Download pattern formation and competition in nonlinear optics - Ino-Cnr

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
1
PATTERN FORMATION AND COMPETITION
IN NONLINEAR OPTICS
F. Tito ARECCHI , Stefano BOCCALETTI, PierLuigi RAMAZZA
Istituto Nazionale di Ottica, Largo E. Fermi, 6, 150125, Florence, Italy
Department of Physics, University of Florence, Florence, Italy
Dept. of Physics and Applied Mathematics, Universidad de Navarra, Irunlarrea s/n, Pamplona, Spain
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
Physics Reports 318 (1999) 1}83
Pattern formation and competition in nonlinear optics
F. Tito Arecchi , Stefano Boccaletti, PierLuigi Ramazza *
Istituto Nazionale di Ottica, Largo E. Fermi, 6, I50125, Florence, Italy
Department of Physics, University of Florence, Florence, Italy
Dept. of Physics and Applied Mathematics, Universidad de Navarra, Irunlarrea s/n, Pamplona, Spain
Received January 1999; editor: I. Procaccia
Contents
1. Introduction
1.1. Optical patterns
1.2. Aspect ratios
1.3. Classi"cation of laser systems depending
on damping rates
1.4. Outline of this review
2. Patterns in active optical systems
2.1. The theory of patterns in lasers
2.2. Experiments with lasers
2.3. Patterns in photorefractive systems
3. Patterns in passive optical systems
3.1. Filamentation in single-pass systems
3.2. Solitons in single-pass systems
3.3. Counterpropagating beams in a nonlinear
medium
3.4. Nonlinear medium in an optical cavity
3.5. Nonlinear slice with optical feedback
3.6. Nonlocal interactions
4. Defects and phase singularities in optics
4.1. Phase singularities and topological defects
in linear waves
4.2. Phase singularities in nonlinear waves
4
4
4
7
9
11
11
14
17
24
24
25
28
32
38
45
51
5. Open problems and conclusions
5.1. Localized structures in feedback systems
5.2. Control of patterns
5.3. Patterns in atom optics
Acknowledgements
Appendix A. A reminder of nonlinear optics
A.1. Nonlinear susceptibility
A.2. The two level approximation
A.3. The s and s nonlinear optics
A.4. The photorefractive (PR) e!ect
Appendix B. Rescaling the Maxwell}Bloch
equations to account for detuning and a large
aspect ratio
Appendix C. Multiple scale analysis of the
bifurcation problem for the non lasing solution
of the Maxwell}Bloch equations
Appendix D. Symmetries and normal form
equations
References
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* Corresponding author.
0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved.
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
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Abstract
Pattern formation and competition occur in a nonlinear extended medium if dissipation allows for
attracting sets, independently of initial and boundary conditions. This intrinsic patterning emerges from
a reaction di!usion dynamics (Turing chemical patterns). In optics, the coupling of an electromagnetic "eld
to a polarizable medium and the presence of losses induce a more general (di!raction-di!usion) mechanism
of pattern formation. The presence of a coherent phase propagation may lead to a large set of unstable bands
and hence to a richer variety with respect to the chemical case. A review of di!erent experimental situations is
presented, including a discussion on suitable indicators which characterize the di!erent regimes. Vistas on
perspective new phenomena and applications include an extension to atom optics. 1999 Elsevier Science
B.V. All rights reserved.
PACS: 05.45.#b
Keywords: Nonlinear optics; Nonlinear dynamics; Pattern formation
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1. Introduction
1.1. Optical patterns
Pattern formation in extended media is the result of the interaction between a local nonlinear
dynamics and space gradient terms which couple neighboring spatial regions. Furthermore,
nonlocal terms may bring interactions from far away either in space or time. Some previous review
papers are available on the matter [1,2].
A recent comprehensive review [3] is mainly devoted to #uid dynamic or chemical patterns,
where the gradient terms result either from momentum transport of from di!usion processes.
Optical patterns, on the other hand, are characterized by a wave transport, mainly pointing in one
direction. Furthermore, use of laser sources and resonant media to enhance the nonlinearities
restricts the time dependence to a quasi-monochromatic behavior. Therefore, the functional terms
we have to deal with in optics are of the type
f (x, y, z, t) eIX\SR ,
(1)
where the exponential term accounts for a plane wave moving along a direction z ((x, y) being the
plane orthogonal to z) and the residual z and t dependence is slow, i.e.
Rf
;k" f ",
Rz
Rf
;u" f " .
Rt
(2)
This is currently called SVEA (slowly varying envelope approximation) and it is a sensible
approximation even when the slow and fast scales di!er by a factor less than 10, as it occurs e.g. for
femto-second pulses.
Based upon this wave transport feature, optical patterns present all classes of relevant phenomena reported elsewhere, plus some ones which are speci"c of optics. It seems then appropriate
to introduce a general classi"cation of optical patterns, within which it is easy to include also
classes of phenomena observed in #uid and chemistry. For convenience, we have collected in
Appendix A some introductory facts on nonlinear optics, together with the corresponding jargon.
1.2. Aspect ratios
In the classi"cation we distinguish between the longitudinal space direction z and the transverse
plane (x, y), since the boundary conditions are usually drastically di!erent for the two cases. We
classify patterns as 0, 1, 2, or 3-dimensional depending on the functional space dependence of the
envelope f. Notice that, at variance with condensed matter instabilities, a 0-dimensional dynamics
(i.e. ruled by an ordinary di!erential equation for f ) still refers to a wave pattern which is
mono-directional and mono-chromatic. Crucial parameters to estimate the dimensionality are the
aspect ratios, which will be de"ned as follows.
Let us con"ne the optical dynamics within a rectangular box of sides ¸ , ¸ , ¸ and take
V W X
¸ <¸ , ¸ .
X
V W
Optical patterns emerge from coupling Maxwell equation to the constitutive matter equations.
The Maxwell "eld e(r, t) induces a polarization p(r, t) in the medium. This polarization acts as
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
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a source for the "eld which then obeys the wave equation (for simplicity, we refer to scalar "elds,
neglecting for the moment the polarizations)
䊐e"!kRp ,
R
(3)
where 䊐,R#R#R, k is a suitable parameter, and we are denoting the partial derivations as
V
W
X
R
"R , etc. We expand the "eld as
V
Rx
e(r, t)"E(x, y, z, t)eIX\SR .
(4)
If the longitudinal variations are mainly accounted for by the plane wave, then we can take the
envelope E as slowly varying in t and z with respect to the variation rates u and k in the plane wave
exponential (Eq. (2)). Furthermore, we shall de"ne P to be the projection of p on the plane wave. By
neglecting second order envelope derivatives in z and t, it is easy to approximate the operator on
E as
1
䊐P2ik R # R #R#R ,
X c R
V
W
(5)
where c is the light velocity. Eq. (5) is usually called the eikonal approximation of wave optics.
Eq. (5) suggests that the comparison between transverse variations along x and y are ruled by the
aspect ratios
¸
F " V,
V j¸
X
¸
F" W ,
W j¸
X
(6)
where j,2n/k is the optical wavelength.
The aspect ratios are called Fresnel numbers, and they have the following heuristic meaning. The
geometric angle of view of an object of linear size ¸ from a distance ¸ is ¸ /¸ . Within this angle
V
X
V X
only details of minimal angular separation j/¸ can be resolved, due to di!raction. Thus,
V
the number of independent resolution elements along ¸ detectable at a distance ¸ is given by F .
V
X
V
The same holds for F .
W
So far we have referred to Eq. (5) in free space, considering p as an external perturbation, thus
introducing a bare aspect ratio. In fact, the constitutive matter equations provide a generally
nonlinear and nonlocal functional dependence p(E).
To de"ne a dressed aspect ratio, we must consider the linear part of the polarization
P"e s*E ,
(7)
where s is the linear susceptibility and the star convolution operator accounts for nonisotropic
e!ects (tensor relations) as well as for nonlocalities in time and space (temporal and spatial
dispersion). Furthermore we must consider the appropriate boundary conditions. Let us con"ne
the dynamics within a volume bound by two parallel mirrors in the longitudinal direction and
consider free lateral boundary conditions. We have the standard longitudinal and transverse
modes well known in laser phenomenology.
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 1. Temporal (a) and spatial (b) frequency space pictures of the active modes in the (i) (1#0), (ii) (1#1) and (iii)
(1#2) dimensional cases.
In the active case, i.e. when the medium provides energy to the "eld, the frequency width *u of
the medium gain line can be compared with the so-called FSR (free spectral range) that is, with the
separation *u
,c/2¸ of the adjacent ith and (i#1)th longitudinal modes. The ratio
GG>
X
*u
(8)
C"
X *u
GG>
de"nes a longitudinal aspect ratio. If C (1 only one longitudinal resonance can be excited, and
X
the cavity "eld is uniform along z, whereas if C '1 many excited longitudinal modes give rise to
X
a short pulse whose spatial length is smaller than the cavity length ¸ .
X
In Fig. 1a we report the frequency position of the modes and the medium line for 0-, 1- and
2-dimensional cases, in Fig. 1b we report the corresponding wavenumbers.
In fact, for the 1- and 2-dimensional cases, we do not consider di!erent wavenumbers, but we use
a pseudo-spectral method consisting in Fourier expanding around the central wavenumber k , and
considering the residual spread as a slow time-space dependence in terms of evolution equations
including the nonlinearities.
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
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Table 1
Classi"cation of optical pattern forming systems depending on the longitudinal and transverse aspect ratios
Dimension
X
>
Z
Systems
0
F (1
V
F (1
W
C (1
X
䢇
Single longitudinal and transverse mode
laser
1
F (1
V
F (1
W
C '1
X
䢇
Single transverse mode, longitudinally
multi-mode laser
Temporal solitons in "bers
䢇
1
F '1
V
F (1
W
C (1
X
䢇
䢇
Linear arrays of semiconductor lasers
Nonlinear optics of slab interferometers
Single mode laser with delayed feedback
Spatial solitons in planar waveguides
䢇
䢇
2
F '1
V
F (1
W
C (1
X
䢇
n.l.o. of planar waveguides
2
F "F '1
V
W
F "F '1
V
W
C (1
X
䢇
Transverse optical patterns in active
media (lasers, PRO) and passive media
(liquid crystals, resonant gases)
F "F '1
V
W
F "F '1
V
W
C (1
X
䢇
3
Co-propagating beam interaction as e.g.
four wave mixing
䢇
Break-up and "lamentation of beams in n.l.o.
Note: n.l.o."Nonlinear optics, PRO"Photorefractive oscillator.
In Table 1 we list optical pattern forming systems of di!erent dimensions. It is understood that
along a dimension there is no evolution whenever the corresponding aspect ratio is less than 1.
1.3. Classixcation of laser systems depending on damping rates
It is well known that a discrete nonlinear dynamical system can undergo a chaotic motion, that
is, at least one of its Liapunov exponents can be positive, only when the number of degrees of
freedom (phase space dimension) is at least 3.
We "nd it convenient to refer to dissipative systems, that is, systems with damping terms for
which the phase-space volume is not conserved. In such systems the sum of all Liapunov exponents
is negative, and initial conditions tend asymptotically to an attractor [4].
For dimensions N"1, the attractor is a "xed point, for N"2 a "xed point or a limit cycle, for
N"3 it can be a "xed point (all the three Liapunov exponents negative), a limit cycle (two
Liapunov exponents negative and one zero), or a torus (one Liapunov exponent negative and two
zero), or even a chaotic attractor (one Liapunov exponent negative, one zero and one positive).
An example of chaotic motion is o!ered by the Lorenz model of hydrodynamic instabilities [5],
which corresponds to the following equations, where the parameter values have been chosen so as
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
to yield one positive Liapunov exponent:
8
x "!10x#10y, y "!y#28x!xz, z "! z#xy ,
3
(9)
where now x, y, z are suitable variables, and dots denote temporal derivatives.
Also the fundamental equations for "eld matter interaction can exhibit chaotic features. Indeed,
if we couple Maxwell equations with Schroedinger equations for N atoms con"ned in a cavity, and
expand the "eld in cavity modes, keeping only the "rst mode which goes unstable, its amplitude E is
coupled with the collective variables P and D describing respectively the atomic polarization and
the population inversion. The resulting equations are
EQ "!kE#gP,
PQ "!c P#gED ,
,
(10)
DQ "!c (D!D )!4gPE .
,
For simplicity, we consider the cavity frequency at resonance with the atomic one, so that we can
take E and P as real variables and we have three real equations. Here k, c , c are the loss rates for
, ,
"eld, polarization and population, respectively, g is a coupling constant and D is the population
inversion which would be established by a pump mechanism in the atomic medium, in the absence
of the coupling. While the "rst equation comes from the Maxwell equation, the two others imply
the reduction of each atom to a two-level atom resonantly coupled with the "eld, that is,
a description of each atom is an isospin space of spin 1/2. The last two equations are like Bloch
equations which describe the spin precession in the presence of a magnetic "eld. For such a reason,
Eqs. (10) are called Maxwell}Bloch equations.
