Moment theory of self-trapped laser beams with nonlinear saturation

Volume 15, number 3

OPTICS COMMUNICATIONS

November/December 1975

MOMENT THEORY OF SELF-TRAPPED LASER BEAMS WITH NONLINEAR SATURATION Juan Francisco LAM, Bernard LIPPMANN Department of Physics, New York University, 4 Washington Place, New York, N. Y. 10003, USA and Frederick TAPPERT* Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N. Y. 10012, USA Received 9 July 1975 The moment theory of the quasi-optical equation is used to obtain simple analytical expressions for the relation between beam radius and power of stationary self-trapped light beams in passive nonlinear media with saturation of the index of refraction. In addition, the stability of these self-trapped beams is determined.

Tile moment theory [1] of the quasi-optical equation is used to obtain simple analytical expressions for the relation between beam radius and power of stationary self-trapped light beams in passive nonlinear media with saturation of the index of refraction. Results are compared to the recent numerical calculations of Piekara et al. [2]. In addition, the stability of these self-trapped beams is determined. We start with the scalar quasi-optical equation for c.w. laser beams without gain or loss, "~0+-~1/.02 + a2~+Q(]~[2)ff=O, x-~ 2k0 \ ~ x 2 3Y 2]

(1)

and we note that eq. (1) has the following invariants:

I o = f l 4 : l 2 dxdy,

(4a)

S

where f(l~OI 2) = ½k o ~

C2n l~Oi2(n+1).

Define the mean square beam radius by

= f ( x 2 +y2)1~12 dxdy/I O.

(5)

where the nonlinear index has the form** Then it can be shown that

oo

0(1412) = ½k 0 ~

C2n I~b12n .

(2) d2(r2) 4 12 d z- I 0

k 4Io(G(lt~12)dxdy'

(6)

The beam power is defined as where

P = (CnO/4rr)

ft~012 dx dy,

(3)

$

* Partially supported by the Energy Research and Development Administration, Contract No. AT(11-1)-3077. ** We use the definition C2n = e2n/eo, where e2n is the coefficient of nonlinearity and it is related to Piekara's C'2n by C2n = C'2n(2n-1)! !/2nn!.

G(I~I 2) = ½k ° ~ n - 1 C2nl~12(n+l); n=l n + l and higher moments satisfy similar equations. This hierarchy of moment equations can be truncated in a simple way by assuming that the intensity of the self-trapped beam be approximately a gaussian. Thus, we get 419

Volume 15, number 3

OPTICS COMMUNICATIONS

November/December 1975

a2

lff[2 = ffo2--~°a2(z) exp [ - ( x 2 +y2)/a2(z)],

(7)

D

J

and note that (r 2) = a2(z). Using this relation in eq. (6) yields an ordinary differential equation for a 2 of the form d2a 2

dz 2 - f(a 2, e]pc),

O

(8)

where

IO

i( 2, e/pc)=

(1 -

~ C2n~ 4 ~n

e/pc)

c9

1 [p~n(

a~)

1

and

Pc = Cno/k2c2'

ao = a(z = 0).

This equation differs from the paraxial ray equation of Wagner et al. [3], who did not use the exact moment equations. Equilibrium solutions of eq. (8) correspond to self-trapped beams and are found by setting f(a 2, PIPe) = 0 and a = a 0 . This yields a relation between a 2 and P/Pc, say

a = a~q(,O/Pc).

(9)

Stability is determined by expanding the right-hand side of eq. (8) around the equilibrium point; the beam is stable if

o ; ( 2 , e/P c ) --

< 0.

Oa2

(10)

la=aeq

Concrete applications of this method are obtained in the following cases: 1) c2n = 0 for n t> 3. If PIPc > 1 and c 4 < 0, one finds that aeq

4X/-} c~__~2 ( F c V ~ _ 1 =

3k---~

1

);

~ \ / ( e / e c) - 1

(lla)

and for c 4 > 0, P/Pc < 1 we have that

aeq

_44~ cV~~X/1 e/pc 3k0 - (P/Pc)"

4

6

p

Pc Fig. l. Beam diameter D = 2a in t~m as a function ofP/Pc for c2 = 1 X 10 -13 esu -2, c4 = -1 X 10 -24 esu -4, c2n = 0 for n >~ 3 and with h 0 = 0.7 tsm, no = 1.5. Solid lines represent analytical results, while open circles are adapted from Piekara's computational work.

results of Piekara. The agreement is seen to be good. It can be easily shown that the solution with P < Pc is unstable and that with P > Pc is stable. Further the minimum stable equilibrium radius occurs when P = 7.Pc and is given by _ 8N/r2 ~ . (aeq)min - 3k-~-O

Ca. c2

(12)

If c 4 is much greater than c 2, then this minimum radius beam is many wavelengths in width. This shows the possibility of determining c 4 by measuring (aeq)min experimentally. 2) c 4 > 0, c 6 < 0 and C2n = 0 for n >/4. In this case, there exists equilibrium solutions only if P > P m i n where Pmin =Pc Cz/(l +&),

(13)

where (1 lb)

Eqs. (11) are plotted in fig. 1 together with numerical 420

2

81 c6c2/c 42 . o~= - .~ If Pmin < P < P ' there is an unstable branch given by

OPTICS COMMUNICATIONS

Volume 15, number 3

November/December 1975

which is given by

no

(M.ml solutions are plotted in fig. 2 together with Piekara's results. Again the agreement is good. The same method may also be applied to other cases. In all cases we have examined, there exists a stable branch of the curves relating the equilibrium beam radius and power. We recommend experiments in which the index saturates below the threshold for induced scattering to verify the existence of stable self-trapped laser beams. Such beams would be useful in transmitting large amounts of power through materials without damaging them. These

IO

0

I

I 2

I

~__L

I

4

.

6 P

Fig. 2. Beam diameter D = 2a in ~m as a function of PIPc for c: = 1 × 10-13esu -z, c4 = 1 X 10-24 esu "4, c6 = - 4 × 10-35 esu-6, c2n = 0 for n ~ 4 and with ho = 0.7 urn, no = 1.5. Solid lines represent analytical results, while open circles are adapted from Piekara's computational work.

aeq

9k 2 c 2 1-(P[Pc) 1 +

+a

1-

. (14a)

References [1] S.N. Vlasov, V.A. Petrishchev and V.1. Talanov, RadioPhysics and Quantum Electronics 14 (1971) 1062; B.R. Suydam, IEEE J. Quantum Electronics QE-10 (1974) 837. [2] A.H. Piekara, J.S. Moore and M.S. Feld, Phys. Rev. A2 (1974) 1403. [3] W.G. Wagner, H.A. Hans and J.H. Marburger, Phys. Rev. 175 (1968) 256.

Further, there is a stable branch for all P > Pmin

421