Interacting Laser Beams in a Resonant Medium

H. Shih, M. Scully, W. H. Louisell, and W. B. McKnight

For potential applications in scanning and measurement, we consider two interacting laser beams in anactive nonlinear medium. The nonlinear medium generates two new beams at the same frequency andin directions different from the incident beams. The Kerr effect is used as a means of phase matching inthe isotropic medium.

1. Introduction

Nonlinearities in matter are usually small but veryfrequency dependent. However, if radiation is inresonance with a dense medium, e.g., a crystal, non-linearities can become so strong that a perturbationtreatment in which the polarization P is expanded ina power series in the field strength E,

Pi = X"'11Ea + X( 2)E.kE + Xiive 3)EaEiEy +

will no longer be valid. Renormalization proceduresshould then be used to describe nonlinear optical ef-fects, not among bare particles but among polari-tons, which are dressed particles composed of pho-tons and elementary excitations of the medium.2 3

Similar enhancement of nonlinear parameters canalso occur for light propagating in gaseous systems asresonance is approached. The preceding perturba-tion expansion can be made and will remain valid aslong as the laser field strength is not too strong andthe Rabi precession frequency pE/h remains smallcompared with the atomic decay rate -y, as we shallassume in the present paper. However, if the re-verse is true, i.e., pE/h > 1, the expansion will notconverge, and alternative procedures must be usedsuch as have been devised to consider self-inducedtransparency,4 high power lasers,5 ' 6 etc. Indicationsare that it will be desirable for application purposesto extend the present calculations into this regime,as we shall show.

Most of the nonlinear phenomena studied to datehave been frequency conversion processes like har-

The first two authors named are with the Physics Department,University of Arizona, Tucson, Arizona 85717; the other authorsare with the Physical Sciences Directorate, U.S. Army MissileResearch, Development and Engineering Laboratory, RedstoneArsenal, Alabama 35809.

Received 18 December 1972.

monic generation, parametric conversion, Brillouinand Raman scattering, etc. In the present paper weshall consider, for potential applications in scanningand measurement, a resonant nonlinear process thatinvolves no frequency shifts, but instead generatesbeams in directions different from that of the inci-dent beams. In particular we shall consider two in-tense laser beams intersecting at an angle 0 inside anonlinear gaseous (active) medium which is boundedby two parallel plane windows and separated by adistance D, as shown in Fig. 1. For simplicity, weshall assume the beams have a Gaussian transversedependence and neglect diffraction. The beamshave a width wo, frequency wo, and wave vectors k1and k2, respectively. We shall assume that ho isapproximately given by Ea - e the separation be-tween the upper and lower single atom energy levels.Due to the nonlinear interactions, beams in the di-rection 2k1 - k2 and 2k2 - k will be generated, andwe shall study the spatial distribution of thesewaves. Although phase matching occurs at smallangles, it appears that by increasing the incidentbeam intensities, phase matching can occur for larg-er angles, a case to be studied later.

Chiao et al.7 have considered the coupling ofStokes and anti-Stokes components propagating atsmall angles with the primary laser beam in an oscil-lator via the molecular orientation Kerr effect. Thepresent study considers an open system (amplifier)and relaxes the plane wave restriction so that thespatial distribution of the laser beams may be ana-lyzed.

Mack8 has observed stimulated light scatteringdue to refractive index changes brought about by opti-cally induced temperature waves. He used two in-tersecting laser picosecond pulses.

In Sec. II we derive the approximate equations forthe incident and scattered waves under the slowlyvarying envelope approximation. In Sec. III we givethe Green's function for the scattered wave. Thescattered wave intensity is obtained in Sec. IV,

2198 APPLIED OPTICS / Vol. 12, No. 9 / September 1973

Fig. 1. Two interacting laser beams in an active medium.

and in Sec. V we consider a numerical example ofthe phase matching condition.

11. Scattering Theory

The electric field that is assumed to be plane po-larized in the y direction (perpendicular to the planeof incidence) obeys the wave equation

(l1c2)(82 E/8t2 - V2 E = (-47/C2)(82 P/8t2) (1)where we neglect diffraction. The field is driven bythe polarization of the homogeneously broadenedmedium in which it propagates. Following Lamb9 itis given by

P(r,t,Awo) = [(.)k + x(M)fEI|E + ... ]

X exp (-iwot) +

where

E(r,t,) = E(r, t) exp(-iwot) + c.c.,XM _ -p 2AN (WL + iTab)

-i (Aco))2 + y2 b'

X3) = X(21)p2Tab

2r 2=ab + (Ao) 2 '

A = - ,

(2)

P 2

hI< YabI/2. (9)

