wave group decomposition of ultrashort light pulses

3
364 OPTICS LETTERS / Vol. 18, No. 5 / March 1, 1993 Wave group decomposition of ultrashort light pulses Marie May and J. F. Morhange Laboratoire de Physique des Solides associe au Centre National de la Recherche Scientifique, Universit6Pierreet Marie Curie, Bolte 79, 4 Place Jussieu, F-75252 Paris Cedex 05, France C. Hirlimann Institut de Physique et Chimie des Mat6riaux de Strasbourg, Groupe d'Optique Non-Lin6aire et d'Opto6lectronique, Universite Louis Pasteur, 5 rue de l'Universit6,F-67084, Strasbourg Cedex, France Received September 24, 1992 We present a new method for analyzing the propagation of short optical pulses through a dispersive medium. The idea is to decompose the pulse Fourier spectrum into an infinite set of elementary wave groups, each propagating with its own group velocity. Such a method may be generalized to the case of nonlinear self-phase- modulation and points out the asymmetry observed in the white-light continua generated in the femtosecond regime. The Fourier theorem is the most classical approach used to describe the propagation of electromagnetic signals through dispersive media introducing a phase shift CF(Do) for each Fourier component of the sig- nal. For long-lasting pulses, say more than 100 fs, having a slowly varying time envelope, the phase D(co) is reduced to the first three terms of its Taylor expansion in the vicinity of the central frequency coo. The resulting pulse propagates through the medium with the group velocity vg(wjo), and its envelope suf- fers an enlargement' 2 characterized by the second- order derivative of F(Do) calculated for c 0 . Today, light pulses with very short duration, less than 10 fs, are experimentally available. 3 For these pulses, the concept of central frequency is no more evident, and higher-order terms in the phase expansion are not negligible. In this situation, a numerical integration of the Fourier integral is necessary, which does not allow a straightforward physical understanding of the envelope propagation. To simplify the computa- tions, an approximation can be made 4 in which the Fourier integral is calculated for the group delays tg(co) = dD(co)/dwo corresponding to each of the fre- quencies of the Fourier sspectrum (stationary phase approximation). The phase term [cwtg(w) - 1F(Do)] is then extremal and reduced to the zeroth- and second- order terms of its Taylor expansion in the vicinity of co. Such an approximation implicitly leads to a time-frequency representation of the propagating pulse, and the physical meaning of the group delay associated to each frequency of the Fourier spectrum is not clear. Moreover, it cannot be used in the case of time-dependent refractive indices. Numerous bidimensional representations of acous- tic and electromagnetic signals have already been suggested. 5 Because of its various mathematical properties, the Wigner-Ville transformation is a way to analyze short optical pulses in both the time and frequency domains. 6 ' 7 Its physical interpretation is obscured by the advent of negative intensities. We present here a method, derived from the Gabor transformation, 8 that allows one properly to cut out wave groups from the pulse amplitude spectrum. 9 Contrary to the Fourier formalism, this approach leads to a time-frequency representation of light pulses, which gives helpful support to physical in- terpretation, and may be generalized to nonlinear media. Moreover, an experimental implementation may be anticipated. Starting with a Fourier-transform-limited signal s(t), characterized by its spectral amplitude S(w), we define the wave group centered at pulsation Q by aS'(Q,t) = 0(Qt)&Q, (1) with O(fl, t) = S(co)exp(iwt)exp[-r- (f - _ )2]dw. (2) The frequency width, fla = 2/ra, of the analyzing function exp[- Ta 2 (fQ _ - ) 2 /4] is chosen much smaller than the spectral width of S(co). It is important to notice that integrating Eq. (2) over fQ gives back the original signal s(t), J +X (3) so that a measure of the wave groups versus Ql would give the knowledge of the full signal, am- plitude, and phase. Incoherent wave groups have already been experimentally realized" 0 by using a white-light source and recording a series of interfero- grams obtained by filtering the white spectrum with a narrow bandpass filter (ra large) at various central wavelengths. The same technique could be used to characterize short light pulses completely without re- sorting to nonlinear photon mixing. Symmetrically, the wave group may also be expressed with a time integral by using 0146-9592/93/050364-03$5.00/0 © 1993 Optical Society of America

Upload: c

Post on 02-Oct-2016

217 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Wave group decomposition of ultrashort light pulses

364 OPTICS LETTERS / Vol. 18, No. 5 / March 1, 1993

Wave group decomposition of ultrashort light pulses

Marie May and J. F. MorhangeLaboratoire de Physique des Solides associe au Centre National de la Recherche Scientifique, Universit6 Pierre et Marie Curie,

Bolte 79, 4 Place Jussieu, F-75252 Paris Cedex 05, France

C. HirlimannInstitut de Physique et Chimie des Mat6riaux de Strasbourg, Groupe d'Optique Non-Lin6aire et d'Opto6lectronique,

Universite Louis Pasteur, 5 rue de l'Universit6, F-67084, Strasbourg Cedex, France

Received September 24, 1992We present a new method for analyzing the propagation of short optical pulses through a dispersive medium.The idea is to decompose the pulse Fourier spectrum into an infinite set of elementary wave groups, eachpropagating with its own group velocity. Such a method may be generalized to the case of nonlinear self-phase-modulation and points out the asymmetry observed in the white-light continua generated in the femtosecondregime.

