Photon kinetic theory of self-phase modulation

1 September 2001

Optics Communications 196 (2001) 285±291

www.elsevier.com/locate/optcom

Photon kinetic theory of self-phase modulation L.O. Silva *, J.T. Mendoncßa GoLP/Centro de Fõsica de Plasmas, Instituto Superior T ecnico, 1049-001 Lisboa, Portugal Received 28 March 2001; received in revised form 28 May 2001; accepted 14 June 2001

Abstract We derive the universal features of self-phase modulation (SPM) of short pulses of radiation, using a photon kinetic theory, equivalent to propagating the Wigner function of the electric ®eld in the short-wavelength limit. The signatures of SPM are discussed within this extended model. We interpret our results in terms of the quasi-classical dynamics of the photons and the concept of photon acceleration. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 42.65. k; 42.15. i Keywords: Photon kinetics; Self-phase modulation; Wigner function; Photon acceleration

Self-phase modulation (SPM) is one of the most important phenomena in Nonlinear Optics, playing a central role in several areas of fundamental and applied laser physics [1]. The SPM of a short laser pulse propagating in a nonlinear medium presents several distinct features: spectral broadening dependent on the laser pulse spatial shape, interference pattern in the phase-modulated spectrum, and pulse steepening (spatial shape distortion) [1±3]. For ultrashort pulses, with durations in the picosecond time scale or below, the spectral broadening can eventually cover the entire visible and near optical spectral range, leading to a supercontinuum of radiation. Furthermore, propagation of a laser pulse for long distances leads to a nonuniform supercontinuum, with regions of maximum

* Corresponding author. Tel.: +351-21-8419-336; fax: +35121-8464-455. E-mail address: [email protected] (L.O. Silva).

spectral energy density followed by zones of destructive interference, clearly showing a spectral interference pattern. Pulse steepening becomes important for longer propagation length scales. Until now, the theoretical description of SPM has been successfully based on the analysis of the evolution of the phase of the laser pulse propagating in a nonlinear medium. In this paper we will employ an alternative description for SPM based on a kinetic equation for a photon distribution in phase space, obtained from the Wigner distribution of the electric ®eld, which is formally equivalent to the collisionless Boltzmann equation, or the Vlasov equation, and it was ®rst developed by Tappert and co-workers [4±6]. The main advantage of this formalism is allowing to clearly separate the phase e€ects from the frequency shift e€ects. Furthermore, it bridges the gap between SPM and frequency upshift processes in plasmas [7,8], or photon acceleration, where the photon kinetic formalism has been applied with success, thus possibly leading to an uni®cation between the

0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 3 6 8 - 2

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several processes of frequency shift and spectral broadening in Plasma Physics and Nonlinear Optics. We stress that in both ®elds the spectral changes are due to the space and time variations in the index of refraction of the electromagnetic (e.m.) ®eld propagating in a nonlinear medium. In this paper, we show that photon kinetic theory [9] can also be used to describe all the features of SPM, whenever the phase of the electric ®eld does not play any role. The photon kinetic theory is based on the semi-classical description of the e.m. ®eld, which is replaced by a photon distribution in the phase space wave vector position. The dynamics of our quasi-particles, or photons, is determined by the ray equations for the photons [10,11]. Thus, a clear and intuitive analogy can be established between photon dynamics, and particle dynamics in the classical limit. Our theoretical approach not only predicts the spectral broadening, but also the chirp, and the asymmetry between the Stokes and anti-Stokes sidebands present in SPM for long propagation distances [12]. More important, since the photon kinetic theory neglects e.m. ®eld phase e€ects but retains the e€ects due to the instantaneous frequency distribution of the ®eld, we can easily point out the e€ects which are due to the phase of the ®eld and those associated with the frequency of the ®eld. This is, in fact, equivalent to comparing the quantum e€ects (associated with the wave function phase) with the classical e€ects (solely associated with the classical dynamical variables: energy, momentum and position). Our results also provide the natural theoretical description of SPM for incoherent [13±15] and random-phased e.m. pulses [16]. Therefore, we can clearly identify the universal signatures of SPM, present for any short pulse of e.m. radiation propagating in a nonlinear medium. Our starting point is the kinetic equation for photons [4±6,17]:   o dr o dk o ‡ ‡ N…r; k; t† ˆ 0 …1† ot dt or dt ok which states e.m. wave action conservation, or photon number conservation, in phase space …r; k†, valid in the short-wavelength high-frequency approximation [4±6,9]. N…r; k; t† describes the photon number distribution and it is related to the

