In order to describe the propagation of the optical laser pulses in semiconductors, we

Transcription

1 In order to describe the propagation of the optical laser pulses in semiconductors, we begin with the wave equation in its most general form. Taking into account both the laser field-induced polarization P (r,t), and the laser-driven current j(r,t), the Maxwell s macroscopic equations can be combined in the form E + 1 c 2 2 E t 2 = 4π c 2 2 P t 2 4π c 2 j t, (1) where E is the transverse electric field ( E(r,t)= 0). In dielectrics, the response (linear and nonlinear)of the medium described by the polarization P (r,t)arises from the bound charges alone while in semiconductors, both the bound charges and free carriers contribute to P (r,t)[1,2]. The conduction current j(r,t)is, of course, entirely due to free carriers. It is in the existence of the conduction current that the semiconductor optics differs essentially from the optics of purely dielectric media where j(r,t)= 0 rigorously. It is accepted that there are two conditions under which the free carrier contribution to the optical properties of semiconductors can be significant: it happens when the material is either heavily doped, or is found to be in a highly excited state. The latter situation is created, for example, when intense optical radiation is used to pump carriers from the valence to the conduction band [1-3]; large densities of excess free carriers can be generated via multiphoton, particularly, the two-photon absorption. However, even in these cases the current j(r,t)in Eq. (1)is always neglected in semiconductor optics: it is assumed to play a minor role in determining the optical properties of semiconductors [1-4]. In this paper, we make a fundamental departure from the conventional approach and reformulate the equations of semiconductor optics so that they remain valid even for those systems in which the induced conduction current is important, in fact, dominant. A laser pulse, for example, not only generates excess free carriers, it also accelerate them, thus, inducing a conduction current. Under certain conditions, this current can be the dominant nonlinear mechanism changing the refractive and absorptive properties of a semiconductor. It has been clearly demonstrated in [5], that a sufficiently intense (I 10 9 W/cm 2 ), short 1

2 (τ L 100 fs)laser pulse tuned near a two-photon resonance in an InSb semiconductor waveguide first generates large densities of excess free carriers (via two-photon absorption), and then accelerates these carries creating a large nonlinear rapidly varying current. This current causes fast and dramatic changes in the optical properties of the medium strongly affecting the pulse dynamics [5]. Our main objective, here, is to construct a wave equation governing the propagation of short, intense laser pulses in a semiconductor medium when the optical frequency of a given pulse falls in the vicinity of a two-photon resonance, particularly, a two-photon absorption resonance. Naturally we will be dealing with third-order intensity dependent dynamic (resonant) optical nonlinearities. The polarization P (r,t)consists of two parts, P = P L + P NL, (2) where P L is the linear polarization induced by the optical field of the laser pulse, and P NL accounts for two-photon-resonant third-order nonlinear polarization. In spite of its appearance, Eq. (2), is not the beginning of some kind of a perturbation theory. With resonance enhancement a particular nonlinearity can become so strong that the perturbative approach is like ly to break down. For instance the Two photon resonance significantly enhances the desired nonlinearity (two-photon absorption), while at the same time the competing linear absorption may be minimized by choosing the optical frequency to be very different from the one-photon resonance frequency [1,3,6]. Thus the peaking of the linear and the nonlinear processes is well separated in the frequency domain (see figure at the end of this Letter), and for the appropriate range, the nonlinear absorption could be dominant. The most general possible linear relation between P L and E in the electric-dipole approximation is given by [3,7] P L (r,t)= χ (1) (t t )E(r,t )dt, (3) where χ (1) (t)is the linear polarization response function of the medium. The chosen time dependence of χ (1) (t)reflects the principle of time invariance, and a causal response is 2