On resonance, x(1) and X(3) are pure imaginary.We would like to consider a case in which two

large incident fields at frequency coo are incidentwith wave vectors k1 and k2 (see Fig. 1) and intersectin a resonant medium and look for the scatteredwaves generated by the nonlinearities in the medi-um. Therefore we let

2

E(r,t) = E: La(r, t) exp(ika r) + e(r, t),a=l

(10)

where the scattered field e is small compared withthe incident fields 8,. When we use this in Eqs. (2)and (1), we neglect e2 (r,t) terms as well a terms in-volving e*(r, t) which will be of lower order in thecoupling and use Eq. (9). We find that the incidentwaves obey the equations (a = 1, 2)

[2t. U(1)2 V] &La exp[i(k,, r - o0t)] = 0, (11)

where the incident waves propagate with velocity

1 1 + 4 XNR + 4X() _ / Y1 + 3l2___ c2

2 co ) (12)

and XNR[>> x)] is the nonresonant contribution tothe susceptibility due to the host background.

The scattered waves generated with wave vectors

k = 2k - k2,

k4 = 2k2 - k,

satisfy

Lt2 V(3)2 V ea(r, t) exp(-icot) =

- 2 a8Pa(3t (r t) exp[i(ka r - coot)],

C2 at2

for a = 3, 4 where

P(3)3(r,t) = X g 12

p(4)3 (r, t) = X(3) 8 281,

(13)

(14)

(15)

and(3) 1 1 87rx(3~ /1 2\/I _+iY2<2 + ci (1 + 12 1211) (183 + Y31(4) V

1,3

2- V 1)2 c2

'/ \ (A) /

(16)

(5) If we assume that the 8, varies slowly in an opticalwavelength and in an optical cycle and let

(6)

Yab = 2(Ya + Yb) + abPeX (7)

and

1 = oYa + Yb'-. (8)Also Ya and ylb are the decay rates of the atoms fromlevels a and b, respectively, is the atomic frequen-cy separation between the levels, p is the atomic di-pole moment, and No is the net rate of pumping ofatoms into level a vs level b. We have made the as-sumption that

= (ka)Ikal r,

Eq. (11) reduces to (a = 1, 2)

+ 4j]a exp[i(ka r - Cwot)], 0

while Eq. (14) reduces to (a = 3, 4)

[v3 at + 8]er, t) exj)(-iwot)V(3) t S

27rioOV(3)c2 Pa(')(r, t) exp[i(ka, r - co~t)].

(17)

(18)

(19)

September 1973 / Vol. 12, No. 9 / APPLIED OPTICS 2199

Equations (18) and (19) are our working equationsfor the incident and scattered waves that are gen-erated, respectively. We assumed that Im[O/v(3)] <<Re[wo/v(3 )]-

111. Green's Function

If the inhomogeneous term on the right of Eq. (19)is called 27riS(r, t), the solution may be written as

e(r, t) = ff dt'dr'G(r - r', t - t')S(r', t'), (20)

provided G is the Green's function that satisfies theequation,

[1 a + a ]Gr t) = 27rib(t)V)(r),

subject to the causality condition

G(r, t) = 0 if t < 0.

(21)

(22)

The solution of Eq. (21) subject to Eq. (22) is givenby

G(r,t) = 2rib( _ t)6(2)[k (k X r)], (23)

where 4 > 0 and t > 0; (2) is the two-dimensional function for the two directions perpendicular to k.

If we use Eqs. (20), (23), and (15), the solution forEq. (19) becomes for a = 3

e,(r,t) exp(-iwot) = , () x(3)

X fdt'fd 3 r' 812(r', t') 82*(r/ t')

X exp[i(k3-r - co0 t)]6(21[k, X (k3 X r)]

X t' - t &(43' -8 3 )], (24)

where we have let Coo/f3 = Rev(3). In this case weshall consider that 81 and 82 are independent of timeso Eq. (24) becomes

e3(r) -T ,B- C )f '3>Sd3 sc -) 2*(r')

X expli[k,' r' + k3 (r - AV(3 )

X 62)[k, X (k3 x r)], (25)

provided 3 > 6'. Therefore, e3(r) is also time inde-pendent.

IV. Simple Case Study

If we assume a Gaussian transverse beam variationfor the incident laser beams, solutions of Eq. (18)may be written as

F(r,t) = A, exp - [y(x + 2D)cscO,

+ (1/W, 2 )Jka X r121. (26)

From Fig. 2 we see that

Ika X rI2 = 7,2 + y2. (27)

Also yi = Im[wo/v(l)]. If we reexpress Eq. (26) interms of (4a, 1, y) from Fig. 2, Eq. (26) becomes

8L(r,t) = Aa exp - [(1/WO2)(17a2 + y)2 .

