Chapter 10. Femtosecond phase spectroscopy

摘要

10 Femtosecond phase spectroscopy By TAKAYOSHI KOBAYASHI Department of Physics Faculty of Science University of Tokyo 7-3-1 Hongo Bunkyo Tokyo 113 Japan 1 Introduction The present study uses a combination of two sophisticated technologies interferometry and femtosecond laser pulse generation. The former was developed many years ago and is used extensively in various applications while the latter only started to develop recently and its use is growing very rapidly. 1.1 Interferometry Interferometry is widely used to measure variations in the phase front of light in several optical methodologies. This technique has been applied to the measurements of dimensions such as length shape displacement and surface roughness in the field of metrology. The minimum detection limit of such displacement measurement is less than one nanometre.Another important application of interferometry is the measurement of the refractive index of materials. Two independent physical variables describing the features of the optical field are phase and amplitude. The real and imaginary parts of the complex refractive index are related to the changes in the phase and amplitude of light in materials respectively. Since optical detectors are basically used to measure the light intensity i.e. the squared amplitude absorbance which is proportional to the imaginary part of the refractive index is easily measured. Phase change or disect;erence is detected by the light intensity change or modulation in interferometry and it can be used to measure change in the real part of the refractive index. 1.2 Femtosecond spectroscopy Short pulse generation utilizing various mode-locked lasers has made it possible to study ultrafast transient processes in condensed materials such as semiconductors and metals molecules in the gas phase and in solutions and biological systems.Among several techniques the pumpndash;probe method with a white-light continuum by which the spectral dependence of transmission changes can be obtained by a single pulse measurement has played a most important role in the progress of femtosecond spectroscopy during the last decade. Data obtained by this method are called disect;erence transmission spectra (DTS) represented by normalized transmittance change *T/T(u,t) at an angular frequency u of probe light at time delay q between pump and 375 probe pulses. The disect;erence absorption spectrum (DAS) is obtained similarly with in this case the absorbance change *A(u) being measured.Femtosecond phase-disect;erence spectroscopy(DPS) is also important as a counterpart for microscopic studies of ultrafast phenomena especially in dispersive materials. The time dependence of the phase change associated with the refractive-index change induced by optical excitation can be measured by means of time-resolved interferometry. Various time-resolved interferometers have been developed and some of them are described in the following subsection. 1.3 Femtosecond phase spectroscopy There are several basic configurations of interferometers which can be used for phase-disect;erence measurement amplitude-division interferometers (ADI),1ndash;4 wavefront- division interferometers polarization-division interferometers and time-division interferometers (TDI).5,6 ADI1ndash;4 and TDI5,6 have been used to measure the transient refractive index change.In the ADI a light beam is split into two which travel along disect;erent optical paths. One is directed through a sample and then recombined with the other beam after travelling through the sample to form an interference fringe. In the TDI the probe and reference are temporally displaced but travel collinearly along a single arm. The time delay of the excitation pulse with respect to the probe is set with a variable delay so that the excitation pulse passes through the sample before and after the arrival time of the probe and reference respectively. In the presence of the excitation pulse only the probe pulse experiences the transient refractive index change.The phase change induced by the optical excitation can be measured from the phase disect;erence between the interference fringes with and without the excitation. The advantage of the TDI is higher visibility of the interference fringes because both the reference and probe pulses transmit through the sample and consequently they have almost the same spectrum. In addition the configuration is more stable to path-length fluctuation because the reference and probe pulses travel along a common path. Even though various time-resolved interferometers had been developed there was no convenient method as in the pumpndash;probe method with a continuum by which DPS could be obtained by a single pulse measurement. It is important to measure the phase spectrum in time-resolved spectroscopy by a single-shot experiment.However most of the previous studies have been performed at single wavelengths of both the excitation and probe.1ndash;5 A frequency-domain interferometer was utilized to obtain DPS by utilizing TDI.6 1.4 The