## Abstract

A new spectral shearing interferometry technique for ultra-short pulse characterization is demonstrated. The method makes use of a spectral shear that varies across the test pulse beam profile to generate a two-dimensional interferogram that allows simultaneous reference phase measurement and pulse-field reconstruction from a single data set. The method uses a new configuration for upconversion of a single (non-spatially chirped) test pulse with two spatially chirped ancillary pulses in a medium with a highly asymmetric phase matching function. This technique is particularly suited for spectral regions where second harmonic is much easier detectable that the fundamental wavelength, such as telecom band around 1.5 *μ*m, since all the data are available from the measurement at the upconverted wavelength. The high degree of redundancy in the two-dimensional interferogram provides a built-in check of the consistency of the reconstruction.

©2007 Optical Society of America

High-energy ultrashort optical pulses are an essential tool in a variety of research and applied fields such as physics, chemistry, biology and medicine. Many new applications in these areas are enabled by the ability to measure and control the shape of short optical pulses with precision and accuracy. Currently, there are several reliable common techniques for ultrashort pulse characterization. Among the most widespread are spectral phase interferometry for direct electric field reconstruction (SPIDER) [1] and frequency resolved optical gating (FROG) [2]. SPIDER is an application of shearing interferometry in the optical frequency domain, that enables measurement of the spectral phase of ultrashort optical pulses [1]. It is one of a broad class of interferometric techniques [3, 4, 5], which are increasingly popular (see e.g. [6, 7, 8]) due to the robustness and simplicity of direct mapping of phase to intensity. In SPIDER, the pulse spectral phase is fully encoded in a one-dimensional (1D) set of data: the interference pattern between two spectrally sheared and temporally delayed replicas of the input pulse. Recently two methods of spectral shearing interferometry making use of a two-dimensional (2D) interference pattern to measure the spectral phase of ultrashort pulses have been reported [9, 10, 11]. These techniques make use of the second dimension to encode the spectral phase, which means that the spectrometer resolution required in these implementations is reduced compared with conventional SPIDER, and they operate at the sampling limit.

In this work we present and demonstrate a novel method of spectral shearing interferometry, referred to as chirped arrangement for SPIDER (CAR-SPIDER), that encodes both the spectral phase and amplitude of ultrashort optical pulses in a self referencing and self calibrating 2D interferogram, in the space-frequency domain. The key feature of this interferogram is that it is generated by interference between two upconverted replicas of the input pulse, for which the spectral shear between the pair varies across the spatial coordinate of the upconverted beams, from +Ω on one side to -Ω on the other (see Fig. 1 and Fig. 4). Each spatial slice of the measured trace is a standard 1D SPIDER interferogram. At a certain position near the center of the beam the spectral shear equals zero and the extracted spectrum at this coordinate is just the spectrum of the pulse. CAR-SPIDER retains all of the useful features of other SPIDER schemes, such as single-shot acquisition, direct phase reconstruction and a lack of moving parts. In addition, all data necessary for complete reconstruction of the test pulse field is measured at the upconverted frequency, and has a high degree of redundancy, allowing stringent test of the consistency of the reconstruction. Finally, the method is not restricted to the encoding of the spectral phase in the frequency domain since the spatially encoded arrangement [9, 10], useful for characterizing ultrabroadband pulses, can naturally be applied by tilting instead of delaying the two spatially chirped replicas.

In conventional SPIDER, the frequency-sheared replicas of the unknown pulse are produced by sum-frequency (SF) generation of a pair of input pulses with a highly chirped pulse. The spectral shear therefore depends on the relative delay between the pulses in the pair. On the other hand it is well known that in SPIDER the determination of the delay between the frequency sheared replicas is an essential calibration step. Usually, this calibration is performed by recording a reference interferogram between the pulse pair with no spectral shear, at the fundamental or the SF wavelength [12, 13], which at the same times eliminates any distortion introduced by the apparatus itself. Note that in HOT-SPIDER or in the 2DSI method, there is no need for a reference, but these methods are intrinsically incapable of single shot operation [11, 14].

In CAR-SPIDER the input beam is mixed in a nonlinear medium with an ancillary quasi-monochromatic *spatial* slice, to generate a replica of the input pulse at a shifted frequency (see Fig. 1). This is in contrast to conventional SPIDER, in which the input pulse is upconverted with a quasi-monochromatic *temporal* slice of an ancillary pulse to generate the frequency shift of the replica. A variation of the frequency across the ancillary beam results therefore in a spatially chirped replica of the unknown pulse [see Figs. 4(a–b)]. The 2D spectral intensity distribution of the spatially chirped replica obviously gives no information about the spectral phase of the input pulse. However, the phase information *is* completely and redundantly encoded in the fringe pattern resulting from the interferences between the chirped replica and its delayed (or tilted) mirror image. For an input pulse with electric field *E*(*x,t*) = *A*(*x*)*∫*
^{+∞}
_{-∞}[*Ĩ*(*ω*)]^{1/2}exp[*i$\tilde{\varphi}$*(*ω*)) - *iωt*]d*ω*, which upconverts locally (at position *x*) with frequency *ω*
_{0} - *αx*, the 2D-interferogram *S̃*(*x, ω*) centered around 2*ω*
_{0} is represented by

