piston e ect in optical astronomical interferometryVisible and infrared interferometry is one of the...

ASTRONOMY & ASTROPHYSICS MARCH 1997, PAGE 553

SUPPLEMENT SERIES

Astron. Astrophys. Suppl. Ser. 121, 553-568 (1997)

Correction of the “piston effect” in optical astronomicalinterferometry

I. Modulus and phase gradient of the visibility function restoration

G. Perrin1

Observatoire de Paris, DESPA, 5 place Jules Janssen, F-92195 Meudon, France

Received November 13, 1995; accepted November 7, 1996

Abstract. An a posteriori method to cancel out atmo-spheric optical path fluctuations to achieve accurate phaserestoration and spectral analysis of interferograms is pre-sented in this paper. The correction algorithm is basedon classical noise reduction methods. The consequence ofpiston noise is twofold: 1) loss of spectral information; 2)loss of visibility phase. A method to measure the modulusand the phase gradient of visibilities is explained and ex-amples of restoration of simulated spectra and visibilitiesfor various S/N ratios are presented.

Key words: atmospheric effects — methods: dataanalysis — technics: interferometric

1. Introduction

Visible and infrared interferometry is one of the mostpromising technics in the coming years to achieve high res-olution imaging in astrophysics. The equivalent techniquein radio wavelengths is providing the astrophysical com-munity with sharp images impossible to obtain with classi-cal single antennas due to fundamental diffraction limita-tions. At optical wavelengths, coherence of light measure-ments are strongly degraded by atmospheric turbulenceand may even become impossible. A partial correction ofturbulence modes can be achieved and is already produc-ing interesting results for imaging. Nevertheless, adaptiveoptics cannot compensate for the 0-order mode of atmo-spheric distortion of wavefronts (the optical path fluctu-ations or “piston effect”) which causes the loss of phase1

information in interferometry cancelling any attempt to

Send offprint requests to: G. Perrin ([email protected])1 The position of the white light fringe measured in opd inwide-band interferometry is roughly, to within a λ

2πfactor, the

phase of the image Fourier transform, i.e. the phase of thecomplex visibility.

reconstruct high resolution images. White fringe positionservoing systems (fringe trackers) make up for the opticalpath fluctuations in real time and lock the system ontothe white fringe to perform low noise acquisition throughlong integration time, but they fail to record a long se-quence containing spectral information. The paper startswith a presentation (Sect. 2) of interferometry and moreespecially Double Fourier interferometry and explains whyoptical path fluctuations effects are undesirable in inter-ferometry. In Sect. 3 the properties of atmospheric opticalpath fluctuations are listed and simulations are presented.The piston correction method is explained in Sect. 4 andresults on simulated interferograms are presented. Section5 is the analysis of the results.

2. General context

The aim of this section is essentially to provide the readerwith a few classical notations as well as basic concepts ofinterferometry. Paragraph 2.1 is a brief theoretical tuto-rial on inteferometry. The purpose of the Double Fourierinterferometric mode probing both temporal and spatialcoherence properties of sources is explained in paragraph2.2. Eventually, paragraph 2.3 adresses the aim of thispaper, that is to say the nature of the disturbances gen-erating optical path fluctuations.

2.1. Brief tutorial on interferometry

Equations derivations are classical and can be found inlitterature (Goodman 1985 for example). Nevertheless, Iintend to give a fundamental subset of equations whoseaccurate knowledge is mandatory for a good understand-ing of the remaining of the article. Let us consider twopupils P1 and P2 of an interferometer and the correspond-ing sampled complex amplitudes of an astronomical wave-front A1(σ, t) andA2(σ, t), where σ is the wavenumber andt is time. The two amplitudes are combined and sent toa detector. The monochromatic energy detected at the

554 G. Perrin: Correction of the “piston effect” in optical astronomical interferometry. I.

output of the combiner is proportional to the averagesquared modulus (on a time scale far greater than theperiod of the wave) of the sum of the two fields:

I(σ, t) = < |A1(σ, t) + A2(σ, t)|2 >t . (1)

For sake of simplicity, all multiplicative coefficients (detec-tor gain, interferometer transmission, etc ...) will be set toone and their expressions will be dropped in the follow-ing equations. The monochromatic energy collected by thetwo pupils (which is proportionnal to the spectrum of thesource) is supposed to be the same and the normalizedtotal energy over the bandwidth is noted B(σ). Equation(1) is then rewritten as:

I(σ, t) = B(σ) + 2 < A1(σ, t)A∗2(σ, t) >t

= B(σ) +B(σ)Re {γ12(σ, t)} , (2)

where γ12(σ, t) is the complex degree of coherence of thetwo beams coming from the source and collected by thetwo pupils of the interferometer. For a monochromatic ra-diation, it has a simple expression as a function of thedelay τ between the two beams (Goodman 1985):

γ12(σ, t) = V12(σ)e−2jπσvt, (3)

where v is the rate of change of opd. V12 is the coherencefactor of the two beams or fringe visibility and is wave-length dependent. The value of visibility is determined bythe geometrical aspect of the source and by the geometri-cal characteristics of the baseline. It is the Fourier trans-form of the intensity distribution of the source at a spatialfrequency equal to the vector baseline over wavelength ac-cording to the Van Cittert-Zernike theorem (Born & Wolf1980; Goodman 1985). Visibility2 is the observable mea-sured by interferometers.It is equivalent to express delay either as a temporal de-lay τ or as an optical path difference x as the ratio of thetwo is the speed of the fringes v. In wide band, Eq. (2)becomes:

I(x) = 1 + Re

{∫band

B(σ)V12(σ)e−2jπσx dσ

}. (4)

It is necessary to compensate delay of one the two beamswith respect to the other one when recording the interfer-ogram because the coherence length, in wide band, is onlya few fringes. Monochromatic waves coherently interferearound the equal optical path position when the opticalpaths of the two beams are matched by a delay line.The first term in Eq. (4) will be left aside in the follow-ing and I will only consider the second one which is the

2 More precisely, visibility moduli are straightforewardlymeasured by interferometers, a more indirect procedure is usedfor measurements of visibility phases, what is measured is thesum of three phases for a three-telescope loop which is calledphase closure, this special technique was designed to cancel outadditionnal phase errors, see references thereafter in the paper.

modulated part of the interferogram. When the complexspectrum is defined as the hermitian part of the spatio-temporal source spectrum (modulus is an even functionand phase is an odd function of wavenumber):

Bc(σ) =1

2(B(−σ)V12(−σ) +B(σ)V ∗12(σ)) , (5)

the modulated part of energy, noted I(x), gets a moreconvenient expression:

I(x) =

∫ +∞

−∞Bc(σ)e2jπσx dσ. (6)

In other words, the Fourier transform of the modulatedinterferometric signal is the complex spectrum of the as-tronomical source.

