rate as 'about 800 kHz' while the central claim is a record value; the main text (likely §4 or experimental results) should report the precise extracted value with uncertainty or error bars to substantiate the record assertion.
  2. A summary table comparing the achieved g_OM, Γ_OM, and relevant figures of merit against cited prior release-free and suspended devices would make the 'nearly twice' and 'comparable' statements more quantitative and verifiable.
  3. Figure captions and legends for mode profiles (optical and mechanical) should explicitly label axes, units, and simulation vs. measurement data to improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work, including the recognition of the record vacuum optomechanical coupling rate achieved in a release-free platform and the broader applicability of the multiphysics inverse-design approach. We appreciate the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation chain consists of a multiphysics inverse-design algorithm followed by device fabrication and experimental extraction of the vacuum optomechanical coupling rate g_OM from measured scattering rates. No load-bearing step reduces by construction to its own inputs via self-definition, fitted-parameter renaming, or a self-citation chain that substitutes for independent verification; the reported 800 kHz value and comparisons to prior release-free devices rest on external experimental data rather than internal normalization or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review limits visibility into parameters or assumptions; relies on standard silicon optomechanics physics and fabrication assumptions.

axioms (1)
  • domain assumption Standard silicon refractive index, Young's modulus, and fabrication process tolerances hold for the designed geometry.
    Implicit in any silicon photonic design and inverse optimization.

pith-pipeline@v0.9.0 · 5539 in / 1086 out tokens · 49007 ms · 2026-05-07T14:13:59.186355+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

55 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    X-HOPE design technique Optomechanical crystal cavities enable strong light–sound interactions by confining both optical and acoustic fields to a wavelength-scale region. The central design challenge is to realize mirror regions, where a bandgap prevents the modes’ propagation, and defect regions, where the modes can interact, simultaneously for near-infr...

  2. [2]

    Specifically, the simulated quality factors are not high enough

    Inverse design of optomechanical structures While the X-HOPE technique developed here im- proves the optomechanical interaction rate, it falls short of delivering a full optomechanical cavity with desirable optical and mechanical properties. Specifically, the simulated quality factors are not high enough. Therefore, we complement X-HOPE, which is based on...

  3. [3]

    We run adjoint simulations, which are equivalent to frequency domain simulations with additional source terms at the eigenfrequency of the respective modes

    (Figure 2). We run adjoint simulations, which are equivalent to frequency domain simulations with additional source terms at the eigenfrequency of the respective modes. The source term is the gradient of the FOM with respect to the displace- ment or electric field. Combining the eigenmode and adjoint fields we obtain the gradient with re- spect to a mater...

  4. [4]

    Aspelmeyer, T

    M. Aspelmeyer, T. J. Kippenberg, and F. Mar- quardt, Cavity optomechanics, Reviews of Mod- ern Physics86, 1391 (2014)

    2014

  5. [5]

    Schliesser, G

    A. Schliesser, G. Anetsberger, R. Rivière, O. Ar- cizet, and T. J. Kippenberg, High-sensitivity mon- itoring of micromechanical vibration using optical whispering gallery mode resonators, New Journal of Physics10, 095015 (2008)

    2008

  6. [6]

    Sansa, M

    M. Sansa, M. Defoort, A. Brenac, M. Hermouet, L. Banniard, A. Fafin, M. Gely, C. Masselon, I. Favero, G. Jourdan, and S. Hentz, Optomechan- ical mass spectrometry, Nature Communications 11, 3781 (2020)

    2020

  7. [7]

    Gavartin, P

    E. Gavartin, P. Verlot, and T. J. Kippenberg, A hybrid on-chip optomechanical transducer for ultrasensitive force measurements, Nature Nan- otechnology7, 509 (2012)

    2012

  8. [8]

    Mirhosseini, A

    M. Mirhosseini, A. Sipahigil, M. Kalaee, and O. Painter, Superconducting qubit to optical pho- ton transduction, Nature588, 599 (2020), pub- lisher: Nature Publishing Group

    2020

  9. [9]

    Jiang, F

    W. Jiang, F. M. Mayor, S. Malik, R. Van Laer, T. P. McKenna, R. N. Patel, J. D. Witmer, and A. H. Safavi-Naeini, Optically heralded mi- crowavephotonaddition,NaturePhysics19,1423 (2023), publisher: Nature Publishing Group

