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1.IntroductionHigh-resolution broadband spectroscopy at near-infrared (NIR) wavelengths (950 to 2450 nm) has been performed using externally dispersed interferometry (EDI) at the Hale telescope at Mt. Palomar, with the TEDI interferometer mounted within the central hole of the 200–in. primary mirror in series with the comounted TripleSpec1 NIR echelle spectrograph. These are the first multidelay EDI demonstrations on starlight. We demonstrated very high () resolution boost and dramatic ( or more) robustness to point spread function (PSF) wavelength drifts in the native spectrograph. A companion paper2 (part 1) describes how to extend single delay spectroscopy3,4 into multiple delay spectroscopy, emphasizing data analysis, results, and instrument noise. EDI theory for radial velocimetry (RV) has been described,5,6 but not for multiple delay EDI used for general spectroscopy. (Single delay EDI has been called dispersed fixed-delay interferometry by other researchers6 using this technique for RV and used to discover a new exoplanet HD 102195b.7) This part 2 describes theoretical photon-limited and readout noise-limited behaviors, for both single- and multiple-delay spectroscopy, using simulated absorption and emission spectra and instrument model with noise added at the detector. The EDI forms Moire patterns by multiplying sinusoidal comb against the input spectrum in a heterodyning effect: where wavenumber has unit . Then, subsequent blurring by the native spectrograph , which removes high frequencies, does not significantly affect the Moire patterns, which are primarily at low frequencies. Equation (1) describes a single exposure () of a set of phase stepped exposures for a given delay, where phase (units of cycles) increments around the circle. Figure 1(b) shows simulated Moire patterns, where phase is splayed vertically continuously, whereas Fig. 2 shows how sinusoidal fitting along columns extracts fringing () and nonfringing () components. (In TEDI, the phase was varied temporally, since transverse detector space was needed to record several light beams. But it is useful to plot phase vertically analogous to the earliest EDIs, which splayed phase spatially across the detector.)The interferometer visibility ranges from 0.85 to 0.95 for TEDI. For simplicity, in the models we assume . Instrumental factors that could reduce include intensity imbalance between the arms (e.g., beamsplitter interface reflectivity changing) and imperfect optical surfaces. (Optics having flatness make this latter factor insignificant.) Heterodyning shifts (beats) high feature frequency () information down to lower frequencies, by the interferometer delay (unit: cm), where it is detected. It is later restored to its original high frequency mathematically. Figure 3 shows heterodyning for a single delay creating a new EDI sensitivity peak (red, ) that is a copy of the native spectrograph sensitivity peak (green, or ): but shifted to higher frequency by delay and having half the amplitude. (Feature frequency space has units of features per , i.e., cm, i.e., same units as delay space.) Lower case denotes space or Fourier transform versions of upper case functions in pixel or space.Under a Doppler shift, the phase of the entire Moire pattern shifts. Thus, EDI can sensitively measure Doppler radial velocity, even when the spectrograph has a very low resolution otherwise insufficient for this task. This was the motivation for inventing EDI. The primary reason for the TEDI project was to demonstrate this in the NIR. It uses the TripleSpec1 spectrograph, which bolts to the Cassegrain output of the Hale telescope (and hence, suffers changing gravimetric drifts). Its resolution of is otherwise insufficient for precision RV measurements. But with the TEDI interferometer, it was able to make precision RV measurements8 of M-stars. An equally useful application of EDI became apparent, which is wide bandwidth (BW) high-resolution spectroscopy. The same Moire pattern produced by the instrument for Doppler velocimetry is studied for its shape rather than its overall phase. The goal is to go backward through the heterodyning process to discover what high-resolution spectra would produce the measured Moire patterns. This is a method of measuring much higher effective resolutions than allowed by the native spectrograph, if the interferometer delay is larger than the width of the native response peak . Since EDI spectroscopy starts with the same Moire data as Doppler measurements, the secondary purpose of the TEDI project was to explore using EDI to make wide BW high-resolution stellar measurements. Single-delay EDI spectroscopy had already been performed,3 and multiple delays promised even greater resolution boost but had been performed only on laboratory sources.4 The TEDI project presented the opportunity to test this concept on stellar measurements. (However, the set of delay values was not ideal for spectroscopy–some were not evenly spaced over the delay range, having been selected for the RV purpose for targets having various rotational broadening.) We discovered that, indeed, wide BW high-resolution spectroscopy is quite practical. Part 1 (Ref. 2) shows the data analysis methods and results of many example high-resolution spectra, having resolution boosts up to (using delays up to 3 cm). Proportionately higher boosts could be achieved by purchasing several more glass etalons to allow a delay range up to 5 cm without gaps. We learned that an important advantage of EDI is not only the resolution boost but that the output spectrum is impressively insensitive to the native spectrograph instrumental distortions (that distort the shape and wavenumber position of the native PSF). Sections 9 and 10 of Ref. 2 describe how we observed a reduction to PSF drift insult using original lineshapes and a reduction with optimized lineshapes. The TEDI data were dominated by severe instrument noise, which was chiefly an irregular and large wavenumber PSF shift, rather than photon-limited or readout-limited noise, and this was the subject of part 1. However, naturally, we are also interested in how EDI responds to low flux environments, where shot noise and readout noise dominate. This is the subject of part 2, which is to study how adding simulated in Eq. (1) at the detector affects the final high-resolution spectrum. A detector noise, such as readout noise, has a constant magnitude (standard deviation), while photon (shot) noise is simulated by scaling by the square root of the noise-free version of . Both absorption and emission spectra have been studied. 2.Phase Stepping and Bandwidth2.1.Sine Fitting Along a ColumnThe process of fitting a sinusoid to a column of data (intensity versus phase at a given ) is called phase stepping arithmetic, and the particular case of four steps is called pushpull arithmetic. Its purpose is to separately extract fringing () and nonfringing () components from the set of phase stepped data , where the exposure number index is also along the phase axis. Consider it to be a sine fit along the column at a particular . Then, the sine amplitude is assigned to the imaginary part of , and the cosine amplitude is assigned to the real part of . The average value of the sinusoid is assigned ordinary spectrum’s value, or , at that wavenumber. 2.2.Generic Expression for N Ideal StepsThe generic expressions for many () regular steps that evenly fit around a circle (, in units of cycles) are where is called a “whirl” and manifests the fringing information we seek. Note the similarity of Eq. (4) to a discrete Fourier transform, which evaluates the sine and cosine amplitudes.2.3.Healing Method of TEDI Handles Many Irregular StepsThe TEDI project used 10 steps that were irregularly spaced around the circle. They were irregular because their value is proportional to the wavenumber, which changes across the band as . The change in delay is the physical constant related to mirror displacement (and glass dispersion causes a small dependence that is corrected for). Hence, the phases may be in an ideal configuration at one particular and then slowly move into an irregular arrangement, including wrapping around the phase circle, as changes across the wide band. To handle this irregularity, a “healing” algorithm was used, which adjusts the weights of each so that effectively one is mixing fractions of other phase steps into a given phase step, to alter its angle and length to bring it into the nearest ideal configuration. Then, Eq. (4) can be applied. 