Nonlinear Optics - Chapter 12 ppsx

Opop-tical damage thus imposes a constraint on the efficiency of many nonlinear optical processes by limiting the maximum field strength E that can be used to excite the nonlinear respon

Chapter 12

Optically Induced Damage and Multiphoton Absorption

12.1 Introduction to Optical Damage

A topic of great practical importance is optically induced damage of opti-cal components Optiopti-cal damage is important because it ultimately limits the maximum amount of power that can be transmitted through a particular op-tical material Opop-tical damage thus imposes a constraint on the efficiency of

many nonlinear optical processes by limiting the maximum field strength E

that can be used to excite the nonlinear response without the occurrence of optical damage In this context, it is worth pointing out that present laser tech-nology can produce laser beams of sufficient intensity to exceed the damage thresholds of all known materials

There are several different physical mechanisms that can lead to optically induced damage These mechanisms, and an approximate statement of the conditions under which each might be observed, are as follows:

• Linear absorption, leading to localized heating and cracking of the optical material This is the dominant damage mechanism for continuous-wave and long-pulse ( 1 μsec) laser beams.

• Avalanche breakdown, which is the dominant mechanism for pulsed lasers (shorter than 1 μsec) for intensities in the range of 109 W/cm2 to

1012W/cm2

• Multiphoton ionization or multiphoton dissociation of the optical mate-rial, which is the dominant mechanism for intensities in the range 1012

to 1016W/cm2

• Direct (single cycle) field ionization, which is the dominant mechanism for

intensities >1020W/cm2

543

FIGURE12.1.1 For a collimated laser beam, optical damage tends to occur at the exiting surface of an optical material, because the boundary conditions on the electric field vector lead to an enhancement at the exiting surface and a deenhancement at the entering surface

We next present a more detailed description of several of these mechanisms

We begin by briefly summarizing some of the basic empirical observations regarding optical damage When a collimated laser beam interacts with an optical material, optical damage usually occurs at a lower threshold on the surfaces than in the interior This observation suggests that cracks and other imperfections on an optical surface can serve to initiate the process of opti-cal damage, either by enhancing the loopti-cal field strength in regions near the cracks or by providing a source of nearly free electrons needed to initiate the avalanche breakdown process It is also observed (Lowdermilk and Milam, 1981) that surface damage occurs with a lower threshold at the exiting surface than at the entering surface of an optical material One mechanism leading

to this behavior results from the nature of the electromagnetic boundary con-ditions at a dielectric/air interface, which lead to a deenhancement in field strength at the entering surface and an enhancement at the exiting surface This process is illustrated pictorially in Fig 12.1.1 Another physical mecha-nism that leads to the same sort of front/back asymmetry is diffraction from defects at the front surface which can lead to significant intensity variation (hot spots) at the exiting surface This effect has been described, for instance,

by Genin et al (2000).

12.2 Avalanche-Breakdown Model

The avalanche-breakdown mechanism is believed to be the dominant dam-age mechanism for most pulsed lasers The nature of this mechanism is that

12.2 Avalanche-Breakdown Model 545

a small number N0 of free electrons initially present within the optical ma-terial are accelerated to high energies through their interaction with the laser field These electrons can then impact-ionize other atoms within the material, thereby producing additional electrons which are subsequently accelerated by the laser field and which eventually produce still more electrons Some frac-tion of the energy imparted to each electron will lead to a localized heating

of the material, which can eventually lead to damage of the material due to cracking or melting The few electrons initially present within the material are created by one of several processes, including thermal excitation, quan-tum mechanical tunneling by means of the Keldysh mechanism (Ammosov

et al., 1986), multiphoton excitation, or free electrons resulting from crystal

defects

Let us next describe the avalanche-breakdown model in a more quantitative

manner We note that the energy Q imparted to an electron initially at rest and

subjected to an electric field ˜E (assumed quasistatic for present) for a time

duration t is given by

Q = e ˜Ed where d =1

2at2=1 2



e ˜ E/m

or

Q = e2 ˜E2t2/ 2m for t  τ. (12.2.2)

This result holds for times t  τ, where τ is the mean time between collisions.

