Spontaneous optical patterns in an atomic vapor: observation and simulation

PHYmA ELSEVIER

Physica D 96 (1996) 230-241

Spontaneous optical patterns in an atomic vapor: observation and simulation W. Lange *, Yu.A. Logvin l, T. Ackemann lnstitut fiir Angewandte Physik, Westfdlische Wilhelms-Universitiit Miinster, Corrensstr. 2/4, D-48149 Miinster, Germany

Abstract

Optical pattem formation in an experiment with single mirror feedback is described. The nonlinear medium is sodium vapor in a buffer gas atmosphere. A microscopic model is given and a stability analysis and numerical simulations are performed. Good agreement between the results of the experiment and the simulation is obtained. By numerical treatment of the model for the case of a plane incident wave (large aspect ratio), the results obtained with a narrow Gaussian beam (small aspect ratio) are traced back to the transition from hexagon to roll formation via 'mixed' states.

1. Introduction

Recently spontaneous pattern formation and more generally spatio-temporal effects in optical systems have found increasing interest [1-8]. In the case of lasers operating far above threshold the occurrence of complicated spatial and spatio-temporal structures has always been of major importance in applications, but now it seems to have been recognized that general methods of nonlinear science may be helpful in studies of these phenomena. Vice versa it might be expected that studies of spatio-temporal effects in optical systems can shed some light on pattern forming processes, since optical systems have advantageous features: light propagation is described by the (linear) Maxwell equations; nonlinearities come into play only via the polarization of the medium induced by the optical field. In favorable cases the po* Corresponding author. 1Permanent address: Institute of Physics, Academy of Sciences, 220072 Minsk, Belarus.

larization can reliably be calculated in a microscopic model, i.e. by calculating the interaction between the medium and the light field in the formalism of quantum mechanics. Thus, in a certain sense, it should be possible to perform ab initio calculations of patterns and spatio-temporal complexity. Obviously low density atomic gases are best suited for this purpose. From the experimental point of view they have the additional advantage of a very good optical quality; the reproducibility of the optical properties is also excellent. Although lasers are prototypes of pattern forming optical systems, they are probably not ideal candidates for studies of pattern formation at present. In most practical lasers the design imposes severe limitations on the spatial patterns which can evolve, with the consequence that a few linear modes of the laser resonator are sufficient to describe the observations in most cases, i.e. the patterns are completely dominated by boundary effects. On the other hand the modelling is extremely hard in the case of a high Fresnel number

0167-2789/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PH S0167-2789(96)00023-1

W. Lange et al./Physica D 96 (1996) 230-241

(a)

nonlinear

I

(b)

Na + N2 laser F E O ~ ~ E L beam ~"--.,)/j~ B ,

mirror

medium

d

I

mirror *---Eb d

--I Nearfield Fig. 1. Scheme of the experiment: thin layer of nonlinear medium with feedback mirror, (a) general, (b) experimental realization (see text).

which corresponds to the situation of a large aspect ratio in hydrodynamics. Recently, however, spatial and spatio-temporal effects in much simpler optical systems have been discussed. One of those is the one shown schematically in Fig. l(a) [9,10]. Here, a light wave passes through a thin nonlinear medium and is reflected by a plane mirror situated in some distance from the medium. Ideally in this system the nonlinearity of the light propagation in the medium can simply be incorporated into a (spatial) phase-modulation of the transmitted wave ('phase encoding'), which gives rise to diffractive effects in the (linear) propagation of the light in the region between the nonlinear medium and the mirror. In this setup the formation of a hexagonal lattice is expected, if the intensity of the incident light field, which is treated as a plane wave, exceeds a threshold value. At higher intensities more complicated spatiotemporal behavior is expected. Also the consequences of replacing the plane wave by a Gaussian beam have been studied [ 12].

