Nonlinear Optics - Chapter 13 ppsx

Chapter 13 Ultrafast and Intense-Field Nonlinear Optics 13.1. Introduction There is currently great interest in the physics of ultrashort laser pulses. Recent advances have led to the generation of laser pulses with durations of the order of 1 attosecond. Ultrashort pulses can be used to probe the proper- ties of matter on extremely short time scales. Within the context of nonlinear optics, ultrashort laser pulses are of interest for two separate reasons. The first reason is that the nature of nonlinear optical interactions is often profoundly modified through the use of ultrashort laser pulses. The next two sections of this chapter treat various aspects of the resulting modifications of the nature of nonlinear optical interactions. The second reason is that ultrashort laser pulses tend to possess extremely high peak intensities, because laser pulse en- ergies tend to be established by the energy-storage capabilities of laser gain media, and thus short laser pulses tend to have much higher peak powers than longer pulses. The second half of this chapter is devoted to a survey of the sorts of nonlinear optical processes that can be excited by extremely intense laser fields. 13.2. Ultrashort Pulse Propagation Equation In this and the following section we treat aspects of the propagation of ultra- short laser pulses through optical systems. Some physical processes that will be included in this analysis include self-steepening leading to optical shock- wave formation, the influence of higher-order dispersion, and space-time cou- 561 562 13 ♦ Ultrafast and Intense-Field Nonlinear Optics pling effects. In the present section we derive a form of the pulse propagation equation relevant to the propagation of an ultrashort laser pulse through a nonlinear, dispersive nonlinear optical medium. In many ways, this equation can be considered to be a generalization of the pulse propagation equation (the so-called nonlinear Schrödinger equation) of Section 7.5. We begin with the wave equation in the time domain (see, for instance, Eq. (2.1.15)) which we express as −∇ 2 ˜ E + 1 c 2 ∂ 2 ˜ D (1) ∂t 2 =− 1  0 c 2 ∂ 2 ˜ P ∂t 2 . (13.2.1) We express the field quantities is terms of their Fourier transforms as ˜ E(r,t)=  E(r,ω)e −iωt dω/2π, (13.2.2a) ˜ D (1) (r,t)=  D (1) (r,ω)e −iωt dω/2π, (13.2.2b) ˜ P(r,t)=  P(r,ω)e −iωt dω/2π, (13.2.2c) where all of the integrals are to be performed over the range −∞ to ∞.We assume that D (1) (r,ω) and E(r,ω) are related by the usual linear dispersion relation as D (1) (r,ω)=  0  (1) (ω)E(r,ω) (13.2.3) and that ˜ P represents the nonlinear part of the material response. By introduc- ing these forms into Eq. (13.2.1), we obtain a relation that can be regarded as the wave equation in the frequency domain and that is given by ∇ 2 E(r,ω)+ 0  (1) (ω)  ω 2 /c 2  E(r,ω)=−  ω 2 / 0 c 2  P(r,ω). (13.2.4) Our goal is to derive a wave equation for the slowly varying field amplitude ˜ A(r,t)defined by ˜ E(r,t)= ˜ A(r,t)e i(k 0 z−ω 0 t) +c.c., (13.2.5) where ω 0 is the carrier frequency and k 0 is the linear part of the wavevector at the