Kramersndash;Kronig relations The Kramersndash;Kronig (Kndash;K) relations have been used to obtain the spectral dispersion of the time-resolved refractive index change from the time-resolved absorbance change. They7 connect the real and imaginary parts of the linear susceptibility s(u) and can be extended to non-linear optics so long as the causality condition is satisfied.8 In particular in steady-state pumpndash;probe spectroscopy the ordinaryKndash;Krelations hold to all orders in the pump field if the pump frequency is fixed. On the other hand the applicability of the Kndash;K relations to time-resolved spectroscopy is not obvious and there has been no detailed work on this subject.Instead without any theoretical support the Kndash;K relations were sometimes used to obtain the refractive index change 376 T. Kobayashi Fig. 1 Experimental set-up of the FDI. from the transient absorption spectrum. It has been dipara;cult until now to study the dispersion relations for time-resolved data because of the experimental dipara;culties in the measurement of time-resolved DPS. We have developed a frequency-domain interferometer (FDI),6 with which both the DTS and DPS can be measured simultaneously. The FDI is therefore suitable for the study of the time-resolved dispersion relations. In the first part of this paper the FDI was applied to carbon disulfide (CS 2 ) to test experimentally for the first time the applicability of the Kndash;K relations in timeresolved spectroscopy.The FDI measurement we made to obtain DPS and DTS simultaneously is described. However only several tens of wavelength points in the spectral region were measured and in that sense it is not a complete spectrum. In the second half of the paper we describe a time-resolved Sagnac interferometer that we have developed to obtain a lsquo;completersquo; transient spectrum of the phase change and the transmittance change. The femtosecond Sagnac interferometer was then combined with a polarization spectroscopy technique which enabled us to separate instantaneous and inertial responses in the time-resolved DPS for CS 2 and CCl 4 . In particular as is shown in Fig. 1 spectral fringes were observed in instantaneous response spectra at negative delay times and the spectral fringe period was nearly inversely proportional to the delay time.The experiment was performed under osect;-resonant conditions and so this cannot be explained by a conventional perturbed free induction decay in which dephasing time plays an important role. In this report we analyse the experimental results numerically and succeed in reproducing the main feature in both instantaneous and inertial response spectra. 2 Experimental 2.1 Frequency-domain interferometer Fig. 1 shows the experimental set-up for the FDI. An amplified colliding-pulse mode locked (CPM) laser was used as the light source for the pump probe and reference. 377 Femtosecond phase spectroscopy Fig. 2 (a) Experimental set-up of the SI BS beam splitter.(b) Time course of the excitation reference and probe pulses at the sample position. The wavelength pulse width and pulse energy of the output of the amplified CPM laser are 620 nm 60 fs and 2mJ at 10 kHz. The output beam is split into two. One beam is used as a pump and the other is focused into an ethylene glycol jet to generate a white continuum. The continuum pulse is further divided into two in a Michelson interferometer to be used as the reference and probe and the two pulses are displaced temporally by a few hundred femtoseconds. They then travel along a common path through a sample and are detected by a spectrometer with a multi-channel photodiode array detector. 2.2 Space-domain interferometer Fig. 2 shows a Sagnac interferometer (SI) used as a space-domain interferometer (SDI).Experiments were performed with laser pulses (780 nm 120 fs 200mJ at 1 kHz) from a Ti sapphire regenerative amplifier (Clark-MXR NJA-4/CPA-1000). The probe and reference pulses were separated from a white continuum generated by focusing the amplified pulses into CCl 4 . The interferometer is composed of a chromium-coated beam splitter and two aluminium mirrors in a triangular configuration. A pair of confocal lenses is inserted in one side of the triangle and the sample was placed at the confocal position of these lenses. The output spatial fringes from the interferometer were detected using a digital charge-coupled device (CCD) camera at each time delay between the excitation and probe pulses after being dispersed with a monochromator. The sample solutions used in both measurements were placed in 1mm thick quartz cells.The excitation for the measurement of time-resolved refractive index change was also the amplified fundamental pulse of each laser system which passed through a variable delay. 