$$+2\sqrt{\tilde{I}\left(\omega -\mathrm{\alpha x}\right)\tilde{I}\left(\omega +\mathrm{\alpha x}\right)}\mid A\left(x\right)\mid \mid A\left(-x\right)\mid $$

$$\times \mathrm{cos}\left[\tilde{\varphi}\left(\omega +\mathrm{\alpha x}\right)-\tilde{\varphi}\left(\omega -\mathrm{\alpha x}\right)+\mathrm{\rho x}+\mathrm{\omega \tau}+\phi \left(x\right)\right].$$

Note that spatial inversion of the beam at a mirror corresponds in this case to the transformation *x* → -*x*. In Eq.1 ρ is the *x* component of the wavevector difference between the two upconverted beams, τ is the delay between them and φ(*x*) denotes the spatial phase of the beams. This latter term can be neglected since at a given position the spatial phase does not vary along the frequency axis. The left picture in Fig. 2 shows a typical CAR-SPIDER 2D interferogram for a linearly chirped input pulse (with τ ≠ 0 and ρ = 0). In this 2D pattern, the intensity profile along the frequency axis at *x* = *x*
_{0} is a conventional SPIDER interferogram, the spectral shear being Ω(*x*
_{0}) = 2*αx*
_{0}. At the position where the two spectra exactly overlap (*x* = 0) there is no spectral shear between the two replicas and the spectral phase difference, extracted from the interferogram, is the reference phase. On the other hand the extracted phase difference at any other position can be used to reconstruct the spectral phase of the unknown pulse, providing that the sampling criterion Ω < 2*π*/Δ*T*, where Δ*T* is the time support of the input pulse, is fulfilled [12]. The consistency of the pulse phase reconstruction can therefore be checked from a single interferogram since, in essence, many different shears are contained in a CAR-SPIDER interferogram. This is fundamentally different from the consistency check in conventional SPIDER. There, the redundancy of data for checking consistency comes from the fact that the spectrum is oversampled, resulting in many spectral samples per shear interval [15]. In CAR-SPIDER, since this is also available, there is a double redundancy that should improve the accuracy and consistency checks dramatically.

The CAR-SPIDER phase-retrieval algorithm is similar to the SEA-SPIDER algorithm [9, 10]. The interferogram is first 2D Fourier transformed with respect to both space and time (see Fig. 2), then one of the two sideband peaks is selected by numerical filtering and Fourier transformed back to the frequency-space domain, yielding *S̃*
^{F}(*x,ω*). At each position *x*, the phase difference * $\tilde{\varphi}$*(

*ω*+

*αx*) -

*$\tilde{\varphi}$*(ω -

*αx*) is obtained by subtraction of the reference phase (arg[

*S*̃

^{F}(0,

*ω*)]) from the argument of

*S̃*

^{F}(

*x,ω*). The spectral phase

*$\tilde{\varphi}$*(

*ω*) is reconstructed by concatenating this phase difference. A nonzero tilt

*ρ*between the two upconverted beams displaces the position of the sideband peaks but does not change the reconstruction algorithm, since at a given position the extra term

*ρx*is a constant. Note that the spectral phase

*$\tilde{\varphi}$*(

*ω*) can also be reconstructed from the difference arg[

*S̃*

^{F}(

*x,ω*)] - arg[

*S̃*

^{F}(

*x′,ω*)] along the frequency axis at two positions

*x*and

*x*′ corresponding to the same spectral shear in absolute value but with opposite sign. Finally, the spectral amplitude is extracted from the spectral intensity distribution at the position where the spectral shear is zero.

In CAR-SPIDER, the *α* coefficient (i.e. the slope of the spatial variation of the spectral shear) and the zero spectral shear position have to be determined so as to be able to extract the pulse phase and amplitude from the 2D interferogram. The *α* coefficient depends only on the geometry of the apparatus and therefore needs to be calibrated only once. This is straightforwardly done by recording the individual 2D spectra (see Fig. 4). The position at which the spectral shear equals to zero can also be determined in this way. Contrary to the *α* coefficient, this position varies with the input beam pointing. However, once *α* is known, the zero shear position can be retrieved from the CAR-SPIDER interferogram.