carrier frequency. We represent ˜ A(r,t)in terms of its spectral content as ˜ A(r,t)=  A(r,ω)e −iωt dω/2π. (13.2.6) Note that E(r,ω) and A(r,ω) are related as in Eq. (7.5.16) by E(r,ω)A(r,ω−ω 0 )e ik 0 z . (13.2.7) 13.2. Ultrashort Pulse Propagation Equation 563 In terms of the quantity A(r,ω) (the slowly varying field amplitude in the frequency domain) the wave equation (13.2.4) becomes ∇ 2 ⊥ A + ∂ 2 A ∂z 2 +2ik 0 ∂A ∂z +  k 2 (ω) −k 2 0  A =− ω 2  0 c 2 P(r,ω)e −ik 0 z , (13.2.8) where k 2 (ω) =(ω)  ω 2 /c 2  . (13.2.9) We next approximate k(ω) as a power series in the frequency difference ω −ω 0 as k(ω) =k 0 +k 1 (ω −ω 0 )+D where D = ∞  n=2 1 n! k n (ω −ω 0 ) n (13.2.10) so that k 2 (ω) can be expressed as k 2 (ω) = k 2 0 +2k 0 k 1 (ω −ω 0 ) +2k 0 D +2k 1 D(ω −ω 0 ) +k 2 1 (ω −ω 0 ) 2 +D 2 . (13.2.11) Here D represents high-order dispersion. We have displayed explicitly the linear term k 1 (ω −ω 0 ) in the power series expansion because k 1 has a direct physical interpretation as the inverse of the group velocity. We now introduce this expression into the wave equation in the form of Eq. (13.2.8), which then becomes ∇ 2 ⊥ A + ∂ 2 A ∂z 2 +2ik 0 ∂A ∂z +2k 0 k 1 (ω −ω 0 )A +2 k 0 DA +2k 1 D(ω −ω 0 )A +k 2 1 (ω −ω 0 ) 2 A =  ω 2 / 0 c 2  P(z,ω)e −ik 0 z , (13.2.12) where we have dropped the contribution D 2 because it is invariably small. We now convert this equation back to the time domain. To do so, we multiply this equation by exp[−i(ω − ω 0 )t] and integrate over all values of ω − ω 0 .We obtain  ∇ 2 ⊥ + ∂ 2 ∂z 2 +2ik 0  ∂ ∂z +k 1 ∂ ∂t  +2ik 1 ˜ D ∂ ∂t +2k 0 ˜ D −k 1 ∂ 2 ∂t 2  ˜ A(r,t) = 1  0 c 2 ∂ 2 ˜ P ∂t 2 e −i(k 0 z−ω 0 t) , (13.2.13) where ˜ D represents the differential operator ˜ D = ∞  n=2 1 n k n  i ∂ ∂t  n =− 1 2 k 2 ∂ 2 ∂t 2 +···. (13.2.14) 564 13 ♦ Ultrafast and Intense-Field Nonlinear Optics We now represent the polarization in terms of its slowly varying amplitude ˜p(r,t)as ˜ P(r,t)=˜p(r,t)e i(k 0 z−ω 0 t) +c.c. (13.2.15) For example, for the case of a material with an instantaneous third-order re- sponse, the polarization amplitude is given by ˜p(r,t)=3 0 χ (3)   ˜ A(r,t)   2 ˜ A(r,t). (13.2.16) We thus find that ∂ ˜ P ∂t =  −iω 0 ˜p + ∂ ˜p ∂t  e i(k 0 z−ω 0 t) +c.c. =−iω 0  1 + i ω 0 ∂ ∂t  ˜p  e i(k 0 z−ω 0 t) +c.c. (13.2.17a) and ∂ 2 ˜ P ∂t 2 =−ω 2 0  1 + i ω 0 ∂ ∂t  2 ˜p  e i(k 0 z−ω 0 t) +c.c. (13.2.17b) By introducing this expression into the wave equation in the form (13.2.13), we obtain  ∇ 2 ⊥ + ∂ 2 ∂z 2 +2ik 0  ∂ ∂z +k 1 ∂ ∂t  +2k 0 ˜ D +2ik 1 ˜ D ∂ ∂t −k 2 1 ∂ 2 ∂t 2  ˜ A(r,t) =− 4πω 2 0 c 2  1 + i ω 0 ∂ ∂t  2 ˜p. (13.2.18) Next we convert this equation to a retarded time frame specified by the coor- dinates z  and τ defined by z  =z and τ =t − 1 v g z =t −k 1 z, (13.2.19) so that ∂ ∂z = ∂ ∂z  −k 1 ∂ ∂τ and ∂ ∂t = ∂ ∂τ . (13.2.20) 13.2. Ultrashort Pulse Propagation Equation 565 The wave equation then becomes  ∇ 2 ⊥ + ∂ 2 ∂z 2 −2k 1 ∂ ∂z  ∂ ∂τ +k 2 1 ∂ 2 ∂τ 2 +2ik 0  ∂ ∂z  −k 1 ∂ ∂τ +k 1 ∂ ∂τ  +2k 0 ˜ D +2ik 1 ˜ D ∂ ∂τ −k 2 1 ∂ 2 ∂τ 2  ˜ A(r,t)=− ω 2 0  0 c 2  1 + i ω 0 ∂ ∂τ  2 ˜p. (13.2.21) We now make the slowly varying amplitude approximation (that