2.2. Double Fourier interferometry

A few different ways of doing interferometry (mostly de-pending on measured quantities) exist (see review by M.Shao 1992). Some interferometers recombine beams in thepupil plane whereas others are image plane oriented. Here,no distinction will be made between these two differentphilosophies. Some interferometers (type I) are operatedin the long exposure mode, that is to say that a ser-voed delay line sets the optical path difference (there-after opd) to a constant value allowing the integration ofthe interferometric signal to achieve high signal to noise(S/N) ratios on fringe modulation. The value of the opdcan be varied to scan through the whole interferogram.Type II interferometers can be considered as short expo-sure interferometers. The interferogram is acquired dur-ing a continuous scan of the opd which is produced eitherby the Earth’s rotation or by a translating retroreflect-ing unit moving at a constant speed or by a combinationof the two. Integration is then no longer possible. TypeI interferometers are not opd fluctuation sensitive as theopd is servoed, whereas many factors can introduce opdfluctuations in type II interferometers (this point will beadressed in Sect. 2.3). Type II interferometers are dedi-cated to spectral data analysis for they naturally reach ahigher spectral resolution while type I data are reducedin direct space. These type II interferometers are calledDouble Fourier Interferometers (hereafter DFI) becausethey both probe spatial and temporal coherence proper-ties of the light emitted by astrophysical sources (Itoh& Ohtsuka 1986; Mariotti & Ridgway 1988). Spatial in-formation is contained in the modulated energy of inter-ferograms as shown in Sect. 2.1 and is wavelength depen-dent. The Fourier transform of the interferogram sequenceyields the source temporal spectrum rescaled by the vis-ibility function. It has been shown (Ridgway et al. 1986)that an accurate knowledge of the visibility-wavelengthrelation would have fruitful astrophysical implications, aswell as visibility as a function of spatial frequency. DFI

G. Perrin: Correction of the “piston effect” in optical astronomical interferometry. I. 555

allows the determination of the visibility as a function ofwavelength and is the interferometric mode that is mostlyconcerned in the following of the paper.

2.3. Optical path fluctuations and phase information loss

As shown in Eq. (6) an interferogram can be consideredas a temporal sequence whose Fourier transform is pro-portionnal to the spectrum of the source. The two con-jugate variables are the opd (or time) in the direct spaceand wavenumber (or frequency) in the Fourier space; oneshifts from one set of conjugate variables to the other bymultiplying or dividing by the speed of the fringes. Thusopd varies linearly with time. This is what one should ex-pect in an ideal interferometer. In reality this occurs ina different way. Opd does not vary linearily with time.Some opd distortions are introduced and have mechani-cal and atmospheric sources. Vibrations of static opticson the beam path change its length and make it oscillate.But they can be lowered to an inoffensive magnitude byincreasing the stiffness of the optics and by absorbing vi-brations. Non-linearities in the motion of the retroreflect-ing stage of the delay line cause opd to fluctuate but theycan also be cancelled out by servoing the speed of thecarriage supporting the retroreflector. Some fiercer fluc-tuations are generated by atmospheric turbulence. Herewe assume that atmospheric perturbations of wavefrontsare reduced to the 0-order ones (the spatially averagedpertubations on the pupil) and that higher orders havebeen filtered out by using monomode fibers for example(Coude du Foresto & Ridgway 1991; Coude du Foresto etal. 1992), by data reduction or with adaptive optics. The0-order term is known as “piston effect” and opd fluctua-tions are due to differential piston between two aperturesof the interferometer. But piston and differential pistonstand for the same physical phenomenon.Opd fluctuations result in the loss of the Fourier relationbetween the spectrum and the interferogram sequence pre-venting physical information recovery from the spectrum.Figure 3 of Sect. 3 shows four simulated low resolutionspectra. The first one is computed from an opd fluctuationfree interferogram whereas the others simulate observedspectra with regular turbulence conditions. It is obviousthat without any piston correction no accurate informa-tion can be extracted from these spectra. Besides, pistonintroduces a random shift of the central fringe of the in-terferogram preventing measurement of the exact phase ofvisibilities, the accurate knowledge of which being essen-tial to reconstruct a high resolution image of the source.If most sources of optical path fluctuations have neg-ligeable effects, atmospheric turbulence effects at opticalwavelengths must be taken into account. It is possible toartificially increase the length of coherent interferogram byscanning at a high speed. Rapid scanning freezes pistonbut it does not reduce its amplitude. Besides, fast scanningreduces the instrumental visibility (hence the S/N ratio of

the modulated signal) as the detector response decreaseswith increasing frequencies. Turbulence opd fluctuationsthus turn out to be a strong limitation to optimum infor-mation extraction in optical astronomical interferometry.In the following Sects. I will expose a method to correct“pistoned” interferograms to retrieve the modulus of thespectrum and the phase to within a constant. This methoddoes not take into account the possibility to retrieve theexact phase of visibilities and does not address the issueof image reconstruction in interferometry.

3. Statistical and temporal properties ofatmospheric optical path fluctuations

This section is dedicated to the characterisitics of piston.The features are used in paragraph 3.6 to simulate pistonand pistoned interferograms.