378 T. Kobayashi 3 Frequency-domain interferometer 3.1 Principle of FDI The FDI set-up employs the time-division scheme for separating the pulses from the reference and the probe. In the FDI,6 the temporal broadening of probe and reference pulses takes place in a monochromator owing to the Fourier transform relation between spectral width and pulse duration. The broadened probe and reference pulses overlap at the detector. The principle of femtosecond time-resolved phase spectroscopy using the FDI is based on the spectrometer which gives the Fourier transform of the incident light pulse.The probe and reference pulses are temporally displaced by T in the interferometer and transmitted through a medium. The Fourier amplitudes of the probe and reference fields are given by E13(u)bsol;FE13(t)bsol;E(uu0 )exp(in#(u)ud/c) (1) E3amp;(u)bsol;E(uu0 )exp(iuTin#(u)ud/c) (2) where u0 is a central angular frequency and E(u) is generally a complex function. d is the thickness of the medium c is the velocity of light in vacuum and n#(u) is the complex refractive index of the medium defined by n#(u)bsol;n(u)ik(u) (3) The probe pulse undergoes a change in complex refractive index upon excitation *n#(u,q)bsol;*n(u,q)i*k(u,q) (4) resulting in the modification of the probe field so that E13(u,q)bsol;FE13(t,q)bsol;E13(u)expi*/(u,q)*K(u,q (5) Here the phase change is given by */{*nud/c (6a) and the amplitude change is determined by *K{*kud/c (6b) The interference signal without excitation measured with a spectrometer is expressed by DE13(u)E3amp;(u) D2bsol;D E13(u) D 2(22cosuT) (7) and with excitation by DE13(u,q)E3amp;(u) D2bsol;D E13(u) D 2M1exp2*K(u,q)2exp*K(u,q) cosuT*/(u,q)N (8) By comparing these two equations */(u,t) and *K(u,t) can be simultaneously determined from the peak shifts and amplitude changes of the fringes respectively with a multi-channel spectrometer.3.2 Results Fig. 3(a) shows probe spectra with and without excitation and the DTS which were 379 Femtosecond phase spectroscopy Fig. 3 Signals for CS 2 without delay-time correction. (a) DTS (upper solid curve) and probe spectra with (dashed curve) and without (lower solid curve).(b) DPS (open circles) interference spectra with (dashed line) and without (lower solid line) excitation and the disect;erence interference spectrum between them (upper solid line). measured by blocking the reference beam.10 Fig. 3(b) shows interference spectra with and without excitation and the DPS calculated from the fringe shift. This was the first measurement of DPS with femtosecond time resolution.6,9,10 The FDI has several advantages first the aparatus is constructed by adding only two more optical components a beam splitter and a mirror to the conventional pumpndash;probe experimental system. Since there is no need for temporal overlap of the pulses before the spectrometer the configuration is much simpler than other set-ups. It is less sensitive to beam deflection because the path disect;erence between the reference and probe is only that of the two very short arms of the Michelson interferometer.Hence we could obtain data with a high signal-to-noise ratio as shown in Fig. 3 even without a feedback electric circuit for stabilization. Secondly optical alignment is much easier for overlapping the reference and probe either spatially or temporally in the set-up. Readjusting the optical path disect;erence between the reference and probe is not needed when the samples are exchanged. Thus FDI provided us with a new possibility for time-resolved phase spectroscopy. To measure continuous DPS by FDI with white-continuum light the delay T between the reference and probe must be swept by the optical oscillation period. Therefore a space-division not a time-division configuration is preferred where the 380 T.Kobayashi Fig. 4 Upper interference spectra observed directly for CS 2 at tbsol;40 fs with excitation (a) without excitation (b) and their disect;erence (c). The time displacement between the reference and probe is 370 fs. Lower (a) (b) and (c) are normalized by the transmitted probe spectrum to obtain (a@) (b@) and (c@) respectively. Open circles (DPS) are calculated from the fringe-valley shifts between (a@) and (b@) as 2n(ji ji 9)/(ji`1 ji) where ji 9 and ji are the ith fringe-valley wavelengths with and without excitation. reference and probe travel along disect;erent paths in an MZ interferometer to be overlapped spatially after the sample with time displacement T. In addition the maximum time delay to obtain *(u,q) is itself limited by T.ForqT the signal gives a disect;erence between */(u,qT) and */(u,q) because the reference is also influenced by the pump. Since the increase in T is limited by the system wavelength resolution the maximum time delay is also limited. In the experiment T was set at 310 fs. These features are improved in the femtosecond SI described in the next subsection. In order to study the time-resolved dispersion relations we measured transparent pure liquid CS 2 the dynamics of which has been extensively studied.11ndash;14 Fig. 4 shows the signal for CS 2 in a 1mm cell at 40 fs time delay to demonstrate how DPS are obtained with the FDI. Open circles show DPS derived from the fringe shifts between a@ and b@. The systematic errors in the DPS due to amplitude fluctuations are estimated as less than 0.01 rad and so can be neglected.Fig. 5 shows both DTS and DPS