2.1.

Sine Fitting Along a Column

The process of fitting a sinusoid to a column of data (intensity versus phase at a given $\nu$) is called phase stepping arithmetic, and the particular case of four steps is called pushpull arithmetic. Its purpose is to separately extract fringing ($\mathbf{W}$) and nonfringing ($B_{\mathrm{ord}}$) components from the set of phase stepped data $B_n(\nu )$, where the exposure number index $n$ is also along the phase axis.

Consider it to be a sine fit along the column at a particular $\nu$. Then, the sine amplitude is assigned to the imaginary part of $\mathbf{W}$, and the cosine amplitude is assigned to the real part of $\mathbf{W}$. The average value of the sinusoid is assigned ordinary spectrum’s value, or $B_{\mathrm{ord}}$, at that wavenumber.

2.2.

Generic Expression for N Ideal Steps

The generic expressions for many ($N$) regular steps that evenly fit around a circle (${\phi }_n=n/N$, in units of cycles) are

2.3.

Healing Method of TEDI Handles Many Irregular Steps

The 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 $\mathrm{\Delta }\phi =\nu \mathrm{\Delta }\tau$. The change in delay $\tau$ is the physical constant related to mirror displacement (and glass dispersion causes a small $\nu$ dependence that is corrected for). Hence, the phases may be in an ideal configuration at one particular $\nu$ and then slowly move into an irregular arrangement, including wrapping around the phase circle, as $\nu$ changes across the wide band.

To handle this irregularity, a “healing” algorithm was used, which adjusts the weights of each $B_n$ 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 Steps

Since readout noise combines in quadrature, the total readout noise grows as the square root of $N$. 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 $\nu$ 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.

Fig. 4

Change in relative noise of fringing component $\mathbf{W}$ due to irregular spacing of the phase steps as they grow across the band, for cases of 4 and 3 steps, and (a) for readout noise and (b) photon dominated noise. For a constant physical delay change as the interferometer mirror is stepped, $\mathrm{\Delta }\tau$, the corresponding phase change varies with wavenumber as $\mathrm{\Delta }\phi =\nu \mathrm{\Delta }\tau$. We define the ideal regular phase configurations (four at 90 deg, three at 120 deg) to occur at $\nu$ of $\mathrm{10,000}{\mathrm{cm}}^{-1}$. When the phases deviate severely from their ideal uniform configuration and cluster in two groups on opposite sides of the phase circle, then the obliquity of a Lissajous plot of $\mathbf{W}$ increases dramatically, increasing the weights needed on data to achieve circularity, increasing the net noise. (a) Vertical axis is readout noise relative to a single readout. An arbitrary definition of practical BW is the noise increasing 20% over 90 deg configuration. Relative BWs of 2.3:1 and 1.8:1 are produced for 4 and 3 exposure cases, respectively. These are sufficiently wide for most kinds of spectroscopy (visible band 400 to 700 nm is 1.75:1).

The results are shown in Fig. 4, which supposes that the ideal configuration is at $\nu =10000{\mathrm{cm}}^{-1}$, 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 $\nu$ is $\sqrt{3}$ and $\sqrt{4}$ 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 $13000{\mathrm{cm}}^{-1}$ 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 $\nu$, 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.)

2.6.

Details on How Irregular Weights Were Calculated

Lets 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:

Fig. 5

How nonideal phase steps can still measure the whirl $\mathbf{W}$, but with magnified noise. (a) Four ideal 90-deg phases, and intensities $B_n$ associated with the vector positions of steps ${\mathbf{S}}_n$. Real and imaginary parts are formed from intensity differences $\mathbf{W}=0.5(B_0-B_2)+i0.5(B_1-B_3)$ associated with vector differences $({\mathbf{S}}_0-{\mathbf{S}}_2)$ and $({\mathbf{S}}_1-{\mathbf{S}}_3)$. For discovering proper weights, “helical” data are used, $B_n=\cos i2\pi (\nu \tau +{\phi }_n)$, which normally creates a circular Lissajous, imaginary part versus real part (red circle). (b) Nonideal phase steps, and $({\mathbf{S}}_0-{\mathbf{S}}_2)$ is no longer orthogonal to $({\mathbf{S}}_1-{\mathbf{S}}_3)$, and Lissajous has obliquity. After weights are adjusted, circularity is achieved. But sum in quadrature of weights has increased, increasing noise. (c) Extremely poor phase step configuration producing very oblique ellipse. Increase of dot product between $({\mathbf{S}}_0-{\mathbf{S}}_2)$ and $({\mathbf{S}}_1-{\mathbf{S}}_3)$, and decrease of their lengths, increase the noise.

