theory and design of two-dimensional spectral shearing interferometry for few-cycle pulse...

1Saeabbnmpematowpamhbmstt

most

Birge et al. Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1165

Theory and design of two-dimensional spectralshearing interferometry for few-cycle pulse

measurement

Jonathan R. Birge,1,* Helder M. Crespo,2 and Franz X. Kärtner1

1Department of Electrical Engineering and Computer Science, Research Laboratory of Electronics, MassachusettsInstitute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

2Instituto de Nanociências e Nanotecnologias and Departamento de Física e Astronomia, Faculdade de Ciências,Universidade do Porto, R. do Campo Alegre 687, 4169-007 Porto, Portugal

*Corresponding author: [email protected]

Received September 11, 2009; revised March 29, 2010; accepted March 30, 2010;posted March 31, 2010 (Doc. ID 117014); published May 10, 2010

We provide a detailed discussion of our recently developed pulse characterization technique for few-cyclepulses, called two-dimensional spectral shearing interferometry (2DSI). Based on spectral phase interferom-etry for direct electric-field reconstruction (SPIDER), 2DSI is relatively simple to implement, but requires nointerferometer delay or scan calibration. We show simulations of 2DSI in the presence of noise and find that itretains the favorable noise performance of spectral shearing methods, performing identically to standard SPI-DER for a given measurement time. The optimal choice of experimental parameters is discussed, with resultsapplicable to any spectral shearing method. Experimental considerations when building and operating a 2DSIare provided, with results shown for a 4.9 fs pulse, verifying the accuracy and precision of the 2DSI. © 2010Optical Society of America

OCIS codes: 320.7100, 120.5050.

ippsScg

psrqstifspDrtaht

qcficr

. INTRODUCTIONpectral shearing interferometry, pioneered by Iaconisnd Walmsley in spectral phase interferometry for directlectric-field reconstruction (SPIDER) [1], is uniquemong pulse measurement techniques in that it unam-iguously measures the spectral phase of an optical signaly directly interfering neighboring frequency compo-ents. Furthermore, by encoding group delay (GD) infor-ation on the phase of a spectral domain fringe, SPIDER

roduces a frequency modulated signal and thus enjoysxcellent immunity from noise and nonlinear phaseatching bandwidth effects. This gives spectral shearing

n advantage for measuring very wide bandwidths, rela-ive to the “amplitude modulated” signals characteristicf other methods, with differences in the efficiency not-ithstanding. Consequently, it has become one of therincipal methods used to measure few-cycle pulses [2],longside frequency resolved optical gating [3], and theany variants of both. As is the case for any method,

owever, challenges arise with standard SPIDER asandwidths approach the single-cycle regime [4]. Ourethod seeks to address these issues by trading single-

hot ability for calibration stability, efficient use of spec-rometer resolution, and simplicity of the implementa-ion.

Spectral shearing methods, by their nature, involveeasuring the spectral GD by observing the interference

f two frequency-shifted copies of the pulse being mea-ured. Any linear phase (delay) that occurs between thewo components will be interpreted as a quadratic phase

0740-3224/10/061165-9/$15.00 © 2

n the reconstruction. The resulting error in the measuredulse width scales linearly with the delay error, multi-lied by a factor proportional to the resolution of the mea-urement [4]. This implies that care must be taken in aPIDER measurement to ensure that any linear phase isalibrated out, especially for pulses in the few-cycle re-ime and beyond.

Our approach avoids the possibility for uncalibratedhase by encoding the spectral phase measurement in aeries of spectra of a single output pulse. Our techniqueequires only the non-critical calibration of the shear fre-uency and does not perturb the pulse before upconver-ion. Rather than encoding the spectral GD in a fringe inhe spectral domain, two-dimensional spectral shearingnterferometry (2DSI) encodes phase in a pure cosineringe along a completely independent dimension, bycanning the relative phase of the two spectrally shearedulse copies. As with spatially encoded arrangement SPI-ER (SEA-SPIDER) [5,6], a closely related method, this

educes the resolution demands on the spectrometer tohat required only for proper sampling of the pulse itself,llowing for complex phase spectra to be measured with aigh accuracy over extremely large bandwidths, poten-ially exceeding an octave.

Because of the reduced spectrometer resolution re-uirements, dispersionless beam path, and lack of criticalalibration, we believe that 2DSI is a cost effective and ef-cient method for accurately and reliably measuring few-ycle and even single-cycle pulses. While the method iselatively recent, it is well-tested and has been success-

010 Optical Society of America

fct

bmfipotwt

2IawFove=tbmc

IcFtp

sdtetidcSsttmtcpase

iqut

iimb

wslmdsc6titdsa

3AIqaht4rotfi(aota

BAbTptt

Fp

1166 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 Birge et al.

ully demonstrated on several different lasers [7–9], in-luding the one producing 4.9 fs pulses with 4.3 fsransform-limited bandwidth.

While we have published our method previously in arief form [7], the aim of the present paper is to provideore detail on the theory of operation, explain how 2DSI

ts into the rubric of pulse measurement methods, androvide detailed guidance in the experimental realizationf 2DSI to assist others in its implementation, followinghe model of [10]. An expanded version of this presentork, with further details on the experimental implemen-

ation, can be found within [11].