10−2

10−1

100

101

102

10−10

10−8

10−6

10−4

10−2

100

Frequency (Hz)

Ene

rgy

(arb

itrar

y un

its)

νν0

ν1 2

-8/3

-17/3

-2/3

Fig. 1. Differential piston power spectrum

3.1. Statistical properties of differential piston

A complete study of statistical properties of turbulenceeffects can be found in Roddier (1981) from which thisparagraph is inspired.Astronomical beams travelling through the turbulent at-mosphere on their way to the aperture traverse a suc-cession of thin independent turbulent layers each with afluctuation statistics. Resulting from the central limit the-orem, the overall statistics are Gaussian and are fully de-termined by the standard deviation. This thus applies tooptical path fluctuations statistics. The standard devia-tion value a priori depends on wavelength λ, fried param-eter r0 and baselength D. Its expression in meters is:

σε =2.62

2πλ

(D

r0

)5/6

(7)


10−3

10−2

10−1

100

101

10−4

10−3

10−2

10−1

100

Sequence length (s)

Nor

mal

ized

sta

ndar

d de

viat

ion

Fig. 2. Effective standard deviation of piston for temporal se-quencies of finite length. The standard deviation is normalizedto unity for a sequence of infinite duration. Curves are plottedfor three different wind speeds: 10 m s−1 (full line), 20 m s−1

(dashed line) and 30 m s−1 (dashed and dotted line)

where the subscript ε refers to the opd fluctuation stan-dard deviation. It is interesting to realize the importanceof this effect. Assuming correct seeing conditions for ob-serving with r0 = 60 cm at 2.2µm and that the interfer-ometer baseline is D = 21.2 m, the standard deviation foropd fluctuations is 18.5µm that is about 9 fringes in theK photometric band (to be compared to the coherencelength of 10 fringes) and an error of 56 radians on visibil-ity phase measurements. Since r0 is proportionnal to λ6/5,σε does not depend on wavelength and piston is an achro-matic effect. The reduction method proposed in this papermainly relies on this critical property of atmospheric dif-ferential piston.

3.2. Temporal properties of differential piston

Let us now focus on the evolution of differential pistonwith time. A theoretical average spectrum is given inConan et al. (1992) for an infinite outer scale. It is a three-slope spectrum in Log-Log representation with cut-off fre-quencies depending on the average speed of turbulent lay-ers v as defined in Roddier (1981), also depending on thebaselength D and the diameter of the pupils d. These fre-quencies have the following expressions:

ν1 = 0.5v

D,

ν2 = 0.3v

d, (8)

and the slopes are typical of a Kolmogorov turbulence:− 2/3, − 8/3 and − 17/3. Considering parameters givenin Colavita et al. (1987) for observations carried out at theMark III interferometer with a 12 m baselength, a 14 m s−1

average wind speed and 2.5 cm apertures, cut-off frequen-cies are 0.6 and 168 Hz which is in good agreement withthe piston power spectra published in this paper (the dataare in the range 0.001−100 Hz and do not display the thirdslope feature).

3.3. The outer scale of turbulence

The idealized temporal and statistical properties given be-fore are correct to a certain extent. They require the outerscale of turbulence (scale at which turbulent energy is in-troduced in the atmosphere) to be infinite. The relation inEq. (7) saturates for baselines larger than the outer scaleand energy in the power spectrum saturates at low fre-quencies. Recent measurements of the outer scale at theSUSI interferometer site in Australia (Davis et al. 1995)for baselines ranging between 5 and 80 m state values ofabout a few meters. Piston measurements show that re-lation (7) departs from linearity in Log-Log coordinates(Fig. 3 of that paper) – for the 20 m baseline, the mea-sured piston standard deviation is 8.4µm whereas the ex-pected value is in the range 16 − 30µm, depending onseeing conditions – and power spectra level out for fre-quencies below ' 0.02 Hz. Although there is a controversyabout the outer scale (references can be found in Daviset al.) it turns out that a finite range is more likely thanan infinite one and that methods based on predictionsfrom the power spectrum at low frequency that found aninfinite outer scale are not reliable because they are con-sistant with a wide range of results. Direct measurementswith different baselines are definitely likely to be a goodbasis for a good understanding of the properties of piston.As a consequence, the power spectrum used in this pa-per, taking into account a finite outer scale, is a four slopespectrum in a Log-Log diagram with a low cut-off freqe-uncy at 0.02 Hz and the two cut-off frequencies predictedby Conan et al. (1992). It is plotted in Fig. 1.

3.4. Differential piston for finite length sequences

Prediction of piston standard deviation by Roddier (1981)applies to sequences of infinite duration. In real life, in-terferometric scans are finite in length and, since pistonspeed is not infinite, standard deviation on a finite lengthis necessarily lower than the prediction. For a sequence ofduration T frequencies below 1

T are atttenuated thus de-creasing standard deviation. In Appendix A the analyticalexpression of standard deviation is given as a function ofscan duration. The result is plotted in Fig. 2 for observingparameters given in Table 1 for three wind speeds (10, 20and 30 m s−1) spread across the range of observed speedsin good astronomical sites (E. Gendron, private communi-cation). Standard deviation for infinite sequence lengths isnormalized to unity. With v = 20 m s−1 and for durationsin the range 0.1− 1 s standard deviation is proportionnal


to T13 whereas it is proportionnal to T

53 for shorter dura-

tions. In short, attenuation of piston variations becomesimportant for sequences shorter than 1

ν2. It is interesting

to derive the coherence time of piston to get a rough esti-mate of the time scale necessary for a piston peak-to-peakvariation. In the same conditions as those used above, thecoherence time is found to be:

τε = 31.5 ms, (9)

which means that piston varies slowly and that it has tobe determined on long sequences to be averaged out.

3.5. Discussion

Both piston amplitude and temporal behavior are wellknown thus allowing realistic simulations as will be shownin the next section. Interferometer sensitivity to piston iswavelength dependent. The shorter the wavelength, thehigher the fringe frequency for a given fringe speed, hencethe higher the effective piston coherence time. But, as faras piston amplitude is concerned, the shorter the wave-length the higher the relative amplitude of optical pathfluctuations. It is of a great interest to know whether anumerical piston correction method is more efficient im-proving amplitude or time coherence. Let us consider sinu-soidal optical path fluctuations as in Brault (1985) of fre-quency ω and amplitude δ. In the spectral domain, thesefluctuations produce two symmetric replicas of the spec-trum shifted by ω and −ω and of relative height δ. Thissimple example shows that it is more profitable to reducepiston amplitude to achieve spectrum restoration as thenoise introduced by piston is proportionnal to its ampli-tude. It also demonstrates that infrared wavelengths aremore suitable for DFI than visible wavelengths.

3.6. Simulations of pistoned interferograms

The simulations presented in this paper are in the K pho-tometric band centered at 2.2µm. The simulated interfer-ometer is the fibered recombination unit FLUOR at thefocus of the IOTA interferometer as explained in Perrinet al. (1995). The characteristics of the inteferometer arelisted in Table 1. The source is supposed to be unresolvedat any wavelength in the band yielding a constant visibil-ity of 1. The spectrum is a spectrum of a blackbody at3500 K seen through aK filter.