3 ensured by demanding χ (1) (t<0)= 0. Since E and P L are both real, the response function must be also real. Generally χ (1) (t)is a second-rank tensor. Using Eqs. (2) (3) we re-write the wave equation (1)in the form E 1 c 2 2 t 2 ε(t t )E(r,t )dt = 4π c 2 2 P NL t 2 + 4π c 2 j t, (4) where we have introduced the dielectric response function, ε(t t )=δ(t t )+4πχ (1) (t t ), (5) which has a Fourier transform (dielectric permittivity) ε(ω) =1+4πχ (1) (ω). (6) The complex-valued function χ (1) (ω) the Fourier transform of the linear response function is the dielectric susceptibility; it determines the linear refractive and absorptive properties of the medium. The imaginary part of χ (1) (ω)describes the linear absorption and arises from the one-photon absorption processes. It vanishes unless the laser frequency is resonant with a transition frequency of the medium. That is the case we will be mostly concerned with. The medium is linearly transparent for a laser light initially tuned near a two-photon resonance. The linear dispersive and absorptive properties are contained in the integral functional in the wave equation (4). An explicit expression for the polarization response function, χ (1) (t) (in the time domain)or the susceptibility, χ (1) (ω)(in the frequency domain)is needed for its evaluation. We shall return to this problem later. Let us now discuss in some detail the processes which give rise to the nonlinear polarization P NL. In semiconductors, the two-photon absorption (TPA)is a nonlinear process in which interband transitions are completed by the absorption of pairs of photons. When the energy, hω, of the incident laser photon lies in the range E g /2 < hω < E g and is well tuned near the two-photon resonance, the laser pulse, propagating through the semiconductor, places 3

4 a large number of free carriers in the conduction band via TPA. In this case the flow of energy from the laser pulse is used in causing real energy transitions of the medium by an absorption process. This is the reason why TPA is classed as an absorptive nonlinearity. The absorptive part of the TPA induced optical nonlinearity is naturally related to the imaginary part of the third-order susceptibility, χ (3), which determines the third-order induced polarization P (3) [2,8]. Since the real and imaginary parts of the susceptibility, governing refraction and absorption, respectively, are linked by the Kramers-Krönig relations [8], the TPA nonlinearity must have a refractive part as well. In order to gain better insight into the physical processes described above, let us discuss in detail the light propagation when it is tuned near the TPA resonance. We will do it in terms of the conventional approach invoking relevant conventional notions and images. In the two-photon interaction, the first photon causes the transition of an electron to a virtual intermediate state. This immediately induces an effective change in the absorption given by α = α 2 I. The virtually excited state is then converted by a second photon to the final exciton state where the electron appears in the conduction band leaving a corresponding hole in the valence band. This completes the two-photon absorption process. The phenomenological equation [1,2], di dz = αi α 2I 2, (7) clearly reflects the spirit of the above-mentioned microscopic picture of the attenuation of light due to the linear and two-photon absorption as it propagates through a semiconductor. Here α = 2(ω/c)I 1+4πχ (1) (ω)is the linear absorption coefficient, α 2 is the TPA coefficient, and I is the light intensity after it has traversed a distance z inside the semiconductor. In the conventional language one would say that two photon-like polaritons merge to form one exciton-like polariton [1]. During their brief existence in the virtual intermediate state, the electrons form an effective background of an excited population which, remaining coherent with the light 4