is, we drop the term ∂ 2 /∂z 2 ) and simplify this expression to obtain  ∇ 2 ⊥ −2k 1 ∂ ∂z  ∂ ∂τ +2ik 0 ∂ ∂z  +2k 0 ˜ D +2ik 1 ˜ D ∂ ∂τ  ˜ A(r,t) =− ω 2 0  0 c 2  1 + i ω 0 ∂ ∂τ  2 ˜p. (13.2.22) This equation can alternatively be written as  ∇ 2 ⊥ +2ik 0 ∂ ∂z   1 + ik 1 k 0 ∂ ∂τ  +2k 0 ˜ D  1 + ik 1 k 0 ∂ ∂τ  ˜ A(r,t) =− ω 2 0  0 c 2  1 + 1 ω 0 ∂ ∂τ  2 ˜p. (13.2.23) Note that two of the terms in this equation depend upon the ratio k 1 /k 0 .This ratio can be approximated as follows: k 1 /k 0 =v −1 g /(nω 0 /c) =n g /(nω 0 ).Ig- noring dispersion, n g = n,sothatk 1 /k 0 = 1/ω 0 . In this approximation the wave equation becomes  ∇ 2 ⊥ +2ik 0 ∂ ∂z   1 + i ω 0 ∂ ∂τ  +2k 0 ˜ D  1 + i ω 0 ∂ ∂τ  ˜ A(r,t) =− ω 2 0  0 c 2  1 + i ω 0 ∂ ∂τ  2 ˜p, (13.2.24) which can also be expressed as  1 + i ω 0 ∂ ∂τ  −1 ∇ 2 ⊥ +2ik 0 ∂ ∂z  +2k 0 ˜ D  ˜ A(r,t)=− ω 2 0  0 c 2  1 + i ω 0 ∂ ∂τ  ˜p. (13.2.25) 566 13 ♦ Ultrafast and Intense-Field Nonlinear Optics This equation can be considered to be a generalization of the nonlinear Schrödinger equation. It includes the effects of higher-order dispersion (through the term that includes ˜ D), space–time coupling (through the pres- ence of the differential operator on the left-hand side of the equation), and self-steepening (through the presence of the differential operator on the right- hand side). This form of the pulse propagation equation has been obtained by Brabec and Krausz (1997). It can be used to treat many types of nonlinear response. For instance, for a material displaying an instantaneous third- and fifth-order nonlinearity, ˜p is given by ˜p =3 0 χ (3) | ˜ A| 2 ˜ A +10 0 χ (5) | ˜ A| 4 ˜ A. This equation can also be used to treat a dispersive nonlinear material. For ultrashort laser pulses, the value of χ (3) can vary appreciably for different frequency components of the pulse. The effects of the dispersion of χ (3) can be modeled in lowest approximation (see for instance Diels and Rudolph, 1996, p. 139) by representing χ (3) (ω) ≡χ (3) (ω =ω +ω − ω) as χ (3) (ω) =χ (3) (ω 0 ) +(ω −ω 0 ) dχ (3) dω , (13.2.26) where the derivative is to be evaluated at frequency ω 0 . Thus, p(ω) can be expressed as p(ω) =3 0  χ (3) (ω 0 ) +(ω −ω 0 ) dχ (3) dω    A(ω)   2 A(ω). (13.2.27) This relation can be converted to the time domain using the same procedure as that used in going from Eq. (13.2.12) to Eq. (13.2.13). One finds that ˜p(τ) = 3 0  χ (3) (ω 0 ) + dχ (3) dω i ∂ ∂τ    ˜ A   2 ˜ A. (13.2.28) This expression for ˜p can be used directly in Eq. (13.2.24) or (13.2.25). How- ever, since Eq. (13.2.26) contains only a linear correction term in (ω − ω 0 ), and consequently Eq. (13.2.28) contains only a contribution first-order in ∂/∂τ, for reasons of consistency one wants to include in the resulting pulse propagation equation only contributions first-order in ∂/∂τ. Noting that  1 + i ω 0 ∂ ∂τ  2 =  1 + 2i ω 0 ∂ ∂τ − 1 ω 2 0 ∂ 2 ∂τ 2  ≈  1 + 2i ω 0 ∂ ∂τ  , (13.2.29) 13.3. Interpretation of the Ultrashort-Pulse Propagation Equation 567 one finds that in this approximation the pulse propagation equation is given by ∇ 2 ⊥ +2ik 0 ∂ ∂z   1 + i ω 0 ∂ ∂τ  +2k 0 ˜ D  1 + i ω 0 ∂ ∂τ  ˜ A(r,t) =  −3/c 2  ω 2 0 χ (3) (ω 0 )  1 +  2 + ω 0 χ (3) (ω 0 ) dχ (3) dω  i ω 0 ∂ ∂τ    ˜ A   2 ˜ A. (13.2.30) Procedures for incorporating other sorts of nonlinearities into the present for- malism have been described by Gaeta (2000). 