There are 1024 samples per sequence. Given the aver-age wind speed v, the scanning frequency νf , I first com-pute a Gaussian sequence of average 0 and of standard de-viation 1. This sequence is filtered with the piston powerspectrum given in Sect. 3.2 and is normalized to a stan-dard deviation of σε attenuated by the factor given bycurves in Fig. 3. The length of each sequence is 400µmcorresponding to a scan duration of 160 ms and yieldingan attenuation factor of 0.57 and a standard deviation of

Table 1. Simulation parameters

FLUOR-IOTA hardware caracterisitics

baselength (D) 21.2 mwavelength (λ0) 2.2µmapertures diameter (d) 0.45 m

Nominal parameters

Fried parameter (r0) 60 cmwind speed (v) 20 m s−1

fringe speed (vf) 5 mm s−1

fringe frequency (νf) 2274 Hzsaturation frequency (ν0) 0.02 Hzlow cut-off Kolmogorov frequency (ν1) 0.45 Hzhigh cut-off Kolmogorov frequency (ν2) 13.33 Hz

10.5µm. This final sequence is the opd fluctuation as afunction of opd x: ε(x). I then compute the nominal in-terferogram I(x) given in Eq. (4) and normalized to anaverage of 1. Eventually, the pistoned interferogram isIp(x) = I(x + ε(x)). In interferometers, the zero opd iscomputed relative to the knowledge of the metrology ofthe system. It is known to within a constant which is theinstrument opd. We assume in the following that the in-strument opd is zero (we only consider the errors due tothe atmosphere).Figure 3 displays a set of four interferograms simulatingan observation of a 3500 K blackbody spectrum star witha resolution of 25 cm−1. On each line the first graph is theinterferogram followed by the modulus of the spectrum,the phase of the spectrum and the piston sequence. Thespeed of the fringes is 5 mm s−1 yielding a fringe frequencyof 2274 Hz. The wind speed is 20 m s−1. The observingconditions are supposed to be quite good with r0 = 60 cmat 2.2µm, hence a seeing of 0.7 arcsec. These parametersare hereafter referred to as the nominal parameters andare listed in Table 1.The first interferogram is not pistoned. When comparingthe pistoned ones to this one it is obvious that the speed ofthe fringes is varying, some fringes being stretched whileothers are compressed (hence the analogy with the me-chanics term “piston”), and that the whole interferogramis shifted with a random position offset. The shape of themodulus of the spectrum has changed. When piston is ac-celerating in the main lobe of the interferogram, fringesare compressed which increases their apparent frequency.The spectrum support is then shifted to higher frequen-cies and enlarged as the envelope of the fringes is tight-ened. This is what can be seen on the second sequence.The opposite behavior is seen on the third sequence wherepiston is decelerating. The first derivative of piston thusshifts the interferogram in the frequency domain. Higherorders even destroy low resolution information as shownby the simulations. In the fourth sequence, the main lobeof fringes is only slightly pistoned as it has been recorded


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Fig. 3. Simulations of interferometric observations with atmospheric optical path fluctuations. Each line corresponds to adifferent piston sequence. For each sequence the interferogram, the modulus of the spectrum, the phase of the spectrum andthe piston perturbation are plotted

during a phase when piston was flat. Its support is in thecorrect range but the spectral information is destroyed.Concerning phase, it is constant in the original interfero-gram whereas it is a function of wavelength in the pistonedones. It is dominated by a linear component because of theposition offset, but some higher order terms are introducedand are due to the asymmetry of the interferograms.These simulations clearly show that there is a poor correla-tion between successive pistoned spectra and that spectralinformation such as line strength cannot be recovered in adirect manner. Let us now focus on a method to retrievespectral information.

4. Statistical reduction of differential piston

The reduction method which is developped in the follow-ing is based on an analytic expression of the fringe packetby means of a Fourier analysis. It does not rely on anymodel and is thus valid for any geometry of the source.The first part of this section is dedicated to the definition,the calculus and the computation of the phase function.In the second part the problem of reduction is addressed.

4.1. The phase function

4.1.1. Definition

The modulated part of the interferogram as defined byEq. (6) can be written as the product of a slowly varying


x�;k =

PNi=1 x

i�;k

N; k = 0; : : : ; 2p � 1

fI i(x)gi=1;:::;N

fBi(�)gi=1;:::;N

f�i(x)gi=1;:::;N

ffxi�;kggi=1;:::;N

corrected interferograms

reduced opd function

pistoned interferograms

spectra

phase functions

opd functions

FFT

FFT

inversion

averaging

resampling

Fig. 4. Algorithm for the correction of the piston effect

positive function, the envelope, by a rapidly oscillatingfunction:

I(x) = ρ(x) cos(φ(x)), (10)

where φ(x) is the phase function. φ can be defined in dif-ferent ways as it is the argument of a cosine function. Forthe purpose of the correction algorithm it will be a mono-tonic, strictly increasing function. It is not continuous andmakes jumps of π when the envelope becomes zero and hasno derivative.

4.1.2. Calculus

The interferogram of Sect. 2 was defined in wide band butis band limited as the light is filtered. For σ0, a wavenum-ber within the support of the one-sided spectrum or the

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function

pistoned interferogram

spectrum

S

E O

phase function

opd function

depistonedinterferogram

averaged opd

Fig. 5. Sequence of functions computed to resample the inter-ferograms with an opd corrected of the piston effect as definedby the algotihm of Fig. 4


spectrum defined for positive wavenumbers, I define theshifted spectrum S by:

S(σ) = B(σ + σ0) V ∗12(σ + σ0). (11)

The complex spectrum is now expressed with the shiftedspectrum:

Bc(σ) =1

2(S(σ − σ0) + S∗(−σ − σ0)) . (12)

This yields a new expression for Eq. (6):

I(x) =1

2F−1 [S(σ)] (x)e2iπσ0x + (13)

1

2F−1 [S∗(−σ)] (x)e−2iπσ0x,

where F is the Fourier transform operator. Let us nowconsider the inverse Fourier transforms of the hermitianand anti-hermitian parts of the shifted spectrum, notedrespectively E and O, and defined as:

E(x) =1

2F−1 [S(σ) + S∗(−σ)] (x), (14)

O(x) =−i

2F−1 [S(σ) −S∗(−σ)] (x),

leading to a new expression for the interferogram:

I(x) = E(x) cos(2πσ0x)−O(x) sin(2πσ0x). (15)

O(x) and E(x) are both functions with real values andare odd and even respectively when the shifted spectrumis real. The expressions of the envelope and the phase func-tion are straightforward:

ρ(x) =√E2(x) +O2(x), (16)

φ(x) = 2πσ0x+ arctan

(O(x)

E(x)

).

