When analyzing the fringe pattern of an interferogram to determine atmospheric wind velocities, inhomogeneities in the optical components and illumination can introduce uncertainty into the results. These variations in the image, which are generally characteristics of the measurement device, are commonly referred to as the “flat-field” of the system. In this work we discuss the effect of this flat-field on measurements made with a Doppler Asymmetric Spatial Heterodyne (DASH) spectrometer. It is found that the flat-field can have a significant effect on any single calculation of the fringe phase, but because the flat-field affects all measurements made with the same system, the uncertainty in the derived wind velocity, which is determined through a comparison of two interferogram fringe phases, typically remains small. Nonetheless, it is recommended to account for the flat-field when analyzing DASH data, if possible. To this end we discuss a method for determining the flat-field using only temperature variations of the system, which is particularly suitable for space-based instruments.
© 2012 Optical Society of America
There is a widely recognized need for global-scale observations of thermospheric winds in order to advance our understanding of thermospheric and ionospheric processes and to improve specification and forecasting of the uppermost part of the atmosphere [1, 2]. Thermospheric winds can be observed by measuring the Doppler shift of naturally occurring atomic oxygen airglow emission lines at 557.7 nm and 630.0 nm or of molecular oxygen emission at 762 nm. To date, two systems have been used to make this type of measurement from a low Earth orbit satellite: the Fabry-Perot Interferometer [3, 4], which is also widely used for ground based observations [5, 6] and the stepped Michelson interferometer [7, 8].
A recently developed technique, Doppler asymmetric spatial heterodyne (DASH) spectrometry, offers a combination of advantages from these conventional techniques and is therefore an attractive candidate for future flight instrumentation . The DASH system, which is derived from the concept of spatial heterodyne spectroscopy (SHS), is similar to a Michelson interferometer, which is commonly used to perform Fourier-transform spectroscopy (FTS), except that the mirrors that terminate the arms of the Michelson interferometer have been replaced with fixed, tilted diffraction gratings . The gratings heterodyne the spatial fringe frequency around the Littrow wavenumber, σL, determined by the tilt and characteristics of the grating. Unlike an FTS interferometer, which sequentially scans the interferogram fringe pattern by changing the optical path length in one interferometer arm, the tilt to the gratings allow the SHS and DASH systems to image the entire interferogram simultaneously in a single exposure by using an array of detectors. The difference between the SHS and DASH concepts is that in the DASH setup an additional optical path difference is introduced in one interferometer arm in order to shift the measurement away from OPD = 0, which maximizes the information on the Doppler shift .
Doppler shifts from thermospheric winds cause a wavelength change of only a few parts in 107, which corresponds to a similarly small change to the spatial fringe frequency of the associated DASH interferogram. Defining the “fringe phase” at a given OPD as the cumulative phase of the fringe pattern starting from the smallest OPD, we find that even for such small frequency changes the fringe phase of a Doppler shifted DASH interferogram will be measurably different from that of an unshifted, or zero-speed, interferogram. Furthermore, this difference increases with increasing path difference. Therefore, the sensitivity of the phase response to the Doppler shift increases with optical path difference. However, an emission line with non-zero thermal width will exhibit an envelope on the interferogram resulting in a decrease in fringe contrast with increasing OPD. Thus, there is an optimal OPD, 2Δd, which depends on the width of the emission line in question, that maximizes the signal to noise for the combined effect of the increasing fringe phase difference and the decreasing fringe contrast . The atmospheric velocity, v, is derived from this phase difference through the following relation,11, 12].