The Fourier theorem is the most classical approachused to describe the propagation of electromagneticsignals through dispersive media introducing a phaseshift CF(Do) for each Fourier component of the sig-nal. For long-lasting pulses, say more than 100 fs,having a slowly varying time envelope, the phaseD(co) is reduced to the first three terms of its Taylor

expansion in the vicinity of the central frequency coo.The resulting pulse propagates through the mediumwith the group velocity vg(wjo), and its envelope suf-fers an enlargement' 2 characterized by the second-order derivative of F(Do) calculated for c0 . Today,light pulses with very short duration, less than 10 fs,are experimentally available.3 For these pulses, theconcept of central frequency is no more evident, andhigher-order terms in the phase expansion are notnegligible. In this situation, a numerical integrationof the Fourier integral is necessary, which does notallow a straightforward physical understanding ofthe envelope propagation. To simplify the computa-tions, an approximation can be made4 in which theFourier integral is calculated for the group delaystg(co) = dD(co)/dwo corresponding to each of the fre-quencies of the Fourier sspectrum (stationary phaseapproximation). The phase term [cwtg(w) - 1F(Do)] isthen extremal and reduced to the zeroth- and second-order terms of its Taylor expansion in the vicinityof co. Such an approximation implicitly leads to atime-frequency representation of the propagatingpulse, and the physical meaning of the group delayassociated to each frequency of the Fourier spectrumis not clear. Moreover, it cannot be used in the caseof time-dependent refractive indices.

Numerous bidimensional representations of acous-tic and electromagnetic signals have already beensuggested.5 Because of its various mathematicalproperties, the Wigner-Ville transformation is a wayto analyze short optical pulses in both the time andfrequency domains.6'7 Its physical interpretation isobscured by the advent of negative intensities.

We present here a method, derived from the Gabortransformation,8 that allows one properly to cut out

wave groups from the pulse amplitude spectrum.9Contrary to the Fourier formalism, this approachleads to a time-frequency representation of lightpulses, which gives helpful support to physical in-terpretation, and may be generalized to nonlinearmedia. Moreover, an experimental implementationmay be anticipated.

Starting with a Fourier-transform-limited signals(t), characterized by its spectral amplitude S(w), wedefine the wave group centered at pulsation Q by

aS'(Q,t) = 0(Qt)&Q, (1)

with

O(fl, t) = S(co)exp(iwt)exp[-r- (f -_ )2]dw.

(2)

The frequency width, fla = 2/ra, of the analyzingfunction exp[- Ta2 (fQ _ - )2 /4] is chosen much smallerthan the spectral width of S(co). It is important tonotice that integrating Eq. (2) over fQ gives back theoriginal signal s(t),

J +X(3)

so that a measure of the wave groups versus Qlwould give the knowledge of the full signal, am-plitude, and phase. Incoherent wave groups havealready been experimentally realized"0 by using awhite-light source and recording a series of interfero-grams obtained by filtering the white spectrum witha narrow bandpass filter (ra large) at various centralwavelengths. The same technique could be used tocharacterize short light pulses completely without re-sorting to nonlinear photon mixing. Symmetrically,the wave group may also be expressed with a timeintegral by using

0146-9592/93/050364-03$5.00/0 © 1993 Optical Society of America

Page 2: Wave group decomposition of ultrashort light pulses

March 1, 1993 / Vol. 18, No. 5 / OPTICS LETTERS 365

properties, we may rewrite Eq. (7) asO(fQ, t) = f s(o)exp[- (t 20) exp[if(t - 0)]dO

O1 n(QI, t) = z jP [P ( lP fF2 P+1(0j - coo)(4) pao P

The signal propagates in the positive z directionof a dispersive and transparent medium, which fillsthe half-space z > 0, and its nonlinear index n2 is, atfirst, neglected. After propagation, the wave group3Sw(t, Q,z) may be written as