- -Yl(4 + 7 cot9O + %2D cscOJ]. (28)

From Eq. (26) we see that the wavefront at r hastraveled a distance (x + Y2D) cscOa in the medium.

We neglect as an unnecessary complication reflect-ed waves at the boundary x = -D and the abruptchange in the phase velocity there.

When we use Eq. (28) the scattered waves [Eq.(25)3 become

e3(r) = [27ri/3(8)](3oo/c)2X3)A,2A2*

x fffd3r'6(2)k3 X 3 (r -

x expi[k, r' - c 0k3 (r - r')/u(3)]

x exp[-,y(1,(3/0A')[x' + (D/2)]

- (1/wo2)(21ki x r'12 + k2 X r'[2)1, (29)

where

310A = [2/(sin9l)] + [1/(sinO2)]. (30)

In the integration, the angles 01 and 02 are fixed.We let

wo _ #3 + i for Ix'J < D/2,V(3 ) lA3 for Ix'l > D/2.

(31)

To perform the integration we choose axes 6', 43', y'as shown in Fig. 3. In these variables 6(2) [3 X (k3x r)] = (y - ')b(n - 13') and d3r' = d'de 1 3'dy'.The limits on 46' are 4. = (2D - 3) X csc03.Then Eq. (29) becomes

e3(r) = pA2A 2* exp[if,34 3 - Y3(2/0A(Y2D + 3 cos93)

- 'YD cscO3 - (3/wo2)(y2 + C 2 3 2) X M(13), (32)

where

M(-q3)Vr W[erf(Q+) - erf(/3)] exp K2, (33)

Fig. 2. Geometry used for incident beam.

2200 APPLIED OPTICS / Vol. 12, No. 9 / September 1973

erf(x) = 2 exp(-t2 )dt,

P3 = [X(3)27ri]//3W(c0/c) 2 ,

W = wl(830B),

(34)

(35)

(36)

K =-iAk + {y3 -yl[(3 inG)/A']- (713/wo2)6(ci,(37)

Ak3 = /3 - 2k, - k21 = /3 - 1(5 - 4k,k 2)1/, (38)= [D/(2W sin03)] - IN 3/(W tan)] - 'AKWI,

(39)

ojj =0i - 0j, (40)

2/OA = (3/OA)X73 /7l) - [1/(sin03 )], (41)

30A' = [2/(sin0j)] + [1/(sin02)1, (42)

2 Y3(2 sin201 3 + sin 20 23 ), (43)

2 -C2 =/3(2 sin20,3 + sin29 2 3), (44)

andC,> = 1 - B .

In the case of phase matching where

Ak = 0,

and small gain (or loss) in the medium

,yjD/(sin~j) <

Case C. Small Interaction Region (D 2 wo)

At the beam center where y = 13 = 0, we have

I3 = IP3 12I (1)/V30BD 2

erf ,w/sinO3W O

(53)

which is plotted in Fig. 4 to show the boundary ef-fects as a function of D/wo.

V. Numerical Example of Phase Matching

In an isotropic medium, the Kerr effect plays animportant role in achieving the phase match condi-tion Ak 3 = 0. From Eqs. (38) and (16), we have

Ak3 = /l(3) - A - 3aAl + 4(1 - k1 k2)] - 1

=- (a),I[1 + 4(1 - k * k2)] ' - 1

+ Re 8 ( / )2 x(3) gl2,

(45)

(46)

(47)

(54)

the output intensity I3 in the k3 direction becomes

I3 e3 12 = IP312 expl-(6/w0 2 )[y2 + ( 3C0)2]I

x IMo( Am) 12, (48)

where

7r wo 3 (4C WO)2 3'kB ex 3

( OB

+ -(COS03 - sin93 )]}

+ (cos93 - sin93 ¢,2)]}).

Case A. Negligible Boundary Effects D/Wo > 1

I3 1P 12I2 27r( o0)2

_ 6

x [y2 + 732(co -0 2)]+ 2OB2(C)2

whereI = IAIJ2,12 = A 212.

Case B. Small Crossing Angles (i.e., Oi << 1 ) andNegligible Boundary Effects (D/wo > 1)

I3 IP3 2I20 2+ 2) exp(23 + 9 23 2

6 2 + 32 [1- (2013 + 023)2]}

2032 + 232

Fig. 3. Geometry used for integration in Eq. (29).

(49)

(50)

(51)

(52)

which is nearly Gaussian in its transverse appearance.