2.4.Low Readout Noise Motivates Use of Three or Four StepsSince readout noise combines in quadrature, the total readout noise grows as the square root of . Hence, in this part 2, when discussing minimizing noise for low flux observations, we are motivated to use the minimum number of step 3 to define a sinusoidal fit. Four steps produce more elegant equations and a slightly wider BW as we will demonstrate. For four exposures, every 90 deg: For three phase steps, symmetrically positioned (trigonal symmetry): 2.5.How Wide is the Bandwidth for Three or Four Steps?As the varies across the band, the phase step will change, gradually bringing it out of ideal configuration into irregularity. An irregular configuration will tend to increase the noise since the weights applied to each phase step to cancel out the nonfringing component, leaving the fringing component behind, tend to increase. The increased weights produce increased net noise (sum in quadrature of all the weights). We therefore have studied (Fig. 4) how the weights and hence net noise varies across the band, for three or four steps, in order to find the useful bandwidth BW. The results are shown in Fig. 4, which supposes that the ideal configuration is at , and that for other wavenumbers, the phase steps are proportionately different. The minima of these curves, i.e., the value of the noise (relative to a single readout) at the ideal is and for the case of readout noise, panel (a). For the case of photon-limited noise (b), there is no dependence on the number of steps since only the total number of detected photons matters; hence, the minima of the curves in (a) merge together in (b). The circular wheel-like diagrams show the configurations at several places across the band. Interestingly, note how the four-step configuration at appears approximately as a three-step configuration, because step 3 overtakes step 0. In this region, we average steps 3 and 0 together and use the three-step formula [Eqs. (9) or (10)]. Then, for larger , step 3 passes counter-clockwise away enough from step 0 that once again a four-step formula [Eq. (6)] produces a smaller noise. (However, note that for the use of the formula, we would relabel the steps so that they increase in index counter-clockwise. We do not relabel them in the figure, so that the reader can track the movement of the steps.) 2.5.1.Bandwidth of three steps is usefully wide, 1.8:1We define the practical BW as where the noise does not increase more than 20% over the minimum four-step value (of ). We observe that the multiplicative BW for four steps is 2.3:1, and for three steps using the same absolute noise threshold is 1.8:1. This is comfortably large, sufficient to handle the visible band (400 to 700 nm, 1.8:1) and most of TripleSpec1 BW (950 to 2450 nm) of 2.6:1. 2.6.Details on How Irregular Weights Were CalculatedLets us describe how we discovered the weights (that multiply each exposure) for the three or four steps configuration, as it becomes irregular. When the phase angles deviate from ideal, the weights deviate from unity. The net noise is the sum in quadrature of these weights. A simple method for discovering these weights is to treat the four data as pair differences, and this style of analysis is called “pushpull,” and it has been used extensively by the first author in analyzing other interferometric data (e.g., Secs. A and B of Ref. 9). Code already developed for pushpull analysis was used. The four steps are divided into pair differences: which are assigned to represent the real and imaginary parts of , as shown in Fig. 5. Then, Vert versus Horiz is plotted to form a Lissajous, which is elliptical generally since the weights are inappropriate for the phase step angles. We desire it to be circular of unity radius to indicate the weights are now correct. (For elegance, we temporarily use a factor of 1/2 instead of 1/4 in Eq. (6), so that the radius can be unity, and then apply another factor of 1/2 later in the processing.)Rather than adjusting individual weights for the four , we adjust other gains that we have created that manipulate the difference horiz and vert, and another gain to apply a linear transformation that corrects for the so-called obliquity of the data. These gains were adjusted until a circular Lissajous was obtained from the data, when test data were used that created a helical . That is, over a small test range of , where the helical data make a single revolution. When the correct weights are found, the Lissajous will be a circle of radius unity. Initially, when unity weights are used for the irregular condition, an elliptical Lissajous is produced, as shown in Fig. 5 having obliquity.The idea is to temporarily replace the actual data with the helical data, find the appropriate gains or weights that make a circular Lissajous, and then apply these gains on actual data. The more irregular the phase configuration, the larger the obliquity. As “Vert” deviates from orthogonality of “Horiz,” the obliquity grows. Also, as the magnitudes of Vert and Horiz decrease, as illustrated by panel (c), where the phase steps are dramatically far from ideal, the size of the Lissajous proportional decreases. Then, the overall gain must be increased to restore Lissajous radius to unity. This increases all the weights, which increase the net noise. The obliquity is related to how far from orthogonality is the Vert compared to Horiz. The gains of the differences and the gain of the obliquity operation are adjusted to produce a circular Lissajous. Then, the equivalent values of the weights (assigned to each ) are calculated and summed in quadrature to yield by what factor the noise increased. As the phase increases from the ideal configuration, the Lissajous deviates from a circle to an ellipse having obliquity different from unity. We define obliquity as the ratio of the major and minor axis of the ellipse, and they align along the and axes when the phases are nonideal. We created an obliquity operation that temporarily transforms the data into two new axes along and , applies a gain that diminishes the length along the axis, and then restores the data to the original axes and . This linear operation has the effect of mixing phase step components: Hence, it is related to the Healing method, which also mixes components. The is the nominal situation not requiring a change in obliquity. One could redefine obliquity to change the minor instead of major axis, by substituting in the equation. Since the obliquity operation changes the radius of the circular Lissajous that is achieved, the final step is to apply an overall gain adjustment that affects all the individual weights the same to bring the circle radius to the desired value of unity. Then, from these operations, we calculate their equivalent effect on the individual weights, and sum these in quadrature to yield the ratio of increase in noise over the ideal value. For configurations that had three inputs, we used Eqs. (8) and (10) to create and . These, of course, produce an elliptical Lissajous for irregular phase steps. To correct this by conveniently using the existing code written for four steps, we map the three steps to four ersatz steps through and adjust the gains to make a circular Lissajous of unity radius.3.Single Delay Noise Behavior3.1.OverviewWe will first discuss a single delay behavior, because multiple delays is a simple extension. 3.1.1.Concrete case of noise reduction for 1× boosting evaluatedFor a concrete example, we will examine the boosting case, where the new information (fringing information) is used to reduce high frequency noise rather than increase the resolution. The motivation is to provide a demonstration and detailed analysis for the leftmost data point of a later graph [Fig. 15(a)] that claims the boosted photon-limited noise for an EDI spectrograph is times less than a conventional spectrograph, which some readers may find surprising. Examining this case also illustrates the relative contributions to a boosting case, since the amount of boost is merely a matter of a different choice in the equalization (EQ) multiplier, which is the final step, and the EQ does not change the signal-to-noise ratio (SNR) curve since both signals and noises are multiplied by the same factor. 