Advances and limitations of phase dispersion measurement by spectrally and spatially resolved interferometry

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Optics Communications 281 (2008) 3051–3061

Advances and limitations of phase dispersion measurementby spectrally and spatially resolved interferometry

A. Borzsonyi a, A.P. Kovacs a, M. Gorbe a, K. Osvay a,b,*

a Department of Optics and Quantum Electronics, University of Szeged, P.O. Box 406, Szeged 6701, Hungaryb Max Born Institute, Max-Born-Str. 2/A, Berlin 12489, Germany

Received 9 October 2007; received in revised form 31 January 2008; accepted 4 February 2008

Abstract

Spectrally and spatially resolved interferometry has become a powerful tool for measurement of relative phase shift of broadbandlaser pulses propagating through any dispersive medium. Since under usual experimental conditions the relative phase shift is due tothe medium only, the values of group delay, group delay dispersion and the third order dispersion provide the common ultrafast disper-sion constants for the given material. In this paper a detailed description of the state of the art of spectrally and spatially resolved inter-ferometry is given. With the consideration of the effects of wavelength calibration, bandwidth of the light source, electrical and opticalnoises, curved wavefronts, visibility of the fringes, and mechanical vibrations of the interferometer, we show that the accuracy for thedetermination of group delay dispersion and third order dispersion can be as high as 0.1 fs2 and 2 fs3, respectively.� 2008 Elsevier B.V. All rights reserved.

Keywords: Ultrafast lasers; Dispersion; Interferometry

1. Introduction

Generation and application of ultrashort laser pulsesrequire the precise knowledge of dispersion values of alloptical elements beginning from the laser oscillator,through the laser system, and the beam transporting ele-ments up to the target [1–3]. To characterize the phase dis-persion of optical elements, one of the usual ways is tomeasure the spectral phase distortion of a broadband lightsource upon propagation through the elements, andexpand the phase shift to Taylor series around the centralfrequency of the pulse. For applications involving trans-form limited pulses not shorter than 20 fs, it is enough togive the coefficients of the first, second and third orderterms, which are called group delay (GD), group delay dis-persion (GDD) and third order dispersion (TOD), respec-

0030-4018/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2008.02.002

* Corresponding author. Address: Department of Optics and QuantumElectronics, University of Szeged, P.O. Box 406, Szeged 6701, Hungary.Tel.: +36 62 544273; fax: +36 62 544658.

E-mail address: [email protected] (K. Osvay).

tively. In case of much shorter pulses or CPA lasers,however, transform limited pulses with high temporal con-trast are obtained only if the fourth and higher order phasederivatives are also considered [4–6].

Besides the direct measurement of spectral group delayof femtosecond pulses propagating through a laser system[7], various interferometric methods have been developedto measure the phase dispersion. Majority of these belongto spectral interferometry, which can be divided into twogroups whether the laser beams are collinear or crossed.First, when the pulses propagate collinearly, a temporaldelay is generally introduced between them to create spec-tral fringes [8–17]. The most widely used procedure for theevaluation of those interferograms is based on the Fourier-processing of the spectral intensity distribution [10–13,16,17]. When a 2D camera is used as a detector, thespectral fringes may provide spatial dependence, as theirevaluation is carried out line-by-line (parallel to the fre-quency axis) to get the spatial dependence of phase disper-sion. Although the experimental technique is similar, inother cases of the 2D fringe system the application of

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3052 A. Borzsonyi et al. / Optics Communications 281 (2008) 3051–3061

temporally overlapping pulses has also proved to be usefulat the adjustment of a stretcher-compressor system, whenthe sign and magnitude of the dispersion coefficients canbe determined from the shape of the spectral fringes [18].

Second, in case of non-collinear beams, the pulses aretemporally overlapped to create spatial fringes resolved inthe frequency domain [19–29]. Upon evaluation a Fou-rier-filter or a cosinusoidal fit for the spatial intensity distri-bution is applied column-by-column (perpendicular to thefrequency axis). Throughout this study we focus on non-collinear spectrally and spatially resolved interferometry,which is named by different groups of authors as spectrallyresolved white light interferometry (SRWLI) [19,25,30],spatial and spectral interference (SSI) [21,22,26], spatialencoded arrangement for temporal analysis by dispersinga pair of light E-fields (SEATADPOLE) [27] and spatiallyand spectrally resolved interference (SSRI) [28,29,31,32]. Inthis paper we use the latter acronym.

The first experimental results obtained by SSRI werepublished by Puccianti in 1901, who studied anomalousdispersion in oxy-hemoglobin [33]. Later he also extendedhis investigations to the anomalous dispersion in metalvapors. The tube containing the metal–vapor was placedin one arm of a two-beam interferometer, a Jamin or aMach–Zehnder interferometer, and in the other arm anempty tube was inserted to compensate the dispersion ofthe end-windows. The bottom of the tube with themetal–vapor was heated, therefore the phase shift causedby the metal–vapor changed along the vertical direction.The shape of the SSRI fringes was very similar to the shapeof the phase function of the vapor. This property of theSSRI is a great advantage compared to other interferomet-ric methods, because it allows a fast, relatively precise,visual dispersion control of the sample.

