Spontaneous optical patterns in an atomic vapor: observation and simulation

ELSEVIER Physica D 96 (1996) 230-241

PHYmA

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 advan- tageous 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. 1 Permanent address: Institute of Physics, Academy of Sci-

ences, 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 quan-

tum 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 candi-

dates for studies of pattern formation at present. In most practical lasers the design imposes severe limi- tations 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) 230-241 231

(a) nonlinear medium

I

mirror

d I

(b) Na + N 2

laser F E O ~ ~ E L

beam ~"--. ,) / j~ *---Eb B , d

mirror

--I Near field

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

which corresponds to the situation of a large aspect

ratio in hydrodynamics.

Recently, however, spatial and spatio-temporal ef-

fects in much simpler optical systems have been dis-

cussed. 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. Ide-

ally 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 spatio- temporal behavior is expected. Also the consequences of replacing the plane wave by a Gaussian beam have been studied [ 12].

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 [14-

18] 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 LCLV-

experiment 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 coun-

terpropagating beams without external feedback. Spa- tial instabilities and hexagonal structures [24,25] have

been observed in this type of experiment.

Recently our group has reported an experiment in-

volving 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 micro-

scopic 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 consid-

erations. 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

232 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 re-

sult from the intensity dependent change of the pop-

ulation 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 varia-

tion as well as a spatial phase modulation. Small vari-

ations of the frequency of the light field in the vicinity

of the atomic resonance transitions allow to find a bal-

ance between large values of Idn /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 Zee-

man 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 con-

verted into a slow diffusion process by adding a buffer

gas. Conventionally a rare gas is used for this pur-

pose, 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 tremen-

dously. (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 broaden-

ing and has the consequence to increase the power re-

quirements for the laser beam. This is a tribute to be

payed for the sake of deducing a simple and yet rea-

sonably 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 ori-

entation 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 orienta-

tion 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 di-

luted gas and cannot change too much in a high inten-

sity 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

W. Lange et aL/Physica D 96 (1996) 230-241 233

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 broaden-

ing 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 photo-

diode D. The distance between the center of the cell

and the mirror is d = 75 mm in the experiments dis-

cussed 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 cru-

cial 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 experi-

ment. In this region of a negative slope of transmis-

E--

1,0

0,8 Z © r.13 0,6

0,4 Z <

0,2

ac(

e i

J

h j

0 0 1 ! ~ J i 0 0 ' ~ - - - ' 0 50 100 150 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 struc-

tures, 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 me-

dia in which the patterns become time-dependent just

after the first or second polygon-like pattern beyond threshold. However, a similar sequence with increas-

ing beam power from a single peak to a triangle and

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

a) b) c) d) e) f) g) h) i) j)

a) z) 8) e) ¢)

Il l / /I l l i l l l l 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 co-

existing in the same power range. Figs. 3(h)-(j) are just

frozen images of patterns which are in permanent mo-

tion. Unfortunately we cannot resolve the full tempo-

ral 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 signifi-

cantly 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

0,36

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

i I I I

235

0,34.

E E '~" 0,32' _¢ t'O ¢J (R

0") r- 0,30,

0,28

t I 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 - DV 2 + P ) m - ~.z P + m × I2. (2)

The components u, v, w of m represent the x, y, z-

components of the expectation value of the magnetic

moment in a volume element. ~, is the collision in-

duced 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 de-

scribe 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 de-

scribes 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 Lar- mor frequencies belonging to the x- and z-component

of the magnetic field, S2x and $2 z, 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 ex-

tra 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 compo-

nent 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 compen-

sation 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 nonlin-

ear optical processes is discussed in more detail in

236

[35]; its role in the present experiment will become

further clarified in Section 3.2.

The pump rate P is proportional to the local in-

tensity 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

Nl#el 2 A + i X -- 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 approx-

imations. We do not only neglect diffraction effects

within the sample, but we also neglect any longitudinal variation of the intensity. This means that we ne-

glect any standing wave effects and replace I Ef + 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 sta- tistical considerations (see [32,33]).