for CS 2 at 50 0 and 50 fs delays together with probe spectra with and without excitation. The excitation density is ca. 1.9 mJ cm~2. Average phase shifts were calculated from the Fourier transform of the interference 381 Femtosecond phase spectroscopy Fig. 5 Transmitted probe spectra with excitation (a) and without excitation (b) DTS (c) and DPS (open circles) for CS 2 at50 0 and 50 fs time delays. signals in the upper part of Fig. 4 to utilize the whole fringes.6 The results are shown in Fig. 6 and are fit to the following equation *n(t)bsol;aI(t)bh(t)expt/T$(1expt/T3) (9) Here the first and second terms represent the electronic and nuclear responses respectively; I(t) is the normalized pulse-intensity envelope; h(t) is the normalized step function; T$ and T3 are the decay and rise times of the nuclear motion and a and b are appropriate constants.This is a phenomenological expression for the femtosecond dynamics of the optical Kerr esect;ect in CS 2 .11ndash;14 A squared hyperbolic secant function of 60 fs half-width is assumed for I(t) a/b is taken as 2 and the values of T$ bsol;1.6 ps and T3 bsol;75 fs from the literature12,14 are used. The fit is reasonably good so that the signal shows typical dynamics for CS 2 . Although the nuclear response consists of several origins with disect;erent kinetics,14 only the orientational relaxation term is used because it is supara;cient for fitting the behaviour up to 200 fs delay. Even when other terms are included the fit is not much disect;erent.382 T. Kobayashi Fig. 6 Average phase shifts (dots) in the probed spectral region as a function of time delay and the fitting function (solid curve). The conditions C D and A are satisfied at 50 50 and 190 fs time delays respectively (see text). The negative sign of the phase change is consistent with a positive non-linear refractive index for CS 2 . Since the excitation is non-resonant there should be no absorption change and dispersion of the refractive index change should be negligible within the observed spectral range. However although there is no net change in transmission DTS shows spectral shifts and broadening of the probe at^50 and 0 fs respectively. This is because the probe phase is modulated by the rapid refractive-index change in the medium (induced phase modulation15,16).The frequency shift of the probe is given by the time derivative of the phase change. The observed spectral change can therefore be explained very well from Fig. 6. For example since at zero delay the probe phase decreases at the leading edge and increases at the trailing edge the probe spectrum shows both red and blue shifts resulting in spectral broadening (self-phase modulation15). In contrast at 190 fs the DPS is constant and the DTS is zero because the probe phase is modulated little at this delay owing to an almost steady phase change. Since DTS and DPS are expressed by the change in the imaginary and real parts of the optical susceptibility respectively the relations between the two can be discussed with the Kndash;K relations. The DTS and DPS at 50 and 190 fs qualitatively satisfy the Kndash;K relations but the Kndash;K relations are not satisfied for the spectra at 0 fs.That is the DTS and DPS are both even functions with respect to the centre frequency of the probe whereas theKndash;Kinversion transforms an even function to an odd function and vice versa. Further the sign of the DTS at 50 fs is opposite to that at 50 fs which is unusual because through the Kndash;K relations the refractive index should increase on the longer wavelength side of the absorption increase. These results can be explained as follows. TheKndash;Krelations are based on the fact that the complex susceptibility function has no poles in the lower half of the complex u plane.7 Physically this is the result of causality that is the polarization P(t) induced in a material by a probe pulse field E(t) is expressed by P(t)bsol;s(t)Oslash;E(t)bsol;P = 0 dt@s(t@)E(tt@) (10) Here Oslash; denotes the convolution operation and s(t) is a response function which is 383 Femtosecond phase spectroscopy zero over tbsol;0 because of causality where time zero is taken as the probe pulse peak.The linear susceptibility function is given by the FT of eqn. (10) as P(u)bsol;FP(t)bsol;s(u)E(u) (11a) s(u)bsol;P = 0 dt expiuts(t) (11b) This function s(u) has no poles in the lower half at the u plane because t0 in expiut. Hence s(u) satisfied the Kndash;K relations. In time-resolved spectroscopy however the causality condition is not satisfied because there is a pump pulse which excites the material to cause the polarization change *P(t)bsol;s(t)Oslash;ME(t)*N(t)bsol;P = 0 dt@ s(t@)E(tt@)*N(tt@) (12) where *N(t) represents the change in the state of the material due to the pump and depends on q such that *N(t)bsol;*N@(tq).Eqn. (12) describes the level population term17 in the perturbation expression whereas the following discussion can also be made for the coherent coupling term. The susceptibility change is then expressed by *s(u)E(u)bsol;F*P(t)bsol;s(u) P = ~= dt expiutE(t)*N(t) (13) Using *N(u)bsol;expiut*N@(u) eqn. (4) is rewritten as *s(u)bsol; s(u)expiuq E(u) Pdu@E(u@)expiu@q*N@(uu@) (14) In general in eqn. (13) this function *s(u) has poles in both halves of the u plane because the integration is over all t. Therefore the Kndash;K relations are not satisfied. However the Kndash;K relations are satisfied in the following special cases A Slow limit process When *N(t) is constant (*N 0 ) *s(u)bsol; s(u) E(u) P = ~= dt expiutE(t)*N 0 bsol;*N 0s(u) (15) B Short probendash;pulse limit When E(t) is a d-function E 0d(t) *s(u)bsol; s(u) E(0) P = ~= dt expiutEd(t)*N(t)bsol;*N(0)s(u)bsol;*N@(q)s(u) (16) C Vanishing non-linearity in negative probe time When *N(t) is zero over tbsol;0 the integration can be over t0 to obtain *s(u)bsol; s(u) E(u) P = 0 dt expiutE(t)*N(t) (17) D Vanishing non-linearity in positive probe time When *N(t) is zero over t0 and s(u) is constant (s0 ) the integration can be over 384 T.Kobayashi tbsol;0 to obtain *s(u)bsol; s0 E(u) P 0 ~= dt expiutE(t)*N(t) (18) Since this function has no poles in the upper half of the u plane it obeys the Kndash;K relations with opposite signs. Among the above cases A and B are equivalent since if the probe is short enough the time dependence of non-linear change is almost constant within the probe pulse.For C and D additional conditions are required because 1/E(u) may generally cause a singularity. First 1/E(u) must not have poles over D u Dbsol;O in the lower and upper half planes for C and D respectively. Secondly *s(u)/(uu0 ) should fall osect; more rapidly than 1/u. Since *s(u) can be rewritten in the following form *s(u)bsol;s(u) Pdu@*N(u@)E(uu@)/E(u) (19) The second condition is given by E(uu@)/E(u)0(ud) with dp0. These conditions are satisfied for e.g. a hyperbolic secant or Lorentzian but not for a Gaussian function. The pulse shape of a mode-locked laser is well approximated by a hyperbolic secant function,18 so that they are usually satisfied. The coherent coupling term to the second order in the pump field and to the first order in the probe field is for example expressed by *P(t)bsol;s(t)Oslash;ME9(t)*N#(t)N (20) *N#(t)bsol;g(t)Oslash;ME9*(t)P(t)Nbsol;P t ~= dt@g(tt@)ME9*(t@)P(t@)N (21) where E9(t) is a pump pulse field * denotes the complex conjugate g(t) is a response function which represents energy relaxation dynamics and is zero for tbsol;0.This term is called the perturbed free induction decay term.17 It is obvious that the conditions A and B hold also for this term but a brief discussion is needed for C and D as follows. If E9(t)bsol;0 for tbsol;0 and t0 the Fourier integration is to be performed only for t0 and tbsol;0 respectively. Further the susceptibility change is expressed by *s(u)bsol;s(u)Pdu@E9(u@)*N#(uu@)/E(u) bsol;s(u) Pdu@E9(u@)g(uu@) PduAE9*(uA)s(uu@uA)E(uu@uA)/E(u) (22) which requires the same additional conditions for E(u).Therefore the conditions C andD also hold. Even for the higher order in the pump field one can readily see that a similar discussion can be made so long as the probe field is weak enough to be limited to the first order. In summary the conditions Andash;D provide strict criteria for applying the Kndash;K relations to time-resolved data in the weak-probe-field limit. Since */(q) in Fig. 6 is considered to be proportional to *N(t) the spectra at each delay in Fig. 5 can be discussed in terms of the conditions above. First at 190 fs condition A is approximately met because */(q) is nearly steady within the probendash;pulse duration. In fact at 190 fs the DTS and DPS are zero and constant 385 Femtosecond phase spectroscopy respectively which is consistent with both condition A and the non-resonant excitation condition *s(u) should be proportional to s(u) under condition A and Ims(u) and Res(u) should be zero and constant respectively in the non-resonant spectral region.Secondly since at50 fs the probe feels the rise in the refractive index change at the trailing edge of the pulse condition C is approximately met. Similarly at 50 fs the probe feels the decay in the refractive index change at the leading edge and condition D is approximately met. In fact the spectra at50 and 50 fs seem to satisfy the ordinaryKndash;Krelations and the relations with opposite signs respectively. Thirdly the spectra at 0 fs violate the Kndash;K relations because none of the conditions are satisfied. In other words the polarization change is not expressed by the first order in the probe field but by a higher order because the pump can be identified with the probe owing to the complete overlap between them.This is the first clear observation of the non-linear dispersion relations. The spectra at^50 fs can be explained in another way. Since s(u) is constant from eqn. (4) *s(u) should be nearly proportional to expiuq. This is the case at ^50 fs where the DPS are approximately expressed by cosuq and the DTS by 2 sinuq. The real part (cosuq) and the imaginary part (sinuq) of expiuq are strictly related to each other through theKndash;K relations but the signs of the relations depend on q. That is for negative q the