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Pattern dynamics and competition in a
photorefractive feedback system
Cornelia Denz, Michael Schwab, Markus Sedlatschek, and Theo Tschudi
Institut fu
¨
r Angewandte Physik, Technische Universita
¨
t Darmstadt, Hochschulstrasse 6, D-64289 Darmstadt,
Germany
Tokuyuki Honda
National Research Laboratory of Metrology, 1-1-4 Umezono, Tsukuba, Ibaraki 305, Japan
Received August 29, 1997; revised manuscript received January 16, 1998
We investigate the temporal dynamics of transverse optical patterns spontaneously formed in a photorefrac-
tive single-feedback system with a virtual feedback mirror. The linear stability analysis for the system is
reviewed and extended to the region of larger propagation lengths. The stationary patterns obtained experi-
mentally are classified as a function of feedback reflectivity and feedback mirror position. Inserting masks
into the feedback path permits pattern selection and control by Fourier filtering. When an asymmetry that is
due to noncollinear pump beams is introduced, the otherwise stationary hexagons show several complex but
periodic rotationlike motions. Furthermore, the competition of hexagonal and square patterns can be ob-
served by the appropriate choice of feedback mirror position and coupling strength. The origin of this behav-
ior is discussed. The temporal evolution of the patterns is illustrated by a method based on unfolding the
angular distribution of the spots in the far field. © 1998 Optical Society of America [S0740-3224(98)01407-6]
OCIS codes: 190.5330, 190.4420, 190.3100.
1. INTRODUCTION
Spontaneous pattern formation in nonlinear optics has re-
cently attracted considerable interest. A generic mecha-
nism that leads to the formation of spatial patterns is the
transverse modulational instability of counterpropagat-
ing pump waves in a third-order nonlinear medium with
respect to the emission of spatial sideband beams with a
certain transverse wave vector. Transverse spatial
structures, most of them hexagonal, have been observed
in a variety of nonlinear media as such as atomic
vapors,
1,2
liquid crystals,
3,4
a liquid-crystal light valve,
5
photorefractive crystals,
6
and organic films.
7
When the
sidebands are emitted at an angle of 60° relative to each
other in the transverse plane, the distance between two
sidebands has the same magnitude as the distance be-
tween a sideband and the pump beam. Thus these side-
bands are resonantly excited, giving rise to an arrange-
ment of the sidebands in a hexagonal spot array.
Vorontsov and Firth
8
explained the hexagon formation as
a nonlinear interaction and mutual support of three ‘‘su-
peractive roll solutions’’ with wave vectors k
i
fulfilling
the resonance condition k
1
1 k
2
1 k
3
5 0.
In this paper our focus of interest is on the dynamic
properties of the transverse patterns that are spontane-
ously formed in a photorefractive single-feedback system
with a virtual feedback mirror. In addition to the con-
ventional hexagonal patterns, square and squeezed hex-
agonal patterns were experimentally observed in this
system,
9
which was to our knowledge the first experimen-
tal observation of such patterns spontaneously formed in
an optical feedback system. Despite the variety of sta-
tionary patterns; in this paper we focus on the dynamic
behavior of the patterns. It was shown by Honda
10
that
a rotation of the far-field and a flow of the corresponding
near-field hexagonal patterns in a photorefractive single-
feedback system similar to the system under consider-
ation here can be induced by transverse intensity gradi-
ents of the pump beams. Mamaev and Saffman
11
observed a more complex rocking motion of the pattern in
the case of noncollinear pump beams. In this paper we
present investigations of pattern dynamics that are due
to two different mechanisms. An induced angular mis-
alignment of the counterpropagating beams leads to peri-
odic rotationlike motions, whereas the competition, i.e.,
the temporal alternation, of different patterns is obtained
without an angular misalignment for an appropriate
choice of feedback mirror position and coupling strength.
This paper is structured as follows. In Section 2 we re-
view the linear stability analysis for the system and dis-
cuss its consequences for our particular system. The
theory is shown to agree well with experimental results
for pattern size and also explains the lack of pattern for-
mation for certain positions of the virtual mirror. We
performed an extension of the analysis to large propaga-
tion lengths, showing leaps of the sideband angle in that
region. In Section 3 the occurrence of different types of
pattern, depending on the system parameters, is experi-
mentally investigated. The suitability of Fourier filter-
ing in the feedback path for pattern selection and control
is shown in Section 4. Because the main concern of this
paper is the temporal dynamics of the patterns, we intro-
duce an appropriate visualization method in Section 5.