To measure the oscillator strength of atomic transitionsin Na-vapor Rozhdestvenskii modified Puccianti’s method.In the lack of precision translators he inserted a thick glassplate in the reference arm of the interferometer. Hook-likeshaped fringes appeared near the absorption line becauseof the large delay caused by the plate [34]. Since the measure-ment of oscillator strength requires only the determinationof the spectral position of the hooks, Rozhdenstvenskii’smethod became very precise and is known as hook-method[35]. SSRI was also used at the measurement of phase dis-persion in metal [36] and multilayer films [37]. However,the use of photographic plates in all of these measurementsdid not allow reaching an outstanding high precision.

The application of CCD cameras made computer pro-cessing of interferograms possible. This technological pro-gress resulted not only in a sudden increase of accuracy ofthe hook method [38], but also opened up the way for mea-suring the phase function of optical elements with high pre-cision practically anywhere in the spectrum. Hence, duringthe last fifteen years the non-collinear SSRI has been usedfor dispersion measurements of chirped laser mirrors[19,21,23] and dispersion compensating saturable absorbermirrors [24], for pump and probe experiments [20], for fine

dispersion adjustment of a stretcher-compressor system ofa CPA laser [18], for the characterization of laser pulses[22,26,27], and for characterization of pulse shapers [29].At the applications mentioned, the delay was changedalong the vertical axis by slightly tilting the reference mir-ror around a horizontal axis or the two beams were fed intothe spectrograph at different positions [27]. Note that spa-tial encoding of the phase information in a spectrallyresolved spatial interferogram has been proved to be alsouseful in a non-linear interferometric pulse characterizationmethod, in the so-called SEA-SPIDER technique, wherethe accuracy of the SPIDER technique is increased signif-icantly by spatial encoding [39–43].

It has to be mentioned that non-collinear SSRI is capa-ble not only to detect angular dispersion (or spectral chirp)but also to distinguish it from material dispersion [44]. Thisfeature especially important in cases when the sample to bemeasured is prismatic [45–47] or the laser beam propagat-ing in the sample arm becomes angularly dispersed dueto the misalignment of the stretcher–compressor systemof a chirped pulse amplification (CPA) laser [48]. If the slitof the spectrograph is in the plane of the angular disper-sion, then the periodicity of the spectrally resolved spatialinterference fringes is not linearly proportional to wave-length any more. Thus, the spectral shape of the fringesis still characteristic to the relative phase difference whilethe change of spectral periodicity along the spatial coordi-nate is a measure of angular dispersion.

As the examples above show that SSRI has become apowerful technique for dispersion measurements. The mostimportant issue is now whether the accuracy can beincreased further, what is the ultimate accuracy at all,and which factors limit it. Collinear Fourier-transformspectral interferometry was carefully analyzed where theeffects of the spectral calibration error, spectral resolution,and sampling frequency were also included [11–13]. For thenon-collinear SSRI, Meshulach and his co-workers havealready given a summary [21]. However, the evaluation ofthe interferograms was presented briefly, and no detailederror analysis was given. The possible reason is that theamount of dispersion measured so far had been relativelyhigh. There are, however, factors which have to be care-fully treated to achieve the highest possible accuracy.

In this paper a rigorous description of the SSRI methodis given with special emphasis on non-collinear geometry.We introduce a novel method for fringe evaluation and ameticulous error analysis, which allows measuring smallamounts of the second and third order dispersion with ashigh accuracy as 0.1 fs2 and 2 fs3, respectively.

2. Theory of spectrally and spatially resolved interferometry

The technique of spectrally and spatially resolved inter-ferometry is based on a two-beam interferometer equippedwith an imaging spectrograph. For the sake of simplicitylet’s assume a Mach–Zehnder interferometer illuminatedby a broadband light source like a femtosecond pulse

A. Borzsonyi et al. / Optics Communications 281 (2008) 3051–3061 3053

(Fig. 1). The arm, where a dispersive object is to be insertedto, is called sample arm. The spectral phase and amplitudeof the sample pulse IS(x) are modified upon propagationthrough the object. The reference pulse IR(x) propagatesundisturbed in the other arm of the interferometer, whichusually equipped with a precision translation stage for cor-rect timing of IR(x) to IS(x). The phase fronts of the sam-ple and reference pulses are tilted with an angle of 2c withrespect to each other. The interference fringes formed bythe temporally overlapped sample and reference pulsesare imaged onto the input slit of an imaging spectrograph.The resulting two dimensional image is hence resolvedspectrally (along the frequency or wavelength) and spa-tially (along the slit).

Depending on the nature of the object, two types of mea-surements are possible. The spectral phase shift and thus thedispersion of the object can be determined with a procedure,when first the SSRI fringe of the empty interferometer isrecorded so that the spectral phase shift of the interferome-ter itself is obtained. Having inserted the object to the sam-ple arm, it is usually necessary to change the delay of thereference pulse to obtain interference fringes again. Fromthe resulting interference fringes and the change of the delayline the spectral phase shift can be determined again. Hav-ing subtracted the spectral phase shift of the empty interfer-ometer from the latter, obviously the spectral phase shiftdue to the object itself can be deducted.

The other type of measurement is when the change ofdispersion is of interest, when one property of the objectis varied, like temperature or pressure [28,29]. The differ-ence to the above described situation is that now the baseinterferogram is recorded when the object is already inthe interferometer. The subsequent interferograms arerecorded upon variation of the given property. Followinga similar process, it is easy to see that the change of spectralphase shift can be ultimately obtained.