231

Pattern formation in a setup involving a sodium cell and feedback by a single curved mirror was first observed by Giusfredi et al. [13]. After the model of Firth and dAlessandro had become available several experiments involving a fiat mirror were performed by means of liquid crystals as the nonlinear medium [1418] and in a system which simulates the setup of Fig. l(a) by means of a liquid crystal light valve (LCLV) [19-21]. Very recently many details in an LCLVexperiment were reproduced in numerical simulations which include the saturation of the medium and polarization effects [22]. These results clearly demonstrate the importance of taking the detailed properties of the nonlinear medium into account. Also an experiment involving rubidium vapor as the nonlinear medium has been reported [23]. Since, however, a polarization instability was involved in this case, the experiment cannot directly be compared with the others. It should be noted that single-mirror experiments in atomic vapors which always require a considerable interaction length have a close relation to corresponding experiments involving the mutual coupling of counterpropagating beams without external feedback. Spatial instabilities and hexagonal structures [24,25] have been observed in this type of experiment. Recently our group has reported an experiment involving sodium vapor as the nonlinear medium in the geometry of Fig. l(a) [26]. In this paper we are going to present further experimental results and to give a theoretical analysis which starts from a microscopic model and does not involve adjustable parameters in principle. From the comparison it will become clear that there is a striking similarity between the experimentally observed scenario and numerical simulations which in turn are backed by analytic considerations. Moreover quantitative comparisons are possible to some extent.

2. The experiment 2.1. The nonlinear medium

In the analysis of [9] it is assumed that the nonlinear medium is a 'Kerr medium', i.e. that there is an

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w. Lange et al./Physica D 96 (1996) 230-241

intensity dependence of the index of refraction of the form n(1) = no + n 2 • I.

(1)

The medium is called 'self-focusing' in the case d n / d l = n2 > 0, while it is 'self-defocusing' in the case dn/ dI < O. In atomic vapors the largest values of l d n / d l l result from the intensity dependent change of the population density of atomic states; since these processes rely on absorption processes the nonlinear dispersion is generally combined with nonlinear absorption and thus the transmitted wave in Fig. l(a) will be partially absorbed and it will contain a spatial amplitude variation as well as a spatial phase modulation. Small variations of the frequency of the light field in the vicinity of the atomic resonance transitions allow to find a balance between large values of I d n / d l I and reasonable absorption losses. The intensity needed to hold a sufficient amount of atoms in the excited state is still too high to be obtained by tunable cw lasers without focusing. Alkali atoms, however, display optical nonlinearities based on optical pumping between Zeeman sublevels of the atomic ground state which is efficiently produced by absorption of the Dl-line. If the light source is a cw dye laser, sodium atoms are most convenient in the experiment. Since the population differences between the Zeeman sublevels, which give rise to an 'orientation' and an 'alignment' of the sample [27], are long-lived [28], the thermal motion of the atoms would prevent any pattern formation based on the resulting nonlinearity in a real experiment. The thermal motion can be converted into a slow diffusion process by adding a buffer gas. Conventionally a rare gas is used for this purpose, but we use nitrogen instead. Nitrogen quenches the fluorescence of the sodium atoms and it is added in order to prevent the diffusion of radiation which would add nonlocal nonlinearity to our problem [29]. Thus the theoretical description is facilitated tremendously. (As a matter of fact the experiment does not work without a quencher.) The buffer gas also introduces line broadening. By using a fairly high pressure (300 hPa) we introduce

a homogeneous linewidth of 3.6 GHz which exceeds the Doppler broadening and the hyperfine splitting of the sodium ground state. Since the experiment relies on nonlinear refraction, the absorption has to be kept low. This requires a large detuning with respect to the atomic resonance due to the large pressure broadening and has the consequence to increase the power requirements for the laser beam. This is a tribute to be payed for the sake of deducing a simple and yet reasonably realistic model for the experiment. The distribution of the population of the Na ground state in the individual Zeeman sublevels is only marginally affected by collisions with the buffer gas. (The decay rate ), describing the effect of collisions is about 6 Hz under the conditions of our experiment.) As a matter of fact the collisional decay time of the orientation is larger than the time it takes a Na atom to diffuse from the region of the laser beam to the walls of the cell which can be expected to destroy any orientation. Since there is no other relaxation mechanism in the experiment, a distribution of orientation would be created which varies smoothly and monotonically from a maximum in the laser beam to zero at the cell walls, if no other loss mechanism is introduced (see below). Due to the long lifetime of the ground state orientation the corresponding optical nonlinearity can easily be saturated. It is necessary to drive the system into the regime of saturation of the nonlinearity in order to achieve sufficient phase modulation of the wave by the nonlinear interaction: It has to be kept in mind that the index of refraction is very close to one in a diluted gas and cannot change too much in a high intensity field. In the present experiment it was even more mandatory to go into the saturation regime, since we had to use a thin medium. Therefore it was clear from the beginning that saturation of the nonlinearity would play its role. It will be shown, however, that it is by no means sufficient to add just suitable saturation terms to Eq. (1). 2.2. Experimental setup

