Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy

Lepetit et al. Vol. 12, No. 12/December 1995/J. Opt. Soc. Am. B 2467 Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy L. Lepetit, G. Chériaux, and M. Joffre Laboratoire d 0 Optique Appliquée, Ecole Nationale Supérieure de Techniques Avancées, Ecole Polytechnique, Unité de Recherche Associée au Centre National de la Recherche Scientifique 1406, F-91120 Palaiseau, France Received March 29, 1995 Although nonlinear methods can provide only the amplitude and the phase of an isolated ultrashort pulse, linear techniques can yield such measurements with a much better sensitivity and reliability when a reference pulse is available. We demonstrate two such methods, dual-quadrature spectral interferometry and Fouriertransform spectral interferometry. These techniques are simple to implement, very sensitive, and provide a complete measurement of the complex electric field, E v, as a continuous function of frequency. 1995 Optical Society of America 1. INTRODUCTION Many concepts and experimental techniques utilized in nonlinear optics have been transposed from the field of nuclear magnetic resonance (NMR). However, although NMR experiments use a complete measurement of magnetization, practical experiments in optics provide only partial measurements of the electric field, such as the total energy or the power spectrum. As a consequence, many experiments involving nonlinear spectroscopy do not provide as much information as a true transposition from NMR would. For example, Weiss et al. recently demonstrated the advantages of a time-resolved photon echo compared with those of a time-integrated photon echo. 1 More generally, a sensitive method that would provide a complete measurement of the electric field, E t, associated with an optical pulse is clearly needed. In conjunction with phase locking 2 it would allow the transposition of all NMR experimental techniques to the optical domain. In this paper, we analyze two such methods, which provide a measurement of the Fourier transform of the electric field, E v, in both amplitude and phase. It must be pointed out that, unlike that in previous work, 3 8 our interest does not lie in the characterization of ultrashort pulses delivered by a femtosecond laser, but in applications to femtosecond spectroscopy. This means that we can assume the availability of an initial reference pulse that is well characterized in amplitude and phase and of sufficient energy. In contrast, the pulse we want to characterize, for example, a coherent transient emission, may be of extremely small energy and complicated spectral phase. Therefore sensitivity and spectral resolution are crucial issues that are addressed in this paper. We also stress that there must be a coherent phase relation between the pulse we want to measure and the initial reference pulse. In this respect, incoherent electric fields, such as those associated with fluorescence emission, are outside the scope of this paper. We address only the case in which the unknown electric field results from a coherent emission triggered by the reference pulse itself or a coherence offspring of it, so that there is a perfect synchronization between the two pulses. In Section 2 we discuss what can be measured by linear-optics techniques. We also point out the distinct advantages of linear techniques in terms of sensitivity, that is, what makes their use highly desirable whenever possible. In Section 3, we compare time-domain with frequency-domain interferometry techniques available up to now. In Sections 4 and 5, we demonstrate two new frequency-domain interferometry techniques, dualquadrature spectral interferometry (DQSI) and Fouriertransform spectral interferometry (FTSI). Finally, we discuss the different spectral interferometry techniques and point out their respective advantages. 2. LINEAR PHASE-MEASUREMENT TECHNIQUES It should now be well known 9 that nonlinear techniques are required for measuring the spectral phase of an isolated pulse, 3 8 e.g., for the purpose of characterizing a short-pulse laser. Indeed, consider the most general time-stationary linear experiment in which only timeintegrated detection is used, as shown in Fig. 1. Basically, the incident beam is sent into a linear device, which is characterized by its response function, R t in the time domain or R v in the frequency domain. The transmitted pulses are then sent into a time-integrating detector. Let us call E t the electric field of the incident light beam and E v je v jexp if v its Fourier transform. E 0 t is the electric field transmitted through the linear device: E 0 t R t 2 t 0 E t 0 dt 0 R v E v exp 2ivt dv. 2p (1) From the assumption of time-integrated detection, it follows that the detected signal is simply the pulse energy, which is also the frequency-integrated power spectrum (Bessel Parseval theorem): 0740-3224/95/122467-08$06.00 1995 Optical Society of America