The presence of loss rates means that the three relevant degrees of freedom are in contact with
a sea of other degrees of freedom. In principle, Eqs. (10) could be deduced from microscopic
equations by statistical reduction techniques [6,365].
The similarity of Eqs. (10) with Eqs. (9) would suggest the easy appearance of chaotic instabilities
in single mode, homogeneous-line lasers. However, time-scale considerations rule out the full
dynamics of Eqs. (10) for most of the available lasers. Eqs. (9) have damping rates which lie within
one order of magnitude of each other. On the contrary, in most lasers the three damping rates are
widely di!erent from one another.
The following classi"cation has been introduced [7].
Class A lasers (e.g. He}Ne, Ar, Kr, dye): c Kc <k. The two last equations of Eqs. (10) can be
,
,
solved at equilibrium (adiabatic elimination procedure) and one single nonlinear "eld equation
describes the laser. Since N"1, only a "xed point attractor is supported in this case.
Class B lasers (e.g. ruby, Nd, CO): c <kKc . Only polarization is adiabatically eliminated
,
,
(middle equation of Eqs. (10)) and the dynamics is ruled by two rate equations for "eld and
population. Fixed points and periodic oscillations are then supported.
Class C lasers (e.g. far infrared lasers): c &c &k. The complete set of Eqs. (10) has to be taken
,
,
into account, hence Lorenz like chaos is feasible.
A series of experiments on the birth of deterministic chaos in CO lasers (Class B) was initially
carried out in various con"gurations, namely: with the introduction of a time dependent parameter
to make the system nonautonomous [8]; with the injection of a signal from an external laser
detuned with respect to the main one [7]; with the use of a bidirectional ring cavity [9], and with
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
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the addition of an overall feedback, besides that provided by the cavity mirrors [10,387]. All these
con"gurations imply a third degree of freedom, besides the two ones intrinsic of a class B laser, thus
making chaos possible.
Lorenz chaos was later extensively investigated in far infrared lasers, corresponding to molecular
(rotational) transitions with damping rates comparable with the cavity decay rate [11].
A few comprehensive reviews on the subject are available in Refs. [12,13].
In the forthcoming sections, Maxwell}Bloch equations will be generalized as follows:
1. introducing a detuning *u"u!u between the laser frequency u and the center u of the
?
?
atomic line, thus E and P must be considered as complex quantities, and Eqs. (10) transform to
a set of 5 real equations;
2. increasing the cavity aspect ratios, transforming the problem from a discrete to an extended one
in the real space. One should carefully distinguish between the dimensions N of the phase space
(N"number of dynamical degrees of freedom) and the dimension D of the real space within
which the physics takes place. As one goes from D"0 to D"1, then the time derivative EQ in the
equation for E will be replaced by R #cR . This extension was "rst considered in connection
R
X
with the propagation of pulses in an excited two-level medium in presence of scattering losses
compensating for the gain, so that soliton pulses (so called n pulses) resulted propagating at the
light speed, the nonlinearity compensating for the dispersion [14]. In large aspect ratio cavities
(D"2) the time derivative in the equation for E will be replaced by R #(ic/2k)
, where
R
,
,R#R is the transverse Laplacian operator. This extension was "rst considered in
,
V
W
"lamentation problems [15].
In fact, the self focusing and self defocusing phenomena imply the whole 3D structure of the "eld,
and hence require the use of the operator R #cR #(ic/2k)
, where the time derivative will be
R
X
,
dropped in case of stationary phenomena [16].
1.4. Outline of this review
As it has become customary, we call active optical devices those ones in which the medium is in
an excited state and it can transfer energy to the light "eld, whereas we call passive optical devices
those in which the medium is in its ground state. As a result, active optics can start from
spontaneous emission processes, whereas passive optics always requires an incident "eld. Laser are
prototypical examples of active optics. In the case of photorefractive oscillators, which are based on
two or four wave mixing (see Appendix A) one or more input "elds will have the role of pumps.
As we will see in this review, evidence of reliable patterns in lasers has been made di$cult by two
reasons: (i) laser cavities have usually small transverse aspect ratios, (ii) the time scales of the
dynamics are so fast that only averaged patterns can be visualized. Both limitations are not present
in a photorefractive oscillator which has then become a very useful testbench to explore active
pattern formation.
Let us detail what we mean by the title of this report. `Pattern formationa refers to the fact that
in an extended nonlinear medium, above a suitable threshold, any uniform amplitude distribution
becomes unstable and the space}time distribution of the amplitude splits into correlated domains.
The "rst symmetry is ruled by the boundary conditions. We call a pure pattern an eigenstate of the
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
propagation problem associated with the linear part of the dynamics, in the absence of nonlinearities. Nonlinearities imply the interaction of many eigenstates, or pure patterns. This may induce
many scenarios of `Pattern competitiona, which, in order of increasing complexity, are respectively:
1. winner-takes-all dynamics: one pure pattern prevails on the others;
2. cooperation: many pure patterns coexist (e.g. hexagons as coexistence of three roll sets at 1203
with each other, mutually coupled by a quadratic nonlinearity);
3. locking of space and time phases of the wavevectors corresponding to di!erent hexagon families;
4. time alternation of patterns, which "ll the whole available region, inducing time chaotic
phenomena and yet keeping a spatial coherence;
5. segregation of di!erent domains in each of which a di!erent stationary pattern is present;
6. space}time chaos with a limited correlation length and correlation time.
Examples of the six cases are discussed in the following.
This report is organized as follows.
In Section 2 we report the theoretical and experimental results on pattern formation and
competition in nonlinear active optics. In Section 2.1 we show how Maxwell}Bloch equations can
be generalized to account for spatial dependence, and how the two homogeneous stationary
solutions (lasing solution and nonlasing solution) bifurcate to patterned states which can be
described by suitable amplitude equations. In Section 2.2 we review the most signi"cant results on
patterns in laser systems, in both the low and high dimensional cases. In Section 2.3 we report one
of the "rst evidences of spatial dependent chaotic regimes in a photorefractive oscillator, and we
show that the main features of its dynamics can be captured by simple normal forms equations
accounting for the symmetry requirements.
In Section 3 we review the case of passive nonlinear optics, mainly referring to Kerr media. We
consider various experimental situations, namely, "lamentation in single-pass systems (Section 3.1),
formation of solitons in single-pass systems (Section 3.2), counterpropagating beams in a nonlinear
medium (Section 3.3), patterned states originated by a nonlinear medium con"ned within an
optical cavity (Section 3.4), patterns in a nonlinear slice with optical feedback (Section 3.5) and the
e!ects of nonlocal interactions in the selection of the pattern shape and of the relevant pattern size
(Section 3.6).
In Section 4 we de"ne what is a phase singularity, or defect, or optical vortex, and we discuss how
the presence of these defects is related to the dynamics of the pattern forming system. In Section 4.1,
we summarize the main properties of phase singularities and topological defects in linear waves,
and we report their scaling laws. In Section 4.2, we analyze the case of phase singularities in
nonlinear waves, and we report the experimental evidence of the dynamical transition from
a boundary dominated regime to a bulk dominated regime, in terms of the defect statistical
properties.
In Section 5 we discuss the perspectives of this area of investigation, referring to some challenging cases, namely, the formation and evolution of localized structures in nonlinear optics (Section
5.1), the problem of stabilizing unstable patterns within space}time chaotic states (Section 5.2), and
the pattern formation in atom optics (Section 5.3), where the di!ractive properties are associated
with the atom Schroedinger "eld, and the coherence requirements have only recently been reached
by the evidence of atomic BEC (Bose}Einstein condensation) [17,388,389], as well as of the atom
laser [18].
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
11
2. Patterns in active optical systems
2.1. The theory of patterns in lasers
The theory of pattern formation in the transverse pro"le of a laser cavity is quite recent. If one
considers the dynamics of the electromagnetic "eld in a cavity with #at end mirrors (Fabry-Perot
con"guration) housing an active medium done by two level atoms, it is described by the Maxwell}Bloch equations [19] (see Appendix B), which read
EQ !ia
E"!pE#pP,
PQ #(1#iX)P"(r!N)E,
NQ #bN"(EHP#EPH) .
(11)
The above equations rule the behavior of the con"ned electromagnetic complex "eld E in the
transverse plane (
,(R/Rx)#(R/Ry); (x, y) being the plane transverse to the direction z of the
cavity axis). They are written in the complex Lorenz notation [20], and derive directly from the
Maxwell equation for the "eld E and from the Bloch equations for the complex atomic polarization
P and the real population inversion N, under the assumption that E and P have a preferred
plane-wave dependence in the z direction, and a slow residual dependence upon the transverse
variables x and y.
In Eq. (11), p and b are respectively the decay rates of E and N scaled to the decay rate of the
polarization, r accounts for the pumping process, a is a real parameter which will be later speci"ed,
and X represents the detuning, i.e. the di!erence between atomic and cavity frequencies.
The Maxwell}Bloch equations possess two homogeneous stationary solutions: E"P"N"0
(nonlasing solution) and E"PO0, NO0 (lasing solution).
The destabilizations of both solutions, leading to di!erent pattern formation, are described by
suitable amplitude equations. For the nonlasing solution, Refs. [21,22] report the application of
weakly nonlinear analysis to Eq. (11), and show that its bifurcation leads to the complex
Swift}Hohenberg equation [23] for class A and C lasers, and to a complex Swift}Hohenberg
equation coupled to a mean #ow for class B lasers.
As for the bifurcation of the lasing solution, amplitude equations have been derived for class B
lasers by linear analysis and symmetry techniques Ref. [24], showing that patterns are here ruled
by a complex Swift}Hohenberg equation coupled to a Kuramoto}Shivasinsky equation [25,366].
The derivation of the amplitude equations for the case of the nonlasing solution is reported
in Appendix C.
Let us begin with the nonlasing solution. It is straightforward to show that such a solution
becomes unstable for pumping coe$cients r larger than a critical value
(X!ak)
,
r "1#
(1#p)
(12)
which is called lasing threshold. In Eq. (12) k is a critical value for the transverse wavenumber k.
Furthermore k "0 as far as X(0, while k"X/a in the case X'0. Therefore, the nature of the
bifurcation depends on the sign of the detuning. In other words, if X(0, the bifurcation occurs at
a homogeneous state (k "0), while for X'0 the bifurcation takes place at a preferred wavenum
ber k O0. The physical reason for this di!erence is that the emergence of a patterned state at kO0
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would imply an o!-axis propagation of the beam, which introduces a k-dependent phase retardation. If we are in presence of a positive detuning, the o!-axis propagation may compensate for it,
and therefore a patterned state with a suitable k (that for which the phase retardation compensates
for the detuning) locates at the center of the atomic gain line, thus having the minimum value for
the instability threshold. In the opposite case (negative detuning), the phase retardation can never
compensate for the detuning, so that the state with minimal threshold is that at k"0.
Refs. [21,22] consider small detuning values, and expand all relevant variables of Eq. (11) in
powers of a small parameter e, by writing
X"eX , (E, P, N)"(E , P , N )#e(E , P , N )#e(E , P , N )#2 .
(13)
Furthermore, the spatial scaling are easily derived from the breadth of the band of the unstable
modes above threshold. The resulting scaling is
X"(r!1)x,
>"(r!1)y .
(14)
If one further assumes r"1#e, it results X"(ex, >"(ey. As for the time scales, Refs.
[21,22] consider ¹ "et and ¹ "et as two di!erent secular time scales. Plugging all those
expressions into the Maxwell}Bloch equations, and identifying the coe$cients of each power of e,
one gets for the "rst order "eld amplitude E "t the equation (for the derivation see Appendix C)
p
p
Rt
(X#a
)t#ia
t!iXpt! "t"t ,
(15)
(p#1) "p(r!1)t!
(1#p)
b
Rt
which is the Swift}Hohenberg equation.
The derivation process (Appendix C) was carried out under the hypothesis that both p and b are
of the order of one, since they have not been expanded as function of the smallness parameter e.
Therefore, one can say that Eq. (15) describes what happens in class C lasers.
For class A lasers, instead, p is small, and a derivation similar to what reported in Appendix
C leads to a Swift}Hohenberg equation where the coe$cient 1/(1#p) in Eq. (15) is expanded in
series of p.
Finally, for class B lasers (bP0), the same procedure with small b and at order four in e leads to
two coupled equations, namely,
p
Rt
(X#a
)t#ia
t!iXpt!pmt,
(p#1) "p(r!1)t!
(1#p)
Rt
Rm
"!bm#"t" ,
(16)
Rt
that is, a Swift}Hohenberg equation and a real mean #ow m.
Similar equations for the bifurcation of the nonlasing solution have been obtained through other
techniques [26]. In particular, several di!erent partial di!erential equations have been derived
from the Maxwell}Bloch equations through center manifold techniques [27].
Ref. [28], even though formally less detailed than the approach summarized in Appendix C, is
heuristically appealing insofar as it starts from the simple naive Ginzburg}Landau (GL) equation
derivable for a class A laser (pP0) with the inclusion of the di!ractive Laplacian term ia
E.