    2023

  10. [10]

    P. K. Shandilya, D. P. Lake, M. J. Mitchell, D. D. Sukachev, and P. E. Barclay, Optomechanical interface between telecom photons and spin quan- tum memory, Nature Physics17, 1420 (2021)

    2021

  11. [11]

    A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, Electromagneti- cally induced transparency and slow light with optomechanics, Nature472, 69 (2011), publisher: Nature Publishing Group

    2011

  12. [12]

    J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. As- pelmeyer, and O. Painter, Laser cooling of a nanomechanical oscillator into its quantum ground state, Nature478, 89 (2011), publisher: Nature Publishing Group

    2011

  13. [13]

    A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, Squeezed light from a silicon micromechanical resonator, Nature500, 185 (2013), publisher: Na- ture Publishing Group

    2013

  14. [14]

    J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meene- han, and O. Painter, Optimized optomechanical crystal cavity with acoustic radiation shield, Ap- plied Physics Letters101, 081115 (2012)

    2012

  15. [15]

    Sonar, U

    S. Sonar, U. Hatipoglu, S. Meesala, D. P. Lake, H. Ren, and O. Painter, High-efficiency low-noise optomechanical crystal photon-phonon transduc- ers, Optica12, 99 (2025), publisher: Optica Pub- lishing Group

    2025

  16. [16]

    A. H. Safavi-Naeini, D. Van Thourhout, R. Baets, and R. Van Laer, Controlling phonons and pho- tonsatthewavelengthscale: integratedphotonics meets integrated phononics, Optica6, 213 (2019), publisher: Optica Publishing Group

    2019

  17. [17]

    S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, Pulsed Excitation Dynamics of an Optomechanical Crys- tal Resonator near Its Quantum Ground State of Motion, Physical Review X5, 041002 (2015), publisher: American Physical Society

    2015

  18. [18]

    Burger, J

    P. Burger, J. Frey, J. Kolvik, D. Hambraeus, and R. Van Laer, Design of a release-free piezo- optomechanical quantum transducer, APL Pho- tonics10, 010801 (2025)

    2025

  19. [19]

    H. Ren, M. H. Matheny, G. S. MacCabe, J. Luo, H. Pfeifer, M. Mirhosseini, and O. Painter, Two- dimensional optomechanical crystal cavity with high quantum cooperativity, Nature Communica- tions11, 3373 (2020), publisher: Nature Publish- ing Group

    2020

  20. [20]

    F. M. Mayor, S. Malik, A. G. Primo, S. Gyger, W. Jiang, T. P. M. Alegre, and A. H. Safavi- Naeini, High photon-phonon pair generation rate 9 in a two-dimensional optomechanical crystal, Na- ture Communications16, 2576 (2025), publisher: Nature Publishing Group

    2025

  21. [21]

    A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, Two- Dimensional Phononic-Photonic Band Gap Op- tomechanical Crystal Cavity, Physical Review Letters112, 153603 (2014), publisher: American Physical Society

    2014

  22. [22]

    L. Qiu, I. Shomroni, P. Seidler, and T. J. Kippen- berg, Laser Cooling of a Nanomechanical Oscil- lator to Its Zero-Point Energy, Physical Review Letters124, 173601 (2020), publisher: American Physical Society

    2020

  23. [23]

    Kolvik, P

    J. Kolvik, P. Burger, J. Frey, and R. Van Laer, Clamped and sideband-resolved silicon optome- chanical crystals, Optica10, 913 (2023), pub- lisher: Optica Publishing Group

    2023

  24. [24]

    Kolvik, P

    J. Kolvik, P. Burger, D. Hambraeus, T. H. Haug, J. Frey, M. B. Kristensen, and R. Van Laer, Op- tomechanical crystal in light-resilient quantum ground state (2025), arXiv:2510.15724 [quant-ph]

    work page arXiv 2025

  25. [25]

    Jensen and O

    J. Jensen and O. Sigmund, Topology optimization for nano-photonics, Laser & Photonics Reviews 5, 308 (2011)

    2011

  26. [26]