3S",(fl,t,z) = 3nflfS((w)

x exp{i[cot - CD (co)]}exp[- 2 (fQ - c)2] d, (5)

where the dephasing term is given by (D(c)) =con(fo)z/C. Ta is large enough to ensure that theanalyzing function has only nonnegligible values overa spectral range lying in the neighborhood of QI.Therefore, the Taylor expansion 4F(Do) in the vicinityof Ql only needs to be performed up to first order.Replacing the expansion in Eq. (5) yields

15S (fl,t,z) = a8f exp{i[ftgn - ID(fl)]}O(fl,t - tfl),(6)

where tgn is the group delay of the fQ wave group,given by tg = (dFD/dw)an. The comparison betweenEqs. (6) and (1) obviously shows that, except for aconstant phase factor, the fQ wave group propagatesthrough the dispersive medium without distortion.It only undergoes a time delay as a result of thegroup-velocity dispersion. The propagation of a shortlight pulse in a linear dispersive medium can be seenas the propagation of a continuous set of transform-limited wave groups, each traveling at its own groupvelocity.

Let us assume now that the intensity I(t) of thepulse is high enough to induce a time variation of therefractive index. If we neglect the linear dispersionof the medium, the phase shift suffered by the pulseat time t after a propagation z is usually"l written asa(t) = (co/c)n2zI(t), where w0 is the central frequencyof the pulse and n2 is assumed to be frequency inde-pendent. The central frequency, which is no longera valid concept for the whole ultrashort pulse, keepsits physical meaning for the fQ wave group. Aftera propagation through a length z, the amplitudecharacterizing the fQ wave group may be written,following Eq. (4), as

Onl(, t) = f s(0)exp[ (t _)2]

x exp[ifl(t - 0)] exp[-in czn 2Is(0)I2]do. (7)

The signal s(0) is given by Af(0)exp(icwo0), where f(0)

is assumed to be real, positive, and normalized andwhere A is the maximum amplitude of the pulse.Using the Taylor expansion and the Fourier inversion

X exp(icot)exp[-j42(fQ - .)2]dw, (8)

with

Y(fl) =- n2(A)2zC (9)

and where F2p+1(w - coo) is the Fourier transformof [f(o)]SP+P exp(iwo0). The linear dispersion is thentaken into account by applying Eq. (6) to the nonlin-ear wave group.

As an example, we describe the propagation of aGaussian pulse of central pulsation cwo and durationr: s(t) = exp(-t2 /T2)exp(iwoot). The amplitude of theincident fl wave group is given from Eq. (5) by

= exp(-2 t2A~wQ, , 0 =exp(-2 + ra)

x exp[- 4(T2 + 2) (f - coo)2]

x exp[i( T2 + Ta2 )t|afI(10)

This wave group is characterized by a Gaussian enve-lope, the temporal width of which is close to 2 ra sincethis decomposition is only valid for those values of rathat are much larger than T. For similar reasons,its carrier frequency is nearly equal to fQ, coo0T

2/Ta

2

being much smaller than fQ. In this case (Fourier-transform-limited signal), the envelopes of all thewave groups peak at t = 0; their respective amplitudemaxima M are fQ dependent and can be consideredindependent of 7a:

M(fQ, 0) = exp[- 4(_2+a2) (Q - 0o)2]

exp[-e (fl -( o) (11)

If we neglect r2 with respect to ra2 , the amplitude3Sw(jf, t, z) of the fQ wave group at any point z of themedium may be written using Eqs. (6) and (8) as

OsOut = M exp[- (t t)] exp{i[flt - (D)]}

(), p Q exp[- 4(2p + 1) 1 (12)

This equation clearly shows the combined action ofthe linear and nonlinear responses of the medium.On the one hand, the maximum M(fQ,z), given by

M(Q,z) = Y ( ) [(wi)]P exp[- T((2 °)2]'

(13)

occurs at time tga, which varies from one wave groupto another and characterizes the action of the linear

Page 3: Wave group decomposition of ultrashort light pulses

366 OPTICS LETTERS / Vol. 18, No. 5 / March 1, 1993

(Al,

Fig. 1. Amplitude of the maxima of the wave groups,seen at a particular point z of the quartz as a functionof the time and the frequency: (a) z = 0, (b) z = 2 cm.

M(Q2)

tFig. 2. Amplitude of the wave groups' maxima for a 3-fspulse calculated for a phase shift at the peak of the pulse.z = 4 cm, T(wo) = v.

dispersion. On the other hand, the value of M(fQ, z)is strongly modified with respect to M(fQ, 0) and re-veals significant frequency broadening as a result ofself-phase-modulation. The duration of each Q wavegroup remains unchanged despite the total frequencyenlargement of the pulse.

