Optical vortices

Volume 73, number 5

OPTICS COMMUNICATIONS

1 November 1989

OPTICAL VORTICES P. C O U L L E T 1, L. G I L and F. R O C C A Laboratoire de Physique Thborique, Parc Valrose, 06034 Nice Cedex, France Received 27 June 1989

It is shown that laser cavities with large Fresnel number can exhibit a state which bears some analogy with a superfluid vortex. This optical vortex is studied in the framework ofa Maxwell-Blochmodel.

Transitions are often associated with spontaneous symmetry changes. Phase transitions [1] in magnetic systems, in liquid crystals, instabilities [2 ] in hydrodynamics as the Rayleigh-Benard or the Couette-Taylor flows provide paradigms of such symmetry breaking transitions. Defects as for example walls in magnets, vortex in superfluid helium, grain boundaries and dislocations in cellular patterns, are natural consequences of the symmetry breaking phenomenon [3]. The aim of this communication is to describe a new laser state which is related with the existence o f a vortex solution of the laser equations. The first part of this communication is devoted to clarify the nature o f the optical vortex on symmetry grounds. In the second part, we propose a quantitative analysis which relies on a Maxwell-Bloch model [ 4 ], which was first formulated to describe the phenomenon of spontaneous transverse spatial pattern formation in lasers, similarly to what was previously done in the case of passive systems [ 5,6 ]. The analysis o f this model provides both numerical and analytical evidences for the existence of the optical vortex. It must be kept in mind however, that these results do not depend on the special features of this model nor on the approximations involved in its derivation [ 6 ]; what is crucial, as it is well known in other kinds of systems, is only the presence of some symmetry property. As a conclusion the possibility of a turbulent state involving a dilute gas of such vortices [ 7 ] is considered. Also Observatoire de la C6te d'Azur. Mont Gros.

Laser transition belongs to the wide class of broken symmetry transitions [8]. Above the laser threshold the phase of the electric field inside the cavity takes a definite but arbitrary value. The appearance o f a laser state breaks a phase invariance of the optical medium. This symmetry property reflects itself as the invariance of the dynamical system describing the laser under a gauge transformation for the electric field E ~ E exp (i¢), provided that other quantities, as for example the polarisation, transform appropriately. To be more explicit, we consider a laser cavity in the case of large Fresnel number, which is similar to the condition of large aspect ratio in hydrodynamical systems [2]. Above threshold, the electric field sustains large scale spatio-temporal phase fluctuations. These fluctuations are reminiscent of the Goldstone modes associated with continuous symmetry breaking as for example in superfluid helium or in X - Y magnets [9]. In nonequilibrium systems the dynamics of these fluctuations is described by a phase equation [ 10,1 1 ]. Recently such an equation has been explicitely derived from the Maxwell-Bloch model of ref. [4] which involves diffraction in the transverse plane [ 12 ]. Below threshold, the electric field inside the cavity is assumed to be a small fluctuating field. In three dimensions, the zeroes of a complex field lie at the intersection of surfaces where the real part and the imaginary part both vanish. Inside the cavity one should then expect to observe fluctuating lines of zeroes. These lines cut a typical transversal plane into points. Let us now, for the sake of clarity, perform the fol-

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lowing G e d a n k e n experiment. At some time 1=0, where the laser is suddenly brought above threshold, one assumes the existence of only one zero of the electric field in the transverse plane (see fig. l a ) . Because of elementary topological considerations, although a null electric field is d y n a m i c a l l y unstable above threshold, such a zero is expected to survive in the long-term dynamics. Far from this singular point, the modulus o f the electric field reaches its asymptotic equilibrium value corresponding to the homogeneous steady state o f the laser (see fig. l b ) , while its phase changes when one moves a r o u n d this point (see fig. l c). More precisely the circulation o f the phase gradient on any loop which encloses the vortex core is equal to _+2Jr. This state, formally analog to a vortex in superfluidity is called optical vortex. In the superfluid transition, the macroscopic state of the helium breaks the rotational invariance o f the wave function, leading to a similar s y m m e t r y breaking p h e n o m e n o n . Generally a given electric field can show several zeroes transversally to the laser cavity.