DASH spectroscopy has several advantages over other techniques for observing thermospheric winds from space or from the ground. First, it benefits from relaxed tolerances for the optical components as compared to Fabry-Perot interferometers (FPI). Secondly, it can measure several emission lines simultaneously, including an on-board calibration source, which allows for constant, simultaneous calibration . However, the need for a detector array, rather than the single element detector that is often employed by FTS, raises concerns about the proper characterization of variations between detector elements and of spatial variations in the optics. These variations are typically grouped together in what is termed the flat-field associated with an optical system . As in SHS, the DASH flat-field consists of two components which will modify the result of any fringe phase analysis: a mutiplicative factor on the measured interferogram (e.g. from variations of the grating performance across the field of view) and sample-to-sample variations in the fringe modulation (e.g. from CCD pixel sensitivities). Fortunately, the flat-field is typically nearly time-invariant for a given spectrometer, so that the same flat-field is applied to every measurement taken with that system. Moreover, the flat-field is expected to have a similar effect on the analysis of both the calibration line fringes and the fringes from the atmospheric signal, so that the total effect on the phase difference is expected to be much smaller than for the individual phases. Another way to picture this behavior is to consider the flat-field effect in the spectral domain. In the spectral domain, a multiplicative flat-field results in a modified instrumental line shape function, which is convolved equally with the calibration line and with the atmospheric line. Thus, though this modified instrumental line shape can result in an effective shift to any measured line position, a similar shift will be imposed on any calibration lines as well, resulting in derived velocities which have only a weak sensitivity to the flat-field.
In this work we study the effect of the flat-field correction and its importance during the analysis of DASH data. The following section addresses in the most general form, the process with which a phase is derived from the measured interferogram. We assess how the flat-field plays into this process and what effects can be predicted from these equations. Section 3 provides a study of the error associated with ignoring the flat-field when using a reference signal to derive atmospheric wind velocities with a DASH interferometer. As expected, we discover that even though a single measurement is corrupted by ignoring the flat-field during analysis, a comparison between two fringe phases is much less affected by the inclusion of a flat-field. Section 4 provides one method for accurately deriving the flat-field correction using three interferograms with different phases. These phase changes are induced through temperature variations of the optics. This method is particularly attractive for satellite instruments, since no additional hardware is required to obtain these measurements. Section 5 summarizes the results of this study and discusses their potential impact on future systems.
We begin our discussion on the effects of a flat-field, F(x), where x is the distance along the detector, by giving a general derivation of the cumulative phase of a monochromatic interferogram fringe pattern. As mentioned above, the difference between the fringe phase measured from an atmosphere in motion and the phase from a motionless atmosphere, is proportional to the Doppler shift induced by the moving particles. A full discussion on the derivation of this phase may be found in Reference  by Englert, Babcock, and Harlander. We begin with Eq. (6) from that work which gives the intensity, I(x), for an interferogram composed of several lines in a single passband.
Now we apply a Fouier Transform, 𝔉(x,σ), to transform from the interferogram domain (pixel space) to the spectral domain (frequency space).Eq. (5) represents a complex function, including both real (ℜ) and imaginary () parts. Taking the reverse transform, 𝔉−1(σ,x), of (5) we find
Let us assume for a moment that F(x) = 1 (i.e. no flat-field). Because S is constant we find the following:Eq. (9) by Eq. (8) we find Eq. (7), if the flat-field only contains low frequency components, the high frequency isolation function gives T(σ)𝔉[F(x)S] = 0. However, even with the isolation function, the flat-field still remains as a multiplicative factor on the oscillating term in the phase calculation.
3. Effect of ignoring flat-field
In this section we examine the error generated when the flat-field correction is ignored. As we saw in the previous section, even with the high-frequency isolation function, the flat-field still modifies the cosine term, ultimately affecting the phase calculation. Thus, we expect that any absolute measurement will be shifted by some amount. However, the error associated with a relative difference in phase between two distinct measurements will be much smaller as the flat-field affects both in a similar way. We anticipate then that a Doppler-shifted measurement and a zero-wind or calibration source signal to which we will compare that measurement, will be (nearly) equally affected by the flat-field correction.