. MOTIVATION AND BACKGROUNDn any spectral shearing method, the net result is the cre-tion of two spectral copies of the pulse under measure,ith a small frequency shift � disposed between them.or the sake of simplicity, in this paper we will ignore theverall offset in frequency caused by the nonlinear upcon-ersion, and will express everything in terms of the low-st frequency upconverted pulse spectrum A��A��ei��. In the conventional SPIDER method, the

wo output pulses are delayed with respect to one anothery a time �, leading to a dense fringe in the spectral do-ain upon which a finite difference of the phase is en-

oded,

I�� = �A��2 + �A�� − ��2

+ 2�A��A�� − ��cos�� + �� − �� − ��. �1�

f the delay is large enough, the argument to the cosinean be isolated by signal processing of the fringe in theourier domain. The contribution of the delay is then sub-racted out, leaving the finite difference of the spectralhase. More details are provided in [1,12].One of the most reliable approaches to calibrating a

tandard SPIDER fringe, introduced by Dorrer et al. [13],oes not involve explicitly computing a delay, but ratheraking two SPIDER measurements with effectively differ-nt shears and subtracting one from the other to isolatehe SPIDER phase net of any delay- or spectrometer-nduced phase. This was done in [13] by the use of homo-yning two separate upconverted pulses with a “local os-illator” pulse; the principle applies to any embodiment ofPIDER. While this approach assumes that the two mea-urements can be done such that the delay is stable be-ween the measurements, this can be ensured by takinghem simultaneously, as in [14]. Another approach is toeasure a reference phase by interfering the fundamen-

al pulses, as proposed in [15] and refined in [16]. In anyase, a measurement of the delay is inherent, whether ex-licit or not. As the main component of the SPIDER fringend the leading term in any calibration error, it makesense to cast any discussion of sensitivity in terms of theffective delay uncertainty.

Any unaccounted for the interpulse delay � is lumpedn with the phase finite difference, and is mistaken for auadratic phase term, potentially one which results in thenderestimation of the true pulse width. Any error in es-imating the interpulse delay results in an absolute error

n the extracted pulse width. Using a Gaussian analysis,t was shown in [4] that in order to keep the pulse width

easurement error below 5%, the delay uncertainty muste kept to

�� 0.64�

��2 , �2�

here �� is the pulse bandwidth. It was also demon-trated that this result is rather conservative when simu-ating the full width at half-maximum (FWHM) measure-

ent of a pulse from an actual few-cycle laser, whicheviates significantly from a Gaussian spectrum and hasignificant high-order dispersion. For a well-compressed 2ycle pulse, even the delay from 30 nm resulted in over% error. The scaling of Eq. (2) with bandwidth meanshat a single-cycle pulse would require delay stage stabil-ty and calibration well below 7 nm. This assumes thathe shear cannot scale with the bandwidth, but this is in-eed the case for pulses directly from oscillators, as thehear is limited by intracavity phase oscillations whichre unrelated to the laser bandwidth.

. PRINCIPLE OF OPERATION. Pulse Chirping and Splitting

n what follows, we refer to Fig. 1 for a conceptual fre-uency domain block diagram of the conceptual process,nd Fig. 2 for our prototypical experimental layout. Aighly chirped pulse is created by picking off a portion ofhe pulse to be measured and dispersing it (A). Roughly% of the short pulse under test is picked off by a Fresneleflection from an uncoated beam sampler. The remainderf the pulse is split in the interferometer, where a 1 in.hick beam splitter, made of BK7 glass (A), provides suf-cient chirping to measure a pulse in the few-cycle regimesee Subsection 5.B). The arms of the interferometer (B,C)re significantly delayed relative to one another (on therder of picoseconds) such that at any given instant inime there is a difference of � between the instantaneousngular frequencies of the chirped beams.

. Upconversionfter the interferometer, the polarization of each chirpedeam is rotated by a half-wave plate, as we are using theype II upconversion. A simple zeroth-order half-wavelate is sufficient since the rotation only needs to occur atwo frequencies that are separated by the shear, which isypically on the order of 5–10 THz.

eiφ

eiτωω OSA

Input

CPU

eiD2ω2/2

eiτsfgΩ

ig. 1. (Color online) Frequency domain block diagram of 2DSIrocess.

p(cpibp

fims

bmsmoa

ctitmbctnp

CTcpb(tcSfts

wmprlbasno

hrfttatfwi

twsniac

FsaawSls


The beams are made parallel and then focused by aarabolic mirror into a thin Type II ��2� -barium borateBBO) crystal (E), roughly 30 m thick. When the twohirped pulses are then mixed with the original shortulse in the nonlinear crystal, the short pulse effectivelynteracts with only a single frequency of each chirpedeam, provided the chirped pulses are sufficiently dis-ersed.As pointed out by Walmsley in [12], the BBO is rather

ortuitous as an upconversion medium for spectral shear-ng methods in the near-infrared, as its Type II phase

atching curve can be engineered to have octave-panning bandwidths in one polarization, with narrow

τcw

τΩ

τsfg

Half-wavePlate

ParabolicMirror

GlassObjective

Type-IIBBO

Spectrometer

CPU

Fig. 2. (Color online) Experimental schematic of 2DSI setup.