3.1.2.Numerical simulator used to study noise propagationTo study the propagation of noise, we use a numerical simulator, which uses similar algorithms for processing the data as the actual TEDI code, after the process of phase stepping or sine fitting. Here, the phase stepping process uses four constant steps at 90 deg because the issues of phase step irregularity and its dependence have been separately studied in Sec. 2. The numerical simulator has been used to study various types of noises (shot, detector) under various spectra types (absorption, emission), using various processing choices (bell weighting yes/no, various types of EQ). The simulator equations are in Appendix A. 3.1.3.Externally dispersed interferometry calculator produces smooth theory curves quicklyA second useful tool is the “EDI calculator,” which is a set of equations that can be quickly evaluated that lack the stochastic variations of the simulation and do not require a specific input spectrum. We have confirmed that these replicate the numerical simulator. The calculator displays the responses of the various components, signals and noises and their ratio, versus feature frequency . The calculator equations in Appendix B are based on theory, except for the case of the smooth change between uncorrelated and partially correlated behavior found empirically by the numerical simulation. In that case, the EDI calculator uses a best fit [Fig. 8(b)] modeling that observed behavior, since we have not yet developed a complete analytical explanation for it. 3.2.Simulator Results for 1× Boosting and Emission SpectroscopyFigure 6 shows a portion of the hypothetical emission spectrum made by spikes of random heights (a) prior to blurring and (b) after blurring to resolution 3725 at and adding simulated shot noise (nonfringing component is shown). (The actual TEDI native resolution was closer to 2700, but we are not trying to strictly reproduce TEDI, but use calculationally or graphically convenient parameters.) When simulating absorption spectroscopy, we used a hypothetical spectrum having a continuum. We simulated photon noise by scaling the noise magnitude as square root of local intensity and detector noise by having fixed standard deviation. The sinusoidal interferometer transmission comb was multiplied against the input spectrum [Eq. (1)] to form four signals in 90-deg phase relation and blurred. The blur and interferometer delays were adjustable. Figure 1(b) shows the appearance of similar signals, but for an absorption spectrum and displaying continuous phase along -axis that produces a smoother appearance. The nonfringing signal was produced by summing the four signals [Eq. (7)] to cancel fringes. The fringing component was obtained using Eq. (6), which subtracted exposure pairs and assigned them to real and imaginary parts of . The computational grid had a spacing of , a sufficiently small spacing to hold high-resolution signals boosted 10 times the native resolution. The boundaries of the Fourier space (Nyquist frequency) were , which is many times greater than the 0.22-cm half width at half max (HWHM) of the native response peak . This comfortably accommodates delays at least up to 5 cm. Figure 7(a) shows the simulation results after the fringing component was shifted in frequency by the delay [via Eq. (22)], to restore the high frequency information to its original high frequencies. This creates a wavelet appearance to the fringing result in red. The source spectrum had a small section 7420 to that was a pure sinusoid of frequency 0.22 cm. This made it easy to visually confirm that the relative component magnitudes from the simulation output agreed with the EDI calculator at a specific of 0.22 in (c), indicated by heights of the colored dots. The difference between the results having added noise at the detector and no added noise was subtracted to produce residuals (b). Note that the EDI result (purple) has less high frequency noise than the conventional result (black dashes), which confirms our claim that adding an interferometer to the spectrograph can reduce high frequency noise, for the same detected flux. We calculate noises on the components “bass” (nonfringing) and “treble” (fringing), and their combination “net.” Fourier transform of the noises shows how residual noises vary with frequency. 3.2.1.Fringing versus nonfringing noises, correlated or uncorrelated?Figure 8 shows how repeated instances of the numerical simulation with photon noise were used to study how the type of noise varied between the two extremes of correlated (sum linearly, open circles), or uncorrelated (sum in quadrature, open squares), depending on the delay value. We discovered that the behavior has a peak, which is well fitted by the native response peak . However, the peak does not reach the perfectly correlated level when the delay approaches zero. Instead, it is 0.7 of this distance. Since this was only recently discovered, we have not yet found an analytical source for the 0.7 factor but suspect that it is . When the type of noise was switched from photon to detector, the peak disappeared and the noise acted uncorrelated for all delays. This result can also be argued analytically. A correlation between nonfringing and fringing-like signals involves integrals over products like , where represents the noise apart from the signal. Hence for the detector case the correlation is approximately zero, because the detector noise variance is nominally the same between the exposures. This argument does not require the heterodyning which involves the delay. For the photon limited case the noise depends on flux, so could differ from . This produces some correlation, confirmed by the peak at small delays in Fig. 4(b). The heterodyning contributes the delay dependence by providing another mechanism for noncorrelation at large delays. 3.3.1× Boost Simulation Result in Frequency SpaceFigure 9 shows stages in the calculation of the SNR curves (versus ) for a delay of 0.4 cm, which is on the wing of the native response peak and a typical placement of the peak. The EQ step is omitted to better show high frequency behavior. Figure 10 is the same as Fig. 9 but with EQ applied, to achieve the same final resolution for the net EDI as in the conventional result, i.e., achieving boost. A different EQ could have produced a boosted resolution of . The EQ step does not change the SNR curve shape since it multiplies both signal and noise. 3.3.1.Photon-limited noise caseThe simulation output in Figs. 9 and 10 is the photon-limited noise case. The SNR for the net EDI is plotted as a thick purple curve in (d) showing a hump at 0.4 cm, where the fringing peak (red curve) contributes. This hump exceeds the conventional SNR (black dotted Gaussian curve) significantly for frequencies above the midpoint of the conventional response, say 0.3 cm and higher. Figure 11 evaluates the root mean square (RMS) average noise in more detail. 3.3.2.Readout noise-limited caseThe simulation uses four steps, whereas a conventional measurement only a single readout (unless the conventional is dithering by two exposures to reduce fixed pattern noise, something that EDI fringing signal is immune to since it already dithers). Then, for the readout noise-limited case, we divide the SNR output by or to simulate the readout noise-limited case for four (thin purple) or three (long dash thin purple) reads [Fig. 9(d)]. Similarly, in Fig. 10(a), we divide the dotted black photon-limited curve by to represent the single readout noise dominated case (black long dashes). Also, a recent paper10 proposes using a readout noise-free electron multiplying charge coupled detector (EMCCD), with fast scanning of interferometer phase to improve Doppler precision. 3.3.3.Fixed pattern noise rejectionIn the presence of significant fixed pattern (FP) noise, it is fair to assume the conventional technique will dither at least two exposures (such as shifting to an adjacent row of pixels) to cancel this artifact. Since EDI automatically rejects FP noise by its uses of differences between exposures, then for three exposures EDI readout noise would be only higher than conventional. 