The
Sagnac Fourier Spectrometer

Basically,
we are looking at a Sagnac interferometer where one of the
mirrors is replaced by a transmission grating. Since the
action of a transmission grating is reversible, both
directions experience the same diffraction under the same
angles at a given wavelength λ_{0}.
At this wavelength, the setup works like a standard Sagnac
interferometer. For a single wavelength (with zero bandwidth)
and the device at rest, there is a homogeneously illuminated
field in the output arm (actually dark, since they are exactly
180° out of phase and therefore interfere destructively). For
wavelengths off this design wavelength however, the
propagation direction (k-vector) is under a small angle to the
optical axis, resulting in a tilted wavefront. When these
tilted wavefronts interfere in the output arm of the
spectrometer, they lead to Fizeau fringes. An odd(!) number of
mirrors causes the two wavefronts in the output arm to be
oppositely tilted and is therefore indispensible.

The sign
convention for transmission gratings is that angles on the
same side of the grating normal have the same sign on both
sides of the grating. Hence, in the figure above, both angles
α and β are positive WLOG. For the wavelength λ_{0}+Δλ,
the spacing Δx of these fringes is:

where g
is the groove density of the transmission grating. The
wavelength λ_{0}+Δλ is diffracted under the angle β,
the angular dispersion is Δβ/Δλ = g/cosβ. That means, the
angular dispersion (and the resolving power of the
instrument) increases with the angle β, obviously only
limited by the clear aperture.

**
References:**

[2] Matthias Lenzner "Sagnac Fourier Spectrometer (SAFOS)" US Patent 10,571,442 ; issued on Feb 25, 2020