Rotating elliptical gaussian beams in nonlinear media

Volume 81, number 3,4

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15 February 1991

Rotating elliptical gaussian beams in nonlinear media A.M. G o n c h a r e n k o , Yu.A. Logvin, A.M. S a m s o n a n d P.S. S h a p o v a l o v B.L Stepanov Institute of Physics, BSSR Academy of Sciences, 220602 Minsk, USSR

Received 25 July 1990; revised manuscript received20 November 1990

An analytical treatment of rotating elliptical gaussian beam propagation in nonlinear media is presented. It is shown that the self-focusingthreshold depends on the parameter specifyingthe beam rotation and is higher than for the beam without rotation. It is established by the followingrule: the greater the differencebetween the light beam and the axiallysymmetricone, the higher the threshold power needed for its self-focusing.

1. Introduction The propagation of light beams is largely determined by their initial geometry. The elliptical gaussian beams (EGB) have a lower symmetry compared to the axially symmetric ones [ 1 ]. They have two perpendicular symmetry planes passing through the axis of beam propagation. It has been found theoretically [2-4 ] and experimentally [ 5 ] that in a nonlinear medium the threshold power for EGB selffocusing increases with increasing ellipticity. In linear optics, beams of a more general form rotating elliptical gaussian beams (REGB) - ' are thoroughly studied [1,6]. REGB is obtained from an ordinary gaussian beam when the latter passes through a system of lenses in which the principal planes of astigmatic surfaces are arbitrarily oriented relative to one another. Therefore, in ref. [ 6 ] REGB are referred to as gaussian beams with general astigmatism. In such beams, the symmetry planes are absent, since the light spot ellipse and the phase front rotate along the axis of beam propagation. This paper considers, as an aberrationless approximation [ 7 ], the REGB propagation in a nonlinear medium as well as in a nonlinear medium with quadratic inhomogeneity in the transverse direction (nonlinear light guide). The equations describing the REGB propagation are reduced to a system of equations for two coupled nonlinear oscillators, which is interesting from the point of view of the theory of dynamical systems [ 8 ]. It is a hamiltonian Ermakov system [ 9-

11 ], it has two integrals of motion and is solved analytically. Analysis of the results shows that the self-focusing threshold for REGB depends on the parameter specifying the beam rotation and is higher than for EGB without rotation. In this case, oscillations of the light spot sizes along the axis of propagation are superimposed on a beam rotation. The main qualitative result of the work can be formulated as follows: the greater the difference between the light beam and the axially symmetric one, the higher the threshold power needed for its self-focusing.

2. Model equations We consider a nonlinear medium whose permittivity in the cartesian coordinate system is ~=~o( 1 - y ( x ~ + y 2) + f l E E * ) .

( 1)

Here Eo is the permittivity on the axis z, y> 0 is the inhomogeneity parameter, fl is the coefficient of medium nonlinearity, E is the electric field strength. The light propagation in the medium ( 1 ) is described by the nonlinear Schr6dinger equation

dU d2U d2U -2ik~z + ~ + dy--T - k2y(xE+yE)U+k2fll UI2U=0,

where

U is the

0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

light beam

amplitude

(2) (E= 225

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U e x p ( - i k z ) ), k is the propagation constant along the z-axis. The case of axially symmetric gaussian beam in the medium (1) was considered in refs. [ 12,13 ]. Let us assume that at the boundary of the nonlinear medium ( z = 0 ) the field is given by

U(x, y)=Uoexp

(

x2

y2

wE

W~o - - i , , o

kxy)-~- .

(3)

Here W~o and wro are the light spot halfwidths along the x- and y-axes respectively, the parameter Ro defines the initial wavefront and specifies the beam rotation. We seek the solution of eq. (2) in the form of the REGB [1,5]: U= Um exp

X2

y2

Xy

w2x

w2

w2

ikxy

ikx 2

iky2

R

2Rx

2Ry

-

iP)

'

(4)

q = x cos q - y sin ~,

(5)

where q(z) =0.5 arctan(w2w2/w2(w 2 - w2) ).

(6)

Let us introduce dimensionless halfwidths a and b of the light spot ellipse in the rotating coordinate system (fig. 1 ). They are related to wx, wy and w by

i~+aa=(4(b/a-x) 1 +3b/a "~/ 2+ 6(b/a) [ l _ ( b / a ) 2 ] 3 j / a v,

k2/b2=cos2~/w 2 + s i n 2 ~ / w 2 - s i n 2~/2w 2 .

l+3a/b ~/-2 + 8(a/b) [ l _ ( a / b ) 2 1 3 ) / o a, (o=8( a2 + b2) / ( a2-b2) 2 ,

(7)

Fig. 1. Light spot ellipse in the rotating coordinate system ~r/z.

