Surfaces: coatings, losses and polarisation¶

Surfaces¶

An interface (mirror, grating) element is the place where a beam meets a surface. The engine computes the s and p amplitude coefficients -- POWER-normalised, so |r|^2 = R and |t|^2 = T including the Re(n cos)/Re(n cos) factor -- from the media on both sides, the angle of incidence (per wavelength when traced) and the coating, and applies them as a Jones matrix in the transported x/y basis.

Convention: exp(-i omega t) physics convention; r_s = r_p at normal incidence (a mirror does not swap x/y labels); a positive d(phase)/d(omega) is a delay (light penetrating a dielectric stack is delayed).

Coating kinds¶

kind	fields
`none` / no coating		bare Fresnel interface (complex indices, so TIR and metals work)
`stack`	`layers: [{material, thicknessM \| qwot}]`, `designWavelengthM`	physical multilayer, transfer-matrix method
`quarterWave`	`design: "(HL)^12 H 2L"`, `designWavelengthM`, `H`, `L`, `M` materials	quarter-wave notation expanded to a stack
`ideal`	`R` (or `Rs`, `Rp`), `T`, `phaseRad`	wavelength-independent
`ar`	`R` (residual reflectance per surface)
`gdd`	`R`, `gddS2`, `todS3`, `fodS4`, `referenceWavelengthM`	ideal dispersive (chirped) mirror
`table`	`wavelengthsM`, `R`/`Rs`/`Rp`, `T`/`Ts`/`Tp`, `phaseRs`... or `groupDelayRsS`...	measured or vendor data; exported OpticStudio coatings
`efficiency`	`s`, `p`	grating diffraction efficiency

All kinds take an optional lidt. Example:

"coatings": {
  "HR1030@45": { "kind": "quarterWave", "design": "(HL)^14 H", "designWavelengthM": 1.13e-6, "H": "Ta2O5", "L": "SiO2_film" },
  "CM": { "kind": "gdd", "R": 0.999, "gddS2": -100e-30, "lidt": { "fluenceJPerCm2": 0.4, "pulseDurationS": 1e-12 } }
}

Substrate echoes¶

A mirror with a substrate adds the echo of the light that leaks through the coating, reflects from the back face of the substrate and returns after 2 n_g d cos(theta) / c (62 / 93 / 124 ps for 6.35 / 9.5 / 12.7 mm of fused silica at 1030 nm). The transfer is exact in n(omega), the leakage follows the coating's T(lambda), the back face may be bare, AR-coated or wedged. Keys, the catalogue of substrates and the physics: Delayed light.

{ "id": "m1", "type": "mirror", "coating": "HR1030", "substrate": "FS-12.7mm" }

Coating ripple¶

"ripple": {"phaseRmsRad": 0.03, "amplitudeRms": 0.01, "maxDelayS": 5e-12, "seed": 7} on a mirror or interface multiplies the reflection by a random phase and amplitude ripple whose pedestal lies within +-maxDelayS; measured ripple is a table coating. See Delayed light.

Delayed scatter¶

"delayedScatter": {"fraction": 1e-4, "delay": {"kind": "uniform", "maxS": 40e-12}, "spectralExponent": 4} returns a fraction of the power as incoherent delayed light instead of losing it (roughnessRmsM below books the lost part). On a plate the default delay is 0 .. 2 n_g d cos(theta)/c. "model": "surface" derives the fraction from the roughness and correlation length (TIS, and the share 1 - exp(-(l/w)^2) that stays in the mode). See Delayed light.

Loss mechanisms¶

mechanism	source
`fresnel`	bare interface reflection/transmission
`coating`	coated surfaces, chirped mirrors
`grating`	diffraction efficiency
`clip`	Gaussian beam on a hard aperture: T = 1 - exp(-2 a^2 / w^2) (circular, round beam; an elliptical beam wx x wy gives the exact integral over the disk, (1 / pi wx wy) ∫ (1 - exp(-a^2 q)) / q dφ with q = 2 (cos²φ / wx² + sin²φ / wy²)), erf product (rectangular); per-wavelength when the beam size is; chief-ray vignetting from OpticStudio
`scatter`	total integrated scatter of rough surfaces: reflection (4 pi sigma n cos(theta) / lambda)^2, transmission (2 pi sigma
`absorption`	material attenuation, internal transmission, absorbing time responses
`filter`, `transmission`, `custom`	filters and tabulated transmissions
`polarization`	polarisers
`raman+numerical`, `numerical`, `harmonics (out of band)`	energy change inside nonlinear segments

summary.json -> lossBudget gives each mechanism's total (J and fraction of the input) and every element's lossJ gives the breakdown per element; together they close exactly on input - output energy.

Damage (LIDT)¶

For surfaces with a beam size, the engine reports the peak fluence F = 2E/(pi wx wy), the coating's (or material's) LIDT scaled to the pulse duration at that point, F_LIDT (tau/tau_ref)^exponent (default 0.5), and lidtMargin = F_LIDT / F, with a warning below 1. It also reports the peak intensity.

Polarisation¶

physics.polarization: "jones" (default) keeps x and y fields and applies 2x2 Jones matrices:

surfaces: J = B^T diag(a_s, a_p) B, where B rotates the transported (x, y) basis onto (s, p);
waveplate (retardance proportional to omega for a zero-order plate), polarizer (extinction ratio), rotator; birefringent group delay via groupDelayYS on sampled elements.

Basis transport. With traced vectors (kIn, kOut, normal -- exported by the Zemax binding) the local basis is carried along the folded 3D path by the rigid rotation that maps (s, p_in, k_in) to (s, p_out, k_out), so "x" at the output is the transported image of the input x axis (a periscope rotates it, as in the lab). Without vectors, each surface's plane of incidence is given by planeOfIncidenceDeg (90 = folding in the x-z plane, the default). physics.polarization: "scalar" averages s and p.

The nonlinear coupling of x and y in isotropic media is the full chi(3) vector form (cross-phase 2/3 plus the coherent term), so circular light correctly sees 2/3 of the linear SPM.