2468 J. Opt. Soc. Am. B/Vol. 12, No. 12/December 1995 Lepetit et al. Fig. 1. General linear experimental setup: the incident pulse is sent into a linear device and then into a time-integrating detector. The linear device is characterized by its response function, R t in the time domain or R v in the frequency domain. S je 0 t j 2 dt je 0 v j 2 dv 2p jr v j 2 je v j 2 dv. (2) 2p Therefore the detected signal is a function of only the power spectrum je v j 2 and is always independent of the spectral phase f v. In short, a linear experiment will always provide the same result whether the incident beam is made of femtosecond pulses or is a random electric field with the same power spectrum delivered by an incoherent source. This is a well-known consequence of the linear superposition principle. The Michelson interferometer is an example of a linear device that can be used in the setup shown in Fig. 1. In this case, the electric field transmitted through the device is a sequence of two pulses. The signal recorded as a function of the time delay t between the two arms is well known to be the first-order autocorrelation of the incident electric field whose Fourier transform is the power spectrum. This result is the basis of Fourier-transform spectroscopy, a standard technique routinely used in the infrared spectral domain. The Michelson interferometer is therefore an optimal linear device, in the sense that it provides the maximum information that can be obtained in a linear device, i.e., the power spectrum. Other examples include the diffraction-grating spectrometer and the scanning Fabry Perot étalon, both also providing a measurement of the power spectrum. Any linear device will be a variation of the above. A parameter that affects the response function R t is usually varied, such as the time delay in a Michelson interferometer or the grating angle in a diffraction spectrometer. Sometimes the experiment is performed in a single-shot geometry in which a linear diode array or a CCD detector is used. In all cases, the following result always holds: no linear device coupled to a time-integrating detector can provide information on the spectral phase associated with a light pulse. Yet, as mentioned in the introduction, this does not mean that linear techniques should be ruled out for femtosecond spectroscopy. On the contrary, they should be of great importance. Consider an interferometer in which we recombine, through a beam splitter, two light beams associated with two different electric fields, E and E 0. According to the above discussion, a time-integrating detector will be sensitive to only the power spectrum of the sum of the two electric fields, which includes a crossed term E 0 v E v. Therefore the signal will be sensitive to the phase difference, f v 2f 0 v. If E and E 0 were derived from an incident pulse through linear devices, the phase of the incident pulse would appear as an additive factor in both f and f 0, so that it would cancel out in the phase difference and would not affect the experimental result. Again we find that it is impossible to characterize the phase of an isolated pulse in a linear-optics experiment. However, if E 0 is a reference pulse whose phase f 0 v is already known, then f v can be recovered from the measured phase difference simply by the addition of the phase of the reference pulse. The unknown electric field E v can therefore be fully characterized through linear interferometry, as long as a known reference pulse E 0 is available. The only requirement on the reference pulse is that its spectrum must overlap all the frequency components of the unknown pulse, because stationary interferences occur only between identical-frequency components. Of course, this only translates the problem to the measurement of the amplitude and the phase of a reference pulse. However, such a task can be easily achieved by the use of one of the many off-the-shelf nonlinear techniques now available for the measurement of ultrashort pulses. 3 8 Furthermore, in most if not all cases, a nearly Fourier-limited pulse can also be used as the reference. Such a pulse is easily characterized by the fact that the measured second-order autocorrelation function is in good agreement with that computed with the experimental power spectrum, assuming a flat spectral phase. Finally, we point out the unique advantage of linear interferometry in terms of sensitivity. Indeed, if we were to use a standard nonlinear technique, the detected signal would be proportional to E 4 for second-order techniques 3 6,8 or even to E 6 for third-order techniques. 7 In contrast, linear interferometry yields a signal in E 0 E that is directly proportional to the first power of the electric field. When one is dealing with light pulses of extremely small energy, down to a few photons per pulse, the difference between a technique sensitive to the fourth power of the electric field and a linear technique can mean many orders of magnitude in the detected number of counts, in favor of the linear method. Furthermore, as in all homodyne detection schemes, the signal detected with linear interferometry can be enhanced by the use of a higher-energy reference pulse. Therefore the linearinterferometry technique will be far superior to nonlinear techniques for the measurement of weak coherent transient emissions encountered in femtosecond spectroscopy experiments. 3. TIME-DOMAIN AND FREQUENCY- DOMAIN INTERFEROMETRY If the phase of E v is independent of frequency, the frequency-integrated interference signal obtained for a fixed delay between the unknown and the reference pulses 10 13 is usually sufficient to provide the required phase. However, when the phase depends on frequency, a full interferogram must be recorded, either as a function of time delay (time-domain interferometry) or as a function of frequency (frequency-domain interferometry). Time-domain interferometry, also known as dispersive Fourier-transform spectrometry, 14 is a well-known technique in linear optics that provides the measurement of the refractive index of a material. It consists of recording, in a scanning Michelson interferometer, the inter-