For longer time durations, the total energy imparted to the electron will be

given approximately by the energy imparted to the electron in time interval τ (that is, by e2 ˜E2τ2/ 2m) multiplied by the number of such time intervals (that

is, by t/τ ), giving

Q = e2˜E2tτ/ 2m for t > τ. (12.2.3) The rate at which the electron gains energy is given in this limit by∗

P =dQ

∗This result can also be deduced by noting that the rate of Joule heating of a conducting material

is given by

NP= 1

2σ ˜ E2, where N is the number density of electrons and σ is the electrical conductivity, which, according to

the standard Drude formula, is given by

σ=(N e2/m)τ

1+ ω2τ2 This result constitutes a generalization of that of Eq (12.2.4) and reduces to it in the limit ωτ 1.

We next assume that the number density of free electrons N (t) changes in

time according to

dN

dt =f N P

where W is the ionization threshold of the material under consideration, P is the absorbed power given by Eq (12.2.4), and f is the fraction of the absorbed

power that leads to further ionization so that 1− f represents the fraction that

leads to heating The solution to Eq (12.2.5) is thus

N (t) = N0 e gt where g=f e2˜E2τ

We next introduce the assumption that optical damage will occur if the

elec-tron density N (T p ) at the end of the laser pulse of duration T p exceeds

some damage threshold value Nth, which is often assumed to be of the or-der of 1018cm−3 The condition for the occurrence of laser damage can thus

be expressed as

f e2˜E2τ T p

The right-hand side of this equality depends only weakly on the assumed

val-ues of Nth and N0 and can be taken to have a value of the order of 30 This result can be used to find that the threshold intensity for producing laser dam-age is given by

Ith= n0 c ˜E2

= 2n0 c W m

f e2τ T p ln(Nth/N0). (12.2.8)

If we evaluate this expression under the assumption that n ≈ 1, W = 5 eV,

τ≈ 10−15 s, T p ≈ 10−9 s, and f ≈ 0.01, we find that Ith  40 GW/cm2, in reasonable agreement with measured values

12.3 Influence of Laser Pulse Duration

There is a well-established scaling law that relates the laser damage threshold

to the laser pulse duration T p for pulse durations in the approximate range of

10 ps to 10 ns In particular, this scaling law states that the fluence (energy per

unit area) required to produce damage increases with pulse duration as T p 1/2, and correspondingly the intensity required to produce laser damage decreases

with pulse duration as T −1/2

p This scaling law can be interpreted as a state-ment that (for this range of pulse durations) optical damage depends not solely

12.3 Influence of Laser Pulse Duration 547

FIGURE12.3.1 Measured dependence of laser damage threshold on laser pulse

du-ration (Stuart et al., 1995).

on laser fluence or on laser intensity but rather upon their geometrical mean

It should be noted that this observed scaling law is inconsistent with the pre-dictions given by the simple model that leads to Eq (12.2.8), which implies that laser damage should depend only on the laser intensity Some possible physical processes that could account for this discrepancy are described be-low Data illustrating the observed scaling law are shown in Fig 12.3.1, and more information regarding this law can be found in Lowdermilk and Milam

(1981) and Du et al (1994).

The T p 1/2scaling law can be understood, at least in general terms, by noting that the avalanche-breakdown model ascribes the actual damage mechanism

to rapid localized heating of the optical material The local temperature

distri-bution T (r, t) obeys the heat transport equation (see also Eq (4.5.2))