Pattern dynamics that are due to an induced noncol-
linearity of the pump beams is investigated in Section 6.
Denz et al. Vol. 15, No. 7/July 1998/J. Opt. Soc. Am. B 2057
0740-3224/98/072057-08$15.00 © 1998 Optical Society of America
The spontaneous competition of different patterns is in-
vestigated, and its possible origin is discussed in Section
7.
2. THEORETICAL ANALYSIS
The interaction geometry of our system is depicted sche-
matically in Fig. 1. A plane wave of complex amplitude
F is incident upon a thick photorefractive medium of
length l, where it interacts with a counterpropagating
feedback beam of complex amplitude B, which is provided
by reflection of a forward wave F from a feedback mirror.
These two beams are coupled by means of a reflection
grating of wave vector 2k
0
n
0
inside the photorefractive
medium, where k
0
is the vacuum wave number of beams
F and B and n
0
is the linear refractive index of the pho-
torefractive medium. In the case under consideration, a
thin biconvex lens is introduced into the feedback path at
distance 2 f from the medium’s back surface and a dis-
tance 2 f 1 L from the feedback mirror. The lens images
the mirror at a distance L from the medium’s back sur-
face and thus creates a virtual mirror near the medium,
which we can move by shifting the feedback mirror. The
advantage of this system compared with other feedback
mirror configurations is that the virtual mirror can be
placed inside the medium, allowing for negative propaga-
tion lengths. During the round trip in the feedback path
the sideband beams experience a phase lag of k
0
u
2
L rela-
tive to the carrier beam, where
u
is the angle between the
carrier and the sidebands. A negative propagation
length thus corresponds to a negative phase lag.
Our linear stability analysis for obtaining the threshold
condition of pattern formation in this system is based on
the transverse modulational instability of two energy-
exchanging counterpropagating beams, which couple
through a single reflection grating (see, e.g., Refs. 12 and
13). A different approach by Kukhtarev et al.
14
based on
pairwise two-wave mixing of the incident beam and the
sidebands that are generated by scattering through trans-
mission gratings is not considered here. In contrast to
ours, that approach would require taking into account the
angular dependence of the coupling coefficient
g
to calcu-
late interactions by means of several transmission grat-
ings in different orientations. Note that, in general, the
result of a linear stability analysis for a nonlinear system
is limited to the calculation of the magnitude of the un-
stable transverse wave vector and thus the emission
angle of the generated sideband beams. The formation of
different pattern types in the system cannot be deduced
from a linear stability analysis but requires a nonlinear
analysis, which has not yet been developed for photore-
fractive systems.
Following the detailed treatment by Honda and
Banerjee
12
for the case of a simple feedback mirror, we ex-
tended this analysis for the case of negative propagation
lengths.
9
We now review this analysis, extend it to
larger propagation lengths, discuss the consequences of a
lower coupling strength on the occurrence of sidebands,
and compare the results with experimental observations.
For details and assumptions see Refs. 9 and 12.
The system under consideration is governed by the
usual equations for contradirectional two-beam coupling
9
:
]
F
]
z
2
i
2k
0
n
0
¹
'
2
F 5 i
g
u
B
u
2
u
F
u
2
1
u
B
u
2
F, (1)
]
B
]
z
1
i
2k
0
n
0
¹
'
2
B 52i
g
*
u
F
u
2
u
F
u
2
1
u
B
u
2
B,
(2)
where z is the propagation direction, ' denotes the trans-
verse direction, and
g
is the complex photorefractive cou-
pling coefficient. The term i/(2k
0
n
0
)¹
'
2
accounts for
variations in the transverse beam profile. To perform a
linear stability analysis we assume that the amplitudes of
the forward and the backward waves are modulated by
weak sidebands according to
F
~
r
!
5 F
0
~
z
!
@
1 1 F
11
~
z
!
exp
~
ik
'
• r
'
!
1 F
21
~
z
!
exp
~
2ik
'
• r
'
!
#
, (3)
B
~
r
!
5 B
0
~
z
!
@
1 1 B
11
~
z
!
exp
~
ik
'
• r
'
!
1 B
21
~
z
!
exp
~
2ik
'
• r
'
!
