 Spectral-density conversion between wavelength and Raman shift

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 $\mathrm{c}{\mathrm{m}}^{-1}$,Let $D_{\lambda }$ denote detected counts per nanometre and $D_s$ counts per unit Raman shift. Conservation of counts under a change of variables givesThe 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 $\mathrm{c}{\mathrm{m}}^{-1}$. A one-nanometre bin there spans approximately $27.78\mathrm{c}{\mathrm{m}}^{-1}$, so its integrated count remains about $1200$. At $800\mathrm{nm}$, the same wavelength density becomes $76.8$ per $\mathrm{c}{\mathrm{m}}^{-1}$. 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.^[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.