5 field, induce intensity dependent changes in the linear refractive index (the optical Kerr effect)[1,2,7] n = n 0 + n 2 I, (8) where n 0 = R 1+4πχ (1) (ω)is the linear refractive index, and n 2 describes nonlinear refraction. Obviously n 2 R[χ (3) ]. When the light is sufficiently intense, the density of the generated electron-hole pairs reaches very high values. The excitons lose their identity and we can no longer treat them as individual quasiparticles. Actually a new collective phase, the excess carrier plasma [1,2,9], is formed. If the light pulse is long (picoseconds or longer), the coherence between the light and the optically generated free carriers is rapidly lost because of scattering [1,2]. In this paper, however, we are concerned with coherent resonant nonlinear processes which take place in the interaction of ultrashort (τ L < 100fs), intense (I 10 9 W/cm 2 )laser pulses with semiconductors. The characteristic time scales of these processes are less than the fastest relaxation times of excess carriers in semiconductors [1,2]. We begin with the most general form of P NL (r,t)describing the third-order nonlinearity [3,7,10], P NL = dt dt dt χ (3) (t t,t t,t t )E(r,t )E(r,t )E(r,t ), (9) where χ (3) is the third-order polarization response function, generally, a fourth-rank tensor. Its Fourier transform χ (3) (ω 1,ω 2,ω 3 )= dt dt dt χ (3) (t,t,t )e iω 1t +iω 2 t +iω 3 t (10) is the third-order susceptibility of the medium. There is no universal or general expression for χ (3). Its explicit form depends on the nonlinear process being investigated. However, the form of χ (3) (ω 1,ω 2,ω 3 )is constrained by causality, and the reality of the corresponding third-order response function χ (3) (t,t,t ). Causality implies that χ (3) (ω 1,ω 2,ω 3 )is analytic when all the frequencies lie in the upper 5

6 half-plane. Because of the reality condition we have [χ (3) (ω 1,ω 2,ω 3 )] = χ (3) ( ω 1, ω 2, ω 3 ), where the symbol * denotes complex conjugation. Equations (9)and (10)also suggest that, as a fourth-rank tensor, χ (3) has the intrinsic permutation symmetry, i.e., χ (3) µσ 1 σ 2 σ 3 (ω 1,ω 2,ω 3 )is invariant under 3! permutations of 3 pairs (σ j,ω j ),j=1, 2, 3. In semiconductors, there are various intensity-dependent nonlinearities such as the fourwave mixing (FWM), hyper Raman scattering (HRS), coherent anti-stokes Raman scattering (CARS), two-photon absorption (TPA), etc. All of them can be described as χ (3) effects [1-4]. Which subset of these processes is relevant depends on a variety of factors: the characteristics of the material as well as of the laser pulse such as intensity, width, frequency content, etc. All these have to be translated into a properly constructed ( tuned ) wave equation with properly chosen initial and boundary conditions. It is obvious that, in addition to satisfying the general requirements, an effective χ (3) describing the TPA must account for: (i)the laser pulse attenuation due to real energy transitions in the medium that result in generation of a real population of free charge carriers, (ii)the corresponding nonlinear intensity-dependent change in the refractive index, (iii)the resonant character of TPA. The expression for P NL that seems to best meet all of the above requirements may be written as, P NL = t (3) χ eff E 2 (r,t ) E(r,t )dt, (11) with χ (3) eff = c2 n π 2 α 2eff, (12) where n 0 is the background index of refraction, and α 2eff is given by 6

7 α2 (ω) E(r,ω) 2 dω α 2eff =. (13) E(r,ω) 2 dω Here E(r,ω)is the time Fourier transform of the laser field E(r,t)and the denominator in Eq. (18)defines the total power of the pulse. By Parseval s theorem (E(r,t)is real), where E(r,ω) 2 dω = E 2 (r,t)dt. (14) The quantity in the angle brackets defines the pulse intensity I through the relation, I = cn 0 4π E 2 (r,t) = cn t+τ0 /2 0 E 2 (r,t )dt, (15) 8πτ 0 t τ 0 /2 2π τ 0 = ω E(r,ω) 2 dω E(r,ω) 2 dω. (16) Taking into account all these, we write the wave equation in the form: E 1 c 2 2 t 2 ε(t t )E(r,t )dt = 4π c 2 t ( (3) χ eff E 2 (r,t) E(r,t) ) + 4π j c 2 t, (17) where the first term on the right-hand reflects the nonlinear polarization given by Eq. (11). This term describes the pulse attenuation due to TPA as well as contributes to the refractive index as an intensity-dependent correction to χ (1). This will become more clear when we apply Eq. (17)to the propagation of a quasi monochromatic wavepacket and, using slowly-varying-envelope approximation (SVEA), derive a modified nonlinear Schrödinger equation for the envelope of the wavepacket and compare it to the existing model equations [11,12]. This term also takes into account the dispersion of the TPA coefficient as well as the finite bandwidths of the frequency spectra of both the laser pulse and the TPA. The frequency dependence of the TPA coefficient α 2 is given by [6] α 2 (ω) = C ) ( hω (2x 1)3/2 F n 2 0Eg 3 2, F 2 (x) =, (18) E g (2x) 5 where C is the material-independent constant [6,8]. This formula has been proved to be in a fairly good agreement with experimental data obtained for a wide class of direct-gap semiconductors [8,13]. According to Eq. (16) α 2 peaks at hω/e g =5/7. 7