13.3. Interpretation of the Ultrashort-Pulse Propagation Equation Let us next attempt to obtain some level of intuitive understanding of the var- ious physical processes described in Eq. (13.2.24). As a first step, we study a simplified version of this equation obtained by ignoring the correction terms (i/ω 0 )∂/∂τ by replacing the factors [1 + (i/ω 0 )(∂/∂τ )] by unity and by in- cluding only the lowest-order contribution (known as second-order disper- sion) to ˜ D. One obtains ∂A(r,t) ∂z  =  i 2k 0 ∇ 2 ⊥ − i 2 k 2 ∂ 2 ∂τ 2 + 3iω 0 2n 0 c χ (3) (ω 0 )   ˜ A   2  ˜ A. (13.3.1) Written in this form, the equation leads to the interpretation that the field am- plitude A varies with propagation distance z  (the left-hand side) because of three physical effects (the three terms on the right-hand side). The term in- volving the transverse laplacian describes the spreading of the beam due to diffraction, the term involving the second time derivative describes the tem- poral spreading of the pulse due to group velocity dispersion, and the third term describes the nonlinear acquisition of phase. It is useful to introduce dis- tance scales over which each of the terms becomes appreciable. We define these scales as follows: L dif = 1 2 k 0 w 2 0 (diffraction length), (13.3.2a) L dis = T 2 /|k 2 | (dispersion length), (13.3.2b) L NL = 2n 0 c 3ω 0 χ (3) |A| 2 = 1 (ω 0 /c)n 2 I (nonlinear length). (13.3.2c) In these equations w 0 is a measure of the characteristic beam radius, and T is a measure of the characteristic pulse duration. The significance of these 568 13 ♦ Ultrafast and Intense-Field Nonlinear Optics distance scales is that for a given physical situation the process with the shortest distance scales is expected to be dominant. For reference, note that for fused silica at a wavelength of 800 nm n 2 = 3.5 × 10 −20 m 2 /Wand k 2 = 446 fsec 2 /cm. Through use of Eq. (13.3.2b) we see that, for a 20-fsec pulse propagating through fused silica, L dis is approximately 0.9 cm. Thus, in propagating through 0.9 cm of fused silica a 20-fsec pulse approximately doubles in pulse duration as a consequence of group velocity dispersion. 13.3.1. Self-Steepening Let us next examine the influence of the correction factor [1 +(i/ω 0 )(∂/∂τ )] on the nonlinear source term of Eq. (13.2.25). To isolate this influence, we drop the correction factor in other places in the equation. Also, for generality, we use the propagation equation in the form given by (13.2.30), which allows the nonlinear response to be dispersive. We also transform back to the labo- ratory reference frame z, t (not the z  ,τ frame in which the pulse is nearly stationary) so that the factor k 1 ∂ ˜ A/∂t = (1/v g )∂ ˜ A/∂t = (n (g) 0 /c)∂ ˜ A/∂t ap- pears explicitly in the wave equation, which takes the form ∂ ˜ A ∂z − n (g) 0 c ∂ ˜ A ∂t = i 2k 0 ∇ 2 ⊥ ˜ A − i 2 k 2 ∂ 2 ˜ A ∂t 2 + i6πω 0 n 