(8) (9)

where a = y / k 2 and the overdot indicates differentiation with respect to z=zk. The parameter x= P/Pc is the ratio of the input beam power P = WxoWyoU2 to the threshold power of self-focusing Pc = 2/k2fl for the cylindrical beam. (In the expressions for powers the immaterial constants are omitted. ) The quantity 82 = k(w2o - w20)/Ro is responsible for beam rotation. The system (8) defines two coupled nonlinear oscillators and is quite interesting from the viewpoint of the theory of dynamical systems [ 8 ]. The system (8) is classified with the Ermakov systems and is written in the standard form [9-11]: /~+ to 2a = f ( b / a ) / a 2b,

b'+to2b=g(a/b)/b2a.

k2/a2=cos2~/w2+sin2~/w 2 +sin 2~/2w 2 ,

226

Substituting expression (4) into eq. (2) and taking into account the aberrationless condition [ 7 ], we obtain a system of eight differential equations describing the EGB dynamics along the z-axis. In this system, the equations defining the light spot are independent of the other equations and are of the following form [ 2-4 ]:

b'+ab=(4(a/b-x)

where Urn, Wx, Wy, W, Rx, Ry, R, P are the real functions of the z-coordinate. The real terms of the exponential describe the shape of the light spot, the imaginary terms describe the shape of the wavefront. We shall further restrict ourselves to the REGB energy characteristics. Let us pass to the coordinate system which is associated with the light spot and rotates with it (fig. 1 ): ~ = x c o s ~ + y s i n ~,

15 February 1991

(10)

It is seen that in our case the function g coincides with f a n d parameter ot plays the role of o92. Besides being classified with the Ermakov systems, the system (8) is also a hamiltonian one. Due to this, two constants of motion exist for it: the energy integral and the Ermakov-Levis-Ray-Reid (ELRR) invariant. In ref. [ 11 ], the system (8) has been analysed from the viewpoint of the theory of hamiltonian systems and its complete integrability has been proved. It should be noted that the system (8) contains the case of propagation of an ordinary EGB in a non-

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linear medium (or and ~ should be assumed to be equal to zero), which was considered in refs. [2,4 ]. However, the system's being classified with the Ermakov systems remained unnoticed and its solution was carried out by numerical methods. Let us demonstrate the analytical solution of the system (8).

15 February 1991

easy to obtain a differential equation for the function Q(z):

[G(T) dQ/dT] 2

= (I-4Q2)(I-2/Q2+4x/Q)-d2/2.

(16)

Let us now introduce a new variable t in eq. (16) using (14) or ( 15 ):

3. Analytical solution

t= i d s / G ( s ) .

(17)

The constants of motion for the systems (8) are the hamiltonian

Eq. (16) transforms to

H = (ti)2/2+ (6)5/2

(dQ/dt) 2= _ 41( Q - Qo ) ( Q - Q~ ) X ( Q - Q2) ( Q - Q 3 ) ,

+ ot(a2+b2)/2+2(a2+b2)/a2b 2 -- 4tc/ab+(~2/2)(a2+b2)/(a2-b2) 2 ,

(11)

and the ELRR-invariant (its construction is described in ref. [ 10 ] )

(18)

where Qo = WxoWyo/(Wxo 2 + Wyo), 2 and Q1, Q2 and Q3 are the roots of the cubic equation Q3+ IQo

4XQ2 +

2xQo - 1 1 2IQ-~o Q - 2 - ~

=0.

I= (,~b-ab)2/2

(19)

+ 2(aE+bE)E/a2b2-41c(aE+b2)/ab + (O2/2)(aE+b2)2/(aE-b2) 2 .

Eq. (18) is transformed to the integral (12)

The ELRR-invariant is independent of the in.homogeneity parameter a, since ot in (8) plays the same role a s o ) 2 in (10) and disappears when the ELRRinvariant is constructed. Let us multiply hamiltonian ( 11 ) by a E + b 2 and subtract the ELRR-invariant. As the result, we obtain a differential equation for the function G(z) =a2+b 2, which defines the rms halfwidth of the EGB:

8 H G - 81- 4otG 2= (dG/dT)z.

(20)

which is reduced to the elliptical ones [ 14 ]. Knowing G and Q, we determine a, b and ¢0by the formulas

a=[(G+2GQ)~/2+(G-2GQ)I/2]/2,

(21)

b=[(G+2GQ)'/2-(G-2GQ)I/2]/2,

(22)

~(T)= i 6ds/[G(s)(1-4Q2(s)],

(23)

0

and thus find the solution of system (8), (9) in the analytical form.