8 The mathematical structure of P NL reflects the resonant character of TPA process: it is obvious from Eqs. (11) (13) that the TPA induced nonlinear term reaches its maximum when the peak of the laser pulse spectrum coincides with the peak of TPA in the frequency domain. At the same time this structure ensures the TPA nonlinearity to be automatically switched off as soon as the pulse is detuned from the TPA resonance. So far we have concentrated on the contributions of the linear and the nonlinear polarizations to the wave equation. However, after generating the excess free carriers via TPA, the powerful pulse is likely to accelerate them setting up a strong, nonlinear, rapidly changing conduction current. How this current is driven and what role does it play in the nonlinear dynamics of the laser pulse is discussed in some detail in Ref. [5]. We believe that this current is primarily created by the free electrons in the conduction band which are accelerated in the field of a laser pulse. But for completeness, we include in this current the possible contribution of the holes-quasi particles-vacancies in the valence band. The explicit form of the current is j(r,t)= e 2 s=e,h m s t dt N s(r,t ) t t t dt E(r,t ). (19) It is natural to assume that, for intrinsic (i.e., undoped)semiconductors, N e = N h N. For doped systems with excess electrons, the hole current can be simply neglected. So we have the generic expression j(r,t)= e2 t dt N(r,t ) t dt E(r,t ), (20) m r t t where m r is the so-called reduced mass of carriers, 1 m r = 1 m e + 1. (21) m h From Eq. (19)it follows that j(r,t) t = e2 N(r,t)E(r,t). (22) m r The modified rate equation determining the density of the newly born excess carriers due to the one- and two-photon absorption processes is 8

9 ( ) ( ) N α t = α2 I + I 2, (23) hω eff 2 hω eff where (...) eff == (...) E(r,ω) 2 dω E(r,ω) 2 dω, (24) and α and α 2 are the one- and two-photon absorption coefficients, respectively. In this form, Eq. (23)naturally ensures the switching on and off of the mechanism of carrier photogeneration depending on the resonant conditions. Note that for the quasi-monochromatic wavepackets or wavetrains this equation reduces to the corresponding conventional expression [1,2]. We are now in a position to explicitly write down (in the wave equation)the two principal pieces of the nonlinear response induced by a propagating short, intense laser pulse tuned near a two-photon resonance. Assuming that the two-photon absorption is the dominant nonlinear process, these two pieces consist of: the nonlinear current of the free charge carriers driven by the pulse in the excess carrier plasma (also created through the resonant TPA process)and the linear and nonlinear refractive and absorption effects arising from the pulse induced polarization. The wave equation becomes E 1 c 2 2 t 2 ε(t t )E(r,t )dt = 4π c 2 t ( (3) χ eff E 2 (r,t) E(r,t) ) + 4πe2 N(r,t)E(r,t), c 2 m r (25) where we have still to provide an appropriate expression for the linear part of the pulse induced polarization represented by ε(t)or χ(t). The linear response is generally modelled phenomenologically by treating the system as a collection of bound, forced oscillators with eigenfrequencies, also called resonance frequencies [1,7,10]. A semiconductor, like any other bulk matter, has many degrees of freedom. Consequently there are many resonance frequencies corresponding, for example, to phonons, excitons, etc. It is impossible to handle them all together. Fortunately, it is not necessary to do so. The frequency content of a light pulse is narrowly localized around some frequency 9

