Ultrafast and intense field nonlinear optics Chapter 13

The second reason is that ultrashort laser pulses tend to possess extremely high peak intensities, because laser pulse energies tend to be established by the energy-storage capabilities

of 1 attosecond Ultrashort pulses can be used to probe the properties of matter

on extremely short time scales Within the context of nonlinear optics, ultra- short 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 energies 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

1 3.2 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 coupling effects In the present section we derive a form of the pulse propagation equa- tion relevant to the propagation of an ultrashort laser pulse through a nonlinear, dispersive nonlinear optical medium In many ways, this equation can be con- sidered to be a generalization of the pulse propagation equation (the so-called nonlinear Schrodinger equation) of Section 7.5 We begin with the wave equa- tion in the time domain (see, for instance, Eq (2.1.14)) which we express

and that P represents the nonlinear part of the material response By introduc- ing these forms into Eq (13.2 I), we obtain a relation that can be regarded as the wave equation in the frequency domain and which is given by

Our goal is to derive a wave equation for the slowly varying field amplitude A(r, t) defined by

where wo is the carrier frequency and ko is the linear part of the wavevector at

the carrier frequency We represent A(r, t) in terms of its spectral content as

Note that E (r, w) and A (r, W) are related by

13.2 Ultrashort Pulse Propagation Equation 535

In terms of the quantity A(r, w) (the slowly varying field amplitude in the frequency domain) the wave equation (13.2.4) becomes

so that k2(m) can be expressed as

Here D represents high-order dispersion We have displayed explicitly the linear term k , (w - oo) in the power series expansion because kl 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

where we have dropped the contribution D~ 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 (w - coo) t ] and integrate over all values of w - coo We obtain

where D represents the differential operator

We now represent the polarization in terms of its slowly varying amplitude

p(r, t) as

B(r, t) = p(r, t)e i(koz-mot) + C.C (13.2.15) For example, for the case of a material with an instantaneous third-order response, the polarization is given by

13.2 Ultrashort Pulse Propagation Equation 537

We now make the slowly varying amplitude approximation (that is, we drop the term a2/azf2) and simplify this expression to obtain

This equation can alternatively be writen as

Note that several of the terms in this equation depend upon the ratio kl / ko

This ratio can be approximated as follows: kl / ko = v,' /(nwo/c) = ng/(nwo) Ignoring dispersion, n, = n, so that kl / ko = l/wo In this approximation the wave equation becomes

which can also be expressed as

This equation can be considered to be a generalization of the nonlinear Schrodinger equation It includes the effects of higher-order dispersion (through the term that includes b), 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 described

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 x "1 I A I A + 1 O x ( 5 ) I A 1 4A

This equation can also be used to treat a dispersive nonlinear material For ultrashort laser pulses, the value of x(~) can vary appreciably for different frequency components of the pulse The effects of the dispersion of x(-" can

be modeled in first approximation (see for instance Diels and Rudolph, 1996,

( 3 )

p 139) by representing x ( ~ ) ( w ) = x (W = w + w - W ) as

where the derivative is to be evaluated at frequency wo Thus p ( w ) can be represented as

This relation can be converted to the time domain using the same procedure

as that used in going from Eq (1 3.2.12) to Eq (13.2.13) One finds that

This expression for j? can be used directly in Eq (13.2.24) or (13.2.25) How- ever, since Eq ( 1 3.2.26) contains only a linear correction term in ( w - w o ) , and consequently Eq (13.2.28) contains only a contribution first-order in / a t , for reasons of consistency one wants to include in the resulting pulse propagation equation only contributions first-order in a/a t Noting that

i a 2 i 3

( 1 + - - ) 2 = ( 1 + - I " ) 2 ( 1 + EL), (13.2.29)

one finds that in this approximation the pulse propagation equation is given by

Procedures for incorporating other sorts of nonlinearities into the present for- malism have been described by Gaeta (2000)