Re(E)-0

/

Ira(E) 0

(a)

~:.

Associated with each o f these zeroes, a topological charge is defined as the gradient circulation around a closed loop which encloses it. As a simple consequence o f the s y m m e t r y o f the d y n a m i c a l equations, the total vorticity, that is the total topological charge of a given field configuration is a dynamically conserved quantity [3]. Vortices a p p e a r and d i s a p p e a r in pair or with the help o f boundaries. Optical vortices are now studied in the framework o f the Maxwell-Bloch theory. Our starting point is a model which includes diffraction in the transverse plane [4 ], 0 , E = - ( i(oc + t c ) E + x P + i a K ~ ' 2 E .

( Ia )

0,P=--(i(0A+)'i )1)+71 E D .

(lb)

0,D=-)',[(D-D0)+

(lc)

1 5.L

[c)

*2> x

-

-

~

-

{> x

3r~/2

K

Fig. 1. Sketch of a vortex associated with a complex field E. (a) The transversal crossing of the real and the imaginary part of the initial field. (b) The modulus of E. (c) A snapshot of the equiphases. 404

½(EP+PE)] ,

where E , P, and D which represent the electric field, the atomic polarization and the inversion population, respectively, have been a p p r o p r i a t e l y scaled [4]. ~,-, ,, L and 7, are their associated decay rates. ~oa and coc are the atomic and the cavity resonances. Do is the p u m p parameter. V 2 = O / O . v 2 + O / O y 2 is the laplacian operator, where the transverse variables are normalized to the transverse dimension of the active region, which is assumed to have a square section b ~ The p a r a m e t e r a, which scales the diffraction term, is given by [4]

a = 27r b 2 T -

(b)

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1 2MVT.

(2)

where 2 is the wavelength, L is the longitudinal length o f the active region, 7" is the transmissivity coefficient o f the mirrors of the cavity and F = b 2 / 2 L is the Fresnel number. Differently from ref. [4], we do not add lateral mirrors and for the sake o f simplicity, we assume periodic b o u n d a r y conditions. Let us mention that our results are not sensitive to this choice o f the b o u n d a r y conditions. Eqs. ( 1 ) are invariant under the transformations E~Eexp(iO)

,

P~Pexp(iO)

,

D---~D.

(3)

The existence o f a vortex solution uses the fact that this invariance is spontaneously broken above the laser threshold. This transition occurs when Do>Do,~,= 1+0 2, where 0 = ((O( --(OA)/(X+;'= ) is the cavity detuning parameter. Pseudo-spectral nu-

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merical simulations o f eqs. (1)~1 have been perf o r m e d on a Cray-2. F o r convenience the initial conditions have been chosen in order to m i m i c the previous Gedanken experiment, namely E, P and D are small and r a n d o m fields. F o u r significant zeroes o f the electric field were present in this initial state. The p a r a m e t e r s values chosen are x/y± =0.04, y ± / 711= 1, a = 4 × 10 -4, 0 = 0 . 6 and (Do-Do,c)/Do,c= 1.2. Hence our laser belongs to class A in the classification of ref. [13]. After 50000 time steps, the resuiting fields have stationary properties. In particular the electric field exhibits four zeroes at stationary positions. One o f this zeroes (see fig. 2 - 6 ) , roughly located at the center o f the cavity enables us to make a careful study of the vortex solution. The polarization also vanishes at the vortex core while the inversion population field shows a maximum. The vortex core radius is about 5% o f the transversal dimension. The vortex presents a striking analogy with the spiral waves often observed in the Belousov-Zhabotinsky reactions [ 10 ]. The reason for this similarity has to be found in the very nature of the transition which is a H o p f bifurcation in

1 November 1989

Y

F(!((((i X Fig. 3. Snapshot of the equiphases of E corresponding to fig. 1. The phase difference between two adjacent lines is 7t/4.

~ Actually eq. ( 1) have been simulated in the so-called rotating frame. IEI

X

Fig. 4. Perspective view of the modulus of the electric field corresponding to fig. 1.