2

e.m. energy density, jE…r; t†j =8p, and the spectral 2 energy density, jE…k; t†j2 =8p, by jE…r; R R t†j =8p ˆ 3 2 dk=…2p† hxk N and jE…k; t†j =8p ˆ dr hxk N, where hxk is the energy of a single photon, being determined from the local dispersion relation of the e.m. ®eld in the optical medium, D…x; k; r; t†  0. N has the properties of a Wigner function [4± 6,9,18], and can be determined from the electric ®eld using one of the several representations of the Wigner function [4±6,9,20,21]. 1 N is also equivalent to the semi-classical representation of the photons occupation number. By employing a kinetic description for the photons, we can not only calculate averaged physical quantities over the distribution function, but also include kinetic effects [22] which are not taken into account by the usual ¯uid-like radiative transport equations. Furthermore, it is also important to mention that a similar approach has been present in the plasma physics literature to address modulational instabilities of phonons, plasmons, and photons [23,24]. The total time operator in Eq. (1), is determined by the ray equations, written in canonical Hamiltonian form [4±6,9±11]: dr oxk ˆ ; dt ok

dk ˆ dt

oxk or

…2†

where the local frequency xk  x…r; k; t† plays the role of the Hamiltonian generating the photon equations of motion. In particular, for a dispersionless Kerr medium, if we retain the ®rst nonlinear correction to the index of refraction n, xk satis®es: kc kc ˆ xk ˆ …3† n n0 ‡ n2 I…r; t† where n0 is the linear index of refraction, n2 ˆ 16p2 =n0 v…3† is the ®rst nonlinear correction to the index of refraction, I…r; t† ˆ jE…r; t†j2 =8p is the e.m. ®eld intensity, and v…3† is the third-order susceptibility. We consider the e.m. ®eld intensity a slowly varying function, when compared with the typical e.m. ®eld fastest time scale (associated with x0 , 1 For the representation of the number of photons N for di€erent e.m. ®eld con®gurations (1D plane wave, beat-wave, e.m. ®eld with random phases), and a detailed discussion of the connection between the number of photons and the randomphase approximation, see Ref. [19].

L.O. Silva, J.T. Mendoncßa / Optics Communications 196 (2001) 285±291

the e.m. ®eld central frequency) and the e.m. ®eld shortest length scale (related to k0 , the e.m. ®eld central wave number) i.e. x0  jo ln I…r; t†=otj and k0  jr ln I…r; t†j. This means that our analysis is valid for pulse durations s satisfying s(fs)  3k0 …lm). We also neglect the dispersion of v…3† within the bandwidth of the ®eld. The formal solution of Eq. (1) is: N…r; k; t†  N…r0 …r; k; t†; k0 …r; k; t†; t0 †

…4†

where (r0 ; k0 ; t0 ) are the initial conditions, written as a function of the dynamical variables (r; k; t). Hence, by solving Eq. (2) with xk obeying Eq. (3), the time evolution of the photon number distribution N …r; k; t† is fully determined. In general, this can only be performed numerically. We now look for approximate solutions of the ray-tracing equations. Assuming weak nonlinearities, we expand the Hamiltonian (3) to ®rst order in the small parameter n2 I=n0  1. Furthermore, we consider one-dimensional propagation along the x-axis, and pulse steepening e€ects are neglected i.e. the laser pulse shape does not change signi®cantly along propagation. This means that I…x; t† ' I…x c=n0 t†. Introducing the new canonical variables …g ˆ x c=n0 t, p ˆ k† [10,11], the new Hamiltonian X generating the equations of motion of the photons in the phase space …g; p† is X…g; p† ˆ x

pc ˆ n0

pUI…g†

…5†

where we have introduced the parameter U ˆ n2 c=n20 , and x ' pc=n0 …1 n2 =n0 I…g††. X is now a constant of the motion along the photon trajectories (no explicit time dependence). We stress that all approximations must be performed over the Hamiltonian (3), in order to guarantee that in the phase space of the new variables the number of photons N…g; p† is still a conserved quantity, obeying a kinetic equation formally equivalent to Eq. (1). The ray equations are now given by: dg oX ˆ ˆ dt op

UI…g†;

dp ˆ dt

oX o ˆ pU I…g† og og …6†

The integration of these equations is straightforward:

Z

g

g0

dg0 ˆ I…g0 †

U…t

 p0 ˆ p exp

t0 †

Z U

287

t0

t

oI…g† dt og

…7†  …8†

clearly showing that g0 is independent of p, i.e. g0  g0 …g; t†. We ®rst examine the frequency chirp of the pulse, which characterizes how the local and instantaneous frequency x changes along the e.m. pulse, for a given time t. The chirp can be expressed as an average over the photon distribution function [9]: R N…g; p; t†x…g; p† dp R hxig;t ˆ …9† N …g; p; t† dp Using the formal solution (4) and the approximate expression for the frequency x…g; p†, the integral in the numerator can be written as:   Z pc n2 N…g0 ; p0 ; t0 † 1 I…g† dp …10† n0 n0 From Eq. (8), p dp is expressed in terms of p0 dp0 , transforming Eq. (10) into   Z p0 c n2 N…g0 ; p0 ; t0 † 1 I…g† n0 n0  Z  oI…g† dt dp0  exp 2U …11† og For a nondispersive e.m. pulse, and neglecting pulse steepening, I…g† ˆ I…g0 †. This approximation can be performed as long as T  Imax U…t t0 †=r, where Imax is the maximum intensity of the electric ®eld, and r is the typical pulse length. Thus, Eq. (9) reduces to:   oI…g† …t t0 † …12† hxig;t ˆ hxi0 exp 2U og Here, we have used the initial spectral chirp distribution hxi0  hxig0 ;t0 . The stationary points of the chirp hxig;t describe not only the maximum/ minimum values of the frequency hximax , but also correspond to the spectral regions where the photons are most likely to bunch, henceforth describing the frequencies for which the maximum

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spectral intensity is observed. The condition for the existence of an extremum is …o=og†hxig;t ˆ 0. From Eq. (12), we obtain

 o o2 I…g† oI…g† hxig;t ˆ 2Uhxi0 t …t exp 2U og og2 og

 t0 † …13†

which means that the stationary points gmax satisfy o2 I…g† ˆ0 …14† og2 gmax and hxigmax ;t  hximax . Our formalism is valid for any pulse shape. We illustrate our results with a 1D Gaussian pulse, such that

shifted, and at the rear the pulse is up-shifted. We also observe a clear asymmetry between the two sides of the spectrum; we will return to this feature below. The chirp extrema are calculated p from Eq. (14), and are located at gmax ˆ r= 2. Thus, hximax is given by " # p 2 2 UI0 e 1=2 …t t0 † hximax ˆ hxi0 exp  …17† r For small arguments, we can Taylor expand the exponential in Eq. (17), and using the explicit expression for U, we obtain: p 2 2 c n2 I0 e 1=2 Dt …18† Dx ' hxi0 r n0 n0

A typical chirp curve, as determined by Eq. (16), is depicted in Fig. 1. As expected, at the front of the e.m. pulse the frequency spectrum is down-

where Dx ˆ hximax hxi0 and Dt ˆ t t0 . This is in agreement with the standard results of the simpler theory for SPM [1±3] (this agreement is more obvious if the propagation time is replaced by the propagation length z, through t t0 ˆ …z z0 †n0 =c). We should notice for propagation times Dt much longer than rn20 =…n2 cI0 †, ( the inverse of the maximum/minimum frequency growth rate), the expansion leading to Eq. (18) is no longer valid, and Eq. (17) must be employed; the Stokes and the anti-Stokes sidebands asymmetry becomes evident. The asymmetry was already clear in Fig. 1, and it is explicitly represented in Fig. 2, where the

Fig. 1. Chirp C…g; T † ˆ …hxig;t hxi0 †=hxi0 of a Gaussian pulse with duration s, for T ˆ 0:25, with T ˆ tUI0 =r, assuming that for t0 ˆ 0 the pulse is transform limited (hxi0 ˆ const ˆ x0 ), showing how the local and instantaneous frequency varies along the pulse extent.

Fig. 2. Instantaneous frequency of the maximum Stokes (S < 0) and anti-Stokes (S > 0) shifts as a function of the dimensionless parameter T : (Ð) exact solution as determined from Eq. (17); (- - -) approximate solution valid for T  1, with t0 ˆ 0. S is de®ned as S…T † ˆ …hximax hxi0 †=hxi0 .