The experimental scheme is shown in Fig. 1(b). It is very similar to the one described in [26], but we have

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reduced the length of the heated zone in the sodium cell from 40 to 15 mm, in order to be closer to the theoretical model. In the experiment a dye laser is closely tuned to the resonance corresponding to the Na D]-line. The detuning A is about - 1 5 GHz, i.e. the laser is red-detuned. As a result the medium is self-defocusing for small intensities. The laser beam is carefully cleaned by a spatial filter. A beam waist with a radius w0 = 1.38 mm is situated in the center of the cell. The laser beam is circularly polarized in order to induce efficient optical pumping. The sodium density is about 1014 cm -3 in the center of the cell, which contains N2 at a pressure of 300 hPa as a buffer gas. (The sodium density and the pressure broadening are determined by fitting Voigt curves to the small signal absorption profile.) The mirror has a reflection of R = 91.5%. The transmitted beam is monitored by a CCD camera or alternatively focused on a photodiode D. The distance between the center of the cell and the mirror is d = 75 mm in the experiments discussed here. The cell diameter is 8 mm. Two pairs of Helmholtz coils produce an oblique magnetic field. The component parallel to the direction of the input beam, the longitudinal component, has a strength B~ between 5 and 50 p~T and the transverse component Bx is chosen in the range 1-10 txT. The y-component of the earth magnetic field is compensated by a third pair of Helmholtz coils. Though the magnetic field is weaker than the earth magnetic field, its role is crucial in the experiment. It will be discussed in detail in Section 3. Here we only mention that the field has the purpose of counteracting the spatial wash-out effects produced by particle diffusion. (The diffusion constant is estimated to be D = 2 x 10 4 m2/s.) 2.3. Experimental results

When the percentage of the spatially integrated transmitted power is measured in dependence on the input power (Fig. 2), it is seen that there is indeed strong saturation for low power. Counterintuitively the transmission has a maximum at finite values of the power of the laser beam and it drops monotonically in the rest of the power range available in the experiment. In this region of a negative slope of transmis-

1,0 e

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i

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0,8

Z © r.13

0,6 ac(

Z <

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0,4

0,2

001!

' 0

~

50

J

100

i

150

00

2

'

~---

250

Ptas/ mW

Fig. 2. Experimental whole-beam transmission of the sodium vapor cell with feedback mirror. The letters correspond to the patterns displayed in Fig. 3. The experimental parameters are: particle density, N _~ 0.9 × 1014 cm -3, cell-mirror distance, d = 75 mm; detuning, A = -14.5 GHz. The magnetic field corresponds to 12x = 2'rr (304-3) kHz, I2: = 211"(2644-4) kHz. sion we observe patterns. It should be noted that the maximum and the patterns occur only if a transverse as well as a longitudinal magnetic field are applied. The reason will become clear in Section 3. For low powers, but above some threshold, we observe the formation of a dark hole in the center (Fig. 3(a)). With increasing power the triangular structure of Fig. 3(b) occurs. It is replaced by more complicated structures, which are also built from equilateral triangles. The structures seem to adjust themselves to fill the whole high intensity region of the input beam, i.e. the figures seem to be cut out from an infinite hexagonal lattice. The edges of the pattern carrying region seem coarsely to be determined by the condition that the intensity surpasses a threshold in the interior of the region. The modulation depth is up to 100% in the near field observation employed. Note that the observation of a sequence of stable patterns in which the number of constituents increases with power is different from the reported numerical [12] and experimental [18] results in Kerr-like media in which the patterns become time-dependent just after the first or second polygon-like pattern beyond threshold. However, a similar sequence with increasing beam power from a single peak to a triangle and

234

W. Lange et al./Physica D 96 (1996) 230--241

a)

b)

a)

Ill/

c)

d)

e)

z)

8)

/Ill

f)

g)

h)

e)