The phase function can be mathematically defined as longas the envelope does not become zero. The problem ofproperly unwrapping this function will be adressed in thenext section. An interesting property can be derived fromEq. (16): the interfringe is constant in the interferogramas long as the O function is a constant and is 0, which isequivalent to saying that the shifted spectrum is an evenfunction. The expression of the phase function does notdepend on the choice of the wavenumber σ0. This prop-erty will be used in Sect. 4.2 and is demonstrated in theappendix at the end of the paper.

4.1.3. Computation

As defined by Eq. (16), the phase function is not continousas the arctan function is not continuous and returns val-ues in the interval ]− π

2 ,π2 [. The phase is first unwrapped

to eliminate discontinuities larger than π. The function

that is obtained is globally strictly increasing. Some dis-continuities still remain that are not due to the arctanfunction. They occur at each zero of the envelope whenits first dervative is not defined. The mathematical func-tion then makes jumps of −π or +π. In reality these jumpsare not resolved when working with finitely sampled in-terferograms. It is not possible to eliminate them: a pis-toned phase function can reproduce a positive unresolvedleap. But a negative unresolved leap-like variation cannotbe due to piston as variations induced by piston are sup-posed to be smaller than variations of the nominal opd.To insure the final phase function to be monotonic, thenegative leaps are rectified. Numerically, when the neg-ative leap has been detected, 2π is added to the phasefunction after the leap, the leap itself is transformed intoa positive leap by a symmetry with respect to the tangentbefore the leap. The error made on the phase functionafter the symmetry is negligeable because the phase func-tion is very close to a straight line when it is continuous(the interfringe is almost constant). After these transfor-mations, the computed phase function is monotonic but isdefined with a random additive constant proportional toπ. The value of this constant is fixed this way: the value ofthe phase function must be in [−π,+π] around the whitelight fringe.

4.2. Correction algorithm

4.2.1. Philosophy

The piston signal ε(x) is a noise on the variable x, the opd.The idea to correct pistoned inteferograms is to reducenoise on the opd as statistical data reductions are usuallyprocessed. The main difficulty is to express the opd as afunction of something, that is to say to invert Eq. (16) orpart of this equation. To do this it is necessary to find aone-to-one function of opd. This function is the computedphase function which will be thereafter mixed up withthe mathematical one, although they are quite different asexplained in Sect. 4.1.3. Let us note εp(x) the p realizationof a piston sequence. The corresponding interferogram,envelope and phase function are indexed with p. From theunicity property of the phase function and the envelope,the expression of the pistoned interferogram is:

Ip(x) = ρp(x) cos(φp(x)), (17)

with:

ρp(x) = ρ(x+ εp(x)), (18)

φp(x) = φ(x+ εp(x)).

φ = φp(x) is invertible as φp is strictly increasing and so isa one-to-one function. The opd as a function of the phasefunction can be defined and is noted xp(φ). The reducedopd function x(φ) is the average of the realizations of thepistoned opd functions. For an infinite number of realiza-tions the reduced opd function converges to the nominal


opd function. For N realizations of the piston the residualpiston in the reduced interferogram is thus reduced by afactor

√N , the reduced inteferogram being eventually:

Ir(x) = ρ(x) cos(φ(x)). (19)

4.2.2. Algorithm

For sake of clarity the correction algorithm is summarizedin Fig. 4 and the functions used in the reduction processare plotted in Fig. 5. Let us assume that N interferogramshave been recorded: {Ii}i = 1, ..., N , each of length 2p cor-responding to a fixed nominal opd window width ∆x orspectral resolution δσ with ∆x . δσ = 1. The samplingfrequency is at least twice the higher spectral frequencyto obey the sampling theorem.

Computation of the phase functions Let us consider thetwo-sided spectrum obtained by discrete Fourier trans-forming the interferogram. The frequency samples are(−2p−1 + 1)δσ, . . . , 0, . . . , (2p−1)δσ where 2p−1δσ is theNyquist frequency. B(σ) V ∗12(σ) is extracted from posi-tive frequencies, B(−σ) V12(−σ) from negative frequen-cies and conjugated. The sum and the difference of thetwo are computed. 2p−2− 1 and 2p−2 long zero sequencesare added before and after the two lists to produce theS(σ)+S∗ (−σ) and S(σ)−S∗ (−σ) functions of Sect. 4.1.2with σ0 = 2p−2δσ (half the Nyquist frequency). After in-verse Fourier transform of these two, the phase functionis computed as explained in Sect. 4.1.3.

Inversion of the phase functions The previous step yieldsN pistoned phase functions. They are all defined inthe same temporal window (corresponding to an opdwindow width ∆x) but they do not vary in the samerange as the piston randomly shifts the interferogramsposition in the window. For each realization i of thephase function, the maximum φimax and minimum φimin

of the function are computed. The common range ofvariation of all the phase functions is determined tobe: [φmin, φmax], where φmin = maxi = 1, ..., N φ

imin and

φmax = mini = 1, ..., N φimax. This is the maximum com-

mon range on which opd functions xi(φ) can be de-fined. The phase functions are linearly interpolated on

{φk = φmin +(φmax−φmin

2p−1

)k}k = 0, ..., 2p−1 producing

a sampled opd function {xiφ,k = xi(φk)}k = 0, ..., 2p−1

for each interferogram i (the index φ means that theopd function definition is relative to the phase functionwhereas the sampled inteferograms {Iik}k = 0, ..., 2p−1 aredefined for the set of nominal opds {xk = ∆x

2p (k − 2p−1 +1)}k = 0, ..., 2p−1). The error introduced by the linear in-terpolation process is very negligeable because the pis-toned phase functions have variations between two sam-pling opds that are linear with a very good approximationas long as the sampling frequency is chosen far greaterthan the first cut-off frequency of piston.