(ρC) ∂ ˜ T

∂t − κ∇2˜T = N(1 − f ) ˜P, (12.3.1)

where f , N , and P have the same meanings as in the previous section, κ is the thermal conductivity, and (ρC) is the heat capacity per unit volume Let

us temporarily ignore the source term on the right-hand side of this equation,

and estimate the distance L over which a temperature rise T will diffuse in

a time interval T p Replacing derivatives with ratios and assuming diffusion

FIGURE12.3.2 Illustration of the diffusion of heat following absorption of an intense laser pulse

in only one dimension, as indicated symbolically in Fig 12.3.2, we find that

(ρC) T

T p = κ T

or that

L = (DT p ) 1/2 where D = κ/ρc is the diffusion constant. (12.3.3) The heat deposited by the laser pulse is thus spread out over a region of

dimen-sion L that is proportional to T p 1/2, and the threshold for optical damage will

be raised by this same factor Although this explanation for the T p 1/2 depen-dence is widely quoted, and although it leads to the observed dependepen-dence on

the pulse duration T p, some doubt has been expressed (Bloembergen, 1997)

regarding whether values of D for typical materials are sufficiently large for

thermal diffusion to be important Nonetheless, detailed numerical

calcula-tions (Stuart et al., 1995, 1996) that include the effects of multiphoton

ion-ization, Joule heating, and avalanche ionization are in good agreement with experimental results

12.4 Direct Photoionization

In this process the laser field strength is large enough to rip electrons away from the atomic nucleus This process is expected to become dominant if the

peak laser field strength exceeds the atomic field strength Eat= e/4π0 a20=

5× 1011V/m Fields this large are obtained at intensities of

Iat=1

2n0cEat2 ≈ 4 × 1016W/cm2= 4 × 1020W/m2.

12.5 Multiphoton Absorption and Multiphoton Ionization 549

For laser pulses of duration 100 fsec or longer, laser damage can occur at much lower intensities by means of the other processes described above Di-rect photoionization is described in more detail in Chapter 13

12.5 Multiphoton Absorption and Multiphoton Ionization

In this section we calculate the rate at which multiphoton absorption processes occur Some examples of multiphoton absorption processes are shown schematically in Fig 12.5.1 Two-photon absorption was first reported ex-perimentally by Kaiser and Garrett (1961)

Some of the reasons for current interest in the field of multiphoton absorp-tion include the following:

1 Multiphoton spectroscopy can be used to study high-lying electronic states and states not accessible from the ground state because of selection rules

2 Two-photon microscopy (Denk et al., 1990 and Xu and Webb, 1997) has

been used to eliminate much of the background associated with imaging through highly scattering materials, both because most materials scatter less strongly at longer wavelengths and because two-photon excitation pro-vides sensitivity only in the focal volume of the incident laser beam Such behavior is shown in Fig 12.5.2

3 Multiphoton absorption and multiphoton ionization can lead to laser dam-age of optical materials and be used to write permanent refractive index structures into the interior of optical materials See for instance the articles listed at the end of this chapter under the heading Optical Damage with Femtosecond Laser Pulses

FIGURE12.5.1 Several examples of multiphoton absorption processes

FIGURE12.5.2 Fluorescence from a dye solution (20 μM solution of fluoresce in

water) under (a) one-photon excitation and (b) two-photon excitation Note that under two-photon excitation, fluorescence is excited only at the focal spot of the incident laser beam Photographs courtesy of W Webb

4 Multiphoton absorption constitutes a nonlinear loss mechanism that can limit the efficiency of nonlinear optical devices such as optical switches (see also the discussion in Section 7.3)

In principle, we already know how to calculate multiphoton absorption rates

by means of the formulas presented earlier in Chapter 3 For instance, the

lin-ear absorption rate is proportional to Im χ ( 1) (ω) Similarly, the two-photon

absorption rate is proportional to Im χ ( 3) (ω = ω + ω − ω) We have already

seen how to calculate these quantities However, the method we used to

cal-culate χ ( 3) becomes tedious to apply to higher-order processes (e.g., χ ( 5)for three-photon absorption, etc.) For this reason, we now develop a simpler

ap-proach that generalizes more easily to N -photon absorption for arbitrary N

12.5.1 Theory of Single- and Multiphoton Absorption and

Fermi’s Golden Rule

Let us next see how to use the laws of quantum mechanics to calculate single and multiphoton absorption rates We begin by deriving the standard result for the single-photon absorption rate, and we then generalize this result to higher-order processes