    R. E. Christiansen and O. Sigmund, Inverse de- sign in photonics by topology optimization: tuto- rial, JOSA B38, 496 (2021), publisher: Optica Publishing Group

    2021

  27. [27]

    Molesky, Z

    S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vuck- ović, and A. W. Rodriguez, Inverse design in nanophotonics, Nature Photonics12, 659 (2018), number: 11 Publisher: Nature Publishing Group

    2018

  28. [28]

    Schul, S

    L. Schul, S. Mason, S. Eun, G. H. Ahn, and J. Vučković, Inverse design for scalable photonic systems, Nature Reviews Materials , 1 (2026), publisher: Nature Publishing Group

    2026

  29. [29]

    Vitali, C

    V. Vitali, C. Lacava, T. Domínguez Bucio, F. Y. Gardes, and P. Petropoulos, Highly efficient dual- level grating couplers for silicon nitride photonics, Scientific Reports12, 15436 (2022), publisher: Nature Publishing Group

    2022

  30. [30]

    Kokhanovskiy, E

    A. Kokhanovskiy, E. Kuprikov, A. Bednyakova, I. Popkov, S. Smirnov, and S. Turitsyn, Inverse design of mode-locked fiber laser by particle swarm optimization algorithm, Scientific Reports 11, 13555 (2021), publisher: Nature Publishing Group

    2021

  31. [31]

    Sigmund and K

    O. Sigmund and K. Maute, Topology optimiza- tion approaches, Structural and Multidisciplinary Optimization48, 1031 (2013)

    2013

  32. [32]

    Jafar-Zanjani, S

    S. Jafar-Zanjani, S. Inampudi, and H. Mosallaei, Adaptive Genetic Algorithm for Optical Metasur- faces Design, Scientific Reports8, 11040 (2018), publisher: Nature Publishing Group

    2018

  33. [33]

    Jiang, M

    J. Jiang, M. Chen, and J. A. Fan, Deep neural networksfortheevaluationanddesignofphotonic devices (2020), arXiv:2007.00084

    work page arXiv 2020

  34. [34]

    J. L. Su, J. W. You, L. Chen, X. Y. Yu, Q. C. Yin, G. H. Yuan, S. Q. Huang, Q. Ma, J. N. Zhang, and T. J. Cui, MetaPhyNet: intelligent design of large-scale metasurfaces based on physics-driven neural network, Journal of Physics: Photonics6, 035010 (2024), publisher: IOP Publishing

    2024

  35. [35]

    Y. Deng, S. Ren, K. Fan, J. M. Malof, and W. J. Padilla, Neural-adjoint method for the inverse de- sign of all-dielectric metasurfaces, Optics Express 29, 7526 (2021), publisher: Optica Publishing Group

    2021

  36. [36]

    Gahlmann and P

    T. Gahlmann and P. Tassin, Deep neural net- works for the prediction of the optical properties and the free-form inverse design of metamaterials, Physical Review B106, 085408 (2022), publisher: American Physical Society

    2022

  37. [37]

    C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, Adjoint shape optimization applied to electromagnetic design, Optics Express 21, 21693 (2013), publisher: Optica Publishing Group

    2013

  38. [38]

    A. Y. Piggott, J. Lu, T. M. Babinec, K. G. Lagoudakis, J. Petykiewicz, and J. Vučković, In- verse design and implementation of a wavelength demultiplexing grating coupler, Scientific Reports 4, 7210 (2014), arXiv:1406.6185 [physics]

    work page arXiv 2014

  39. [39]

    Michaels and E

    A. Michaels and E. Yablonovitch, Inverse design of near unity efficiency perfectly vertical grating couplers, Optics Express26, 4766 (2018), pub- lisher: Optica Publishing Group

    2018

  40. [40]

    A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer, Nature Photonics9, 374 (2015)

    2015

  41. [41]

    Shang, J

    C. Shang, J. Yang, A. M. Hammond, Z. Chen, M. Chen, Z. Lin, S. G. Johnson, and C. Wang, Inverse-Designed Lithium Niobate Nanophoton- ics, ACS Photonics10, 1019 (2023), publisher: American Chemical Society

    2023

  42. [42]