13.3 lnterpretation of the Ultrashort Pulse Propagation Equation 539

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, let us study

a simplified version of this equation obtained by ignoring the correction terms (i /wo)d/a t by replacing the factors [l + (i /wo) (813 t ) ] by unity and by in- cluding only the lowest-order contribution (known as second-order dispersion)

to D One obtains

Written in this form, the equation leads to the interpretation that the field ampli- tude A varies with propagation distance 2' (the left-hand side) because of three physical effects (the three terms on the right-hand side) The term involving the transverse laplacian describes the spreading of the beam due to diffraction, the term involving the second time derivative describes the temporal spreading of the pulse due to group velocity dispersion, and the third term describes the non- linear acquisition of phase It is useful to introduce distance scales over which each of the terms becomes appreciable We define these scales as follows:

1 Ldif = 3 ko wi (diffraction length), (13.3.2a)

Ldis = T '1 1 k2 1 (dispersion length), (1 3.3.2b)

not - - 1 LNL = (nonlinear length) (1 3.3.2~)

6nwo~(3)IA12 (w/c)n21

In these equations wo is a measure of the characteristic beam radius, and T is

a measure of the characteristic pulse duration The significance of these dis- tance 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 800 nm nz = 3.5 x 10-l6 cm2/W and k2 = 446 fsec2/cm Through use of Eq (13.3.2b) we see that, for a 20-fsec pulse propagating through fused

silica, Ldis is approximately 0.9 cm Thus in propagating through 0.9 cm of

fused silica a 20-fsec pulse approximately doubles in pulse duration because

of group velocity dispersion

Self-steepen ing

Let us next examine the influence of the correction factor [ l + (i/wO) (a/at)]

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 laboratory reference frame Z, t (not the z', t frame in which the pulse is nearly station- ary) SO that the factor k l a A / a t = ( i / v g ) a A / a t = ( n f ) / c ) a A / a t appears

explicitly in the wave equation, which takes the form

Note that in the absence of dispersion yl = y2 In terms of these quantities,

Eq (1 3.3.3) can be expressed more concisely as

Next note that the time derivative in the last term can be written as

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 ( 1 3.3.6) to express

13.3 Interpretation of the Ultrashort Pulse Propagation Equation 541

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

In the last form of this relation, we have introduced the coefficient of the intensity dependence of the group index as

We thus see that the last term in Eq (13.3.5) leads an intensity dependence

of the group index n, as well as to the last term of Eq (13.3.7), which as

mentioned above 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 f ' 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- scribed by DeMartini et al (1967), by Yang and Shen (1984), and by Gaeta (2000) Note also that we can define a self-steepening distance scale analogous

to these of Eqs (13.3.2) by

For the usual situation in which n?' ;t: n2, Lss is much larger than LNL (because,

except for extremely short pulses, cT >> 1/ ko), and thus self-steepening tends

to be difficult to observe

Space- Time Coupling

Let us now examine the influence of space-time coupling, that is, the influ- ence of the differential operator [l + (i/wo) a/as]-' 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

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 sig- nificance of this mathematical form, it is convenient to rewrite this equation

as

Let us first consider the somewhat artificial example of a field of the form A(r, t ) = a (r)e-isot- , such a field is a monochromatic field at frequency wo +

601 We substitute this form into Eq (13.3.12) and obtain

which can alternatively be expressed as

where Sk = ko (6w/wo) This wave thus diffracts as a wave of frequency

oo + 6w rather than a wave of frequency wo More generally, for the case of an ultrashort pulse, the operator [ I + (i/wo)a/a t ] describes the fact that different frequency components of the pulse diffract into different cone angles Thus, after propagation different frequency components will have different radial dependences These effects and their implications for self-focusing have been described by Rothenberg (1992)

Supercontinuum Genera tion

When a short intense pulse propagates through a nonlinear optical medium, it often undergoes significant spectral broadening This effect was first reported

by Alfano and Shapiro (1970) The amount of broadening can be very sig- nificant For instance, using an 80-fsec pulse of peak intensity -loL4 w/cm2 propagating through 0.5 mm of ethylene glycol, Fork et al (1 983) observed a broadened spectrum extending from 0.4 wo to 3.3 wo, where coo is the central frequency of the input laser pulse Supercontinuum generation has also been

observed in gases (Corkum et al., 1986) Many models have been introduced

over the years in attempts to explain supercontinuum generation At present,

13.4 Intense-Field Nonlinear Optics 543

it appears that pulse self-steepening (Yang and Shen, 1984) leading to opti- cal 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 as a power series in the applied electric field amplitude This relation in its simplest form is given by

