1. Early Development

2. Configuration

3. Analysis

4. SHS interferometers
    SHIMMER STPSat-1 

5. References

Spatial Heterodyne Spectroscopy (SHS) is a technique which uses an interferometer for the spectroscopic, remote measurement of light from emission sources. SHS interferometers have been used in large-aperture telescopes^[1] and to measure the wind speed of oxygen in the upper atmosphere.^[2] The SHS technique uses an interferometer with arms terminated by tilted diffraction gratings to create a fringe pattern on a spatially extended detector array. The primary benefits of the SHS technique are that there are no moving parts, compact size, the relative ease of attaining high spectral resolution over a limited spectral range, and the ability to sample the entire fringe pattern in a single measurement. Changes in measured fringe patterns due to the motions of the emitting particles (Doppler Shift) can be analyzed to derive line-of-sight particle velocities.

SHS typically includes a measurement of the fringe pattern at zero optical path difference (OPD) between the two arms of the interferometer. This allows for an absolute phase measurement of the fringe. One technique derived from SHS is Doppler Asymmetric Spatial Heterodyne (DASH) spectroscopy which samples OPDs away from zero path length differential in order to maximize both signal throughput and the change in fringe pattern due to particle motion.^[3] Both techniques derive from the Michelson interferometer.

Early Development

The SHS concept was originally developed by Connes with the SISAM interferometer in 1958. Unfortunately, neither multiple-element detector arrays nor high-powered computers were available at that time which limited the practicality of the SISAM interferometer. It required a single channel scanned detector to sample the entire fringe pattern, essentially negating the benefit of having fixed gratings in the interferometer. Later, the entire fringe pattern would be imaged on films^[5], but without high-powered computers or the ability to easily digitize these films it was difficult to Fourier transform the imaged pattern.

It wasn't until the 80's and early 90's that SHS was rediscovered as a viable option for high-resolution spectroscopy. The advancement of computers and detector arrays, in addition to the use of field widening prisms to increase signal collection, allowed the development of SHS systems for the study of diffuse emission lines originating from the interstellar medium and atmospheric airglow.

Configuration

An SHS interferometer consists of a beamsplitter and two diffraction gratings. It can also include field-widening prisms which allow for the collection of more light (larger entrance aperture) without loss of coherence. Certain versions of SHS interferometers consist of a beamsplitter, only one field-widening prism, and only one grating. In this configuration two parallel light paths utilize half of each optical component.

During operation, light from an emitting source enters through an aperture into the beamsplitter (typically a double prism) and is evenly distributed between two paths. The light then travels through field-widening prisms and impinges upon the gratings. The gratings are tilted away from normal to the Littrow angle, which is the angle at which a certain wavelength of light is diffracted back along the path from which it came. Often this tilt angle is chosen to reflect one of the higher orders of the incident light (e.g. n=7 for the 630.038 nm emission line of oxygen) which eases the requirements on the number of lines per inch of the gratings and allows multiple wavelength ranges to be imaged simultaneously (e.g. also the n=8 order of 557.7 nm emission line of oxygen). This tilt can be thought of as transforming the gratings into a series of very small mirrors each of which has a different optical path distance from the beamsplitter. Thus, the system samples many optical path differences simultaneously, rather than sequentially like a conventional Michelson interferometer. As the light recombines in the beamsplitter and exits towards a detector array, the different optical path lengths cause either constructive or destructive interference, which manifests as a fringe pattern characteristic of the difference between the Littrow wavelength and the wavelength of the incident light.

Analysis

One method for analyzing interferograms obtained from an SHS interferometer is as follows:

The Doppler shift of a measured emission from a particle in motion will be evidenced as a change in frequency of the fringe pattern. For expected wind speeds in the atmosphere, this frequency shift is so small as compared with the total frequency that, at the optical path differences considered, it appears simply as a change in phase in the fringe pattern. Thus by examining the phase from a single interferogram, or fringe pattern from an interferometer, we can deduce the particle's line-of-sight velocity assuming that the zero velocity phase of the emission is already known.^[2]

Once an interferogram is obtained, the first step is to apply a Hann function to apodize the signal in order to reduce edge effects from non-periodic boundary conditions. Next, a Fourier transform is applied to change to frequency space. For an image with multiple wavelengths, there will be several components in the transform, including one near zero frequency representing the DC offset of the fringe pattern. Each line of interest will have a positive and a negative component in the frequency spectrum. To focus on only one line of interest, an isolating function is applied to the spectrum which suppresses all other lines but the positive component of that line.

Now the inverse transform is applied to the isolated component, which results in a signal with real and an imaginary parts and half the intensity of the original signal once the Hann function is removed. The phase of the original signal can be determined by taking the inverse tangent of the ratio of the imaginary part to the real part. This method has the benefit of removing any envelope function which may be affecting the interferogram. In atmospheric studies, this envelope may be a function of the temperature of the emitting oxygen, reducing fringe contrast with increasing optical path difference.

The resulting phase vs OPD is mod 2π, but the absolute phase at zero OPD is known since the phase at this location is unchanged by the emitting particle's velocity. A cumulative phase from zero OPD can be calculated by adding 2π at the phase discontinuities. This cumulative phase can be compared to that obtained from the zero velocity measurement (or from any other measurement for a relative velocity) to determine the Doppler shift and thus the velocity of the emitting particles. The measurements can be compared point by point or, because the resulting phase is nearly linear, either by slope or by an average cumulative phase. The benefit to using the average is that it is simple to calculate (compared to fitting a line) and incorporates all the values from all the detectors in the array, resulting in a smaller total uncertainty.

SHS interferometers

SHIMMER STPSat-1

Having proved the viability of the SHS concept with SHIMMER Middeck, NRL worked with the United States Air Force Research Laboratory (AFRL) to put SHIMMER on the STPSat-1, a STP demonstration satellite that launched March 9th, 2007 on an Atlas-5-401 from Cape Canaveral. SHIMMER flew for almost two and a half years on STPSat-1 and successfully collected data on UV emission (307.9 - 309.4 nm) between 30 - 100 km altitude. ^[1]

References

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