In the limit where Ta is large compared with theemerging pulse duration that is governed by thegroup delay dispersion, the first exponential term inEq. (12) is a slowly varying envelope that can be con-sidered as a constant close to 1. Therefore, even ifthe wave group decomposition is not mathematicallyunique, the signal obtained by integrating Eq. (12)versus fI is adequately described by the envelope ofthe maxima M(fQ, z), which is Ta independent and canbe experimentally measured.

Numerical calculations have been performed thatconsider an ultrashort pulse (r = 3 fs) centered at25 000 cm-' propagating through a quartz slab. Thevariations of the ordinary refractive index versuswavelength are taken from literature,"2 and the non-linear index n2 is assumed to be that of the fusedsilica, i.e., 1 0 -13 esu.13,14 First, it has been verifiedthat M(f, 0) and therefore M(f, z) is independentof ra as soon as Ta is larger than 2 0T. To describethe propagation of the pulse, we consider only thepropagation of the maximum of each wave groupin a three-dimensional representation: time [tg(fQ)],frequency (fI), amplitude M(fQ).

The incident pulse is shown on Fig. 1(a), and, asexpected, all the fI wave group maxima simultane-ously occur at time t = 0. The effect of the singlelinear dispersion is shown in Fig. 1(b). After propa-gation of the pulse through 2 cm of quartz, the de-lay between low- and high-frequency wave groups isstrongly evident. This delay variation between themaxima of each wave group generates the wideningand the distortion of the pulse propagating through

a linear medium. The temporal deformation experi-enced by the pulse can be estimated by

(At), = tg[coo + 2 J] - tg [co - 2 ( I] (14)

where (AfI)e is the l/e maximum width measured onthe curve. For z = 0.5 cm we obtain (At), = 700 fs,whereas for z = 2 cm, (At), = 3000 fs.

Let us consider now the self-phase-modulation ef-fect. After propagation of the pulse through a quartzlength corresponding to a phase shift T(coo) = wr (themaximum intensity of the pulse is 5 TW/cm2), onecan observe the frequency oscillations characteristicof the self-phase-modulation process (Fig. 2). Thecurve is strongly asymmetric on its high-energy sideowing to the increase of the linear dispersion withfrequency. This has to be compared with the experi-mental fact that, in the femtosecond regime, white-light continua are always observed to be asymmetricon the high-energy side.i5

We have studied the propagation of an ultrashortlight pulse through a nonlinear dispersive materialby decomposing this pulse into an infinite set of long-lasting Fourier-transform-limited wave groups. Anexperiment is being set up that will make use of thewave group concept to retrieve the amplitude of thelight pulse. By converting the time-frequency rep-resentation into a space-wave vector representation,such a decomposition can be directly applied to thetreatment of Fresnel diffraction and self-focalization.

References

1. E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454(1969).

2. C. G. B. Garrett and D. E. McCumber, Phys. Rev.A 1, 305 (1970).

3. R. L. Fork, C. H. Brito-Cruz, P. C. Becker, andC. V. Shank, Opt. Lett. 12, 483 (1987).

4. J. D. Jackson, Classical Electrodynamics (Wiley, NewYork, 1975).

5. J. M. Combes, A. Grossmann, and Ph. Tchamitchian,Wavelets, Time-Frequency Methods and Phase Space(Springer-Verlag, Berlin, 1987).

6. C. Hirlimann and J.-F. Morhange, Appl. Opt. 31, 3263(1992).

7. J. Paye, in Digest of Ultrafast Phenomena VIII (EcoleNationale Supdrieure des Techniques Avancdes, Paris,1992), p. 112.

8. D. Gabor, J. Inst. Electr. Eng. 93, 429 (1946).9. C. Hirlimann, M. May, and J-F. Morhange, Ann. Phys.

Colloq. 2 (Suppl. 16), 181 (1991).10. W. H. Knox, N. M. Pearson, K. D. Li, and

C. A. Hirlimann, Opt. Lett. 13, 574 (1988).11. S. L. Shapiro, ed., Ultrashort Light Pulses, Vol. 18

of Topics in Applied Physics (Springer-Verlag, Berlin,1977), p. 156.

12. E. Washburn, ed., International Critical Tables ofNumerical Data Physics, Chemistry and Technology(McGraw-Hill, New York, 1929), Vol. VI, pp. 341-344.

13. L. F. Mollenenauer, R. H. Stolen, and J. P. Gordon,Phys. Rev. Lett. 45, 1095 (1980).

14. H. Nakatsuka, D. Grischkowsky, and A. C. Ballant,Phys. Rev. Lett. 47, 910 (1981).

15. R. L. Fork, C. V. Shank, C. Hirlimann, R. T. Yen, andW. J. Tomlinson, Opt. Lett. 8, 1 (1983).