Evaluating a Raman response

A delayed response needs the field history over its memory time. The temporal window, response sampling and causal convolution each affect the result. This chapter derives the field/envelope drive, gain sign and numerical convolution used with the Raman models.

How the convolution is evaluated

The numerical response depends on which quadratic form of the field drives it, why the transform is taken on a doubled grid, which sign of the kernel spectrum means gain, and how the kernel is sampled and updated along a pressure gradient.

The drive, and why it differs between the grids

The coherence is driven by the intensity, and which real quadratic form that is depends on the grid, exactly as the Kerr cubic does. solprop.nonlinear.raman.raman_drive() implements

grid	thg	drive
field	True	\(E^2\)
field	False	\(\tfrac12\lvert E_a\rvert^2\)
envelope	—	\(\tfrac12\lvert A\rvert^2\)

The factor of one half is the cycle average of \(\cos^2\): \(E^2\) and \(\tfrac12\lvert E_a\rvert^2\) agree in their slowly varying part and differ only by the \(2\omega_0\) beat. When that beat lies well above the Raman lines, the response suppresses it strongly. For a 1030 nm pulse in nitrogen the two field-grid forms give polarisations differing by 2.4 × 10⁻⁴. The thg switch therefore matters far less for Raman than for Kerr, for that narrowband optical example. At longer wavelengths or with broad mixing bands, compare the beat frequency with the actual Raman spectrum instead.

On both grids the drive is real, and so is \(h\), so the convolution needs only a real transform pair, on an envelope grid as on a field grid.

The doubled grid

The convolution is evaluated as a product of transforms, and a transform product is a circular convolution: without padding, the tail of the response wraps back onto the head of the drive. Writing the response and the drive into the first half of an array of twice the length makes the product the linear convolution exactly.

This is not a refinement. Hydrogen’s rotational coherence at one bar lives for about 2.9 ns against a time window of tens of picoseconds; the wrapped tail would be almost the whole response. It costs one real transform of length \(2n_{to}\) in place of one of length \(n_{to}\).

The sample spacing that turns the discrete sum into the integral is folded into the kernel spectrum by solprop.translate.raman.kernel_spectrum(), so a backend computes

\[ \tilde P = \tilde H \cdot \mathrm{rfft}(\text{drive padded}), \]

and the inverse transform’s \(1/N\) supplies the rest. Folding \(\delta t\) into the kernel applies it once per density change, rather than once per column on the other side of the product.

The sign that means gain

In the internal FFT convention (conventions, §2) a Stokes component of an envelope sits at \(A \sim \mathrm{e}^{-\mathrm{i}\Omega t}\). The drive term at that offset is \(\tfrac12 A_p^{*}A_s\), the convolution multiplies it by \(\tilde H(-\Omega) = \tilde H(\Omega)^{*}\), and with \(\partial_z A_s = -\mathrm{i}\,\omega_s/(2\varepsilon_0 c n)\,P_s\) the growth rate is proportional to \(-\mathrm{Im}\,\tilde H(\Omega)\). So

\[ \boxed{\ \mathrm{Im}\,\tilde H(\Omega) < 0 \ \text{is gain on the Stokes side.}\ } \]

Analytically \(\tilde H(\Omega) \to -\mathrm{i}K\tau_2/2\) at resonance for a positive line weight \(K\), which is a minimum of the imaginary part. The test suite asserts it at the vibrational line of nitrogen and at every strong rotational line of hydrogen.

The kernel spectrum is an internal-convention array and must never be passed through solprop.fourier.to_optics(): that conjugation would silently exchange the Stokes and anti-Stokes sides, turning gain into loss without any error.

Sampling the kernel, and the density it depends on

solprop.translate.raman turns a response into the arrays a kernel consumes.

The tail is tapered by the grid’s own collar. A coherence that outlives the window has to be cut somewhere, and cutting it with a step puts a sinc ringing across the whole kernel spectrum. solprop.translate.raman.kernel_taper() gives the falling edge the width of the grid’s temporal collar, so the kernel is cut with the same profile, over the same fraction of the window, as everything else the grid absorbs: one statement of how much of the window is boundary, not a second one for Raman. A coherence that outlives the window is noted in the debug log, once per density, naming the dephasing time and the span that would hold it.

The padding prevents circular wraparound but does not remove this taper’s bias: a delayed probe samples an attenuated kernel if its lag behind the pump falls in the collar. Extend the window until the pump–probe lags that matter are untapered, and check convergence in the window separately from convergence in the time step. A long-lived coherence need not decay inside a short-pulse window when only shorter, untapered lags contribute to the observable. The size of the bias depends on the medium: for fused silica on a 9.1 ps window, whose response has decayed long before the taper, the tapered polarisation equals the untapered convolution bit for bit; for hydrogen on a 228 fs field window, whose rotational coherence outlives it by four orders of magnitude, it differs by 12 % at its peak.

The dephasing is applied per term, per density. The compiled response keeps separate undamped rows because the dephasing depends on the density, which varies along a pressure gradient, and a molecular gas has two dephasing times: at one bar hydrogen’s rotational and vibrational coherences decay 3.5 times apart, and nitrogen’s 17 times apart. Summing the models first and damping the sum once is wrong for every gas with both a rotation and a vibration.

Backends, and the drive form

Raman responses run on the supported NumPy, Numba and native routes. Check backend support for the geometry and execution space of your problem. Backend agreement measures numerical implementation of the same response, not agreement with a material outside its validated range.

On a field grid the Kerr and Raman responses must agree about thg; a Raman response without a Kerr response is refused on that route. On an envelope grid the Raman drive is \(\tfrac12|A|^2\), independent of thg, and an explicitly constructed Raman response can stand alone. Convenience constructors may impose additional joint-response choices described in the Raman workflow.

Two physical Cartesian components require a tensor-capable response. A scalar-only declaration or scalar silica model is refused for vector use. See vector Raman for the contraction and material assumptions.

The kernel contract states the kernel layout and backend mechanics. The qualification record retains the gain, soliton-shift and conservation checks and their measured errors. For a new run, vary the time spacing, window and longitudinal tolerance separately; a previous comparison does not bound the error of a different pulse or material.