To verify this postulate we generated three sample synthetic interferograms using the specifications for the DASH interferometer system described in Reference . These interferograms corresponded to a zero wind signal (V0) and two Doppler shifted measurements constructed to exhibit positive and negative wind velocities (V+ = 20 m/s and V− = −50 m/s respectively). For reference, expected atmospheric winds are expected to reach speeds of up to several hundred meters per second. As stated above, it is prudent to use a source of known and stable wavelength (e.g. a neon lamp at approximately the same brightness as the emission line) to provide a simultaneous calibration of the measurement in order to track thermal and other potential drifts both during and between observations [11, 12]. For completeness then, we also include in our study a simulated calibration line to which we can reference all other measurements. The phase analysis of these four cases (“Reference”) gives the truth values to which subsequent tests were compared. Next, we selected three different types of flat-field, (Flat I, II, and III) and multiplied these to each of the four interferograms (cases Cal, V0, V+, and V−). The first test (Flat I) was a simple parabolic factor applied to the interferogram. This could be caused by an imbalance in the intensity of the light passing through each interferometer arm or some misalignment of the optics. The second flat-field (Flat II) was ∼1% noise applied across the CCD representing variations in the sensitivities of the CCD pixels. The final test (Flat III) was an actual flat-field as measured in the laboratory for a simple breadboard setup of a DASH spectrometer. This test includes elements of both previous tests. The zero wind interferogram, V0, can be seen in Fig. 1 for each of the three tests.
The results of the study are presented in Table 1. To simplify, we aggregate the fringe phase information into a single value by taking the mean of the fringe phase across the CCD . Even though there may be significant distortion in the linearity of the cumulative phase to OPD relationship for any given interferogram, the desired information is not contained in any single fringe phase, but rather in the comparison between two phases. Furthermore, it has been found that nearly identical distortion is evidenced in any other measurements made with the same system. Thus the averaging process ultimately reduces any uncertainty from statistically independent pixel-to-pixel noise . The average phases are given in the second column of Table 1. When the mean values of the cumulative phases from Tests I, II, and III are compared to their corresponding values from the Reference test (third column), we confirm that the flat-field has an effect on the absolute phase calculation. As it is practice to “calibrate” each atmospheric measurement against a known source, the phase differences between the atmospheric emission lines and the calibration line are given in the fourth column of Table 1 for each of the four scenarios.
To determine the wind speed through the Doppler shift of the emission, the calibrated mean phase of any atmospheric emission (ϕV+,− − ϕCal) must be compared to that of a calibrated zero-speed emission (ϕV0 − ϕCal). It is the phase difference between these two values that ultimately leads to the desired velocity. For reference, we have highlighted the calibrated zero-speed values in bold for each of the four tests. We find that these relative phase shifts are much less affected by the flat-field than are the absolute phases. This can be seen in the fifth column of Table 1 where we have converted this relative phase difference into a derived velocity by assuming an optimal path difference in the DASH spectrometer of 2ΔdOPT ≃ 4.2 cm for emission that comes from atmospheric oxygen (rest wavelength = 630.03 nm) at T = 1000K. As expected, we see in the final column that for all cases, we have less than 4% error between the expected velocity and the velocity derived from an interferogram with an associated flat-field.
The fact that the real flat-field case has the largest uncertainty emphasizes that any given flat-field may contain frequency components near lines of interest that would not eliminated by the use of an isolation function. Furthermore, we note that none of the calibrated zero-speed mean phases (bold values, column 4) are exactly equal. This implies that each flat-field has components that affect the analyses of the emission line and of the calibration line differently. Thus, we aver that, if possible, the flat-field should be accounted for during analysis, especially if there is limited opportunity to make zero-wind measurements and a reasonable probability for changes to the optical system through component degradation, internal relative motion, or damage. To that end we discuss in the next section one method of determining the flat-field through a controlled change in temperature of the system.
