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HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
CHAPTER
24
HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
24.1 The Kramers-Heisenberg Formula The processes of absorption and emission described in the previous chapter are one-photon processes since in every transition the field either loses or gains a photon while the atom undergoes a corresponding change of state to keep the total energy (atom plus field) constant. There are other important interactions between an atom and a radiation field which involve a change of more than one photon. In a general scattering event, for example, the field loses an incident photon characterized by k2 and gains another photon—the scattered one—characterized by kT. With laser sources, nu merous multiphoton processes have been observed although the present discussion will be restricted to several aspects of two-photon interactions. In the absence of an external magnetic field the Hamiltonian describing the interaction of an electron with a radiation field is Jt> int — Ή
1
i J€ 2 ·>
$ex = (e/mc)A · p,
Jf2 = (e2/2mc2)A2
(24.1-1)
where A is given by (22.2-17). As in (22.3-8) let with je{~] and Jf [+) given by (22.3-9). For j f 2 we have ne2h ^ ^ ^2 = — v Σ Σ
1 ,
^ (ekA · er*')
x (flue**r + ahe~ik'τ)(α^λ^''r 536
+ d^e'^'r)
(24.1-2)
24.1 THE KRAMERS-HEISENBERG FORMULA
537
ne2h,:
Σ Σ , (Skit-er*') mV kA kA' ^ ω ^ α ν x { 4 Α 4 Ά ' e x p [ - i ( k + k ) · r] + ^ Α ^ Ά ' e x p [ i ( k + k ) · r]
+ αίλ^Ά' e x p [ - i(k - k ) · r] + ^ Α 4 Ά ' exp[j(k - k') · r]} = &ü*eül>
+ ^ί-χ^ί^
+ J T Ü ^ W + jrtÄ^fö
(241-3)
in which J f ü ^ f ö refers to the term containing αΙλα{λ>\ Jf^Jf^ to the term with akA^kA', etc. Matrix elements of α^λ and α^ may be obtained from (22.2-8):
I^AKA)
= yjnu + 1,
= V"kÄ
(24.1-4)
which lead directly to
1|4Α4'Α'|%Α,^Ά'>
= V ( n ^ + l)(Wk'A' + 1),
= V("kA + l)^k'A',
.
= V ^ K ^ T T ) , = V^kA^k'A'.
Since the photon modes are independent we also have [flkA,flk'A'] = [ökA,4'A'] = [ 4 A , 4 ' A ] = 0. The only nonvanishing matrix elements of αίχ and akA are those which involve a change of one photon. For all bilinear combinations of atx and and two photon modes with occupation numbers nkA and nk>x> are present. The total system is in the state |I> = |i;nkA,nk'A'>
(24.1-6)
with energy (not including the zero point energy) Ei = Ex + nkXh(uk + Mk'A'fotfk'.
|f>
(24.1-7)
·
If)
Scattering
II> li;n k x # f c r x ,>
lF>=lf;nkx-1,nkV+1>
FIG. 24.1 Nomenclature for photon scattering. Id and kT refer to the incident and scat tered photons, respectively.
538
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
After scattering has occurred, ηκλ> is increased by one photon and nkA is decreased by one photon so that the total system is now in the state |F> = | f ; n k A - l , n k ^ + l>
(24.1-8)
E¥ = Ef + (nk, - l)fccok + {nwx. + l)ftü*.
(24.1-9)
with energy The energy E{ relative to E{ is arbitrary and the scattered photon may have more or less energy compared with the incident photon. This represents the general case of Raman scattering and includes elastic scattering as a special case. From (24.1-5) it follows thatjrfö|i> jrWJi> =
**h mV
M » - + 1 ) ( g k A . g k , A , ) < % i ( k - k '»- 1 i > .
yJ
cOkCOk'
Therefore
= 2 ^ mV
K(n„ yj
+
1)
cOkCtv
k>
|ei(k-k,,.r|
The scattering process is regarded as a two-photon process in the sense that an incident photon of specified energy, polarization, and propagation direction symbolized by the indices kA is scattered so that the emerging photon may have a different energy, polarization, and direction of propaga tion symbolized by the indices kT. Hence a kA photon has been lost and a k T photon has been gained. Matrix elements of Jf i will be of type (24.1-4) in which there is a change of one photon. Therefore Jf i cannot contribute to scattering in the first order of perturbation theory but may contribute in a higher order. The matrix elements of Jif2, on the other hand, are nonvanishing when there is a change of two photons; first order contributions from 34?2 are therefore expected. The contribution of 2tf\ in second order may be obtained from (14.4-7): Wkm = (2n/h)\
(24.1-11)
with to second order given by
<
W
= <*,.>>+ I.
(24,.12)
Now let |m> = |I>,
|/c> = |F>
where |I> and |F> are given by (24.1-6) and (24.1-8). For the intermediate
24.1 THE KRAMERS-HEISENBERG FORMULA
ID
IL,>
IF)
IL2>
IF)
539
FIG. 24.2 Contributions from p · A term in second order to photon scattering [see Eqs. (24.1-15) and (24.1-16)].
states |/> there are two possibilities 1}
~
(24.1-13)
[|L2> = |/ 2 ;n k A ,n k ^ + 1>.
Hence there are two pathways from |I> to |F>:
pathl: lO-lLO-lF), path 2:
(24.1-14)
|I>-^ |L 2 >-+|F>.