3.3.4.Effective flux increaseWe saw in Fig. 10(a) that the EDI (purple) suppresses high frequency noise relative to the conventional (dotted) photon noise. Figure 11 shows RMS averages evaluating noise fraction relative to the conventional photon-limited noise, for various regions of integration related to the number of pixels per resolution element in space. The latter sets the Nyquist frequency, which is the right-hand limit. (These plots are similar to Fig. 10(a) but using the EDI calculator curves.) Figures 11(a) and 11(b) show that the net EDI has less noise than the conventional, 60% (or 73%) of noise, for 3 (or 2) pixels per resolution elements. Figures 11(c) and 11(d) show that noise fractions are even smaller (34% and 46% for 3 or 2 pixels) when only high frequencies above 0.22 cm (HWHM the native response) are considered. This is justified since science signals typically require the highest resolution of an instrument (e.g., detecting and locating presence of smaller neighboring peaks). This amount of noise reduction is conventionally accomplished by increasing flux by a factor times. Hence, this an effective flux benefit. In the readout noise-limited case, the conventional noise is either or times smaller. Assuming four reads for EDI, then the 0.34 fraction becomes 0.68. This means it still has less noise than the conventional, when considering just the higher frequencies that typically manifest the science. This virtual flux benefit of EDI must be weighed against the decrease in flux due to insertion of the interferometer from parasitic reflections. The topic of how to best construct an ultralow insertion loss interferometer is interesting and saved for a future paper. We speculate that the monolithic prism interferometer, similar to Refs. 11 and 12, is a fruitful avenue. 3.3.5.Conclusions on single delay caseWe show that a single interferometer delay can be used to reduce the high frequency noise at the original resolution (“ boost” case), and that except for delays much smaller than the native response peak half width, the fringing and nonfringing noises act uncorrelated and add in quadrature. This is due to the frequency shifting of the noise due to the heterodyning effect. 4.Multiple Delay Noise Behavior4.1.Uniform Exposure ScheduleHaving explored the single delay case, we can extend our understanding on multiple delays. The total input flux will be subdivided into delays (Fig. 12). We will show below that the peaks add in quadrature to a fixed total sum. Thus, for the case of uniform exposure times for all delays, the height 0.5 of the single fringing peak is subdivided to a height of . Note how the higher delay peaks extend above a hypothetical conventional spectrograph (cross hatching above blue dashed curve) having the goal boosted native resolution. This shows that the EDI can produce SNR that exceeds the classical at the very highest frequencies, which are the most important frequencies. This figure and discussion assume that the grating throw length is fixed in size, which is relevant for airborne and spaceborne platforms, where volume and mass are critically limited. The blue dashed comparison curve was calculated as if it was an ideal conventional spectrograph with no intrinsic lens blur, achieved by reducing the slitwidth by and decreasing the flux by for an extended source (also requiring more pixels). This is just an artifice to remember EDI behavior, and the native spectrograph could be, for example, an adaptive optics (AO) enhanced spectrograph without a slit. For an AO spectrograph to increase its resolution conventionally, it would increase its throw length, since its focal spot size cannot be made smaller. If the instrument throw length grows by a factor 10, then native spectrograph (AO or not) has its weight and volume grow by roughly (ignoring material properties and if all dimensions scaled). In this case, the comparison response curve would have the same width as the blue dashed curve, but have the same height as the native, i.e., unity on that figure, not . Then, the EDI has less SNR than the AO result by , but it needs the pixels and the volume to accomplish this, a heavy cost to pay. 4.2.Gaussian Exposure ScheduleFigure 13 illustrates a Gaussian schedule of exposure time (flux) per delay. This is an optimal distribution of exposure time to produce a Gaussian final lineshape for the signal, while producing a white (uniform versus frequency) distribution for the noise. This is useful because a Gaussian lineshape lacks ringing and simplifies comparison to conventional spectroscopy, which typically has approximately Gaussian lineshape. The sum rule for peak heights squared was used to redistribute the flux in a Gaussian schedule while preserving the total flux. Due to the square root relationship between noise and flux, this means the exposure time Gaussian is narrower than the desired SNR Gaussian (blue dashes). 4.3.Velocimetry Exposure ScheduleFor certain kinds of spectroscopy, such as Doppler RV, or for elucidating the most narrow features of a spectrum, neither a Gaussian nor a uniform flux schedule is optimal. Instead, it is best to concentrate the exposure time for a certain range of high delays (frequencies), where the most Doppler science lies [see gold dashes of Fig. 13(a)]. The optimal frequencies for Doppler velocimetry are found by taking the derivative of the stellar spectrum and finding the maximum in its Fourier transform. For sunlight, this is a broad peak between 0.5 and 1.5 cm (see Fig. 9 of Ref. 5). 4.4.Sum in Quadrature Rule for PeaksLet us demonstrate that when we use bell-shaped weighting, the net SNR versus curve produced by combining fringing components of different delays and the native nonfringing component is a sum in quadrature. The SNR is a ratio. For uniform flux, the noise denominator is the same magnitude between different delays. And for other flux schedules, we normalize to force the denominators to be the same magnitude. Hence, we need only to discuss the numerator, which is the EDI sensitivity plots. Therefore, we use plots, such as Fig. 12, that show the fringing response peaks to also represent the photon SNR peaks. Consider two overlapping peaks 1 and 2, which could include the native peak or different delays. We calculate net SNR by summing the signal S linearly, but combine the noise in quadrature. We use weightings and associated with each peak. Hence, By bell-shaped weighting, we mean that the weight has the same shape as the signal: and . Because we normalize the noise denominators to be the same, . Hence, and similarly for multiple peaks. Thus, we have shown the SNR sum in quadrature. (We calculate that the net noise level is not very sensitive to the weighting shape. Rectangular weightings of 1.2 to 1.8 times peak full width at half max (FWHM) produce of bell weighting case.)The total exposure time may be allocated among the delays in various schedules, which affects their heights while representing the . Since the square of SNR peak height is proportional to the number of photons detected for a delay, and the sum of these is fixed to the total exposure flux, then we have a sum rule for peak heights [Figs. 12 and 13(a)]. The sum rule also works for the area under the curves. 4.5.Distribution of Noise, White or Pink?Suppose we are not doing velocimetry and thus desire Gaussian final frequency response. Then what is the final frequency distribution of the noise, after any EQ step? Figure 14 answers this question, showing Fourier transforms of the noise (residuals from ideal) from a numerical simulation, for cases of (a) uniform or (b) and (c) Gaussian flux scheduling, and also comparing (c) photon and (b) detector types of noise. 4.5.1.Uniform exposure scheduleFor the case of (a) uniform exposure time for each peak, the noise distribution is initially uniform (white). However, a Gaussian EQ is eventually applied to produce the desired Gaussian behavior in the sensitivity. Since the noise is embedded with the signal, the noise also receives this EQ shaping. Hence, the final frequency distribution of the noise in the uniform schedule is Gaussian. It could be called pink noise, having more noise energy at lower frequencies. 4.5.2.Gaussian exposure scheduleFor the case of a Gaussian exposure time schedule versus peak #, the signal distribution is initially Gaussian, and the noise is the square root of that Gaussian. We apply an EQ that divides by this square root Gaussian. This leaves the signal as a Gaussian having a wider width and leaves the noise uniform (white noise), as shown by Figs. 14(b) and 14(c). (Hence, the exposure time Gaussian is narrower than the blue dashed curves of Fig. 13.) This simulation also shows that (c) photon noise, which involves the square root of the flux, and (b) detector noise, which is independent of flux, produce similar overall magnitude of noise, after compensating for the average continuum level of the native spectrum. Hence, there are no surprises for EDI when estimating photon noise by inspecting the average continuum level. 