Rather than adjusting individual weights for the four $B_n$, 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 $\mathbf{W}$. That is,

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 $B_n$) 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 $y=x$ and $y=-x$ axes when the phases are nonideal. We created an obliquity operation that temporarily transforms the data into two new axes along $y=x$ and $y=-x$, applies a gain $g$ that diminishes the length along the $y=x$ axis, and then restores the data to the original axes $y$ and $x$. This linear operation has the effect of mixing phase step components:

Hence, it is related to the Healing method, which also mixes components. The $g=1$ is the nominal situation not requiring a change in obliquity. One could redefine obliquity to change the minor instead of major axis, by substituting $g\to 1/g$ 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 $\mathbf{W}$ and $B$. 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

3.1.

Overview

We will first discuss a single delay behavior, because multiple delays is a simple extension.

3.2.

Simulator Results for 1× Boosting and Emission Spectroscopy

Figure 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 $7450{\mathrm{cm}}^{-1}$ 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.

Fig. 6

(a) Small portion (50 out of $450{\mathrm{cm}}^{-1}$) of emission source spectrum $S_0$ for numerical simulation for studying photon noise, consisting of spikes having no continuum background, using calculational pixels of $0.05{\mathrm{cm}}^{-1}$. (b) Blurred version with added simulated shot noise (native resolution $\sim 3725$ at average $7450{\mathrm{cm}}^{-1}$). For modeling absorption spectroscopy, a spectrum having a continuum was used.

The sinusoidal interferometer transmission comb was multiplied against the input spectrum $S_0$ [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 $Y$-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 $\mathbf{W}$.

The computational grid had a spacing of $0.05{\mathrm{cm}}^{-1}$, a sufficiently small spacing to hold high-resolution signals boosted 10 times the native resolution. The boundaries of the Fourier space (Nyquist frequency) were $0.5(1/0.05)=10\mathrm{cm}$, which is many times greater than the 0.22-cm half width at half max (HWHM) of the native response peak ${\mathrm{psf}}_0(\rho )$. 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 $\tau$ [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.

Fig. 7

(a) Components of the numerical simulation plotted versus $\nu$, using simulated shot noise at the detector, delay $\tau =0.4\mathrm{cm}$, and EQ designed to achieve $1\times$ boost, i.e., same resolution as conventional. Outputs include “Bass” (nonfringing, green), “Treble” (fringing, red), “Net EDI” (purple), and conventional (black dashed). Unblurred input emission spectrum (gray) has a portion of 7420 to $7436{\mathrm{cm}}^{-1}$ as sinusoid with a frequency of 0.22 cm, making convenient linkage of amplitudes to height of dots in Fourier response plot (c). (b) Residuals of conventional (black dashes) and net EDI (purple) when in photon-limited regime. EDI has less noise at high frequency. Treble peak in (c) does not appear centered at $\text{delay}=0.4\mathrm{cm}$ due to EQ multiplication, which diminishes high frequencies (both signal and noise). [Fig. 9(c) shows same curves as (c) but with no EQ.]

The source spectrum had a small section 7420 to $7436{\mathrm{cm}}^{-1}$ 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 $\rho$ 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 $1\times$ 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 ${\mathrm{psf}}_0(\rho )$. 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 $\sqrt{0.5}$.

Fig. 8

(a) Example instance of numerical simulation (purple curve) versus frequency with photon noise probes whether the noise type is correlated (sums linearly, dotted black) or uncorrelated (sums in quadrature, dashed black). Fringing (treble, red) and nonfringing (bass, green) components. (b) For photon noise, repeated instances (purple dots) while changing delay fits a peak (gray curve) having same shape as the native ${\mathrm{psf}}_0(\rho )$ or “bass” peak, but with relative height 70% between perfectly correlated (open dots) and uncorrelated (open squares). Detector noise produces uncorrelated behavior independent of delay. (c) Instrument lineshapes in frequency space, small (0.05 cm) delay (left) producing partial correlation, and large (0.6 cm) delay (right) producing noncorrelation. (d) Same instrument lineshapes in wavenumber space. The small delay has similar lineshape components for fringing (red) and nonfringing (black dash). Large delay produces orthogonal lineshapes because of oscillation under peak envelope. Dimensionless delay horizontal axis is normalized by 0.44 cm FWHM of native ${\mathrm{PSF}}_0(\nu )$. Dashed red curve in (c) is wing of conjugate treble peak on the negative frequency branch (only significant for small delays).