In the first path the transition |I> -► |Li> involves the loss (absorption) of a Id photon accompanied by an atomic transition |i> -► |/i>; the transition |Li> -> |F> involves the gain (emission) of a k T photon with an atomic transition \l{y -► |f >. In the second path there is an emission of a k T photon with an atomic transition |i> -► |/2> in the step |I> -► |L2> and an absorption of a k2 photon with an atomic transition |/2> -► |f > in the step |L2> -* |F>. It is important to note that in each path the intermediate state differs from the initial and final states by just one photon. These steps may be symbolized by diagrams as in Fig. 24.2. In Section 22.3 we saw that Jf(1~) was associated with the absorption of a single photon and Jif[+) with the emission of a single photon. Therefore the matrix elements in the two pathways are the following.
ForlO^lM-lF):|L 1 >|I> =
+ l\^\+%;nkX
-
l,nk.x,}
+)
x 2ne2h m
"kA(«k'A· + 1) COkCOk'
x.
(24.1-15)
540
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
For|I>-+|L2>-+|F>:|L 2 >|I> = r 2
2ne h
+
l\3*?[ \i;nkX,nk>x>}
\nkX{nk>X' + 1)
= ^2TT
+ 1>
+)
/
,
,
< f \e
(ekA'P)/2>
x.
(24.1-16)
Both pathways contribute to the second-order term in (24.1-12). On setting Ei - ELi = Ei-
Eh + k o k ,
£i - £ L 2 = E{ - Eh - hcok>, (24.1-17)
the second-order term becomes 2ne2h
lnkk{nk>k> + Γ)
m 2V
\j y
|
cokcok'
yRf\e~ik''r(ek^ ι\_
ρ)|ίι> Ei — Ei + ha>k
Ei — Ei — ha>k'
(24.1-18)
in which the general summation index / indicates the sum over all inter mediate states accessible to one-photon transitions. The first-order contribution due to the Hamiltonian J f 2 was given by (24.1-10) and is to be added to (24.1-18) to obtain the total matrix element for the scattering process. We shall also adopt the dipole ap proximation whereby all exponentials are set equal to unity. Thus /irl^
Ιτ\
2ne2h
\
= —y- 7
n
^
n
^ + !) L
ω ι ^,
*
x*
f - · ^)«n
, 1 yr
Ei — Ei -\- h(Dk
(24.1-19)
This matrix element may now be inserted into (24.1-11) with the δ func tion replaced by a density of final states (23.1-7). The transition probability per unit time that an incident kA photon has been scattered into the element
24.1 THE KRAMERS-HEISENBERG FORMULA
541
of solid angle dQ as a k T photon then becomes (e k A-e k 'A')^fi
\mcz I V cok
^ "
1
pl/>"" 2 + E[- Ei- hcok>
(24.1-20)
Equation (24.1-20) is known as the Kramers-Heisenberg dispersion formula; it may also be expressed as a differential scattering cross section, do dQ
number of k T photons scattered/sec sr number of incident k/l photons/sec cm 2
w/dn nkc/V = r 0 2 ^ ( n k ^ + l) cok
(e k A-e k 'A')c>fi
1 y Rek-A- - Pri] )(ekA - pii) (ekA * Pn)(ek-A' * Pa) E E + fta>k Ei — Ei — ftcok' L i~ i
+m r
(24.1-21)
where r 0 = e2/mc2 = classical radius of the electron, Pn =
Ρϋ = .
Formula (24.1-21) for the differential cross section contains the factor (ηΐιΆ' + 1); hence the cross section is a sum of two terms one of which is proportional to nkA', the number ofscattered photons. This term corresponds to stimulated Raman scattering; it contributes negligibly at ordinary inten sities, but produces important effects at high intensities. The differential cross section (24.1-21) becomes infinite when one of the denominators vanishes. This is a consequence of the assumption of infinitely sharp atomic states. Near resonance it is therefore necessary to include a damping factor Γ as was done in Section 23.5. The differential cross section, including damping, is do -j^=r02 ail
Ί
cok'
"k'A'+ 1)
(ekA -ek'A')c>fi
cok 1 y Γ (ek'A' Pn)(ekA · PH) + m^lEi-l Ei + ftcok + τϊΓ
(ek;i · p«)(ie k 'A"Pn) 11
+ Ei — Ei —ha>v ha>\ + ? / r j |
(24.1-22)
542
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
The intermediate states |/> which appear in the Kramers-Heisenberg dispersion formula, as well as the initial state |i> and the final state |f > are all eigenstates of the atom, i.e., solutions of the atomic Schrödinger equa tion. By virtue of the existence of matrix elements between |i> and |/> and |/> and |f >, the two states |i> and |f > can interact via two-photon processes. However, the transitions |i> -> |/> and |/> -> |f > are not observable, and it is because of this feature that the states |/> are sometimes called "virtual". We prefer to avoid such terminology because it may give the impression of states whose existence is nebulous and whose properties are not defined which, of course, is not the case at all. 24.2
Scattering—Special Cases
A number of applications of the differential cross section formulas (24.1-21 or 24.1-22), not including the stimulated portion, will now be given. If a>k = cok' in (24.1-21) and hco » \E[ — Et\9 the sum over intermediate states may be neglected and the differential cross section is da/dQ = r02 Sn(ekX - ek'i>)
= r o 2 cos 2 0<5 fi
(24.2-1)
where Θ is the angle between the two polarizations. Since the incident and scattered photons are of the same energy (elastic scattering), the scattering atom remains in the same state and <5fi = 1. The total cross section then becomes σ
= ^(άσ/άΩ) dQ =
8ΤΓ/3Γ02.