10 and If this frequency is sufficiently close to one of the medium resonances, the corresponding degree of freedom will resonate with light, and will be the only which will affect the pulse propagation through the semiconductor. This behavior is reflected in the following formula for the complex dielectric permittivity, ) f ε(ω) =ε b (1+, (26) ω0 2 ω 2 iωδ where ω 0 is the resonance frequency, δ is the damping rate, f is the oscillator strength, and ε b ε( ), the so-called background (or optical) dielectric constant, summarizes contributions of all higher resonances, ω 0j ω 0, j =1, 2,.... The factor f can be expressed as f = ω 2 L ω 2 T, (27) where ω T = ω 0. The frequency ω L characterizes the so-called longitudinal mode which is related to ω T through the Lyddane-Sachs-Teller (LST)relation [1], ε(0) ε( ) = ω2 L. (28) ωt 2 If we go back to the Lorentz (or Lorentz-Sellemeyer)model [1,14] of oscillating bound charges then we can show that in the electric-dipole approximation (local in space but not local in time)we derive the following differential equation governing the macroscopic linear polarization P L (r,t)of the medium subject to the optical field E(r,t)of the laser pulse, 2 P L t 2 + δ P L t + ω 2 0P L = ω 2 0χ 0 E(r,t), (29) where χ 0 is related to the oscillator strength, and the optical dielectric constant by the formula χ 0 = ε( ) f 4π which, with the LST relation (Eq. (28)), becomes ω 2 0, (30) ε(0) ε( ) χ 0 =. (31) 4π Typical values for resonance frequencies and the dielectric constants for GaAs are given in Table I [15]. 10

11 TABLE I. Near the lattice (vibration)frequencies at 300K (GaAs) ω T (experimental) ω L (LSD relation)static dielectric Optical dielectric constant ε(0)constant ε( ) s s Near the exciton resonances at low temperature (GaAs) ω T (experimental) ω L (LSD relation)static dielectric Optical dielectric constant ε(0)constant ε( ) s s Summary of the problem and an Algorithm for the Solution The set of equations E E c 2 t 2 t 2 4π 2 P L c 2 t 2 = 4π c 2 t ( (3) χ eff E 2 (r,t) E(r,t) ) + 4πe2 N(r,t)E(r,t), (32) c 2 m r 2 P L t 2 + δ P L t + ω 2 0P L = ω 2 0χ 0 E(r,t), (33) 11

12 ( ) ( ) N α t = α2 I + I 2, (34) hω eff 2 hω eff together with definitions (12), (13), (15) and (24) gives a full description of the propagation of a ultrashort intense laser pulse through a semiconductor when the pulse is tuned near the TPA resonance. It should be noted that in this model we have a built-in capability for taking care of the interplay between the two- and one photon absorption processes; this feature is crucial for a proper treatment of the problem when the nonlinear current of the free carriers induces large frequency upshifts in the incident pulse [5]. We now suggest a possible algorithm (schematically)for solving the set of equations (33) (35). Initially we specify the field of an incident laser pulse at some interface, say at z=0, which marks the boundary of the semiconductor. At the same time we give the spatial distribution of the optical field of the laser pulse which is, initially, zero inside the sample. Applying these initial and boundary conditions we solve Eqs. (34-35)to determine N and P L. Then we propagate the pulse inside the semiconductor solving the wave equation (32)with known N and P L. For calculating E for subsequent times we return to equations (34) (35) and begin a new cycle of calculations. This is very schematic description of the algorithm. Now we are working on the details of this scheme. At the same time we try to introduce a physically meaningful model for describing χ 3 eff completely in the time domain. Frequency dependence of the one- and two-photon absorption coefficients, curves 1 and 2, respectively. The quantities normalized to their maximum values are shown. The functional forms are: for the One-photon absorption (OPA)coefficient, α(ω) = C 1E g n 0 F 1 ) ( hω, F 1 (x) = E g (x 1)1/2 x (35) and the Two-photon absorption (TPA)coefficient, α 2 (ω) = C ) 2 ( hω F n 2 0Eg 3 2, F 2 (x) = E g (2x 1)3/2 (2x) 5, (36) 12