I…g† ˆ I0 e

g2 =r2

…15†

where I0 and r ˆ cs=n0 are constants, being s the pulse duration. Even though we are considering a Gaussian pulse, we note that no assumptions have been made regarding the pulse spectral content. The chirp temporal evolution is determined from Eq. (12) h i g 2 2 hxig;t ˆ hxi0 exp 4U 2 I0 e g =r …t t0 † …16† r

L.O. Silva, J.T. Mendoncßa / Optics Communications 196 (2001) 285±291

289

time evolutions of the instantaneous frequency of the Stokes and anti-Stokes maximum frequency shifts are plotted. This asymmetry was ®rst identi®ed by Yang and Shen [12]. In order to make connection with their work, we have determined the frequency modulation and the spectrum asymmetry using the same intensity pro®le as in Ref. [12] i.e. I…g† ˆ

I0 cosh g=r

…19†

which gives for the temporal evolution of the chirp:   oI…g† hxig;t ˆ hxi0 exp 2U …t t0 † …20† og For I0 Ujt mated by:

Fig. 3. Distribution of photons N in phase space …g; p† of a 1D transform-limited Guassian pulse for T ˆ 0 (left), and N for T ˆ 0:25 (right). The initial distribution N is determined from the Wigner function of the electric ®eld (see Refs. [4±6,9,19]).

t0 j=r  1, Eq. (20) can be approxi-

hxig;t ˆ hxi0 1

2I0 sinh…g=r† U …t r cosh2 …g=r†

! t0 †

…21†

which exactly reproduces Eq. (7) in Ref. [12], if we note that the parameter Q in Ref. [12] is represented in our variables as Q ˆ 2I0 U…t t0 †=r. For the maximum Stokes and anti-Stokes shifts we obtain from Eq. (12), and using the condition (14):   I0 hximax ˆ hxi0 exp  U …t t0 † …22† r which reduces, in the limit of Q  1, to the same result as in Ref. [12] (right after Eq. (8)), and demonstrates the same type of asymmetry towards the anti-Stokes shift. In order to check our results, we have determined numerically the complete evolution of the number of photons N, corresponding to a transform-limited Gaussian pulse, with the intensity described by Eq. (15), by solving Eqs. (2)±(4); the numerical results include the temporal evolution of the shape of the pulse. In Fig. 3 we present two snapshots of the complete photon distribution, where it is clear for T ˆ 0:25 the average behavior determined by Eq. (9), and plotted in Fig. 1. We also present in Fig. 4(a) the spectral intensity for the same time step. The asymmetry between the Stokes and anti-Stokes shifts is present once again. Also, we point out that the interfer-

Fig. 4. (a) Instantaneous spectral intensity, and (b) e.m. ®eld energy density, calculated from N, for T ˆ 0:25. The dashed lines represent the initial spectral intensity and energy density.

ence pattern usually observed in SPM by laser pulses is absent, since this e€ect is connected with the interference of di€erent ®eld components, and therefore only properly described by the phase of the ®eld. In Fig. 4(b), the spatial distribution of the

290

L.O. Silva, J.T. Mendoncßa / Optics Communications 196 (2001) 285±291

intensity is plotted along with the initial spatial distribution, showing pulse self-steepening, an effect discarded in our analytical estimates but which is described by the photon kinetic numerical calculations. The time dependence of the spectral intensity and the energy density is represented in Fig. 5, denoting the predicted evolution of the

Fig. 5. (a) Temporal evolution of the instantaneous spectral intensity (a.u.), clearly showing the dominant behavior of the anti-Stokes spectral components; (b) temporal evolution of the e.m. ®eld energy density (a.u.), where self-steepening is evident for T ' 0:1.

spectral sidebands, and showing that pulse shape distortion starts to be important for T ' 0:1. Our results allow us to immediately establish the universal features of SPM, which are independent of the phase structure of the e.m. ®eld and should be present for pulses of coherent, incoherent or random-phased radiation: spectral broadening, Stokes/anti-Stokes asymmetry, and self-steepening. These e€ects are a natural consequence of our theoretical approach, and, therefore, should be present whenever the rather relaxed approximations and assumptions considered here are met. Our description shows that the phenomena usually associated with SPM can also be seen as a particular example of photon acceleration in a nonstationary medium. In this case, the nonstationarity is produced by the pulse of radiation itself (or by the photon distribution associated with the pulse), which changes the index of refraction due to the nonlinear response of the medium. In terms of the photon acceleration mechanisms, the photons in the front of the pulse propagate in regions of negative index of refraction gradient, thus meaning that they are decelerated (k_ < 0 in Eq. (2), photons are frequency down-shifted), while in the trailing edge of the pulse, the gradient of n is positive, k_ > 0, and the photons are accelerated (frequency up-shifted). The velocity of the photons (_r) located near the pulse peak is also lower than the velocity of the photons in the leading and trailing edges, thus leading to a sharpened trailing edge i.e. self-steepening of the pulse [25]. In conclusion, SPM of a short pulse of radiation in an optical medium can adequately be predicted by a photon kinetic theory. This result is quite important conceptually, since the phase of the e.m. ®eld is not an ingredient of our formalism. We were able to identify the universal features of SPM, present for any short pulse of radiation propagating in a nonlinear medium: supercontinuum generation, Stokes±anti-Stokes asymmetry, and pulse steepening. In the photon kinetic theory, the phase of the ®eld is not taken into account, only the instantaneous frequency and spatial distribution of the ®eld. Therefore, short pulses of incoherent radiation or short laser pulses with a random distribution of phases propagating in a