(24.2-2)
This is known as the Thomson cross section; it is due entirely to the A2 term in the interaction Hamiltonian. Since the atomic states do not appear in the cross section, Thomson scattering may be regarded as the scattering of photons from free electrons. Formula (24.2-2) is the total cross section when the incident light is unpolarized. If this is not the case, a more detailed geometrical analysis is required (Louisell, 1973). The Kramers-Heisenberg formula (24.1-21) may be converted to another form which is more adaptable for discussions of Raman and Rayleigh scattering (Dirac, 1958). The notation will first be simplified by writing ekA = e
ek'A' = e'
nkA = n
nWk> = ή
cok = ω
cok' = ω '
(24.2-3)
24.2 SCATTERING—SPECIAL CASES
543
Now consider the commutator [e'-r,e.p]. Since [rhPk\
= ihdjk,
(24.2-4)
the expansion of (24.2-4) into its components results in [ e ' . r , e - p ] = ifte'-e
(24.2-5)
= ift = ihe · eSn.
(24.2-6)
from which one obtains
But = X [
£,)
(24.2-7)
(24.2-8)
with a corresponding expression for , we have Σ [(£i - £i)
- (£ f - £,)
(24.2-9)
Also, Σ [- = = 0
(24.2-10)
since the components of r commute with one another. When (24.2-10) is multiplied by hco' and added to (24.2-9) the result is ffl F
2
[(£,-£
i
+
- (Et -E,
feB') + fttt/)
(24.2-11)
544
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
We now replace the matrix elements in (24.1-21) by expressions of the type (24.2-8) a n d e' · e<5fi by (24.2-11). T h e result is „2
do ail
2
ω m ~ ( « ' + l ) ^ co
£|
(£, -Ej
+ fao')(£i - E, + tuo) - (£ f - £,)(£, - £Q £i - Ei + \ux>
■
dn
= r02wco'\ri
+ l)w 2
f
|
E[ — Ei — ίιω'
(24.2-13)
This is the general form for R a m a n scattering. T h e energy of the scattered radiation (ha>') can be lower or higher than the energy of the incident radiation (ftco). The process is, therefore, inelastic a n d involves energetic changes in internal degrees of freedom. Stokes a n d anti-Stokes shifts are terms used to describe scattering in which \ιω' < hco a n d hco' < hco, respectively. R a m a n scattering from molecular vibrational modes is discussed in Section 27.4. Rayleigh scattering occurs when ω' = co and hco « \Ei — Et\. Inserting these conditions into (24.2-13), with |i> = |f >, do_ = r 0 2 m 2 co 4 dQ
' . <ί|8τ|Ζ)<Ζ|8'τ|ί)
Ei-El
+■
Ei-Ei
(24.2-14)
which yields the familiar result that at long wavelengths the scattering cross section varies inversely as the fourth power of the wavelength. 24.3
Diagrams
Starting with the Fermi golden rule (24.1-11) in which the general form of the matrix element is = Rkm = Rü> + Äg> + · · ·
(24.3-1)
24.3 DIAGRAMS
545
where R% =, nm_ymm>
^km
I? 17 Üm — £/
ZJ
/
'
*e=Σ Σ;
(24 3.2)
(Em — Ev)(Em — Ei) it is possible to symbolize these expressions by means of diagrams which are helpful in writing the pertinent matrix elements associated with a particular process. We use the simplified notation (24.2-3). Single photon absorption. |I> = |i;n> |F> = | f ; n - 1> \
Ätf> = = |I>
ll>
IF)
e I2nhn m
Single photon emission.
, ik
(24.3-3)
,
|I> = |i;n> |F> = |f;n + l> RW = = < F | M + ) | I >
(24.3-4)
1) f .e _Ikr .... I2nh(n + 1) _e \2%h{n i k . re_· m yj ω¥ Processes involving two photons (second order processes) are scattering, two-photon absorption and two-photon emission. Scattering has already been discussed in Section 24.1; the pertinent diagrams are shown in Fig. 24.2. Two-photon absorption. n>
IF>-
|I> = |i;n,w'>, |F> = jf;n - Ι,π' - 1>, R(2)^
(FiM-lLxXL!^-^) Li
Ei — ELl
(
|Li> = \h;n - l,n'>, |L 2 > = \l2;n,n' - 1>,
y|L2> Ei — ELl
L2
= M3 + M 4 ,
(24.3-5)
2
M
* 2π6 ^ ^ < Γ Κ " ^ - ρ ) | / ι > < / ι Κ - ' ( 8 - ρ ) | ί > 3 = —2 -ΓΓ / 7 Σ — ^— r—ΓΛ ' mz V yj ωω if E{- Eh + ftco
\
ll>
IL,>
y
IF)
(24.3-6)
546
M4 =
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
f ! M p 7 y
^
(243 _ 7)
^
ID
IL2>
IF)
Two-photon emission. |I> = |i; η,η'), |F> = |f;n + Ι,η' + 1>, D(2)_V
+
|Li> = |/i; n + l,n'>, |L2> = j/ 2 ;n,n' + 1>,
'|L1> ,
v
= M5 + M6, =
^2πδ
(24.3-8)
/(»+ l)(n' + 1) y
t
Me =
k r
/? II)
r
(24.3-9)
/A IL,>
IF>
e^ 2nh Hn + l)(n' + 1) ^ m V yj ωω' Ex — Et2 — Ηω' l2 (24.3-10)
ll>
'//
IL2>
f
IF>
All the matrix elements considered thus far were of the operator Jf t in (24.1-1); for Jf2 (24.1-2) and (24.1-3) we have the following: Scattering |I> = |i;n,n'>,
|F> = |f;n - ί,η' + 1>,
Ktf' = = 2 = 2 = ί! 2π*
^ ρ ^ . ^
ll>
t _ /(k _ k0 . r]|i>>
IF)
(24Μ1)
24.4
OPTICAL SUSCEPTIBILITY AND NONLINEAR EFFECTS
547
Two-photon absorption. |I> = |i;n,n'}, RW =
|F> = |f;n - Ι,η' - 1>,
=
= T- ljv- J^-> ®' s')· 2m V v ωω
ll>
( 24 · 3 - 12 )
IF)
Two-photon emission. |I> = |i;n,n'>,
|F> = |f;n+l,n' + l>
KÖ = = Jft^|I> , e2 2πδ /(n + \)(ri + 1) Ί| ' (e-e). (24.3-13) 2m V v (oo)' k' IF)
ID
Note that in dipole approximation there is no contribution from the A2 term to two-photon absorption or emission; as shown previously the contribution to scattering occurs only in the elastic case. 24.4 Optical Susceptibility and Nonlinear Effects For a system of N identical particles (e.g., atoms, molecules, etc.) let the Hamiltonian be JiT = J^o + Mf" (24.4-1) with JT = - E ( r , i ) D (24.4-2) where D is the dipole moment operator eR and Jf' = 0 at t = t0. In terms of the individual particles
#o = Σ ^n\ n
D = X D„ n
(24.4-3)
with the assumption that operators associated with different particles com mute. The macroscopic polarization is defined by P = = TrpD.