To emphasize the difference between the collinear andthe non-collinear spectrally and spatially resolved interfer-ometry, we start with the mathematical description of theircommon origin, that is with two-beam spectral interferom-etry, that is when the sample and reference pulses reach theslit collinearly (c = 0). It is feasible to suppose that thephase shift in each arm is independent of the spatial coor-dinate y. We may write the spectral phase of the referenceand sample pulses in the usual form of Taylor series

Broadbandlight source

Sample arm

Reference arm

Object ( Obj)

Delay

Broadbandlight source

Sample arm

Reference arm

Object ( Obj )

lObj

nMed

Delay

ϕ

Fig. 1. Formation of spectrally and spa

uiðxÞ ¼ uiðx0Þ þGDiðx0ÞDxþ 1

2GDDiðx0ÞDx2

þ 1

6TODiðx0ÞDx3 þ . . . ; i ¼ R; S; ð1Þ

where the coefficients noted by the usual symbols of GD(group delay), GDD (group delay dispersion), and TOD(third order dispersion) are the phase derivatives at the cen-tral frequency of x0 and Dx = x � x0.

In the case of empty interferometer, the phase shift forboth arms can be originated from the optical componentsin the given arm (uOpt

R , uOptS ) and from the medium where

the interferometer is placed, which is usually air (uMedR ,

uMedS ). Assuming the reference and the sample pulses propa-

gate a distance of LR and LS in their arms, respectively, onecan write

uMedi ðxÞ ¼ uMed

i ðx0Þ þ gdMedðx0ÞLiDx

þ 1

2gddMedðx0ÞLiDx2 þ . . . ; i ¼ R; S; ð2Þ

where gdMed, gddMed, . . . are the specific dispersion coeffi-cients of the given medium, i.e. the dispersion value perunit length of propagation. According to the theory of lin-ear propagation of laser pulses [1], the linear phase shiftsare additive, so are their coefficients, hence they can bewritten in the form of

GDiðx0Þ ¼ GDOpti ðx0Þ þ gdMedðx0ÞLi;

GDDiðx0Þ ¼ GDDOpti ðx0Þ þ gddMedðx0ÞLi;

i ¼ R; S

. . .

ð3Þ

Spatial interference fringes are obtained when the phasefront of the reference and sample pulses are not parallelto each other, that is c 6¼ 0. Assuming that the phase frontsare plane, the interference of the reference and sample pulseresults in an intensity pattern depending also on the spatialcoordinate y in the form of

Iðy;xÞ ¼ IRðy;xÞ þ ISðy;xÞ þ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIRðy;xÞISðy;xÞ

p

� cos DuðxÞ þ xnMedðxÞc

2cðy � y0Þ� �

; ð4Þ

where Du(x) = uR(x) � uS(x), c is the velocity of light invacuum, nMed is the refractive index of the medium. Theheight y0, at which the propagation times at the central fre-quency in the two arms are equal (see Eq. (5)), is

Inte

nsity

2D im

agin

gsp

ectr

ogra

ph

Tilted mirrors

2

Inte

nsity

2D im

agin

gsp

ectr

ogra

ph

Tilted mirrors

y

y0

2γ

tially resolved interference fringes.


DGDðy0;x0Þ ¼ GDRðx0Þ �GDSðx0Þ¼ DGDOptðx0Þ þ gdMedðx0ÞDL ¼ 0; ð5Þ

where DL = LR � LS. It means that the dispersion of theempty interferometer, that is the phase difference betweenthe reference and sample arm can be characterized by therelative coefficients as

Duð0Þðy;x0Þ ¼ DuOptðx0Þ

þ x0nMedðx0Þc

½DLþ 2cðy � y0Þ�

DGDð0Þðy;x0Þ ¼ DGDOptðx0Þþ gdMedðx0Þ½DLþ 2cðy � y0Þ�

DGDDð0Þðy;x0Þ ¼ DGDDOptðx0Þþ gddMedðx0Þ½DLþ 2cðy � y0Þ�

. . .

ð6Þ

The values of Du(0),DGD(0), DGDD(0), etc. can be then ob-tained from recording the interference pattern I(y,x), andthe constituting intensity distributions IR(y,x) and IS(y,x).Ways of spectral phase computation from the interfero-gram are analyzed in the next section.

It is worth noting that the values of Du(0), DGD(0), andDGDD(0) are independent of condition Eq. (5). In orderto highlight the connections to classical textbook interfer-ometry, please note that Eq. (5) is one of the basic condi-tions of interference describing that the pulses need totemporally overlap in a certain spatial position at the samefrequency. In case of a polychromatic light source like laserpulses, if Eq. (5) holds for each constituting frequency at thesame spatial position, then interference fringes with maxi-mal visibility are obtained at the output of the interferome-ter. This is the core equation of time of flight interferometry,where spectrally tunable light pulses were used to measurethe spectral group delay of optical components [2,49,50].

As a next step, let’s insert an object into the sample armand assume that its spectral phase shift uObj(x) can beexpanded into similar Taylor series as Eq. (1). In generalit can be also presumed that the object decreases the pathlength in the surrounding medium nMed of the samplearm by ‘Obj. In order to keep the reference and samplepulses overlapped, the length of the reference arm has tobe changed by dL. Accordingly, the relative dispersioncoefficients can be written in the form of

Duð1Þðy;x0Þ ¼ DuOptðx0Þ � uObjðx0Þ

þ x0nMedðx0Þc

½DLþ ‘Obj þ dLþ 2cðy � y0Þ�

DGDð1Þðy;x0Þ ¼ DGDOptðx0Þ �GDObjðx0Þþ gdMedðx0Þ½DLþ ‘Obj þ dLþ 2cðy � y0Þ�

DGDDð1Þðy;x0Þ ¼ DGDDOptðx0Þ �GDDObjðx0Þþ gddMedðx0Þ½DLþ ‘Obj þ dLþ 2cðy � y0Þ�

. . .