#
, (4)
where k
'
is the transverse wave vector of the sidebands,
r
'
denotes position in the transverse plane, and F
61
and
B
61
are the relative amplitudes of the sidebands. The
boundary conditions for sideband generation with a feed-
back mirror are
F
61
~
0
!
5 0, (5)
B
61
~
l
!
5 exp
~
2ik
d
n
0
L
!
F
61
~
l
!
, (6)
where k
d
[ k
0
u
2
/2n
0
is the wave number corresponding
to the phase lag between the carrier and the sideband per
unit length in the photorefractive medium. By inserting
Eqs. (3) and (4) into two-beam coupling equations (1) and
(2), linearizing with respect to the amplitudes of the weak
Fig. 1. Principle of the interaction geometry: SB’s, spatial sidebands; M, mirror; v.M., virtual mirror; L, propagation length; l, crystal
length;
u
’s, sideband angles; p.r., photorefractive.
2058 J. Opt. Soc. Am. B/Vol. 15, No. 7/July 1998 Denz et al.
sideband beams and, using boundary conditions (5) and
(6), we obtain the threshold condition for sideband gen-
eration as
12
cos wl cos k
d
l 1
g
I
2w
sin wl cos k
d
~
l 1 2n
0
L
!
1
g
R
1 2k
d
2w
sin wl sin k
d
l
2
g
R
2w
sin wl sin k
d
~
l 1 2n
0
L
!
5 0,
(7)
where
g
R
and
g
I
are the real and the imaginary parts, re-
spectively, of photorefractive coupling coefficient
g
and
w 5 (k
d
2
1
g
R
k
d
2
g
I
2
/4)
1/2
. In our experiments we
used KNbO
3
as a photorefractive medium, which shows a
nonlocal photorefractive response and performs pure en-
ergy coupling between the beams. Thus we assume that
g
R
5 0 in what follows.
13
For photorefractive media
with local response, a detailed linear stability analysis
was published recently.
15
Threshold function (7) can be plotted to express
g
l as a
function of k
d
l/
p
for various normalized propagation
lengths n
0
L/l (see Fig. 2 of Ref. 9 for typical examples).
The angle
u
of sideband emission corresponds to the value
of k
d
l/
p
that gives a minimum threshold of
g
l. Then
u
can be calculated from
u
5 (2k
d
n
0
/k
0
)
1/2
for each value of
n
0
L/l. In Fig. 2, the angle of sideband generation is plot-
ted as a function of the normalized virtual mirror position
n
0
L/l for a wide range of positions. Note that the side-
band angle is symmetrical about the crystal center at
n
0
L/l 520.5 and that the plot is limited to the param-
eter region shown for visibility only. The sideband angle
decreases for positive virtual mirror positions, corre-
sponding to the simple mirror configuration, which has al-
ready been confirmed experimentally by Honda and
Banerjee.
12
However, for larger positive values of n
0
L/l
the behavior changes significantly, and the sideband
angle shows a sharp increase for certain large positive
values of n
0
L/l, which has not yet been confirmed experi-
mentally. This behavior can be explained by the particu-
lar shape of the threshold curve, which is shown in Fig. 3
for n
0
L/l 5 1.4. For mirror positions near or in the crys-
tal, the global minimum of the threshold curve is the first,
i.e., the left, local minimum of the curve. Figure 3 repre-
sents the situation when the global minimum leaps from
the first to the second local minimum, causing a leap of D
u
in the sideband angle. This situation corresponds to the
sharp increase of
u
in Fig. 2 at n
0
L/l 5 1.4. For larger
values of n
0
L/l this condition is repeated, with other local
minima providing the absolute minimum.
In Fig. 4 the angle
u
of sideband generation is plotted
for 21 < n
0
L/l < 0 (virtual mirror inside the crystal).
The filled circles indicate the sideband angles of the ex-
perimentally observed hexagonal far-field patterns. An
example of such a pattern is shown in the inset of Fig. 4.
To permit a theoretical investigation appropriate to ex-
perimental conditions, we shall present these experimen-
tal results in this section, whereas the experimental setup
is explained in Section 3. For 20.75 < n
0
L/l < 20.3, no
patterns were observed in the experiment, but good
agreement with the theoretical pattern size is apparent
for the left-hand and right-hand regions of the crystal.