4.6.Selecting Your Delays at Blue End of BandFor a fixed resolving power , the width in -space of the fringing peaks decreases as one moves to the blue (increasing ), since increases and thus decreases. Hence, there is a danger that gaps may open up between the different delay peaks in the blue while being adequately overlapped in the red. Since there is little penalty for having too much overlap (other than excessive number of delays and hence readout noise), but a severe penalty for a gap (which causes a divide by zero blow up in the EQ and hence increases noise), we recommend selecting the delay positions at the blue end of the band, by subdividing the delay range needed to produce a certain final resolution by the width of . 4.7.Readout Noise Case for MultipleAnalogous to the single delay case, the calculation output here for multiple delays is for photon-limited noise, neglecting readout noise. Then to account for readout noise relative to conventional single readout, we divide the EDI SNR peaks by factors of or for three or four phase steps. We also have to create two different versions for the native peak, one assembled from three or four exposures, and hence having a SNR reduced by or . (This would be the EDI one.) The other would be the original native peak, which would represent the conventional measurement done in a single read but four times longer exposure. The final factor to consider is the increase in readout noise versus due to changing phase step size, which is from Fig. 4(a). 4.8.Actual Delay Positions for TEDIIn contrast to the uniform spacing of delays used in this theoretical discussion, the actual delay values used in the TEDI interferometer were irregularly positioned across delay space and had gaps. This was due to the delays being primarily chosen for precision RV, anticipating different sources having different rotational broadening. This required different delay positions over a wide range, and we had only eight positions in our rotary “filter” holder that held the glass delay etalons. Figure 21 of part 12 plots the fringing peak positions in delay space for the TEDI instrument and indicates the TripleSpec native peak as a green peak at the origin. This plot for modulation transfer function is essentially a plot for SNR, since the noise denominator is uniform. The eight delay values for September 2010, labeled E1 to E8, are 0.083, 0.34, 0.66, 0.96, 1.27, 1.75, 2.92, 4.63 cm. In June 2001, the E1 position was swapped for a new delay called E6.5 having 2.4 cm to fill the gap between E6 and E7 (1.75 to 2.92 cm). This still left a gap between E7 and E8. In principle, a 10-position rotary holder holding two more delays at 3.5 and 4.0 cm could have made a contiguous coverage. This would have allowed minimal ringing (Gaussian) resolutions up to 36,000 (at ), rather than the 27,000 we produced. (Processing 36,000 with a delay gap would have produced significant ringing in the lineshape.) 5.Quantifying Performance Relative to ClassicalFigures 13(b) and 13(c) red curves show that in the limit of numerous, heavily overlapped, multiple delays having a Gaussian flux schedule, the asymptotic behavior of the EDI is similar to a classical spectrograph of boosted resolution, but having a reduced SNR as , consistent with either more read noise from the required more pixels (to maintain fixed number of pixels per resolution element), or more fractional photon noise from less flux (as if the flux was decreasing from a reduced slitwidth acting on an extended source). Let us quantify the performance of the EDI result by the fraction of this “classical” result. Let the effective Gaussian height (EGH) be the -intercept (SNR), where the Gaussian intersects as the number of delay increases to asymptotically produce a smooth curve, and let FC be the fraction relative to the classical result . For example in (b), the boost is , and the classical value would have EGH at SNR of . The net EDI (red curve) is fitted by a Gaussian shape (blue dashes) of fixed resolution having an SNR EGH of 0.25. This is the classical result diminished to height, so FC is 0.78. Then, the EGH intercept 0.25 is plotted in Fig. 15(a) as a purple dot at boost = 10. 5.1.Plotting Fraction of Classical versus BoostHow does the EDI compare to the classical result, and how does that vary with boost? Figure 13(b) shows that for a boost, we achieve 78% of the classical SNR, and [c] for boost, we achieve 95% of classical SNR. Figure 15(a) shows in purple dots how the EGH or -intercept for SNR () varies with a variety of boost values. The green dashed line is the “classic” standard, which has a square root dependence because we are assuming its flux decreases as 1/boost. Note the crossover near boost of . For larger boosts, the asymptotic behavior is that the EDI has a square root dependence like the classical, but a factor worse, and for smaller boosts than 4, the asymptotic behavior is a factor better than classical. The reason EDI can be slightly better than classical for lower boosts is that then the native sensitivity peak is much more included in the Gaussian fit of the combination. For very high boosts, the native peak is so much higher than the EDI fringing peaks that it is not effectively included in the Gaussian that must fit both, in the manner described in the caption of Fig. 15(c), which requires the RMS of an EQ curve to be unity. 5.2.Including Focal Blur in Native ModelA more realistic spectrograph will act differently from our classic standard. Namely, it will have a minimum focal blur (FB) that convolves with the slitwidth to put a ceiling to the resolution even while the flux decreases because of the decreasing slitwidth. This behavior is shown in Fig. 16 in the green curve, in comparison to the EDI result in black diamonds. This is a theoretical comparison between EDI and conventional photon noise behaviors versus final resolution, when the native spectrograph used for both is operated near its resolution limit, and where the slitwidth is summed in quadrature with a constant FB to calculate the native resolution. It illustrates the key point that at some point, every dispersive spectrograph will reach a resolution limit, which is controlled by the FB or the detector pixel density. 6.Comparing TEDI Instrument and Photon Noises6.1.TEDI Instrument Noise Roughly 3%For TEDI instrument noise, we observe a large shift (along wavenumber) of the native PSF versus time, as much as in the A-order, as shown by Fig. 35 of part 1.2 Since the resolution is about in this order, this is a very large relative instrumental insult. Because it varies in magnitude and even polarity across the band, it cannot be removed by a simple monolithic shift. To express this as a vertical (intensity) noise to compare it to the photon noise, we subtract each of the individual spectra from the average spectra. Figure 38(c) of part 12 shows the residual is between 2 and 5%, which we call roughly 3%. 6.2.TEDI Photon Noise Roughly 0.1%Figure 11 of part 12 shows that a typical single exposure of the phase stepping set has of order 10,000 to 30,000 counts per pixel for the continuum portion. (It represents an average single exposure, not yet summed over 10 steps. This may not have been clear from that figure caption.) Summing over the 10 exposures per phase set, the total count is 1 to , and since there are 3.8 photons per count, this is 4 to photons per pixel. Since there are per native resolution element, there are about 1 to photons per native resolution element. Taking the square root, this yields a photon noise of order 0.1%. 