0.6 0.8 1.0 1.2 1.40.7

0.8

0.9

1.0

Pulse Wavelength (μm)

PhaseMismatch

Factor

0.6

0.8

1.0

1.2

1.4

0.6 0.8 1.0 1.2 1.4

Upconversion

Wavelength(um)

ig. 3. (Color online) Top: 2DSI phase matching plot for Type IIum frequency generation for a 50 m thick BBO cut to measuretypical few-cycle Ti:sapphire laser, assuming a beam crossing

ngle of 17°. The lined areas denote the phase matched regions,ith the lines denoting efficiency differences of 10%. Bottom:lices of the phase matching curves for two upconversion wave-

engths separated by 6 THz, showing the efficiency of upconver-ion for the two spectrally sheared components.

andwidths in the other. In Fig. 3 we show a phaseatching curve for a typical 2DSI geometry. Numerical

imulations show that the Type II bandwidth is maxi-ized when the narrowband beams (and therefore the

utput) are collinear. 2DSI thus improves upon the favor-ble phase matching of standard SPIDER.If the chirped pulses are collinear, the two frequency

omponents will have the same propagation vector andhus we can regard the output as a single pulse, render-ng the system immune to interpulse delay errors. Whilehis neglects the small difference in the transverse photonomentum between the two chirped pulses, there should

e no possibility for the delay to occur so long as thehirped beams are parallel. A given wavelength is imagedo the same pixel regardless of the output angle, and thuso phase difference should arise according to Fermat’srinciple.

. Controlo observe the phase difference between the two pulseomponents at a given wavelength in a robust way, thehase of one of the quasi-continuous-wave (quasi-CW)eams is scanned over a few optical cycles prior to mixingblock B in Fig. 1), which results in a phase � being addedo the upconversion. The net effect is to add a variableonstant phase shift to one of the two output components.canning this phase allows one to directly observe inter-

erence between the two spectral components as a func-ion of wavelength, providing a direct observation of thepectral GD of the original pulse.

Mirror C is controlled manually by a translation stageith a relatively long travel (at least millimeters). Thisirror is used to control the delay between the chirped

ulse copies, which determines the shear. The other mir-or (B) is controlled by a short-throw piezoelectric trans-ation stage. Which mirror performs each function is ar-itrary, of course, and both the large manual adjustmentnd the small piezo oscillation can be performed on theame mirror. A third delay stage (D), while not strictlyecessary, is a convenience for adjusting the temporalverlap between the short pulse and the chirped pulses.

Simulations show that the nonlinearity of the fringeas no effect on the accuracy, although it does affect theeconstruction in the presence of noise by broadening theringe in the frequency domain. Empirically, we foundhat even 50% nonlinearity, defined as the maximum rela-ive deviation from the nominal scan rate, results in only

halving of the signal-to-noise ratio (SNR). Nonlineari-ies below 10% were not found to have an appreciable ef-ect on the noise performance. The linearity of the scanill thus not be an issue for most commercial stages, and

t can thus be handled via an open loop controller.To take a single measurement, a computer controlling

he piezoelectric stage moves mirror (B) over severalavelengths, recording a dozen or so spectra during the

can. While technically only four spectra are actuallyeeded for the reconstruction, taking more spectra results

n a more intuitively understandable trace and one thatllows for a simplified reconstruction algorithm, as dis-ussed in Section 4.

DTa(

waptSantrwptsllltbeas

ECd

cp�sattcnampp

4ATigs

tttscmspmr

adfs

Fwfsaestpt

FawTiur


. Form of Interferogramhe spectrum of the upconverted signal is then recordeds a function of the phase �, yielding a two-dimensional2-D) intensity function that is given by

I��,�� = �A�� + A�� − ��ei��2

= 2�A��A�� − ��cos�� + �� − �� − ��g��−�/2��+O��2�

� + D . C . ,

�3�

here, as before, A�� is the upconverted pulse spectrumnd �� is its spectral phase. The under-bracketed ex-ression can be viewed as a finite difference approxima-ion of the GD scaled by the shear. This term is what allPIDER variants measure, although in 2DSI it is avail-ble directly without any filtering, as discussed in theext section, since there are no other terms dependent onhe optical frequency. A simple 2-D raster plot of I�� ,��eveals the up-shifted pulse spectrum along the �-axis,ith fringes along the �-axis that are locally shifted inroportion to the GD at the given frequency (as illus-rated in Fig. 4). The user can thus immediately ascertainalient properties of the complex spectrum simply byooking at the raw data: the cosine fringe at each wave-ength is vertically shifted in proportion to its actual de-ay in time, with the fringe amplitude roughly propor-ional to the power spectral density (neglectingandwidth effects). The ability to use the raw spectrom-ter data when optimizing a laser yields information notvailable from the reconstructed pulse data, such as mea-urement noise and laser stability.

. Relation to the SPIDERomparison of Eqs. (1) and (3) shows that the fringes pro-uced by 2DSI and conventional SPIDER are mathemati-

λ (nm)

GD(fs)

GD(fs)

80

60

40

20

0

τcw( fs)

5.4

4.0

2.7

1.3

0

τcw( fs)

5.4

4.0

2.7

1.3

0

80

60

40

20

0

(a)

(b)

650 750 850 950 1050550

ig. 4. (Color online) 2DSI interferograms from 5 fs laser, plotith respect to the wavelength of the fundamental pulse (i.e., be-

ore upconversion): (a) extracted spectral GD overlaid to demon-trate the interpretation of fringe offset, and (b) the same pulsefter dispersion by approximately 1 mm of fused silica. The pres-nce of extra dispersion is evident in the interferogram. The colorcale is arbitrary. The y-axis is shown calibrated in terms of bothhe actual mirror translation, as well as the corresponding im-lied spectral GD computed by dividing the phase of the fringe byhe shear frequency � (taken from [7]).

ally identical, save for the fact that the fringe in 2DSI isroduced by a phase occurring in a separate domain (thedomain), whereas in SPIDER the fringe oscillates in the

pectral domain. In either case, the fringe can be vieweds creating sidebands in the respective Fourier domain ofhe fringe. In SPIDER, the fact that the sidebands are inhe optical frequency domain results in a significant in-rease in the required spectrometer resolution over thateeded to simply resolve the fundamental pulse spectrums illustrated in Fig. 5. In 2DSI, the extra dimensioneans that the spectrometer resolution required is sim-

ly that required by the pulse’s complex spectrum to beroperly sampled.

