Two-photon absorption of optical and x-ray quanta

Volume

8, number

4

August

OPTICS COMMUNICATIONS

TWO-PHOTON

ABSORPTION

OF OPTICAL

AND X-RAY

197 3

QUANTA

Isaac FREUND Department

of Physics, Bar-llan University, Ramat-Gan, Israel Received

29 May 1973

Two-photon absorption of an optical (laser) and an x-ray quantum is considered within the context of the Born approximation. A simple expression relating the two-photon and one-photon photoionization absorption coefficients is obtained. If the upper state is one of the conduction bands of a solid, the two-photon absorption coefficient is shown to peak at points in the zone for which the curvature of the band is a maximum. Numerical estimates show that with a 100 kW Nd laser, the intensity of the two-photon absorption process is a few percent of that of the one-photon process.

Two-photon absorption [I] has become in recent years an important spectroscopic tool. The most prevalent experimental technique involves the study of the change in the absorption coefficient at a near ultraviolet frequency from a spontaneous emission source due to the simultaneous irradiation of the sample by an intense optical laser. Here we develop a first approximation to the theory of a similar experiment in which the near ultra-violet frequency is replaced by one in the x-ray region, Since, in general, absorption in the x-ray region corresponds to photoionization, we treat the final state of the electron within the framework of the zeroth-order Born approximation [2]. This enables us to obtain a simple relationship between the one- and two-photon absorption coefficients at the x-ray frequency ox, to examine for solids the qualitative dependence of the process on the band structure, and to assess the feasibility of experimentation. The two-photon absorption coefficient ZJ(~)(W,) may be written @)(w

x

) = grr3cr2 Ym& Nti

QlMfi12>

where m(A .A) arises in first-order perturbation theory from the A2 term in the interaction hamiltonian, while m(p.A) arises in second order perturbation theory from the correspondingp.A term. A convenient form for these matrix elements which is consistent with the present treatment has been given previously [4]. For transitions between discrete, non-degenerate states it may be shown that

mwX

lim

m&.A)

Wo"O

(3)

so that when w. is small compared to the frequencies of all characteristic (x-ray) resonances, M, goes to zero with w,. However, if the final state is degenerate or is part of a continuous set, the small u. limit is non-zero and may be directly evaluated. If the continuum of states is labeled by the wave vector K, and the final state by K = k, we have*

Mki= +iix.ii-o)wo x

(1)

where w, is the frequency a:d Z, the intensity of the optical laser, N is the number density of absorbing atoms in the sample and CYthe fine structure constant. Q is the “Q” of the transition and is given as Q = w,p(w,), with dmx) the density of final electronic states. The matrix elementsMn [3] we write as

mfi(A-A) =

lim Wo'O

s

(k(zlK)(KI(a/az)li)[o,

(4) tA(k,K)]-ldK.

where cj is a unit polarization vector along the field at Wj, A(~,K) is the transition frequency from the state

l

In obtaining eq. (4) we consider the system to be enclosed in an impenetrable container so that the surface integral of I K) vanishes. We have also set exp(&, -r) equal to unity on the basis of the justification given in ref. [4].

Mfi = y&A .A) + ms(p.A), 401

Volume

8, number

4

I& to the state sidered

OPTICS (‘OMMUNI<‘ATIONS

the matrix

IK),

to be a sharply

spherical

symmetry

We evaluate

element

peaked

function

(klzl~)

August

is conT--

of k ~--K, and

T---r7

has been assumed.

eq. (4) within

Born approximation

the framework

for a K-shell

--r/a

Ii)

= (1/7r’V)e

A(~,K)

= (fi/.Im)(~*

of the

electron,

i.e. we write

/

(5b)

-- k’).

(5c)

I’

_-*-’

= (2mn/h)k

P&J

Eq. (5d) implies directions

sin 8 de.

(54

that eq. (1) is to be integrated

of emission,

Rather that1 display we present, instead,

0, of the photoionized

127&a

!J’%+J

5

01

directly the results for p(*)(w,) the ratio of this quantity to the one-

10

p(l)(wy)

nlw<;

In obtaining

eq. (6) we neglect terms of order 2 lo-*O and ke2 5 IO-l8 in comparison with W/mu , - 5 x lo-l6 for a Nd laser. We compute +(ka)2]

,u(*)/p(*) details

because,

as is immediately

of the calculation,

ficiences

of both

many

apparent

from the

of the well-known

the Born approximation

initial state wave function

I

5.0

o

1

1

IO

Fig. 1. Two-photon, M(2) (-Irand one-photon, ,u(‘)(), absorption coefficients for Cu versus wavelength in A. The twophoton curve is computed from the one-photon one by means of cq. (6). and an arbitrary relative intensity scale is employed. The results displayed arc per K- or L-shell electron, the total L.-shell abwrption is, of course, a factor of four times larger.

a direct test of the utility of the analysis presented We turn now to an assessment of the feasibility

iJo

Here n is the frequency of the absorption edge of the electron shellt (K, L, etc.), the x-ray source is assumed to be unpolarized, and the optical laser to be planepolarized.