X Fig. 2. Snapshot of the real part of the electric field in the transverse plane. The grey scale follows linearly the value of the field.

both cases. As the spiral wave which is not a stationary solution, the optical vortex rotates a r o u n d its core with a constant angular velocity o f the order o f the laser frequency. The next step in our analysis consists in reducing eqs. ( 1 ) to a simpler one. This reduction is reminiscent o f the Newell-Whitehead-Segel analysis [ 14,15 ], of the convection onset in Rayleigh-Benard flows. Close enough to threshold, a solution o f eqs. ( 1 ) is found as 405

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I November 1989

......................

IEI

,---L p-I...........

,, y i'

,

X

//J i

i

J Fig. 6. Radical sketch of the optical vortex. X

e=x//)i;-~D0,<, m e a s u r e s the d i s t a n c e from threshold. A represents the c o m p l e x a m p l i t u d e of the order p a r a m e t e r . Eq. ( 4 ) is a solution o f e q s . ( 1 p r o v i d e d ,4 ~2 obeys the following e q u a t i o n

/+

•

/j/

!

~--

0,,1 ( OT = _ l + q + i 0 o

Re(P)!

+ia ~

|

i

0

+cA exp(--i~Orl)

\Do/ -~21AI2

+ O ( ~ 3) .

I¢O)A + 7 ± COo:. COc,+ tioga --

406

,<+ ]'±

l +q

)

V2..t. (5)

(,)

where 00= ( e ) C - - O A ) / 7 ± is the cavity d e t u n i n g par a m e t e r in the good carlO, limit ~ 0 K T=~ 2 - ( 1 +r/)2+.{00[ ( q - 1 ) / ( r / + l ) ]}2 l , a n d V2=O/OX2+O/OY2, with

1 iO 0

X= e

(4)

T h e p a r a m e t e r r/ is d e f i n e d as r/=~c/7'± a n d e;R is given by the well k n o w n m o d e p u l l i n g f o r m u l a

(0 R

20o,1

l-il+~l+O~[(l_~l)/(l+q)]2

/

Fig. 5. Zoom of the optical vortex located at the center of the cavity. (a) The modulus of the electric field. (b) The inversion population. (c) The spiral of the real part of the polarisation.

=

-

/

X

(!) (°/

(

1 71 , l+r/)(.,i-l,.ll2A)

X

I1 + r/+ 02( ( 1 -- t / ) / ( 1 + q ) 2 ) ],/2

a n d a s i m i l a r scaling relation holds b e t w e e n Y and y. It is i n s t r u c t i v e to r e m a r k that in the good cavity limit eq. ( 5 ) reduces to [4,12]

8TA= (1 + i 0 0 ) ( A - - ] A I 2 A ) - i - i a V 2 A .

(6)

This e q u a t i o n only involves a purely imaginary. d i f ,2 As usual in bifurcation analysis eq. (5) is found as a solvability condition.

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fusion term. In particular, in the limit of large detuning, eq. (4) becomes the nonlinear Schr6dinger equation. This equation is known in two dimensions to give finite time blowing up [ 16 ]. From a mathematical point of view eq. (6) is structurally unstable since any small positive real diffusion constant suppresses this blowing up. As seen from eq. ( 5 ) such a small diffusion constant is naturally provided, in the case of positive detuning, by the first order correction to the good cavity approximation. The existence of a positive real diffusion constant in eq. (5) is crucial to explain the regularity of the vortex solution of the Maxwell-Bloch equations. In order to make contact with previous studies on eq. (5), it is convenient to introduce a new complex amplitude, new temporal and spatial scales. For positive 0o eq. (5) can be cast in the form 0rA = A - (1 + i o t ) I A I 2 A + ( 1 +ifl) V2A.