¢)

ill

i)

j)

ll

Fig. 3. Examples of the patterns in the experiment (lst row, parameters as in Fig. 2) and in the simulation (3rd row, see Section 3) and their Fourier transforms (2nd and 4th rows, resp.). The power levels used in the experiment are marked in Fig. 2, the ones used in the simulation are marked in Fig. 6 (see Section 3). The DC-component is suppressed in the Fourier spectra. than to a rhombus has been found in numerical studies of passive cavities with plane mirrors [30]. With increasing beam power the patterns are less stable. They may switch between different species coexisting in the same power range. Figs. 3(h)-(j) are just frozen images of patterns which are in permanent motion. Unfortunately we cannot resolve the full temporal evolution at moment and thus the correlation time is unknown. It is, however, certainly less than 1 ms. The patterns obtained at the highest power levels are no longer built from triangles. The dark holes are not regularly ordered, but one may observe a tendency of the holes to arrange in parallel (straight or curved) lines, i.e. there seems to be a tendency to form rolls (see Fig. 3(h) or 3(i)). Some information can be obtained from the Fourier transforms of the patterns, i.e. from the spectra of spatial frequencies. They are displayed in the second row of Fig. 3. Due to the build-up of the ordered patterns from regular triangles the corresponding Fourier transforms are regular hexagons. The diameter of the hexagons in the Fourier plane defines

the 'wavelength' of the patterns. We prefer the term 'characteristic length' instead. In Fig. 4 it can be seen that the characteristic length does not depend significantly on the laser power. In the case of the irregular patterns the power density in the Fourier plane is still maximum on a ring whose diameter is the same as the diameter of the hexagons at lower power levels, i.e. there is still a characteristic length in the system (see Fig. 4). It does not change significantly in the transition from the regular to the irregular region.

3. Theoretical description 3.1. Model of the experiment

In the theoretical description of the experiment we use the approach described by Firth [9,10], but replace the assumption of a Kerr medium by a microscopic model of the experiment which has been found to describe other nonlinear optical experiments involving sodium vapor in a very satisfying way [31,32]. It is based on the following equation of motion for a Bloch

235

W. Lange et al./Physica D 96 (1996) 230-241 0,36

i

I

I

I

0,34.

E E '~" 0,32'

t I

_¢

t'O ¢J (R

0") r- 0,30,

0,28

regular

irregular

I

I

i

I

I

0,0

0,2

0,4

0,6

0,8

reduced pump power

(P-Pthres)/Pthres

Fig. 4. Length scale of patterns in dependence on the normalized distance from threshold. Experiment, open circles (parameters as in Fig. 3); simulation, full circles (see Section 3).

vector m = (u, v, w) which is built from components of the density matrix of the sodium ground state Otto = - ( 7 - D V 2 + P ) m - ~.z P + m × I2.

(2)

The components u, v, w of m represent the x, y, zcomponents of the expectation value of the magnetic moment in a volume element. ~, is the collision induced relaxation of m. D is the diffusion constant, V 2 is the Laplacian, P denotes the optical pump rate./2 is a torque vector. The first term on the right hand side of Eq. (2) contains losses of the magnetic moment by relaxation by diffusion and also by a power dependent contribution. The diffusive term has been added to describe the thermal motion of the atoms whose mean free path is very small in comparison with the length scales found in the experiment. The second term describes the creation of a z-component of m due to the optical pumping process and the third term describes a precession of m around the vector I2. The vector /2 = ([2x, O, [2 z - P A ) is not only built from the Larmor frequencies belonging to the x- and z-component of the magnetic field, S2x and $2z, but it also contains the term P A , i.e. it depends on the pump rate and on the detuning A, which is normalized to the relaxation

constant F2 of the polarization of the medium. The extra term describes a light-induced shift of the Zeeman sublevel m = - ½ , i.e. a 'light shift' [31]. The light shift is obviously equivalent to a longitudinal component of the magnetic field. It was already mentioned in Section 2.1 that the transverse component of the magnetic field Bx is needed in order to counteract the wash-out produced by the atomic motion. It provides a mechanism which destroys the longitudinal component of S2, and thus is a substitute for relaxation processes which could prevent a spatially uniform saturation of the medium. Formally the role of Bx is described by the presence of S2x in ~ . Since the influence of this component on w is most pronounced, if the third component of /2 vanishes, i.e. if m processes around the x-axis, the role of B z in the experiment is immediately clear: it serves for compensating the light shift. This compensation occurs for a well-defined intensity only. In this way a strong intensity dependence of w is introduced which survives the diffusion processes. It is translated into spatial dependence during the process of pattern formation. The influence of the light shift on nonlinear optical processes is discussed in more detail in

236

W. Lange et al./Physica D 96 (1996) 230-241

[35]; its role in the present experiment will become further clarified in Section 3.2. The pump rate P is proportional to the local intensity which is given by superimposing the field strengths Ef and Eb of the forward and the backward wave P = (lEe + Ebl2)ltzel2/4h2F2( A2 + 1).