Noticing that such GL equation would be structurally unstable, Ref. [28] releases the strong
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
13
Fig. 2. From Ref. [26]. Stability analysis of the non lasing solution: perturbation growth exponents vs. perturbation
wavenumber for three di!erent laser models. Squares: classical complex Ginzburg}Landau equation, circles:
Swift}Hohenberg equation, triangles: full Maxwell}Bloch description.
adiabatic elimination hypothesis by expanding the polarization in terms of a small parameter
gJp. The "rst contribution amounts to correcting the Laplacian term with a di!usive constribution d E, where, however, d depends on the sign of the detuning, and would provide an
unphysical antidi!usion in the case of a negative detuning. Introducing the successive higher order
terms in g, Ref. [26] arrives to the Swift}Hohenberg equation. Fig. 2 shows the stability analysis of
the nonlasing solution for the Ginzburg}Landau equation, for the Swift}Hohenberg equation and
for the fully Maxwell}Bloch equations.
In Ref. [29], a Ginzburg}Landau equation was found near the lasing threshold, when considering the interaction of an electromagnetic "eld with matter in a laser cavity without the assumption
of a "xed direction of the transverse electric "eld.
Pattern formation and evolution in a single longitudinal mode two-level and Raman laser with
#at end mirrors, with uniform transverse pumping was investigated numerically [30], and analyzed
at and beyond threshold [31].
Other theoretical analyses of transverse pattern forming instabilities in lasers are given in Refs.
[32}38]. In particular, the roles of symmetries and symmetry breaking mechanisms have been
investigated, with the aim of extracting simple normal form equations for the dynamics of the "rst
order transverse modes arising from the primary bifurcation of the nonlasing state [39}42].
Further theoretical analysis has regarded the study of tilted and standing waves in class A lasers
[43], and that of polarization e!ects in the transverse mode dynamics [44], leading to the so called
vector complex Ginzburg}Landau equation [45]. Also the case of a laser with injected signal has
been the object of theoretical investigation [46].
Recently, the problem of boundary-driven selection of patterns has been studied in class B lasers
with reference to Eqs. (16) [47]. Furthermore, analytic treatments of Maxwell}Bloch equations for
fully 3D lasers have been provided [48].
Finally, analysis of a coherently optically pumped three level laser has pointed out the possibility
of spiral formation by a mechanism similar to that occurring in excitable media, but here uniquely
due to nonlinear di!ractive interactions [49].
A completely di!erent scenario characterizes the bifurcation of the homogeneous lasing solution
[24,50]. In this case, when performing a linear stability analysis of a lasing homogeneous solution
14
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
E of Maxwell}Bloch equations in Fourier space and for class B lasers, the dynamics is governed
by a real "eld u and a complex amplitude A. In Ref. [24], it is shown that the bifurcation is
described by
a
1#X
"E"
K1! 1#
u!i(AeSR!AHe\SR)#O["A", (
u)] ,
4
"E "
"E "
(17)
where u is the frequency of the bifurcation (uK(2pb"E "/(1#X)).
Linear properties and symmetry arguments lead to equations for u and A of the form
c
R u,!k
u! u#c (
u)#c "A" ,
R
2
R A"kA!il(q#
)A!a(q#
)A![b "A"#b (
u)]A#b u ) A ,
R
(18)
where c , c , c , b , b , b are suitable parameters, kKpXa/2, k ;q , q is the well de"ned
wavenumber of the bifurcation, k, l and a are parameters directly derived from linear stability
analysis arguments.
The "rst of Eq. (18) is the Kuramoto}Shivasinsky equation, while the second is the Swift}
Hohenberg equation. A similar scenario has been observed in recent hydrodynamical experiments
[51,52]. The above theoretical predictions have been experimentally tested for a CO laser with
large transverse aperture, thus allowing a large number of possible transverse modes [24], with an
experimental setup similar to that already used in Ref. [53].
2.2. Experiments with lasers
Even though the theory provides accurate amplitude equations, no global solutions of them are
available in 2D, thus no detailed comparison can be drawn with the existing experiments on
pattern formation and competition in laser systems. Only recently, evidence of vortices, shocks,
domains of tilted waves and cross-roll patterns on a photorefractive oscillator has been compared
with the corresponding numerical solutions of the complex Swift}Hohenberg equation [54].
At this point, we should distinguish between two very di!erent experimental conditions. As we
have pointed out in the previous subsection, patterns arise in the plane transverse to the cavity axis,
that is, to the propagation direction of the light within the optical cavity. Therefore, there is an
important role played by the transverse aspect ratio. Namely, when the transverse "nite size e!ects
allows only few transverse modes to survive, we will generally speak of a low-dimensional situation.
In this case, usually, boundary symmetries are crucial in selecting the pattern shape and in
imposing suitable couplings in their dynamics.
On the contrary, when the aspect ratio is su$ciently large to allow for a very large number of
independent modes, the phase space is high dimensional, and the role of the boundary conditions
will be irrelevant.
A clear dynamical transition between these two regimes will be presented in Section 4.2. At
this stage, this di!erence has been presented only qualitatively, so as the reader might be able to
understand the jargon that we will use hereafter.
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
15
Fig. 3. From Ref. [57]. Transverse pattern arising from a cooperative frequency locking of two modes, each one
possessing a cylindrical symmetry.
Even though patterns in lasers have been observed since their discovery [55], one of the "rst
systematic experimental evidences of their competition in a laser was realized in Ref. [56]. In this
case, the competition between the "rst orders transverse modes compatible with a cylindrical
symmetry was studied in a helium}neon laser. Ref. [57] reports a stable spatial pattern emerging
from a cooperative frequency locking of the "rst two modes. Such a pioneering work dealt with one
of the most simple situations, wherein the transverse dynamics was ruled by the competition of two
modes (each one of them possessing a cylindrical symmetry). The arising patterns are reported in
Fig. 3.
Similar investigations on a Na ring laser have been reported in Ref. [57]. Also in this case
boundary regulated patterns have been studied in a low-dimensional con"guration.
Following previous experimental studies [58], and theoretical speculations on spontaneous
symmetry breaking phenomena [59], the appearance of traveling waves in the transverse intensity
pro"le of a CO laser was reported in Ref. [60]. These observations have con"rmed that the
spatiotemporal behavior of a laser in the low-dimensional regime crucially depends on symmetries,
and that most of the dynamical features can be directly captured as consequences of spontaneous
symmetry breaking mechanisms.
The theoretical study of the transverse boundary e!ects on a positively detuned laser system is
the subject of Ref. [61]. Further theoretical investigation on standing and travelling waves in
homogeneously and inhomogeneously broadened lasers is contained in Ref. [62].
More recently, the attention has moved from boundary e!ects (small transverse sizes) to the high
dimensional situation, where the large aspect ratio allows for many di!erent modes. Patterns in
a highly pumped high dimensional CO laser has been the object of Refs. [24,63]. In these
conditions, the observed pattern (Fig. 4) is characterized by a high degree of complexity and by
the absence of zeroes in the intensity pro"le. This last feature has supported the claim that, in the
high dimensional case, patterns are generated by a di!erent bifurcation with respect to the
low dimensional case. Namely, the experimental observations seem to support the hypothesis
that patterns are here obtained as a modulation of the lasing state, as opposed to the previous
cases, wherein the boundary symmetries induced a bifurcation of the nonlasing state toward
suitable modes.
16
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 4. From Ref. [24]. (a) Pattern of intensity observed on an infrared image plate. The pattern refers to the transverse
intensity distribution in a highly pumped CO laser with large Fresnel number. (b1) Intensity distribution vs. transverse
coordinate, obtained through a measurement wherein a rotating mirror de#ects the beam onto a fast detector. The
pattern is here characterized by a high degree of complexity and by the absence of zeroes in the intensity pro"le. (b2)
Temporal #uctuations of the intensity on a point in the transverse section of the beam.
Fig. 5. From Ref. [64]. Experimental patterns (a)}(c) and pattern reconstructed in terms of superpositions of Hermite}Gauss modes (d)}(f). The experiments are carried out with a CO laser with large aspect ratio.
In another experimental study of a CO laser with a large aspect ratio [64], many complicated
patterns have been observed in various experimental con"gurations (Fig. 5) and heuristically
described in terms of superpositions of Hermite}Gauss modes.
In the same years, after many theoretical [65] and experimental [66,67] studies on bidirectional
ring lasers, it has been shown that the cavity resonances for transverse modes depend crucially on
the number of mirrors housed within the cavity, thus implying that di!erent cavity con"gurations
may lead to di!erent dynamical behaviors [68].
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
17
Another important subject of recent investigation is the polarization dynamics in the low-order
transverse modes of a CO laser in a Fabry-Perot con"guration [69]. While polarization e!ects in
single mode lasers had been modeled in optically pumped far infrared lasers [70,71], "ber lasers
[72], edge-emitting semiconductor lasers [73] and vertical cavity semiconductor lasers [74,75],
the corresponding experiments [72,76,77] were often sensitive to anisotropies. In Ref. [68] the
polarization properties of the lowest order transverse modes of a CO laser have been thoroughly
investigated.
Finally, we summarize the main results on semiconductor lasers. They represent the most
convenient source of high-power optical coherent radiation, therefore the dynamics of a single laser
and of an array of them has been object of a huge body of investigations. One of the "rst theoretical
approaches to spatiotemporal behavior of broad-area semiconductor lasers discovered the presence of "lamentation phenomena in the transverse pro"le [78,79]. This pioneering work stimulated
many other theoretical analyses, which have considered the e!ects of spatial inhomogeneities in the
active medium on the onset of a spatiotemporal dynamics [80,81].
On the other side, spatiotemporal phenomena were studied in large arrays of spatially coherent
semiconductor lasers. Following the idea of chaos synchronization [82], it has been numerically
shown that a large array of semiconductor lasers can cluster in many subsets of synchronized
chaotic systems, and the disintegration of such clusters leads to the appearance of space}time chaos
[83], which is signaled by a spatial symmetry breaking mechanism. Also phase locking phenomena
were studied in such a kind of system [84,85]. The dynamics of large arrays of lasers was modeled
under various laboratory conditions [86,87].
2.3. Patterns in photorefractive systems
In the late 1980s spatial e!ects in optical oscillators di!erent from lasers have received serious
consideration for the reasons listed in Section 1.4. We report here the investigation of a PRO
(photorefractive oscillator) consisting of a photorefractive crystal pumped by a laser and emitting
within a ring cavity [88].
Control of the number of transverse modes which can oscillate (that is, the aspect ratio) is
performed by varying the aperture of a cavity pupil. The setup is shown in Fig. 6. The photorefractive crystal is a 5;5;10 mm BSO (bismuth silicon oxide) to which a dc electric "eld is applied. The
medium is pumped by a cw Argon laser with intensity around 1 mW/cm. The pump beam shines
the rear part of the crystal forming an angle h with the cavity axis, thus inducing an intensity
grating by interference with the cavity "eld. The cavity is made by four high-re#ectivity dielectric
mirrors plus a lens L (500 mm focal length), whose role is to enhance the cavity mode stability by
providing a near-confocal con"guration.
A pinhole is inserted within the optical path between two confocal lenses L of short focal length.
The displacements of the pinhole along the optical axis yield a continous change of the ratio
between the aperture and the spot size. As a consequence, a di!erent number of transverse modes is
inhibited. The e!ective Fresnel number F is the ratio of the area of the di!racting aperture that
limits the system (pupil) to that of the fundamental Gaussian-mode spot in the plane where the
aperture is placed. Changing the position of the pupil along the optical axis means varying F in the
range from 0 to approximately 100. The corresponding number of transverse modes that can
oscillate is proportional to F, so that it ranges from 0 to 10 000.
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 6. From Ref. [88]. Experimental setup for the photorefractive oscillator. VD is a videocamera recording the
wavefront pattern, PM is a photomultiplier measuring the time evolution at a "xed point selected by the "ber OF. BSO is
the photorefractive crystal, M are mirrors delimiting the ring cavity, A is a pupil controlling the Fresnel number, L and L
are lenses of suitable focal length so as to make the cavity quasi confocal, BS is a beam splitter, V is an applied voltage on
the BSO.
From the above de"nition, F is the ratio of the area of the pinhole (a) to that of the fundamental
mode spot in the pinhole plane (w(z )):
N
a
.
(19)
F"
w(z )
N
On the other hand, the ABCD matrix method for propagation of Gaussian beams provides w (z) at
each z position. The overall size of the Gaussian mode of order n is w K(nw . This implies that
L
the highest allowed mode (of order n "n
and size w "aK(n w ) will be such that
L
n w
(20)
FK "n .
w
Eq. (20) tells us that the Fresnel number gives the maximum order of the transverse modes that can
oscillate.
The experimental intensity patterns observed by increasing Fresnel numbers F are shown in
Fig. 7, together with the spatial correlation functions.