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 $\sim (B_0+B_{180})$ and fringing-like signals $\sim (B_0-B_{180})$ involves integrals over products like $(N_0+N_{180})(N_0-N_{180})=(N_0^2-N_{180}^2)\approx 0$, where $N$ represents the noise apart from the signal. Hence for the detector case the correlation is approximately zero, because the detector noise variance $N^2$ 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 $N_0^2$ could differ from $N_{180}^2$. 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 Space

Figure 9 shows stages in the calculation of the SNR curves (versus $\rho$) 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 $1\times$ boost. A different EQ could have produced a boosted resolution of $\sim 2$. The EQ step does not change the SNR curve shape since it multiplies both signal and noise.

Fig. 9

(a) and (b) Noise, (c) signal, and (d) SNR curves versus feature frequency, $\rho$, for readout dominated noise and photon dominated noise cases, for a delay of 0.4 cm. (a) The average of 10 simulations (black dotted) with noise confirms the analytical result (b) from “EDI calculator.” The conventional photon dominated noise (dashed) in (a) and (b) is nonfringing noise without bell weighting, and hence is uniform with $\rho$. (c) Signals after bell weighting, for both bass (nonfringing) and treble (fringing). Weighting improves SNR by deleting noise from the high frequency portions of the bass curve from overlapping the 0.4-cm region of the treble. Bell weighting shape is same as original peaks, hence reduces widths of bass and treble signal peaks to 70%, and halves height of the treble. Shaping effects of weighting are normally compensated during EQ process. But EQ step is skipped to better show high frequency behavior. (EQ does not change SNR since it multiplies both noise and signal.) (d) SNR for EDI is higher in the 0.4 cm and higher region than conventional, even when output is divided by $\sqrt{3}$ (long dashes) or $\sqrt{4}$ (thin purple) to account for multiple readouts of EDI relative to conventional, when readout noise dominates.

3.3.1.

Photon-limited noise case

The 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.

Fig. 10

Simulation output when EQ is applied to achieve “$1\times$” boost, i.e., same final resolution for net EDI as conventional. (a) The photon-dominated conventional noise (short dashes) is divided by $\sqrt{4}$ to estimate single readout noise (long dashes), since simulation uses four exposures. (a) Net EDI noise (purple), bass (green), treble (red). EQ’ing deletes high frequency noise, so net EDI has less noise than conventional photon-limited (or single readout noises), for $\rho$ above 0.2 (or 0.3 cm). (b) Net EDI signal (purple) overlays the conventional signal (black dashes) due to the EQ’ing. (c) $\mathrm{EQ}(\rho )$ used.

Fig. 11

Fraction of average EDI noise (area under purple) compared to conventional (black dashed line) for case of Fig. 10(a). A root mean square (RMS) integration up to a Nyquist imposed limit set by pixel density. (a) 73% for 2 pixels per resolution element (FWHM in $\nu$-space) or (b) 60% for 3 pixels per resolution elements. The 2 pixel per resolution elements definition is approximately same as 5% signal height definition used in Fig. 15. Thus, the $1.4\times$ improvement of leftmost datum of EDI to conventional in Fig. 15 is justified. (c) and (d) Noise fractions are even smaller (46% and 34%) when only high frequencies are considered, this is where science signals typically reside. The low $\rho$ integration limit is set at 0.22 cm by the HWHM of ${\mathrm{psf}}_0(\rho )$. Note that a $0.34\times$ photon-limited noise reduction is conventionally obtained by increasing flux $1/{(0.34)}^2=8.6$ times.

4.1.

Uniform Exposure Schedule

Having explored the single delay case, we can extend our understanding on multiple delays. The total input flux will be subdivided into $M$ 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 $0.5/\sqrt{M}$.

Fig. 12

Multiple EDI sensitivity peaks (black, red is net) of the same height, when each delay has same exposure time, so total flux is evenly divided among $M$ delays. Native spectrograph (green peak at origin) defines unity SNR. An equal-area rule for ${\mathrm{SNR}}^2$ causes peak height to be $0.5/\sqrt{M}$ for $M$ delays. Blue dashed Gaussian response is for grating with $10\times$ resolution increase and $10\times$ less flux (hypothetically due to slit narrowing $10\times$). In this figure, the spectrograph throw length is considered fixed, so that avenue for increasing resolution for AO and classical spectrographs is not shown here.