Lepetit et al. Vol. 12, No. 12/December 1995/J. Opt. Soc. Am. B 2469 Fig. 2. Experimental setup for spectral interferometry, based on a Mach ehnder interferometer. An unknown electric field, E v, is generated in an arbitrary experiment (EXP), either linear or nonlinear. It is set up to interfere in a spectrometer with a reference E 0 v. BS s, beam splitters; SP, spectrometer; CCD, charge-coupled device array detector. ference signal as a function of the time delay between a reference pulse and the unknown pulse. The same technique can also be directly applied to nonlinear optics, in which the unknown pulse is generated through nonlinear processes in a sample placed in one arm of the Michelson interferometer. This has been applied to the measurement of photon-echo signals, although it was shown that, for practical reasons, it had to be complemented with other measurements. 15 Indeed, one difficulty with timedomain interferometry is that the signal oscillates very quickly with the optical path, the oscillation period being the central wavelength. This means that a large number of data points must be acquired, unless one makes use of the Shannon principle. Even then the optical path must be known with a very good accuracy, requiring a reference laser beam going through the interferometer, as is done in Fourier-transform spectrometers. The large number of fringes that must be recorded to obtain a high spectral resolution also precludes the use of single-shot techniques because of the limited number of pixels in standard imaging optics and CCD arrays. However, time-domain interferometry is still a promising technique for femtosecond spectroscopy, especially in the infrared spectral domain. The converse technique of spectral interferometry was worked out more than 20 years ago by Froehly and coworkers. 16,17 As shown in Fig. 2, the beam to be characterized is set up to interfere with the reference beam in a spectrometer, yielding spectral fringes of a period inversely proportional to the optical path difference between the two beams, which in turn depends on the phase difference. The signal detected at the output of the spectrometer reads S v je 0 v 1 E v j 2 je 0 v j 2 1 je v j 2 1 2ReE 0 v E v, (3) where the last term is proportional to cos f v 2f 0 v, the cosine of the phase difference, Df, between the two beams. The great advantage of frequency-domain interferometry over time-domain interferometry is that the whole interferogram can be recorded simultaneously by the use of either photography or a multichannel analyzer such as a CCD detector. Unlike in time-domain interferometry, small fluctuations of the optical path do not invalidate the interferogram, although they do reduce the fringe contrast if their time scale is shorter than the exposure time. Spectral interferometry has been applied to the measurement of the spectral phase introduced by optical fibers, either linearly 17 or nonlinearly. 18 More recently, the same technique has been applied to phase locking 2 and to phase-resolved pump-probe experiments. 19,20 One drawback of spectral interferometry is that it provides a measurement of the phase through its cosine only. The standard technique for obtaining Df v f v 2 f 0 v is to point the frequencies that correspond to fringe maxima. However, this yields the phase for only discrete values of frequencies for which the phase difference is a multiple of 2p. The usual means of measuring a phase smaller than 2p and improving spectral resolution is to delay the reference pulse by a time delay t, adding a large linear phase, vt, to that of the reference pulse. This provides more data points but also degrades the signal-tonoise ratio. Therefore standard spectral interferometry is appropriate only when the variation of phase with respect to frequency is slow enough that a high spectral resolution is not required. To obtain the phase as a continuous function of frequency, one could think of simply taking the arccosine of cos Df v. However, such a technique presents three major drawbacks. First, the arccosine is a two-valued function, because Df and 2p2Dfhave the same cosine. Second, the experimental data, je 0 v E v jcos Df v, must be divided by je v j before the arccosine can be taken. The amplitude je v j can be obtained only from the square root of the measured power spectrum. This means that the result is no longer sensitive to E but to E 2, inducing a loss in sensitivity. Indeed, dividing by the square root of the power spectrum introduces a lot of noise into the experimental data. Third, the arccosine function presents an infinite derivative at Df 0, so that small changes in the cosine will appear as large variations in the phase. This means that a small noise in cos Df will be amplified to an unacceptable phase-noise level near Df 0. This technique is therefore not practical for actual experimental data, especially for weak photon-noiselimited signals. In Sections 4 and 5, we demonstrate new techniques that provide the measurement of the amplitude and the phase as continuous functions of frequency, with no compromise on either the spectral resolution or the signalto-noise ratio. Because the spectral resolution is limited only by that of the spectrometer, our techniques are ideally suited for the measurement of rapidly varying spectral phases, such as those encountered around narrow spectral resonances. 