The low F regime (F44) is characterized by a time alternation between pure cavity modes
(Fig. 7a), with a spatial correlation length m covering the whole transverse size D of the beam. On
the contrary, for large F (FK15), the signal is a complicated pattern obtained by the superposition
of many modes irregularly evolving in space and time (Fig. 7c). It is characterized by a short
correlation length (m/D(0.1). The transition between these two regimes is marked by a continuous
variation of the ratio m/D. An intermediate situation is shown in Fig. 7b. Fig. 8 shows an example of
alternating pure mode con"gurations (low F regime).
Since here m/D&O(1), the temporal evolution of the pattern is coherent in space, so that studying
the global features of the dynamics is fully equivalent to analysing the local temporal behavior of
the intensity in an arbitrary point on the wavefront.
Let us interpret the above results. The cylindrical geometry of the cavity constrains the symmetry
of the output "eld to be O(2). However, the pumping process breaks the O(2) symmetry by
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
19
Fig. 7. From Ref. [88]. Intensity distribution of the wavefront (left column) and spatial autocorrelation function (right
column) for increasing Fresnel numbers. (a) F"5, one single mode at a time is present, and the ratio between
the correlation length m and the transverse size of the system D is of the order of one. (b) F"20, m/DK0.25.
(c) F"70, m/DK0.1.
introducing a privileged plane, de"ned by the propagation vectors of pump and signal "elds. The
oscillator generates "eld patterns varying in time. Two di!erent dynamical regimes arise by
changing the size of the cavity pupil. For large pupils, the "eld displays a complex pattern made of
a large number of solutions of the free propagation problem (the so called cavity modes). For small
pupils, the "eld is made of a single mode at any time, but a small number of modes (from two to
about ten) can alternate. The alternation is an ordered sequence of quasi-stationary modes.
Depending on some control parameter (tiny adjustments of F or of the pump), the time of
persistence of each mode is either regular (periodic alternation, PA) or irregular (chaotic alternation, CA). Apart from the short switching time from one mode to another, the amount of mode
mixing is here negligible.
This kind of dynamics is not peculiar of the system of Ref. [88]. Indeed, a phenomenon similar to
CA, called chaotic itinerancy, was reported theoretically for a one-dimensional laser [89], and
for an array of coupled lasers [90]. Later, the phenomenon was found also in globally coupled
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 8. From Ref. [88]. An example of periodic alternation obtained for low Fresnel number. The pictures show the
intensity patterns of the pure modes in their order of consecutive appearance in a cycle of periodic alternation obtained
for F"5.
iteration maps [91] and in nonequilibrium neural networks [92]. While chaotic itinerancy implies
erratic jumps among the available quasi-stationary states, here CA keeps the ordering sequence.
By increasing the Fresnel number F, a new regime, called spatio-temporal chaos (STC) is
observed, where a large number of modes coexist. From a statistical point of view, this regime has
been characterized by Hohenberg and Shraiman [93] in the following way. Suppose a generic "eld
u(r, t) be ruled by a PDE including nonlinear and gradient terms, and take the "eld of deviations
away from the local time average
du(r, t)"u(r, t)!1u(r, t)2 ,
(21)
(122 denotes time average). Under very broad assumptions, the leading part of the correlation
function is an exponential,
C(r, r)"1du(r, t)du(r, t)2Ke\P\PYK .
(22)
Whenever the correlation length m exceeds the system size ¸(m'¸), low dimensional chaos is
observed. This means that the system can be chaotic in time, but its evolution is coherent in space,
that is, the system is single mode, in a suitable mode expansion, and the corresponding chaotic
attractor is low dimensional. On the contrary, in the limit m;¸, any local chaotic signal cannot be
con"ned within a low dimensional space. In this case, a new outstanding feature appears.
The transition between these two dynamical regimes (low and high F) was observed in Ref. [94],
and will be reported in Section 4.2. Herewith we summarize the theoretical description of the low
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
21
Fresnel number regime, where boundary e!ects are dominant, and therefore the dynamical
equations can be retrieved by looking at the symmetry properties imposed by the boundary
conditions.
Periodic and chaotic alternation is a dynamics proper of systems with imperfect O(2) symmetry
[95]. Let us consider a situation involving three transverse modes competing nonlinearly according
to the symmetry constraints.
The considered modes are a central one, with complex amplitude z , and two higher order ones
(rotating and counter rotating along an azimuthal coordinate h) with respective complex amplitudes z and z and angular momenta $1. The cavity "eld can be expressed as
(23)
E"f (r)(z eF#z e\F)eSR#f (r)z eSR ,
where f and f are the space distributions of the modes. The optical frequencies u and u are in
general di!erent. The slow time dependence due to the dynamics is accounted for in the amplitudes
z (t) (i"0, 1, 2).
G
The zero intensity situation is described by z "z "z "0, the central mode by z "z "0
and an azimuthal standing wave by z "0, z "z .
The time sequence of these three con"gurations is the simplest case experimentally observed in
Ref. [88], and model equations having the above sets of z values as "xed points can be built for
reproducing such a behavior. The experimental observations tell us that all quasi-stationary points
persist only for a "nite time, thus each of the "xed points will be considered to have at least an
unstable direction. These general rules are compounded now with the symmetry requirements. The
cylindrical geometry of the cavity imposes the following constraints on mode amplitudes [96]
H : (z , z , z )P(eFz , e\Fz , z ) ,
(24)
K : (z , z , z )P(z , z , z ) ,
(H being the rotation operation, and K the re#ection operator around a privileged plane). If one
considers the modes as born from Hopf bifurcations, then there is an additional time symmetry
(25)
B : (z , z , z )P(e@z , e@z , e@z ) .
The normal form for the nonlinear interaction among the three modes, assuming it to be invariant
under the above symmetries is [97,98] (dots denote time derivatives)
z "j z #(a("z "#"z ")#b"z ")z ,
z "j z #(c"z "#d"z "#e"z ")z #ez ,
(26)
z "j z #(d"z "#c"z "#e"z ")z #ez ,
(j , j , a, b, c, d, e being complex coe$cients and e"o ePC being a symmetry breaking parameter).
C
The parameter e is reminiscent of the breaking of the cylindrical symmetry induced by the pumping
procedure which privileges a de"ned plane, thus breaking the rotational invariance. The analysis of
Eq. (26) is reported in Appendix D. The numerical solutions of Eq. (26) (Fig. 9) shows and example
Both Ref. [97] as well as the application of the symmetry arguments for the patterns of a CO laser [98] refer to a two
mode interaction and are concerned with the steady solutions whereas this is a three mode case.
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 9. From Ref. [95]. (o , o ) projection of the solution of Eqs. (26) for aP"3, bP"1, cP"!1, dP"!2, eP"!1,
jP "0.5, jP "!1.5, o "0.5, u "0, cG!dG"1. P is marginally stable, and any initial condition generates a periodic
C
C
orbit. The closer a periodic orbit is to the three "xed points (O, C, SW), the larger the period. The imaginary parts of the
coe$cients j , b, a and e only contribute to the dynamical evolution of u and u .
of periodic alternation, in the case of Fig. 9, (Re b/Re e)'(2Re a/Re (c#d)) (see Appendix D). For
any initial condition a periodic solution exists passing through it, and this suggests the existence of
integrals of motion. There is an heteroclinic solution connecting the three "xed points on the axes,
and therefore the period of a periodic solution is larger the closer it is to the heteroclinic solution.
As already discussed, the experimental evidence of PA is, however, largely limited, whereas the
natural evolution of the dynamics seems to lead to a chaotic alternation among the available
con"gurations. This feature can be explained by invoking a frequency degeneracy u "u .
A resonance between the three states gives rise to an additional symmetry. Precisely, the time
symmetry now becomes [96]
B : (z , z , z )P(e@z , e@z , e@z ) .
and additional terms survive in the normal form equation, which becomes
(27)
z "j z #(a("z "#"z ")#b"z ")z #fz z zH ,
(28)
z "j z #(c"z "#d"z "#e"z ")z #ez #gzzH ,
z "j z #(d"z "#c"z "#e"z ")z #ez #gzzH ,
( f and g being complex coe$cients). The additional terms due to resonance act as forcing terms
with frequency 2u !(u #u ). They induce dramatic changes in the structure of the solutions.
Indeed, if the additional terms are `turned ona when the other parameters are close to an
heteroclinic solution, Melnikov like arguments [99] suggest that some of the periodic solutions can
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
23
Fig. 10. From Ref. [95]. (o , o ) projection of the solution of Eqs. (28) for the same parameters as in the caption of Fig. 9,
plus f "!0.01, gP"0.02, f G"0 and gG"0. The result is a chaotic alternation between modes O, C and SW.
P
Fig. 11. From Ref. [54]. Vortices separated by shocks. Numerically (a)}(c) and experimentally (d)}(e) obtained "eld
distributions: near "eld amplitude (a), phase (b), far "eld amplitude (c). (d) and (e) are experimental images of the near "eld
and far "eld plane. Experiments are carried out with a photorefractive oscillator.
disappear, some can bifurcate to solutions of di!erent periodicity, and even chaotic behavior can be
expected. Fig. 10 reports the integration of Eq. (28), and highlights the presence of a chaotic
solution getting close to the pure modes (CA). It is crucial to notice that both PA and CA are
structurally stable, in so far as they persist over wide ranges of parameter values.
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
In summary, by symmetry arguments, one can construct a model for PA and CA in a dynamical
system with a broken O(2) symmetry. Even though the model has been referred to a particular
system, a similar behavior can be expected in other systems with the same symmetry properties.
Recently, rather than limiting to a low dimensional dynamics described by a small set of
ordinary di!erential equations, a numerical solution of a Swift}Hohenberg equation has shown
qualitative agreement with experiments on a photorefractive oscillator [54]. Fig. 11 reports
numerical as well as experimental evidence of vortices separated by shocks for an aspect ratio
F"10.
3. Patterns in passive optical systems
3.1. Filamentation in single-pass systems
A laser beam propagating through a nonlinear optical medium, perhaps the simpler system one
can think of in nonlinear optics, can undergo spatial instabilities leading to spot formation. This
phenomenon has been already described in the early times of nonlinear optics development
[16,100,101].
Consider a medium the optical response of which is described by a Kerr nonlinearity. Its
polarization has a s term, P"sEEE. In the paraxial approximation the slowly varying
envelope A(z, q"t!z/v) of an optical wave propagating through the medium obeys the following
stationary equation
iv
RA
"k
A"a"A"A ,
,
Rz
(29)
where v is the light velocity in the medium, ,(R/Rx)#(R/Ry) denotes the transverse
,
laplacian, and a gives the sign and the strength of the nonlinearity. Eq. (29) admits the stationary
solution:
A (z, q)"A (q) exp (ia"A (q)"z/v).
Let us introduce a spatially dependent perturbation:
(30)
(31)
A(z, x , q)"(A (q)#f (z, q)eqx,) exp(ia"A (q)"z/v)
,
with x ,(x, y). By substituting this expression in the wave equation (29) we obtain, at "rst order
,
in f
iv
Rf
!kqf#a"A "f#"A "f H"0 .
Rz
(32)
The eigenvalues of the associated system of linear equation governing the evolution of f, f H along
z are
q
j "$i (kq!2aI k .
!
c
(33)
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
25
Fig. 12. Growth rate j of the transverse perturbations at spatial frequency q for the "lamentation in a focusing Kerr
O
medium (see Eq. (33)). Lower curve: aI "1; upper curve: aI "2.
Fig. 13. From Ref. [148]. Filamentation of a wide radially symmetric beam passing through a photorefractive SBN
crystal, for increasing voltage < applied to the crystal. (a) <"0, (b) <"600 V, (c) <"1200 V, (d) <"1500 V. The small
horizontal modulation of the beam observed in (a) is due to striations in the crystal.
The spatial growth of the perturbation f requires the existence of a real, positive j, which is possible
only for a'0, i.e. for focusing media. In this case, a whole band of wavenumbers q is unstable
(Fig. 12). The maximum growth rate occurs for
a
q " I
k (34)
this corresponds to a wave that at any value of z is exactly phase-matched with the medium
nonlinear polarization.
This instability is referred to as "lamentation because eventually light "laments of various
diameters, small compared to the initial beam size, propagate through the medium.
Recent investigations have been devoted to "lamentation in non-Kerr media [102}104,148], e.g.
in photorefractives and in liquid crystals. Fig. 13 shows the "lamentation of the light beam passing
a photorefractive crystal. Though the instability mechanism is still based on the self-focusing
properties of these materials, other phenomena (beam bending, longitudinal beam undulations)
related to the anisotropic nature of the materials considered are also observed in these cases
(see Fig. 13).
3.2. Solitons in single-pass systems
A phenomenon strictly related to "lamentation in self-focusing media is beam self-trapping, i.e.,
the propagation of light beams that conserve a limited transverse size by a balance between the
26
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 14. From Ref. [107]. Intensity and phase as functions of coordinate x for bright (a), dark (b) and gray (c) solitons.
tendency to spread due to di!raction, and the focusing properties of the nonlinear medium. Such
beams are usually called solitons, even though in many cases they do not satisfy all the properties
required by the mathematical de"nition of a soliton [105].
