Xu et al. Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. B 2363
Analysis of the measurement ofpolarization-shaped ultrashort laser pulses bytomographic ultrafast retrieval of transverse
light E fields
Lina Xu,1,* Philip Schlup,2 Omid Masihzadeh,2 Randy A. Bartels,2 and Rick Trebino1
1School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA2Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, Colorado 80523, USA
. INTRODUCTIONolarization-varying complex ultrashort laser pulses wererst used in quantum control [1–5] and are now playingoles in many areas of research. Such “polarization-haped” pulses have been considered for the generationnd measurement of high-harmonic pulses [6] and for theontrol of two-dimensional lattice vibrations in crystals7]. Polarization-shaped pulses have been generated byarious means, mostly based on Fourier-domain pulsehaping of individual polarization components [3,8–11].n the other hand, only a few methods exist to measure
hem. Time-resolved ellipsometry (TRE) [12,13] was onef the first technologies used to characterize the polariza-ion evolution of ultrashort pulses. It involves measuringll four Stokes parameters of the pulse, but it is labor in-ensive. A simpler technique is polarization-labeled inter-erence versus wavelength for only a glint (POLLIWOG)14], which uses spectral interferometry [15] toharacterize—successively or simultaneously—the tworthogonal polarization components relative to a well-haracterized reference pulse. Other approaches involveeasuring the spectrum and the cross correlation of the
olarization components or the cross-phase modulation,oth combined with iterative numerical algorithms16,17]. POLLIWOG is the most commonly used tech-ique, and it works well; but it requires careful phase sta-ilization and measurement of the relative phase betweenhe two polarizations, and it requires a separate self-eferenced technique for measuring the required refer-nce pulse.
Recently, we introduced a self-referenced technique for
easuring polarization-shaped pulses, which we calledomographic ultrafast retrieval of transverse light E fieldsTURTLE) (Fig. 1) [18]. It does not require a separateell-characterized reference pulse and is based on mea-
uring the electric field versus time at three different lin-ar polarizations, obtained by making such measure-ents after a polarizer for three different polarizer
ngles. Two of the measurements characterize the electriceld for mutually orthogonal field components, and thehird—measured at an arbitrary angle in between (typi-ally 45°)—is used to determine the phase relationshipetween these components, which yields the full vectorolarization evolution of the pulse. Any establishedethod that determines the complex field E��� of a single
inear polarization can, in principle, be used in TURTLE.n addition the pulse energy or the average power muste measured for each polarization. No modifications tohe standard pulse-measurement apparatus are needed.
Here we study TURTLE technique using second-armonic-generation frequency-resolved optical gatingSHG FROG) by performing detailed simulations. Weimulate TURTLE’s performance using SHG FROG forimple and complex polarization-shaped pulses and findhat it works very well, even for very complex pulses. Ourimulations show that an error minimization algorithmsing the SHG FROG trace performs robustly—even inhe presence of added noise. We attribute this robust be-avior to the well-known overdetermination of the pulseomplex electric field afforded by a FROG trace.
We chose FROG because it is the most mature self-eferenced pulse-measurement technique available and
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2364 J. Opt. Soc. Am. B/Vol. 26, No. 12 /December 2009 Xu et al.