ð7Þ

Subtracting Eq. (6) from Eq. (7), the absolute dispersioncoefficients can hence be obtained as

uObjðx0Þ ¼ Duð0Þðy;x0Þ � Duð1Þðy;x0Þþ x0nMedðx0Þ

c f‘Obj þ dLgGDObjðx0Þ ¼ DGDð0Þðy;x0Þ � DGDð1Þðy;x0Þ

þgdMedðx0Þf‘Obj þ dLgGDDObjðx0Þ ¼ DGDDð0Þðy;x0Þ � DGDDð1Þðy;x0Þ

þgddMedðx0Þf‘Obj þ dLg. . .

ð8Þ

When the change of dispersion properties of the object is tobe measured, the situation is somewhat different. Let’smark the object phase shift in the initial phase conditionuObj

a and that at a different dispersion state uObjb . After a

similar treatment as was derived above, the change of theobject’s dispersion DuObj ¼ uObj

b � uObja can be obtained

from Eq. (8) with ‘Obj = 0.Eq. (8) means that the absolute dispersion coefficients of

the object can be obtained from recording two interfero-grams (without and with the object), precise measurementof the change of the reference arm length dL, knowledgeof the dispersion of the medium, and from the geometricallength of the object itself ‘Obj. For the determination of thechange of dispersion properties of the object, however, theknowledge of ‘Obj, which is difficult to measure in manycases, is not necessary.

3. Evaluation methods of the interference patterns

3.1. Simple cosine fit

The most evident evaluation method of the interferencefringes [19] is obtained directly from the theory, when Eq.(4) is rearranged as

cos DuðxÞ þ xnMedðxÞc

2cðy � y0Þ� �

¼ Iðy;xÞ � IRðy;xÞ � ISðy;xÞ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIRðy;xÞISðy;xÞ

p : ð9Þ

It can be seen from Eq. (9) that we generate a so-called nor-malized interferogram by recording the intensity distribu-tion of the SSRI pattern (I(y,x)), the reference (IR(y,x))and the sample (IS(y,x)) arm. Note that the intensity distri-bution of the normalized interferogram is described by acosine-function whose argument contains the phase to bemeasured. To obtain the frequency dependence of thephase, cosine-functions are fitted for each columns of thenormalized interferogram. From the retrieved spectralphase values the phase derivatives are determined by apolynomial fit of the sufficient order (Fig. 2).

The advantage of this method is that the evaluation pro-gram can be easily performed e.g. in MathCad, Matlab,etc. similar to the Fourier transformation evaluation meth-ods [10,27]. A further advantage is that it detects and mea-

Polynomial fit at ω0 :

Cosine-fit to a column:

Spa

tial a

xis

(y)

Optical frequency

Assign the retrievedphase to a frequency:

Sp

atia

laxi

s (y

)

Intensity

GD(ω0)

GDD(ω0)

TOD(ω0)

Optical frequency

Spatialaxis (y)

Phase

Assign the retrieved phase toa frequency at a spatial point:

Step-by-step cosine fit:

Polynomial fit for each row:

Sp

atia

laxi

s (y

)

Intensity

Spatial axis (y)

GD

D

Spatial axis (y)

TOD

Simple cosine fit Phase surface mapping

Optical frequency

Pha

se

Polynomial fit at ω0 :

Cosine-fit to a column:

Spa

tial a

xis

(y)

Optical frequency

Spa

tial a

xis

(y)

Optical frequency

Assign the retrievedphase to a frequency:

Sp

atia

laxi

s (y

)

Intensity

GD(ω0)

GDD(ω0)

TOD(ω0)

Optical frequency

Spatialaxis (y)

Phase

Optical frequency

Spatialaxis (y)

Phase

Assign the retrieved phase toa frequency at a spatial point:

Step-by-step cosine fit:

Polynomial fit for each row:

Sp

atia

laxi

s (y

)

Intensity

Spatial axis (y)

GD

D

Spatial axis (y)

TOD

Simple cosine fit Phase surface mapping

Optical frequency

Pha

se

Fig. 2. Retrieval of phase derivatives from the SSRI pattern by using thesimple cosine fit method and phase surface mapping.


sures the phase front angular dispersion in the samplebeam, since the fit also gives the periodicity of the fringes(k/2c) as a function of frequency [44]. If there is no angulardispersion, that is the angle between the reference and sam-ple phase fronts is constant (2c(k) = 2c), the periodicity ofthe SSRI fringes is proportional to k.

So far we have supposed that the phase fronts of thesample and reference beam are plane waves. If, however,the phase fronts are curved, the periodicity of the SSRIfringes will depend on the coordinate y. In this case the pre-cision of the method may decrease significantly, since theretrieved spectral phase values are a kind of average overthe spatial coordinate. Considering that femtosecondpulses propagate in Gaussian beams so their wave frontis curved, it is essential to find a novel method of evaluationwhich is suitable for pulses with both plane and curvedphase fronts.

