Fused silica#

Fused silica (silica glass, SiO2) has linear and nonlinear models. This page covers:

  • its complex refractive index from the extreme ultraviolet to the far infrared;

  • its nonlinear index n2 with the Raman part;

  • how well both are known, and how they compare with the published alternatives.

The manual for solids is Solids: conventions and models; the register entries are TC-321 to TC-324 and TC-399 (the linear index) and TC-393 (n2).

from optmat.solid.models.registry import default_model, models
from optmat.solid.kerr import kerr_models

silica = default_model("SiO2")  # fit/SiO2, at 20 degrees C
n = silica.refractive_index(800e-9)  # complex n + ik
beta2 = silica.gvd(800e-9)  # s^2/m
deps = silica.dpermittivity_domega(120e-9)  # exact d eps / d omega

cold = models("SiO2")["leviton_2008/sellmeier"].at_temperature(120.0)  # 30-300 K

kerr = kerr_models()["kerr/SiO2"]
kerr.n2(1030e-9)  # long-pulse n2, m^2/W
kerr.n2_effective(1030e-9, 50e-15)  # what a 50 fs pulse measures

The complex index#

The recommended model is fit/SiO2 (TC-399). It starts from Franta et al.’s universal dispersion model of bulk Schott Lithosil Q2 (franta_2016/udm_lithosil_q2), refitted to the absolute refractometry of the transparent window.

  • Range: 24.8 nm to 125 µm, at 20 degrees C.

  • Physics: it is causal, passive and normalised to its transition strengths, with an exact d eps/d omega.

  • What it rests on:

    • in the window, the absolute indices of Leviton & Frey (Corning 7980, 2008) and Malitson’s three prisms (1965), which agree with each other to 1e-5;

    • Franta’s measured absorption at the band edge (146-225 nm) and in the infrared, and his infrared n;

    • in the vacuum ultraviolet, Franta’s model, which above 8.5 eV was fitted to Philipp’s tables in Palik’s handbook.

  • What it changed:

    • the interband strength (-2 %) and the higher-energy term (-6 %, and 16.1 eV against 17.0 eV);

    • the band edge (8.355 against 8.363 eV) and the Urbach width (0.188 against 0.189 eV);

    • one factor each on the excitons (x1.59), the phonon bands (x1.0006 and x1.003) and the multiphonon bands (x1.12).

  • What it keeps of Franta’s model (rms):

    • the band-edge k to 1.7 %;

    • the vacuum-ultraviolet n and k to 0.008 and 1.7 %;

    • in the infrared, k to 6-9 % and n to 2e-3.

In the vacuum ultraviolet the band edge (E_g = 8.36 eV, 148 nm, with an Urbach tail) and the excitonic structure near 10-12 eV dominate:

lambda (nm)

n

k

50

0.8080

0.300

80

1.1935

0.650

100

1.4305

0.767

110

1.8046

0.739

120

2.2399

0.969

130

2.0202

0.0684

150

1.7061

0.000880

160

1.6490

6.68e-05

180

1.5859

9.12e-07

200

1.5511

2.98e-08

Extrapolating a Sellmeier formula through a pole does not describe absorption. For example, evaluating Malitson’s printed formula at vacuum wavelengths with a complex square root, outside its stated range, gives Re n = 0.000 at 110 nm and 2.94 at 120 nm, where the recommended complex model gives 1.80 and 2.24. The discrepancy is not an uncertainty estimate for either model.

In the transparent window the absolute refractometry agrees to 1e-5, once each measurement is referred to vacuum:

At 800 nm

Leviton 2008 (20 C)

Malitson 1965

Arosa 2020

Franta 2016

fit/SiO2

n

1.45371

1.45371

1.45337

1.45355

1.45371

group index

1.46757

1.46756

1.46727

1.46726

1.46755

GVD (fs^2/mm)

36.31

36.23

35.86

35.76

36.20

  • Relative to air or to vacuum. Refractometry by minimum deviation in room air gives the index relative to that air, 2.7e-4 below the absolute index at room conditions (TC-398).

    • Leviton & Frey measured in vacuum (the CHARMS refractometer), from 30 to 310 K, to +-1e-5 (leviton_2008).

    • Malitson’s 1965 indices are relative to air. The paper does not say so, but Leviton & Frey read them so, and converted to vacuum they agree with Leviton’s to 4.5e-6 rms from 0.4 to 2 µm. malitson_1965/sellmeier now returns the absolute index, through Edlen’s air at 20 degrees C (TC-399).

    • Rodney & Spindler’s 1954 indices, from the same apparatus, are relative to air too (rodney_1954).