4. DUAL-QUADRATURE SPECTRAL INTERFEROMETRY To circumvent the fact that only the phase cosine is measured in standard spectral interferometry, we want to acquire the two quadratures of the interference signal, i.e., both cosine and sine. One possibility is to do two sequential measurements of the spectral interferogram, which is associated with a change in the optical path of l 2 that corresponds to a phase shift of p 2, so that the first measurement provides the cosine and the second one provides the sine. Such a technique could be implemented, but at the expense of losing the main advantage of spectral interferometry that we discussed above, which corresponds to the fact that the whole spectrum measurement is done

2470 J. Opt. Soc. Am. B/Vol. 12, No. 12/December 1995 Lepetit et al. Fig. 3. DQSI experimental setup for a phase sensitive measurement of an unknown electric field E v : BS s, beam splitters; LP s, linear polarizers; QWP, quarter-wave plate; WP, Wollaston prism; SP, spectrometer; CCD, charge-coupled-device array detector. simultaneously, making the technique largely insensitive to small optical path fluctuations. If two sequential measurements must be made, then it will be necessary either to stabilize or to measure the exact optical path difference between the two recordings. Instead, we prefer to measure the two quadratures simultaneously, using polarization multiplexing and arranging our experimental setup so that one linear polarization of the reference field varies in quadrature with the other polarization. This can be easily achieved by the use of a circularly polarized reference beam. 21 As shown in Fig. 3, DQSI is based on a frequencyresolved Mach ehnder interferometer. The experimental setup is placed in one arm of the interferometer, generating the unknown optical pulse, linearly polarized at a 45 ± angle with respect to the vertical axis. 22 We write the two polarization components of the pulse as E v, E v. The reference pulse, going through the other arm of the interferometer, is circularly polarized by a quarter-wave plate, so that its two linear polarization components are E 0 v, ie 0 v. Both beams are recombined in a collinear geometry through a beam splitter, and the two perpendicular linear polarizations are analyzed independently and spectrally resolved on two tracks of a two-dimensional CCD array detector. The signals measured for the two polarization components are given by je 0 v 1 E v j 2 je 0 v j 2 1 je v j 2 1 2ReE 0 v E v, jie 0 v 1 E v j 2 je 0 v j 2 1 je v j 2 1 2ImE 0 v E v. (4) During data acquisition, shutters on both arms of the interferometer are operated in order to extract the noninterferometric part of the signal. After subtraction of these terms, only the cross terms in Eqs. (4) remain, i.e., Re E 0 v E v and Im E0 v E v. Combining these two measurements, we obtain the complex quantity E 0 v E v, from which we can deduce the complex value E v, as the reference pulse E 0 v is known in both amplitude and phase. The pulse phase f v, and even the time-dependent electric field E t (through the use of fast Fourier-transform routines), can be displayed in real time. To demonstrate the technique, we use a 20-fs Ti:Sapphire laser, 23 whose Fourier-transform-limited pulses are used as the reference. We first consider the case in which the unknown pulse is simply the reference pulse shifted by a time delay t. In this case the spectral phase f v is equal to vt and should vary linearly with respect to frequency. The experimental result is shown in Fig. 4. The solid and the dotted curves show the two quadratures of the interferogram that correspond to the two linear polarization components. They exhibit spectral oscillations, shifted by p 2, as expected for cos vt and sin vt. The dashed dotted line shows the phase retrieved from these two polarization components. As expected, it varies linearly with respect to frequency, and the slope corresponds to a time delay of 500 fs, in agreement with the delay indication on our stepper-motor controller. The slight oscillations barely observable on the low-frequency side of the spectrum result from the fact that the quarter-wave plate phase shift is exactly p 2 only for a wavelength of 800 nm. However, the actual phase introduced when the frequency is far from the corresponding frequency (375 THz) can be calculated or measured, so that the instrument can be fully calibrated. To demonstrate our technique further, we generated an electric field of rapidly varying spectral phase by transmission of the reference pulse through an interferential filter. Because