as been shown to measure accurately the full intensitynd the phase of an arbitrary complex ultrashort pulse19], subject only to trivial ambiguities. Specifically wehose SHG FROG due to its high sensitivity and preva-ence. Also, it has minimal ambiguities and then onlyrivial ones. Trivial ambiguities of standard SHG FROGnclude the direction of time (DOT) of the pulse; that is,HG FROG cannot distinguish between a reconstructedeld E��� and its complex conjugate, E����. For pulseshat are well-separated in either optical frequency orime, an additional ambiguity arises in their relativehases �� [20,21]. For example, relative phases of both� and ��+� yield the same SHG FROG trace for doubleulses in time. However, it has been shown that theserivial ambiguities can easily be removed using simpleechniques that involve minimal additional effort. Addingknown spectral dispersion (chirp) and performing a sec-
nd FROG measurement removes the DOT ambiguity.ven better, replacing the FROG beam splitter with antalon generates identical trains of overlapping (and de-aying) pulses in both arms of the device; such waveformso not experience such ambiguities, and the original pulsean be retrieved from them easily and unambiguously22], except for the usual absolute-phase and arrival-timembiguities common to all self-referenced pulse-easurement techniques. Thus FROG and its variations
ield the best-posed (least ambiguous) set of self-eferenced pulse-measurement techniques currentlyvailable.While these remaining two ambiguities are generally
onsidered trivial, and they are for most purposes, theyre not so trivial for the measurement of polarization-haped pulses. Nonmeasurement of the absolute phasend time preclude the determination of key quantities ofhe full vector field. Specifically, what distinguishes mono-hromatic 45° linear polarization from circular polariza-ion is the relative phase of the horizontal and vertical po-arizations, which is the difference between the two pulsebsolute phases, which are not measured in self-eferenced pulse-measurement techniques in general.nd what distinguishes 45° linear polarization from twoell-separated pulses of orthogonal polarization is, of
ourse, their relative arrival times. Thus these two other-ise trivial ambiguities are not so trivial for polarization-
haped pulses and thus become the principal unknowns
ig. 1. (Color online) Schematic visualization of the TURTLErinciple. The time-evolving electric field vector E�t� (not shown)s characterized by measuring linear projections Ex���, Ey���,nd E���� in the frequency domain using an existing ultrashort-ulse characterization technique. The algorithm establishes theelative amplitude r, delay �, and phase � between the projec-ions to retrieve the full vector field.
hat TURTLE aims to determine. It is the third FROGrace that accomplishes this. The only case we have foundn which TURTLE, as described above, does not work ishe trivial case in which the two polarization componentsre identical, and the polarization thus does not actuallyvolve, but this ambiguity can easily be removed with onedditional measurement.
. THEORYn the frequency domain, we write the polarization-haped vector field as
E��� = Ex���x + rEy���e−i���+��y, �1�
here the optical angular frequency ���−�0, and Ex���nd Ey��� represent the complex frequency-domain polar-zation components along the Cartesian axes, with theeam propagating along z. We use this formulation be-ause ultrafast polarization shapers typically operate byndependently shaping orthogonal polarization compo-ents. To obtain the full polarization information, weeed to know not only the fields Ex��� and Ey��� but alsohe relative amplitude r, the relative delay �, and theelative phase � between the components. No existingelf-referenced single-polarization pulse-measurementechnique is able to provide absolute time or phase infor-ation, but it is easy to measure the relative amplitudesing a simple energy detector as given below. The mea-urement technique that we call TURTLE determineshese relative quantities from an additional SHG FROGrace of the polarization component, projected here at 45°etween x and y. In the following, we label this projectionngle �.The easiest ambiguity to resolve is the relative ampli-
ude ratio r. We can determine it experimentally by mea-uring the average power P for each linear projectioneasurement. The power can be written as
P �−
�r�E����2d� = r�2�−
�E����2d�, �2�
here r� is a scaling factor that relates the reconstructedrbitrarily normalized field E��� to the physically presenteld. So, if we normalize the retrieved fields accordingo ��E����2d�=1, then we can find r in Eq. (1) from r�Py /Px. As we show below, the power measurements areritical for the trivial case of pure elliptical polarization ashown below, where the reconstructed fields are identicalxcept for the amplitude factors. In the following simula-ions, we set r=1 without loss of generality.
TURTLE aims to determine the relative delay � andhe relative phase � in Eq. (1) using an additional polar-zation projection at angle �. We denote this projectedeld as E����, and it can be written as
E���� = cos � Ex��� + r sin � Ey���e−i���+��. �3�
choice of �=45° will usually give the best results be-ause Ex��� and Ey��� contribute equally to the projectedeld E ���. We choose this angle in the simulations below.