(3)

Calculating from Eq. (2) the z-component of the Bloch vector, w and inserting it into the expression for the complex susceptibility

X --

Nl#el 2 A + i 2hEoF2 ,42 q - ~ (1 - w) ~ Xlin(1 - W)

(4)

with N being the sodium particle density, we obtain a self-consistent system of equations for field and medium. In its solution we have to introduce some approximations. We do not only neglect diffraction effects within the sample, but we also neglect any longitudinal variation of the intensity. This means that we neglect any standing wave effects and replace IEf + Eb 12 by lEd + IEb] 2. This may not be unreasonable, since standing wave-effects can be expected to be washed out by thermal motion. Moreover we replace Ef by the incident field E0 and calculate Eb from the transmitted one Et = Eoe -ixkl/2 after its propagation in free space between the cell and the mirror with reflection coefficient R. It should be emphasized that the model presented here up to now completely neglects the nuclear spin of the sodium atom (I = 3). Even if the spectral width of the incident light or the homogeneous width of the transition exceeds the hyperfine splitting, the hyperfine interaction still has consequences. It has been shown that the Lande factor gj of the magnetic interaction has to be replaced by IgFI = gL/4 [32]. Moreover the efficiency of the pumping process producing orientation is reduced by the hyperfine coupling. This effect is very roughly taken into account by introducing a correction factor 3 in Eq. (3) which results from statistical considerations (see [32,33]). As a consequence of all these approximations we cannot expect quantitative agreement with the exper-

iments, but we can still hope to explain their main features. 3.2. Stability analysis

The steady-state plane-wave solution of Eq. (2) is given by the nonlinear algebraic equation for the orientation Ws w~ -

Ps

- -

($2z - APs) 2 + (Y + Ps) 2

}" ÷ Ps (a"2z -- APs)2 -4- (y ÷ Ps) 2 ÷ a"~x 2

(5)

with Ps = PO(1 + Rle(-iklzunO-ws)/2)12) and the successive substitution of Ws into the expressions for the other Bloch vector components

Us = Vs =

(t'-2z _ A P s ) ~ x W s (~z - APs)2 ÷ (2" ÷ Ps) 2'

(1 + Ps)S2xWs (s2z - ` 4 P s ) 2 + ( y + p , ) 2

(6) (7)

In Eq. (5), P0 denotes the pump rate introduced by the forward beam (P0 ~ [E0[ 2) which is used as a control parameter. The structure of Eq. (5) permits to explain the main features of the dependence ws(P0) presented in Fig. 5. Beginning from zero, Ws increases very quickly because of saturation of the first factor (V is small and IS2x[ is small compared to I~z - PAl). With further increasing P0 the second factor dominates and displays a 'resonance' behavior at Y2z = Ps A. More exactly, the shape of the curve is akin to a nonlinear resonance [34] with an overlapping part of characteristic that corresponds to the phenomenon of optical bistability. Obviously for a given detuning g the condition I2z - PA = 0 just defines the intensity needed for compensating the Zeeman splitting introduced by the longitudinal magnetic field component. It can be seen from Eq. (5) that Ws rapidly decreases with IS2xl for a given value of P~, if the condition is met, while otherwise the influence of IS2x] is small. Due to Eq. (5) and the relation n = 1 + Re(;()/2 between susceptibility and index of refraction the nonlinear medium behaves in a different manner in the intervals with positive or negative slope of the characteristic ws(P0): since Re(xli,) is positive for negative detuning `4, the index of refraction decreases with intensity at the intervals with positive slope of ws and

237

w. Lange et aL/Physica D 96 (1996) 230-241

the medium is defocusing in the corresponding intensity range, whereas it is self-focusing in intervals with negative slope. Thus the deep minimum in the graph describing the power dependence of Ws has the consequence of introducing self-focusing in a certain power range, though the experiment is performed on the 'self-defocusing side' of the resonance line. The next stage of investigation is the analysis of stability against a spatial inhomogeneous perturbation proportional to cos(k±r±), which yields the marginal stability condition

:Fe. APs-- ~:

APt :Felt"

o

-S2x

(us- Avs),

I

¢ ¢.,.

0.6

o ¢-.