of the pole in the response function, the spectral phase should exhibit a shift of p when the frequency crosses the resonance of the Fabry Perot cavity. Figure 5(a) shows the experimental result obtained after the insertion of a commercial unblocked interferential filter at normal incidence in one arm of the interferometer. Surprisingly, the observed spectral phase undergoes a 2p phase jump. This is, however, easily explained when one notes that this particular filter is made of two cavities, each contributing half of the 2p phase jump. This is verified when the angle of incidence is changed with the appearance of two separate resonances, as shown in Fig. 5(b). This rotation makes the two cavity resonance frequencies shift by different amounts, and the spectral phase then shows two individual p phase jumps, each occurring at the new resonance frequency of the corresponding tilted cavity. Figure 5 also demonstrates the good sensitivity of the technique. Indeed, the phase is measured extremely far out into the wings of the spectrum, where the intensity is weak. As discussed in Section 2, Fig. 4. Spectral interferograms obtained from both polarization components, horizontal (solid curve) and vertical (dotted curve), when the unknown pulse is the same as the reference pulse but has undergone a time delay shift t. The dashed dotted line shows the phase retrieved from the cosine and the sine parts of the interferogram.

Lepetit et al. Vol. 12, No. 12/December 1995/J. Opt. Soc. Am. B 2471 Fig. 5. Spectral phase (solid curve) obtained when the unknown pulse results from transmission through an interferential filter, either (a) at normal incidence or (b) when the filter is tilted. The dashed dotted curve shows the spectral intensity. this is possible only because the signal is proportional to the field amplitude, so that the signal-to-noise ratio remains fairly good even when the intensity is extremely weak. 5. FOURIER-TRANSFORM SPECTRAL INTERFEROMETRY Femtosecond pump-probe experiments performed in materials that present long dephasing times have been shown to produce spectral oscillations similar to those found in spectral interferometry. 24,25 In these experiments, the oscillations result from the spectral interference between the incident probe pulse and the pump-perturbed polarization decay. 26 By using these spectral oscillations, we have shown that it was possible to recover the full time-dependent third-order polarization through the Fourier transform of a single spectrum. 27 This technique, known as Fourier-transform dynamics, makes use of the causality principle. In this section we describe a related technique applied to spectral interferometry, FTSI. As shown in Fig. 2, the experimental setup is now a straightforward Mach ehnder interferometer, so that the reference pulse precedes the unknown pulse by a fixed time delay t. The use of shutters on both arms of the interferometer allows us to subtract out background signals and to record the interference part only, i.e., t. f t is the correlation product between the reference and the unknown electric field. In the case of a coherent transient emission, E t will obey the causality principle, in the sense that there will be no emission before the excitation. If the exciting pulses present clean fast leading edges, E t will present the same property, although it may extend arbitrarily far for positive times. Given a proper choice of time origin (corresponding, for example, to the center of the last exciting pulse), it is a fairly good approximation to assume that f t is equal to 0 for t,2t, for T larger than a few times the exciting pulse duration. Under this assumption and choosing t larger than T, we find that the two terms in Eq. (6) do not overlap in time. Therefore we can single out the relevant term f t 2t by multiplying the equation by the Heaviside function Q t. Finally, a Fourier transform allows us to recover f v, in both amplitude and phase, and the complex electric field can be obtained by using the following expression: E v F Q t F 21 S v exp 2ivt E 0 v. (7) The fact that applying above the causality principle allows us to obtain the full complex electric field from the knowledge of its real part only is of course closely related to the Kramers Kronig relation. Our technique is illustrated by the simulation shown in Fig. 6, in which E t is simply equal to the time-delayed reference pulse. The figure shows the frequency-domain interference, the time-domain data obtained through an inverse Fourier transform, and finally the phase in the spectral domain S v 2Ref v exp ivt, (5) where f v E 0 v E v. By Fourier transforming the experimental spectrum, we obtain F 21 S v f t 2t 1f 2t2t. (6) Because the measured signal S v is real, i.e., the sum of f v exp ivt and its complex conjugate, the inverse Fourier transform is the sum of two terms, time reversed from each other, whose separation in time increases with Fig. 6. (a) Simulated spectral interferogram between two Gaussian pulses; (b) inverse Fourier transform of the above interferogram, computed in a rotating frame; (c) spectral phase (solid curve) and amplitude (dotted curve) obtained by taking the Fourier transform of the above curve where the points at negative times have been set to 0.