Note how the higher delay peaks extend above a hypothetical conventional spectrograph (cross hatching above blue dashed curve) having the goal $10\times$ 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 $10\times$ and decreasing the flux by $10\times$ for an extended source (also requiring $10\times$ 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 $1000\times$ (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 $1/\sqrt{10}$. Then, the EDI has less SNR than the AO result by $\sqrt{10}$, but it needs $10\times$ the pixels and $1000\times$ the volume to accomplish this, a heavy cost to pay.

4.2.

Gaussian Exposure Schedule

Figure 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 $1.4\times$ narrower than the desired SNR Gaussian (blue dashes).

Fig. 13

(a) Net photon SNR behavior (red) when a Gaussian distribution of exposure time versus delay # is used. This is superior to uniform exposure time if white noise behavior is desired, such as in general spectroscopy. For Doppler spectroscopy, concentrating flux in a few high delays (gold dashes) is better. The native spectrograph (green peak at origin) defines unity photon SNR. (b) Asymptote of many overlapped delays, for $10\times$ boost, is $\sim 78\%$ of a “classical” spectrograph having resolution a factor boost larger, and height $\sqrt{\mathrm{boost}}$ smaller, as if slitwidth and flux boost times smaller. (c) Asymptote for $4\times$ boost is 95% of classical. Gold line at 0.02 is photon SNR of Fourier transform spectrometer, reduced from native peak by square root of the number ($\sim 2500$) of native resolution elements, since in Fourier transform spectroscopy (FTS), fringes of different phases sum on a single pixel. In this figure, the spectrograph throw length is considered fixed, so that avenue for increasing resolution for AO and classical spectrographs is not shown here.

4.3.

Velocimetry Exposure Schedule

For 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 Peaks

Let us demonstrate that when we use bell-shaped weighting, the net SNR versus $\rho$ 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 $N$ in quadrature. We use weightings $k_1$ and $k_2$ associated with each peak. Hence,

By bell-shaped weighting, we mean that the weight has the same shape as the signal: $k_1(\rho )=S_1(\rho )$ and $k_2(\rho )=S_2(\rho )$. Because we normalize the noise denominators to be the same, $N_1=N_2=N_0$. Hence,

The total exposure time may be allocated among the delays in various schedules, which affects their heights while representing the $\mathrm{SNR}(\rho )$. 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 ${\mathrm{SNR}}^2$ peak heights [Figs. 12 and 13(a)]. The sum rule also works for the area under the ${\mathrm{SNR}}^2$ 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.

Fig. 14

Fourier transforms of the residuals, after EQ, for different schedules of exposure time (flux) per delay, showing propagation of simulated noise injected at the detector for the case of $10\times$ resolution boost. (a) Uniform exposure schedule (Fig. 12) produces pink noise since a Gaussian EQ is later applied to produce a Gaussian lineshape. (b) and (c) Gaussian schedule of exposures (Fig. 13) using (b) detector and (c) photon noises. After an EQ divides by square root of the Gaussian, the noise becomes a uniform (white) distribution, and signal becomes Gaussian with $1.4\times$ wider width. (d) Native spectrum from one of eight delays (1/8th flux), consistent with rule of thumb that EDI noise is roughly conventional noise at 1/boost flux, i.e., 1/10th flux. (a)–(c) Low noise section ($<0.25\mathrm{cm}$) due to the cleaner contribution of the native averaged over eight exposure times. Simulated noise was 3% of continuum on four exposures on $0.05{\mathrm{cm}}^{-1}$ pixels.

4.6.

Selecting Your Delays at Blue End of Band

For a fixed resolving power $R=\nu /\mathrm{\Delta }\nu$, the width in $\rho$-space of the fringing peaks decreases as one moves to the blue (increasing $\nu$), since $\mathrm{\Delta }\nu$ increases and thus $\mathrm{\Delta }\rho$ 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 ${\mathrm{psf}}_0(\rho )$.

4.7.

Readout Noise Case for Multiple

Analogous 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 $\sqrt{3}$ or $\sqrt{4}$ 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 $\sqrt{3}$ or $\sqrt{4}$. (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 $\nu$ due to changing phase step size, which is from Fig. 4(a).

4.8.

Actual Delay Positions for TEDI

In 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 $\sim 1\hspace{0.17em}\mathrm{cm}$ 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 $7450{\mathrm{cm}}^{-1}$), rather than the 27,000 we produced. (Processing 36,000 with a delay gap would have produced significant ringing in the lineshape.)

5.1.

Plotting Fraction of Classical versus Boost

How does the EDI compare to the classical result, and how does that vary with boost? Figure 13(b) shows that for a $10\times$ boost, we achieve 78% of the classical SNR, and [c] for $4\times$ boost, we achieve 95% of classical SNR.

