= Spectral-density conversion between wavelength and Raman shift
\b[Spectral-density conversion between wavelength and Raman shift] preserves the integrated physical quantity while changing its horizontal coordinate. Fix the excitation wavelength $\lambda_0$, and let $s$ be the <Raman shift>. Using numerical vacuum wavelengths in nanometres and shifts in $\mathrm{cm^{-1}}$,
$$
s=10^7(\lambda_0^{-1}-\lambda^{-1}),\qquad
\frac{ds}{d\lambda}=\frac{10^7}{\lambda^2}.
$$
Let $D_\lambda$ denote detected counts per nanometre and $D_s$ counts per unit Raman shift. Conservation of counts under a <change of variables> gives
$$
D_s(s)\,ds=D_\lambda(\lambda)\,d\lambda,\qquad
\boxed{D_s(s)=D_\lambda(\lambda)\frac{\lambda^2}{10^7}}.
$$
The conversion factor carries the corresponding density-unit change. The same argument applies to calibrated power densities, provided both sides describe the same quantity.
For an original narrow bin centred at $600\,\mathrm{nm}$, a count density of $1200$ per nanometre becomes $43.2$ per $\mathrm{cm^{-1}}$. A one-nanometre bin there spans approximately $27.78\,\mathrm{cm^{-1}}$, so its integrated count remains about $1200$. At $800\,\mathrm{nm}$, the same wavelength density becomes $76.8$ per $\mathrm{cm^{-1}}$. Relabeling bin positions without adjusting the density would conceal this coordinate dependence.
For finite bins, use their exact transformed edges and preserve each bin's integral. <Raman spectral rebinning with conserved integrated counts> distinguishes this operation from interpolating peak heights onto a new grid. Instrument response correction is another step; the Jacobian alone does not calibrate detector sensitivity or remove background.
The primary FSRS apparatus calibrated Raman shifts using known solvent lines and dispersed the probe spectrum onto an array.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Instrumentation/Figure 2, spectral detection and absolute-shift calibration]{ref} Its dimensionless <Raman pump-on pump-off gain ratio> is a different observable: a ratio of matched spectral densities can cancel their common coordinate factor. It should not be treated as a count density merely because it is plotted against Raman shift.
Back to article page