0.4

0.2

H/t MS1R MS2 U H0

0

10

2o

3o

4o

I

(Vs + A u s ) S + SZ,-I=0, :Fetr -

(1

I

where gefr = Y + Ps + D k 2 and Re Xlinkl le I-ik/xli°(1

Po/y * 10 .3

i

-

(8)

3 ---- - R P o

0.8

u,,))12

(d is the distance between the cell and the mirror, k is the wave number of the light). Analysing the condition (8), we find the unstable domain on the steady-state characteristic in Fig. 5. Because of the self-focusing property of the medium on the interval with negative slope, the structures born here have a different, i.e. larger, spatial period than the patterns on the increasing, self-defocusing, part of the characteristic in accordance with the predictions of the theory for a Kerr medium [9,10]. 3.3. Numerical simulations

Two numerical codes were developed. The first one was designed to simulate the real experiment for the input beam with a Gaussian intensity profile (Pin(r±) = 2Po/zrw 2 e x p ( - Z r Z / w ~ ) ) and the Diricfilet boundary conditions (m Is = 0), i.e. it is assumed that the Bloch vector components vanish at the walls of the cylindrical cell. The explicit difference scheme was applied to integrate Eq. (2) and the fast-Fourier-transform was used to solve the paraxial wave equation describing the light propagation in free space between the cell

Fig. 5. Steady-state characteristic for orientation versus external pump rate. The states between the outer dashed lines are unstable against spatial perturbations. The dashed lines separate domains of existence of different pattern: H'rr, negative hexagons; MSI, MS2, 'mixed states' (see text); R, rolls; U, spatial ultra-harmonics of primary patterns; H O, positive hexagons. Parameters as in Fig. 2. In addition F2 = 3.6 GHz, y = 6 Hz, D = 2 x 10-4 m2/s and /Ze = 1.728 x 10-29 cm are used.

and the mirror. The second code makes use of periodic boundary conditions and has the purpose of checking the predictions of the plane-wave analysis. The simulations were carried out on a Cartesian grid (256 x 256). The incident intensity distribution was perturbed by random noise of small amplitude in the calculation. 3.3.1. Gaussian beam simulations The first feature we examined in the simulations was the dependence of the whole-beam transmission coefficient on the laser power (Fig. 6). Just as in the experiment it has a large slope for small values of the laser power, passes a maximum and then decreases slowly (cf. Fig. 2). It can be concluded from the calculation that the peculiar negative slope of T for large values of the laser power is a consequence of the shape of the characteristic discussed in Section 3.2, i.e. it is caused by the magnetic field. It is also revealed that in the region of negative slope the medium is self-focusing, at least in the central part of the beam. This is of special importance, since the characteristic length is expected to be different for self-focusing or -defocusing media

W. Lange et aL/Physica D 96 (1996) 230-241

238

I-

0.8

Z 0 (/)

0.6

z

0.4 0.2 0

50

100

150

200

250

300

Pp./roW

Fig. 6. Whole-beam transmission found in the simulations for a Gaussian beam. The letters correspond to the patterns displayed in Fig. 3 (Parameters as in Fig. 5).

[9] and since we observe the formation of patterns in the region of negative slope. The first structures appearing on the profile of the transmitted beam (Figs. 3(00 and (~)) represent one (symmetry 02) or three (symmetry D3) dark filaments in the beam center. The further scenario of the pattern development depends on its beginning: if the onehole pattern ( Fig. 3(a)) emerges as the first, the next structure with increasing intensity is a hexagon with symmetry D6 shown in Fig. 3(8). If the three-hole pattern (Fig. 3(/~)) is formed at the beginning we observe formation of the six-hole structures presented in Fig. 3(X), which are followed by the 12-hole pattern (Fig. 3(e)). One can see that new patterns are obtained by the formation of additional constituents at the edge of the structure leaving the central part without changes. Obviously, the appearance of new holes is explained by the fact that with increasing intensity a larger area of the beam exceeds threshold. The intensity nonuniformity of the Gaussian beam profile, however, is the reason that the constituents in the beam center have a different internal structure in comparison with the ones near the edge. For higher intensities, beginning from that corresponding to Fig. 3(e), we find the disordered patterns presented in Fig. 3(q~) and (2/). For Fig. 3(~) and (y), the local hexagonal arrangement is destroyed and the structure of the individual spot is also complicated. In Fig. 3(y) one can see the tendency of the spots in the

center to be merged into a line. Similar behavior was found in the experiment (see Section 2.3). Just as in the experiment we calculate the Fourier transform, of course, and the results are also given in Fig. 3. From the Fourier transform we can again determine the characteristic lengths and the results are incorporated into Fig. 4. It is our feeling that the agreement between the observations and the results of the numerical simulation is remarkably good: not only the scenario seems to be the same, but also the transmission curve, the power range of the occurrence of the individual patterns and many experimental details are very similar. It would be desirable to compare the irregular patterns obtained at high powers in the beam between experiment and simulation, but we run into the problem that it can evidently not be expected to find exactly the observed patterns in the simulations, since not even the experimental patterns are reproducible in detail, of course. Here quantitative measures of characterization are needed. We used just the 'characteristic length', while quantities like the (spatial) autocorrelation function did not prove useful due to the small aspect ratio.