    • Arosa & de la Fuente’s formula (2020) is levelled by an Abbe refractometer, which also reads relative to air. Taken to vacuum it agrees with Leviton’s to 5e-5 from 0.4 to 2 µm; the table shows it as published.

  • Franta’s published model lies 3.4e-4 below Leviton’s at 400 nm and 0.8e-4 below at 2 µm, and its GVD is 1.5 % low at 800 nm. It was the recommended model until this comparison (TC-324); the refit keeps everything else about it.

  • Uncertainty of the recommended model in the window (0.21-2.6 µm):

    • n to about 2e-5: the fit’s residuals against Malitson and Leviton are 1-2e-5, and Malitson’s twelve specimens of three grades scatter by 1-3e-5 about it;

    • the group index to about 3e-5 (the refit is 2.3e-5 below Leviton’s formula and 0.6e-5 below Malitson’s at 800 nm);

    • GVD to about 0.3 %: Leviton’s formula gives 36.31 fs^2/mm and Malitson’s 36.23 at 800 nm, against the fit’s 36.20.

  • Near the 2.73 µm OH band and beyond grades differ: the fit, with Lithosil Q2’s OH band and Franta’s infrared, departs from Malitson’s glasses by up to 5e-5 at 3.3 µm.

  • The dispersion near 2 µm carries Lithosil Q2’s OH bands. The fit keeps Franta’s bands of the plate unscaled (Nph36-52; TC-399), and their Kramers-Kronig partners shape the dispersion well before the bands themselves:

    • the 2.73 µm fundamental (Nph36-41);

    • a narrow group at 2.21-2.25 µm, 51-293 cm⁻¹ wide, with k up to 1.0e-5 (0.6 cm⁻¹) at 2212 nm (Nph45-47);

    • weaker bands at 1.90-2.59 µm and 1.38 µm.

    At 2 µm the fit’s GVD is 1.7 % above Malitson’s formula’s. The bands account for +2.1 % of that (the fundamental 1.3 %, the 2.2 µm group 0.8 %); the refit alone would be 0.3 % below. The 2.2 µm group alone changes β3 by 7 % at 2 µm, and near 2.2 µm the GVD swings through +700 fs²/mm. A finite-difference stencil a few nm wide resolves this. It is not an artefact of the fit. It is the dispersion of absorption measured in one sample:

    • Franta et al. do not assign the 2.2 µm bands;

    • their n swing, 1.2e-5 peak to peak, is about the precision of the absolute refractometry, whose wavelengths are 0.1-0.2 µm apart;

    • Malitson’s three grades and Leviton’s Corning 7980 agree with the fit to 1e-5 from 1.5 to 2.44 µm. They neither confirm nor exclude bands of this strength in their glasses.

    A grade with less OH has weaker bands. Malitson’s and Leviton’s Sellmeier formulas have none (TC-841).

  • Evaluation noise: the UDM’s closed forms lose digits to cancellation below a term’s characteristic energy (TC-321). n carries 1e-15 to 6e-15 of relative rounding noise from 0.8 to 3.5 µm, and dn/dω (a complex step, exact) 1e-13 to 6e-13. That is far below any physical effect, but it bounds a derivative differenced from them (TC-841).

  • Temperature: Leviton & Frey’s temperature-dependent formula (Corning 7980, 30-300 K, 0.4-2.6 µm) is the model leviton_2008/sellmeier; at_temperature evaluates it at any temperature in that range. Their dn/dT (Table 6) is curated as datasets: 8.6e-6 per K at 800 nm and 295 K.

In the infrared the model’s phonon bands are Franta’s measurement of Lithosil Q2. Philipp’s older tables (Kramers-Kronig of reflectance) differ by up to 0.5 in n on the reststrahlen bands (TC-324).

The nonlinear index#

The model (kerr/SiO2; TC-393) gives the long-pulse n2 and its Raman fraction f_R:

\[ n_{2,\mathrm{meas}}(\lambda, \tau) = n_2(\lambda)\left[(1 - f_R) + f_R\,\eta(\tau)\right], \qquad n_2(\lambda) = N\left(1 + a E^2\right), \]

  • E is the photon energy (eV);

  • eta(tau) is the share of the delayed Raman response a Gaussian pulse of duration tau sees, from Hollenbeck & Cantrell’s 13-mode response;

  • eta is 0.96 at 100 fs, 0.86 at 47 fs, and 1 for picosecond and longer pulses.