(24.4-4)
In order to calculate P it is necessary to calculate the density operator p in the presence of thefieldE. For this purpose we refer to (13.5-9) which gives
548
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
the expansion of the density operator in the interaction representation: Px(t) =
pf°>(i) + p['\t) + p[2\t) + ■■■
(24.4-5)
where Pf0)(0 = Piihl p\1\t) = (i/h)^odt1\_Pl(to),jel'(t1)-] p\2\t) = (i/h)2 j!o dh £ ' dt2 [[p,(io), Jf ,'(i2)], Jfi'(h)l
etc., (24.4-6)
and = - E(r, f) · eije°'lh De ~ **<*' = - E(r, i) · D,(t).
(24.4-7)
If the system is in thermal equilibrium at the time the perturbation is applied, pi0)(r) = P,(io) = PT where, according to (13.4-7), p T = e-*oikTiYte-*olkT = (i/z) e -*o/*r
(24.4-8) (244.9)
One may now expand P to various orders: P = P<0) + P (1) + P (2) + · · ·
(24.4-10)
P (0) = Trp T D.
(24.4-11)
P(1,(i) = Trp (1) D = Tr[e-üro,/*pT)eijro,/*D]
(24.4-12)
with the leading term The next term is or, in component form, P$\t) = Tr[e_ljro"*pl1)ew'0,/* D J
(24.4-13)
= Tr j e-^0"" [' dh E(r, h) · [D,^),p T ]e üro,/ *D # ,. (24.4-14) Expanding the scalar product, PV\t) = ^ T r e - « - " ' » { £ Ä! Σ£.(r,t 1 )[A.(tiXpT]|e«'o' / »I) M = l- £ d tl Σ £.(r, t d T r i e - ^ I T M t ! ) , ρ τ ] ^ < " / Α D„}.
(24.4-15)
24.4 OPTICAL SUSCEPTIBILITY AND NONLINEAR EFFECTS
549
The expression inside the trace may be simplified somewhat since = [Αα(ίι - ή,Ρτ] = [Α.(ί'),Ρτ] (24.4-16) where t' = ii - t.
(24.4-17)
Also, since the trace of a product of operators is invariant under a cyclic permutation of their order, Tr{[DIa(0, p T ] Dß} = Tr{pT[D„, Αα(ί')]}.
(24.4-18)
The electric field Ea(r, ii) may be expressed in terms of its Fourier com ponents. Suppressing the dependence of the field on the spatial coordinates, E*(t1) =
f"aoEa(co)e-imtidto
= f ^ Λϋ£ β (ω)β- ίβΛ β-' ω(ί| "°
= Ρ ^ dcoEa((D)e-i(0te-i(ut'.
(24.4-19)
Inserting (24.4-18) and (24.4-19) into (24.4-15) with t0 = - oo and t = 0, Wit) = $"„ d«> Σ |"(i/Ä) J!«, Λ'Τ Γ { Ρτ [ϋ μ ,ϋ Ια (0]}^ ίωί Ί£:α(ω)β- ίωί . (24.4-20) This expression may be compared with (14.6-29) written in component form; Ρ(Λή = Γ
Σή1Λω)Εα(ω)άωβ-ίωΐ
(24.4-21)
which then permits us to identify the quantity in the square brackets in (24.4-20) as a component of the second-rank susceptibility tensor, i.e., jrfi' = (iß) $°_χ ^ T r ^ T ^ D , ^ ' ) ] } ^ " 0 ' ' ·
(24.4-22)
Let us now choose a basis set which consists of the eigenfunctions of Jf70, i.e., let (24.4-23) jTo\ky = Ek\k>. In this basis set Tr{pT[D„DIa(i')]} = Σ <*|pT[D,.,A,(f)]|*> k
= Σ Σ
(24.4-24)
550
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
But
(i/ZKk\e-^'kT\l}
=
= (1/Ζ)<Λ|«Γϋ,'°'*ψ>δ„ =
(24.4-25)
= Σ = Z k
= Σ Σ [exp[i(£i - Ek)t'/h] k
I
- exp[i(£fc - Ε,Κ/Α]].
