Nonlinear Optics - Chapter 13 ppsx

Ultrashort Pulse Propagation Equation In this and the following section we treat aspects of the propagation of short laser pulses through optical systems.. In the present section we deri

of nonlinear optical interactions The second reason is that ultrashort laserpulses tend to possess extremely high peak intensities, because laser pulse en-ergies tend to be established by the energy-storage capabilities of laser gainmedia, and thus short laser pulses tend to have much higher peak powers thanlonger pulses The second half of this chapter is devoted to a survey of thesorts of nonlinear optical processes that can be excited by extremely intenselaser fields.

13.2 Ultrashort Pulse Propagation Equation

In this and the following section we treat aspects of the propagation of short laser pulses through optical systems Some physical processes that will

ultra-be included in this analysis include self-steepening leading to optical wave formation, the influence of higher-order dispersion, and space-time cou-

shock-561

pling effects In the present section we derive a form of the pulse propagationequation relevant to the propagation of an ultrashort laser pulse through anonlinear, dispersive nonlinear optical medium In many ways, this equationcan be considered to be a generalization of the pulse propagation equation(the so-called nonlinear Schrödinger equation) of Section 7.5 We begin withthe wave equation in the time domain (see, for instance, Eq (2.1.15)) which

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)

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 ˜Prepresents 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 asthe wave equation in the frequency domain and that is given by

∇2E(r, ω) + 0 ( 1) (ω)

ω2/c2

E(r, ω)= −ω2/0c2

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(k0z −ω0t ) + c.c., (13.2.5)

where ω0is the carrier frequency and k0is the linear part of the wavevector atthe 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 − ω0)e ik0z (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

1(ω − ω0)2A=ω2/0c2

P (z, ω)e −ik0z , (13.2.12)

where we have dropped the contribution D2because it is invariably small Wenow convert this equation back to the time domain To do so, we multiply thisequation by exp[−i(ω − ω0)t ] and integrate over all values of ω − ω0 Weobtain

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

˜p(r, t) as

˜

P (r, t) = ˜p(r, t)e i(k0z −ω0t )+ 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

Next we convert this equation to a retarded time frame specified by the

coor-dinates zand τ defined by

13.2 Ultrashort Pulse Propagation Equation 565The wave equation then becomes

∂τ − k2 1

We now make the slowly varying amplitude approximation (that is, we drop

the term ∂2/∂z 2) and simplify this expression to obtain

Note that two of the terms in this equation depend upon the ratio k1/k0 This

ratio can be approximated as follows: k1/k0= v g−1/(nω0/c) = n g /(nω0)

Ig-noring dispersion, n g = n, so that k1/k0= 1/ω0 In this approximation thewave equation becomes

This equation can be considered to be a generalization of the nonlinearSchrödinger 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), andself-steepening (through the presence of the differential operator on the right-hand side) This form of the pulse propagation equation has been obtained byBrabec and Krausz (1997) It can be used to treat many types of nonlinearresponse For instance, for a material displaying an instantaneous third- andfifth-order nonlinearity, ˜p is given by ˜p = 30χ ( 3)| ˜A|2A˜+ 100χ ( 5)| ˜A|4A˜.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,

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

How-∂/∂τ, for reasons of consistency one wants to include in the resulting pulse

propagation equation only contributions first-order in ∂/∂τ Noting that

13.3 Interpretation of the Ultrashort-Pulse Propagation Equation 567one finds that in this approximation the pulse propagation equation is givenby

i ω0

∂

∂τ

˜A 2A.˜(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 ious physical processes described in Eq (13.2.24) As a first step, we study asimplified version of this equation obtained by ignoring the correction terms

var-(i/ω0)∂/∂τ by replacing the factors[1 + (i/ω0)(∂/∂τ )] by unity and by cluding only the lowest-order contribution (known as second-order disper-sion) to ˜D One obtains

in-∂A(r, t)