This lack of pattern formation in the central region can be
explained if we assume that the photorefractive coupling
Fig. 2. Sideband angle
u
as a function of normalized virtual mir-
ror position n
0
L/l according to the linear stability analysis.
Fig. 3. Threshold curve for n
0
L/l 5 1.4, indicating the leap in
the sideband angle. The minimum of the threshold curve is
passed over from the first to the second branch, resulting in a
leap of D
u
of the sideband angle.
Fig. 4. Sideband angle
u
as a function of virtual mirror position
for the case of a virtual mirror position inside the crystal. Filled
circles, experimental points; solid curve, theoretical curve. In-
set, experimentally obtained hexagonal pattern in the far field,
including second- and third-order spots.
Denz et al. Vol. 15, No. 7/July 1998/J. Opt. Soc. Am. B 2059
constant
g
was too small for the generation of patterns in
that region. Figure 5 shows the minima of the threshold
curves as a function of the virtual mirror position. This
parameter curve starts for n
0
L/l 521 (point A) and in-
creases via point B (n
0
L/l 520.7). The turning point is
point C, with a parameter value of n
0
L/l 520.5. Then
the direction is reversed, passing point B at n
0
L/l
520.3 and point A at n
0
L/l 5 0 and ending at point D
at n
0
L/l 5 1. If we set
g
l 5 6, for example, a parameter
region 20.7 < n
0
L/l < 20.3 exists, for which the thresh-
old for sideband generation is above this value of
g
l.
Thus pattern formation is not possible for this parameter
region with the virtual mirror near the central region of
the crystal. With
u
max
5 0.9° (see Fig. 4) the estimated
photorefractive coupling strength is
g
l 5 5.5.
3. STATIONARY PATTERNS
The experimental setup is depicted in Fig. 6. A
Fe:KNbO
3
crystal doped with 2400 parts in 10
6
of Fe,
measuring l 5 5.6 mm along the c axis, was illuminated
by a frequency-doubled Nd:YAG laser (l 5 532 nm, P
5 100 mW), which was isolated from retroreflections by
an optical diode. The crystal’s c axis was chosen to be in
the direction of the pump beam, leading to depletion of
the input and amplification of the feedback beam. To re-
duce the influence of reflections from its surfaces we tilted
the crystal slightly by '6°. The beam diameter inside
the crystal was approximately 350
m
m, and the power in-
cident upon the crystal was 22 mW. We chose the polar-
ization to be in the direction of the crystal’s a axis to take
advantage of the large r
13
component of the electro-optic
tensor. Lens 1, with a focal length of f 5 100 mm, was
used to focus the beam into the crystal with the beam
waist at its back surface. A double-passed 2 f –2 f imag-
ing system (f 5 100 mm) provided the feedback beam.
Beam splitter 1 permitted observation of the feedback
beam, which carried the transverse structures, and beam
splitter 2 separated this beam for observation of the far-
field and, by means of a Fourier-transform lens, the near-
field patterns via cameras CCD 1 and CCD 2, respec-
tively. Furthermore, beam splitter 1 permitted
measurement of the input beam diameter without dis-
turbing the feedback system.
In the inset of Fig. 4 a typical far-field hexagonal pat-
tern is depicted that incorporates second- and third-order
sidebands, whereas the corresponding near field is a hon-
eycomb structure. The reflectivity of the optical compo-
nents in the feedback path was r 5 0.95. In this configu-
ration with a coupling strength of
g
l ' 5.5, only patterns
with hexagonal symmetry were observed. The relative
intensity of the entire first-order hexagon with respect to
the central spot was '12% but declined for larger trans-
verse scales to '5%. Note that the square and squeezed
Fig. 5. Minima of threshold curves as a function of virtual mir-
ror position for the parameter region 21 < n
0
L/l < 1. Selected
parameter values are n
0
L/l 521, 0 (point A), n
0
L/l
520.7, 20.3 (point B), n
0
L/l 520.5 (point C), and n
0
L/l 5 1
(point D). The largest observable sideband angle
u
max
is indi-
cated, corresponding to a coupling strength of
g
l ' 5.5.
Fig. 6. Experimental setup: OD, optical diode; L’s, lenses; M’s,
mirrors; BS’s, beam splitters; MLS, microscope lens system.
Fig. 7. Dependence of pattern type on feedback reflectivity and mirror position. I, no observable pattern; II, weak hexagonal pattern;
III, pattern without geometric symmetry; IV, washed-out hexagonal pattern with emphasis on two spots opposite each other; V, static
hexagonal pattern.