. RECONSTRUCTION ALGORITHM. Fast Fourier Transform Harmonic Inversionhe inversion process for 2DSI is rather simple. The only

nformation we need to extract from the 2-D interfero-ram is the phase of the fringes along the direction of thecan delay (i.e., the vertical direction in Fig. 4).

In SPIDER, the Takeda algorithm [17] is used to isolatehe spectral phase finite difference term. In 2DSI, the facthat no information is encoded in each fringe other thanhe spectral GD means that we can use simpler recon-truction algorithms, however. In essence, the 2DSI re-onstruction is an issue of a very simple single-term har-onic inversion [18] (i.e., characterization of an unknown

inusoid from a time series) rather than the decoding of ahase modulated known carrier typical of other shearingethods. In this sense, 2DSI is not really a form of Fou-

ier transform spectral interferometry.A simple, computationally efficient, and direct way to

ccess the fringe phase is to compute a series of one-imensional fast Fourier transforms (FFTs) along theringe axis. According to Eq. (3), the fringes generate well-eparated sidebands in the Fourier domain and their

-200 -100 0 100 200 150 250 350f (THz) t (a.u.)

1D FFT

-200 0 200t

kφ-200 0 200f (THz)

φ

2D FFT

ig. 5. Illustration of the interference fringes in 2DSI (top left)nd SPIDER (bottom left), for a hypothetical Gaussian pulseith second- and third-order dispersions and 1% satellite pulses.he Fourier transforms of both fringes are shown on the right,

llustrating that the delay scanning in 2DSI serves to move thesual SPIDER sidebands into a new dimension, thus minimizingequired spectrometer resolution.

pnscoep

wcemtrtslsd

BHlttpae

dtfaptrwsnlssvptst

CIcfooifpm

sootFs

5ATptaw

WrtAmwsostmiFa

kisvttsssm2

thsiww

BTtqpiu


hase angle represents the spectral GD. Windowing is notecessary so long as a sufficient number of fringes are ob-erved. Because we are actually measuring the GD, anyonstant offset is meaningless, which mitigates the effectsf windowing. The worst-case relative error in the recov-red pulse spectral phase �rec can be shown (see the Ap-endix) to be approximately limited by

��r��

n + ��, �4�

here n is the index of the discrete Fourier transformomponent (the harmonic number) and �� is the differ-nce in the frequency between the fringe and the har-onic, normalized relative to the fundamental, such that

he worst-case scenario is ��=0.5. As can be seen, the er-or will be relatively small if the fringe frequency is closeo the FFT harmonic, or if n is large. As a rule of thumb,o long as three or more fringes are visible, or the scanength is within 10% of the fringe period, the worst pos-ible error will be no more than a few percent and win-owing will not be necessary.

. Finite Difference Inversionaving determined the phase of the fringe at each wave-

ength, dividing by the shear yields the finite difference ofhe spectral phase as shown in Eq. (3). The final step,hen, is to compute the spectral phase of the measuredulse from these finite differences. This step is common toll spectral shearing methods and has been discussedlsewhere in the literature [19].

In the case of a single shear measurement, the finiteifference operation is simply a linear discrete time filter,he effects of which can be completely inverted (asiderom noise issues). To do so, the data must first be anti-liased by filtering out all �-domain oscillations with aeriod shorter than �. Then, sinc interpolation of the fil-ered data can be used to compute the fringe phase � on aegular grid of points spaced by �. Simple concatenationill then yield the spectral phase of the pulse. It may

eem that this approach is not optimal in terms of theoise performance, as the data will be concatenated at a

ower resolution than that provided by the spectrometer,eemingly throwing out data points that are beingkipped over. However, the antialiasing filter step pro-ides for the optimal combining of all data points. By sup-ressing all the content beyond the shear’s Nyquist limit,he final filtered phase data will be made internally con-istent such that the concatenation operation will yieldhe same result regardless of the starting point.

. Multiple Shearingn [4], it was very briefly proposed that the calibration ac-uracy of spectral shearing measurements could benefitrom the use of multiple shears that are integer multiplesf the finest shear used. Recently, Austin et al. [20] devel-ped a rigorous general theory of multiple spectral shear-ng, proving that multiple shears improves the noise per-ormance for a given total measurement time androviding an algorithm for optimally combining the infor-ation from all shears.

By its nature, 2DSI is well-suited to the use of multiplehears. The shear can be adjusted without affecting anyther aspect of the measurement, simply by translatingne of the interferometer mirrors a distance large enougho shift the local frequency of one of the chirped pulses.or further details on how to optimally analyze multiplehear measurements, we refer the reader to [20].