(24.4-27)
When this expression is substituted in (24.4-22), the integration may be carried out keeping in mind the adiabatic approximation which implies that the perturbation vanishes at t' = — oo. Writing Ei- Ek = hcoik,
Ek- Ει = \ιωη,
^
+
/i\n..\k\l
(24.4-28) ω^ + ω which is another form of the Kramers-Heisenberg dispersion formula (24.1-21). The next term in the expansion of (24.4-10) is P (2) whose general form is h ki
|_
ω/Λ -
ω
P(2)(co! + ω2) = χ(2)(ωι, ω2)Ε(ωι)Ε(ω2). (2)
(24.4-29)
The susceptibility χ (ω1,ω2) is a tensor of rank 3 and is associated with processes involving the mixing of two light waves such as parametric ampli fication and second harmonic generation. The symmetry of the medium imposes important restrictions on the sus ceptibility functions. Thus, for a medium with a center of symmetry χ(2) = 0. More generally χ(2) must be invariant under any operation of the symmetry group of the medium. χ{3) is the first nonlinear term for atoms and molecules that have centers of inversion. χ(3) is a tensor of rank 4 and encompasses numerous interactions which may occur among three distinct frequency components (Bloembergen, 1965; Butcher, 1965; Loudon, 1973).
24
HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
24.1 The Kramers-Heisenberg Formula The processes of absorption and emission described in the previous chapter are one-photon processes since in every transition the field either loses or gains a photon while the atom undergoes a corresponding change of state to keep the total energy (atom plus field) constant. There are other important interactions between an atom and a radiation field which involve a change of more than one photon. In a general scattering event, for example, the field loses an incident photon characterized by k2 and gains another photon—the scattered one—characterized by kT. With laser sources, nu merous multiphoton processes have been observed although the present discussion will be restricted to several aspects of two-photon interactions. In the absence of an external magnetic field the Hamiltonian describing the interaction of an electron with a radiation field is Jt> int — Ή
1
i J€ 2 ·>
$ex = (e/mc)A · p,
Jf2 = (e2/2mc2)A2
(24.1-1)
where A is given by (22.2-17). As in (22.3-8) let with je{~] and Jf [+) given by (22.3-9). For j f 2 we have ne2h ^ ^ ^2 = — v Σ Σ
1 ,
^ (ekA · er*')
x (flue**r + ahe~ik'τ)(α^λ^''r 536
+ d^e'^'r)
(24.1-2)
24.1 THE KRAMERS-HEISENBERG FORMULA
537
ne2h,:
Σ Σ , (Skit-er*') mV kA kA' ^ ω ^ α ν x { 4 Α 4 Ά ' e x p [ - i ( k + k ) · r] + ^ Α ^ Ά ' e x p [ i ( k + k ) · r]
+ αίλ^Ά' e x p [ - i(k - k ) · r] + ^ Α 4 Ά ' exp[j(k - k') · r]} = &ü*eül>
+ ^ί-χ^ί^
+ J T Ü ^ W + jrtÄ^fö
(241-3)
in which J f ü ^ f ö refers to the term containing αΙλα{λ>\ Jf^Jf^ to the term with akA^kA', etc. Matrix elements of α^λ and α^ may be obtained from (22.2-8):
I^AKA)
= yjnu + 1,
(24.1-4)
which lead directly to
1|4Α4'Α'|%Α,^Ά'>
= V ( n ^ + l)(Wk'A' + 1),
.
Since the photon modes are independent we also have [flkA,flk'A'] = [ökA,4'A'] = [ 4 A , 4 ' A ] = 0. The only nonvanishing matrix elements of αίχ and akA are those which involve a change of one photon. For all bilinear combinations of atx and
(24.1-6)
with energy (not including the zero point energy) Ei = Ex + nkXh(uk + Mk'A'fotfk'.
|f>
(24.1-7)
·
If)
Scattering
II> li;n k x # f c r x ,>
lF>=lf;nkx-1,nkV+1>
FIG. 24.1 Nomenclature for photon scattering. Id and kT refer to the incident and scat tered photons, respectively.
538
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
After scattering has occurred, ηκλ> is increased by one photon and nkA is decreased by one photon so that the total system is now in the state |F> = | f ; n k A - l , n k ^ + l>
(24.1-8)
E¥ = Ef + (nk, - l)fccok + {nwx. + l)ftü*.
(24.1-9)
with energy The energy E{ relative to E{ is arbitrary and the scattered photon may have more or less energy compared with the incident photon. This represents the general case of Raman scattering and includes elastic scattering as a special case. From (24.1-5) it follows that
**h mV
M » - + 1 ) ( g k A . g k , A , ) < % i ( k - k '»- 1 i > .
yJ
cOkCOk'
Therefore
K(n„ yj
+
1)
cOkCtv
k>
|ei(k-k,,.r|
The scattering process is regarded as a two-photon process in the sense that an incident photon of specified energy, polarization, and propagation direction symbolized by the indices kA is scattered so that the emerging photon may have a different energy, polarization, and direction of propaga tion symbolized by the indices kT. Hence a kA photon has been lost and a k T photon has been gained. Matrix elements of Jf i will be of type (24.1-4) in which there is a change of one photon. Therefore Jf i cannot contribute to scattering in the first order of perturbation theory but may contribute in a higher order. The matrix elements of Jif2, on the other hand, are nonvanishing when there is a change of two photons; first order contributions from 34?2 are therefore expected. The contribution of 2tf\ in second order may be obtained from (14.4-7): Wkm = (2n/h)\
(24.1-11)
with to second order given by
<
W
= <*,.>>+ I
(24,.12)
Now let |m> = |I>,
|/c> = |F>
where |I> and |F> are given by (24.1-6) and (24.1-8). For the intermediate
24.1 THE KRAMERS-HEISENBERG FORMULA
ID
IL,>
IF)
IL2>
IF)
539
FIG. 24.2 Contributions from p · A term in second order to photon scattering [see Eqs. (24.1-15) and (24.1-16)].
states |/> there are two possibilities 1}
~
(24.1-13)
[|L2> = |/ 2 ;n k A ,n k ^ + 1>.
Hence there are two pathways from |I> to |F>:
pathl: lO-lLO-lF), path 2:
(24.1-14)
|I>-^ |L 2 >-+|F>.