(14)

where Go = k 2( Wxo 2 + Wyo). 2 For the case of a nonlinear inhomogeneous medium (or ~ 0) we obtain G ( r ) = H / a + (Go - H / a ) cos 2 x / ~ z .

= 2x/~t,

(13)

The solution of eq. (13) will be different depending on the value of the parameter or. In the case of a nonlinear homogeneous medium, ot = 0 and the solution of eq. (13) is given by

G( Q = 2Hz2 + Go ,

Q

f S d S / [ ( S - Q o ) ( S - Q1 ) ( S - Q 2 ) ( S - Q3) ],/2 t2o

(15)

Note that the quantity Q = ab/(a 2+ b 2 ) defines the beam eUipticity and varies from zero to 1/2 (the case of cylindrical beam). From the ELRR-invariant it is

4. Results and discussion

4.1. Medium without inhomogeneity (or=O) A convenient characteristic for analyzing the REGB behavior is the quantity G. For the sake of 227

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simplicity, we assumed the input beam (3) to be parallel, i.e., ff~(0)=0, fly(0)=0. Proceeding from formula (14), we obtain for REGB in nonlinear medium without inhomogeneity - by analogy with the case of an ordinary EGB [2-4 ] - three regimes of propagation.

The quantity Q characterizing the beam ellipticity is a periodic function of the variable t with period

1. If H > 0, the diffraction spread of the beam prevails (fig. 2a). The quantity G increases by the parabolic law, a and b undergoing oscillations. Let us analyze in more detail the character of these oscillations.

(24)

~d. 2~

3t (a)

3~/2

tT=

2 [I(Ql - Q 2 ) (Qo - Q 3 )

11/2

× [ ( Q i - Q 3 , H ( r c , ~ ,Qo m ) -+QQ, 3 F ( n , m ) ] ,

where mm [ ( Q I - Q o ) ( Q 2 - Q 3 ) / ( Q I - Q 2 ) ( Q o - Q 3 ) ] , / z , F(O, m) and H(~, a, m) are elliptical integrals of the first and third type, respectively. For the roots of the cubic equation (19) the condition 1 / 2 > Q~ > Qo > 0 > (22 > Q3 is fulfilled and the value of Q varies from (20 to Q~. According to (14), (17), the variable t is related to the real distance z = z k by

r= (2H/Go)1/2 tan[ (2HGo)l/2t] . X/2

0 3

I

2

3

~"

0

(b)

0

0

3~¢2

~2 0

0

Fig. 2. REGB propagation along the z-axis (3= kz) in a nonlinear medium without inhomogeneity (ct=0). a and b are the dimensionless halfwidths (G=a2+b 2 determines the rms halfwidth) of the light spot in the rotating coordinate system. ~ is the angle of rotation of coordinate system 0/z. ( a ) P < P~, diffraction spread prevails ( x = l, t~=0.7). (b) P=P~, the self-trapping regime (t¢=1.32, t~=0.7). (c) P>P¢,, self-focusing regime ( r = l . 4 , ~=0.7).

228

(25)

As z is varied from 0 to oo, the value t is varied from 0 to 7r/(8HGo) t/2. Therefore, the number of oscillations for Q cannot exceed 7r/tT(8HGo) l/z. The dependence (25) between z and t explains also the zdependence of a and b which becomes smoother with increasing z. At should be noted that unlike the ordinary EGB in a nonlinear medium [2-4 ], REGB never becomes cylindrical (a is never equal to b). 2. Self-trapping takes place at H = 0 (fig. 2b). From the expression ( I 1 ) for H we find the threshold value of power Per for REGB self-focusing

pcr/Pc=l/2Qo + w 2~ w ~2k 2/8QoRo. 2

(26)

For the axially symmetrical cylindrical gaussian beam expression (26) is equal to unity. For the ordinary EGB which has two symmetry planes the value l / 2 Q o > l points to an increase in the threshold power [2-5 ]. The last term in expression (26) points to an additional increase in the threshold due to the EGB rotation. The following rule is observed: as the degree of symmetry of the gaussian beam decreases, there is an increase in the threshold power for its selffocusing. In the case of self-trapping, the integral I is equal to zero, as is H. Due to this fact expressions ( 1 8 ) (20) are of a somewhat different form. The value of Q is found from

Volume 81, number 3,4

(2 ~ SdS/[S-Qo)(Q,-S)(S-Q2)

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15 February 1991

3 '

] '/2

2~

(2o

(27)

=4x/~t ,

76

where Q~ and Q2 are roots of the quadratic equation Q2

2rQo-1 Q+ 1 8Q----~o ~o =0,

(28)

and at integration the condition 1/2>Q~>Q> Qo > 0 > Q2 is fulfilled [14 ]. Now z= Got and the periodic z-dependence of Q, a and b is illustrated in fig. 2b. The REGB rotation in a linear medium is limited by the angle ~ [ 1,6]. In a nonlinear medium this limitation is lifted and the angle of beam rotation (0, according to (23 ), monotonously increases (fig. 2).