We have shown recently that in a properly chosen long nonlinear crystal, the phase matching greatly simplifies the generation of a sheared replica of the test pulse. [16, 17, 18]. This is achieved by tailoring the phase matching function (PMF) for sum-frequency generation in a type-II interaction so that the ordinary wave has a large acceptance bandwidth, whereas the extraordinary wave has a narrow bandwidth. This asymmetry is the result of a group velocity match between the ordinary polarized fundamental and the upconverted pulses together with a group velocity mismatch between the two fundamental pulses. Thus, the highly asymmetric PMF itself selects the ancilla frequency with which the test pulse spectrum is mixed, resulting in its replication at the upconverted frequency. This particular ancilla frequency is determined by the angle between the propagation direction of the *e*-wave and the crystal optic axis. A spatially chirped replica can therefore easily be generated by sending a focused beam into the crystal. In this case each plane wave component of the focused beam is efficiently mixed with a different ancilla frequency, depending on its direction of propagation. After the crystal, each direction of propagation, corresponding to a unique exit angle is mapped onto a specific position by a lens, resulting in a spatially-chirped SF replica. The CAR-SPIDER arrangement thus dramatically simplifies the calibration compared to LX-SPIDER [16, 18].

The experimental setup is shown in Fig. 3. The test pulse passes first through a zero order half-waveplate and a 10 mm thick quartz plate. The half-waveplate rotates the polarization of the vertically polarized input pulse by 45°, while the quartz plate splits the input pulse into two orthogonally polarized pulses. The birefringence in the quartz plate predelays (by 317 fs at 830 nm) the vertically polarized pulse (i.e. the *e*-wave in the nonlinear crystal) with respect to the horizontally polarized pulse (the *o*-wave). This predelay ensures that in the nonlinear crystal, whose optic axis is in the drawing plane, the fastest *e*-polarized pulse completely walks through the slowest *o*-pulse, enabling the replication of the *o*-pulse at the upconverted frequency [18]. A 50 mm focal lens then focuses the beam into a type II potassium dihydrogen phosphate (KDP) crystal cut for maximum collinear upconversion at 830 nm (8 mm thick, *θ* = 68°, optic axis in the drawing plane). The nonlinear crystal is placed before the focus to minimize the effect of spatial walk-off. At the output of the crystal each exit angle is mapped onto position by a 200mm lens. The resulting spatially chirped beam is directed into a modified Michelson interferometer arrangement that spatially laterally inverts the beam in one of the two arms, leading to an inversion of the spatial chirp direction (see arrows in Fig. 3), and sets the delay required for recording the SPIDER interferogram. Finally, the two beams are directed into the entrance slit of a 2D-imaging spectrometer with a 1800 lines/mm grating. Note that to avoid space-time coupling due to diffraction through the system, the spectrometer slit has to be in the rear focal plane of the second lens such that the image of the spatially dispersed upconverted spectrum is formed on the slit.

A typical example of the spatial distribution of the spectra of the two replicas is shown in Fig. 4(a–b). Figure 4(c) displays the frequency shear between the two replicas versus position obtained from the separately recorded 2D spectra, showing a linear variation of the spectral shear from -20 mrad/fs to +20 mrad/fs across the SF beams. The zero shear position and the a coefficient can be extracted from this curve. As a test of the CAR-SPIDER technique we have characterized 70 fs pulses delivered by a Kerr-lens mode-locked Ti:sapphire laser with a pulse repetition rate of 80 MHz. The average power at the input of our setup was attenuated down to 100 mW. The retrieved spectral phase is plotted in Fig. 5(a) for five different spectral shear values from 4 mrad/fs to 8 mrad/fs, after taking into account the dispersion experienced by the pulse in the quartz plate and the nonlinear crystal. The extracted spectral intensity is also shown. As can be seen all the reconstructions agree well, indicating that the correct reference phase profile has been subtracted. Finally, the apparatus accuracy has been tested by adding a known phase to the test pulse. The spectral phase difference acquired by the test pulse by passing through 10 cm of BK7 glass is plotted in Fig. 5 (b) and shows good agreement with the theoretical phase difference calculated from the Sellmeier equation. The root-mean-square phase error is 1.8% [15]. Figure 4(d) displays the CAR-SPIDER interferogram of the pulse after passing through the glass block. In our experiment, the parameters have been set to *ρ*= 0 and *τ*=1.8 ps.

In conclusion CAR-SPIDER is a novel spectral shearing interferometric method to characterize ultrashort pulses that encodes both the pulse amplitude and phase in a single 2D interferogram. This 2D interferogram is generated by interference between two spatially chirped replicas of the unknown pulse. The CAR-SPIDER technique has true single shot capabilities in the sense that the reference phase, essential in SPIDER, and the spectral amplitude are encoded in the 2D interferogram together with the spectral phase. The spatially chirped replicas can be generated in a simple way by sending a focused laser beam into a properly chosen long nonlinear crystal cut for type II sum-frequency, but other schemes are also possible.

This work was supported by the Engineering and Physical Sciences Research Council, grant No. EP/D503248/1. S.-P. Gorza acknowledge the support of the Belgian Science Policy office under Grant No. IAP-VI10 and the Fonds de la Recherche Scientifique (FRS-FNRS, Belgium).

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