complex integration of expiuq/(uu0 ) can be performed in the lower half of the plane to obtain the ordinary relations while for positive t the integration path must go round the upper half of the plane to obtain the relations with opposite signs.The sign reversal of the Kndash;K relations is considered to be inherent in timeresolved spectroscopy where the change can occur before applying the probe light field. To meet the condition D s(u) should be nearly constant or in other words s(t) should be a d-function in the time domain. This is fulfilled when the excitation is non-resonant as in this study or when the phase-relaxation time is much shorter than the pulse duration in the case of resonant excitation. In other cases s(u) has poles in the upper half of the plane such that the complex integration leaves residues. Hence it is usually dipara;cult to observe the relations with opposite signs for resonant materials. In fact the signals for CdSxSe 1~x doped glass were shown to satisfy the ordinary relations qualitatively even for positive delays.6 On the other hand the conditionC has no such restriction.It is therefore concluded that in time-resolved spectroscopy the Kndash;K relations are approximately applicable for negative time delays. This fact is important because an optical Stark esect;ect (OSE) in time-resolved spectroscopy which is caused by the perturbed free induction decay term appears dominant for negative delays to obtain the net change in the refractive index at the absorption peak. The esect;ect of rapid phase change on chirped continuum pulses is studied with an FID. Because of the chirp temporal evolution of the optical Kerr response in CS 2 is projected into DPS. The chirped continuum shows spectral shifts that are due to induced phase modulation even when the continuum has a flat spectrum.19 Pumpndash;probe spectroscopy of amorphous semiconductor and glass samples has been performed.The spectral shift of a continuum probe because of induced phase modulation was observed when the continuum had a sharp peak in the spectrum. When a fundamental pulse was used for the probe oscillatory structures in the DTS were observed for both negative and positive time delays. This behaviour can be reproduced by a numerical simulation with the assumption of induced amplitude and 386 T. Kobayashi phase modulations of the probe pulse. Both the real and the imaginary parts of s(3) were simultaneously estimated from the modulations.9,19 4 Sagnac interferometer 4.1 Principle of SI To achieve high stability one of the common-path interferometers an SI was used.20ndash;22 Furthermore zeroth order interference of white light can be automatically found in SI because of negligible optical path disect;erence between the probe and reference pulses.The essential point for the femtosecond SI is that the sample is not placed at the centre.20ndash;22 The time course of the excitation reference and probe at the sample position is shown in Fig. 2(b). The counter-clockwise propagating reference pulse in the triangle passes through the sample first and the clockwise probe pulse arrives at the sample almost after one round trip of the triangle. Since the probe and reference are temporally separated in the SI the SI is also used in a similar way to the TDI.5,6 Even though the probe and reference pulses propagate anti-collinearly in the interferometer and the arrival time at the sample is displaced they come out of the interferometer coincidentally.The maximum time delay was 2.2 ns determined by the time displacement between the reference and probe. This time displacement can be much longer than that for the FDI. In the previous studies using SI transient change in either the refractive index20,22 or absorbance21 was measured at a fixed wavelength. However it is important to measure simultaneously the DTS and DPS using a multi-channel detector as described above. The femtosecond SI described here is most suited to the measurement of continuous dispersion of complex non-linear optical susceptibility using a femtosecond white continuum in a single shot.23 One of the mirrors in the interferometer is slightly tilted to the vertical direction to measure the nearly parallel spatial fringes due to interference of equal inclination between the probe and reference pulses.These recombined pulses are spectrally dispersed in a monochromator in the horizontal direction. Finally the two-dimensional image with the vertical and horizontal axes of the optical path disect;erence and wavelength respectively is detected with a twodimensional detector. Hereafter this two-dimensional image will be denoted as a lsquo;spectro-interferogramrsquo;. Fig. 7 shows the two-dimensional spectro-interferogram with time delays of the excitation of2000,300 0 and 300 fs. The spectro-interferogram at2000 fs Fig. 7(a) is the same as that without excitation. Fringe deformation due to the refractive index change in the sample is remarkably observed around the centre wavelength in the image at300 0 and 300 fs corresponding to Fig.7(b)ndash;(d). 