The Spatial Heterodyne Spectrometer


SHS_classic.jpg

Basically, an SHS is a Michelson interferometer, in which the mirrors are replaced by diffraction gratings, placed under Littrow angle for a certain design wavelength. The two wave fronts of an infinitely narrow emission line at Littrow wavelength returning from the two arms are parallel, leading to a bright or dark field in the output arm, depending on the difference in length of the two arms. For such a spectral line at an off-Littrow wavelength however, the wave fronts are tilted in opposite directions as shown in the figure above. To ensure this opposite tilt, the two gratings have to be operated in the same diffraction order (sign and value). One of the returning wave fronts is reversed by the reflection at the beam splitter while the other one is transmitted and keeps its orientation.

To acquire the spectrum from the interferogram, one has to know the fringe spacing s in dependence on the wavelength (derivation here):

fringe spacing

where λ0 is the Littrow wavelength, Δλ the deviation of the fringe-forming wave from the Littrow wavelength, d the groove spacing of the grating and θ0 the Littrow angle. The superposition of all these Fizeau fringes forms an interferogram, which contains information about the spectrum of the source. This interferogram will be recorded by a CCD or CMOS camera.

The achievable resolution of the SHS when the grating is used in first order was given (without a derivation) in [1]:


with the illuminated grating width W = N d, where N is the total number of illuminated grooves.  A derivation published in [2] assumes that the resolving power is determined by "the smallest frequency ... that can be resolved on the detector. Ignoring any optical magnifications, one obtains the smallest frequency (1/W) when only one fringe is recorded by the detector." This is simply wrong. One has to assume instead, that for two wavelengths, the number of fringes over the width of the aperture has to differ by at least 1/2 in order to be recorded as separate. This is a Rayleigh-like criterion in the following sense: If the fringes line up on one end of the interferogram, maximum and minimum coincide on the other end. With this, one would get a resolving power that is twice the amount quoted above. A derivation and discussion of this latter result is given in [3].

However, it's not worth scrutinizing this further, since there is a different phenomenon limiting the resolution of the SHS: The tilt of the energy front of the interfering wave packets, shown in the following figure:

tilt

This is a footprint of the SHS, in which the optical paths of the two arms are projected onto a single optical axis. The left side shows the incoming wave packet with a coherence length coh. The right side shows the situation in the output arm, where - due the relatively large Littrow angle of the gratings - the exiting wave packets have an opposite tilt γ (which is different from the tilt of the wave fronts Δα!).

The actual disadvantage for the resolving power of the SHS is now that this tilt of the energy front causes a lateral confinement of the overlap region, i.e. the region in which Fizeau fringes can be observed. In other words, the number of grating grooves that contribute to the interferogram is decreased, leading to a smaller resolving power:

resolution

The tilt γ becomes larger if one tries to increase the resolving power of the device by increasing the groove density N of the utilized gratings. Therefore, choosing the correct grating constant for a given light source is clearly an optimization problem.

References:

[1]   J. Harlander, R. J. Reynolds, F. L. Roesler Spatial Heterodyne Spectroscopy for the exploration of diffuse interstellar emission lines at far-ultraviolet wavelengths The Astrophysical Journal, 396 (1992) 730

[2]  S. Chakrabarti, D. M. Cotton, J. S. Vickers, B. C. Bush Self-compensating, all-reflection interferometer Appl. Opt., 32 (1994) 2596

[3]  M. Lenzner and J.-C. Diels "Concerning the Spatial Heterodyne Spectrometer" Opt. Express, 24 (2016) 1829  ( view )