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Xu et al. Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. B 2365
The expression for the SHG FROG trace of a single lin-arly polarized pulse temporal electric field E�t� is [23]
IFROG��,T� = ��−
E�t�E�t − T�e−i�tdt�2
. �4�
he FROG trace is collected by recording the second-armonic spectra generated as the delay T between twoeplicas of E�t� is varied. From the SHG FROG trace, anstablished generalized-projections algorithm reliablynds the full intensity and phase of an arbitrary ul-rashort laser pulse [23]. Thus, Ex��� and Ey��� caneadily be determined experimentally without the needor additional reference pulses.
Having found Ex��� and Ey���, we use a minimizationlgorithm to find the relative delay � and the relativehase � for which Ex��� and Ey��� yield the projectedeld E����. The algorithm can find these parameters us-
ng the additional information contained in the-projected SHG FROG trace. We sample the FROGraces onto regularly spaced optical frequency �i and de-ay Tj axes in an N�N grid. We calculate the projectedROG trace from E���� using Eq. (4), and TURTLE in-olves minimizing the difference between the calculated
�calc�i , j� and the measured I�
meas�i , j� traces. We use theriterion of the RMS error defined as [24]
e =
�i,j=1
N
I�meas�i,j� − I�
calc�i,j��2
�i,j=1
N
I�meas�i,j��2
, �5�
hich describes the difference between the two FROGraces I�
meas and I�calc divided by the nonzero area. The er-
or e is then minimized with respect to the iterated valuesf � and �, with the optimal values corresponding to thosealues of � and � that minimize e. In the simulations, wealculate the error surface e�� ,�� about the optimal val-es.We must also ensure that TURTLE uses the correct
elative DOT between the components Ex,y���, since arong DOT in one projection and the correct DOT in thether no longer corresponds to the vector field being mea-ured. This ambiguity is easily resolved in two ways: wean determine the overall correct DOT for both fields,˜
x��� and Ey���, by placing an etalon or adding a knownmount of material chirp in one of more of the SHGROG measurements. This is the standard method for re-olving the time ambiguity in SHG FROG. In TURTLE,nowing the DOT of one component, Ex��� or Ey���, isufficient to determine that of the other and hence that ofhe entire polarization-shaped pulse. In other words, ifnly the shape of the vector field—but not its absoluteOT—is needed, we can calculate the error e�� ,�� sepa-
ately for both combinations of relative DOTs: Ex���Ey��� and Ex���+ Ey
����. The TURTLE trace for non-rivial vector pulse shapes is sensitive to the relativeOT, so the minimum error in e will be lower for the cor-
ect relative DOT.
To simulate the practical environment, we added 0.5%oisson noise to each SHG FROG trace. In this approach,
he measured trace with such an additive noise [23] atach pixel is
IFROG��� ��i,�j� = IFROG��i,�j� + �ij�/�, �6�
here �ij is a pseudorandom number drawn from a Pois-on distribution of mean � and � is the noise fraction. Weerified that the maximum trace value at the edges of therray is less than 0.5% of the peak value of the FROGrace. Suppressing the background noise is important inHG FROG measurements. Any nonzero average back-round (due to noise) in a FROG trace implies spuriousonzero intensity at large times and with high frequency
n the pulse, that is, spurious pulse wings with high fre-uency noise. So, in practice, before running the pulse re-rieval program, background subtraction is always per-ormed. Several methods are available, and they includeourier low-pass filtering, corner suppression, and mean-ackground subtraction. In our simulations, we chose toerform only simple mean-background subtraction (al-hough performing the others as well would likely haveurther improved the performance beyond what we ob-erve). The mean of the noise was obtained by averaginghe values in the 5�5 pixel squares in the four corners ofhe FROG trace (i.e., far from the center of the trace,here the most important pulse information is located).fter subtracting this constant background from alloints in the trace, we set all the resulting negativeoints to zero (as is usually done).We found the values of the relative phase and delay us-
ng the 45° polarized FROG trace and the fields deter-ined from the x- and y-polarized traces, using aATLAB simplex minimization routine for multidimen-
ional unconstrained optimization [25]. This routine isdeal for the TURTLE technique because, while simplexoutines are known to be slow, TURTLE involves only awo-parameter minimization, and so it converges rela-ively quickly (typically 1 min or so for a 256�256 tracen a laptop). Also, simplex routines are less likely thanerivative-based routines to fall into possible localinima.