As a consequence of all these approximations we cannot expect quantitative agreement with the exper-

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

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

Ps ($2z - APs) 2 + (Y + Ps) 2 w~ - - - (5)

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

with Ps = PO(1 + Rle(-iklzunO-ws)/2)12) and the suc-

cessive substitution of Ws into the expressions for the

other Bloch vector components

(t'-2z _ A P s ) ~ x W s Us = (6)

(~z - APs)2 ÷ (2" ÷ Ps) 2'

(1 + Ps)S2xWs Vs = (7)

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

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 domi- nates and displays a 'resonance' behavior at Y2 z = 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 I2 z - PA = 0 just defines the intensity needed for compensating the Zeeman splitting intro-

duced 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 charac-

teristic 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

w. Lange et aL/Physica D 96 (1996) 230-241 237

the medium is defocusing in the corresponding in-

tensity 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. APt (us - Avs), I APs-- ~: :Felt" (Vs + A u s ) S + SZ, - I=0,

I

o -S2x :Fetr - (1 - I i

(8)

where gefr = Y + Ps + D k 2 and

3 ---- - R P o Re Xlinkl le I-ik/xli°(1 u,,))12

¢ ¢. , .

o

¢ - .

0.8

0.6

0.4

0.2 H/t MS1R MS2 U H0

0 10 2o 3o 4o

Po/y * 10 .3

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 do- mains 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. Pa- rameters 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.

(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 increas-

ing, self-defocusing, part of the characteristic in ac-

cordance 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 in-

put 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 cylindri- cal 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

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 sim-

ulations 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 coef-

ficient on the laser power (Fig. 6). Just as in the exper-

iment 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

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

I - 0.8

Z 0 0.6 (/)

0.4 z

0.2

0 50 100 150 200 250

Pp./roW

300

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 pat-

tern development depends on its beginning: if the one-

hole 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 pat-

tern (Fig. 3(e)). One can see that new patterns are ob-

tained 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 ob-

servations 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 ex-

periment 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 func-

tion 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 fol-

lows 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).

a) b)

i i

W. Lange et aL/Physica D 96 (1996) 230-241 239

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

C) reflects an universal scenario occuring in pattern forming systems. In our case the transition is mediated by

i 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. Ac-

cording to [36] mixed states are not stable in an ideal

pattern. 'Mixed states' in a more general sense, how-

ever, have been found, e.g. in numerical [37] and ex-

perimental [38] studies of chemical reaction-diffusion

systems. With further increasing pump rate P0, the roll inter-

val is followed by another 'mixed state' region MS2

_ _,_ _,_ (see Fig. 5). During the approach to the minimum of the steady-state characteristic (the interval U), har-

monics begin to dominate in the spectra of spatial fre-

quencies and determine the size of the patterns. The

predictions of the linear stability analysis about the

18 20 pattern size lose their validity in this regime completely, of course. The resulting pattern is a hexag-

onal lattice of bright spots. The hexagons might be called 'ultra-hexagons', since the length scale corre-

sponds to the spatial harmonics. This pattern gives way to another hexagonal pattern of slightly differ-

ent characteristic length, which belongs to a defocusing nonlinearity. (The region is labeled HO in Fig.

5.) Finally the system settles down in the homoge-

neous 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 assump-

tion of a Gaussian beam and the experiment, though

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

0.6

E E

r , .

4 )

0.5

0.4

0.3 i - -

regular ] disorder i

0.2 ~ 8 10 1'2 14 16

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 ex- ample 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 domi-

nant 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

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

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 dis-

crepancy which is worst near threshold may be surpris-

ing on first sight. It has to be kept in mind, however,

that the patterns are always very far from the homo-

geneous 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 be-

gin 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 cal-

culations 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 exper-

iment 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 exper-

iment the application of an oblique magnetic field proved to be crucial. It allows to change the prop-

erties 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 system-

atically. Moreover the transverse and the longitudinal component of the magnetic field are important param-

eters, 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 alterna-

tive 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 ultra- hexagons 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) 230-241 241

Section 3.3.2. Again the agreement with s imulat ions

assuming a Gauss ian beam is very satisfactory.

As a quanti tat ive measure of compar ison 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 l inear stability analysis. In other parame-

ter ranges which have been studied less systematical ly

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 fundamenta l hexagon pattern by ul tra-hexagons

has been found by means of the characteristic length

[41]. Thus this quantity, though it is not very specific,

can give some informat ion on the system under regard,

provided that a suitable model is at hand.

Acknowledgements

Yu. A.L. was supported by the Deutscher Akademi-

scher Austauschdienst . The help of A. Heuer and B.

Berge in the measurements , in the evaluat ion and in

the preparation of figures is gratefully acknowledged.

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Spontaneous optical patterns in an atomic vapor: observation and simulation