13 where C 1,2 are the material-independent constants. The figure shows that in the frequency region where TPA is maximum, we have no OPA, while at the frequencies where the OPA resonance reaches its maximum α 2 is significantly damped to nearly one tenth of its peak value. 13

14 Semiconductor Optics with Ultrashort Intense Laser Pulses II. Description of the Dynamics of Laser Pulses under TPA nonlinearity in Semiconductors within Modified Nonlinear Schrödinger Equation We begin this section with the full wave equation (32)(see previous section)let us express the spatio-temporal dependences of E by the formula, E(r,t)=êA(r,t)exp(ik 0 z iω 0 t)+ c.c., (37) where ω 0, and the corresponding wavenumber k 0 (ω 0 )are related by an appropriate dispersion relation. this is, of course, the representation of a laser pulse as a wavepacket with a central frequency ( also called the carrier frequency) ω 0 and a central wavenumber k 0 of a laser pulse spectrum. For the time being, we are not imposing any condition or assumption restricting the space-time behavior of the envelope A(r,t). The vector ê ê = 1 2 ( x + iɛŷ), (38) specifies the polarization of the optical field. The parameter of ellipticity takes the value ɛ = 0 (ɛ = ±1)for a linearly (circularly)polarized optical field. Without any loss of generality, we assume that the laser pulse propagates along the z axis, and is polarized in the perpendicular plane. Substituting (37)into Eq. (32), we obtain the following equation for the amplitude A(r,t): 2 A + 2 A z +2ik A 2 0 z k2 0A + 1 ( ω c i ) 2 A + 4π ( ω t c i ) 2 ( χ (1) ω 0 + i ) A = t t 2π(1 + ɛ 2 ) c 2 { ( (3) χ eff t A 2 A ) } iω 0 χ (3) eff A 2 A + 4πe2 NA. (39) c 2 m r In the representation (37), the first term on the right-hand side of Eq. (39) is split into the two terms contained in the curly brackets in Eq. (39). From the structure of these terms it is clear that they are different in nature: the first term contributes to nonlinear changes in the refractive index while the second one has a well pronounced dissipative (absorptive) 14

15 nature and describes the attenuation of the laser pulse due to the two-photon absorption. The last term on the right hand side of Eq. (39)is the laser driven nonlinear current of free carriers. In deriving Eq. (39), we have made use of the following relations: E(r,ω)=ê exp(ik 0 z)a(r,ω ω 0 )+ê exp( ik 0 z)a (r,ω+ ω 0 ), (40) and P L (r,ω)=χ (1) (ω)e(r,ω), (41) where the symbol denotes complex conjugation. Both Eqs. (40)and (41)follow from the properties of the Fourier transform. Substituting (40)into (41)and applying the inverse Fourier transform to P L (r,ω), we find, ( P L (r,t)= χ (1) (t t )E(r,t )dt = ê exp(ik 0 z iω 0 t)χ (1) ω 0 + i ) A(r,t)+ c.c.. (42) t Note that on the left-hand side of (42) χ (1) denotes the linear polarization response function of the medium whereas on the right-hand side χ (1) is its Fourier Transform, i.e., the linear susceptibility. The derivation of Eq. (39), using Eqs. (40) and (41), is a nontrivial technical step which implies prerequisites important for the ideological foundation of the nonlinear Schrödinger equation. Since χ (1) (ω)is complex, it is convenient to introduce a complex wavenumber κ = ω c 1+4πχ (1) (ω), (43) and split it into its its real and imaginary parts: κ = k + i α 2, (44) where k and α (both real)are the propagation constant and the absorption coefficient, respectively. 15

