Raman response models

A response function describes how a molecular vibration, rotation or silica network continues to affect the field after excitation. This chapter gives the oscillator strengths, dephasing laws and normalisations used by the Raman workflow. Keep the response model, its thermodynamic state and the temporal resolution consistent.

The models

Each model specifies an undamped response and a dephasing law. The latter sets how quickly molecular coherence decays and can depend on the density along a pressure gradient. solprop.nonlinear.raman.Dephasing represents that dependence. A built-in gas’s response is optmat’s, mapped onto the same terms (Raman data); the vibrational mode and the rotational ladder below are what a caller’s own molecular constants, a MolecularRaman on a custom species, are built into.

The damped oscillator

Every gas-phase line is the impulse response of a driven oscillator in the weak-damping limit, solprop.nonlinear.raman.DampedOscillator:

\[ h(t) = K \sin(\Omega t)\, \mathrm{e}^{-t/\tau_2}, \qquad t > 0 . \]

Its integral is \(K\Omega/(\Omega^2 + \tau_2^{-2})\), so solprop.nonlinear.raman.normalised_damped_oscillator() scales \(K\) by the reciprocal of that factor and gives a line of a stated total strength — the form a fibre GNLSE states its Raman contribution in.

The vibrational mode

solprop.nonlinear.raman.vibrational() builds one oscillator at the mode frequency with

\[ K = \frac{\left(4\pi\varepsilon_0\, \partial\alpha/\partial Q\right)^2} {2\mu\Omega_v\,\varepsilon_0} , \]

the \(4\pi\varepsilon_0\) converting the tabulated volume polarisability derivative to an SI one, and the trailing \(\varepsilon_0\) the response convention. The constants are the species’ own, a VibrationalRaman.

Here \(Q\) is a displacement coordinate (the bond length for N2), \(\mu\) its effective mass and \(\Omega_v\) the angular frequency of the mode. The induced-dipole energy is \(-\alpha_{\rm SI}(Q) E_{\rm real}^2/2\), so the force is \((d\alpha_{\rm SI}/dQ) E_{\rm real}^2/2\); solving the driven oscillator and multiplying its displacement by the same derivative gives the denominator two. The cycle average belongs to the drive, not to \(K\): the envelope drive is already \(|A|^2/2\) (Raman numerics), and a further half in \(K\) — a denominator of four — would count it twice and halve every vibrational amplitude. An independent quantum-oscillator impulse test verifies the strength. It checks the common formula, not the calibration of a supplied derivative and effective mass.

For a mass-weighted normal coordinate the derivative and kinetic energy use a different convention: convert both consistently before constructing the model.

The rotational ladder

A linear molecule’s levels are \(E_J = 2\pi\hbar c\,[B J(J+1) - D (J(J+1))^2]\) and the Raman-active transitions are the \(S\) branch, \(J \to J+2\). solprop.nonlinear.raman.rotational_nonrigid() returns a solprop.nonlinear.raman.RotationalLines holding the levels, their Boltzmann populations \(p_J\) with the nuclear-spin statistical weights, the transition frequencies and the line weights

\[ K\,\frac{(J+1)(J+2)}{2J+3} \left(\frac{p_{J+2}}{2J+5} - \frac{p_J}{2J+1}\right), \qquad K = -\frac{(4\pi\varepsilon_0)^2}{\varepsilon_0}\,\frac{2}{15}\, \frac{\Delta\alpha^2}{\hbar} . \]

\(K\) is negative and so is the bracket, because the population per magnetic sublevel falls with \(J\); the lines therefore come out positive, with the same sign as a vibration. The ladder is truncated wherever \(E_J\) stops increasing with \(J\): beyond that the two-term expansion has left its range of validity and the levels it predicts are an artefact. For hydrogen’s constants (\(B = 58.9\), \(D = 0.05\) cm⁻¹), whose centrifugal distortion is large, that is \(J(J+1) > B/2D = 589\), so \(J \le 24\). The ladder otherwise runs from MolecularOptions.min_j to max_j (default 0 and 50), and the populations are normalised over the levels retained.

One linewidth is used for every \(J\). The \(J\) dependence is real and not modelled, and this is the model’s weakest point.