3.2. Phase surface mapping

To avoid the decrease in accuracy caused by the curvedphase front, the above simple method is modified in orderto create a complete 2D phase map from the interferogram,

that is, the phase is determined not only in the function offrequency, but also of the spatial coordinate y.

To do that, first a step-wise cosine fitting is applied toeach column (Fig. 2) of the normalized interferogram.The length of one step should be short enough to determinethe proper local phase value but sufficiently long to recog-nize the cosinusoidal shape. From each fitting step theparameters characterizing the fitted cosine function (phase,periodicity of the fringes, amplitude, and offset value) areobtained and assigned to the middle point of the interval.Then the short interval steps by only one point and anew fitting starts. This process provides the phase in everypoint except the edges of the current column. After all, thealgorithm jumps to the next column, so that the entireinterferogram is scanned through.

The values of GD, GDD and TOD are then obtainedfor each row by fitting of third-order polynomial for themeasured phase values along a given row. Areas with lowsignal to noise ratio provide phase with more uncertainty.During the fitting, correlation factors were calculated tomark these points. Because of the sensitivity of high ordercoefficients of polynomial fitting, the points above a manu-ally chosen (uncertainty) threshold were not considered.

The advantages of the phase surface method can besummarized as follows. First, it gives the opportunity tovisualise the entire phase surface. Then, from qualitativepoint of view, a high number of phase data can be obtainedwhile the simple cosine fitting considers only one phasevalue by column. More data provides obviously the possi-bility of a more precise error calculation. The algorithm isspecialized to evaluate not only the GDD and the TOD,but also their standard deviations. These standard devia-tions characterize the accuracy of the measurement dis-turbed by various effects. Last but not least, the newalgorithm allows a fast and automatic processing of anentire interferogram.

4. Error analysis and optimization

The SSRI with phase surface evaluation can be in theoryan appropriate method to measure small dispersions withhigh accuracy. However, the possible limitations in achiev-ing the best results have to be addressed, including theexamination of the experimental conditions as well as ofthe evaluating algorithm.

To study the effect of each possible parameters, interfer-ograms are computed to simulate a 10 bit camera with apixel resolution of 1024 � 1280, which are evaluated bythe developed code based on the method of surface map-ping. Standard deviations of dispersion coefficients arethen calculated to characterise the effect of various distor-tions. As the core parameters of all simulations, the spec-tral and spatial resolution is assumed to be 0.11 nm/pixeland 6.7 lm/pixel, respectively. The illuminating pulses areassumed to have a Gaussian spectral profile, while the rel-ative dispersion values between the reference and samplepulses are GDD = 300 fs2 and TOD = 2000 fs3. Note that


almost the same values of standard deviation have beenobtained for each error source when GDD and TOD werevaried between 10 and 10,000 fs2 and fs3, respectively.Among the variables are the number and the visibility ofthe interference fringes, and the bandwidth of the pulses.Unless they are concerned, their values were fixed at 20fringes, at unit visibility, and at a full width at half maxi-mum (FWHM) of 70 nm.

4.1. Noise of the CCD camera

One of the most dominant and hardly avoidable effectson the captured interferograms is the noise of the CCDcamera. Under noise we mean the entire effect of manykind, like differences in pixel sensitivity, charge level, pho-ton noise, dark current, and also inhomogeneous pixel dis-tribution. Such resultant noise follows with goodapproximation a Gaussian distribution. It is customaryto characterize the achievable highest signal-to-noise ratiowith the so called noise RMS, which by definition is theroot mean square of the ratio of the integrated grey valuesover the full width half maximum bandwidth of the noiseGaussian to the highest possible signal level [51]. Toaccount for the effect of noise, all the simulated interfero-grams were computed at 5 different noise RMS levels from0.05% up to 5%. For comparison, the noise RMS of a typ-ical non-cooled and cooled CCD camera can be in practicereduced below 1% and 0.1%, respectively.

4.2. Computation error

The other inevitable group of inaccuracy is formed bytwo types of computation error, which occur upon evalua-tion of an interferogram. The one is like a digital-analogueconversion problem, as the interference fringes are coded in10 bit while the functions to fit the fringe cross sections arecontinuous. The second one is a so called rounding error,which occurs in the further steps of data processing. Tominimize the latter, a data size of 8 bytes was chosen.Under our standard circumstances described above, thestandard deviation of GDD was below 0.02 fs2, while theerror of TOD was around 0.7 fs3. These numbers can beregarded as the achievable lowest errors of the measure-ment of dispersion coefficients.

4.3. Dynamic range – visibility of the interference fringes

Having established the basic limiting factors as noiseand computation, in the next step the effect of fringe visibil-ity, which is by definition the quotient of the difference andsum of the minimal and maximal intensity values, is ana-lyzed. The visibility itself cannot affect the accuracy ifone ensures that the fringes are always resolved on thehighest greyscale allowed by the camera, which is in ourcase 10 bit. However, various effects as non-equal lightintensities from the arms of the interferometer, blur ofthe fringes due to non-avoidable vibration of optical ele-

ments (see also 4.4), and tilt of phase fronts [52] result ina lower visibility and hence a lower signal-to-noise ratio.Visibility of the fringes is then a good measure of the prac-tical data resolution. As it can be seen on the Fig. 3, a highlevel of noise seriously reduces the accuracy of dispersionretrieval even at high visibilities. At normal noise levels,however, it is recommended to keep the visibility of theinterference fringes above 0.5. When the noise is extremelylow, then the error of the retrieved values stays below theunbelievably low values of 0.03 fs2 and 0.8 fs3 as far asthe visibility is 0.4 or higher.