The possibility of a beam self-trapping has been pointed out already in the early days of
nonlinear optics [15,16,101,106]. Since then, soliton physics has been investigated extensively from
a theoretical as well as from an experimental point of view. The richness of results and the
rami"cations of this matter has rendered prohibitive a satisfactory coverage of this subject in the
present review. We will therefore limit to survey rapidly some of the most important ideas and
experimental results about solitons, addressing the interested reader to speci"c review articles fully
devoted to this topic [107].
For the very special case in which propagation in a Kerr medium is considered and the
transverse geometry is limited to one dimension x, it is possible to give an analytic form for the
electric "eld distribution. The evolution is described in this case by Eq. (29), the general solution of
which in the one-dimensional case and for a focusing medium is [105,107}111]
A eTV\T\X
,
(35)
A(x, z)" cosh A (x!vz)
where A is the soliton amplitude and v is its velocity (all variables are suitably rescaled to a non
dimensional form). For v"0 the fundamental bright soliton (Fig. 14a) is de"ned as
A eX
.
A(x, z)" cosh A x
In the defocusing case, the general localized solution reads [105,107,110}112]
(36)
(37)
A(x, z)"A (b tanh h#ia)eX ,
where h"A b(x!aA z). The parameters a and b satisfy a#b"1, hence a single parameter
u can be introduced such that a"sin u, b"cos u. The value A sin u has the meaning of soliton
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
27
velocity along x. For uO0, Eq. (37) represents a gray soliton shown in Fig. 14c. For u"0, the "eld
distribution (37) reduces to
(38)
A(x, z)"A tanh(A x)eX .
That is referred to as the fundamental dark soliton (Fig. 14b).
Linear stability analysis [107,108,112,113] shows that these solitons are stable in the (1#1)
geometry (one transverse coordinate plus one longitudinal coordinate). Observation of bright
solitons in this geometry has been reported in nonlinear liquids [114}117] and in planar waveguides [118], while dark solitons have been observed in photorefractives [119].
In the (2#1) dimensional case, both the solitons described by Eqs. (36) and (38) are unstable.
Instability of bright solitons is a generic feature in (2#1) dimensional Kerr media [120}122]. It
has been shown in general [113] that a soliton propagating in a (D#1) dimensional geometry
through a focusing medium described by a nonlinearity of the type *nJIN is stable for p(2/D.
Hence, in a Kerr medium, bright solitons are stable in (1#1), but unstable in (2#1) dimensions.
From a physical point of view, the dependence of the soliton stability upon the transverse
dimensionality D is due to the fact that, while spreading by di!raction acts in a way independent of
D, focusing in a Kerr medium depends on intensity, i.e. on the ratio P/gD, where P is the total beam
power, and g is its size.
Bright solitons in (2#1) dimensions can nevertheless exist, and have been described in many
circumstances. Saturation of the nonlinearity has been proposed [123,124] as a "rst mechanism
able to stabilize these beams. Early experimental observations of bright solitons in sodium vapours
[125] have con"rmed the validity of this suggestion. More recently, bright solitons have been
observed in photorefractive crystals both operating in a regime in which the nonlinearity is local
and saturable [126}133], and in a regime in which it is intensity independent but nonlocal
[134}136]. In this latter case, solitons exist only in the transient regime.
Finally, bright solitons have been shown to exist in media described by a s nonlinearity, both
in regime of second harmonic generation [137,138], and in regime of parametric ampli"cation
[139,367}374]. The limitation and stability derived for the Kerr medium do not apply in this case,
in which instead there is a mutual trapping between the three optical waves at di!erent frequencies
coupled via the s in the crystal.
A mutual guiding is also at the base of the existence of vectorial solitons, formed by a pair of
solitons that can be either one bright and one dark, or having di!erent polarizations [140}142].
As for the dark solitons in (2#1) geometry, the planar ones described by Eq. (38) are known to
be unstable with respect to long wavelength transverse modulations [143]. This mechanism,
resulting in a modulation of the initial dark stripe and successive creation of pairs of vortices of
opposite polarity [144,145], has been observed both in atomic vapours [146] and in photorefractives [147,148] (Fig. 15).
However, dark stripes can be stabilized by the medium di!usion, or be observed even if unstable,
when the transverse size of the beam is small compared to the typical scale of the instability. Several
observations of stable dark stripes have indeed been reported in absorbing liquids, [149,150] in
semiconductors [151,152] and in photorefractives [153].
Dark solitons with circular symmetry in (2#1) dimensions have been considered for long time
stable in defocusing media [121,122], even though recent results [107] show that some instabilities
may occur also in this case. A particularly interesting structure is the vortex soliton, consisting of
28
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 15. From Ref. [146]. Transverse instability of a dark stripe soliton in a Rb cell. The vapor concentration increases
from vanishingly small (a) to the order of 10 cm\ (f). The inset shows the intensity distribution along a line through the
center of the upper vortex in (f).
a "eld distribution characterized by a dip reaching the zero in its center, and a helical structure of
the phase distribution. These vortices appear to be unstable in focusing media, in which they break
resulting in a pair of bright solitons [154}156]. On the contrary, vortices have been shown to
constitute stable solitons in defocusing media [157,158]. Several experimental con"rmations of this
fact have been given, using as materials absorbing liquids [159], vapours [160,161] (see Fig. 16) or
photorefractive crystals [162,163].
3.3. Counterpropagating beams in a nonlinear medium
Let us consider a pair of optical beams of wavenumber k and amplitudes E , E counterN N
propagating inside a Kerr medium of length l (Fig. 17a). For low values of the wave intensities,
a stationary, plane standing wave is found in the medium. Perturbations to this situation are taken
into account by introducing the small amplitude probe beams E , E propagating at a small angle
h with respect to E , and E , E counterpropagating to E , E (Fig. 17b). In the limit of undepleted
N
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
29
Fig. 16. From Ref. [161]. Vortex soliton formation in a Rb cell. Experimental beam pro"les recorded at the input
window of the cell (a); and at the output window in the regime of linear propagation (b), and nonlinear propagation
regime (c); interference pattern at the output of the cell in the nonlinear propagation regime (d).
Fig. 17. Two pump beams E and E interact with a nonlinear Kerr medium of length ¸ (a). Spontaneous coherent
N
N
emission can appear in a direction making an angle h with respect to the pump beam axis (b).
pumps E , E , the stationary evolution of E is governed by [164}166]
N N
RE
"!im[(1#a)EYH#qEHeIFX#(1#a)E eIFX#qE ] ,
Rz
(39)
where m is proportional to E E (m is negative in the focusing case and positive for a defocusing
N N
medium), q""E "/"E ", a (04a41) weights the wavelength-scale refractive index gratings due
N N
to the counterpropagating "elds. In media in which the Kerr excitation is very mobile, e.g. by
di!usion, these short scale gratings are washed out, resulting in a"0.
Each term at the r.h.s. of Eq. (39) represent a four-wave mixing (4WM) contribution to the
evolution of E ; similar equations hold for E , E and E . The "rst term is associated with the
backward 4WM gain, i.e. with absorption of one photon in each pump and emission of photons in
beams E , E . The second term describes forward 4WM, due to the annihilation of two photons of
30
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 18. From Ref. [165]. Instability threshold curves "m"¸ vs. h. (a) focusing case, a"0; (b) defocusing case, a"0;
(c) focusing case, a"1; (d) defocusing case, a"1. In all plots the solid curves are associated to equal pump beams
("E """E "), the dashed curves to "E ""0.85"E ". The horizontal lines are the thresholds predicted by a model in which
N
N
N
N
also the backward 4WM is accounted for [169].
the beam E and emission of photons in the beams E , E . The third term describes the parametric
N
process in which one photon is absorbed in beams E , E and one photon is emitted in beams
N E , E . The last term describes nonlinear self modulation of E . Eq (39), together with the
N analogous equations for E , E and E , forms a set of four linear di!erential equations that must be
supplemented with the boundary conditions E (0)"E (0)"0, E (¸)"E (¸)"0. Solutions of
this boundary-value problem [164}168] gives rise to the marginal stability curves of Fig. 18.
Several important features are visible in these curves. The horizontal lines at "m" ¸"n/2 (a"0)
and "m" ¸"n/4 (a"1) correspond to the instability without any spatial structure predicted in
[169] for the case in which only the h-independent backward 4WM is retained. We notice that the
inclusion of h-dependent 4WM terms leads both to a selectivity in h, and to a lowering of the
instability threshold, with the exception of the defocusing case for a"0. At the physical origin of
these phenomena lies the fact that the inclusion of h-dependent 4WM processes results in an
increasing of net gain for some h component, with respect to the case in which these processes are
neglected. This gain enhancement is e!ective, however, only if the various 4WM processes are
phase-matched along the medium length ¸. This constraint leads to the selection of transverse wavenumbers q,kh" O(2k/¸) [164}168]. In the self focusing case, phase-matching is
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
31
Fig. 19. From Ref. [170]. Shape of one transmitted beam in the far "eld of a sodium cell pumped by counterpropagating
beams. The medium nonlinearity is of focusing type in these experimental conditions. The picture (a) corresponds
to a situation where the pump beams are well aligned at counterpropagation, while they make a small angle
(K5;10\ rad.) in picture (b).
achieved through a balance between di!raction and nonlinear self modulation of each probe wave.
This balance is not possible in a defocusing material, since the e!ects of these two mechanisms here
add instead of subtracting. For this reason, as shown in Fig. 18b, the h-dependent threshold is
higher than the h-dependent one for defocusing media in the case a"0. If aO0, however, cross
phase modulation among the counterpropagating waves can balance for the loss of e$ciency of the
o!-axis gain [166,167], so that again instabilities at a "nite value of h is predicted.
From an experimental point of view, instabilities of counterpropagating waves have been
observed in atomic vapors [170}172] and in photorefractive crystals [173}180]. In some circumstances, it has been theoretically discussed and experimentally demonstrated that instabilities
similar to the ones here discussed occur also in copropagating beam geometries [168,171,181].
Fig. 19 displays far-"eld hexagons observed in sodium vapors pumped with counterpropagating
beams of light at frequency u'u , u being the frequency of the D resonance line. In these
conditions the nonlinearity is of focusing type. The symmetry of the experimentally observed
patterns is often hexagonal, though also rolls, squares and other less generic structures have been
reported. For the Kerr medium hexagons have been theoretically and numerically shown to be
generic close to the instability threshold in the focusing case [182}184], while rolls or rhombs occur
in the focusing case.
We notice that neither atomic vapours nor photorefractives are adequately described by
a Kerr-like nonlinearity [164,185]; this does not seem to pose a serious limitation to the predictions
of the above reported model in comparison with the experiments. The reason lies in the fact that on
32
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
one side what is important is the ability of the material to support four-wave mixing interactions
[167,185}187]. On the other side, the geometrical constraint given by the longitudinal boundary
conditions results in a scale selection mechanism that is largely independent of the details of the
nonlinearity. It is a reasonable guess that the speci"c kind of nonlinear material in#uences more
strongly the pattern symmetry selection rules. This topic however has not been investigated in
detail.
Fig. 20 reports the patterns observed in the far and near "eld for counterpropagating beams in
a photorefractive SBN crystal to which an external electric "eld is applied. The anisotropic nature
of the photorefractive nonlinearity accounts for selection of rolls rather than hexagons for low
values of the applied "eld.
3.4. Nonlinear medium in an optical cavity
Optical resonators containing a passive nonlinear medium have been extensively investigated
earlier in the context of optical bistability [188], and more recently as systems capable to display
pattern formation. Two typical geometries for these systems, namely, a ring resonator partially
"lled with the medium and a Fabry-Perot resonator entirely "lled with the medium, are shown
in Fig. 21.
Let us consider the case in which the cavity is uniformly "lled with a Kerr medium, and driven by
a coherent external "eld E [189}194]. If only one longitudinal mode of the cavity is excited, it is
possible to neglect the z dependence of the "eld envelope E. This approximation is sometimes
referred to as the mean "eld limit [190]. In these conditions, the evolution of the cavity "eld is ruled
by:
RE
"!E#E #ig("E"!h)E#ia
E ,
,
Rq
(40)
where q represents a scaled time t/t , t being the mean cavity lifetime for the photons;
a,1/4nF, F being the Fresnel number of the cavity of transverse size b; h is the detuning between
"eld and cavity frequencies; g"$1 determines whether the nonlinearity is of focusing or
defocusing type.
Eq. (40) is known to give rise to optical bistability in the transverse uniform limit. The relation
between input and output optical intensities in this case reads
I "I (1#(h!I )) ,
(41)
where I ""E ", I ""E " in the steady state. It follows from Eq. (41) that the cavity is bistable for
h'(3, independently of the sign of the nonlinearity. By introducing the variable transformation
E"E (1#A)
(42)
Eq. (40) can be rewritten as [187,194]
RA
"![1#ig(h!I )]A#ia
A#igI (A#AH#A#2"A"#A"A") .