1.0

,,’

w* ~~R (6)

a2/[1

0.5

_-’

over all electron.

photon photoionization absorption coefficient computed with the same set of approximations: /-J2+JxI

1973

are thereby

de-

[2] and our

experimentation. From eq. (6) it would appear that a very small value for wg would greatly enhance the intensity of the two-photon absorption. Our treatment neglects damping, however, so that a practical lower limit for wg is the inverse In the x-ray region direct

a transition

from,

lifetime

this lifetime

recombination

tinuum lifetime

cancelled,yielding

here of

of the upper

is not determined

since the oscillator for example,

to the K-shell is determined

the bottom

strength

state. by for

of the con-

is approximately ew4 [6]. The rather by the filling of the K-shell

an ultimately more reliable estimate for PC*). In fig. 1 we plot both the one-photon [5] and our calculated two-photon [eq. (6)] absorption coefficients for Cu on

vacancy by an L-shell electron. This implies that a suitable damping factor may be obtained from a mcasure-

an arbitrary

by Das Gupta and Welch [7], and confirmed by Eisenbelger et al. [8], the appropriate damping factor is, in

differences immediately

relative

intensity

scale. The major

may be seen to be two-fold: rising to a maximum

qualitative

first, instead

at an absorption

of edge

as does p(l), AL(*)starts at rero. Second, whereas the maxima in p(l) scale approximately as R-’ [6] , so that for Cu the maximum absorption /L-shell electron is approximately IO times that for a K-shell electron, for P 0) the L- and K-shell maxima arc more nearly the same height. A measurement of this spectrum would serve as t Smce the parameters

in eq. (6), this equation from different shells.

402

of the initial state, Ii), no longer appear is applicable to two-photon absorption

ment

of the natural

width

of the Ka lines. As discovered

fact, the classical value [9] 2e2fin/33r?zc3 = 9 x 1014sec-1 for Cu-Ka. A Nd laser, whose frequency is twice this value, provides, therefore, a suitable source. Because experimentation is greatly facilitated by a high repetition rate, we assume here the very modest value I, = 1 O7 W/cm* This may be conveniently obtained from a (commercial) 1o- 100 pps, - 100 kW Nd-YAG laser focused to a 1 nln? spot size. Since the laser is necessarily Q-switched, a matching pulsed x-ray source may be conveniently constructed as described by Passner = 1.4 x 1019sec-1 further wy = ?R i

[lo] Assuming for the Cu K-shell

Volume 8, number 4

OPTICS COMMUNICATIONS

in the x-ray region will prove to be of special significance to solid state spectroscopy.

we have from eq. (6)

/.I(~)//..&~)- 2-3s.

(7)

Considering this comparatively mild experimental conditions that observation within

large value and the very we have assumed, it appears

of the process

the realm of current

described

the particular

not necessarily

are no longer very suitable.

tion of the details

of the derivation

An examina-

reveals, however,

that

,(2)//.L’1’ = (u. . V/p(k) I2) where E(k) is the conduction photon points

absorption

process

in the zone for which

band energy. is thus expected

The twoto peak at

the band curvature

Since such points are normally

is a

not emphasized in one-photon absorption, and since a determination of regions of maximum and minimum curvature suffices to define the major features of the band, we anticipate that measurements of two-photon absorption

maximum.

Phys. Rev. Letters 7 (1961) 229. [ 21 W. Heitler, The quantum theory of radiation (Oxford Univ. Press, London, 1954) sec. 2 I. [3] M. Goppert-Mayer, Ann. Physik 9 (1931) 273; N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965) ch. 2.8. [4] I. Freund, Opt. Commun. 6 (1972)421. ]5] International tables for x-ray crystallography (Kynoch Press, Birmingham, England, 1965). [ 61 R.W. James, The optical principles of the diffraction of xrays (Cornell University Press, Ithaca, 1965) ch. 4. [7] K. Das Gupta and H. Welch, Phys. Rev. Letters 21 (1968) 651. [ 81 P. Eisenberger, N.G. Alexandropoulos, and P.M. Platzman,

[l]

technology.

form of eq. (6), though

References

here is well

If the final state of the electron is one of the conduction bands of a solid, eqs. (5~) and (5d) and hence its magnitude,

August 1973

W. Kaiser and C.G.B. Garrett,

Phys. Rev. Letters 28 (1972) 1519. [9] A.H. Compton and S.K. Allison, X-rays in theory and experiment (D. Van Nostrand, Princeton, 1935) ch. 4. [lo] A. Passner, Rev. Sci. Instr. 43 (1972) 1640.

403