2060 J. Opt. Soc. Am. B/Vol. 15, No. 7/July 1998 Denz et al.
hexagonal patterns that we reported previously
9
were ob-
tained for larger coupling of
g
l ' 11.5.
Reducing the feedback reflectivity by reducing the feed-
back mirror’s reflectivity induced remarkable pattern
transitions. The dependence of the pattern type on the
feedback reflectivity r is shown in Fig. 7; the forbidden
area, without pattern formation around the central region
of the crystal, is indicated. Beginning with the unaltered
high reflectivity, for feedback reflectivities r
.
0.35 (re-
gion V), only hexagonal patterns were observed. For
0.22
,
r
,
0.35 (region IV), the hexagonal pattern
showed an emphasis on two spots opposite each other, al-
though no misalignment of the pump beams was present.
Introducing a small misalignment led to a competition be-
tween roll and hexagonal patterns in this region. In re-
gion III no clear pattern was observable; it is obviously a
transition region between regions IV and II. In region II
a hexagonal pattern was observed again, and for r
, 0.13 (region I) no spatial sideband beams could be ob-
served.
4. PATTERN CONTROL BY FOURIER
FILTERING
Fourier filtering techniques for the manipulation of two-
dimensional image information in linear optical systems
are well known. Degtiarev and Vorontsov used Fourier
filtering in the feedback path of a nonlinear optical liquid-
crystal light-valve feedback system to suppress phase
distortions.
16
Other theoretical studies have recently
considered Fourier filtering techniques for the stabiliza-
tion and selection of optical patterns.
17,18
A Fourier fil-
tering method for the selection of optical patterns with
the filter directly inserted into the feedback path was ex-
perimentally realized by Mamaev and Saffman.
19
Their
feedback path consisted of a double-passed f –f system in
which a Fourier filter plane was created in the common
focal plane of the two lenses.
Without having to alter our 2 f –2 f feedback system we
were able to realize a similar control scheme. The inser-
tion of different spatial filters near the lens into the feed-
back path (where the far-field pattern was observable) al-
tered the geometrical structure of the system in such a
way that the stable hexagonal solution was suppressed
and formerly underlying unstable nonhexogonal solutions
of the system were excited. For a slit-shaped spatial fil-
ter, roll patterns were obtained, whereas for a cross-
shaped filter square patterns were the corresponding so-
lutions. Finally, for a large-aperture slit, squeezed
hexagons were observed. It should be kept in mind that,
without a spatial filter, hexagonal patterns are the only
stable and thus observable solutions in this parameter re-
gion. In the case of roll and square patterns we were
able to rotate the patterns continually around the central
spot by slowly rotating the Fourier filter. We observed
that no orientation of these patterns was favored, and
thus arbitrary orientations of the patterns can be ad-
justed by a corresponding orientation of the Fourier filter.
Roll patterns have the interesting property that the sys-
tem control by the filter (i.e., the power absorbed) declines
to zero in the equilibrium state. In contrast, in the case
of hexagonal or square patterns higher-order spots are
emitted during readout of the grating in the photorefrac-
tive medium by the forward beam and are observable at
the filter. These higher-order spots are blocked by the
Fourier filter, leading to small but nonzero power losses
in the system for the latter case, even in the equilibrium
state.
5. ILLUSTRATION OF PATTERN
DYNAMICS
Our particular interest in the dynamics of the patterns
that arise in this system requires an appropriate method
for the illustration of such pattern dynamics. The most
appropriate method for our purposes was proposed by
Thu
¨
ring et al.
20
Figure 8 gives a survey of this simple
but effective illustration method. The distribution of far-
field spots upon a circle with radius k
1
is projected upon a
linear axis with the polar coordinate
f
as the horizontal
axis and time as the vertical axis. As a result, every
movement of the spots on the circle leads to an appropri-
ate pattern change in Cartesian coordinates. For ex-
ample, a static hexagon causes six stripes traveling par-
allel in the positive y direction (time axis); a hexagon
rotating with constant angular velocity
v
leads to six
tilted parallel stripes as shown in Fig. 8, with the slope
indicating the angular velocity. The major advantage of
this method is that a single image provides us with all the
Fig. 8. Method for illustrating pattern dynamics based on the
unfolding of the angular spectrum of the far-field pattern.