It is fitted to eleven sources’ measured self coefficients, each with its own calibration scale (a 10 % prior), from 350 nm to 3.5 µm, and to Stolen & Ippen’s measured Raman gain of fused quartz, which sets f_R (TC-696):

lambda (nm)

n2 (1e-20 m^2/W)

sigma

electronic

400

3.05

0.18

2.39

532

2.78

0.13

2.17

800

2.59

0.11

2.02

1030

2.53

0.11

1.97

1550

2.47

0.11

1.93

2000

2.46

0.11

1.92

The Raman fraction is f_R = 0.22 +- 0.03, set by Stolen & Ippen’s gain, 1.5e-11 cm/W at 330 cm⁻¹ for a 526 nm pump, predicted as 2 omega_s n2 f_R Im H_R / c with Hollenbeck & Cantrell’s response (TC-695, TC-696). It agrees with Hellwarth et al.’s nuclear share of n2 from Raman spectra, 0.21 +- 0.03, which is compared, not fitted. Stolen et al.’s 0.18, the prior until then, is left out: it was derived from these two measurements. The electronic part is 0.78 of n2; a 50 fs pulse measures 0.97 of it. The Raman response itself, with this f_R and two literature forms, is on The Raman response.

  • Compared with the literature:

    • Milam’s much-used recommended values (2.74e-20 m^2/W at 1053 nm) are 8-11 % higher. His assessment rests largely on nanosecond measurements, which may include electrostriction.

    • Santran et al.’s ten grades are 37 % higher, beyond their stated absolute accuracy (10-28 %). Their pump-probe isolates the electronic part, which the larger f_R lowers.

    • The other nine sources lie within -13 % and +4 % of the fit. Milam & Weber’s 1976 interferometric value at 1064 nm (125 ps) is 4 % above it (TC-594).

  • Below 350 nm two-photon absorption sets in (2 hbar omega above the 8.4 eV gap). DeSalvo et al.’s 266 nm value is twice this dispersion’s extrapolation, and the model warns there.

  • Two-photon absorption, multiphoton and field ionisation, and laser damage: their data are curated (TC-325 onwards). Their models are next.

The nonlinear bundle#

A propagation code in the \(n_0 = 1\) convention of the nonlinear Schrödinger equation takes the response \(R(t) = (1 - f_R)\,\delta(t) + f_R\,h_R(t)\), with \(\int h_R\,dt = 1\) and the total \(n_2\): the triple \((n_2, f_R, h_R)\). optmat.solid.bundle.silica_bundle gives it as one object whose parts cannot be mismatched (OEP 1, sections 4.7 and 6.9). Every bundle holds the Kerr fit’s total long-pulse \(n_2(\lambda)\), the quantity the data measure, and its shape’s own \(f_R\). Its electronic part is \((1 - f_R)\,n_2\), so a scalar and a vector run agree in the long-pulse limit.

  • "default": Hollenbeck and Cantrell’s shape with the Kerr fit’s \(f_R = 0.219\), paired, copolarised only.

  • "tensor": Lin and Agrawal’s shape with its own \(f_R = 0.245\) (TC-734), which has isotropic and anisotropic parts. Its electronic part is 3.3 % smaller than the default’s, and its Raman part 12 % larger.

  • One bundle for each silica response, under the response’s name.

The silica response’s raman_coefficient and electronic_susceptibility include \(n_0^2\); a code in the \(n_0 = 1\) convention takes the bundle’s parts instead.

Each bundle carries its gain check against Stolen and Ippen’s measured gain, \(1.5 \times 10^{-11}\) cm/W ± 10 % at 330 cm⁻¹ for a 526 nm pump (gain_check). The Kerr fit’s \(f_R\) was set by this gain through Hollenbeck’s shape, so for another shape it is a consistency check, not an independent validation:

Bundle

\(f_R\)

Gain at 330 cm⁻¹ (10⁻¹¹ cm/W)

Against the measurement

default (Hollenbeck, paired)

0.219

1.35

−9.7 %

tensor (Lin and Agrawal, their \(f_R\))

0.245

1.38

−7.9 %

blow_1989 (Blow and Wood, their \(f_R\))

0.222

1.46

−2.5 %

 
import numpy as np

from optmat.solid.bundle import silica_bundle

bundle = silica_bundle("tensor")
n2 = bundle.n2(1030e-9)  # m^2/W, the Kerr fit's total
fraction = bundle.raman_fraction
t = np.linspace(0.0, 1e-12, 2001)
h = bundle.h_r(t, "b")  # 1/s, the anisotropic part
assert abs(float(bundle.n2_electronic(1030e-9)) - (1 - fraction) * float(n2)) < 1e-30
assert abs(bundle.gain_check.deviation) < 0.1