. DESIGN CONSIDERATIONS. Shear Frequencyhe shear frequency ultimately becomes the “samplingeriod” of the phase in the spectral domain, and most ofhe other parameters can be derived from it. By the Sh-nnon sampling theorem, this determines the temporalindow, over which we can reliably measure

T 2�

�. �5�

e will refer to this time period as the shear Nyquist du-ation. It is not adequate to simply pick a shear sufficiento include a region of interest, such as the main pulse.ny satellite pulses or pedestal structure will still beeasured, but will be aliased into our chosen temporalindow, resulting in errors. Thus, the shear must be cho-

en so as to allow resolution of all spectral features. More-ver, satellite pulses which may seem negligible can haveignificant effects when aliased onto the main pulse dueo the fact that aliases add coherently. For example, nu-erical simulations show that a satellite pulse with an

ntensity of only 2% of the main pulse can change theWHM of the main pulse by up to 5.5% should it beliased on top of it.The required temporal measurement window T is not

nown a priori, by definition, in a pulse that we are seek-ng to characterize, nor is it possible to conclude that theampling was sufficient by taking a measurement anderifying that the reconstruction is well-contained withhe time window. Once a signal is sampled, one cannotell the difference between correctly or insufficientlyampled signals. The only reliable way to assure that theampling rate is sufficient is to take measurements at aeries of decreasing shears and verify that the measure-ents converge to the required precision. The fact that

DSI can use multiple shears is an advantage here.In practice, for sub-2-cycle pulses produced by oscilla-

ors with dispersion compensating mirrors (DCMs), weave found that a shear of around 4–5 THz is required toufficiently resolve the satellite pulses and pedestal. Thiss to be expected given the penetration depths achievableith DCMs, and this will be independent of the band-idth achieved.

. Chirping Dispersionhe signal of the final measurement is inversely propor-

ional to the amount of dispersion used to create theuasi-CW beams, and thus the chirp of the ancillaryulses should be the minimal amount required to resultn an accurate measurement. Over-chirping results in annnecessarily weak signal, and under-chirping results in

attssr

amwsawsppqtt

App5pb

CTazwaIroel

6TiFatgu

2stigTtowf

0nmbma

mdinFocuwcb

tnebsivbms

Fsosorttso


complicated blurring of the measurement in the spec-ral domain as the upconversion occurs with a range of in-eracting wavelengths. Having determined the requiredpectral resolution with the shear, the chirping should beelected such that the associated blur is smaller than thisesolution.

In most cases, we only care about accurately measuringmain pulse. This is consistent with our previous state-ent that all satellite structure must be resolved; whilee may not be interested in the accurate measurement of

atellite pulses, we still must ensure that they do notlias. If a well-separated satellite pulse is upconvertedith different frequencies than the main pulse, it may re-

ult in a local error in the reconstruction of that satelliteulse, but it will not affect the reconstruction of the mainulse. Thus, we simply require that the frequency of theuasi-CW beam change by no more than the shear overhe temporal extent of the main pulse. Taking this widtho be Tp, this gives

D2 �Tp

�. �6�

safe configuration capable of measuring few-cycleulses can be found by assuming a shear of 5 THz and aulse width of no more than 25 fs, yielding a dispersion of000 fs2. This is a small enough amount that it can berovided by the cube beam splitter used in the chirpedeam interferometer (as in Fig. 2).

. Delay Scan Lengthhe scan must be long enough such that the sidebandsre well-separated from the central peak centered at theero frequency. This distance will vary depending onhether or not we are performing windowing, and howccurately the scan can be matched to the fringe period.n general, at least three fringes must be visible, as de-ived in Section 4. This implies scan lengths on the orderf 1 or 2 m, at most, allowing very stable short-throw pi-zoelectric stages to be used, and enabling high scan ratesimited only by signal levels.

. SENSITIVITY TO NOISEhe sidebands in 2DSI are spectrally compact; the fringe

s simply an impulsive line in the Fourier domain (seeig. 5). This spectral compactness means that the bulk ofny noise will not interfere with the signal, maximizinghe SNR. A significant amount of noise immunity is thusained by isolating the sideband, assuming the noise isniformly distributed over all frequencies.An important question to answer is whether or not

DSI’s need to scan over a delay renders it more or lessensitive to noise than standard SPIDER. 2DSI essen-ially just takes the SPIDER sidebands and moves themnto another dimension, so it stands to reason that for aiven total measurement time, they would be comparable.o illustrate the noise immunity of 2DSI (and in fact spec-ral shearing in general) we simulated the measurementf a pulse with a rectangular spectrum of 300 THz band-idth, whose transform-limited pulse width is roughly 3

s. Additive Gaussian noise was simulated, with a SNR of

.5. The SPIDER method was simulated using the sameoise source, with an integration time equal to the totaleasurement time of the 2DSI trace. The SPIDER cali-

ration method used was that of [16], where a calibrationeasurement with exactly zero shear was assumed to be

vailable that exactly preserved the delay.The results are shown in Fig. 6. For a given measure-ent time, 2DSI has half the variance of SPIDER. The

ifference between the two is entirely due to uncertaintyntroduced into the SPIDER calibration by the detectoroise. Furthermore, it is apparent from the middle plot inig. 6 that the SPIDER error is predominantly composedf a second-order term, validating our consideration of thealibration as largely manifesting as an effective delayncertainty. Both methods recover the spectral phaseell, despite the low SNR. Note, also, that the 2DSI re-

onstruction involved no knowledge of the fringe spacingy the algorithm.If the SPIDER calibration noise issue is ignored, the

wo methods perform identically, regardless of the type ofoise used (i.e., additive noise or shot noise). This is to bexpected, given that 2DSI simply takes the SPIDER side-ands and moves them into another dimension. From aignal processing point of view, a SPIDER measurements equivalent to a 2DSI measurement where the upcon-ersion phase is constant and the sidebands are createdy a temporal delay. It is thus to be expected that the twoethods would perform similarly outside calibration is-

ues.