3.3.2. Plane-wave simulations While it is not possible in the experiment to increase the 'aspect ratio' drastically, we can easily do so in the simulation, and we can hope that this gives us some clues for the interpretation of the features found in the 'small aspect ratio' case. Following this strategy we switch immediately to the plane wave case, of course. When the intensity of the incident wave exceeds the value marked by the first dashed line in Fig. 5, the transmitted intensity subcritically takes the form of a honeycomb hexagon pattern with a minimum in the center of each hexagon as shown in Fig. 7(a). The appearance of such negative (or H-rr-) hexagons above threshold is determined by the sign of the quadratic nonlinearity of the system and is in accordance with the results of a weakly nonlinear analysis which follows the approach described in [10]. With increasing pump rate P0 the peaks in Fourier space achieve different height, i.e. one of the rolls forming the hexagons begins to dominate (not shown).

W. Lange et aL/Physica D 96 (1996) 230-241

a)

C)

b) i

i i i

Fig. 7. Calculated patterns in the plane-wavecase in regions (a) H~', (b) MSI and (c) R of Fig. 5 and their Fourier transforms. 0.6

E E

0.5

0.4

_

_,_ _,_

r,. 4)

0.3

i-regular

]

disorder

i

0.2 ~ 8

10

1'2

14

16

18

20

Po/'f* 10 .3

Fig. 8. Length scale derived from the wavelength of maximum Lyapunov exponent (broken line) and from the numerical simulation (dots) for the plane wave case. Abscissa is the normalized pump rate. The solid line is the boundary curve. The dashed line limits the region of stationary patterns (parameters as in Fig. 5). At further increasing power, defects are developing and the patterns become strongly disordered. An example is presented in Fig. 7(b). Yet inspection of the Fourier spectra reveals that there are two pronounced maxima in the spectra which indicate that one dominant roll pattern is still present. Further increasing of P0 results in the emergence of the roll-like pattern shown in Fig. 7(c). It should be noted that the rolls in our simulations are never stationary and perfectly parallel and the patterns always possess a residual disorder and a small scale structure that can be seen in Fig. 7(c). The intensities used in the calculation of Fig. 7(b) and (c) are exactly the

239

same as those in the beam center of Fig. 3(¢) and (y), respectively. The intervals of existence of different patterns on the axis of the stress parameter P0 are depicted in Fig. 5. The sequence (hexagons H~r ~ rolls R) found here reflects an universal scenario occuring in pattern forming systems. In our case the transition is mediated by a disordered state which does not seem to be stationary, i.e. we did not reach a stationary pattern in the calculations. Refraining from analysing the structure of particular defects we associate it with a mixed state formed by a superposition of rolls and hexagons. According to [36] mixed states are not stable in an ideal pattern. 'Mixed states' in a more general sense, however, have been found, e.g. in numerical [37] and experimental [38] studies of chemical reaction-diffusion systems. With further increasing pump rate P0, the roll interval is followed by another 'mixed state' region M S 2 (see Fig. 5). During the approach to the minimum of the steady-state characteristic (the interval U), harmonics begin to dominate in the spectra of spatial frequencies and determine the size of the patterns. The predictions of the linear stability analysis about the pattern size lose their validity in this regime completely, of course. The resulting pattern is a hexagonal lattice of bright spots. The hexagons might be called 'ultra-hexagons', since the length scale corresponds to the spatial harmonics. This pattern gives way to another hexagonal pattern of slightly different characteristic length, which belongs to a defocusing nonlinearity. (The region is labeled H O in Fig. 5.) Finally the system settles down in the homogeneous state (full saturation of the nonlinearity); the transition is indicated by an arrow in Fig. 5. Overall the lattice of holes at threshold (negative hexagons) is replaced by lattices of intensity maxima (positive hexagons) at the right end of the instability interval. The transition from negative to positive hexagons or vice versa has also been predicted in other systems [ 11,39,40]. In the plane wave case we determine the characteristic length just as before. The results depicted in Fig. 8 are similar to the ones obtained under the assumption of a Gaussian beam and the experiment, though