,
Rq
(43)
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
33
Fig. 20. From Ref. [180]. Far "eld (left) and near "eld (right) patterns observed in a photorefractive SBN crystal pumped
by counterpropagating beams, for increasing values of the voltage applied to the crystal (reported in volts in the upper
corner of each "gure).
Notice that no linearity approximation has been used in passing from Eqs. (41) to (43), hence the
two equations are fully equivalent. Eq. (43) shows that a perturbation (not necessarily small) to the
uniform steady solution can undergo gain via the four-wave mixing (4WM) term igI AH. This
mixing describes annihilation of two photons of the steady uniform solution, and creation of one
34
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 21. Passive optical cavities with nonlinear medium and injected signal: (a) ring con"guration; (b) Fabry-Perot
con"guration.
photon in mode A and one in mode AH. Since these modes can be spatially dependent in the
transverse plane, the 4WM term can lead to the destabilization of the uniform solution and
eventually to the formation of transverse structures.
Linear stability analysis of the system formed by Eq. (43) and its complex conjugate is performed
by introducing the ansatz
(A, AH)JeHReqr .
(44)
The boundary between stable and unstable perturbations is given by the curve j"0 in
the hyperplane (I , aq, h, g). By "xing the values of h and g, we obtain marginal stability curves
in the (aq, I ) plane of the kind of the ones shown in Fig. 22 [194]. Let us consider in more
detail the focusing case g"#1. The minimum of the marginal stability curve for h "xed has
coordinates
I "1,
aq"2I !h"2!h .
(45)
Eq. (45) determines the most unstable transverse mode, i.e. the one that will "rst bifurcate when
increasing the input pump intensity I . The selection of the critical wavenumber q arises from
a balance between the di!ractive phase modulation aq, the nonlinear phase modulation 2I , and
the cavity detuning h. Exact compensation among these three e!ects, expressed by Eq. (45), results
in a perfect phase-matching of the four-wave mixing interaction that enhances the perturbation,
and hence in an optimum gain. Notice that physical values of q are positive, so that Eq. (18) is
meaningful only for h(2.
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
35
Fig. 22. From Ref. [194]. Domains in the (aq, I ) plane in which the transversally homogeneous stationary solution of
Eq. (43) is unstable to the growth of inhomogeneous perturbations. (a) focusing case, h"1 (right curve), h"5 (left curve);
(b) defocusing case, h"5.
The bistability boundary for the uniform solution, h"(3, correspond to tangency of the
marginal stability curve to the vertical axis. For h((3, the steady input-output characteristic of
the cavity is monostable, and an instability with respect to a "nite transverse q whose critical
value is given by Eq. (45) exists. Fig. 23a shows the regions of stability of the transverse
homogeneous solution. For (3(h(2 the system is bistable in the transverse homogeneous case,
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 23. From Ref. [194]. Transverse homogeneous stationary solutions of Eq. (41). (a): focusing case, h"1; (b) focusing
case, h"5; (c) defocusing case, h"5. The dotted portions of the curves are unstable with respect to homogeneous or
modulated perturbations.
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
37
Fig. 24. From Ref. [195]. Examples of the change in the patterns numerically observed in a model of a cavity "lled with
a purely absorptive medium, for increasing input intensity from (a) to (c).
but instabilities with respect to a "nite wavenumber exist for both the lower and the upper stability
branches. For h'2 the bistable uniform solution is stable with respect to any q in its lower branch,
but unstable with respect to "nite transverse wavenumbers in its upper branch (Fig. 23b).
In the self defocusing case, the marginal stability curves are as the one plotted in Fig. 22b. The
coordinates of the minimum are
I "1,
aq"h!2I "h!2 .
(46)
A physically meaningful minimum of the curve exists only for h'2, i.e. fully in the region of
bistability de"ned by h'(3. Fig. 23c shows the stability of the uniform solution in this case. Since
the upper branch is now stable with respect to any perturbation, it is to be expected that
perturbations in the instability region of the lower branch lead the system to switch to the upper
uniform solution. Hence, no pattern forming instabilities occur in the defocusing case.
The model here discussed, in which the nonlinearity is of Kerr type, is also valid for a cavity "lled
with two-level atoms in the limit of large detuning between input "eld and atomic line and fast
atomic relaxation times [190]. For this model, both nonlinear stability analysis [193,194] and
numerical simulations predict the bifurcation of subcritical hexagons close to threshold. This is
a general feature related to the existence of quadratic nonlinearities in Eq. (43). Other mean "eld
models have considered the situations in which either the medium nonlinearity is purely absorptive
[195] (see Fig. 24), or polarization e!ects are taken into account. In these cases, the occurrence of
rolls and negative hexagons has been predicted. It has also been shown that polarization e!ects
lead to the occurrence of transverse instabilities for a defocusing Kerr nonlinearity, at variance with
the scalar model.
In mean "eld models, considering a cavity "lled with two-level atoms instead of a Kerr medium,
temporal and spatiotemporal instabilities have been predicted as well as the purely spatial ones
here discussed. Similar results have been found when the mean "eld limit approximation is not
adopted, though the material response is assumed to be fast with respect to the "eld dynamics, and
the system evolution is described by an in"nite-dimensional map [196}200]. This last case was
actually historically the "rst in which spatial instabilities in nonlinear interferometers were
described.
In the region of nascent bistability for the homogeneous wave transmitted through the cavity,
the amplitude equation for the optical "eld is a real Swift}Hohenberg equation [201,202].
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
In a model of ring cavity partially "lled with two level atoms, relaxation of the mean "eld
approximation as well as the assumption of instantaneous material response has led to the
prediction of multiconical emission [203}205], i.e. the instability of many di!erent transverse
wavenumbers having approximately the same threshold value.
Despite the relevant amount of theoretical and numerical studies about nonlinear passive
cavities, the experimental results in this "eld are not abundant. Experiments involving the use of
Fabry-Perot resonators "lled with sodium vapors have demonstrated the occurrence of both time
oscillations and spatial pattern formation in the output beam [206}211]. Though, a direct
comparison between the experimental results and the theoretical predictions is di$cult. Among the
reasons of these di$culties are the low values of the atomic nonlinearities, that imposes the use of
a narrow laser beam and consequently a low Fresnel number system, the fast response of the
nonlinearity, that introduces serious di$culties for an experimental detection resolved both in
space and time, the mismatch between the two-level description of the atoms and their real
behaviour [212], the motion of atoms, resulting in di!usion processes.
3.5. Nonlinear slice with optical feedback
In the previous paragraphs we dealt with optical systems in which medium nonlinearity and
wave propagation take place in the same physical location. Here we report investigations about
setups in which the action of these two mechanisms occurs at distinct regions in space. As
a consequence of this spatial separation, an easier identi"cation of the speci"c role played by
nonlinearity and propagation on pattern formation is possible.
Let us consider the system shown in Fig. 25 [213}216], formed by a thin slice of Kerr material
and a mirror. A plane wave of intensity I ""E " is sent through the material, propagates up to the
mirror and is re#ected back onto the Kerr slice, which induces a phase retardation on the plane
wave, given by u(r , t)"aI (r , t), where I is the total intensity impinging on the material, a gives
,
R ,
the sign and strength of the Kerr nonlinearity, and r denotes the coordinates transverse to wave
,
propagation. If we neglect the small scale interference gratings due to the superposition of
the forward and backward waved, which is a reasonable assumption if the di!usion length of the
Fig. 25. Kerr medium with a purely di!ractive feedback.
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
39
material is much larger than the optical wavelength, I is given by the sum of I with the intensity
I of the backward feedback "eld.
Since we are interested in pattern formation, i.e. in instabilities involving "nite wavenumbers, the
role of the spatially uniform intensity I in determining the evolution of the phase u can be
neglected, because I just give rise to a uniform additive variation of u. The evolution of the phase
retardation induced by the medium on the incoming plane wave is then ruled by [214}216]
u(r , t)
Ru(r , t)
, "! , #D
u(r , t)#aI (r , t) ,
, ,
,
q
Rt
(47)
where q and D are, respectively, the local relaxation time and di!usion constant of the medium. In
the simple geometry here considered, the feedback "eld distribution is due to di!ractive propagation of the "eld E(r , z"0)"E ePP, exiting the Kerr slice due to the propagation of the incoming
,
wave through the slice. In the usual paraxial approximation, E(r , z) evolves following
,
i
RE((r , z, t)
,
" E(r , z, t) ,
(48)
2k , ,
Rz
where k ,2n/j is the optical wavenumber. The formal solution of this equation is
(49)
E(r , z, t)"e
I,XE(r , z, t)
,
,
substituting this expression into Eq. (47) and calling ¸ the propagation length, the equation
governing the phase evolution becomes
u(r )
Ru(r )
, "! , #D
u(r )#aI "e
I,BePr," .
, ,
q
Rt
(50)
For small phase perturbations u;1, Eq. (50) can be linearized, giving
Ru
u
O"! O#aI "e\OI* (1#iu )" ,
O
Rt
q
(51)
where the evolution of a single spatial Fourier component at spatial frequency q has been singled
out. Eq. (51) highlights the speci"c role of di!raction and nonlinearity in destabilizing the
plane wave solution u(r )"0. Indeed if di!raction is absent (¸"0), the feedback term in Eq. (51)
,
reduces to
I "I "1#iu "KI
(52)
O
at "rst order in u . Thus no feedback is e!ective at this order, and the plane wave solution is stable.
O
Indeed, in such a case, the feedback beam is phase but not intensity modulated. This is ine!ective
on the phase evolution, since the Kerr nonlinearity responds to intensity. When free propagation
over a length ¸ is introduced, the Fourier component at frequency q is dephased by a factor
e\IOB with respect to the continuous component at q"0, represented by the factor `1a in Eq. (52).
The feedback intensity reads now
I "I "1#iu e\OI*" .
O
(53)
40
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
For q¸/2k "(n/2)#2Kn or q¸/2k "n#2Kn, K integer, the feedback intensity is respec
tively I "I (1#2u ), I "I (1!2u ) at "rst order in u . The phase modulation has been in
O O
O
this case completely converted into intensity modulation by the di!ractive propagation, hence an
e$cient feedback action on the phase via the Kerr nonlinearity is possible.
For generic values of the parameters, the linear growth rate j for a mode at spatial frequency
O
q is
q¸
1
j "! !Dq#2aI sin
O
2k
q
the curve j "0 gives the marginal stability in the (q, aI ) plane [216]:
O
1#lq
qaI "
2sin(q¸/2k )
(54)
(55)
where the material di!usion length l ,(Dq has been introduced.
In Fig. 26 we report the marginal stability curves for two di!erent values of l . The upper and
lower half planes correspond respectively to focusing and defocusing nonlinearity (a'0, a(0).
The extrema of each "nger-like branch correspond to q¸/2k K(n/2)#2Kn, q¸/2k K
n#2Kn for the focusing and defocusing case respectively in the limit l<j¸. The extrema
corresponding to successive values of the integer K lie, in the same limit, on the curve qaI "
(1#lq)/2. Di!usion is seen to lift the degeneracy of the threshold condition for the successive
bands, leading to lower threshold values for the ones occurring at low spatial frequencies.
The predictions of this simple model have been experimentally veri"ed in systems using various
kinds of nonlinear materials, among which liquid crystals are the one that "t better the Kerr-like
description of the nonlinearity. Liquid crystals have been used both in simple layers [217}223], and
in a hybrid electrooptical device named Liquid Crystal Light Valve (LCLV) [214,224}251]. This
device is still adequately described by an optical Kerr model for a broad range of experimental
parameters, and by the use of an auxiliary electric "eld it provides nonlinear susceptibilities much
larger than a simple liquid crystal layer, thus allowing low intensity operation and the consequent
possibility of investigating large Fresnel number systems.
The typical patterns observed close to the instability threshold are hexagonal (Fig. 27)
[217,220,223,234,245}248], due to the superposition of three Fourier modes oriented at 1203 one
with respect to the other. These hexagons bifurcate subcritically [241], in agreement with the
predictions of a nonlinear stability analysis. Simple roll solutions appear instead to be always
unstable [252]. The scaling of the selected wavenumber q with the free propagation length ¸ has
also been veri"ed [217,234]. Competition and cooperative phenomena among patterns with
closeby threshold values have been predicted. In particular, it has been shown how spatial mode
locking among wavevectors belonging to di!erent bands in Fourier space can result in singledomain, multiscale patterns of the kind shown in Fig. 28. The transverse boundary conditions have
been shown to play an important role on symmetry selection when the system size is of the order of
few unstable wavelength [218,222,231,236,253]. In this case, polygons di!erent from hexagons are
observed. For values of the input intensity far above the instability threshold, broadening of the
excited spectral bands leads to appearance and motion of defects in the hexagonal structure, and
"nally to a space}time chaotic situation (Fig. 27) [216,223,248].