0 c χ (3) (ω 0 )   ˜ A   2 ˜ A + i3ω 0 2n 0 c χ (3) (ω 0 )  2 + ω 0 χ (3) (ω 0 ) dχ (3) dω  i ω 0 ∂ ∂t   ˜ A   2 A. (13.3.3) We now introduce nonlinear coefficients γ 1 and γ 2 defined by γ 1 = 3ω 0 2n 0 c χ (3) (ω 0 ) and γ 2 = 3ω 0 2n 0 c χ (3) (ω 0 )  1 + 1 2 ω 0 χ (3) dχ (3) dω  . (13.3.4) Note that in the absence of dispersion γ 1 = γ 2 . In terms of these quantities, Eq. (13.3.3) can be expressed more concisely as ∂ ˜ A ∂z − n (g) 0 c ∂ ˜ A ∂t = i 2k 0 ∇ 2 ⊥ ˜ A − i 2 k 2 ∂ 2 ˜ A ∂t 2 +iγ 1 |A| 2 A −2γ 2 1 ω 0 ∂ ∂t    ˜ A   2 A  . (13.3.5) Next note that the time derivative in the last term can be written as ∂ ∂t    ˜ A   2 ˜ A  = ∂ ∂t  ˜ A 2 ˜ A ∗  = ˜ A 2 ∂ ˜ A ∗ ∂t +2 ˜ A ∗ ˜ A ∂ ˜ A ∂t = 2   ˜ A   2 ∂ ˜ A ∂t + ˜ A 2 ∂ ˜ A ∗ ∂t . (13.3.6) 13.3. Interpretation of the Ultrashort-Pulse Propagation Equation 569 The first contribution to the last form can be identified as an intensity- dependent contribution to the group velocity. The second contribution does not have a simple physical interpretation, but can be considered to represent a dispersive four-wave mixing term. To proceed we make use of Eq. (13.3.6) to express Eq. (13.3.4) as ∂ ˜ A ∂z − n (g) eff c ∂ ˜ A ∂t = i 2k 0 ∇ 2 ⊥ ˜ A − i 2 k 2 ∂ 2 ˜ A ∂t 2 +iγ 1 |A| 2 A − 2γ 2 ω 0 ˜ A 2 ∂ ˜ A ∗ ∂t , (13.3.7) where n (g) eff =n (g) 0 + 4γ 2 c ω 0   ˜ A   2 ≡n (g) 0 +n (g) 2 I. (13.3.8) In the last form of this relation, we have introduced the coefficient of the intensity dependence of the group index as n (g) 2 = 48π 2 n 2 0 c χ (3) (ω 0 )  1 + 1 2 ω 0 χ (3) (ω 0 ) dχ (3) dω  . (13.3.9) We thus see that the last term in Eq. (13.3.4) leads to an intensity dependence of the group index n g as well as to the last term of Eq. (13.3.7), which as just mentioned is a dispersive four-wave mixing contribution. We also see from Eq. (13.3.9) that the intensity dependence of the group index depends both on the susceptibility and on its dispersion. The intensity dependence of the group velocity leads to the phenomena of self-steepening and optical shock wave formation. These phenomena are illus- trated in Fig. 13.3.1. Note that for the usual situation in which n (g) 2 is positive, the peak of the pulse is slowed down more than the edges of the pulse, leading to steepening of the trailing edge of the pulse. If this edge becomes infinitely steep, it is said to form an optical shock wave. Self-steepening has been de- FIGURE 13.3.1 Self-steepening and optical shock formation. (a) The incident optical pulse is assumed to have a gaussian time evolution. (b) After propagation through a nonlinear medium, the pulse displays self-steepening, typically of the trailing edge. (c) If the self-steepening becomes sufficiently pronounced that the intensity changes instantaneously, an optical shock wave is formed. 570 13 ♦ Ultrafast and Intense-Field Nonlinear Optics scribed by DeMartini et al. (1967), Yang and Shen (1984), and Gaeta (2000). Note also that we can define a self-steepening distance scale analogous to these of Eqs. (13.3.2) by L ss = cT n (g) 2 I . (13.3.10) For the usual situation in which n (g) 2 ≈ n 2 , L ss is much larger than L NL (because, except for extremely short pulses, cT  1/k 0 ), and thus self- steepening tends to be difficult to observe. 