    Vercruysse, N

    D. Vercruysse, N. V. Sapra, K. Y. Yang, and J. Vučković, Inverse-Designed Photonic Crystal Circuits for Optical Beam Steering, ACS Photon- ics8, 3085 (2021), publisher: American Chemical Society

    2021

  43. [43]

    M. B. Giles and N. A. Pierce, An Introduction to the Adjoint Approach to Design, Flow, Turbu- lence and Combustion65, 393 (2000)

    2000

  44. [44]

    Albrechtsen, B

    M. Albrechtsen, B. Vosoughi Lahijani, R. E. Christiansen, V. T. H. Nguyen, L. N. Casses, S. E. Hansen, N. Stenger, O. Sigmund, H. Jansen, J. Mørk, and S. Stobbe, Nanometer-scale photon confinement in topology-optimized dielectric cav- ities, Nature Communications13, 6281 (2022), publisher: Nature Publishing Group

    2022

  45. [45]

    Lee, and E

    V.Akçelik, G.Biros, O.Ghattas, D.Keyes, K.Ko, L.-Q. Lee, and E. G. Ng, Adjoint methods for electromagnetic shape optimization of the low- loss cavity for the International Linear Collider, Journal of Physics: Conference Series16, 435 10 (2005)

    2005

  46. [46]

    Toader and C

    A.-M. Toader and C. Barbarosie, 6. Optimization of eigenvalues and eigenmodes by using the ad- jointmethod,inTopological Optimization and Op- timal Transport: In the Applied Sciences, edited by M. Bergounioux, E. Oudet, G. Rumpf, Martin adn Carlier, T. Champion, and F. Santambrogio (De Gruyter, 2017) pp. 142–158

    2017

  47. [47]

    R. L. Fox and M. P. Kapoor, Rates of change of eigenvalues and eigenvectors., AIAA Journal 6, 2426 (1968), publisher: American Institute of Aeronautics and Astronautics

    1968

  48. [48]

    T. H. Lee, An adjoint variable method for struc- tural design sensitivity analysis of a distinct eigen- value problem, KSME International Journal13, 470 (1999)

    1999

  49. [49]

    D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization (2017), arXiv:1412.6980 [cs]

    work page internal anchor Pith review arXiv 2017

  50. [50]

    Y. Ota, R. Katsumi, K. Watanabe, S. Iwamoto, and Y. Arakawa, Topological photonic crystal nanocavity laser, Communications Physics1, 86 (2018), publisher: Nature Publishing Group

    2018

  51. [51]

    Sigmund, Manufacturing tolerant topology op- timization, Acta Mechanica Sinica25, 227 (2009)

    O. Sigmund, Manufacturing tolerant topology op- timization, Acta Mechanica Sinica25, 227 (2009)

    2009

  52. [52]

    E. W. Wang, D. Sell, T. Phan, and J. A. Fan, Robust design of topology-optimized metasur- faces, Optical Materials Express9, 469 (2019), publisher: Optica Publishing Group

    2019

  53. [53]

    A. H. Safavi-Naeini and O. Painter, Design of optomechanical cavities and waveguides on a si- multaneous bandgap phononic-photonic crystal slab, Optics Express18, 14926 (2010), publisher: Optica Publishing Group. V. ACKNOWLEDGEMENTS We acknowledge support from the Knut and Al- ice Wallenberg foundation through the Wallen- berg Centre for Quantum Technolog...

    2010

  54. [54]

    Optomechanical coupling We account for two types of optomechanical interactions: one from the moving boundary and the other is from the photoelastic effect. The moving boundary contribution to the vacuum optomechanical coupling is computed as [50] gmb = ωo 2 ∫ ∂Ω(q·ˆn)(∆ϵE2 ||−∆(ϵ−1)D2 ⊥) dS∫ ϵE2 dV xzpf, whereωo is the optical frequency,q is the normaliz...

  55. [55]

    This is fundamentally different from frequency or time domain simulations, which are both simply solving a linear system of equationsAx=b

    The adjoint method for resonant structures Simulation of resonant structures with high quality factors is often done with an eigenvalue solver. This is fundamentally different from frequency or time domain simulations, which are both simply solving a linear system of equationsAx=b. An eigensolver instead finds solutionsv,λto equations like(Mλ2 +Dλ+K)v= 0....