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
41
Fig. 26. Marginal stability curves for the Kerr medium with purely di!ractive feedback. The upper half plane refers to
a focusing medium, the lower one to a defocusing one. The curves are obtained for ¸"30 cm, j,2n/k "514 nm.
l "20 lm (solid line), l "50 lm (dashed line).
Fig. 27. From Ref. [248]. Large aspect ratio structures observed in a LCLV with di!ractive feedback. Top: hexagonal
structures, bottom: space}time chaotic state observed for higher values of pumping intensity.
The use of photorefractive crystals instead of liquid crystals or LCLVs as nonlinear media has
led to the observation of both patterns with hexagonal symmetry [178,254], as well as squeezes
hexagons and squares [255]. The reason for this symmetry breaking seems to be due to the
anisotropic nonlinear response of photorefractives. Investigations of photorefractives [254] and
organic materials [256] have clari"ed that the instability mechanism here discussed is not limited
to the case in which the medium nonlinearity is purely dispersive, but basically holds also for
absorptive or absorptive-dispersive nonlinearities.
Other studies have been devoted to the use of atomic vapors as the nonlinear medium
[213,257}269]. The nonlinearity in this case can be either focusing or defocusing, depending on the
experimental conditions, and is poorly described by the Kerr model [261,265]. Furthermore,
important polarization e!ects come into play when atomic vapors are used [260,261,265,269]. In
particular, transitions between stable square or rolls and stable hexagons have been predicted
[260] and experimentally observed [269] when varying the polarization of the light incident on
a cell "lled with sodium vapors.
In other experimental conditions, for "xed input polarization and varying pump intensity,
a variety of di!erent structures have been reported [261,264}266,269], including polygons, positive
42
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 28. From Ref. [237]. Near "eld (left) and far "eld (right) images of multiscale frequency locked patterns in a LCLV
with di!ractive feedback close to threshold.
and negative hexagons, turbulent states and multipetal patterns in#uenced by the symmetry and
size of the input beam. Most of these structures have been reproduced in simulations using suitable
models for the sodium nonlinear response [259,260,265,266] (see Fig. 29). In the case of atomic
vapors, the occurrence of simultaneous instability of several Fourier bands [257], as well as the
instability of the basic hexagonal patterns with respect to its own harmonics or subharmonics
spatial frequencies [268] have been predicted.
The mechanism governing the instabilities here discussed can be conceptually split in two steps.
First, a conversion of phase #uctuations into intensity #uctuations is needed. Second, intensity
#uctuations must give rise to phase #uctuations, in order to close the feedback loop. Up to now we
considered the situation in which the "rst process take place via optical free propagation, however,
this is not the only possibility. An easy way to convert phases into amplitudes is by means of an
interferometric process. Let us consider the setup shown in Fig. 30. The Kerr medium is operating
only in re#ection (in practice, this is actually the case if a LCLV is used). The light fed back to the
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
43
Fig. 29. From Ref. [265]. Experimental (top) and corresponding simulated (bottom) patterns observed in a laser beam
transmitted through a Na cell for di!erent experimental parameters.
Fig. 30. Setup for a Kerr medium operating in re#ection with di!ractive and interferential feedback. The linear operator
M can account for di!raction, imaging, or nonlocal operators.
medium is a superposition of the wave re#ected by the medium itself, and a reference wave re#ected
by mirror A. A generic linear operator M is included in the loop.
Historically, a similar device was introduced with the aim of correcting the phase distorsions of
the input beam, and a very rich fenomenology of pattern forming instabilities was early reported
using this system [214,224}227]. If the arm containing the mirror A is blocked and the operator
M represents simply di!raction, this setup is fully equivalent to the basic one of Fig. 25. Another
limit case is that in which the arm containing A is unblocked, and M represents a one to one
imaging of the front face of the medium onto its rear face. The evolution of the phase #uctuations
u(r , t) follows
,
u(r , t)
Ru(r , t)
, "! , #D
u(r , t)#aI [1#m cos(u(r , t)!u )] ,
(56)
, ,
,
q
Rt
44
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 31. (a) Graphical solution of Eq. (57), showing the onset of optical bistability in the Kerr medium with interferential
feedback. (b) A typical uniform steady state characteristic u vs. I for the same problem.
where m is a modulation factor depending on the medium and mirror re#ectivities, and u is the
phase di!erence between the in-axis waves travelling in the two arms of the interferometer. Eq. (56)
admits the uniform stationary solution
u/aqI "1#m cos(u!u ) .
(57)
This displays bistability (or multistability) for high enough values of I (Fig. 31). A linear stability
analysis of this solution shows that the mode with maximum temporal growth rate is the one at
zero spatial frequency. This is to be expected, since di!usion is the only space-dependent phenomenon that comes into play. Hence, as I is varied, jumps between di!erent uniform stable solutions
of Eq. (57) are expected, but no pattern formation takes place.
A completely di!erent situation occurs if both interference and di!raction are at work
[227,228,232,251]. The evolution equation of the phase #uctuations in this case is [251]
u(r )
Ru(r )
, "! , #D
u(r )#aI "e
I,*(A#BePr,\P" ,
, ,
q
Rt
(58)
where A and B are the normalized amplitudes of the waves travelling the two arms of the
interferometer. Fig. 32 shows the marginal stability curves on the (q, I ) plane in the focusing case
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
45
Fig. 32. Marginal stability curves for the Kerr medium with di!ractive plus interferential feedback (see Eq. (58)), for
increasing values of the modulation parameter m,2AB/(A#B). (a) m"0, (b) m"0.42, (c) m"0.68.
(a'0), for di!erent values of the modulation parameter m,2AB/(A#B). It can be seen that for
m"0 the pure di!ractive behavior is recovered (Fig. 32a). For increasing values of m, the curves
are progressively distorted, till touching the q"0 axis. In these conditions the instability at a "nite
q is accompanied by instability of the uniform background state with respect to uniform perturbations. Bistability of the uniform solutions and bifurcations at "nite q thus occur together, resulting
in a complicate dynamical behaviour about which is not captured by the linear stability analysis. In
Fig. 33 we show some of the spatial structures that have been numerically predicted for this regime
[229,215]. From an experimental point of view, though a rich variety of results have been reported
in pioneering experiments [214,224}227], systematic observations in the regime in which both
di!raction and interference are present are rather recent [232,250]. Among the main results of these
studies there is the observation of localized structures [250].
3.6. Nonlocal interactions
The introduction of nonlocal interactions in a system formed by a nonlinear material slice with
optical feedback leads to qualitatively di!erent scenarios of pattern forming instabilities as
compared to the ones above discussed.
Nonlocality is introduced via rotation [213,225}230,234,235,237,238,240}243,270], translation
[219,226,244}246,262,263] or magni"cation [213,226] in the optical feedback loop. In some cases,
the e!ects of a global feedback have also been considered [239].
46
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 33. From Ref. [229]. Examples of patterns numerically predicted for a Kerr medium with di!ractive plus
interferential feedback, for various values of free propagation length, and relative phase between the beams coming from
the two interferometer arms.
In the system with purely interferential feedback the discrete transport due to rotation or
translation can destabilize the uniform steady states both in the monostable and in the bistable
regimes. This occurs if the local feedback of these states, given by the term aI [1#m cos(
!
)]
in Eq. (56) is negative and strong enough. Suppose that, in these circumstances, a translation of an
amount *x is introduced in the feedback loop. Perturbations of spatial period 2*x can be
destabilized since the feedback on these waves is now positive, resulting from the spatial shift by
n of the original negative feedback in the absence of translation. This instability mechanism is the
spatial counterpart of the well known temporal oscillatory behavior in an electronic ampli"er with
delayed negative feedback.
The same mechanism operates if rotation rather than translation is introduced in the feedback
loop, resulting in the bifurcation of spatial Fourier bands limited only by the material di!usion.
The occurrence of this kind of instabilities have been experimentally veri"ed in systems based on
LCLV [226,230,240,241]. For increasing pump parameters, secondary instabilities and the "nal
occurrence of space}time chaotic regimes [230,240,241] (Fig. 34) have been reported. The introduction of nonlocal interactions has important consequences also in systems with purely di!ractive
feedback. In this case an e$cient conversion of phase into amplitude #uctuations via di!raction
over a length ¸ is required in order to have a strong feedback, leading to the scale selection rules
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
47
Fig. 34. From Ref. [241]. Patterns observed in a LLV with purely interferential feedback, and a rotation of L in the
feedback loop. The control parameter is the voltage < applied to the LCLV, that controls the sensitivity of this Kerr like
device. < increases from (a) to (j).
Fig. 35. From Ref. [242]. Marginal stability curves for the LCLV with purely di!ractive feedback and a rotation of
D,2n/N in the feedback loop, evaluated for ¸"75 cm and l "15.5 lm, j"633 nm.
q(¸/2k )K(n/2) or q(¸/2k )K3n/2 for focusing and defocusing materials respectively. It is not
expected that these rules are drastically changed by the discrete transport introduced, though it is
possible to convert an e$cient negative feedback into an e$cient positive one via a spatial shift or
rotation, just the same way we discussed for systems with interferential feedback. For these reasons,
in presence of transport, a defocusing medium can be unstable with respect to spatial frequencies
that are proper of a focusing medium in conditions of local feedback, and vice versa
[234,238,242,244,245]. In Fig. 35 we plot the marginal stability curves for a defocusing medium in
a di!ractive feedback loop, when a rotation of an angle D"2n/N, N integer is introduced. The
bistability band at q(¸/2k )K3n/2 (band II) is the one that exists also for local feedback. The one
at q(¸/2k )Kn/2 (band I), that would exist only for a focusing medium if the feedback was local,
48
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 36. From Ref. [242]. Crystalline and quasi crystalline patterns observed in the LCLV with purely di!ractive
feedback and rotation of D,2n/N in the feedback loop. Left: near "eld, right: far "eld. The integer N is reported close to
each picture.
for N even has always a minimum lower than the band II. For N odd, still band I exists, but its
instability threshold is lower than that of band II only for rather high values of N.
The in#uence of the rotation on the selection of the symmetry of the patterns is even stronger.
Instead of the hexagons formed in the case of local feedback, it is now possible to observe structures
displaying crystalline or quasicrystalline symmetries [242,270] close to the instability threshold
(Fig. 36). Competition between band I and II has been studied for "xed N"7, by increasing the
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
49
Fig. 37. From Ref. [243]. Transition from patterns on band II to patterns on band I, observed for increasing the input
intensity (from (a) to (c)) in a LCLV with di!ractive feedback rotated of D"2n/7. (a)}(c): near "eld, (d) and (e): far "eld.
Notice the coexistence of patterns at di!erent scales in di!erent spatial domains in (b).
input pump intensity I [243]. For low values of I , only band II bifurcates, and quasicrystalline
patterns are observed (Fig. 37a). High values of I lead to saturation of the gain for band II and
domination of patterns belonging to band I (Fig. 37b). At intermediate values of I , irregular time
alternation between structures belonging to the two bands occur. These features can be inferred
from Fig. 38, showing the time series of the quantity
g(t)"S (t)/[S (t)#S (t)] ,
(59)
where S (t) denotes the total power in band j at the time t. In this time alternation regime,
H
coexistence of patterns belonging to the two bands occur via a spatial segregation of scales
resulting in formation of domains, in each one of which only patterns at one scale is present
(Fig. 37b).
If the nonlocality is of translational type, a "rst e!ect expected is the occurrence of drifting
instabilities [244}246,262,263,271]. Consider for simplicity a 1-dimensional system in which
a discrete transport *x is introduced in the feedback loop. If patterns exists at a spatial frequency q,
a phase shift "q*x arise between the material excitation and the optical intensity fed back to
V
the medium [263]. To compensate this phase shift, a drift of the pattern is expected, leading to
a wave varying as cos(qx!ut) instead of cos qx, since in the time dependent case the medium
steady state excitation exhibits a phase shift K
t, i.e. uKq*x/q. The occurrence of drifting
R
instabilities and the agreement of the drift frequency with the prediction of the above heuristics
50
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 38. From Ref. [243]. Temporal evolution of the normalized power spectrum g(t) (see Eq. (59)) for the LCLV with
di!ractive feedback rotated of an angle D"2n/7, for di!erent values of the above threshold parameter e. Lower curve:
e"1, middle curve: e"2.1, and upper curve: e"4.1.
Fig. 39. From Ref. [245]. Near "eld (above) and far "eld (below) patterns observed in the LCLV with purely di!ractive
feedback translated by an amount *x. The input intensity I and free propagation length ¸ are kept "xed
(I "72 lW/cm, ¸"26 cm). (a,a): *x"0; (b,b): *x"50 lm; (c,c): *x"180 lm; (d,d): *x"220 lm; (e,e):
*x"400 lm.
have been veri"ed in experiments based both on atomic vapours [263], and on the LC [219] or
LCLVs [244,245].