13.3.2. Space-Time Coupling Let us now examine the influence of space-time coupling, that is, the influ- ence of the differential operator [1 + (i/ω 0 )∂/∂τ] −1 on the left-hand side of Eq. (13.2.25). We can see the significance of this effect most simply by considering propagation through a dispersionless, linear material so that the wave equation becomes  1 + i ω 0 ∂ ∂τ  −1 ∇ 2 ⊥ ˜ A(r,t)+2ik 0 ∂ ∂z  ˜ A(r,t)=0. (13.3.11) The first term is said to represent space-time coupling because it involves both temporal and spatial derivatives of the field amplitude. To examine the signif- icance of this mathematical form, it is convenient to rewrite this equation as ∇ 2 ⊥ ˜ A(r,t)+  1 + i ω 0 ∂ ∂τ  2ik 0 ∂ ∂z  ˜ A(r,t)=0. (13.3.12) Let us first consider the somewhat artificial example of a field of the form ˜ A(r,t)=a(r)e −iδωt ; such a field is a monochromatic field at frequency ω 0 + δω. We substitute this form into Eq. (13.3.12) and obtain ∇ 2 ⊥ a(r) +  1 + δω ω 0  2ik 0 ∂ ∂z  a(r) = 0, (13.3.13) which can alternatively be expressed as ∇ 2 ⊥ a(r) +2i(k 0 +δk) ∂ ∂z  a(r) = 0, (13.3.14) where δk =k 0 (δω/ω 0 ). This wave thus diffracts as a wave of frequency ω 0 + δω rather than a wave of frequency ω 0 . More generally, for the case of an ultrashort pulse, the operator [1+(i/ω 0 )∂/∂τ ] describes the fact that different frequency components of the pulse diffract into different cone angles. Thus, [...]... process of HHG by propagating the laser beam through a gas-filled capillary waveguide 13. 7 Nonlinear Optics of Plasmas and Relativistic Nonlinear Optics 579 13. 7 Nonlinear Optics of Plasmas and Relativistic Nonlinear Optics A plasma is a partially or fully ionized gas Plasmas play an important role in nonlinear optics in two different ways: (1) Nonlinear optical processes such as multiphoton ionization... to the 33rd harmonic with laser intensities as large as 1 013 W/cm2 using Ar, Kr, and Xe gases (Fig 13. 6.3) Kulan- 578 13 ♦ Ultrafast and Intense-Field Nonlinear Optics F IGURE 13. 6.3 Experimental data of Ferray et al (1988) illustrating high-harmonic generation der and Shore (1989) presented one of the first successful computer models of high-harmonic generation L’Huillier and Balcou (1993) observed... conjunction with Eq (13. 7.8) to determine the value of γ to be used in Eq (13. 7.9) We next calculate the nonlinear coefficient n2 by determining the lowestorder change in refractive index The relativistic factor γ is given by the square root of expression (13. 7.10), which to lowest order becomes γ2 =1+ γ =1+ 2 1 e2 E0 ≡ 1 + x, 2 m2 ω 2 c 2 (13. 7.11) 13 ♦ Ultrafast and Intense-Field Nonlinear Optics 582 where... 10−34 cm2 /W (13. 8.9) We saw in Chapter 7 that strong self-action effects are expected only if the power of a laser beam exceeds the critical power for self-focusing Pcr = λ2 8n0 n2 (13. 8.10) We find by combining Eqs (13. 8.9) and (13. 8.10) that at a wavelength of 1 μm, Pcr = 4.4 × 1024 W, (13. 