Computation of the reduced opd function The opd func-tions are then reduced by processing:

¯xφ,k =

∑Ni = 1 x

iφ,k

N, k = 0, . . . , 2p − 1, (20)

yielding an estimation of the opd fluctuations for eachinterferogram:

δik = ¯xφ,k − xiφ,k, k = 0, . . . , 2p − 1. (21)

The reduced set of opds width ∆x = ¯xφ,2p−1 − ¯xφ,0 issmaller than the nominal width ∆x which means thespectral resolution has been decreased by the reductionprocess.

Computation of the “unpistoned” interferograms The re-sult of the three previous steps is the estimation of theopd as a function of the nominal opd: xφ = xφ(x). Thereduced interferograms or unpistoned interferograms arethe following sets of samples:

{(xφ(xk), Iik)}k = 0, ..., 2p−1, (22)

where the xφ(xk) are interpolated from the samples ¯xφ,k.These sequences are then interpolated to yield regularlysampled interferograms.

4.3. An example of correction

The correction algorithm that has just been presented isused to reduce simulated pistoned interferograms. Thesource has a 3500 K blackbody spectrum and it is seenthrough a K filter. It is unresolved with a constant visi-bility modulus of one and with a constant visibility equalto zero. Most of the energy is in the range σ = 4000to 5000 cm−1. The number of samples is 1024 and theparameters are those listed in Table 1. The length ofthe sequences and the spectral resolution are the sameas in Sect. 3.6. 100 interferograms have been simulated.Figure 6 shows the result of the reduction. The left viewof Fig. 6 is the corrected interferogram. The spectrum ofthe corrected interferogram (full line), the original spec-trum (square dots), and the spectrum of the pistoned in-terferogram (dashed line) are on the right view. In Fig. 7are plotted one of the simulated piston sequences (dashedline) and the approximation of the error signal (full line)as a result of the reduction process. See Sect. 5 for com-ments.

4.4. Influence of noises on the correction of theinterferorams

4.4.1. Multiplicative noise

The expression of the interferogram given in Eq. (2) re-quires to assume that the intensity in the two arms doesnot fluctuate. This situation prevails when using single


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Fig. 6. Corrected interferogram and spectrum for an infiniteS/N . Top view: corrected interferogram. Bottom view: spec-trum of the corrected interferogram (full line), original spec-trum (open circles)

−200 −150 −100 −50 0 50 100 150 200−2

−1

0

1

2

3

4

5


Pis

ton

(opd

/lam

bda)

Fig. 7. A simulated piston sequence (dashed line) and the errorsignal output by the reduction program (full line)

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spe

ctru

m e

nerg

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Fig. 8. Spectrum after reduction and summation of 100 inter-ferograms with S/N = 100 (the dashed curve is the originalspectrum)

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nerg

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Fig. 9. Simulated interferogram in a narrow K band andspectrum


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Fig. 10. Spectra of interferograms reduced with the nar-row band method for three different S/N ratios. Top view:S/N = 100. Middle view: S/N = 50. Bottom view: S/N = 20.(The dashed curve is the original spectrum)

mode fibers (Coude du Foresto & Ridgway 1991; Coude duForesto et al. 1992) because all coherent photons injectedin fibers do interfer and because fibers allow to momitorthe fluctuations of photometry in the two arms. The pho-tometric fluctuations are corrected during data reductionand interferograms are renormalized. These variations arelow frequency variations (usually of the order of 100 Hz)and the interferograms can be scanned at higher frequen-cies allowing for a correct filtering to remove these varia-tions if fiber optics are not used. Yet, the visivility transferfunction is allowed to vary with time as long as the varia-tions are achromatic. This then only changes the amountof modulated energy but it does not change the phasefunction. In other words, Eq. (2) can be supposed reliablefor the correction algorithm.

4.4.2. Additive noise

The correction simulation of the previous section was car-ried out assuming an infinite signal-to-noise ratio. TheS/N ratio is defined as in Brault (1985): assuming a whitenoise with rms fluctuations εx in the temporal domain, theS/N ratio is the ratio of the white fringe amplitude and εx:

S/N = I(0)εx

. The local S/N ratio is thus rapidly decreas-ing in the interferogram with distance to the white lightfringe. For the spectrum simulated in this paper, the S/Nratio at the top of the first side lobes is only 15% of theBrault signal-to-noise ratio. Figure 8 shows the modulusof the spectrum after correction of 100 interferograms withS/N = 100. After correction of piston, spectra have beenaveraged to reduce the additive noise. The spectrum cor-rection quality is very degraded because the phase func-tion computed for a local S/N ratio less than 1 is notreliable. Besides, the phase function is very sensitive inthe zones where the envelope of the interferogram is zero.As a matter of fact the sign of the interferogram can ran-domly change there producing random π phase shifts. Themore random phase shifts the more different phase func-tions from one interferogram to the other. Although themain lobe is corrected with a good quality, additive noiseis a disaster to correct other lobes when the S/N ratiois not very high, hence the great difference between thecorrected spectrum and the original brightness density.Nevertheless, the correction can be improved. The inter-ferometer has two outputs which are theoretically comple-mentary because the input energy is conserved. The de-tection of the complementary interferogram can be donedifferently to use it as a correction interferogram. The useof a narrower K filter for the second output will increasethe coherence length of the beams, thus widening the en-velope of the second interferogram, and will take the firstside lobes away, the first lobe width being proportionnal tothe inverse of the spectrum width. Assuming the amountof noise is the same in the two arms of the interferome-ter, the S/N ratio for the narrow filter interferogram isthe S/N ratio times the ratio of collected energy through


the K and narrow K filters. The spectrum seen throughthe narrow K filter with a 25 cm−1 spectral resolution isshown in Fig. 9. The S/N ratio for the narrow band inter-ferogram is approximately degraded by a factor of 5 withrespect to the S/N ratio for the regular K filter inter-ferogram. Series of 100 simulated pistoned interferogramshave been corrected and averaged for S/N ratios of 100,50 and 20. The resulting spectra are presented in Fig. 10.It is to be noticed that if filters of adjustable width areavailable then it is possible to adapt the width of the nar-row filter to optimize the final spectral resolution. As amatter of fact, for large S/N ratios in wide band the widthof the narrow band can be chosen so that the S/N ratio ofthe narrow band signal is for example 20 (or larger). The

width of the narrow filter is then S/N20

times smaller thanthat of the wide filter and the resolution of the wide bandspectrum is increased by the same factor. If the method issystematically applied then the final resolution is inverselyproportional to the S/N ratio of the wide band signal.