A temperature along z. The populations are a Boltzmann ladder fixed for the run: the undamped rows the kernel samples are built once per grid, so populations that followed a temperature varying along the fibre would need a new kernel at every position, which solprop does not do. A run whose gas temperature is a path therefore computes them at one reference temperature: MolecularOptions(temperature=...) if given, otherwise the mean over the run \([0, L]\) weighted by that species’ own number density,

\[\bar T_s = \frac{\int_0^L \rho_s\,T\,\mathrm{d}z}{\int_0^L \rho_s\,\mathrm{d}z}\]

(raman_temperature(), by adaptive quadrature across the breakpoints of both). The mean warns with a RamanTemperatureWarning naming the species, the reference and the spread of \(T(z)\) over the run, since the populations do not follow \(T(z)\); an explicit MolecularOptions(temperature=...) is the caller’s own choice and logs the same message at INFO instead. The populations are the species’ own, so the mean weights the positions where that species is: in a gradient it leans to the dense, high-pressure end, and in independent populations each species takes its own — nitrogen fed from one end (1 to 0.1 bar) and helium from the other, the gas warming from 294 K to 400 K over a centimetre, holds nitrogen at 330.4 K, where the total density would give 352.9 K. The average never reaches past the fibre, even for a constant pressure (whose density has no end) or a temperature path longer than the run. A molecule without a rotational ladder, or a run with rotation=False, is not warned. The linewidths are unchanged: they follow the density only, as at a constant temperature, and their own temperature dependence is not modelled either way. A constant temperature is used as it is.

Fused silica

A glass has no resolved rotational ladder. Its scalar Raman response is the shape \(h_R\) of optmat’s silica bundle (SilicaBundle): Hollenbeck and Cantrell’s fit, of unit integral, with the bundle’s own Raman fraction (materials), sampled on the run’s delays (SampledShape). The shape has no density dependence: a glass is a fixed medium.

A caller’s own line set of damped, Gaussian-broadened lines (IntermediateBroadeningRaman) is evaluated by solprop.nonlinear.raman.IntermediateBroadening,

\[ h(t) \propto \sum_i A_i\, \mathrm{e}^{-\gamma_i t}\, \mathrm{e}^{-\Gamma_i^2 t^2/4}\, \sin(\omega_i t) . \]

The widths are the dephasing, so this model has no density dependence either. The amplitudes are relative, so the model is normalised to a stated integral, and the normalisation is analytic:

\[ \int_0^\infty \mathrm{e}^{-\gamma t}\mathrm{e}^{-\Gamma^2t^2/4}\sin(\omega t)\,dt = \mathrm{Im}\left[\frac{\sqrt{\pi}}{\Gamma}\, w\!\left(\frac{\omega + \mathrm{i}\gamma}{\Gamma}\right)\right], \]

with \(w\) the Faddeeva function scipy.special.wofz, the numerically stable spelling of \(\mathrm{e}^{z^2}\mathrm{erfc}(z)\) in the half plane these arguments occupy.

Fused silica, as a vector response

The orientation average, tensor channels and envelope conventions shared by every vector Raman response are derived in polarisation-dependent Raman; this section gives silica’s own tensor model. A second silica model lives beside the scalar one and does not replace it: solprop.nonlinear.raman.SilicaTensorResponse, the delayed tensor response of Lin and Agrawal, which a material region of a general modal guide declares beside its MaterialKerr (see general modal propagation). By default it is optmat’s tensor bundle, SilicaBundle.resolve(tensor=True); Lin and Agrawal’s own parameters are a LinAgrawalRaman, given explicitly. It is a different fit of the same physics — two normalised shapes rather than one,

\[ h_a(t) = \frac{\tau_1^2+\tau_2^2}{\tau_1\tau_2^2}\, \mathrm{e}^{-t/\tau_2}\sin(t/\tau_1), \qquad h_b(t) = \frac{2\tau_b - t}{\tau_b^2}\,\mathrm{e}^{-t/\tau_b} , \]

with \(\tau_1 = 12.2\) fs, \(\tau_2 = 32\) fs and \(\tau_b = 96\) fs — and it carries what a scalar fit cannot: the isotropic response \(a = w_a h_a\) and the anisotropic one \(b = w_b h_b + w_{ab} h_a\) are separate functions of delay, so a cross-polarised probe sees \(a\) alone where a co-polarised one sees \(a + b\). The seam’s two channels are the trace and the traceless rank-two one, whose shapes integrate to \(w_a + (w_b + w_{ab})/3\) and \(2(w_b + w_{ab})/3\) — 5/6 and 1/6 for Lin and Agrawal’s weights, because \(w_a = 0.75\), and something else for any others — and sum to one. The scalar limit is the ordinary fibre model with \(h_R = a+b\) integrating to one. The Raman fraction is \(f_R = 0.245\) of the region’s own \(\chi^{(3)}\), and the instantaneous Kerr term keeps \(1-f_R\) of it.