(7)

This equation, known as the complex Ginzburg-Landau equation [10], describes the weakly nonlinear development of the oscillatory instability. Eq. (7) possesses spiral wave solutions [ 17 ] in the form

A=R(r) exp[io)sT+iTS(r) ± i O ] ,

(8)

where r and (9 are the polar coordinates, R(r) and ~U(r) are two functions which obey ordinary differential equations in r and such that R(r)~2sr for small r and R ( r ) ~ l and ~(r)~ksr for large r. cos and ks are function of ot and fl and can be obtained from matching conditions. A comparison of the vortex solution of eq. ( 1 ) with the spiral solution of eq. (7) together with a detailed derivation of the Ginzburg-Landau equation will be given elsewhere. We have shown in this communication the existence o f a new laser state which consists in spiral waves rotating around the zeroes of the electric field. This solution, which can be seen as a phase singularity of the electric field, appears naturally in numerical simulations performed on the Maxwell-Bloch equations. Laser cavities with large Fresnel number could be the ideal experimental candidates in order to isolate optical vortices. This phenomenon requires small values of the parameter a in order to allow the inter play o f a large number o f transverse modes. In our simulation we had a = 4 × 10 -4, but we expect that the phenomenon persists for a ~ 10- 2.

1 November 1989

Because a = 1/2zcFT, if we choose T = 10 -2 F h a s to be of the order 10 3. Recent study on the Ginzburg-Landau equation have emphasized the existence of a type of turbulence associated with the presence of a dilute gas of vortices [ 7 ]. This state is shown to be a consequence of the instability of phase fluctuations. Direct simulations of the Maxwell-Bloch equations in the parameter range where phase instability occurs are in progress.

Acknowledgements N. Abraham, T. Arecchi, J. Lega, P. Lallemand, L. Lugiato, L. Narducci, J. Tredicce and D. Walgraef are acknowledged for a number of fruitful discussions. L. Lugiato is also acknowledged for his considerable help in re-writing parts of this paper. L. Lugiato, L. Narducci and J. Tredicce are acknowledged for their communicative enthousiasm which helped a lot to complete this work. One of us (P.C) want specially to thank N. Abraham and P. Lallemand for conveying his generic interests into optics. We acknowledge the CCVR (Centre de Calcul Vectoriel pour la Recherche) where the numerical simulations presented in this paper have been performed, the NCAR, and the D R E T (contract N o 88CO145 ) (Direction des Recherches Etudes et Techniques) for a financial support and the EEC twinning contract

Spatio-temporal chaos in extended system,

References [ 1] L.D. Landau and E.M. Lifshitz, Statistical physics, Part 2 (Pergamon Press, 1980). [ 2 ] Cellular structures in instabilities, eds. J.E. Wesfreid and S. Zaleski, Lecture Notes in Physics 210 (Springer-Verlag, Berlin, 1984). [ 31 Physicsof defects, eds. R. Balian, M. Kleman and J.P. Poirier (North Holland, Amsterdam, 1980). [4] L.A. Lugiato, C. Oldano and L.M. Narducci, J. Opt. Soc. Am. B5 (1988) 879. [5] L.A. Lugiato and R. Lefever, Phys. Rev. Lett. 58 (1987) 2209. [6] L.A. Lugiato and C. Oldano, Phys. Rev. 37A (1988) 3896. [7] P. Coullet, L. Gil and J. Lega, Phys. Rev. 62 (1989) 1619. [8 ] H. Haken, Synergetics: an introduction (Springer Verlag, Berlin, 1977). 407

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[9 ] D. Forster, Hydrodynamic fluctuations, broken symmetry, and correlation functions, Frontiers in Physics 47 ( 1975 ). [10] A.T. Winfree, The geometry of biological time (Springer, Berlin, 1980). [ 11 ] Y. Pomeau and P. Manneville, J. Phys. Letters 40 ( 1979 ) 609. [12] R. Lefever, L.A. Lugiato, Wang Kaise, P. Mandel and N. Abraham, Phys. Lett. 135 (1989) 254, [13]F.T. Arecchi, Instability and chaos in single-mode homogeneous line lasers, in: Instabilities and chaos in

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1 November 1989

quantum optics, eds. F.T. Arecchi and R.G. Harrisson, Springer Series in Synergetics 34 (Springer-Verlag, 1987 ). [ 14 ] A.C. Newell and J.A. Whitehead, J. Fluid Mech. 38 ( 1969 ) 279. [ 15 ] L.A. Segel, J. Fluid Mech. 38 ( 1969 ) 203. [ 16] A.C. Newell, Solitons in mathematics and physics, SIAM, CBMS, 48 (1985). [17]P.S. Hagan, 1982, Spiral waves in reaction-diffusion equations, SIAM J. Appl. Math. 42, 762.