The calculation uses procedures similar to those used in Section 3.2 to cal-culate the nonlinear optical susceptibility We assume that the atomic

wave-12.5 Multiphoton Absorption and Multiphoton Ionization 551

function ψ(r, t) obeys the time-dependent Schrödinger equation

i ¯h ∂ψ(r, t) ∂t = ˆH ψ(r, t), (12.5.1) where the Hamiltonian ˆH is represented as

ˆ

Here ˆH0is the Hamiltonian for a free atom and

where ˆμ = −eˆr, is the interaction energy with the applied optical field For

simplicity we take this field as a monochromatic wave of the form

that is switched on suddenly at time t= 0

We assume that the solutions to Schrödinger’s equation for a free atom are known, and that the wavefunctions associated with the energy eigenstates can

be represented as

ψ n (r, t) = u n (r)e −iω n t , where ω n = E n / ¯h. (12.5.5)

We see that expression (12.5.5) will satisfy Eq (12.5.1) (with ˆH set equal

to ˆH0) if u n (r )satisfies the eigenvalue equation

ˆ

We return now to the general problem of solving Schrödinger’s equation in the presence of a time-dependent interaction potential ˆV (t):

i ¯h ∂ψ(r, t) ∂t = ˆH0+ ˆV (t)ψ(r, t). (12.5.7) Since the energy eigenstates of ˆH0 form a complete set, we can express the solution to Eq (12.5.7) as a linear combination of these eigenstates—that is, as

ψ(r, t)=

l

a l (t)u l (r)e −iω l t (12.5.8)

We introduce Eq (12.5.8) into Eq (12.5.7) and find that

i ¯h

l

da l

dt u l (r)e

−iω l t + i ¯h

l ( −iω l )a l (t)u l (r)e −iω l t

l

a l (t)E l u l (r)e −iω l t+

l

a l (t) ˆ V u l (r)e −iω l t , (12.5.9)

where (since E l = ¯hω l) clearly the second and third terms cancel To simplify

this expression further, we multiply both sides (from the left) by u∗

m (r)and integrate over all space Making use of the orthonormality condition



u∗

m (r)u l (r) d3r = δ ml , (12.5.10)

we obtain

i ¯h da dt m =

l

a l (t)V ml e −iω lm t , (12.5.11)

where ω lm = ω l − ω mand where

V ml=



u∗

m (r) ˆ V u l (r) d3r (12.5.12) are the matrix elements of the interaction Hamiltonian ˆV Equation (12.5.11)

is a matrix form of the Schrödinger equation

Oftentimes, as in the case at hand, Eq (12.5.11) cannot be solved exactly and must be solved using perturbation techniques To this end, we introduce

an expansion parameter λ which is assumed to vary continuously between zero and one; the value λ= 1 is taken to correspond to the physical situation

at hand We replace V ml by λV ml in Eq (12.5.11) and expand a m (t)in powers

of the interaction as

a m (t) = a ( 0)

m (t) + λa ( 1)

m (t) + λ2a m ( 2) (t) + · · · (12.5.13)

By equating powers of λ on each side of the resulting form of Eq (12.5.11)

we obtain the set of equations

da m (N )

dt = (i ¯h)−1

l

a (N −1)

l V ml e −iω lm t , N = 1, 2, 3, (12.5.14)

12.5.2 Linear Absorption

Let us first see how to use Eq (12.5.14) to describe linear absorption We set

N= 1 to correspond to an interaction first-order in the field We also assume

that in the absence of the applied laser field the atom is in the state g (typically

the ground state) so that

a g ( 0) (t) = 1, a l ( 0) (t) = 0 for l = g (12.5.15)

for all times t Through use of Eqs (12.5.3) and (12.5.4), we represent V mgas

V mg = −μ mg



Ee −iωt + E∗e iωt