4.2 Data analysis of SI The vertical section of the spectro-interferogram corresponds to the interference fringe at a particular wavelength which is called the lsquo;interferogramrsquo;. Fig. 8 shows the interferograms at 590nm (a) without and (b) with excitation. The interferogram without excitation in (a) is also depicted in the same frame of (b) with excitation. A fringe phase shift is clearly observed in Fig. 8(b). The interferogram at every 387 Femtosecond phase spectroscopy Fig. 7 Spectro-interferograms at time delays of (a) 2000 (b) 300 (c) 0 and (d) 300 fs detected with aCCDcamera. The bright and dark lines represent the fringe peak and valley respectively.wavelength is a sine function in principle but it is slightly distorted by imperfect conditions such as wave-front distortion of the white continuum light. Hence the interferogram was Fourier transformed to determine the amplitude and period of the interference fringe. From the phase shift of the fringe at each wavelength we finally obtained the time-resolved DPS shown in Fig. 9(a). The shift of the DPS peak is observed to go from blue to red and is due to the spectral chirp in the probe white continuum. If the imaginary part of the index change is not zero the intensities of the probe and reference become unbalanced and as a result the visibility of the interference fringe changes. The DTS can be individually determined from the spectro-interferograms as shown in Fig.9(b). They are calculated from the Fourier amplitude of the main spatial-frequency component. The time dependence of the phase change was fitted with a sum of two exponential functions A 1 exp(t/q1 )A 2 exp(t/q2 ). Fig. 10 shows the time dependence of the 388 T. Kobayashi Fig. 8 Interferograms at 590nm (a) without and (b) with excitation. The dashed curve at the bottom represents the same interferogram as in the upper figure. phase change at 590 nm. The two time constants of exponential decay were determined to be t 1 bsol;360^80 fs and t 2 bsol;1.7^0.6 ps with relative amplitude A 1 bsol;0.76 and A 2 bsol;0.24. These values are close to the time constants previously reported.1 This result shows that the time dependence of the phase change at particular wavelengths can be successfully measured.At disect;erent wavelengths (j) the time-zero is observed to shift owing to the chirp. By means of this principle the group delay in the white continuum pulses can be measured. Fig. 11 shows the tndash;j plot of the chirped continuum. The fitted curve represents a linear chirp induced by 22mm thick BK7. This value is reasonable when all the optical components are considered including CCl 4 and CS 2 . 4.3 Minimum detection limit of the phase change A high-resolutionCCD camera with 1000(horizontal)1018(vertical) pixels was used for the SI experiments. The wavelength resolution on the camera was ca. 389 Femtosecond phase spectroscopy Fig. 9 Time-resolved DPS (a) and DTS (b) of CS 2 measured at several time delays. The spectral shift with the time delay is due to the chirp of the white continuum.0.2nmpixel~1 but the resolution of the whole system was limited to ca. 1nm by the slit width of the monochromator. The spectral range on the camera was ca. 200nm (0.2nmpixel~1 1000 pixels) but the esect;ective area was limited to ca. 530ndash;650nm with a centre wavelength of 590nm because of the strong stray light at the edges of the image. The minimum detectable phase change is limited by the stability of the interferometer the spatial resolution of the CCD camera and error in determination of the sine function in the interferogram. In the present measurement the frame-to-frame fluctuation was estimated to be ca. 1/110 which causes an error in the determined phase shift. The sensitivity in the phase disect;erence of the whole system was evaluated from the standard deviation of the phase shift over the interferogram to be ca.1/69. Further improvement is expected by careful optimization of the spatial fringe. When the fringe spacing is larger the precision of the fringe phase which is determined by the spatial resolution of the camera becomes higher. On the other hand the error in determination of the phase becomes smaller when the fringe spacing is smaller consequently the number of fringe peaks is larger. By adjustment of the equal inclination the fringe spacing can be easily changed. In this way we consider that a resolution of ca. 1/200 will be attained. 