4.4. Change of path length in the interferometer

In practice the beam path lengths in the two arms of theinterferometer continuously change around their meanvalue because of three main reasons. The one is themechanical vibrations of the optical elements of the inter-ferometer. The second is due to ambient air in the labora-tory, which can flow slowly and its refractive index can alsovary locally due to local temperature variations. The thirdis in connection to beam pointing stability, but it becomessignificant only for asymmetric interferometers (see at Sec-tion 4.9).

The change of relative beam path lengths lies in the rangeof few tens of nanometers and seldom reaches a micron, as itwas experienced in our experiment [31]. This small changehas basically one effect, that is, the relative carrier envelopephase (Du(0) in Eq. (6)) varies. Consequently, the spatialposition and slope of the fringes also differs from shot toshot. Since the exposition time of a CCD camera is relativelylong, it can capture many interferograms on top of eachother. Eventually an interferogram is obtained withdegraded visibility (Fig. 4), but the shapes of the fringesare unaffected [31,32]. The degradation of visibility, how-ever, may decrease the accuracy of phase retrieval.

As the phase difference is proportional to the frequencyat a certain reference arm – sample arm path difference, thevisibility is degraded towards the blue side of the spectrum,as is seen on the curves corresponding to different wave-lengths on Fig. 4.

The degradation of visibility, however, may decrease theaccuracy of phase retrieval, as seen in Section 4.3. Therequired accuracy of dispersion measurement determinesa minimal visibility level, which can be read from Fig. 3.If the visibility of the interference fringes is not sufficient,then a tight isolation of the optical setup against laboratoryair flow is recommended. If it is still not satisfactory, it maybe necessary to adopt more advanced anti-vibrationtechniques.

4.5. Spatial resolution – number of interference fringes

At a given camera, the number of fringes determines thespatial resolution of the fringes and hence the number ofavailable points for the cosine fitting procedure. The num-ber of fringes can be experimentally set by the relative tilt c

0 0.2 0.4 0.6 0.8 1

Visibility

0.01

0.1

1

10

100

Err

or o

f GD

D [f

s2 ]

Noise RMS:<0.05%0.5%1%2.5%5%

0 0.2 0.4 0.6 0.8 1

Visibility

0.1

1

10

100

1000

10000

Err

or o

f TO

D [f

s3 ]

Fig. 3. Standard deviations of GDD (a) and TOD (b) plotted versus visibility of the fringes.

0 100 200 300 400

Beam path fluctuation (nm)

0.2

0.4

0.6

0.8

1

Vis

ibili

ty

λ=775nmλ=800nmλ=835nm

Fig. 4. The visibility of the fringes as a function of random beam pathchange in the interferometer. In the simulation 128 interferograms wereoverlapped.


of the phase fronts from the two arms. The simulationsshow (Fig. 5) that there is an optimal range of fringes atwhich the error of GDD and TOD is minimal. On theone hand, when the fringe number is too low, then thenumber of cosine cycles is also limited so that the fit canhave a large error. On the other hand, if the fringe densityis too high, then the available number of points within acosine cycle becomes too low so that a high accuracy fitcannot be achieved. One can conclude, that the optimalnumber of fringes is slightly below 20, while satisfactoryresults can be anticipated between 10 and 80 fringes.Regarding the camera modeled, the optimum number offringes corresponds to a fringe distance of 68 pixel.

4.6. Spectral resolution – bandwidth of the illuminating

pulses

Having analyzed the resolution problems associated withthe cosine fit to each columns, let us regard the challenge of

0 20 40 60 80 100

Number of fringes

0.01

0.1

1

10

100

Err

or o

f GD

D [f

s2 ] Noise RMS<0.00.5%1%2.5%5%

Fig. 5. Standard deviations of GDD (a) and TOD (b)

polynomial fit to the retrieved phase values along the entirespectral range. Since the dispersion coefficients are directlyresulted from this polynomial fit, it is easy to see thatthe spectral range itself, along which the fit is done, is ofhighest importance. Therefore a high accuracy is expectedwhen pulses with as broad bandwidth as possible areused. Fig. 6 shows the results of the simulations when thebandwidth of the pulse was varied from 20 nm to 120 nm.For noisy cameras the error of GDD and TOD start ris-ing below a bandwidth of 70 nm, while the accuracy ofGDD and TOD measurement with use of virtually noisefree cameras is nearly independent of bandwidth above50 nm.

4.7. Error of wavelength calibration

Similarly to the case of collinear spectral interferometry[11], the error of the spectral calibration is expected tospread further to the dispersion coefficients. Let us supposethat the interference fringes are evaluated with the use ofthe measured calibration of kM(i) = kM + iDkM, while thereal wavelength distribution is k(i) = k + iDk, where i isassociated to the ith column. If the dispersion coefficientsare obtained from a third order polynomial fitting, thenthe measured GDDM and TODM differ from the realGDD and TOD as

GDD � GDDM þ TODM2pc1

kM

� 1

k0

� �Dk

DkM

k2M

k2

� ��

� 1

k� 1

k0

� ��DkM

Dk

� �2 kkM

� �4

; ð10Þ

0 20 40 60 80 100

Number of fringes

:5%

0.1

1

10

100

1000

Err

or o

f TO

D [f

s3 ]

as function of the number of interference fringes.

