A Blazed Gratings and Littrow Mounting

[fsonnichsen]

TL;DR Summary

How do blazed gratings improve efficiency in spectrometer

I cannot for the life of me find any concise information on how blazed gratings improve the spectral throughput of a spectrometer system. The blazed grating structure and how it redirects spectral reflection is itself understandable- but any information I have indicates that it somehow "concentrates" the diffracted light energy into a higher order spectrum. What I see is that the specular output is indeed reflected away from the m=0 order, but beyond this I do not see how the specular content leads to an enhanced diffraction/ spectrum. In fact it would seem as those this light would obscure the higher order into which it is reflected.
I have consulted the ubiquitous Born & Wolfe on this (no information at all!), Hecht (meager information with no mathematical analysis) and Jenkins & white, not to mention a half dozen books on spectrometers. All simply state that the blaze, when mounted at the Littrow angle for the wavelength, mysteriously enhances the signal. I need the math here, or at least a ray-trace sketch to figure out how this happens.

I am trying to track down the 1916 paper by Wood and Trowbridge but meantime does anyone know a textbook (or even an online white paper) working through this problem?

Thanks
Fritz

 

[Charles Link]

The diffraction intensity is given by the formula for the interference of $N$ sources, $I=I_o \frac{\sin^2(N \phi / 2)}{\sin^2(\phi / 2)}$, ($\phi=\frac{2 \pi d ( \sin{\theta_i}+\sin{\theta_r})}{\lambda}$), multiplied by the diffraction pattern of a single slit, (or Huygens reflector). It is this second term where the blazed grating offers higher efficiency. The diffraction pattern from a single reflector peaks where angle of incidence=angle of reflection. Here it is a reflective type grating (rather than transmissive). The blaze makes it so that this peak occurs at some other angle than the m=0 maximum.

 

[Andy Resnick]

Summary:: How do blazed gratings improve efficiency in spectrometer

I cannot for the life of me find any concise information on how blazed gratings improve the spectral throughput of a spectrometer system.

Richardson Grating Labs has a 'handbook' that may be of help:
https://www.gratinglab.com/Information/Handbook/Handbook.aspx

In particular, there is a chapter on designing gratings for maximum efficiency. That handbook references a paper by Loewen:

https://www.osapublishing.org/ao/abstract.cfm?uri=ao-16-10-2711

that may have some useful details.