4.4.3. Unknown absolute optical path difference

The reduction algorithm has to be adapted when a com-mon origin independent of the interferograms is not knownwith a precision better than one tenth of a wavelength. Acommon origin is required for all interferograms to definethe position functions. It is very convenient to set the posi-tion where the phase function is zero as the origin becauseit can be very accurately determined. The same algorithmcan then be applied, the only difference being that the vis-ibility phase is determined to within a constant and doesnot allow image reconstruction. Accurate phase recoverywill be adressed in another paper.

4.5. Irregularly sampled corrected spectra computation

The correction algorithm of Sect. 4.2.2 has led to the ir-regularly sampled interferograms of Formula (22). Thederivation of spectral information from these sequencesis not straightforward. The usual way is to interpolate theirregularly sampled sequences at regularly spaced opds toget regular samples and compute their discrete Fouriertransform. The interpolating process introduces an ex-tra noise all the more important as the sampling fre-quency is lower. This numerical noise can be reducedby oversampling the original sequences to get better es-timates when interpolating. The corrected low resolu-tion spectra presented in this paper were obtained us-ing this method. The method fails when high resolutioninformation is mandatory. Oversampling would increasethe number of samples without increasing resolution, andvery long sequences cannot be oversampled indefinitely.Besides, the residual noise would destroy part of the highresolution information. A method overcoming interpola-tion drawbacks is necessary. An algorithm has been de-velopped by Feichtinger et al. (1994) to compute spectra

from non-uniform samples. The algorithm is iterative andthus slower than the Fast Fourier Transform algorithm.25 iterations, which is roughly 107 floating point opera-tions, are needed to reconstruct the spectrum from 2300samples with a high accuracy (the normalized error is lessthan 10−13). The time required by the restoration algo-rithm increases with the size of the gaps in the samples.The number of iterations would thus increase with thestrength of piston.

5. Results

5.1. Reduction of interferograms with an infinite S/Nratio

From the corrected spectrum and interferogram calculatedin Sect. 4.3 and presented in Fig. 6, the modulus and thephase of the complex visibility are computed in the range4000−5000 cm−1 (Fig. 11). The average measured visibil-ity is 1.006 with a standard deviation of 1.3% (Table 2).The phase is almost perfectly zero, the error being lessthan 0.01 rad. The oscillations in the visibility modu-lus account for most of the noise. The closer the edgesof the spectrum, the bigger the oscillations. The phase isleft unaffected by these oscillations. This is the Gibbs phe-nomenon (Bracewell 1986). As shown in Fig. 7 the injectederror signal is well reconstructed by the reduction processon more than 100µm on each side around the zero opdposition. The difference between the injected error sig-nal and the correction process output error signal is, asexpected, of the order of λ2 rms. It is a smooth and coher-ent noise which explains the good reconstruction of thespectrum. A fast varying noise with the same rms fluctua-tions would produce a spectrum with few similarities withthe original one. Before −108µm and after 107µm the re-constructed piston signal is shifted by 1 λ (or the phasefunction is shifted by π) with respect to the signal recon-structed in the interval [−100,+100]. The interpolationsnecessary to compute the interferograms with fluctuatingopds generate errors in the interferograms that are simi-lar to a noise. The π-leap in the output piston is due tothis noise. As a consequence, although the reduced inter-ferogram is defined on a width ∆x ' 360µm, the actualspectral resolution is determined by a width ∆x ' 200µmand is δσ ' 50 cm−1. It is possible to increase spectral res-olution with an infinite S/N ratio if the method explainedat the end of Sect. 4.4.2 is extrapolated. The width of thenarrow band interferogram is the reverse of the bandwidthand the resolution of the reconstructed wide band spec-trum is then of the order of the bandwidth of the narrowband spectrum.

5.2. Reduction of interferograms with finite S/N ratios

More realistic simulations are necessary to determine howgood the correction can be when data are noisy and whatshould be expected with this method. Three S/N ratios


4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000

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Fig. 11. Visibility modulus (left view) and visibility phase (right view) for an infinite S/N ratio

Table 2. Reduction results

S/N +∞ 100 50 20

V 1.006 1.021 1.044 1.153

σV 0.013 0.045 0.047 0.424

have been simulated: 100, 50, 20. The spectrum displayedin Fig. 8, which is the result of the reduction and sum-mation of 100 interferograms with S/N = 100, shows thelimits of the method in its basic use. For such a S/N ratiothe spectrum reconstruction starts to be of poor qualityand the inferred visibility function gets less and less re-liable. Spectra obtained with the narrower filter methodof Sect. 4.4.2 and displayed in Fig. 10 lead to the visi-bility functions moduli and phases of Fig. 12. As for thecase of the infinite S/N ratio, the Gibbs phenomenon isvisible. Spectral fluctuations frequencies decrease with de-creasing S/N ratios indicating that spectral resolution isalso decreasing, and fluctuations amplitudes increase withnoise. The statistics of the moduli of the measured visibil-ity functions are presented in Table 2. Visibility functionsare measured with a quite good accuracy for S/N ratiosof 100 and 50 with average precisions of 2.1% and 4.4%respectively. Because of the Gibbs phenomenon, local er-rors can be larger than 10% on the edge of the spectrum.For S/N = 20 the correction algorithm fails to recon-struct workable visibility moduli. As far as phases are con-cerned, the reconstruction is easier. The maximum erroris 0.01 rad for S/N = 100 and 50 and it reaches 0.02 radfor S/N = 20.