. SIMULATIONSelow we give several examples of using the TURTLE
echnique to find the polarization state of an ultrashortulse. While the majority of TURTLE measurements arenticipated to be used to characterize complex pulses withomplex polarization evolution, we begin with someimple cases since the extremely simple case of nonevolv-ng polarization with identical x and y components re-ealed the only ambiguity we encountered in our study.he ambiguity disappears in the presence of even slightolarization complexity and so is unlikely to present prob-ems for the use of TURTLE.
The first example pulse comprises two identical x and yomponents consisting of transform-limited Gaussianulses with a full width at half-maximum (FWHM) dura-ion of 30 fs so that Ex�t�=exp�−2 ln 2t /30 fs�2 and
�t�=exp�−2 ln 2�t+� /2� /30 fs�2 . The relative delay and
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2366 J. Opt. Soc. Am. B/Vol. 26, No. 12 /December 2009 Xu et al.
he relative phase between the polarization componentsere �=170 fs and �=� /3. The resulting SHG FROG
races for the projected fields Ex���, Ey���, and E���� arehown in Figs. 2(a)–2(c), respectively. In Figs. 2(d) and(e), we show the pulse intensity and the temporal phaseeconstructed from the SHG FROG traces for Ex�t� andy�t�. The peak intensity is set at t=0. The zero order andrst order spectral phases, which correspond to the rela-ive phase and delay in the evolution of the polarization,re not reflected in these two retrieved fields. The relativehase and delay are obtained from the SHG FROG tracef E����. Since the pulses are symmetrical in time andrequency, the reconstructed field projections closelyatch the generating fields, in particular with regard to
he absolute time and phase of each component.
ig. 2. (Color online) TURTLE retrieval steps for a vector fieldonsisting of two transform-limited Gaussian components sepa-ated by �=170 fs. (a),(b),(c) Simulations of measured SHGROG traces for Ex���, Ey���, and E����, respectively, with �45°. (d),(e) Pulse fields Ex�t� ,Ey�t� obtained using the standardeconstruction algorithm (dots), compared with the generatingelds (solid curve). (f) The error surface; the two minima indicatehe � rad phase ambiguity arising from the SHG FROG trace ofwo pulses well separated in time. (g) Sketch of the full vectoreld E�t�.
Figure 2(f) shows the error surface e, about the targetalues of � and �, calculated using the reconstructed
˜x,y��� and the �-projected FROG trace I� from Eq. (5).n this error surface, the parameter minimization re-
rieved a relative delay of 169.65 fs and a relative phase of.3344� or 1.3316� rad, depending on the initial guess.his is the expected � rad phase ambiguity that arises
rom SHG FROG traces for pulses separated in time; asiscussed above, an additional measurement by addingdditional chirp on either one of the pulses or both pulseso make Ex�t� and Ey�t� overlap in time can be used toliminate this ambiguity. A three-dimensional representa-ion of the vector field E�t� is sketched in Fig. 2(g). Exami-ation of the �-projected SHG FROG trace of Fig. 2(c) re-eals spectral intensity modulations at a FROG delay of=0. As discussed in Section 2, these fringes correspond
o spectral interferometry fringes, and their spacing is in-ersely proportional to �, and the peak locations relativeo �=0 are given by �.