2472 J. Opt. Soc. Am. B/Vol. 12, No. 12/December 1995 Lepetit et al. Fig. 7. (a) Simulated spectral interferogram between a Gaussian pulse and a monoexponential pulse oscillating at a central frequency of 370 THz; (b) inverse Fourier transform of the interferogram; (c) amplitude (dotted curve) and phase (solid curve) of the monoexponential pulse, showing the p phase jump at resonance. ratio. Conversely, the signal-to-noise ratio could be improved by the use of a smaller time delay between the two pulses, but at the expense of a lower spectral resolution. On the contrary, both DQSI and FTSI provide good results, with no compromise on either the spectral resolution or on the signal-to-noise ratio. DQSI provides the amplitude and the phase in real time, with virtually no calculation except division by the reference complex electric field. FTSI requires the computation of two Fourier transforms, which makes it slightly slower, but the experimental setup is simpler than that for DQSI. However, FTSI requires the assumption that the electric field is 0 before some rising time. This is usually easily verified in experiments that involve coherent light emission by a material, in either linear or nonlinear optics. For example, in a three-pulse photon-echo experiment, performed in a noncollinear geometry, the causality principle implies that the light emitted in the echo direction will rise only after the last exciting pulse excites the sample. In such experiments, the assumption required for the use of FTSI will cause no limitation. In other experiments, however, such as the measurement of the electric field transmitted through a zero-dispersion line with a spatial mask in its Fourier plane, the electric field might present a leading edge that is as long as its trailing edge. 28 In this case, the alternatives will be between FTSI with a large time delay, obtained by multiplication by the Heaviside function and Fourier transformation back into the frequency domain. Figure 7 shows a simulation that corresponds to an exponentially decaying E t. Again, the amplitude and the phase can be recovered by the use of Eq. (7). Note that the reference pulse must precede the unknown pulse in order to avoid any overlap between the two pulses. Only then can we separate the two contributions in Eq. (6). Finally, Fig. 8 shows an experimental result obtained when the unknown pulse is simply equal to the timedelayed reference pulse. The experimental data are the same as in the previous section (Fig. 4), except that we now use only one quadrature of the interference. Thus both methods can be compared. The perfect agreement in the spectral phase recovered by either DQSI or FTSI demonstrates the validity of the latter technique. 6. DISCUSSION We now compare the various available spectral interferometry techniques, using the simulation shown in Fig. 9(a), which corresponds to an exponentially decaying E t. The figure shows the spectral phase obtained through the use of three techniques: pointing the maxima of the spectral interference fringes (b), DQSI (c), and FTSI (d). As was mentioned above, pointing the maxima of the interference fringes provides the amplitude and the phase for only discrete values of the frequency. Furthermore, the noise level is much higher than in the other two techniques. If one increases the time delay with the reference pulse, more data points will be available, but at the expense of degrading the signal-to-noise Fig. 8. (a) Experimental spectral interferogram between femtosecond pulses delayed by a time t 500 fs. (b) Amplitude of the inverse Fourier transform of the above interferogram. The direct pulse is centered at 500 fs, while the time-reversed one is centered at 2500 fs. (c) Spectral phase obtained from the Fourier transform of the above time-domain response, after multiplication by the Heaviside function, i.e., by the use of FTSI (solid line). Experimental result obtained by DQSI (dotted line) is shifted up for clarity.