16 Now on our way to the modified nonlinear Schrödinger equation we have to make one more important step: we assume that the dispersion relation that connects k 0 and ω 0 is determined by the linear refractive properties of the medium so that the propagation constant k at ω = ω 0 is to be that of an electromagnetic wave of frequency ω 0 in the linear medium, k 0 = k(ω 0 )= ω 0 c R 1+4πχ (1) (ω 0 ). (45) This assumption is clearly perturbative in nature, and is consistent with and, in some sense, follows from the slowly-varying envelope approximation (SVEA)which implies, A t ω A 0 A, z k 0 A. (46) With this ordering, we can formally write Eq. (39)as, 2 A + 2 A z +2ik A 2 0 z k2 0A+ [ + k 0 + i α ( k 2 + i ω + i ) α ( 2 ω ω 0 t + 1 m k m=2 m! ω + i m ) ( α i m ] 2 A = m 2 ω t) m ω 0 2π(1 + ɛ 2 { ) ( (3) χ c 2 eff t A 2 A ) } iω 0 χ (3) eff A 2 A + 4πe2 NA. (47) c 2 m r From Eq. (47), it is clear that all dispersion effects such as the group velocity dispersion (GVD)( 2 k/ ω 2 )or other higher order dispersion emerge as perturbations to a ground state in which, to leading order, the pulse propagates with the group velocity v g =( ω/ k) ω0. Consistently following the perturbative approach, and ignoring terms higher order than the third derivative of the envelope A with respect to time, Eq. (47)is reduced to the following modified nonlinear Schrödinger equation: A z +1v A g t + i 2 β 2 A 2 t β 3 A 3 t + α 3 2 A i 2 2k A = 0 i 2ω 0 t (α 2 A 2 A) α 2 2 A 2 A i 2 k ωp (N) 2 0 A, (48) ω0 2 where, to facilitate comparison of Eq. (48)with the existing model equations [3,7,11,12,14], we have introduced the following notation conventional in nonlinear optics: 16

17 ( 2 ) k β 2 = 1 v g, (49) ω 2 ω 0 vg 2 ω 0 is the group velocity dispersion, and ( 3 ) k β 3 = 2 ω 3 ω 0 vg 3 ( ) 2 vg 1 ω 0 v 2 g 2 v g. (50) ω0 2 It may very well be the case that β 2 is zero at some wavelength λ D which is referred to as the zero-dispersion wavelength. However, the total dispersion does not vanish at λ D, and in this case the third-order term with the coefficient β 3 provides the leading order dispersion. This term can also become important for relatively short pulses even when the wavelength λ is far away from the zero-dispersion wavelength. The fifth term on the left-hand side of (47)accounts for the pulse attenuation due to linear absorption. Since we assume a weakly absorbing medium in the linear regime, this is the only α dependent term that will be retained. In obtaining Eq. ((47)), we normalized the pulse amplitude A so that A 2 represents the pulse intensity, A We have omitted the symbol for prime for notational simplicity. n0 c 8π (1 + ɛ2 )A. (51) Let us now discuss the nonlinear terms appearing on the right-hand side of Eq. (48). The first two terms result from the nonlinear polarization describing the TPA process. In our model it is natural to expect these two terms. The second term on the right-hand side is purely absorbing and describes TPA within the SVEA. Up to this point, our equation (48) is in full agreement with the existing models [11,12,14]. The first term on the right-hand side of Eq. (48)also originates from the TPA induced nonlinear polarization, but it has an obviously different nature: it contributes to the nonlinear refraction. This is the first important departure of our model from the conventional models. Moreover, this term not only give phenomenological description of the TPA induced nonlinear refraction but also predicts the change of sign that has been observed in experiments [2,8]. This is another signature 17