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they seem to lie somewhat higher systematically: This might indicate a tendency of the Gaussian beam to 'compress' the patterns. We have also incorporated the boundary curve and those values of the wavelength of perturbation which give the largest Lyapunov exponent in the stability analysis. It might be expected that these wavelengths define the characteristic length, but the results of the simulation are smaller by up to about 20%. This discrepancy which is worst near threshold may be surprising on first sight. It has to be kept in mind, however, that the patterns are always very far from the homogeneous state, since we observe nearly 100% modulation. Thus the basic assumption of the linear stability analysis is not valid, once the patterns have developed. This argument is supported by the temporal evolution of the patterns. The simulation reveals that in the beginning perturbations with the length scale given by the linear stability analysis grow, but in the process of growing and interacting the spatial frequencies shift to larger values, until the value observed in the developed pattern is reached. We conclude that the scale of the characteristic length may coarsely be determined by a combination of physical quantities like wavelength of the light and mirror distance, but the exact value is the result of the pattern forming process itself. When the plane wave is substituted by a Gaussian beam, finite size effects come into play. When we begin to increase the laser power from zero, then first in the center of the beam the intensity is reached which would yield hexagons in the plane wave case and this region expands with increasing power. The calculations reveal that hexagonal structures occur in the whole region of sufficient intensity; the characteristic length is only marginally changed with respect to the plane wave case. If the power is further increased, then the intensity in the central part would finally require rolls in the plane wave case. In the outer part of the beam, however, the intensity is not sufficient for roll formation and this seems to prevent the formation of clear rolls in the case of the Gaussian beam and in the experiment. In both cases, however, there is still an indication of a dominant system of rolls in the Fourier spectra.

4. Conclusions Formation of regular and irregular patterns can be observed in the very simple optical scheme discussed by d'Alessandro and Firth [9], if sodium vapor in a buffer gas atmosphere is used as the nonlinear medium. In a theoretical description of our experiment the Kerr medium used in the discussion of [9,10] has to be replaced by a more refined model which can be deduced microscopically. In the experiment the application of an oblique magnetic field proved to be crucial. It allows to change the properties of the system tremendously, and - by means of the microscopic model - it is possible to a large extent to tailor them corresponding to experimental requirements. The model is capable of describing the observed scenario of pattern formation quite well. In the present paper we discussed the results obtained for a single set of fixed parameters only, using just the laser power as a control parameter. It is the set documented best at present. In a more complete treatment the distance d certainly should be varied systematically. Moreover the transverse and the longitudinal component of the magnetic field are important parameters, since the shape of the characteristic (Fig. 5) can be varied in this way. The availability of computing time imposes some restrictions on exploring the full parameter space by simulations. In the experiment we cannot easily increase the power of the laser beam in order to reach a range with more dramatic effects. As an alternative we can reduce the longitudinal component of the magnetic field and move the minimum of the characteristic to smaller intensities in this way. In an experiment of this type we also increased the mirror distance to d ---- 175 mm in order to obtain clearer patterns. With this large value of d the number of constituents of the patterns is reduced even further (very small aspect ratio), but we observed ultrahexagons and bright spots indeed [41]. At first sight the latter ones have some similarity to the structures observed by Grynberg et al. in rubidium vapor [23]. In the present case, however, they can be interpreted to be remnants of the positive hexagons discussed in

W. Lange et al./Physica D 96 (1996) 230-241

Section 3.3.2. A g a i n the agreement with simulations a s s u m i n g a G a u s s i a n b e a m is very satisfactory. As a quantitative measure of c o m p a r i s o n we always used the characteristic length. In the material presented here the characteristic length is constant within the margins of error. It revealed, however, that the size of the patterns is not in agreement with the expectation based on the linear stability analysis. In other parameter ranges which have been studied less systematically up to now considerable changes in the characteristic length can be observed. These can be attributed to switching from the self-focusing to a self-defocusing part of the characteristic [41]. Also the replacement of the f u n d a m e n t a l hexagon pattern by ultra-hexagons has been found by m e a n s of the characteristic length [41]. Thus this quantity, though it is not very specific, can give some information on the system under regard, provided that a suitable model is at hand.

Acknowledgements Yu. A.L. was supported by the Deutscher A k a d e m i scher Austauschdienst. The help of A. Heuer and B. Berge in the measurements, in the evaluation and in the preparation of figures is gratefully acknowledged.

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