Denz et al. Vol. 15, No. 7/July 1998/J. Opt. Soc. Am. B 2061
information on the dynamics instead of a sequence of
many single-pattern images.
6. DYNAMICS OF HEXAGONS OWING TO
NONCOLLINEAR PUMP BEAMS
Shifting the lens used to focus the beam into the crystal in
the transverse plane (lens L1 in Fig. 6), will induce a
slight asymmetry. The counterpropagating forward and
backward beams are no longer collinear and intersect at a
small tilt angle. As a result, the near-field pattern starts
to flow similarly to the behavior reported by Honda
10
; i.e.,
the honeycomb structure moves within the Gaussian
beam envelope. The corresponding far-field hexagonal
pattern performs a variety of different complex motions,
which are the subject of investigation in this section.
The situation favored for a maximum feedback reflectivity
of r 5 0.95 is depicted in Fig. 9. Two stable spots and a
rotation with a fast leap back of the other four spots of the
hexagon (called a rocking motion
11
) can be observed, caus-
ing us to call it rock ’n’ roll motion. The axis defined by
the two stable spots is always perpendicular to the lens
shift and, as a consequence, to the periodic flow in the
near field. As Fig. 10 indicates, the frequency of the mo-
tion depends linearly on the tilt angle introduced. This
result is in agreement with previously reported experi-
ments that used a system consisting of rubidium vapor
and a feedback mirror.
21
In the special case of larger tilt
angles (.0.225°) the rocking motion of the other four
spots vanishes and a roll pattern is left in the far field.
Another type of motion that can be observed in this sys-
tem is a rotation of the entire hexagon, followed by a fast
leap back to the initial position. This motion was ob-
served previously in a different system and was named
rocking motion.
11
All the types of motion that we have described in this
section are completely periodic in time. In the case of the
rock ’n’ roll motion, changes in the tilt angle result only in
a change of the motion frequency, whereas for the rocking
motion a slight change in the tilt angle gives rise to more-
complex motions such as the one depicted in Fig. 11. The
situation shown here consists of a rocking motion with
two different time scales. The regular rocking motion is
evident, but all spots also oscillate periodically on a
shorter time scale. Numerous other complex motions
situations were also found, but they are beyond the scope
of this paper. The origins of these different types of mo-
tion are currently under investigation.
When the feedback reflectivity is significantly reduced
in this region of noncollinear pump beams, the temporally
periodic dynamics of one pattern vanish. For r ' 1/3, al-
ready a small tilt angle ('0.1°) between the pump and
the feedback beam leads to completely new behavior. In
contrast to the dynamics of a single pattern, a competition
between two different patterns, i.e., a hexagonal and a
roll pattern, was observed, with the roll solution being
preferred for larger tilt angles. The competition of differ-
ent patterns without asymmetry is the subject of Section
7.
7. PATTERN COMPETITION
Besides the pattern dynamics discussed in Section 6,
which are induced by noncollinear pump beams, a spon-
Fig. 9. Rock ’n’ roll motion. Two spots, opposite each other, are
nearly stable; the other four spots exhibit a rotation and a fast
leap back to the initial position.
Fig. 10. Frequency of the rock ’n’ roll motion as a function of the
tilt angle (full angle between the incoming and the feedback
beam outside the crystal).
Fig. 11. Rocking motion with two time scales. All six spots
show a rotation and a fast leap back on a long time scale,
whereas on a shorter time scale the spots oscillate periodically.
2062 J. Opt. Soc. Am. B/Vol. 15, No. 7/July 1998 Denz et al.
taneous temporal alternation of patterns of different ge-
ometries without an induced noncollinearity, a Fourier
filter, or any other external inference can be observed in
this system, which is due to pattern competition.
22
In
the experiments reported in this section the alignment of
the pump beams was controlled, and thus noncollinearity
can be excluded as a possible origin of the dynamics.
We previously reported the observation of square pat-
terns in this system for large values of the coupling
strength
g
l ' 11.5 and a virtual mirror position inside
the crystal.
9
Note that the pattern collapses in this re-
gion for smaller values of
g
l, as was discussed in the pre-
vious sections. When the virtual mirror was placed at a
position between regions where stable hexagonal or
square patterns were observed (n
0
L/l ' 20.25; see Fig. 6
of Ref. 9), hexagonal and square patterns (shown in Fig.