Figure 15(a) shows in purple dots how the EGH or $Y$-intercept for SNR ($\text{FC}/\sqrt{\mathrm{boost}}$) 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 $4\times$. For larger boosts, the asymptotic behavior is that the EDI has a square root dependence like the classical, but a factor $\sim \sqrt{2}$ worse, and for smaller boosts than 4, the asymptotic behavior is a factor $\sim \sqrt{2}$ 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 Model

A 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.

Fig. 16

Theoretical comparison of photon noise-to-signal ratio (arb. units) between the two techniques versus boost (final/native resolution), for EDI (black diamonds), and native-alone (green curve). Hypothetical native spectrograph has a slitwidth (SW) and a FB of 0.70, that sum in quadrature. Red dashed line having square root behavior is idealized “classic” spectrograph standard, whose flux decreases as 1/SW but without any FB. We suppose native spectrograph operating point is at knee of curve (solid green dot), where $\mathrm{SW}=\mathrm{FB}$. From operating point, we drop $1.4\times$ to set net EDI noise behavior at boost of $1\times$, and the rest of the EDI curve follows based on this instance of native spectrograph. The factor of 1.4 is based on reciprocal of Fig. 15(a), and confirmed by simulation Fig. 11. The EDI for low boosts has lower net noise than native-alone because it combines native signal with fringing signal, and they have uncorrelated noises.

6.1.

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 $0.6{\mathrm{cm}}^{-1}$ in the A-order, as shown by Fig. 35 of part 1.² Since the resolution is about $2{\mathrm{cm}}^{-1}$ 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 $3\times {10}^5$, and since there are 3.8 photons per count, this is 4 to $11\times {10}^5$ photons per pixel. Since there are $\sim 3\text{pixels}$ per native resolution element, there are about 1 to $3\times {10}^6$ photons per native resolution element. Taking the square root, this yields a photon noise of order 0.1%.

6.3.

Noise Contribution Plot

From 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.

Fig. 17

Estimated relative contributions of various noise contributions for absorption spectroscopy in the A-order of TEDI, compared to native spectrograph (if it could increase its resolution), and how these two vary with boost ratio (EDI resolution over native). Green denotes the conventional “native” spectrograph and red denotes EDI. Circles are instrument noises and squares are photon noises.

The red and green symbols denote EDI and native spectrograph. When boost varies for EDI, it uses the same $1\times$ 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 $1.4\times$ 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 $\sim 3000$, it is in between these values.

EDI instrumental: For low boosts of $\sim 1\times$, 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 $1\times$ 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 $6\times$, 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 $20\times$ less horizontal reaction to a PSF drift, (and $350\times$ less using a more sophisticated “crossfading” process that modifies the lineshapes. See Sec. 10 of Ref. 2). But at a $6\times$ higher boost, the slope of lineshape that connects between vertical and horizontal errors has increased by $6\times$, so the net downward movement of the red dot is $20/6=3\times$. This reduces the EDI instrumental noise to a 1% level, $20\times$ less than it would be with the conventional alone. Instrument noise is still dominant over estimated EDI photon noise of 0.2% at $6\times$ boost.

7.1.

Comparison to Dispersive Spectrographs

The purely dispersive method (a) has a peak at the origin whose width is proportional to spectral resolution. Let the FWHM be ${\tau }_{\max }$ (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:

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 $0.5/\sqrt{M}$ 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 Spectroscopy

The 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 ${10}^3-{10}^4$, the EDI can have a better photon SNR over the FTS by a factor of $30\times$ to $100\times$.

7.3.

Comparison to Dispersed-Fourier Transform Spectroscopy

In a dispersed-FTS^16,17 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 Interferometers

The internally dispersed interferometer techniques called SHS^12,18 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 demonstrated^18,20 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 Imaging

Related 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 Light

Because 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.

8.1.

Uncertainty Principle Followed

Some 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.

Fig. 19

Apparent increase in coherence length of a grating when viewed through an interferometer having delay $\tau$. Any object viewed through a Michelson interferometer appears twice (with 50% intensity for each image), and with the second image delayed. The two images of the grating appear as a single grating with a longer coherence length. Since the spectral resolution is proportional to the net grating coherence length, the resolution increases.

This increase incoherence length causes greater ambiguity for time scales of $\tau /c$ or $0.05/3\times {10}^8$ 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 Spectrographs

The 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 Spectrographs

EDI 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 $350\times$, 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 Cost

The 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.