In the first path the transition |I> -► |Li> involves the loss (absorption) of a Id photon accompanied by an atomic transition |i> -► |/i>; the transition |Li> -> |F> involves the gain (emission) of a k T photon with an atomic transition \l{y -► |f >. In the second path there is an emission of a k T photon with an atomic transition |i> -► |/2> in the step |I> -► |L2> and an absorption of a k2 photon with an atomic transition |/2> -► |f > in the step |L2> -* |F>. It is important to note that in each path the intermediate state differs from the initial and final states by just one photon. These steps may be symbolized by diagrams as in Fig. 24.2. In Section 22.3 we saw that Jf(1~) was associated with the absorption of a single photon and Jif[+) with the emission of a single photon. Therefore the matrix elements in the two pathways are the following.
ForlO^lM-lF):
+ l\^\+%;nkX
-
l,nk.x,}
+)
x 2ne2h m
"kA(«k'A· + 1) COkCOk'
x.
540
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
For|I>-+|L2>-+|F>:
2ne h
+
l\3*?[ \i;nkX,nk>x>}
\nkX{nk>X' + 1)
= ^2TT
+ 1>
+)
/
,
,
< f \e
(ekA'P)/2>
x.
(24.1-16)
Both pathways contribute to the second-order term in (24.1-12). On setting Ei - ELi = Ei-
Eh + k o k ,
£i - £ L 2 = E{ - Eh - hcok>, (24.1-17)
the second-order term becomes 2ne2h
lnkk{nk>k> + Γ)
m 2V
\j y
|
cokcok'
yRf\e~ik''r(ek^ ι\_
ρ)|ίι> Ei — Ei + ha>k
(24.1-18)
in which the general summation index / indicates the sum over all inter mediate states accessible to one-photon transitions. The first-order contribution due to the Hamiltonian J f 2 was given by (24.1-10) and is to be added to (24.1-18) to obtain the total matrix element
Ιτ\
2ne2h
\
n
^
n
^ + !) L
ω ι ^,
*
x*
f - · ^)«n
, 1 yr
Ei — Ei -\- h(Dk
(24.1-19)
This matrix element may now be inserted into (24.1-11) with the δ func tion replaced by a density of final states (23.1-7). The transition probability per unit time that an incident kA photon has been scattered into the element
24.1 THE KRAMERS-HEISENBERG FORMULA
541
of solid angle dQ as a k T photon then becomes (e k A-e k 'A')^fi
\mcz I V cok
^ "
1
(24.1-20)
Equation (24.1-20) is known as the Kramers-Heisenberg dispersion formula; it may also be expressed as a differential scattering cross section, do dQ
number of k T photons scattered/sec sr number of incident k/l photons/sec cm 2
w/dn nkc/V = r 0 2 ^ ( n k ^ + l) cok
(e k A-e k 'A')c>fi
1 y Rek-A- - Pri] )(ekA - pii) (ekA * Pn)(ek-A' * Pa) E E + fta>k Ei — Ei — ftcok' L i~ i
+m r
(24.1-21)
where r 0 = e2/mc2 = classical radius of the electron, Pn =
Ρϋ = .
Formula (24.1-21) for the differential cross section contains the factor (ηΐιΆ' + 1); hence the cross section is a sum of two terms one of which is proportional to nkA', the number ofscattered photons. This term corresponds to stimulated Raman scattering; it contributes negligibly at ordinary inten sities, but produces important effects at high intensities. The differential cross section (24.1-21) becomes infinite when one of the denominators vanishes. This is a consequence of the assumption of infinitely sharp atomic states. Near resonance it is therefore necessary to include a damping factor Γ as was done in Section 23.5. The differential cross section, including damping, is do -j^=r02 ail
Ί
cok'
"k'A'+ 1)
(ekA -ek'A')c>fi
cok 1 y Γ (ek'A' Pn)(ekA · PH) + m^lEi-l Ei + ftcok + τϊΓ
(ek;i · p«)(ie k 'A"Pn) 11
+ Ei — Ei —ha>v ha>\ + ? / r j |
(24.1-22)
542
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
The intermediate states |/> which appear in the Kramers-Heisenberg dispersion formula, as well as the initial state |i> and the final state |f > are all eigenstates of the atom, i.e., solutions of the atomic Schrödinger equa tion. By virtue of the existence of matrix elements between |i> and |/> and |/> and |f >, the two states |i> and |f > can interact via two-photon processes. However, the transitions |i> -> |/> and |/> -> |f > are not observable, and it is because of this feature that the states |/> are sometimes called "virtual". We prefer to avoid such terminology because it may give the impression of states whose existence is nebulous and whose properties are not defined which, of course, is not the case at all. 24.2
Scattering—Special Cases
A number of applications of the differential cross section formulas (24.1-21 or 24.1-22), not including the stimulated portion, will now be given. If a>k = cok' in (24.1-21) and hco » \E[ — Et\9 the sum over intermediate states may be neglected and the differential cross section is da/dQ = r02 Sn(ekX - ek'i>)
= r o 2 cos 2 0<5 fi
(24.2-1)
where Θ is the angle between the two polarizations. Since the incident and scattered photons are of the same energy (elastic scattering), the scattering atom remains in the same state and <5fi = 1. The total cross section then becomes σ
= ^(άσ/άΩ) dQ =
8ΤΓ/3Γ02.