6.3.Noise Contribution PlotFrom what we have learned about TEDI noise from this and the companion paper (part 1),2 we assemble a very approximate picture in Fig. 17 of the various types of noise contributions and how they might vary versus resolution boost ratio. For TEDI, the native spectrograph instrumental noise is about 3% due to drifts of the PSF. By contrast, the photon shot noise is about 30 times smaller at about 0.1%. This plot is suggestive and not rigorous — useful for identifying which issues to further investigate. The red and green symbols denote EDI and native spectrograph. When boost varies for EDI, it uses the same behavior of the native but with different delay arrangements. But when boost varies for the native, it is used alone and the EDI behavior is not recomputed to use the higher resolution of the new native. Readout noise is neglected here since the photon flux of TEDI was very high. Native photon: The native classic photon noise would grow as square root of resolution as the flux is assumed to decrease linearly with resolution (we are considering the case in which the spectrograph throwlength is fixed). EDI photon: The EDI photon noise has the S-shaped behavior, crudely similar to the square root classic behavior, but better than classic by at low boosts and worse than classic at high boosts (from Fig. 15). Native instrumental: Measured vertical error from TEDI’s native spectrograph PSF drift in A-order is 2% to 5%. Regarding its dependence on the boost, this approximately follows a power law of 3/2 for blended lines (hence low res) and of power of 1 for isolated lines (at high res). For TEDI data at resolution , it is in between these values. EDI instrumental: For low boosts of , the interferometer comb in the wavelet is as nearly as large as the wavelet envelope. This makes the EDI result more susceptible to native PSF changes. Hence, we place the EDI dot near but slightly below the native dot—below it because the EDI also eliminates fixed pattern noise, so that counts for something. By contrast, at higher boosts such as , the interferometer comb period is so much finer than the wavelet envelope that it is insensitive to the PSF drifts. Calculations artificially shifting the Moire data of a ThAr line show that TEDI has at least less horizontal reaction to a PSF drift, (and less using a more sophisticated “crossfading” process that modifies the lineshapes. See Sec. 10 of Ref. 2). But at a higher boost, the slope of lineshape that connects between vertical and horizontal errors has increased by , so the net downward movement of the red dot is . This reduces the EDI instrumental noise to a 1% level, less than it would be with the conventional alone. Instrument noise is still dominant over estimated EDI photon noise of 0.2% at boost. 7.Zoology of Different Spectroscopy MethodsFigure 18 is a notional plot comparing several methods of spectroscopies in Fourier space (delay space), including FTS. The general goal is essential to map the Fourier information of the source spectrum over a delay range. The higher the maximum delay, the higher the achieved resolution. Apparatus photos and example data of several different kinds of dispersed interferometers are shown in Ref. 13. Different techniques accomplish the mapping in different manners. Either (a) all at once (purely dispersive), (b) subdividing the delay space in a series of discrete chunks (EDI), (c) one chunk that is continuously scanned (dispersed FTS), (d) spatially recording the delay range at once [spatial heterodyning spectroscopy (SHS)], or (e) an extremely narrow spike that is scanned (pure FTS). 7.1.Comparison to Dispersive SpectrographsThe purely dispersive method (a) has a peak at the origin whose width is proportional to spectral resolution. Let the FWHM be (which is also the approximate the rightmost extent of the delay range for a FTS or EDI to map out to the wing of the Gaussian). Then, from the uncertainty principle: which can be easily remembered as the number of wavelengths that fit into . The , wavenumber when wavelength is in cm, and maximum interferometer delay in cm. So, a Res 50,000 dispersive spectrograph at wavelength has a peak 2.5 cm in half width, or to the wing of the Gaussian.Panel (b) shows the EDI, which maps delays space in chunks (one peak per delay) set by the native peak (green), which now can be narrower (lower resolution) than in the purely dispersive case (a). The heights of the peaks are relative to the native. Importantly, and the subject of the companion paper,2 the center of each peak is the most stable region against PSF translations, and the EDI places these at high frequencies, where the science information resides. By contrast, the classic spectrograph has this stable region at zero frequencies, which does less good for the science signals. Hence, the EDI can be an order of magnitude more robust to PSF drifts for the important high frequencies. 7.2.Comparison to Fourier Transform SpectroscopyThe purely interferometric method (e, FTS) scans the interferometer delay continuously over the delay range, recording the Fourier information with an extremely narrow peak. It then Fourier transforms this into a spectrum. The scanning delay requires a time responsive detector capable of recording high frequencies and prevents use of integrating detectors. By contrast, the EDI does not scan a delay continuously but sits in several discrete positions. It can use the slow (integrating) but sensitive CCD detectors already present in astronomical spectrographs. Thus, EDI can be an add-on unit to enhance existing spectrographs. For measuring single shot or rapidly changing phenomena, we have designed an EDI using multiple delays in parallel on different detector regions to make snapshot measurements (see Figs. 12A and 12B of Ref. 14). The EDI signal has lower photon noise15 than the FTS by the square root of the number of native spectral resolution elements, because the disperser isolates adjacent wavelengths on the detector having independent phases. For an echelle spectrograph that is of order , the EDI can have a better photon SNR over the FTS by a factor of to . 7.3.Comparison to Dispersed-Fourier Transform SpectroscopyIn a dispersed-FTS16,17 shown in Fig. 18(c), a FTS is in series with a disperser. The latter increases fringe visibility, so its photon limited SNR is intermediate between purely dispersive and purely interferometric cases. It is sufficiently high to allow it to measure stellar spectra and Doppler velocities, such as the spectroscopic binaries measured at the Steward Observatory 2.3 m Bok telescope.16,17 The scanning delay requires time responsive detectors. 7.4.Comparison to Internally Dispersed InterferometersThe internally dispersed interferometer techniques called SHS12,18 or heterodyning holographic spectroscopy19 is related to an FTS, but where the delay range is recorded at once splayed spatially, and thus can use an integrating detector rather than being scanned over time [Fig. 18(d)]. Similar to the other hybrids, the photon-limited SNR is generally intermediate between purely dispersive and purely interferometric extremes, because there is some overlap of signal between pixels, which degrades SNR by a square root effect, as described in Eq. A40 of Ref. 19. The SHS technique is known for its very high resolution at high etendue of extended objects such as the atmospheric glow. Recording the delay range at once makes the instrument very rugged and well suited for aerospace platforms, such as described in Ref. 12, for measuring upper atmospheric wind by emission lines. The one-dimensional (noncross-dispersed) internally dispersed interferometer can have a significantly reduced BW, because it produces a fringe comb, whose period varies strongly with wavenumber (due to the internal grating that changes the angle of interference), and thus, the fringe frequency can exceed the pixel pitch outside of a BW. Within this band, the resolution can be extremely high. (By contrast, the EDI has an almost uniform interferometer comb period. This allows a much larger BW, limited only by the native spectrograph.) However, newer cross-dispersed SHS have been demonstrated18,20 that produce a two-dimensional interferogram, and these have a much wider BW than the noncross-dispersed type. 7.5.Comparison to Super-resolution Techniques in 2-D ImagingRelated mathematical methods of enhancing image resolution by increasing its width in Fourier space have been developed in other fields. For example, the microscopy method of “structured illumination”21 creates Moire patterns with spatial grids at various orientations. We caution the reader not to confuse EDI with the “superresolution”22 technique of photo enhancement that relies on the alias signal developed when the signal is undersampled and translated in subpixel displacements. Our technique does not use the alias signal and excludes it from the processed signal by filtering. We avoid using the alias signal because it is susceptible to irregular placement of the pixels, which can occur on a subpixel level, say, 0.1 pixel. This level may not be of concern to ordinary imaging but is significant to spectroscopy. 