4. Use of temperature to derive flat-field
In the previous section we demonstrated that ignoring the flat-field correction caused only a small error in relative phase measurements from a DASH spectrometer. Nonetheless, it is prudent to account for the flat-field of the spectrometer, if possible. The method of determining the flat-field by sequentially blocking light from each arm of the interferometer  is often not feasible for space-based experiments because this process typically requires moving parts. On a satellite this corresponds to additional weight, power, and complexity. In Reference , Englert and Harlander discuss alternate methods of deriving the flat-field. Here we examine the potential of the three-phase method for use on space-based experiments employing DASH spectrometers. In that work several options are presented for obtaining the various phases for this analysis, including tilting the optics, changing the length of one arm of the interferometer, or inducing pressure changes in the optical path. Another method available to DASH interferometers specifically (because they do not sample OPD = 0, where the phase does not change) is the use of temperature variation in the optical components themselves to achieve a phase change in the signal. If successful, this method would be favorable for space-based applications as heating elements are already often employed on satellite systems to regulate equipment temperature.
As before, changes to the frequency of the measured emission line in a DASH interferometer are evidenced by a change in the fringe frequency of the interferogram recorded by the detector array. For very small changes in signal frequency, the slight change to the interferogram is most visible at large optical path differences and appears as a phase shift of the fringe pattern. Changes to the temperature of the system similarly affect the fringe frequency of the imaged interferogram and therefore also cause a change in phase for the fringe pattern in the OPD interval observed by the DASH spectrometer. Like the FPI, Michelson, and SHS systems before it, it is common for the DASH system to employ temperature controls to reduce thermal variations [11, 12] and to measure a calibration line to account for stray thermal effects in the system . However, it is possible to use this thermal sensitivity to determine the flat-field of the system. By changing the temperature we can sweep the fringe phase across every sample in the interferogram. In this way, we characterize the individual response to the strength of the incident signal for each detector element in the array. In practice, we compare two distinct phase shifted measurements of the interferogram, which allows us to separate the signal into a modulated and a non modulated part. A third distinct phase shifted interferogram is used to supplement the locations where the first two interferograms intersect and the original comparison is invalid . Knowledge of the absolute temperatures and phases is not necessary for this process as long as three well-separated phase patterns are obtained.
To demonstrate the efficacy of this method, we prepared a laboratory setup of a DASH interferometer with uncoupled optical components, thermal probes, and a nearby heat source. The system temperature was raised roughly one degree over the course of an hour and a half. Every five minutes, the camera recorded a 20 s exposure of the fringe pattern from a neon lamp of the type that might be used as a calibration source to track thermal drifts. As the system temperature slowly increased, the phase of the neon line changed. Figure 2a shows a sample interferogram, including an associated flat-field factor. Figure 2b displays a close up of a small feature of this interferogram caused by the flat-field, with the two additional phase-shifted interferograms shown in red and blue. These three interferograms were taken roughly eleven minutes apart.
Using the determined phase, Φi, and signal intensity, Ii, from each interferogram, i, j = 1, 2, 3, Englert and Harlander derive in  the following equations for the modulated, Mi(x,κ0), and non-modulated, Ni(x,κ0), parts of an interferogram, where κ0 is the frequency of the emission source:
Figure 3 shows the results of using the measured neon phases and intensities seen in Fig. 2 in these equations. For comparison, we include the flat-field as derived by the method of blocking light in each arm of the interferometer. Note that approximately 30 pixels at each end of the plot should be ignored due to the edge effects of the Fast Fourier Transform. Finally, Fig. 4 displays the original interferogram with the flat-field now removed. Notice that the feature seen earlier has been completely removed.
In this study, we demonstrated that because the flat-field error caused by variations in optical components is a characteristic of the system it affects all measurements roughly equally. This means that a comparison between two measurements, rather than a single absolute measurement, would be nearly unaffected by the flat-field. Also, we have shown that although the uncertainty introduced by a typical flat-field is limited to a few percent of the derived wind speed, it is possible to derive the flat-field even for systems with limited power, complexity, and size allowances like a space flight instrument. With only a modest change in temperature to the system, the phase variation needed to derive a flat-field for the system can be achieved. This temperature change can be achieved by a small adjustment of the interferometer temperature . The authors would like to thank Pat Bell for his support. This work is supported by the Office of Naval Research.
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