However, for sufficiently large field strengths, this power series expansion need not converge We saw in Chapter 6 that under resonant conditions this power- series description breaks down if the Rabi frequency S2 = p b , E / f i associated

with the interaction of the laser field with the atom becomes comparable to

1/ T I , where TI is the atomic excited-state lifetime Even under highly non- resonant conditions, Eq (13.6.1) can become invalid This breakdown will certainly occur if the laser field amplitude E becomes comparable to or larger than the atomic field strength

which corresponds to an intensity oft

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 consider briefly the conceptual framework one might use to describe intense-field nonlinear optics Recall that the quantum-mechanical calculation

of the nonlinear optical susceptibility presented in Chapter 3 presupposes that the Harniltonian of an atom in the presence of a laser field is of the form

where tio is the Harniltonian of an isolated atom and P ( t ) = - p i ( t ) represents the interaction energy of the atom with the laser field Schrodinger's equation

* The photorefractive effect of Chapter I I being an obvious exception

t Here we take the peak field strength of the optical wave, which we assume to be linearly polarized,

to be E,t

is then solved for this Hamiltonian through use of perturbation theory under the assumption V(t) << Ho 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 Ho 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

Let us initially ignore both relativistic effects and the influence of the magnetic field associated with the laser beam We assume the laser beam to be linearly polarized and of the form ~ ( t ) = I?(t)i, where E(t) = ~ e - ' ~ ' + C.C The equation of motion of the electron is then given by

m 2 = - e E ( t ) or m 2 = - eEe-i"t + c.c., (13.5.1) which leads to the solution

2 (t) = xe-'Of + C.C (13.5.2) where

The time-averaged kinetic energy associated with this motion is given by K =

Im 2 (i (t)2) or, since

This energy is known as the jitter energy (as it is associated with the oscillation

of the electron about its equilibrium position) or as the ponderomotive energy (Kibble, 1966) This energy can be appreciable By way of example, consider

a laser field of wavelength 1.06 pm One finds by numerical evaluation that the ponderomotive energy is equal to 13.6 eV (a typical atomic energy) for

I = 1.3 x 1014 w/cm2, is equal to 4.2 keV for I = I,, (which is given by

Eq (13.4.3)), and is equal to mc2 = 500 keV for I = 4.8 x 1018 w/cm2 The equation of motion (1 3.5.1) and its solution (1 3.5.2) are linear in the laser field amplitude Both magnetic and relativistic effects can induce nonlinearity

in the electronic response Let us first consider briefly the influence of magnetic

13.5 Motion of a Free Electron in a Laser Field 545

effects; see also Problem 1 at the end of this chapter for a more detailed analysis The electric field of Eq (13.5.1) has a magnetic field associated with

it Assuming propagation in the z direction, this magnetic field is of the form

~ ( t ) = ~ ( t ) ? , where ~ ( t ) = ~ e ' ~ ' + C.C and where, assuming propagation

in vacuum, B = E Since according to Eq (13.5.4) the electron has a velocity

in the x direction, it will experience a magnetic force F = (v/c) x B in the z

direction The equation of motion for the z component of the velocity is thus

The right-hand side of this equation consists of terms at zero frequency and at frequencies k 2 w When Eq (13.5.6) is solved, one find that the z-component

of the electron motion consists of oscillations at frequency 2w and amplitude

beam of peak field strength Eo, i.e., k = Eo c o s ( ~ r - u z / c ) , the electron moves in a figure-8 pattern superposed on a uniform translational motion in the z-direction In the reference frame moving with the uniform translational velocity, the electron motion can be described the equations

For circularly polarized radiation described by E y = Eo cos(wt - wz/c),

Ex = Eo sin(wt - wzlc), the electron moves with uniform angular velocity in

* This conclusion arises, for instance, as a generalization of the results of Problem 7 of Chapter 4