18 showing that our model for P NL (r,t)proposed in Eqs. (11) (13)is physically meaningful. It not only gives the correct form of the TPA term within the nonlinear Schrödinger equation, but also extends the existing models in the right direction. The second important distinction of our model is the last term on the right-hand side of Eq. (48). This term represents the laser pulse induced current in the free carriers. It is not hard to imagine that with the increase in the density of the TPA generated free carriers, this current could become the dominant nonlinearity of the system [5]. In order to stay within the validity of the nonlinear Schrödinger equation, i.e., within the SVEA, we are restricted by the condition: ω 2 p ω 2 0 1, (52) where ω p defines the effective free-carrier plasma frequency, and ε 0 is the linear dielectric permittivity, ωp 2 = 4πe2 N, (53) ε 0 m r ε0 = k 0c ω 0 =Re 1+4πχ (1) (ω 0 ). (54) This last type of nonlinearity exhibits a rather interesting feature: its effect accumulates as the pulse propagates through the medium as was demonstrated within a simplified but physically meaningful model in Ref. [5]. Under conditions more stringent than SVEA, the nonlinearity embodied in the conduction current gives rise to a number of interesting phenomena: frequency upshift of a laser pulse, its splitting into forward and backward propagating pulses, and non-absorptive damping. In order to fully exploit the rich variety of new nonlinear phenomena one should deal directly with the full wave equation (32) [with Eqs. (34) (35)]. The full system will allow us to study the TPA nonlinearities for a wide range of practically interesting physical parameters. Thus, starting from the full wave equation [Eq. (32)] we have derived the modified nonlinear Schrödinger (NLS)equation [Eq. (48)] which describes the nonlinear dynamics of 18

19 a laser pulse tuned near the TPA resonance within the SVEA. Besides the effects accounted for by previous models, it contains new effects giving a more complete and correct physical description of the processes driven by the TPA nonlinearity. 19

20 References 1. C.F. Klingshirn, Semiconductor Optics, (Springer-Verlag, Berlin, 1995). 2. A. Miller, Semiconductors, in Nonlinear Optics in Signal Processing, Eds. R. W. Eason, A. Miller, (Chapman & Hall, London, 1993). 3. P.N. Butcher and D. Cotter, The Elements of Nonlinear Optics, (University Press, Cambridge, 1990). 4. R.L. Sutherland, Handbook of Nonlinear Optics, (Marcel Dekker, New York, 1996). 5. V.I. Berezhiani, S.M. Mahajan, I.G. Murusidze, A photon accelerator: Large blueshifting of femtosecond laser pulses in semiconductors, Phys. Rev. A, 56(6), (1997). 6. B.S. Wherrett, Scaling rules for multiphoton interband absorption in semiconductors, J. Opt. Soc. Am. B, 1(1), 67-72, (1984). 7. A.C. Newell, J.V. Moloney, Nonlinear Optics, (Addison-Wesley, Redwood City, CA., 1992). 8. M. Sheik-Bahae et al., Dispersion of Bound Electron Nonlinear Refraction in Solids, IEEE J. Quantum Electronics, 27(6), (1991). 9. W. Sha, A.L. Smirl, W.F. Tseng, Coherent Plasma Oscillations in Bulk Semiconductors, Phys. Rev. Lett. 74(21), (1995) 10. B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, (Wiley, New York, 1991). 11. G.P. Agrawal, Effect of two-photon absorption on the amplification of ultrashort optical pulses, Phys. Rev. E, 48(3), (1993). 12. Y. Silberberg, Solitons and two-photon absorption, Optics Letters, 15(18), (1990). 20

21 13. A. Villeneuve et al., Two-photon absorption in In 1 x y Ga x Al y As/InP waveguides at communications wavelength, Appl. Phys. Lett. 56(19), (1990). 14. G.P. Agrawal, Nonlinear Fiber Optics, (Academic Press, Inc., San Diego, 1995). 15. I.H. Campbell, S.K. Kirbly, and P.M. Fauchet, Temporal Reshaping of Ultrashort Pulses Reflected by GaAs, in: Photonics: Nonlinear Optics and Ultrafast Phenomena, Eds. R.R. Alfano, L. Rothberg, (Nova Science Publ., New York, 1991), pp

22 hω 5 E g 1