5.3. Concluding remarks

From these results a few conclusions can be drawn on whatcan be expected from the piston reduction process. First,

in the simulations the fringes were scanned at 5 mm s−1

corresponding to a fringe frequency of 2274 Hz at 2.2µm.For a scan length of 400µm it has been shown that thestandard deviation of piston is 10.5µm for average atmo-spherical conditions on the IOTA interferometer and forthis fringe speed. Each set of simulated data is a record of100 interferograms. After reduction the strength of pistonis thus attenuated by a factor of 10 and the standard devi-ation of optical path fluctuations is λ

2at 2.2µm. The good

quality of the correction obtained with the proposed algo-rithm in realistic conditions of observation shows that itis workable for real observations. Second, very good S/Nratios are necessary if visibility measurements with preci-sions better than 5% are required. It turns out that themaximum level of noise that makes this goal achievablecorresponds to a S/N ratio close to 50. Third, this methodnecessarily degrades the initial spectral resolution. Aftercorrection, the estimated resolution for an infinite S/Nratio is about 50 cm−1 and is of the order of 100 cm−1 ormore for S/N ratios of 100 and 50. These resolutions arepoor relative to the resolutions needed for a fruitful anal-ysis of CO lines in the K band for example. Nevertheless,although it is theoretically possible to indefinitely increasespectral resolution in the case of an infinite S/N ratio asexplained in Sect. 5.1, with these resolutions it is possi-ble to derive gradients and higher order terms from thecomputed visibilities that should contain interesting in-formations.

6. Conclusion

Interferometric measurements of fundamental parametersof stars are common applications of astronomical interfer-ometers in the optical range. What is measured is the aver-age value of parameters across a spectral band. Knowledgeof the dependence of these parameters with wavelengthwould be precious and would bring rich informations on


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se (

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Fig. 12. Visibility moduli (left views) and phases (right views) for S/N of: 100 (top views), 50 (middle views) and 20 (bottomviews)


the physical conditions of the sources. A fundamental limi-tation to achieve this goal is the atmospheric phenomenonof optical path fluctuations or piston effect. It has beenshown that the method presented in this paper allows torecover the visibility function with a low spectral resolu-tion. Visibility moduli are reconstructed with a precisionthat can be as good as a few percent between 2 and 2.5µmfor S/N ratios greater than 50. Visibility phase gradientsare also very well extracted from the simulated data witherrors that are less than 0.01 rad.This reduction algorithm for correction of piston fails torecover the true phase of visibilities. This is not prejudi-cial to the spectral analysis of visibilities. A complementto this algorithm is needed if one wants to remove atmo-spheric phase errors from interferometric data to achievehigh resolution imaging and will be presented in a forth-coming paper.

Appendix A

Supposing a piston sequence of duration T . Let us definethe standard deviation of piston during that sequence:

σ(T ) =

√√√√ 1

T

∫ +T2

−T2

(ε(t)− ε)2dt, (23)

where ε is the average value of piston:

ε =1

T

∫ + T2

−T2

ε(t) dt. (24)

Let us consider the T -periodic function:

γ(t) = [ε(t) . ΠT (t)] ? ttT (t), (25)

where ΠT and ttT are, respectively, the window functionof width T and the Dirac comb with spacing T . Then, γequals ε on the interval

[−T2 ,+

T2

]and the moving average:

a(t) =1

T

∫ t+T2

t−T2

γ(x) dx (26)

is a constant equal to ε by definition of γ. Besides, themoving average acts as a low pass filter with transfer func-

tion sin(πωT )πωT

on γ. It is thus straightforward to derive thespectrum of the periodic function Γ(t) = γ(t)−a(t) whichmatches ε(t)− ε on

[−T2 ,+

T2

]:

Γ(ω) =

(1−

sin(πωT )

πωT

). γ(ω), (27)

with:

γ(ω) =

[ε(ω) ?

sin(πωT )

πωT

]. ttT (ω). (28)

Γ has discrete values with spacing 1T

and Γ(0) = 0. Hence,for ω 6= 0:

Γ(ω) =n=+∞∑n=−∞

[ε(ω) ?

sin(πωT )

πωT

] (nT

). δ(ω −

n

T) (29)

and the spectrum is zero except for harmonics:

hn =

∫ +∞

−∞ε(n

T− ω) .

sin(πωT )

πωTdω. (30)

Computing the standard deviation for a piston sequenceof duration T thus amounts to computing the standard de-viation of the periodic function Γ which is the square rootof the infinite sum of the squared harmonics. Eventually:

σ(T ) =

√√√√2n = +∞∑n = 1

hn2. (31)

Appendix B

For two wavenumbers σ1 and σ2 I define the correspondingenvelopes ρ1 and ρ2, the phase functions φ1 and φ2, theshifted spectra S1 and S2, the odd and even functions O1

and O2, E1 andE2. By definition, there is a simple relationbetween S1 and S2:

S2(σ) = S1(σ + σ2 − σ1). (32)

Let us derive the expression of E2:

E2(x) =1

2F−1 [S2(σ) + S∗2 (−σ)] (x) (33)

=1

2F−1 [S1(σ + σ2 − σ1) + S∗1 (−σ + σ2 − σ1)] (x)

=1

2F−1 [S1(σ)] (x)e2iπ(σ1−σ2)x +

1

2F−1 [S∗1 (−σ)] (x)e−2iπ(σ1−σ2)x

= E1(x) cos(2π(σ1 − σ2)x)−O1(x) sin(2π(σ1 − σ2)x).

In the same way it can be shown that:

O2(x) = O1(x) cos(2π(σ1−σ2)x)+E1(x) sin(2π(σ1−σ2)x).(34)

As a consequence, it is obvious that the envelopes are thesame:

ρ1(x) =√E2

1(x) + O21(x) =

√E2

2(x) +O22(x) = ρ2(x). (35)

Let us show that the two calculi lead to the same phasefunction:

φ2(x) = 2πσ2x+ arctan

(O2(x)

E2(x)

)(36)

= 2πσ2x+

arctan

(O1(x) cos(2π(σ1 − σ2)x) + E1(x) sin(2π(σ1 − σ2)x)

E1(x) cos(2π(σ1 − σ2)x)− O1(x) sin(2π(σ1 − σ2)x)

)= 2πσ2x+ arctan

(O1(x)E1(x) + tan(2π(σ1 − σ2)x)

1− O1(x)E1(x)

tan(2π(σ1 − σ2)x)

)= 2πσ2x+ arctan (tan [φ1(x)− 2πσ1x+ 2π(σ1 − σ2)x])

= φ1(x).


The previous lines thus demonstrate that the envelopeand the phase function do not depend on the choice ofthe shifting wavenumber. The derivation is still valid withσ0 = 0. In this case I(x) = E(x) and:

ρ(x) =√E2(x) +O2(x), (37)

φ(x) = arctan

(O(x)

E(x)

).

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