8.11) which is considerably larger than the power of any laser source currently contemplated 586 13 ♦ Ultrafast... the assumption V (t) H0 For the case of intense-field nonlinear optics, the nature of this inequality is the reverse—that is, the interaction energy V (t) is much larger than H0 This observation suggests that it should prove useful to begin our study of intense-field nonlinear optics by considering the motion of a free electron in an intense laser field 13. 5 Motion of a Free Electron in a Laser Field.. .13. 4 Intense-Field Nonlinear Optics 571 after propagation different frequency components will have different radial dependences These effects and their implications for self-focusing have been described by Rothenberg (1992) 13. 3.3 Supercontinuum Generation When a short intense pulse propagates through a nonlinear optical medium, it often undergoes significant... introduced over the years in attempts to explain supercontinuum generation At present, it appears that pulse self-steepening (Yang and Shen, 1984) leading to optical shock-wave formation (Gaeta, 2000) is the physical mechanism leading to supercontinuum generation 13. 4 Intense-Field Nonlinear Optics Most nonlinear optical phenomena∗ can be described by assuming that the material polarization can be expanded... Another mechanism of nonlinearity in plasmas is relativistic effects (Wagner et al., 1997) In a sufficiently intense laser beam (I 1018 W/cm2 ) a free electron can be accelerated to relativistic velocities in a half optical period This conclusion can be reached by equating the ponderomotive energy K of 13. 7 Nonlinear Optics of Plasmas and Relativistic Nonlinear Optics 581 Section 13. 5 with the value... highly nonresonant conditions, Eq (13. 4.1) can become invalid This breakdown will certainly occur if the laser field amplitude E becomes comparable to or larger than the atomic field strength Eat = e/4π 2 a0 0 = e/4π 0 (4π 0 h2 /me2 )2 ¯ = 6 × 1011 V/m, ∗ The photorefractive effect of Chapter 11 being an obvious exception (13. 4.2) 13 ♦ Ultrafast and Intense-Field Nonlinear Optics 572 which corresponds to... (13. 4.3) In fact, lasers that can produce intensities larger than 1020 W/cm2 are presently available (Mourou et al., 1998) In this chapter we explore some of the physical phenomena that can occur through use of fields this intense Let us begin by considering briefly the conceptual framework one might use to describe intense-field nonlinear optics Recall that the quantum-mechanical calculation of the nonlinear . beam through a gas-filled capillary waveguide. 13. 7. Nonlinear Optics of Plasmas and Relativistic Nonlinear Optics 579 13. 7. Nonlinear Optics of Plasmas and Relativistic Nonlinear Optics A plasma. −k 1 ∂ 2 ∂t 2  ˜ A(r,t) = 1  0 c 2 ∂ 2 ˜ P ∂t 2 e −i(k 0 z−ω 0 t) , (13. 2 .13) where ˜ D represents the differential operator ˜ D = ∞  n=2 1 n k n  i ∂ ∂t  n =− 1 2 k 2 ∂ 2 ∂t 2 +···. (13. 2.14) 564 13 ♦ Ultrafast and Intense-Field Nonlinear Optics We. might use to describe intense-field nonlinear optics. Recall that the quantum-mechanical calculation of the nonlinear optical susceptibility presented in Chapter 3 pre- supposes that the Hamiltonian