Lepetit et al. Vol. 12, No. 12/December 1995/J. Opt. Soc. Am. B 2473 Fig. 9. Simulated spectra that corresponds to the spectral interference between a Gaussian pulse and a monoexponential pulse oscillating at a central frequency of 370 THz. In (b), (c) and (d), the dotted curve shows the actual spectral phase. (a) The two quadratures of the spectral interference, as would be measured in a DQSI setup. Noise was added to the spectra in order to simulate typical experimental conditions. (b) Spectral phase (filled diamonds) as obtained from pointing the fringe maxima in the dotted-line curve of (a). (c) Spectral phase as obtained from DQSI. (d) Spectral phase as obtained from FTSI with only the dotted curve in (a). or DQSI, in which case a small, or even zero, time delay can be used. Finally, we discuss the possible applications of our phase-measurement techniques. First, linear-optics applications should not be neglected, as white-light interferometry is a powerful technique that provides the measurement of a material refractive index. 29 Furthermore, many essential components used in femtosecond lasers are purely linear, such as pulse shapers or stretchercompressor systems. In such a case, it is much better to characterize the linear device itself by the use of DQSI or FTSI rather than to characterize the pulse transmitted through the device by the use of a nonlinear phase-measurement technique. The linear technique is not only easier to implement, it is also more reliable. Indeed, pulse shapers can produce intricate pulses, in which case the convergence of the iterative algorithms required in nonlinear techniques is no longer granted. Such techniques should be reserved for the measurement of well-behaved pulses, which can then be transmitted in a linearly characterized pulse shaper. The case of nonlinear optics is even stronger. Many recent nonlinear-optics experiments have proven that interesting results can be obtained when a more careful measurement of the electric field is performed. 1,30 Still, these experiments measured only the intensity profile of the unknown pulse, using cross correlation through upconversion with a reference pulse. One important benefit of up-conversion is that a strong signal at t 0 can be rejected with a good dynamic range, a unique advantage for example in the case of collinear photon-echo experiments. In a noncollinear geometry, in which this advantage is not essential, spectral interferometry techniques would provide more information with a better sensitivity. Again, direct nonlinear phase-measurement techniques should be ruled out on the grounds of sensitivity and reliability of the phase measurement. For all the above applications, we expect that DQSI or FTSI will result in important improvements in terms of either spectral resolution or sensitivity. More generally, we think that such techniques should ease the way for the transposition of any NMR technique to the domain of optics. As an example, we have thus implemented the transposition to optics of two-dimensional NMR. 31 7. CONCLUSION We have developed two new methods, DQSI and FTSI, to measure the amplitude and the phase of the electric field of an unknown optical pulse, provided an adequate reference pulse is available. Unlike other frequency-domain techniques, these methods provide a measurement of the phase as a continuous function of frequency, with a spectral resolution limited only by that of the spectrometer. As homodyne detection schemes, our techniques are sensitive and are thus ideally suited for the detection of the very weak signals often encountered in nonlinear-optics experiments. Finally, these phase-measurement methods involve no iterative algorithm, making them real-time techniques with good reliability even for arbitrarily com-