7.6.Relation to Amplitude Squeezed LightBecause the noise in the fringing and nonfringing signals is uncorrelated, when the two signals are combined, the net SNR can be better than the native nonfringing signal used alone. A related sub-shot-noise behavior has been previously observed by other researchers23 (a topic called “amplitude-squeezed light”), using a conceptually similar experimental arrangement to EDI, but without the spectrograph. Namely, they have an interferometer with detectors on both (complimentary) outputs. This detects a photon both by summing the complementary outputs (the classical way) while also simultaneously subtracting the two complementary outputs; combining both signals produces a sub-shot-noise level of net SNR. 8.Concluding RemarksWe show that a single interferometer delay can be used to reduce the high frequency noise at the original resolution (“ boost” case), and that except for delays much smaller than the native response peak half width, the fringing and nonfringing noises act uncorrelated and add in quadrature. This is due to the frequency shifting of the noise due to the heterodyning effect. We study the change between uncorrelated and partially correlated noise as the delay goes to zero. We find a sum rule for the noise variance for multiple delays. The multiple delay EDI using a Gaussian distribution of exposure times has a noise-to-signal ratio (NSR) similar to a classical spectrograph with a proportionately reduced slitwidth to achieve the boost in the classical manner, but without the focal spot limitation and pixel spacing Nyquist limitations. That is, . At low resolution boosts (), the EDI has slightly smaller () noise than the conventional, and at higher than four boosts, the EDI has slightly larger () noise than conventional. The better than conventional noise at low boost is due to combining fringing and nonfringing components while their noises are uncorrelated due to heterodyne shifting. The worse than conventional noise at high boosts is due to the factor 2 smaller height of the single delay fringing peak relative to the native, and the sum rule that spreads this over several delays. The readout noise decreases as the square root of number of reads, motivating use of three or four reads instead of the 10 used in TEDI. With the irregular phase steps stemming from using three or four phase step exposures at changing wavenumbers, the BW is still comfortably large () sufficient to handle, for example, the visible band (400 to 700 nm, 1.8:1). 8.1.Uncertainty Principle FollowedSome readers may find it nonintuitive that including an interferometer can boost resolution (or equivalently, decrease noise at constant resolution). Consider that the coherence length of the light passing through a high-resolution grating spectrograph is broadened more than passing through a low resolution spectrograph (imagine a single perfectly short input pulse). The output will consist of a train of pulses, one per grating groove. Thus, by including an interferometer of significant delay with a grating, the output will also have a broadened coherence length, comprising the convolution of the grating pulse train with a two-pulse impulse response (Fig. 19). Following the uncertainty principle, the increased coherence length is consistent with a spectral resolution increase. This increase incoherence length causes greater ambiguity for time scales of or or 166 picoseconds. This is not a problem for astronomy which typically measures much slower phenomena. The increased ambiguity applies to any method of increasing spectral resolution including those in conventional dispersive spectrographs. Operating a high-resolution spectrograph system is then a business of producing very long coherence lengths, and doing it in a very controlled manner. The dithering of an EDI interferometer delay is very controlled, and the interferometer has only three degrees of freedom. By contrast, the grating has a multitude of degrees of freedom, at least one per grating groove, and many of these are uncontrolled by environmental insults. 8.2.Externally Dispersed Interferometry Can Benefit Adaptive Optics SpectrographsThe EDI can be useful in boosting the resolution and stability of an AO spectrograph, whose resolution is limited by the number of detector pixels. Ordinarily, to increase the resolution of an AO enhanced spectrograph (or classical spectrograph limited by FB), one increases the throw length, and the number of pixels, by a factor boost. This has the disadvantage of increasing the volume and weight of the instrument by a power law having an exponent between 2 and 3, with the expense growing nonlinearly as well. Many airborne and spaceborne platforms have severe weight and volume constraints. Hence, using EDI with an AO spectrograph is an attractive means of achieving higher resolution and stability without exceeding the financial weight and volume limits. 8.3.Externally Dispersed Interferometry Can Benefit Integral Field SpectrographsEDI can benefit integral field spectrographs that strive to produce spectra for each point on an image, because these systems are typically starved of pixels. Even if AO is used to produce a diffraction-limited focal spot, the spectral resolution is still limited by the paucity of pixels in the spatial dimensions. Consider an objective prism creating a rainbow for each object. One cannot spread the rainbows over very many pixels without danger of overlapping rainbows of adjacent objects. The EDI can boost the spectral resolution by inserting an interferometer along the beam path. By taking a series of measurements at different delays and combining the results, the EDI boosts the final resolution to supersede what pixels allow in a single exposure. Provided a fairly light efficient interferometer is used, we believe it would benefit spectrographs to have an EDI as a front end. The suppression of fixed pattern noise and the enhancement of the stability of the PSF through the sinusoidal fiducial comb are just as important of advantages as the resolution boost, especially for precision radial velocity, which requires an extremely stable PSF. New data analysis methods that can further improve the PSF stability, potentially up to , by crossfading overlapped pairs of delays by reshaping the lineshape is an exciting new development (Sec. 10 of part 1).2 8.4.Interferometer Fiducial Comb Lowers Spectrograph CostThe EDI enhances PSF stability through its sinusoidal fiducial comb, which is embedded with the input spectra and shifts along with it under a PSF shifting insult. Thus, the Moire (which depends on difference between spectrum and comb) is largely robust to a PSF shift. By enhancing the robustness of the native spectrograph with the fiducial comb, the structural and optical tolerances of its design can be relaxed, saving cost and weight. It is possible that the bulky and heavy vacuum tank enclosing some spectrographs could be eliminated. Optical mounts can be made lighter. Thermal expansion can be less worrisome not requiring special materials. The diffraction grating and other optics could be optimized to maximize throughput rather than reduction of aberrations. Some lens elements or mirrors may be eliminated, increasing throughput and decreasing weight. Hence, the EDI presents a new and potentially useful leverage on the design trade study, which could potentially improve the SNR achieved at a given needed resolution. AppendicesAppendix A:Numerical Simulator EquationsThe equations used for the numerical simulation of the EDI are below. The simulator begins at Eq. (1) with a computation in wavenumber space, using a hypothetical spectrum that is multiplied by a sinusoid and then blurred. Calculational pixels are wide. The blur is 40 pixels () FWHM at for a native resolution of 3725. The added random noise is of either two types, detector (magnitude independent of flux) or shot noise (magnitude scales as square root of prior to noise). We used a noise standard deviation amount of 3% in some simulations and 30% of continuum in others. The calculational pixels were . The fringing () and nonfringing () components are separated by idealized four step Eqs. (6) and (7). The variation of phase step with across the band is ignored here and treated elsewhere. This is justified since EDI works at a local level as small as an individual resolution element. Since it is easy to forget a factor of 2 somewhere in the math chain, especially if sometimes we use a Fourier transform displaying both frequency branches and at other times a single branch, we confirm the correct relative sizes of the and components in the simulation results by inspecting the output of the purely sinusoidal test section of the input spectrum, 7420 to in Fig. 7(a) with the expected magnitudes from the theory in frequency space indicated by red and green dots of panel (c). Prior to heterodyning, reversal of some filtering is performed to kill off high frequencies that contain mostly noise and