12) appeared alternately in time in a nonperiodic manner.
In Fig. 13 a time sequence of such a competition is
shown, illustrated by the method explained above. Be-
cause the six spots of a hexagonal pattern all lie upon a
circle of a specific radius k
1
around the central spot but
only four spots of the square pattern lie upon this radius
(cf. Fig. 12), the competition is also shown for the four
spots in the corners of the square at a distance
A
2k
1
from
the central spot. Nevertheless, the hexagon and the
square have the same transverse scale k
1
. The top part
of Fig. 13 shows the temporal evolution of the spots of dis-
tance k
1
, and the bottom part shows the same for the
spots of distance
A
2k
1
, i.e., the corners of the square.
Note that both parts of Fig. 13 refer to same time se-
quence of the same competition scene. Of course, only a
limited excerpt of the competition can be shown in Fig.
13. It should be mentioned that different excerpts of the
same competition sequence do not differ significantly.
When longer time sequences of a competition sequence
are looked at, it becomes more obvious that the pattern
alternation during the competition is irregular in time
and is thus nonperiodic.
A number of significant conclusions can be drawn from
the two parts of Fig. 13. It is clearly apparent from the
top part of the figure that two spots remain stable during
the competition; they are in common for the hexagonal
and the square patterns and are thus emphasized by the
visualization method. This behavior may be due to a roll
pattern coexisting with both of the temporally alternating
patterns. The figure also shows that a coexistence of a
hexagonal and a square pattern does not occur for a sig-
nificant time interval. It is also apparent from the bot-
tom part of Fig. 13 that the occurrence of spots with dis-
tance
A
2k
1
almost completely coincides with the
appearance of the other four spots of the square pattern,
as one can see by comparing the two parts of the figure.
Furthermore, the fact that the vertical lines in Fig. 13
formed by the temporal evolution of the spots are not
tilted proves that the hexagonal and the square patterns
do not rotate during their existence. This result in turn
confirms that an angular misalignment of the pump
beams was not present and that the beams were in fact
collinear.
The nonperiodic spontaneous alternation of hexagons
and squares can obviously be explained in the following
way. On the border between the regions for the existence
of stable hexagonal and stable square solutions, both so-
lutions are possible, and the system itself spontaneously
oscillates between them.
Fig. 12. Experimentally observed far-field pattern of (left) hex-
agonal and (right) square symmetry. For strong coupling of
g
l
' 11.5 and an appropriate choice of the virtual mirror position
(e.g., n
0
L/l ' 20.25), the patterns alternate in time.
Fig. 13. Angular distribution of spots during competition of
hexagons and squares. Top, radius k 5 k
1
; bottom, radius k
5
A
2k
1
.
Denz et al. Vol. 15, No. 7/July 1998/J. Opt. Soc. Am. B 2063
A theoretical description of this behavior should in-
clude both transverse dimensions and time. Although
the treatment of this problem is extremely complicated
and has not yet been fully developed, first approaches to
contradirectional two-beam coupling without feedback in
one transverse dimension and time have been
developed.
23,24
8. CONCLUSION
We have investigated the dynamics of patterns formed in
a photorefractive single-feedback system with a virtual
feedback mirror. The linear stability analysis for the
system was reviewed and extended to the region of larger
propagation lengths. The pattern size predicted by the
theory was shown to be in good agreement with the ex-
perimental results. In addition, within the framework of
linear stability analysis, we were able to explain the lack
of pattern formation for certain feedback mirror positions
at lower coupling strengths. The patterns that can be ob-
served for reduced feedback reflectivities were discussed.
Moreover, the possibility of controlling the type and ori-
entation by a Fourier filter in the feedback path was
shown. Complex rotationlike periodic motions of the
hexagonal pattern were induced by an angular misalign-
ment of the counterpropagating pump beams, and a lin-
ear dependence of the frequency of motion on the magni-
tude of the misalignment was obtained. Finally, we
described the spontaneous oscillation of the system be-
tween hexagonal and square patterns.
ACKNOWLEDGMENTS
The authors acknowledge fruitful discussions with B.
Thu
¨
ring, R. Neubecker, and T. Rauch. This study was
supported by the Deutsche Forschungsgemeinschaft,
Sonderforschungsbereich 185 ‘‘Nichtlineare Dynamik.’’
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