20 45 70 95 120

Bandwidth [nm]

0.01

0.1

1

10

100

Err

or o

f GD

D [f

s2 ]

Noise RMS:<0.05%0.5%1%2.5%5%

20 45 70 95 120

Bandwidth [nm]

0.1

1

10

100

1000

Err

or o

f TO

D [f

s3 ]

Fig. 6. Standard deviations of GDD (a) and TOD (b) plotted against the bandwidth of the light source.

R1

R2

R1

R2

y

z

δ (y)

Fig. 7. Gaussian beams with different phase front curvatures.


TOD � TODM

DkM

Dk

� �3 kkM

� �6

; ð11Þ

where k0 is the central wavelength. Studying our calibra-tion method, the accuracy of the intercept and the slopeis 0.01% (�0.1 nm) and 0.1% (�10�4 nm/pixel), respec-tively. Considering the values of the GDD (300 fs2) andthe TOD (2000 fs3), the inaccuracies arising from calibra-tion are lower than 0.13% and 0.25%, respectively.

4.8. Temporal delay between the sample and the reference

pulses

Regarding the alignment of the interferometer, an errormay arise from the non-complete temporal overlapping ofthe reference and sample pulses, that is, from the violationof condition Eq. (5). In practice the slope of the interfer-ence fringes may not be completely horizontal at the point(y0, x0) but inclines a small angle [30]. It means that thelength of the reference arm differs by e from the length ofequal propagating time. Thus, it can be taken into accountin Eq. (8) by changing the value of dL to dL + e.

It is easy to see that among the dispersion coefficientsGDObj depends mostly on e since it is determined fromdL. In the usual laboratory practice, however, e is smallerby two- or three orders of magnitude than dL. At the mea-surement of extreme small dispersion, when dL = 0, thevalue of e can be determined from the slope of the fringes.Even in this case, the higher order coefficients, however,remain practically insensitive to e.

4.9. Measurement with subsequent pulses – the effect of

carrier envelope phase

The phase constant ui(x0) of Eq. (1) is in a direct rela-tion to the carrier envelope phase of the illuminating wave-packet, and determines the absolute position of the fringesalong axis y. In the case of a symmetrical interferometer,when the propagation time in both arms are equal, the car-rier envelope phase difference between the sample and ref-erence pulses is always constant, that is, the spatial positionof the fringes is unchanged.

In many experiments the geometrical length of theobject to be measured is so long that the length of the ref-erence arm exceeds the pulse separation of the pulse train.

In these cases it is advantageous to create an asymmetricMach-Zehnder interferometer, when the propagationlength difference between the reference and sample armcorresponds to the round trip of the oscillator length[18,28,29,53]. So, the interference fringes are created bysubsequent laser pulses having different carrier envelopephase. It means that the resulting interference patternsare shifted along the vertical direction, leading to a degra-dation of the visibility of the recorded pattern. Since theshot-to-shot variation of carrier envelope offset is relativelysmall [54], the smearing out of the fringes is not severe.Note, that the slope of the fringes, which is characteristicto the dispersion, is not affected. It is worth mentioning,moreover, that a method has been recently developed todetermine the carrier envelope offset of the pulses fromthe measurement of the visibility of the non-collinear SSRIfringes [31,32].

4.10. Gaussian pulses (non-plane wave approximation)

If there is a propagation length difference between thesample and reference arms, like in the case of asymmetricalinterferometers in 4.9, then the beam radii will be differentat the entrance slit of the spectrometer. It can be easily cal-culated that for a vertical position y taken from the opticalaxis the wave fronts with radii R1 and R2 have a distanced(y) = (R1 � R2)/2R1R2 � y2 along the propagation direc-tion z (Fig. 7). This obviously results in a spatially depen-dent phase shift, so that the left side of Eq. (9) has to bemodified accordingly.

To give a pessimistic estimation for the error caused bythis effect, the phase shift between the beam surfaces istaken as 2p � d(ymax)/k, where ymax is the maximum value


of y at the top edge of the CCD-chip. The simulations(Fig. 8) show that both the GDD and TOD have a rela-tively flat range below the phase shift difference of 4 rad.In laboratory practice, however, this value is already rarelyachievable. For instance, the phase shift difference of 4 radoccurs when the radius of beam waist is 3 lm, the beamwaist is at the first beamsplitter of the interferometer, andthe necessary propagation length difference is 3.2 m. Inthe laboratory practice when the beam waists is around3 mm, the maximum of phase shift difference is only0.2 rad. Note, that this effect can be calculated with themethod of phase surface evaluation only.

4.11. Optical noise

Finally, but not at least, we need to account for a prac-tical disturbing source, which is avoidable but often embit-ters the everyday laboratory life, though. This is diffractionon dustmotes and scratches being on the surface of opticalelements, which introduces light and dark spots to the spa-tial intensity distribution of the interfering beams.

This effect was simulated by placing spots on the inter-ference picture in a random manner. The size, the numberand the bit depth of the spots were also chosen randomly.The sign of the bit depth determines that the spot is eitherdark or light. The diameter of the spots was varied from100 to 300 pixels in such a way that the larger sizesoccurred with less probability than the smaller ones. Thevariable quantity in this case is the average depth of thespots. The depth values were selected according to a nor-mal distribution with a user given mean. Fig. 9 shows the

0 42 6

Phase difference [rad]

0

0.2

0.4

0.6

0.8

Err

or o

f GD

D [f

s2 ]

Fig. 8. Standard deviation of GDD (a) and TOD (b) as a function of

0 40 80 120

Average spot depth

0

2

4

6

Err

or o

f GD

D [f

s2 ]

Fig. 9. Effect of optical noise on the standar

standard deviations of GDD and TOD plotted againstthe average depth. There is an almost linear connectionbetween the error of the coefficients and the average depth.