Y (ARBITRARY UNITS)

Ya er X,

I I I I , f , . I M. x0 2

2 2D 2SIN 93 1+SIN 32

°O SIN2 93

Y - 3/ (Wo/(b 0)2

Fig. 4. Intensity of scattered wave as a function of D/wo to dem-onstrate the influence of boundary effects.

September 1973 / Vol. 12, No. 9 / APPLIED OPTICS 2201

( erf{ 1i2B [

+ erf {l 3B[ Dsin9 3 L w0

I

where we assume II 12. From Eq. (5) forsmall angles such that 1 - «l * 2 < 1, Eq. (54) be-comes

Ak, - (,2(1- k 2)

- Re [l)J 167rco02pE 1 abRL;-_/V(l] c21t2r[yab2 + (A(o)2 ]

If we use Eq. (4) and note that lel < la, we havep2AN 0

Ak3 - -26()(1- klk 2) + h

X )[A1 4w + Y(I)Yab]

[fD 2 + ][Yab 2 + () 2

x 167r021 pEl2YbC2 hr[yb 2 + (AW)fj

The first term represents the difference in ampli-tudes between the wave vectors 2k - k2 0Sl,2h,- h2 and f(l)kl, whereas the second term representsthe Kerr effect which attains its maximum value atAw@ = Va/3.

To obtain a rough order of magnitude estimate, letr - Tab/V 3 and F = Yab. Then,

Ak3 = -2#(,f(1 - k, *k2)

3 p 2 ANo [fl( + 3y(1)] 12 02 ¾ pEl2

4 'Yab [#(i 4 (1)23 C2hayb

If (l) -cwo/c and y(1) < f(l), this may be written

k( =-2 kI 2 2 pfAk(3) -=-2fl()( -k ) 1-6 ( b - 2)2' (58)

where

27r /[Ii(' ] = (V3/4)[(p2ANo)/IaYb]1m)2w. (59)

Consider the special case of CO2 laser radiation at10.6 in gaseous SF6 at a partial pressure of 0.05Torr. The absorption length 'C(1) can be estimatedfrom the known absorption coefficient a 0.344cm1 TorrL, viz,

1(l)[1/(O.344 x 0.05)]cm 57cm. (60)

For (1) - coo/c = 2r/10.6 x 10-4 cm, we have byEqs. (60) and (59)

/3 p 2 AN 0 c 10.6 X 10-4

Yai, =C )°kO0 = 27r x 57 _3 X 10' (61)

For such a dilute gas, the Doppler line width dom-

(55)

(56)

inates and has a value of approximately 30 MHz.The estimated dipole moment for SF6 is p 3 10-20 esu-cm. Then for

(p /yD) 2 _ 0.1, (62)

a laser intensity of approximately 500 W/cm- 2 is re-quired. In this case, if we use Eqs. (60) and (62), weobtain

Ak3 _ [47r/X1(l)](9l22/2) - 3r X 0.1 X 2 10-

(63)

where

912 = COS'lklk 2 << 1.

For phase match, we set Ak3 = 0 which implies

12 - 6 X 10- 3 rad. (64)

Although this angle is small, it can be increased ap-preciably by increasing the pressure and the pump-ing to increase (AN)o. In addition, the field intensi-ty may be increased sufficiently so that pS/h'YD - 1,in which case a more elaborate investigation must bemade because the approximations we have used be-come inadequate.

Part of the work reported here was done by W. H.Louisell while he was on sabbatical leave at BellTelephone Laboratories, Holmdel, New Jersey. Hethanks J. P. Gordon, C. K. N. Patel, and S. J.Buchsbaum of Bell Laboratories for this opportunity.

H. Shih is supported by the U.S. Air Force, Kirt-land AFB Laser Laboratories.

W. H. Louisell is on leave from the Department ofPhysics and Electrical Engineering of the Universityof Southern California.

References1. N. Bloembergen, Nonlinear Optics (W. W. Benjamin, New

York, 1965).2. L. N. Ovander, Sov. Phys. Usp. 8, 337 (1965).3. F. T. Arrecchi, Nuovo Cimento 1, 181 (1969).4. S. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 (1969).5. S. Stenholm and W. E. Lamb, Jr., Phys. Rev. 181, 618 (1969).6. B. J. Feldman and M. S. Feld, Phys. Rev. Al, 1375 (1970).7. R. Y. Chiao, P. L. Kelley, and E. Garmire, Phys. Rev. Lett.

17, 1158 (1966).8. M. E. Mack, Phys. Rev. Lett. 22, 13 (1969) and Ann. N. Y.

Acad. Sci. 168, 419 (1970).9. W. E. Lamb, Jr., Phys. Rev. 134A, 1429 (1964).

2202 APPLIED OPTICS / Vol. 12, No. 9 / September 1973