Combining the identical fields Ex��� and Ey��� fromhe previous example with �=0 and �= ±� /2 yields a cir-ularly polarized pulse shown in Fig. 3. In this case, wean relate the field components by Ey���= Ex���e−i� sohat the �-projected SHG FROG trace for �=45° will beiven by
hus, the projections x and y yield identical SHG FROGraces [Fig. 3(a)] with the � projection being qualitativelyhe same but scaled by an intensity-weighting factor of�1+e−i��2�2= 2�1+cos ���2. Since this factor does not de-end on the sign of �, which determines the handedness ofhe vector field, TURTLE cannot distinguish between leftnd right circularly polarized fields. This can be seen inhe error surface shown in Fig. 3(b), which indicates twoymmetric minima at ±�. The TURTLE fitting algorithmetrieved a relative delay of �=0.0174 fs and a relativehase of �=0.5014� or −0.5015� rad, depending on thenitial guess. Further, in this case of indistinguishableHG FROG traces, the normalization of Eq. (5) and that
nherent in the standard FROG reconstruction algorithmeans that the ellipticity, determined by the relative am-
litude r of the x and y components, cannot be directlyetermined. An independent power or pulse energy mea-urement is thus necessary to determine the ellipticity.
The handedness ambiguity can be resolved by introduc-ng different chirps to the two components as shown inig. 4. Here, we added ��2����= ±200 fs2/rad quadraticpectral phase to each of Ex,y���. The SHG FROG tracesor these components are still indistinguishable [Fig. 4(a)]ut, due to the knowledge of the signs of the added chirps,e can correctly reconstruct the fields as shown in Figs.(b) and 4(c). The �-projected SHG FROG trace, shown inig. 4(d), is now distinct and its shape uniquely deter-ines the correct phase since the error surface [Fig. 4(e)]
xhibits only a single minimum. The retrieved relativeelay and phase were �=−1.0242 fs and �=0.5183� rad,dentifying the pulse as right circularly polarized. Alter-atively, we could characterize the pulses transmitted
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Xu et al. Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. B 2367
hrough a circular polarizer, analogously to the determi-ation of one of the four Stokes vectors needed to fully de-ne the vector field. The ambiguity in the sign of � arisesnly in the case of transform-limited temporally symmet-ic pulses and is not expected to occur for polarization-haped pulses.
We show in Fig. 5 results from a randomly generatedore complex pulse. We simulate an arbitrary vector field�t� by creating random complex field components for
˜x��� and Ey��� and by applying Gaussian amplitude fil-
ers in both time and frequency domains. For the pulsehown in Fig. 5, the temporal and frequency filter FWHMidths were �t=180 fs and ��=0.3 rad/fs. The resulting
ime–bandwidth products (TBPs) were 17.8 for the x- and3.0 for the y-polarized components. We chose relative de-ay and phase values of �=60 fs and �=� /3 rad, respec-ively. The SHG FROG traces of Figs. 5(a)–5(c) show aapid structure characteristic of nontrivial pulses. Ashown in Figs. 5(d) and 5(e), we verified that the FROGeconstructions were in good agreement with the generat-ng fields, and from the error surface [Fig. 5(f)] the mini-
ig. 3. (Color online) (a) Simulation of measured SHG FROGraces; in this case all three projections yield the same trace. (b)he error surface; the two minima indicate an ambiguity in thehirality of the vector field E�t�, which is shown in (c).
ization algorithm retrieved relative delay and phasealues of 62.74 fs and 0.3349� rad, respectively.