little or no signal. This can improve the photon-limited SNR by a factor of because it prevents two frequency branches of noise, positive and negative, from combining into the final signal. At this point, the signal lies in the neighborhood of zero, so we optionally apply low pass filtering. While rectangular lowpassing is the easiest to code and is the minimal amount done, it is optimal to apply a filter passband shape that has the same shape and magnitude as the expected signal response, which has a shape , which we model as a Gaussian. We call this “bell-shape weighting” to be more generic, denoted by . This is really meaningful in the region, where two different signals overlap (bass and treble, or trebles that belong to multiple delays), since for an isolated component, the EQ function would undo the effect of any weighting. The net noise changes slowly from the use of a nonideal shape or magnitude of weighting. Gaussian shaped (bell-shaped) weightings are applied: or if no weighting is used. The factor 0.5 for weighting is because its expected response is half that of the nonfringing component.Reversal of heterodyning is more accurately computed in pixel or wavenumber space on , even though we discuss it as being shifted in frequency space by an amount of delay : The polarity of the exponent is chosen to restore the Moire signals to the high frequencies they originally had in the input spectrum . A continuum in the input spectrum makes a sinusoidal comb in the measured . Our convention is that we assign that a positive frequency. Hence, during heterodyning reversal, we apply a negative exponent to shift that down to zero frequency to recover the constant continuum. The EQ is chosen to reshape the hump like shape of : into an ideal Gaussian of a user selected width (choosing the final resolution). The EQ is the ratio of desired shape divided by actual shape, which is where to prevent divide by zero blow ups.The factor 0.5 is because the fringing response peak is half of the native. This includes the contribution of the conjugate (or “twin”) treble peak in the other (negative) frequency branch, which can bleed over into the positive branch for very small delays relative to the native width. Appendix B:Code for Externally Dispersed Interferometry CalculatorThe EDI calculator is a faster way to visualize EDI behavior than the numerical simulator, and it lacks the statistical variations. It differs from the simulator by not using a specific input spectrum. It shows frequency response (-space). It is based on equations that we have confirmed reproduce the numerical simulation results. Script written in Wavemetrics Igor data analysis application. Functions below are in Fourier space, which is feature frequency space having variable rho(units cm). ================ Gaussian Peak maker ======================== function GaussRsp(Res,aveWn,tau) // Calculates gaussian blurring vs rho, of grating variable Res, aveWn, tau // Res is resolution, aveWn is in cm-1, tau is delay in cm return exp(-(((x-tau)*1.133*aveWn)/(0.6006*Res))^2); // This is wavenumber savvy end ================= Signals prior to EQ, and weighting ======== //<Bass signal, called hSigBassFinal> hBass = GaussRsp(3725,7450,0); // Makes Gaussian peak centered at delay (3rd parameter), widthfrom resolution in //first parameter, and average wavenumber in 2nd parameter. //<Treble signal, called hSigTreb> hTreble = 0.5*GaussRsp(3725,7450,tauG); hTrebleTwin = 0.5*GaussRsp(3725,7450,-tauG); // This is the conjugate treble peak, on the otherfrequency branch, //that can bleed over when delay is small. ================ Make Weighting: Bell, or none ============== k0 = hBass // Make weighting have same bell shape as expected signal. kedi = hTreble // Make weighting have same bell shape as expected signal. kediTwin = hTrebleTwin // Make weighting have same bell shape as expected signal. or if desiring no weighting, make all weights = 1 ================ Signals after weighting, prior to EQ ======= hBassFinal = hBass*k0 = hBass*hBass HTreble2 = hTreble*kedi = kTreble*hTreble HTrebleTwin2 = hTrebleTwin*kediTwin = hTrebleTwin*hTrebleTwin hTrbBoth = HTreble2+HTrebleTwin2 //< Net EDI signal, called hFinal> hAll = hBassFinal + hTrbBoth ================ Make EQ multiplier ========================= // Currently the net signal has a hump and thus does not have a ideal Gaussian shape. The purposeof the EQ is to force // the final net signal into a Gaussian shape. The final width is chosen by user. Boost = 1 // We are examining the particular case where final resolution is same as native. hBassWider = GaussRsp(3725*Boost,7450,0) // This one differs from hBass by having resolutionincreased by Boost hEq = hBassWider/(hAll+1e-10) // The 1e-10 useful for preventing blowups ================ Signals after EQ ========================== hSigBassFinal = hBassFinal*hEq // Equalization hSigTreb = hTrbBoth*hEq // Equalization hFinal = hAll*hEq // Equalization =============== Noises ===================================== //<Bass Noise, called hBassN> hBassN = 1*k0 = 1*hBass // The noise (constant) times the bell weighting If no bell weighting, then hBassN = 1 hBassN *= hEq // Equalization //<Treble Noise, called hTrbBothN> hTrbBothN = sqrt(kedi*hTreble + kediTwin*hTrebleTwin) = sqrt(hTreble^2 + hTrebleTwin^2)// Add conjugate from other //frequency branch in quadrature, after applying weightings If no bell weighting, then hTrbBothN = 1.414 hTrbBothN *= hEq // Equalization //<Net EDI Noise, called hPurpleN> //We will calculate both linear and quadrature versions, then mix them appropriate to the delay.Unless delay is very //small, quadrature is the usual result. hAllN = sqrt(hBassN^2 + hTrbBothN^2) // <Sum in quadrature version> hAllN *= hEq // Equalization hAllNL = hBassN + hTrbBothN // <Linear version> hAllNL *= hEq // Equalization if (YesShotTypeNoise) // If Shot noise, then its somewhere between Quadrature and Linear(uncorr and corr). // Use Fig. 8(b) (noise vs delay), and MaxCorr*hBass(tauG) to estimate where in between thetwo asymptotic behaviors. // The peak is empirically found to fit a shape that is same as hBass, but only 70% high, //not the full 100% between the linear (upper) and quad (lower) behaviors. MaxCorr = sqrt(0.5) // Empirically found to be ~0.7, guessing its 0.707 hPurpleN = hAllN*(1-MaxCorr*hBass(tauG)) + MaxCorr*hBass(tauG)*hAllNL else hPurpleN = hAllN // If detector noise, then purely quadrature endif ================ SNR ===================================== hPurpleSNR = hFinal/hPurpleN <For Net EDI, no readnoise case (photon limited)> hPurpleSNR2 = hPurpleSNR/2 <For Net EDI, simulate four reads by dividing by sqrt(4)> hTrebSNR = hSigTreb/hTrbBothN <For Treble component, fringing> hConvSNR = hBass/1 <Bass is same as hBass because noise is uniform> AcknowledgmentsThis material is based upon work supported by the National Science Foundation under Grant Nos. AST-0505366, AST-096064, PAARE AST-1059158, NASA under Grant No. NNX09AB38G and by Lawrence Livermore Nat. Lab. under Contract No. DE-AC52-07NA27344. Thanks to Ed Moses for his valuable support during the genesis years. Thanks to Y. Ishikawa, E. McDonald, W. V. Shourt, A. M. Vanderburg, J. Wright, D. Harbeck, S. 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BiographyDavid J. Erskine has been an experimental physicist at Lawrence Livermore National Laboratory since 1987 and has experience in femtosecond lasers, semiconductor physics, superconductivity, diamond anvil cell high pressure physics, shock physics, high-speed recording techniques, Doppler interferometry, white light interferometry, digital holography, Fourier signal processing, image reconstruction, and phase stepping algorithms for interferogram analysis. Since 1998, he has collaborated with astronomers to innovate interferometric techniques for the Doppler planet search and high-resolution spectroscopy. He is a member of SPIE. Edward H. Wishnow received his PhD in physics from the University of British Columbia. He is now a research physicist at the Space Sciences Lab at UC Berkeley. He is working on stellar interferometry and spectroscopy in the midinfrared and visible. Martin Sirk has over 30 years of experience in the design, construction, calibration, and science analysis of astronomical instrumentation. This has included working with Digicon detectors on the Hubble Space Telescope, CCD detectors on ground-based telescopes, microchannel plate detectors and optics on six NASA missions (EUVE, FUSE, ORFEUS, CHIPS, SPEAR, ICON), and photographic plates at Lick Observatory. |