-5

0

5

10

φ(rad)

2DSISPIDERIdeal

150-150 -100 -50 0 50 1000

0.5

1

f (THz)

φStd.Dev.(rad)

f (THz)−200 −100 0 100 200

0

1

2

3

4

5

φcw(πrad)

ig. 6. (Color online) Simulated 2DSI spectrogram (top) mea-ured with 64 phase steps for a sinc pulse with second- and third-rder dispersions and a satellite, in the presence of additive andhot noise such that the resulting SNR per sample is 0.5. A line-ut showing the fringe at a single frequency is inset. A sampleeconstruction, including comparison with SPIDER is shown inhe middle frame. The bottom frame shows the standard devia-ion of the phase measurement for both SPIDER and 2DSI,howing that the lack of delay calibration in 2DSI yields a factorf 2 improvement in noise performance.

7AT(mFbsccguwmcnalnl

aOmtcmtisp7p4

tBssor

mmoatsets

BTsrsptl

afdm

btmafioiwwectTtp[o

Fs


. EXPERIMENTAL DEMONSTRATION. Precision Testo gauge the relative precision of the method, a few-cycle5 fs FWHM) pulse from a prismless Ti:sapphire laser waseasured using a 2DSI setup similar to that shown inig. 1, using a shear of 18 THz. The pulse was measuredoth before and after dispersion by a nominal 1 mm fusedilica plate [Fig. 7(b)]. It is apparent from the spectral GDurves that the pulse is initially slightly negativelyhirped, and the positive dispersion introduced by thelass plate is evident. The phase of the plate was foundsing published Sellmeier coefficients for fused silica,ith the precise overall thickness (about 1.18 mm) deter-ined by a least squares fit. While this introduces a cir-

ular nature to the measurement, and means that we can-ot rule out a constant error factor, this was unavoidables 2DSI itself is the most precise measure of the phase de-ay we have, available in our laboratory; the plate thick-ess could not be determined to within attoseconds of de-

ay in any other way.The sharp roll-off in the GD below 650 nm is genuine

nd caused by phase distortion from the output coupler.scillations in the spectral GD, caused by the chirpedirrors, are also clearly visible. Despite these perturba-

ions in the individual spectra, the oscillations completelyancel in the difference between the phases of the twoeasurements, which matches well to that predicted by

he known Sellmeier equations for fused silica as shownn Fig. 7(c). In fact, in terms of the phase delay, the 2DSIystem measured the glass dispersion to within 30 as ofhase delay over a bandwidth from 600 to 1000 nm [Fig.(d)]. This precision was achieved despite the absolutehase delay of each measurement ranging over more than0,000 as.These results are only intended to demonstrate that

he 2DSI apparatus is capable of precise measurements.ecause the glass thickness was calibrated using theame apparatus, and because the final phase is found byubtracting two measurements from each other, this dem-nstration does not rule out the possibility of constant er-ors occurring that are consistent between measure-

0

0.2

0.4

0.6

0.8

GD(fs)

GD

700600 800 900 10

020406080

Pulse w/glassPulse alone

(a)

(b)

Wavelength (nm)

ig. 7. (Color online) (a) Spectrum of the 5 fs laser used in the telide as measured by 2DSI and as predicted by known glass disp

ents, nor a constant multiplicative factor. Thiseasurement thus only verifies that no significant spuri-

us high-order dispersion terms are present and providesn estimate of the noise-limited precision. For example, ifhere were an unknown linear phase included in the mea-urement somehow, such a systematic error would not bevident from these dispersion measurements. We attempto verify low-order dispersion accuracy in the followingubsection.

. Accuracy Testo qualitatively demonstrate the absolute accuracy of theystem and rule out the existence of systematic errors, weecently performed a measurement on an octave-panning sub-2-cycle pulse [21,22] from another Ti:sap-hire oscillator, and compared the 2DSI measurement tohat obtained with a standard interferometric autocorre-ations (IACs, Fig. 8).

In Fig. 8(d), we show the reconstructed pulse envelopend phase for the pulse, measured to have a FWHM of 4.9s. As shown in Fig. 8(b), the measured IAC and that pre-icted by the 2DSI measurement show fairly close agree-ent.We attribute most of the difference between the two to

and-limiting effects on the IAC, which is not well-suitedo measuring a 4.9 fs pulse. Thus, we do not present thiseasurement as further evidence of the precision of 2DSI,

s an IAC is not a particularly reliable measurement ofne detail, but rather a demonstration of at least low-rder accuracy to complement the precision measurementn Subsection 7.A. However, the fact that the overall pulseidths predicted by both are consistent does suggest thate are correct in assuming that no significant hidden lin-ar phase can occur in 2DSI, and that the method is ac-urate at least to low-order. This validates the assertionhat 2DSI does not require a separate calibration step.his has also been corroborated by detailed simulations of

he laser cavity in question, which also predict a 4.9 fsulse with a large sub-pulse 9 fs away from the peak (see23] for details). This result, combined with the high-rder verification provided in Subsection 7.A indicates

600 700 800 900 1000

(d)

(c)

0246802

Sellmeier2DSI

0000000

Wavelength (nm)

xtracted GD both with and without glass slide; (c) phase of glass; (d) net phase delay error in glass dispersion measurement.

00

11

-3-2-1

123

Phase(radians)

Phasedelayerror(attosec)

st; (b) eersion

tdc

8TmtrsMssattesqsc

sttittbpvc

9Tibmlttltt

AFWccp�wqvtwngow

Pct

Fp


hat the less than 30 as phase measurement precision in-icated in Fig. 7 may be representative of the actual ac-uracy.

