July 2022

Externally Dispersed Interferometry
Lawrence Livermore Nat. Lab.    &    Space Sciences Lab. UC Berkeley

EDI is a novel technique for the Doppler planet search
and high resolution spectroscopy

A method for boosting the stability & performance of existing dispersive spectrographs, by adding a small interferometer prior to the slit, and processing the moire patterns

With the EDI, your inexpensive & compact spectrograph can now perform:
Precision stellar Doppler radial velocimetry (paper1)(paper2) (planet detection! (Ge_Virgo_ApJ2006.pdf))
High resolution spectroscopy: EDI boosts the spectral resolution of a dispersive spectrograph 2 - 20x over a wide simultaneous bandwidth by a process of heterodyning against the extremely periodic transmission comb of an interferometer, and combining the so-produced moire signals from one or more delays. (JATIS paper 2016 part 1, part 2).

EDI can also boost the stability of a spectrograph against uncontrolled instrumental wavelength drifts by 100 to 1000x. Firstly, the fine wavelength discrimination is performed by the interferometer, not the disperser, which is already an improvement over the disperser used alone. This is because under an instrumental drift such as a spectrograph camera focal point shift relative to the detector, both the comb and the stellar spectrum shift by the same amount and thus tend to cancel the moire phase change, which is relative between them.

Furthermore, we have recently learned how to combine pairs of interferometer signals ("crossfading") to cancel their net reaction to a wavelength drift-- this produces an even further stability improvement, for a net stability boost into the 100x - 1000x range compared to the disperser used alone (defined as wavelength drift in disperser divided by net wavelength drift in interferometer processed output.) We envision the EDI technique applied to facility spectrographs to boost their Doppler and high resolution spectroscopy performance. The Doppler crossfading-EDI would require at least two delays instead of the previous single-delay EDI. (Having more delays is not necessary for Doppler but useful for measuring lineshapes to higher resolution.)

Our latest JATIS paper (2021) describes this crossfading-EDI technique. The key to the method is that two slightly different delays can produce opposite phase shifts under the same wavelength drift, and by combining them with appropriate weights can cancel their net reaction. The simulation below uses data of a ThAr spectral lamp line measured by the T-EDI at Mt. Palomar in 2011. This project used a handful of delays to study the benefit of multiple delays on boosting spectral resolution in measuring the *shape* of a spectrum. We were not anticipating that having multiple delays would also be so beneficial in preventing the *shift* of the spectrum to unwanted drifts in the conventional spectrograph, but that is what we eventually figured out how to do, and which we are interested in exploring further.

How it measures Doppler shifts

Shifts are 500x smaller than feature linewidths!
When the exoplanet pulls the star it creates a Doppler shift in the spectrum vs wavelength. This shift is difficult to measure with a grating spectrograph because it is a small shift easily confused with irregularities in the spectrograph dispersion. The typical linewidth of a stellar absorption line is equivalent to 6000 m/s, so that a Jupiter-class exoplanet Doppler shift of 12 m/s is 500 times smaller than the width of the line!
This is a very difficult measurement for a conventional spectrograph, and can be easily overwhelmed by uncontrolled distortions in the instrument response.

The interferometer is the "vernier", the grating the "coarse" readout

The EDI technique uses an interferometer for the fine wavelength discrimination and a grating spectrograph for the coarse discrimination. It is like the vernier and coarse readouts on a precision instrument. The interferometer has a much simpler instrument response, robust to many distortions that plague the grating.

The EDI Method
An unequal-path Interferometer is added to a dispersive spectrograph

• The interferometer causes a periodic fringe vs wavelength

• The fringe pattern beats with the spectrum

• Heterodyning creates a moire pattern

• High-resolution features are measurable through the moire

• Nyquist limit and slit blurring are defeated

• The fringe fiducial allows error removal

• Phase stepping nulls instrumental errors

Solar spectrum without fringes (upper) and with fringes (lower).

Doppler wavelength shifts are measured through phase shifts of "fringes" which form moire patterns seen in the spectrum. (The interferometer can't be used effectively alone because otherwise the fringes of many different wavelengths having many different phases would fall on the same detector pixels and wash itself out.) The grating spectrograph separates the fringes so that the fringe visibility can be high. But otherwise the key role of measuring a Doppler shift is with the interferometer.

The advantage of this is that the interferometer is independent of the thermomechanical instrument distortions that cause large velocity errors in the grating spectrograph.