(24.2-2)
This is known as the Thomson cross section; it is due entirely to the A2 term in the interaction Hamiltonian. Since the atomic states do not appear in the cross section, Thomson scattering may be regarded as the scattering of photons from free electrons. Formula (24.2-2) is the total cross section when the incident light is unpolarized. If this is not the case, a more detailed geometrical analysis is required (Louisell, 1973). The Kramers-Heisenberg formula (24.1-21) may be converted to another form which is more adaptable for discussions of Raman and Rayleigh scattering (Dirac, 1958). The notation will first be simplified by writing ekA = e
ek'A' = e'
nkA = n
nWk> = ή
cok = ω
cok' = ω '
(24.2-3)
24.2 SCATTERING—SPECIAL CASES
543
Now consider the commutator [e'-r,e.p]. Since [rhPk\
= ihdjk,
(24.2-4)
the expansion of (24.2-4) into its components results in [ e ' . r , e - p ] = ifte'-e
(24.2-5)
(24.2-6)
from which one obtains
But
£,)
(24.2-7)
(24.2-8)
with a corresponding expression for , we have Σ [(£i - £i)
- (£ f - £,)
(24.2-9)
Also, Σ [
(24.2-10)
since the components of r commute with one another. When (24.2-10) is multiplied by hco' and added to (24.2-9) the result is ffl F
2
[(£,-£
i
+
- (Et -E,
feB')
(24.2-11)
544
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
We now replace the matrix elements in (24.1-21) by expressions of the type (24.2-8) a n d e' · e<5fi by (24.2-11). T h e result is „2
do ail
2
ω m ~ ( « ' + l ) ^ co
£|
(£, -Ej
+ fao')(£i - E, + tuo) - (£ f - £,)(£, - £Q £i - Ei + \ux>
■
dn
= r02wco'\ri
+ l)w 2
f
|
(24.2-13)
This is the general form for R a m a n scattering. T h e energy of the scattered radiation (ha>') can be lower or higher than the energy of the incident radiation (ftco). The process is, therefore, inelastic a n d involves energetic changes in internal degrees of freedom. Stokes a n d anti-Stokes shifts are terms used to describe scattering in which \ιω' < hco a n d hco' < hco, respectively. R a m a n scattering from molecular vibrational modes is discussed in Section 27.4. Rayleigh scattering occurs when ω' = co and hco « \Ei — Et\. Inserting these conditions into (24.2-13), with |i> = |f >, do_ = r 0 2 m 2 co 4 dQ
'
Ei-El
+■
Ei-Ei
(24.2-14)
which yields the familiar result that at long wavelengths the scattering cross section varies inversely as the fourth power of the wavelength. 24.3
Diagrams
Starting with the Fermi golden rule (24.1-11) in which the general form of the matrix element is
(24.3-1)
24.3 DIAGRAMS
545
where R% =
^km
I? 17 Üm — £/
ZJ
/
'
*e=Σ Σ
(24 3.2)
(Em — Ev)(Em — Ei) it is possible to symbolize these expressions by means of diagrams which are helpful in writing the pertinent matrix elements associated with a particular process. We use the simplified notation (24.2-3). Single photon absorption. |I> = |i;n> |F> = | f ; n - 1> \
Ätf> =
ll>
IF)
e I2nhn m
Single photon emission.
, ik
(24.3-3)
,
|I> = |i;n> |F> = |f;n + l> RW =
(24.3-4)
1) f .e _Ikr .... I2nh(n + 1) _e \2%h{n i k . re_
IF>-
|I> = |i;n,w'>, |F> = jf;n - Ι,π' - 1>, R(2)^
(FiM-lLxXL!^-^) Li
Ei — ELl
(
|Li> = \h;n - l,n'>, |L 2 > = \l2;n,n' - 1>,
y
L2
= M3 + M 4 ,
(24.3-5)
2
M
* 2π6 ^ ^ < Γ Κ " ^ - ρ ) | / ι > < / ι Κ - ' ( 8 - ρ ) | ί > 3 = —2 -ΓΓ / 7 Σ — ^— r—ΓΛ ' mz V yj ωω if E{- Eh + ftco
\
ll>
IL,>
y
IF)
(24.3-6)
546
M4 =
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
f ! M p 7 y
^
(243 _ 7)
^
ID
IL2>
IF)
Two-photon emission. |I> = |i; η,η'), |F> = |f;n + Ι,η' + 1>, D(2)_V
+
|Li> = |/i; n + l,n'>, |L2> = j/ 2 ;n,n' + 1>,
'|L1>
v
= M5 + M6, =
^2πδ
(24.3-8)
/(»+ l)(n' + 1) y
t
Me =
k r
/? II)
r
(24.3-9)
/A IL,>
IF>
e^ 2nh Hn + l)(n' + 1) ^
ll>
'//
IL2>
f
IF>
All the matrix elements considered thus far were of the operator Jf t in (24.1-1); for Jf2 (24.1-2) and (24.1-3) we have the following: Scattering |I> = |i;n,n'>,
|F> = |f;n - ί,η' + 1>,
Ktf' =
^ ρ ^ . ^
ll>
t _ /(k _ k0 . r]|i>>
IF)
(24Μ1)
24.4
OPTICAL SUSCEPTIBILITY AND NONLINEAR EFFECTS
547
Two-photon absorption. |I> = |i;n,n'}, RW =
|F> = |f;n - Ι,η' - 1>,
=
= T- ljv- J^-> ®' s')
ll>
( 24 · 3 - 12 )
IF)
Two-photon emission. |I> = |i;n,n'>,
|F> = |f;n+l,n' + l>
KÖ =
ID
Note that in dipole approximation there is no contribution from the A2 term to two-photon absorption or emission; as shown previously the contribution to scattering occurs only in the elastic case. 24.4 Optical Susceptibility and Nonlinear Effects For a system of N identical particles (e.g., atoms, molecules, etc.) let the Hamiltonian be JiT = J^o + Mf" (24.4-1) with JT = - E ( r , i ) D (24.4-2) where D is the dipole moment operator eR and Jf' = 0 at t = t0. In terms of the individual particles
#o = Σ ^n\ n
D = X D„ n
(24.4-3)
with the assumption that operators associated with different particles com mute. The macroscopic polarization is defined by P =
(24.4-4)
In order to calculate P it is necessary to calculate the density operator p in the presence of thefieldE. For this purpose we refer to (13.5-9) which gives
548
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
the expansion of the density operator in the interaction representation: Px(t) =
pf°>(i) + p['\t) + p[2\t) + ■■■
(24.4-5)
where Pf0)(0 = Piihl p\1\t) = (i/h)^odt1\_Pl(to),jel'(t1)-] p\2\t) = (i/h)2 j!o dh £ ' dt2 [[p,(io), Jf ,'(i2)], Jfi'(h)l
etc., (24.4-6)
and = - E(r, f) · eije°'lh De ~ **<*' = - E(r, i) · D,(t).