The kernels carry \(\tfrac32 f_R\), not \(f_R\). \(f_R\) and \(n_2\) are envelope quantities: \(f_R\) is the share of the effective nonlinearity on the fundamental the delayed term carries. In the field domain the delayed polarisation’s drive is the cycle average \(|A|^2/2\), so a kernel of integral \(X\) contributes \(\tfrac12 X|A|^2A\) against the instantaneous cubic’s \(\tfrac34\chi^{(3)}|A|^2A\) — it acts like a Kerr term of \(\tfrac23X\) — and carrying the share \(f_R\) of one \(\chi^{(3)}\) needs \(X = \tfrac32 f_R\chi^{(3)}\) (SILICA_TENSOR_AMPLITUDE, applied in kernels and nowhere else). It is the same \(\tfrac32\), for the same reason, as the scalar mapping under Raman in a fibre GNLSE, and with it the two shares add to exactly one \(\chi^{(3)}\) on the fundamental: \(\tfrac23\cdot\tfrac32 f_R + (1-f_R) = 1\).

The endpoint quadrature is the trapezoid. \(h_b(0^+) = 2/\tau_b\) is not zero — this is the one kernel in the package that does not vanish at zero delay — so a kernel sampled on \(k\,\delta t\) and integrated by the rectangle-rule transform product misses \(\delta t\,h(0^+)/2\) and converges only to first order. The sample at the jump therefore carries the mean of its two one-sided limits, which is half \(h(0^+)\); that turns the sum into the trapezoid rule, restores second order, and costs no causality, because the half-weighted sample multiplies \(Q(t)\) itself and no earlier one. The late lobe of \(h_b\), where \(t > 2\tau_b\) and it goes negative, is left unclipped: it is part of the response and not a gain. Every other kernel in this package vanishes at zero delay, where the two rules coincide, so nothing else is affected.

This model is reached only through a material region of a general modal guide. response_for() does not return it: "SiO2" there is the scalar Hollenbeck–Cantrell shape of optmat’s default bundle, which is what a fibre GNLSE uses.

Dephasing, and what happens at zero density

solprop.nonlinear.raman.Dephasing turns a solprop.physdata.raman_types.Linewidth — full width at half maximum \(C + A/n + Bn\) in hertz, with \(n\) the density in amagat — into the factor \(\mathrm{e}^{-t/\tau_2}\), \(\tau_2 = 1/(\pi\,\mathrm{FWHM})\). The middle term is Dicke narrowing, which makes lines narrower as the density rises until collisional broadening takes over.

A coherence decaying as \(\exp(-t/\tau_2)\) has a Lorentzian spectral line with FWHM \(1/(\pi\tau_2)\) in Hz, or \(1/(\pi c\tau_2)\) in inverse metres. Its HWHM is half that number. The intensity of a CARS signal proportional to that coherence squared decays twice as fast, so an intensity lifetime is \(\tau_2/2\). These molecular linewidth conventions are separate from the angular-frequency Lorentzian/Gaussian parameters of the silica model.

At zero density there are three cases, and which applies is fixed by the coefficients:

linewidth	\(\tau_2(0)\)	meaning
\(A > 0\)	\(0\)	the Dicke term diverges: no coherence at all
\(A = C = 0\)	\(\infty\)	the linewidth vanishes: an undamped coherence
\(A = 0,\ C > 0\)	\(1/(\pi C)\)	the density-independent term survives

All three are handled without special cases in a run (a collapsed coherence is given a finite decay rate that kills it within one sample; see Raman numerics). The density multiplies the polarisation anyway, so at zero density the choice is invisible in the result and matters only for whether the result is a finite number.