. FUTURE WORKhe geometry of 2DSI is unique in that the pulse to beeasured never encounters a dispersive element, and yet

he arrangement is still collinear. Were the spectrometereplaced with an imaging spectrometer, a spatially re-olved 2DSI measurement could be taken along one axis.oreover, if a full 2-D imaging spectrometer were used,

uch as that available with imaging Fourier transformpectroscopy, one could make a full 2DSI measurement at2-D array of points along both transverse axes. By spa-

ially filtering the chirped beams so that they were spa-ially coherent, they would provide a constant phase ref-rence across the beam profile, enabling a fullpatiotemporal reconstruction up to a trivial constant anduadratic spatial phase (focusing). This would allow for aelf-referenced three-dimensional measurement of few-ycle pulses, work that is currently underway.

By using a nanostructured stepped mirror, a single-hot version of 2DSI might be implemented. The differen-ial upconversion phase would then be encoded as a func-ion of space, and the fringe could be read using anmaging spectrometer. This arrangement would result inhe mixing of the spatial profile with the fringe, essen-ially creating a collinear variation of SEA-SPIDER [5],ut with a spatially varying zeroth-order phase as op-osed to the linear phase. This could provide similar ad-antages to SEA-SPIDER, but with potentially simpleralibration and alignment.

-300

-200

-100

0

100

200

300

(a)

-0

-0

0

0

0

600 700 800 900 1000 1100 12000.0

0.2

0.4

0.6

0.8

1.0

Wavelength (nm)

PowerDensity(a.u.)

(c)

600 700 800 900 1000 1100 1200

ig. 8. (Color online) (a) Raw 2DSI data; (b) comparison of IAChase (dashed curve); (d) reconstructed pulse (solid curve), simu

. CONCLUSIONwo-dimensional spectral shearing interferometry (2DSI)

nvolves a relatively simple optical setup with a little cali-ration required, and yet is capable of spectral phaseeasurements to within tens of attoseconds of phase de-

ay over octaves of bandwidth. The lack of dispersion ofhe pulse to be measured, the absence of delay betweenhe sheared pulses, and the relaxed spectrometer reso-ution requirements make 2DSI extremely well-suited forhe measurement of wide-bandwidth pulses, includinghose with potentially complicated spectral phase.

PPENDIX: DERIVATION OF WORST-CASEFT HARMONIC INVERSION ERRORe derive the relative error in recovering the phase of a

osine fringe by taking the phase of the dominant FFTomponent, as discussed in Section 4. The FFT will be ap-roximated by a continuous integral over the space0,2��, as it is the windowing, not the discretization, thate are concerned with. We consider a fringe with a fre-uency of n+��, where n is an integer and �� is the de-iation from that integer frequency. In other words, n ishe approximate number of complete fringes containedithin the scan and corresponds to the index of the domi-ant FFT component we are to consider. Without loss ofenerality we will consider a cosine fringe with a phaseffset �. The complex value fn of the nth FFT componentill be approximately given by the continuous integral,

fn �0

2�

dx cos��n + ��x + ��e−inx. �A1�

erforming this integral, computing the argument of theomplex value, and simplifying the expression give the ex-racted fringe phase,

-20 -10 0 10 20012345678

IAC2DSI

Delay (fs)

(b)

4.9 fs FWHM

-40 -20 0 20 40

Retrie

vedIntensity(a.u.)

0.0

0.2

0.4

0.6

0.8

1.0

Time (fs)

(d)

at predicted from the 2DSI measurement; (c) extracted spectralulse (dotted curve), and temporal phase (dotted curve).

NormalizedIntensity

-2

-1

0

1

2

.4

.2

.0

.2

.4

Phase(2prad.)

and thlated p

TeGtwtfis

Stt

TpitF

ATvHfi0DiSTaf

R

1

1

1

1

1

1

1

1

1

1

2

2

2

2


�ext = arctanIfn

Rfn�A2�

=arctan�n tan�� + ��

n + �� . �A3�

he above expression gives us the absolute phase recov-red from the FFT, which is proportional to the spectralD of the pulse we are measuring. Since constant offsets

o the GD are irrelevant, we are really only concernedith the slope of the relation between �rec and �. This is

he factor by which fringe phase changes will be magni-ed. It is thus the relative error in our extracted disper-ion at a given wavelength,

�r�� = 1 − ��ext

��

�

�A4�

=1 −n�n + ��sec�� + ��2

�n + ��2 + n2 tan�� + ��2 . �A5�

ince we cannot control the offset of �, we must considerhe worst-case error over all �. We thus solve for the sta-ionary point of Eq. (A5). This yields the equation

2n��n + ��2n + ��sec�� + ��2tan�� + ��

��n + ��2 + n2 tan�� + ��2�2 = 0.

�A6�

here are multiple roots of Eq. (A6), but the one that ap-lies for �� near zero is �=−��. Plugging this solutionn Eq. (A5) and simplifying gives us a final expression forhe worst-case relative measurement error due to theFT harmonic inversion,

��r��

n + ��. �A7�

CKNOWLEDGMENTShe authors acknowledge funding from the Defense Ad-anced Research Projects Agency (DARPA) through theyperspectral Source Program via the U.S. Air Force Of-ce of Scientific Research (AFOSR) grants FA9550-06-1-468 and FA9550-07-1-0014, and would also like to thankr. Richard Ell who designed and built the 5 fs laser used

n the precision testing. H. M. Crespo acknowledges grantFRH/BPD/27127/2006 from Fundação para a Ciência eecnologia (FCT-MCTES), Portugal. The authors wouldlso like to thank the anonymous reviewers for their help-ul suggestions and feedback.