(24.4-7)
If the system is in thermal equilibrium at the time the perturbation is applied, pi0)(r) = P,(io) = PT where, according to (13.4-7), p T = e-*oikTiYte-*olkT = (i/z) e -*o/*r
(24.4-8) (244.9)
One may now expand P to various orders: P = P<0) + P (1) + P (2) + · · ·
(24.4-10)
P (0) = Trp T D.
(24.4-11)
P(1,(i) = Trp (1) D = Tr[e-üro,/*pT)eijro,/*D]
(24.4-12)
with the leading term The next term is or, in component form, P$\t) = Tr[e_ljro"*pl1)ew'0,/* D J
(24.4-13)
= Tr j e-^0"" [' dh E(r, h) · [D,^),p T ]e üro,/ *D # ,. (24.4-14) Expanding the scalar product, PV\t) = ^ T r e - « - " ' » { £ Ä! Σ£.(r,t 1 )[A.(tiXpT]|e«'o' / »I) M = l- £ d tl Σ £.(r, t d T r i e - ^ I T M t ! ) , ρ τ ] ^ < " / Α D„}.
(24.4-15)
24.4 OPTICAL SUSCEPTIBILITY AND NONLINEAR EFFECTS
549
The expression inside the trace may be simplified somewhat since = [Αα(ίι - ή,Ρτ] = [Α.(ί'),Ρτ] (24.4-16) where t' = ii - t.
(24.4-17)
Also, since the trace of a product of operators is invariant under a cyclic permutation of their order, Tr{[DIa(0, p T ] Dß} = Tr{pT[D„, Αα(ί')]}.
(24.4-18)
The electric field Ea(r, ii) may be expressed in terms of its Fourier com ponents. Suppressing the dependence of the field on the spatial coordinates, E*(t1) =
f"aoEa(co)e-imtidto
= f ^ Λϋ£ β (ω)β- ίβΛ β-' ω(ί| "°
= Ρ ^ dcoEa((D)e-i(0te-i(ut'.
(24.4-19)
Inserting (24.4-18) and (24.4-19) into (24.4-15) with t0 = - oo and t = 0, Wit) = $"„ d«> Σ |"(i/Ä) J!«, Λ'Τ Γ { Ρτ [ϋ μ ,ϋ Ια (0]}^ ίωί Ί£:α(ω)β- ίωί . (24.4-20) This expression may be compared with (14.6-29) written in component form; Ρ(Λή = Γ
Σή1Λω)Εα(ω)άωβ-ίωΐ
(24.4-21)
which then permits us to identify the quantity in the square brackets in (24.4-20) as a component of the second-rank susceptibility tensor, i.e., jrfi' = (iß) $°_χ ^ T r ^ T ^ D , ^ ' ) ] } ^ " 0 ' ' ·
(24.4-22)
Let us now choose a basis set which consists of the eigenfunctions of Jf70, i.e., let (24.4-23) jTo\ky = Ek\k>. In this basis set Tr{pT[D„DIa(i')]} = Σ <*|pT[D,.,A,(f)]|*> k
= Σ Σ
(24.4-24)
550
24. HIGHER ORDER ELECTROMAGNETIC INTERACTIONS
But
(i/ZKk\e-^'kT\l}
=
= (1/Ζ)<Λ|«Γϋ,'°'*ψ>δ„ =
(24.4-25)
= Σ Σ [exp[i(£i - Ek)t'/h] k
I
- exp[i(£fc - Ε,Κ/Α]].
(24.4-27)
When this expression is substituted in (24.4-22), the integration may be carried out keeping in mind the adiabatic approximation which implies that the perturbation vanishes at t' = — oo. Writing Ei- Ek = hcoik,
Ek- Ει = \ιωη,
^
+
/i\n..\k\l
(24.4-28) ω^ + ω which is another form of the Kramers-Heisenberg dispersion formula (24.1-21). The next term in the expansion of (24.4-10) is P (2) whose general form is h ki
|_
ω/Λ -
ω
P(2)(co! + ω2) = χ(2)(ωι, ω2)Ε(ωι)Ε(ω2). (2)
(24.4-29)
The susceptibility χ (ω1,ω2) is a tensor of rank 3 and is associated with processes involving the mixing of two light waves such as parametric ampli fication and second harmonic generation. The symmetry of the medium imposes important restrictions on the sus ceptibility functions. Thus, for a medium with a center of symmetry χ(2) = 0. More generally χ(2) must be invariant under any operation of the symmetry group of the medium. χ{3) is the first nonlinear term for atoms and molecules that have centers of inversion. χ(3) is a tensor of rank 4 and encompasses numerous interactions which may occur among three distinct frequency components (Bloembergen, 1965; Butcher, 1965; Loudon, 1973).