L.O. Silva, J.T. Mendoncßa / Optics Communications 196 (2001) 285±291

nonlinear medium also undergo SPM. However, for these pulses the interference pattern observed in the SPM spectrum of short laser pulses should not appear. We stress that our formalism can also describe SPM of chirped pulses, and can be generalized in a straightforward way to consider twoor three-dimensional e€ects, such as self-focusing, or a ®nite response time of the nonlinearity. Our results demonstrate how powerful a photon kinetic theory can be, allowing for an almost complete description of di€erent e.m. ®eld propagation problems in the high-frequency long-wavelength approximation within an extremely intuitive framework, which is also numerically easy to implement within the ``particle-in-cell'' framework [26]. Furthermore, photon kinetic theory also allows for the study of modulational instabilities in Kerr media by incoherent radiation sources: this is particularly relevant in what concerns the pulse stability. This problem will be addressed in a future publication. The main features of SPM have been identi®ed and described by a photon kinetic theory, and these signatures should be observed for any short pulse of radiation, independently of its phase structure.

Acknowledgements One of the authors (LOS) acknowledges fruitful discussions with Profs. J.M. Dawson and W.B. Mori. This work was partially supported by FCT (Portugal).

291

References [1] R.R. Alfano (Ed.), The Supercontinuum Laser Source, Springer Verlag, New York, 1989, and references therein. [2] G.P. Agrawal, Nonlinear Fiber Optics, second ed., Academic Press, San Diego, 1995, pp. 89±128, Chapter 4 and references therein. [3] Y.R. Shen, The Principles of Nonlinear Optics, InterScience, New York, 1984, pp. 324±329, and references therein. [4] F.D. Tappert, SIAM Rev. (Chronicle) 13 (1971) 281. [5] I.M. Besieris, F.D. Tappert, J. Math. Phys. 14 (1973) 704. [6] I.M. Besieris, F.D. Tappert, 17 (1976) 734. [7] W.B. Mori (Ed.), IEEE Trans. Plasma Sci. 21 (1) (1993), special issue on Coherent Radiation using Plasmas, and references therein. [8] S.C. Wilks, J.M. Dawson, W.B. Mori, et al., Phys. Rev. Lett. 62 (1989) 2600. [9] L.O. Silva, J.T. Mendoncßa, Phys. Rev. E 57 (1998) 3423. [10] L.O. Silva, J.T. Mendoncßa, IEEE Trans. Plasma Sci. 24 (1996) 14. [11] J.T. Mendoncßa, L.O. Silva, Phys. Rev. E 49 (1994) 3520. [12] G. Yang, Y.R. Shen, Opt. Lett. 9 (1984) 510. [13] J.T. Manassah, Opt. Lett. 15 (1990) 329. [14] J.T. Manassah, Opt. Lett. 16 (1991) 1638. [15] B. Gross, J.T. Manassah, Opt. Lett. 16 (1991) 1835. [16] Y. Kato, K. Mima, Appl. Phys. B 29 (1982) 186. [17] J.T. Mendoncßa, N.L. Tsintsadze, Phys. Rev. E 62 (2000) 4276. [18] E. Wigner, Phys. Rev. 40 (1932) 749. [19] L.O. Silva, R. Bingham, J.M. Dawson, W.B. Mori, Phys. Rev. E 59 (1999) 2273. [20] J. Paye, IEEE J. Quant. Electron. 28 (1992) 2262. [21] M.J. Bastiaans, Opt. Commun. 25 (1978) 26. [22] R. Bingham, J.T. Mendoncßa, J.M. Dawson, Phys. Rev. Lett. 78 (1997) 247. [23] R.Z. Sagdeev, A.A. Galeev, Nonlinear Plasma Theory, W.A. Benjamin, New York, 1969. [24] P.K. Shukla, L. Sten¯o, Phys. Lett. A 237 (1998) 385. [25] F. DeMartini, C.H. Townes, T.F. Gustafson, P.L. Kelley, Phys. Rev. 164 (1967) 312. [26] L.O. Silva, et al., IEEE Trans. Plasma Sci. 28 (2000) 1202.