Mathematically simple instrument response
The interferometer can provide superior fine wavelength discrimination because it has a mathematically simple and accurate response, a sinusoid with only three degrees of freedom: phase, amplitude, and intensity offset. In contrast the instrument response of a grating has hundreds of degrees of freedom: one for each groove of the grating, which can vary with temperatue and time and create significant velocity errors.

Interferometer has sinusoidal transmission

The interferometer "sorts" light into two outputs depending on very slight wavelength differences. The interferometer has two unequal arms so there is a relative delay between the two paths of light. If a whole number of wavelengths fits in the delay, then there is constructive interference and the light passes out the main output perfectly. If there is an odd number of half wavelengths, then there is destructive interference and the light passes totally out the complementary output and zero out the main output. (Every photon that enters the interferometer leaves the interferometer to be detected, except for minor mirror reflectivity losses. The only issue is *which* output the photon leaves by.) Thus for each output, the transmission function is sinusoidal versus wavelength (actually, vs wavenumber which is 1/wavelength).

Beats or moire patterns formed
When this transmission pattern is dispersed by the spectrograph it creates a periodic grid. This grid multiplies the input stellar spectrum. A consequence is that beats are formed between features and the grid, for the features that have similar size as the grid. These beats are also called moire patterns.

The Doppler effect causes phase shifts
In the EDI the wavelength shift creates a phase shift in the moire pattern. Often the interferometer phase is arranged to vary vertically so that a moire phase shift is a vertical shift. The phase shift is multiplied by 15,000 m/s per fringe proportionality (for a 1 cm delay in green light) to find the Doppler velocity.

This phase shift can be more easily measured than the horizontal shift because the moire patterns are broader than the narrow features that created them, and less susceptible to instrument distortions.

You can confirm this by looking at this animation from far away, or by looking at the lower animation which is blurred to simulate spectrograph blurring.

Note that you can easily see the moire pattern move up and down even though you cannot resolve the individual vertical lines.

Bottom line: with the EDI method you do not need to resolve the individual absorption lines. You can obtain ~1 meter/sec Doppler velocities with a small low resolution spectrograph and inexpensive interferometer by measuring the moire patterns.

Notable results include:
[1] demonstrated ~1 m/s Doppler precision;
[2] the 12 m/s pull of the moon on the Earth measured in sunlight;
[3] first stellar fringes Dec 1999 at Lick Obs. 1 meter;
[4] Exoplanets have been detected using EDI.

More explanation on how EDI works for Doppler velocimetry is at this page.

How EDI measures high resolution spectra

A second important application of EDI has been demonstrated by our group: boosting the spectral resolution of an existing spectrograph by factors of several.

Moire patterns formed by heterodyning
The moire patterns represent originally high spatial frequency information (that is, narrow features in the spectrum) beaten down to low spatial frequencies (broad moire patterns). These broad moire patterns are more easily measured by small inexpensive and light efficient spectrographs. This is called a heterodyning process, and it shifts the spatial frequency of the input to a lower value by a known amount, set by the interferometer delay.

Heterodyning is reversed during data analysis
This optical heterodyning process is reversed numerically during data analysis to recover high spatial frequencies that would ordinarily not be resolved by the spectrograph.

Ordinary spectrum also recovered
The ordinary spectrum is also recovered from the data simply by adding all the phase channels vertically so that the moire patterns cancel. Hence the original spectrograph capabilities are retained. The moire signal is thus new information above and beyond the original spectrum. A composite spectrum is then formed from the ordinary + moire components. This composite spectrum has higher effective resolution than the native spectrograph, by factors of several. This is a method of defeating limits imposed by slit blurring, spectrograph optical aberrations, and the CCD Nyquist criterion.

Bottom line: we can boost the spectral resolution of any spectrograph by demonstrated factors of 2x to 6x and beyond by inserting a small interferometer near the slit and processing the moire fringes with our special algorithm.

Notable results include:
[1] Six-fold resolution boost, from 25,000 to a final 140,000,
demonstrated on the iodine spectrum;
[2] 2-fold resolution boost, from 50,000 to a final 100,000,
demonstrated on stellar echelle spectra at Lick Obs.;
[3] 2.5-fold resolutionboost, from 20,000 to a final 50,000,
demonstrated on sunlight

More explanation on how EDI boosts spectral resolution is at this page.

Site Index

Cover art
Contact info
How it Works (Doppler)
How it Works (Res Boost)
What is Heterodyning?
Graphical Demos
Photon noise simulation
Experimental Demos

Early History of EDI
History ii
Apparatus Schemes
Apparatus Photos
Spectral Astrometry

EDI to go to Palomar (TEDI)
First light at Palomar
10x res boost!
6x res boost

Planets detected!