5. Cosine fitting vs. Fourier processing

As it was mentioned in the introductory section, nowa-days most of the evaluation methods of interferograms arebased on a variant of Fourier processing. Since they pro-vide reasonable accuracy within very short processing timeand easy to perform, a question may arise what are thebenefits of cosine fitting we used throughout this study.

In most of the aspects we analyzed in the previous sec-tion both methods provide practically the same absoluteaccuracy but at different sets of parameters. For instance,Fig. 10a shows the error of the two methods in the functionof the number of the fringes, where the plotted data forcosine fit is identical with the plot belongs to 1% noiseRMS on Fig. 5a, in Section 4.5. As one can see, the similarlevel of accuracy is performed by the cosine fit method andFFT around 15 fringes and above 50 fringes, respectively.

In one of the major aspects, however, when a pulsepropagates according to Gaussian laws (detailed in Section4.10), the FFT method seemed considerably less reliable.The Fig. 10b compares the FFT accuracy to cosine fit-related data on the Fig. 8a, but on a larger scale. By alter-nation of interference fringe periods, as expected, the stan-dard error of Fourier processing is decreased.

From this comparison we can hence conclude that ifthe near real time data acquisition and processing withreasonable accuracy is the major concern at a certain

0 42 6


0

4

8

12

16

20

Err

or o

f TO

D [f

s3 ]

the difference between the phase fronts at the edge of the beams.

0 40 80 120

Average spot depth

0

40

80

120

160

200

Err

or o

f TO

D [f

s3 ]

d deviations of GDD (a) and TOD (b).

0 20 40 60 80 100

Number of fringes

0.1

1

10

100

Err

or o

f GD

D [f

s2 ]

Evaluation type:cos-fit

FFT

0 642


0

1

2

3

4

St.

dev.

of G

DD

[fs2 ] Evaluation

type:cos-fitFFT

Fig. 10. Comparison of standard deviation of GDD obtained from a spectrally and spatially resolved interferogram using cosine fit and FFT methods inthe function of (a) the number of interference fringes, and (b) the phase front curvature.


measurement, then Fourier processing method is the rightsolution. If the pulses used in an experiment are close toGaussian and the highest possible accuracy is to beachieved, then the cosine fit method is the ultimate choice.This latter provides an evaluation error of almost an orderof magnitude lower level at the expense of a three-fourtimes longer processing time than those with FFT.

6. Discussion

From the above analysis we can conclude that the noiseof the camera is one of the most crucial factors. If the noiselevel is kept below the extremely low value of 0.05%, thenthe ultimate accuracy of determination of GDD and TODcan be as high as 0.02 fs2 and 0.7 fs3, respectively. Forachievement of such unpreceded precision it is requiredthat the laboratory standards should be high, as the con-tamination of the optics surface is kept a very low level,and the vibration of the optical bench as well as the airmovement are avoided. The experimental conditions haveto be chosen that the bandwidth of the light source shallbe broader than 50 nm, the vertical distance between theneighbour interference fringes shall be around 50 pixels,and the visibility of the interference pattern shall be at least0.4.

When an affordable camera with a noise level of 0.5% isused, then the accuracy can be kept still high with trade infew experimental conditions. If it is ensured that the visibil-ity is around unity and the bandwidth of the light source isat least 100 nm, then the error of GDD and TOD is stillbelow 0.1 fs2 and 2 fs3, respectively. If the bandwidth ofthe available pulses cannot exceed 60 nm, then no onecan expect a more accurate determination of GDD andTOD than 1 fs2 and 10 fs3, respectively. Please note thatthe experimental errors established in our previous mea-surements [19,28,29,44] are in full agreement with resultsof the above simulations.

Finally but not at least we have to draw the readers’attention, that although all simulations in Section 4 havebeen carried out for crossed beam (or non-collinear) inter-ferometry, most of the results and conclusions, particularlythose of Sections 4.2, 4.6 and 4.7 are directly adaptable forcollinear interferometry, too.

7. Summary

In summary we described a detailed theory of the two-dimensional spectrally and spatially resolved interferome-try. A new fringe evaluation technique, the so-called phasesurface mapping method was introduced, which providesthe full phase surface along spectral and spatial coordi-nates. Using computer simulations, we studied the effectof the various error sources on the accuracy of the disper-sion measurement. It has been shown that the ultimateaccuracy of determination of GDD and TOD can be lessthan 0.1 fs2 and 2 fs3, respectively. For CCD cameras andlaser sources commonly available in ultrafast laboratories,the corresponding error is an order of magnitude higher.We believe that still such accuracy of 1 fs2 and 10 fs3 placesthe technique of non-collinear SSRI among the mostadvanced ones ever developed for dispersion measurement.

Acknowledgements

This work was supported by OTKA under grantT047078 and NKFP 1/00007/2005. Financial support bythe Access to Research Infrastructures activity in the SixthFramework Programme of the EU (Contract RII3-CT-2003-506350, Laserlab Europe) for conducting the researchis gratefully acknowledged.

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Advances and limitations of phase dispersion measurement by spectrally and spatially resolved interferometry

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