∂z =



i 2k0∇2

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 ofthree physical effects (the three terms on the right-hand side) The term in-volving the transverse laplacian describes the spreading of the beam due todiffraction, the term involving the second time derivative describes the tem-poral spreading of the pulse due to group velocity dispersion, and the thirdterm describes the nonlinear acquisition of phase It is useful to introduce dis-tance scales over which each of the terms becomes appreciable We definethese scales as follows:

Ldif=1

2 k0w02 ( diffraction length), (13.3.2a)

Ldis= T2/ |k2| (dispersion length), (13.3.2b)

LNL= 2n0c

3ω0χ ( 3) |A|2 = 1

(ω0/c)n2I ( nonlinear length) (13.3.2c)

In these equations w0 is a measure of the characteristic beam radius, and T

is a measure of the characteristic pulse duration The significance of these

distance scales is that for a given physical situation the process with theshortest distance scales is expected to be dominant For reference, note that

for fused silica at a wavelength of 800 nm n2 = 3.5 × 10−20 m2/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 approximatelydoubles 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/ω0)(∂/∂τ )]

on the nonlinear source term of Eq (13.2.25) To isolate this influence, wedrop 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 allowsthe 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 k1∂ ˜ A/∂t = (1/v g )∂ ˜ A/∂t = (n (g)

0 /c)∂ ˜ A/∂t pears explicitly in the wave equation, which takes the form

Note that in the absence of dispersion γ1= γ2 In terms of these quantities,

Eq (13.3.3) can be expressed more concisely as

13.3 Interpretation of the Ultrashort-Pulse Propagation Equation 569The first contribution to the last form can be identified as an intensity-dependent contribution to the group velocity The second contribution doesnot have a simple physical interpretation, but can be considered to represent adispersive four-wave mixing term To proceed we make use of Eq (13.3.6) toexpress Eq (13.3.4) as

ω0

χ ( 3) (ω0)

dχ ( 3) dω

We thus see that the last term in Eq (13.3.4) leads to an intensity dependence

of the group index n gas well as to the last term of Eq (13.3.7), which as justmentioned 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 onthe susceptibility and on its dispersion

The intensity dependence of the group velocity leads to the phenomena ofself-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 infinitelysteep, it is said to form an optical shock wave Self-steepening has been de-

FIGURE13.3.1 Self-steepening and optical shock formation (a) The incident opticalpulse is assumed to have a gaussian time evolution (b) After propagation through anonlinear medium, the pulse displays self-steepening, typically of the trailing edge.(c) If the self-steepening becomes sufficiently pronounced that the intensity changesinstantaneously, an optical shock wave is formed

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 tothese of Eqs (13.3.2) by

˜

A(r, t) = a(r)e −iδωt ; such a field is a monochromatic field at frequency ω0+

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

∇2

⊥a(r) + 2i(k0+ δk) ∂

∂za(r) = 0, (13.3.14)

where δk = k0(δω/ω0) This wave thus diffracts as a wave of frequency ω0+

δω rather than a wave of frequency ω0 More generally, for the case of anultrashort pulse, the operator[1+(i/ω0)∂/∂τ] describes the fact that differentfrequency components of the pulse diffract into different cone angles Thus,

13.4 Intense-Field Nonlinear Optics 571after propagation different frequency components will have different radialdependences These effects and their implications for self-focusing have beendescribed by Rothenberg (1992).

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

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

it appears that pulse self-steepening (Yang and Shen, 1984) leading to opticalshock-wave formation (Gaeta, 2000) is the physical mechanism leading tosupercontinuum generation

13.4 Intense-Field Nonlinear Optics

Most nonlinear optical phenomena∗ can be described by assuming that thematerial polarization can be expanded as a power series in the applied electricfield amplitude This relation in its simplest form is given by

˜

P (t) = χ ( 1) ˜E(t) + χ ( 2) ˜E(t)2+ χ ( 3) ˜E(t)3+ · · · (13.4.1)However, for sufficiently large field strengths, this power series expansionneed not converge We saw in Chapter 6 that under resonant conditions this