0

I

2

3

0

I

2

3

q~

3. At H < 0 (P>P=) REGB collapses at the point z¢ ~ ( - Go/ 2H ) ' /2 / k .

(29)

As z¢ is approached, the frequency of oscillations of Q (as well as of a and b) increases without bound, the beam rotates at an increasing speed (at z-,z¢, ¢--, +oo in fig. 2c).

~'-~"~

t 'f~~&

mo

O! 0

I

2

3

4

'[

// I

(d)

4.2. Nonlinear light guide (a # O)

Let us consider briefly the REGB behavior in a nonlinear medium with transverse inhomogeneity. Compared to the previous case, there are differences. Since the medium exhibits waveguide properties, the regime of diffraction spread is absent. According to (15), the rms quantity G varies harmonically, and a and b periodically depend on z (figs. 3a3c). It is also seen from (15) that the regime corresponding to the self-trapping in the previous case now takes place at H/aGo= 1 or, in terms of power, at Pw: 2 2 2 Pw/P~ = l /2Qo + w xowyok /8QoRo2

2

2

4

-aWxoWyok /8Qo.

(30)

The dependences of G, a and b on the longitudinal coordinate for this regime are given in fig. 3b. The self-trapping threshold is not affected by the medium inhomogeneity, and the threshold condition for P~r coincides with expression (26). Fig. 3c shows the

'"i mt['[o

0

0

0,5

I

"~

.

Fig. 3. REGB in a nonlinear medium with transverse inhomogeneity ( a # 0 ) . Notations are as in fig. 2. (a) P
regimecorrespondingto the self-trappingin fig.2b (d=0.6, ~= 1, J=0.7). (c) PwPc, self-focusingregime (a=0.6, s:= 1.4,J=0.7). case where the REGB power satisfies the condition Per > P> Pw. At power P> Pc, REGB collapses at the point zc z¢ = arccos [ H / ( H - ago) ] / 2 x / ~ k .

( 31 )

As in the previous case, here, as the point zc is approached, the angle ¢ and the frequency of oscillations of a and b increase beyond all bounds (fig. 3d). 229

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All above-mentioned results are obtained within the scope o f an aberrationless approximation. Since this approxim__ation is rather r o u g h t h e relevance o f system (8) and the justifiability o f the results obtained should be discussed. Besides the aberrationless approximation there are other more precise analytical methods in the beam propagation theory: a method making use o f the invariants o f eq. (2) [ 15 ], a moments method [ 16 ], a variational approach [ 2 ]. These methods make a good evaluation o f the selffocusing threshold (the deviation from numerical experimental results equals to several per cent only). A system like system (8) can be obtained by these methods. A main divergence is manifested in the value o f parameter K describing the m e d i u m nonlinearity. Other terms in the system (8) are responsible for linear effects (including the beam rotation) which are not distorted by the aberrationless approximation. It gives reason to hope for correct qualitative description o f the rotating beam propagation in nonlinear media. Let us confirm a hypothesis about the beam symmetry influence on the self-focusing threshold with the help o f the invariants o f eq. (2) [ 15 ]: I,=$5 0

[ U[2 d x d y ,

(32)

0

12=i~(IViU,2-kfl[U[4/2)dxdy 0

(33)

0

(the inhomogeneity is not considered for simplicity, ? = 0). The self-focusing occurs at 12 < 0, i.e., when a nonlinear compression predominates:

0

0

0

0

(34) One oppose the nonlinear compression at I~ = const, by increasing the integral in the left-hand side o f inequality (34). The value o f this integral is the least for the axially symmetric beam. Symmetry breaking o f the beam leads to increasing the U-field gradient (the left,hand side o f inequality ( 3 4 ) ) , which confirms o u r hypothesis.

230

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5. Conclusion The paper ¢on~idem the behaviour o f R E G B i n nonlinear media. Compared to the axially symmetric gaussian beam and EGB without rotation, for REGB the threshold power for self-focusing is greater by a value depending on the parameter specifying the beam rotation. This property o f ERGB can be used to prevent an optical system from being destroyed by powerful radiation fluxes. Proceeding from the above-mentioned rule, the following assumption can be made: if the gaussian beam is made even more asymmetrical, the threshold power for its self-focusing will increase. In this paper the found rule is considered using the aberrationless approximation, which is rather rough. Furthermore we plan a more detailed investigation o f symmetry beam effect on its propagation, including a numerical experiment.

References

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