Erskine's background
Erskine's public. list

www.SpectralFringe.org site maintained by
David Erskine

10x resolution boost demo'd on starlight at Mt. Palomar 200 inch Hale telescope

Externally Dispersed Interferometry (EDI) was invented by David Erskine at LLNL in 1997. See
Early History.


Demonstration of interferometric resolution boosting to 10-fold, which is the highest demonstrated to date on starlight (June 2011). The resolution and lineshape accuracy of a conventional spectrograph can be increased by factors of several by combining it in series with a small interfemeter, taking data in multiple exposures, then recombining the exposures after special data processing. Normally, the limiting spectrograph resolution scales with spectrograph size, since it is usually limited by detector pixel size, or aberrations in spectrograph lens optics, rather than slit width. The instrument lineshape accuracy is also stabilized by the mathematically simpler behavior of an inteferometer. Hence, EDI is a method of achieving the high performance of a much large spectrograph at the low size and cost of a smaller spectrograph. The tradeoff is slightly lower photon signal to noise and more complicated data analysis. However, the improved instrument lineshape stability often makes the net performance a winner over the naked spectrograph, since many spectrographs are limited by their poor lineshape stability (affected by air convection, thermal drifts etc.) rather than photon noise performance.

Here is reconstructed spectrum of starlight (kappa CrB) to a resolution 10x higher than the native spectrograph used without the EDI interferometer. The spectrograph was the IR "Triplespec" spectrograph of bandwidth 4000 to 10,000 cm-1 and resolution ~2700 at 4000 cm-1, mounted on the Cassegrain output hole of the 200 inch telescope at Mt. Palomar Observatory. The green dashed curve is the Triplespec spectrum without benefit of the interferometer, at a resolution of 2700. It cannot resolve many of the narrow features caused by atmospheric absorption (telluric lines). The red curve is the spectrum reconstructed by combining 7 delays worth of fringing data and special Fourier processing, in a technique called ISR (interferometric spectral reconstruction) or resolution boosting. It has an effective resolution of 27,000 and agrees well with a model of telluric features calculated by Henry Roe and artificially blurred by us to that resolution. A ThAr spectral lamp provides calibration emission lines. This graph shows only a small (110 cm-1) section of the reconstructed bandwidth which extends 6000 cm-1 from ~4000 cm-1 to ~10000 cm-1. The ISR technique is described in these 3 papers: Ten-fold SPIE, Scot16Bppr2.pdf, BoostApJ1693b.pdf. See our latest page on Res boosting.

Website host
David J. Erskine

An Astr. Soc. Pacific Astronomy Beat article describes some history of the EDI.


(Above) Simulation of a cross-fading EDI using real EDI data of a ThAr spectral calibration lamp measured while at Mt. Palomar Observatory in 2011, and software recently improved by the PI. Each interferometer delay measures a moire pattern which, after numerical processing, creates a wavelet in the ouput spectral space. The wavelet envelope is defined by the dispersive spectrograph, and its phase by the interferometer. (The horizontal axis is wavenumber which is reciprocal of wavelength in cm). The simulated insult drift ∆x in the dispersive spectrograph can be seen in the shift of the wavelet envelope (slanted dashed lines). The sum of the wavelets forms the output peak (red curve), which is narrower than the envelope (hence there is resolution boosting). By choosing weights of each wavelet strategically, we can make the location of the output peak (dotted line) virtually independent of the disperser insult drift ∆x along the horizontal axis. (Actually it's a 1000 times smaller, which is so small you cannot see it by eye here.) This is really cool!!


The same spectral freature produces moire patterns having opposite polarity slopes, from two interferometer delay combs (having slightly different periodicities). An unwanted sideways translation (disperser drift ∆x) would produce opposite polarity errors, which can cancel when we combine the signals.

<-----Our January 2022 iPoster using fun animations of Moire fringes in our Crossfading technique, prepared for the AAS 239th meeting.


-------- Latest two posters July 2022 --------
(left) Demos of single-delay Crossfading technique, and (right) preliminary EDI measurements diagnosing the Keck Planet Finder spectrograph's blur width and shape (in engineering tests at UC Berkeley Space Sciences Lab), prepared for the SPIE Astronomical Instruments meeting in Montreal 2022 July.


---- Other interesting recent graphics ----