= μ ba E/ ¯h

as-sociated with the interaction of the laser field with the atom becomes

compa-rable to 1/T1, where T1is the atomic excited-state lifetime Even under highlynonresonant 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

which corresponds to an intensity of∗

Iat=1

20cEat2 = 4 × 1016W/cm2= 4 × 1020W/m2. (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-mechanicalcalculation of the nonlinear optical susceptibility presented in Chapter 3 pre-supposes that the Hamiltonian of an atom in the presence of a laser field is ofthe form

ˆ

H = ˆH0+ ˆV (t), (13.4.4)where Hˆ0 is the Hamiltonian of an isolated atom and ˆV (t) = −μ ˜E(t)

represents the interaction energy of the atom with the laser field Schrödinger’sequation is then solved for this Hamiltonian through use of perturbation theory

under the assumption V (t) 0 For the case of intense-field nonlinear tics, the nature of this inequality is the reverse—that is, the interaction energy

op-V (t) is much larger than H0 This observation suggests that it should proveuseful to begin our study of intense-field nonlinear optics by considering themotion 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 magneticfield associated with the laser beam We assume the laser beam to be linearlypolarized and of the form ˜E(t) = ˜E(t) ˆx, where ˜E(t) = Ee −iωt + c.c Theequation of motion of the electron is then given by

m ¨ ˜x = −e ˜E(t) or m ¨˜x = −eEe −iωt + c.c., (13.5.1)which leads to the solution

˜x(t) = xe −iωt + c.c., (13.5.2)where

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

polar-ized, to be Eat

13.5 Motion of a Free Electron in a Laser Field 573

The time-averaged kinetic energy associated with this motion is given by K=1

2m  ˙˜x(t)2 or, since

˙˜x(t) = (−iωx)e iωt + c.c., (13.5.4)by

K=e2E2

mω2 =e

2Epeak2

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 μm One finds by numerical evaluation that

the ponderomotive energy is equal to 13.6 eV (a typical atomic energy) for

I = 1.3 × 1014W/cm2= 1.3 × 1018 W/m2, is equal to 4.2 keV for I = Iat

(which is given by Eq (13.4.3)), and is equal to mc2= 500 keV for I = 4.8× 1018W/cm2= 4.8 × 1022W/m2

The equation of motion (13.5.1) and its solution (13.5.2) are linear in thelaser field amplitude Both magnetic and relativistic effects can induce non-linearity in the electronic response Let us consider briefly the influence ofmagnetic effects; see also Problem 1 at the end of this chapter for a moredetailed analysis The electric field of Eq (13.5.1) has a magnetic field as-

sociated with it Assuming propagation in the z direction, this magnetic field

is of the form ˜B(t) = ˜B(t) ˆy, where ˜B(t) = Be iωt+ c.c and where, assuming

propagation in vacuum, B = E/c Since according to Eq (13.5.4) the electron

has a velocity in the x direction, it will experience a magnetic force F= v × B

in the z direction The equation of motion for the z component of the velocity

of the electron motion consists of oscillations at frequency 2ω and amplitude eEB/m2ω3 superposed on a uniform drift velocity The velocity associated

with this motion leads to a magnetic force in the x direction at frequency 3ω.

In similar manner, all harmonics of the laser frequency appear in the atomicmotion.∗

As just noted, relativistic effects also lead to nonlinearities in the atomicresponse The origin of this effect is the relativistic change in electron mass

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

to describe intense-field nonlinear optics Recall that the quantum-mechanicalcalculation of the nonlinear optical susceptibility presented in Chapter pre-supposes that the Hamiltonian... self-steepening (Yang and Shen, 1984) leading to opticalshock-wave formation (Gaeta, 2000) is the physical mechanism leading tosupercontinuum generation

13. 4 Intense-Field Nonlinear Optics< /b>... 2A.˜ (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