EFERENCES1. C. Iaconis and I. Walmsley, “Self-referencing spectral inter-

ferometry for measuring ultrashort pulses,” IEEE J. Quan-tum Electron. 35, 501–509 (1999).

2. L. Gallmann, D. Sutter, N. Matuschek, G. Steinmeyer, andU. Keller, “Techniques for the characterization of sub-10-fsoptical pulses: a comparison,” Appl. Phys. B 70, S67–S75(2000).

3. D. Kane and R. Trebino, “Characterization of arbitrary fem-tosecond pulses using frequency-resolved optical gating,”IEEE J. Quantum Electron. 29, 571–579 (1993).

4. J. R. Birge and F. X. Kärtner, “Analysis and mitigation ofsystematic errors in spectral shearing interferometry ofpulses approaching the single-cycle limit [invited],” J. Opt.Soc. Am. B 25, A111–A119 (2008).

5. E. Kosik, A. Radunksy, I. Walmsley, and C. Dorrer, “Inter-ferometric technique for measuring broadband ultrashortpulses at the sampling limit,” Opt. Lett. 30, 326–328 (2005).

6. A. S. Wyatt, I. A. Walmsley, G. Stibenz, and G. Steinmeyer,“Sub-10 fs pulse characterization using spatially encodedarrangement for spectral phase interferometry for directelectric field reconstruction,” Opt. Lett. 31, 1914–1916(2006).

7. J. R. Birge, R. Ell, and F. X. Kärtner, “Two-dimensionalspectral shearing interferometry for few-cycle pulse charac-terization,” Opt. Lett. 31, 2063–2065 (2006).

8. H. M. Crespo, J. R. Birge, M. Y. Sander, E. L. Falçao-Filho,A. Benedick, and F. X. Kärtner, “Phase stabilization of sub-two-cycle pulses from prismless octave-spanning Ti:sap-phire lasers,” J. Opt. Soc. Am. B 25, B147–B154 (2008).

9. K. F. Lee, K. J. Kubarych, A. Bonvalet, and M. Joffre,“Characterization of mid-infrared femtosecond pulses [in-vited],” J. Opt. Soc. Am. B 25, A54–A62 (2008).

0. P. Baum and E. Riedle, “Design and calibration of zero-additional-phase spider,” J. Opt. Soc. Am. B 22, 1875–1883(2005).

1. J. R. Birge, “Methods for engineering few-cycle mode-lockedlasers,” Ph.D. dissertation (MIT, 2009).

2. F. X. Kärtner, ed., Few-Cycle Laser Pulse Generation and itsApplications (Springer, 2004).

3. C. Dorrer, P. Londero, and I. A. Walmsley, “Homodyne de-tection in spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 26, 1510–1512 (2001).

4. S.-P. Gorza, P. Wasylczyk, and I. A. Walmsley, “Spectralshearing interferometry with spatially chirped replicas formeasuring ultrashort pulses,” Opt. Express 15, 15168–15174 (2007).

5. C. Iaconis and I. A. Walmsley, “Spectral phase interferom-etry for direct electric-field reconstruction of ultrashort op-tical pulses,” Opt. Lett. 23, 792–794 (1998).

6. C. Dorrer, “Influence of the calibration of the detector onspectral interferometry,” J. Opt. Soc. Am. B 16, 1160–1168(1999).

7. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transformmethod of fringe-pattern analysis for computer-based to-pography and interferometry,” J. Opt. Soc. Am. 72, 156–160(1982).

8. V. A. Mandelshtam and H. S. Taylor, “Harmonic inversionof time signals and its applications,” J. Chem. Phys. 107,6756–6769 (1997).

9. C. Dorrer and I. Walmsley, “Accuracy criterion for ul-trashort pulse characterization techniques: application tospectral phase interferometry for direct electric field recon-struction,” J. Opt. Soc. Am. B 19, 1019–1029 (2002).

0. D. R. Austin, T. Witting, and I. A. Walmsley, “High preci-sion self-referenced phase retrieval of complex pulses withmultiple-shearing spectral interferometry,” J. Opt. Soc. Am.B 26, 1818–1830 (2009).

1. H. M. Crespo, J. R. Birge, E. L. Falçao-Filho, M. Y. Sander,A. Benedick, and F. X. Kärtner, “Non-intrusive phase-stabilization of sub-two-cycle pulses from a prismlessoctave-spanning Ti:sapphire laser,” Opt. Lett. 33, 833–835(2008).

2. J. R. Birge, H. M. Crespo, M. Sander, and F. X. Kärtner,“Non-intrusive sub-two-cycle carrier-envelope stabilizedpulses using engineered chirped mirrors,” in Conference onLasers and Electro-Optics/Quantum Electronics and LaserScience Conference and Photonic Applications SystemsTechnologies, OSA Technical Digest (CD) (Optical Society ofAmerica, 2008), paper CTuC3.

3. M. Y. Sander, J. R. Birge, A. Benedick, H. M. Crespo, and F.X. Kärtner, “Dynamics of dispersion managed octave-spanning titanium:sapphire lasers,” J. Opt. Soc. Am. B 26,743–749 (2009).

theory and design of two-dimensional spectral shearing interferometry for few-cycle pulse...

Documents