2474 J. Opt. Soc. Am. B/Vol. 12, No. 12/December 1995 Lepetit et al. plicated phase profiles. Therefore they should be preferred over their nonlinear counterparts. ACKNOWLEDGMENTS This work originated from discussions with J. Paye on the new possibilities opened up by phase-resolved techniques. His contribution is gratefully acknowledged. We have also benefited from many fruitful discussions with D. Hulin, J.-P. Likforman, A. Migus, F. Salin and L. Sarger. REFERENCES AND NOTES 1. S. Weiss, M.-A. Mycek, J.-Y. Bigot, S. Schmitt-Rink, and D. S. Chemla, Phys. Rev. Lett. 69, 2685 (1992). 2. N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991). 3. K. Naganuma, K. Mogi, and H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989). 4. J. L. A. Chilla and O. E. Martinez, IEEE J. Quantum Electron. 27, 1228 (1991). 5. T. F. Albrecht, K. Seibert, and H. Kurz, Opt. Commun. 84, 223 (1991). 6. J.-P. Foing, J.-P. Likforman, M. Joffre, and A. Migus, IEEE J. Quantum Electron. 28, 2285 (1992). 7. D. J. Kane and R. Trebino, Opt. Lett. 18, 823 (1993); R. Trebino and D. J. Kane, J. Opt. Soc. Am. A 10, 1101 (1993). 8. J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, Opt. Lett. 18, 1946 (1993); J. Paye, IEEE J. Quantum Electron. 30, 2693 (1994). 9. V. Wong and I. A. Walmsley, Opt. Lett. 19, 287 (1994); J. Opt. Soc. Am. B 12, 1491 (1995). 10. M. J. LaGasse, K. K. Anderson, H. A. Hasue, and J. G. Fujimoto, Appl. Phys. Lett. 54, 2068 (1989). 11. N. Finlayson, W. C. Banyai, C. T. Seaton, and G. I. Stegeman, J. Opt. Soc. Am. B 6, 675 (1989). 12. K. Minoshima, M. Taiji, and T. Kobayashi, Opt. Lett. 16, 1683 (1991). 13. L. Sarger, P. Segonds, L. Canioni, F. Adamietz, A. Ducasse, C. Duchesne, E. Fargin, R. Olazcuaga, and G. Le Flem, J. Opt. Soc. Am. B 11, 995 (1994). 14. P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry, Vol. 83 of Chemical Analysis Series (Wiley, New York, 1986), p. 369, 15. D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, and W. Schafer, Phys. Rev. B 50, 8439 (1994). 16. C. Froehly, A. Lacourt, and J. C. Vienot, J. Opt. (Paris) 4, 183 (1973). 17. J. Piasecki, B. Colombeau, M. Vampouille, C. Froehly, and J. A. Arnaud, Appl. Opt. 19, 3749 (1980). 18. F. Reynaud, F. Salin, and A. Barthelemy, Opt. Lett. 14, 275 (1989). 19. E. Tokunaga, A. Terasaki, and T. Kobayashi, Opt. Lett. 18, 370 (1993). 20. J.-P. Geindre, P. Audebert, A. Rousse, F. Falliès, J. C. Gauthier, A. Mysyrowicz, A. Dos Santos, G. Hamoniaux, and A. Antonetti, Opt. Lett. 19, 1997 (1994). 21. P. Juncar and J. Pinard, Rev. Sci. Instrum. 53, 939 (1982). 22. When the state of polarization of the unknown pulse must be measured, one may repeat the experiments for two positions of a linear analyzer placed on the unknown pulse beam. 23. Ch. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, IEEE J. Quantum Electron. 30, 1100 (1994). 24. B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, Phys. Rev. Lett. 59, 2588 (1987). 25. C.-H. Brito-Cruz, J. P. Gordon, P. C. Becker, R. L. Fork, and C. V. Shank, IEEE J. Quantum Electron. 24, 261 (1988). 26. M. Joffre, D. Hulin, A. Migus, A. Antonetti, C. Benoit àla Guillaume, N. Peyghambarian, M. Lindberg, and S. W. Koch, Opt. Lett. 13, 276 (1988). 27. M. Joffre, D. Hulin, J.-P. Foing, J.-P. Chambaret, A. Migus, and A. Antonetti, IEEE J. Quantum Electron. 25, 2505 (1989). 28. J. Paye and A. Migus, J. Opt. Soc. Am. B 12, 1480 (1995) 29. C. Sainz, P. Jourdain, R. Escalona, and J. Calatroni, Opt. Commun. 111,632 (1994). 30. E. J. Mayer, G. O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, S. W. Koch, P. Thomas, R. Hey, and K. Ploog, Phys. Rev. B 51, 10909 (1995). 31. L. Lepetit, M. Joffre, G. Chériaux, A. Migus and D. Hulin, in Quantum Electronics and Laser Science, Vol. 16 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QWG1.