Another case with a more complex pulse is shown inig. 6. The method to generate these two complex pulses
s the same as the previous case. The temporal and fre-uency filter FWHM widths in this case were �t1800 fs and ��=0.3 rad/fs. The resulting TBPs were69.7 for the x- and 180.4 for the y-polarized components.ue to the limitation of the computer memory, these are
he most complicated pulses generated. We chose relativeelay and phase values of �=500 fs and �=� /3 rad, re-pectively. The SHG FROG traces of Figs. 6(a)–6(c) show
rapid structure characteristic of highly nontrivialulses. As shown in Figs. 6(d) and 6(e), we verified thathe FROG reconstructions were in good agreement withhe generating fields, and from the error surface [Fig. 6(f)]he minimization algorithm retrieved relative delay andhase values of 504.96 fs and 0.35� rad, respectively.Table 1 shows some cases with different pulse com-
lexities. All these x and y components are generatedrom random pulses filtered by a clean Gaussian pulseith FWHM widths of �t=900 fs in the time domain and�=0.3 rad/fs in the frequency domain. We chose rela-
ive delay and phase values of �=500 fs and �0.33� rad, respectively, for all cases. Without any noisedded, the exactly correct relative delay and relative
ig. 4. (Color online) Establishing the chirality of a circularlyolarized field by adding a known chirp; cf. Fig. 3. (a) Simulationf measured SHG FROG trace for Ex���; an identical trace is re-orded for Ey���. (b),(c) Pulse fields Ex�t� ,Ey�t� obtained using thetandard reconstruction algorithm (dots), compared with theenerating fields (solid curve). (d) SHG FROG trace for the-projected component. (e) The error surface that shows only aingle minimum at �=+� /2. (f) Sketch of the full vector field E�t�.
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2368 J. Opt. Soc. Am. B/Vol. 26, No. 12 /December 2009 Xu et al.
hase can be reconstructed in each case. With 0.5% Pois-on noised added, the retrieved values are varied by atost 1.3% in the relative delay and 7.5% in the relative
hase.
. CONCLUSIONSe have analyzed the performance of TURTLE using
HG FROG for the self-referenced measurement of theomplete vector field intensity and phase of polarization-haped ultrashort laser pulses. Our simulations showhat TURTLE works very well, robustly yielding the com-lete vector polarization even of very complex pulses andn the presence of noise. Indeed, the SHG FROG TURTLE
inimization also reliably distinguishes the relativeOTs of the polarizations. We found no nontrivial ambi-
ig. 5. (Color online) TURTLE retrieval steps for a randomlyenerated vector field. (a),(b),(c) Simulations of measured SHGROG traces for Ex���, Ey���, and E����, respectively. (d),(e)ulse fields Ex�t� ,Ey�t� obtained using the standard reconstruc-ion algorithm (dots), compared with the generating fields (solidurve). (f) The error surface and (g) sketch of the full vector field�t�.
uities. We expect this success to extend to TURTLEased on other FROG nonlinearities. We conclude thatHG-FROG-based TURTLE is a reliable technique for
ig. 6. (Color online) TURTLE retrieval steps for a randomlyenerated very complex vector field. (a),(b),(c) Simulations ofeasured SHG FROG traces for Ex���, Ey���, and E����, respec-
ively. (d),(e) Pulse fields Ex�t� ,Ey�t� obtained using the standardeconstruction algorithm (dots), compared with the generatingelds (solid curve). (f) The error surface and (g) sketch of the fullector field E�t�.
Table 1. Different Pulses with TheirReconstructed Relative Delays And Relative
Phases
TBPx component)
TBP(y component)
Reconstructed�
Reconstructed�
84 121.8 501.23 0.329�
103.3 95 493.65 0.359�
57.7 89.5 501.59 0.325�
70.5 37.5 497.52 0.329�
100.4 79.2 499.54 0.333�
83.2 110.7 504 0.318�
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Xu et al. Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. B 2369
elf-referenced measurements of polarization states ofven very complex polarization-shaped pulses.
CKNOWLEDGMENTShe authors at the Georgia Institute of Technology ac-nowledge the Georgia Research Alliance for financialupport. The portion of this work performed at Coloradotate University (CSU) was partially supported by theonfort Family Foundation and National Science Foun-
ation (NSF) ECCS-0348068.
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