390 T. Kobayashi Fig. 10 Time dependence of the phase change in CS 2 at 590 nm. The solid line represents the decay profile fitted with a sum of two exponential functions. The decay time constants of these components are 360^80 fs and 1.7^0.6 ps.Fig. 11 Peak wavelength of the chirped continuum plotted against delay time which was obtained from the DPS in Fig. 6 by plotting the time delays against the wavelengths at the peak of the DPS. The solid curve is fitted with the Selmeier equation of BK7. Recently we analysed numerically the experimental results of DPS taken with the use of the SI and succeeded in reproducing the main features both in instantaneous and inertial response spectra. The equation used for instantaneous response due to the coherent coupling esect;ect24,25 is given by */(E)PERe A 1 E13(E) FME16(t,q)B(t)Oslash;E*16(t,q)E13(t)N B (23) where,Oslash; and F mean convolution and Fourier transformation respectively and the fields are given by 391 Femtosecond phase spectroscopy Fig.12 (a) Experimental results of instantaneous response spectrum for CS 2 . (b) Results of the numerical simulation. E13(t)bsol;E(t){E 0 exp(t2/q2 0 iu0 t) (24) and E16(t,q)bsol;E(tq) (25) The response function is given by b(t)bsol;#(t)exp(t/q$) (26) 392 T. Kobayashi Fig. 13 (a) Coherent oscillations at negative delay times in TRDPS (solid curve) and fitting (dashed curve). (b) Observed and calculated oscillation period plotted against delay time. 393 Femtosecond phase spectroscopy The parameters used are q0 bsol;100/J4ln2 fs u0 bsol;2.40 fs~1 q$ bsol;1.6 ps. Thus the molecular response function due to orientational relaxation is determined to be q$ bsol;1.6 ps.26 The experimental and numerical results for the instantaneous response spectrum are shown in Fig. 12 from which we can verify that the coherent coupling esect;ect is dominant in the instantaneous response spectrum.In particular spectral fringes at negative delay times are also well reproduced and the period coincides fairly well with the experimental results (Fig. 13). The enhancement of such a spectral fringe at negative delay time can be explained by the overlap between b(t)Oslash;E*16(t,q)E13(t) and E16(t,q) in eqn. (23) that is the greater the overlap between these two terms at negative delay time then the greater the amplitude of the spectral fringe. 5 Conclusion Novel femtosecond interferometers the frequency-domain Michelson interferometer and the space-domain Sagnac interferometer were developed for a single-shot measurement of DPS over a whole spectral region of white light continuum.The present methodology interferometric spectroscopy provides full information of timeresolved complex non-linear optical susceptibility. These femtosecond interferometers are useful for single-shot measurements of the time-resolved disect;erence transmission and phase spectra. Simultaneous determination of the real and imaginary parts of the complex susceptibility using the two interferometers was successfully demonstrated. By using FDI we could for the first time show the break-down of the Kramersndash;Kronig relations in time-resolved spectroscopy and theoretically explain this by utilizing SI we observed for the first time spectral fringes in time-resolved DPS at negative delay times and they were numerically verified to be mainly caused by the coherent coupling esect;ect. Acknowledgments The author acknowledges experimental collaboration with Dr.E. Tokunaga on FDI and Mr. H. Kanou Dr. K. Misawa and Mr. A. Ueki on SI. This work is supported by a Grant-in-Aid for Specially Promoted Research from the Ministry of Education Science and Culture (No. 05102002). References 1 J.M. Halbout and C. L. Tang Appl. Phys. Lett. 1982 40 765. 2 N. Finlayson W. C. Banyai C. T. Seaton and G. I. Stegeman J. Opt. Soc. Am. B 1989 6 675. 3 D. Cotter C. N. Ironside B. J. Ainslie and H. P. Girdlestone Opt. Lett. 1989 14 317. 4 K. Minoshima M. Taiji and T. Kobayashi Opt. Lett. 1991 16 1683. 5 M. J. LaGasse K. K. Anderson H. A. Haus and J. G. Fujimoto Appl. Phys. Lett. 1989 54 2068. 6 E. Tokunaga A. Terasaki and T. Kobayashi Opt. Lett. 1992 17 1131. 7 For example L. D. Landau and E. M. Lifshitz Electrodynamics of Continuous Media Addison-Wesley Reading MA 1960; A.Yariv Quantum Electronics Wiley New York 3rd edn. 1988. 8 F. Bassani and S. Scandolo Phys. Rev. B 1991 44 8446 and references therein. 394 T. Kobayashi 9 E. Tokunaga A. Terasaki and T. Kobayashi Phys. Rev. A 1993 47 4581. 10 E. Tokunaga A. Terasaki and T. Kobayashi Opt. Lett. 1993 18 370. 11 J. Etchepare G. Grillon J. P. Chambaret G. Hamoniaux and A. Orszag Opt. Commun. 1987 63 329. 12 C. Kalpouzos D. McMorrow W. T. Lotshaw and G. A. Kenney-Wallace Chem. Phys. Lett. 1988 150 138; Comment Chem. Phys. Lett. 1989 155 240. 13 S. Ruhman and K. A. Nelson J. Chem. Phys. 1991 94 859. 14 T. Hattori and T. Kobayashi J. Chem. Phys. 1991 94 3332. 15 R. R. Alfano and P. P. Ho IEEE J. Quantum Electron. 1988 24 351.16 T. Hattori A. Terasaki T. Kobayashi T. Wada A. Yamada and H. Sasabe J. Chem. Phys. 1991 95 937. 17 C. H. Brito-Cruz J. P. Gordon P. C. Becker R. L. Fork and C. V. Shank IEEE J. Quantum Electron. 1988 24 261. 18 H. A. Haus IEEE J. Quantum Electron. 1975 11 736. 19 E. Tokunaga A. Terasaki T. Wada K. Tsunetomo Y. Osaka and T. Kobayashi Opt. Soc. Am. B 1993 10 2364. 20 Y. Li G. Eichmann and R. R. Alfano Appl. Opt. 1986 25 209. 21 R. Trebino and C. C. Hayden Opt. Lett. 1991 16 493. 22 M.C. Gabriel J. N. A. Whitaker C.W. Dirk M.G. Kuzyk and M. Thakur Opt. Lett. 1991 16 1334. 23 K. Misawa and T. Kobayashi Opt. Lett. 1995 20 1550. 24 L. Palfrey and T. F. Heinz J. Opt. Soc. Am. B 1985 2 674. 25 D. J. L. Oudar IEEE J. Quantum Electron. 1983 19 713. 26 D. McMorrow Opt. Commun. 1991 86 236. 395 Femtosecond phase spectroscopy