As in the case of rotation, translation induces important di!erences in the pattern selection rules
with respect to the case of local feedback. In particular, series of transitions among di!erent
structures for increasing transport length *x have been reported both for systems using focusing or
defocusing materials [245,246]. In Fig. 39 we report the near and far "eld patterns observed in
a LCLV (defocusing medium) in which the e!ective feedback is translated of an increasing amount
of *x. All these structures are fairly accounted for by the linear stability analysis. There is a striking
similarity between some of these patterns and similar ones observed in hydrodynamical systems
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
51
subjected either to an increase of the `pump parametera, or to an inclination of the gravity "eld.
Clearly transport processes play a central role in all these phenomena, though a real understanding
of the similarities and di!erences among pattern forming instabilities in these di!erent systems has
not yet been achieved.
4. Defects and phase singularities in optics
4.1. Phase singularities and topological defects in linear waves
A topological defect (or phase singularity) is a point of the space where the circulation of the
phase gradient around any closed path surrounding it is equal to $2mn. The integer m is called
topological charge. For the wave equation it has been demonstrated that only $1 charges are
stable [272]. Let us recall the Berry's de"nition of a defect [272]:
`Singularities, when considered in the modern way as geometric rather than algebraic structures
are morphologies, that is form rather than matter; and waves are morphologies too (it is not matter,
but form that moves with a wave). Therefore singularities of waves represent a double abstraction
} forms of forms, as it were } and so it comes as something of a surprise to learn that they represent
observable phenomena in a very direct waya.
The topological defects represent one of the most important features of wavefronts, insofar as
they correspond to singularities of the phase function U(r, t). The nature of these singularities is
determined by the fact that the "eld E is a smooth single valued function of r and t. Single
valuedness implies that U may change by 2mn along a circuit C in space}time. When m is not zero
and C is shrunk to a very small loop, then C encloses a singularity, because U is varying in"nitely
fast. The smoothness of E now entails "E""0. The vanishing of "E" requires to satisfy simultaneously two conditions (RE"IE"0). As a consequence, phase singularities are lines in space or
points in the plane.
Consider a random complex "eld E(r) de"ned in the r"(x, y) plane. If it arises from the
interference of a large number of independent components, RE and IE are two independent
random functions with Gaussian statistics.
The zeroes of the functions RE(x, y) and IE(x, y) determine a number of curves in the (x, y) plane,
and the intersections of curves of one family with those of the other give a set of points where
"E(x, y)""0. It is relevant to address the problem of the propagation of such zeroes along the
direction z. According to the wave equation, these points are converted into lines. Those last ones,
in general, do not intersect in three dimensional space. Moreover, a given line cannot appear singly
at same plane z"const, nor it can disappear singly. Topological defects must appear or be
annihilated in pairs.
All topological dimensional arguments do not depend, however, on the nature of the interfering
"elds.
The unbalance N !N (di!erence between the numbers of zeroes with positive and negative
>
\
charges) is conserved along the propagation. On the average, in a cross section of a random "eld
(called speckle "eld) N "N because the beam is statistically homogeneous.
>
\
52
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Refs. [273,375] derives the expression of the total number of zeroes (N"N #N ) in an area
>
\
S of the transverse (x, y) plane:
NS"
dx dyd(E (x, y))d(E (x, y)) " R(E , E )/R(x, y)"
.
(60)
Here E"E #iE . The angle brackets denote an ensemble average process over a set of random
"elds. Each defect of the "eld gives a contribution unity to the right hand side of Eq. (60). Moreover,
positive and negative defects correspond to positive and negative signs of the Jacobian
G"R(E , E )/R(x, y). Eq. (60) can be rewritten as (denoting the gradients as RE /Rx"E , etc.)
V
1
N" dx dy dE dE dE dE dE dE = (E , E , E , E , E , E )d(E )d(E )"G"
V V W W V W V W
S
" dE 2dE = (0, 0, E , E , E , E )"G"
V
W V W V W
(61)
(= being the joint probability of the quantities E , E and their gradients at the given point). The
Gaussian assumption makes it possible to factor out = in terms of correlations of the complex
"eld. Precisely, the Van Cittert}Zernicke theorem tells us that the correlation is
1EH(r )E(r )2"I j(h)eIFP\P dh ,
(62)
where j(h) is the normalized angular spectrum and h"(h , h ). Furthermore
V W
RE(r )
EH(r )
"ik1h 2"ik j(h)h dh .
(63)
Rx
G
G
PP
G
1h 2 and 1h 20 can be made equal to zero by a suitable rotation of the z-axis. This corresponds to
V
W
choosing a z axis in the direction of the center of gravity of the angular distribution. This way, the
complex gradients RE/Rx and RE/Ry are independent of the "eld E(r) itself at each point (x, y).
The correlation matrix
1 REH(r ) RE(r )
1h h 2"
" j(h)h h dh
(64)
G I
G I
Rx
Rx
k
G
I PP
can be transformed to principal axes by a rotation of axes in the x, y plane. All three complex
quantities E, RE/Rx and RE/Ry are mutually independent at the given point. In this framework, with
1h2"0 and with 1h h 2 diagonal, = becomes:
G I
= (E , E , E , E , E , E )
V W V W
1
(E #E )
(E #E )
V exp ! W
W exp[!(E#E)/I]
"
exp ! V
(65)
nk1h21h2I
k1h2I
k1h2I
V
W
V
W
and Eq. (61) gives
k
N" (1h21h2) .
W
2n V
(66)
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
53
Now, the radius of the correlation of the speckle "eld (the transverse speckle size), is proportional
to j/*h (*h being the angular divergence of the beam). Therefore Eq. (66) shows that the density of
dislocations coincides with the number of speckles per unit area. This was veri"ed in a series of
experiments [273,375] where the "eld was produced by transmitting a laser beam through
a distorting phase plate. The structure of the speckle "eld wavefront was investigated by interference with a plane reference wave directed at a certain angle. The fringe separation identi"es the
tilting angle between speckle and reference "elds. Bending of fringes corresponds to the curvature
of the wavefront, while termination or birth of a fringe is a signature of a negative or positive phase
singularity respectively. From Eq. (66), N scales like the square a of the diaphragm diameter on the
phase plate, that is linearly in the Fresnel number F (see Eq. (19) for de"nition). The total number of
dislocations NS,Na scales like F, since S is proportional to F.
4.2. Phase singularities in nonlinear waves
In nonlinear physics, the dynamical role of a topological defect emerges by mediating the
transition between two di!erent types of symmetry.
More generally, one can state that the appearance of a defect marks a symmetry breaking. When
the defect is a space structure (as grain boundaries or point defects in crystals), it is called
`structural defecta. In wave patterns, defects appear in space}time, and they are called `topological defecta [274]. An important point here is that not all phase singularities should be identi"ed
by defects, but only those ones localized at the edge between two patterns with di!erent symmetries.
Defects and their role in mediating turbulence have been widely investigated in hydrodynamic
systems with large aspect ratios as #uid thermal convection [275,276,376], in nematic liquid
crystals [277,377,378], in surface waves [278], and in analytic treatments and numerical simulations [279,379] of partial di!erential equations in 2#1 space}time dimensions. The presence and
the role of defects in nonlinear optics has been discussed theoretically in Refs. [28].
The nucleation and the evolution of phase dislocations in a laser beam interacting with
a photorefractive medium has also been studied in Refs. [280}282].
The appearance of defects has been largely investigated, both theoretically and experimentally, in
many other optical systems, such as a self-defocusing Kerr nonlinear medium [283], a nonlinear
Fabry-Perot resonator [284], a class A laser [43], a saturable self-focusing medium [285]. In this
last case, three dimensional bright spatial solitons have been observed as a result of a modulation
instability of an optical defect [286].
The evolution of the optical defects and their propagation in a three dimensional self-defocusing
medium has been studied in Ref. [287]. Motion and propagation of defects have been investigated
in various experimental conditions [288,289].
The transformation of the topological charge during free-space propagation of light beams
carrying phase singularities, and the conservation of the associated total angular momentum
during the propagation have been studied in Ref. [290].
The possibility for a phase singularity to produce particle trapping, due to the absorption of the
angular momentum has been shown in Refs. [291,292].
Finally, the motion of optical phase singularities has been theoretically and experimentally
compared to that of a #uidodynamical system [293,294].
54
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 40. From Ref. [295]. (a) Single optical vortex in a doughnut mode, with a spiral wave around, which, in the course
of the time changes its sense of rotation (left, clockwise; right, anticlockwise). (b) Reconstruction of the instantaneous
phase portrait for an optical vortex, perspective (left) and equi-phase (right) plots. The phase has been experimentally
reconstructed via the algorithm of Ref. [296].
We here summarize the "rst experimental evidence of the existence of defects in nonlinear optics,
provided in Ref. [295], with an experimental setup similar to that discussed in Section 2.3 [88].
Since a defect implies a phase singularity, in order to detect it experimentally, one needs to
perform a phase measurement. In the case of an optical "eld, a phase measurement can be provided
by heterodyning against an external reference, that is by beating the signal with a reference beam
onto a CCD videocamera. Ref. [296] suggests a suitable algorithm, by whom the instantaneous
surfaces of phase can be reconstructed.
Fig. 40 shows that the phase surface of a doughnut mode is a helix of pitch 2n around the core
(vortex).
When many vortices are present, in order to count each vortex, a tilted reference beam is sent to
the CCD, so that the video signal is now given by
I(x, y)"A#B#2AB cos(Kx#U(x, y)) ,
(67)
where A and B are the amplitude of reference and signal "eld, K the fringe frequency due to tilting,
x the coordinate normal to the fringes and U the local phase. A phase singularity appears as
a dislocation in the system of fringes, so that the topological charge can be visually evaluated.
This way, one can measure the mean number of defect, their mean distance, and the mean value
of the unbalance (di!erence between positively and negatively charged defects) as a function of
some extensive parameter (in Ref. [295] it was the Fresnel number).
In Ref. [94], the defect statistic was measured in order to mark a transition between patterns ruled by the boundary constraints and patterns ruled by the bulk properties of the active
medium.
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
55
In a system far from equilibrium, indeed, two types of patterns can arise. When the symmetries
are imposed by the bulk parameters of the active medium, the pattern is called `spontaneousa.
Instead, when the symmetries are dominated by the boundary, either by the geometry of the system
or by an external driving force, the pattern is called `forceda.
A prototype of spontaneous pattern is the Turing instability [297]. An example of forced
patterns is given by `dispersivea patterns, where the linearized dynamics of the system provides
a dispersion relation f (u, k)"0. In those systems, when u is real, an external forcing frequency
u makes the choice of the length scale k\ by constrain of dispersion. An example is the case of
capillary patterns in #uid layers submitted to a periodic vertical force (Faraday instability)
[298,380]. Experimental evidence of the transition from a dispersive length to a dissipative one was
given for a liquid-vapor interface close to the critical point [299]. When u is imaginary, then
f (u, k)"0 provides an interval of possible unstable k values, and selection of one (or a few)
particular k is provided by the boundary constraints [300]. In both cases, `dispersivea patterns are
dominated by an external in#uence.
The evidence of boundary independent patterns in nonlinear optics was given by Ref. [94]. In
that reference it was reported the transition from dispersive patterns, dominated by the geometric
parameters, to dissipative patterns whose scale length is imposed by the bulk properties of the
medium.
The second patterns are not usual in optics, insofar as the patterns properties depend in general
upon the Fresnel number F"a/j¸, which accounts for competition between geometric acceptance and di!raction phenomena. A "rst attempt to overcome such a limitation was given
theoretically in Ref. [189].
In the experiment of Ref. [94] a fundamental geometric parameter is the spot size of the central
mode constrained by the quasi-confocal con"guration to be [301]
w "
j¸
.
n
(68)
If the mirror size a is larger than w (that is, the Fresnel number is larger than 1), the cavity houses
higher order modes, made of regular arrangements of bright spots (in cylindrical geometry they are
Gauss}Laguerre functions) separated by
w
DK .
(F
(69)
The overall spot size of a transverse mode of order n scales as (nw , so that n"F represents the
largest order mode allowed by the boundary conditions (that is, "lling all the aperture area). It is
worth to remind that patterns built by superposition of Gauss}Laguerre functions have an average
separation 1D2 of zeroes approximately equal to the average separation D of bright peaks [301].
Exploring very large F regimes, a plot of 1D2 versus F (Fig. 41a) shows that Eq. (69) is not valid
everywhere, but it is restricted up to a critical value F , above which D is almost independent of F.
Similarly, the total number N of phase singularities scales as F or F, respectively below and above
F . This transition in the scaling law properties is reported in Fig. 41b.
The di!erence in scaling laws for 1D2 and N is a signature of the transition of the system
from a dynamical regime to another. This transition has the following root. Assume that the
56
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 41. From Ref. [94]. (a) Mean nearest neighbor separation 1D2 (the scale is in lm) between phase singularities and
(b) average total number 1N2 of phase singularities as functions of the Fresnel number F of the cavity. Dashed lines
refer to best "ts for the boundary dependent scaling laws 1D2&F\ and 1N2&F. Solid lines are best "ts for the
boundary independent scaling laws 1D2&F and 1N2&F. The transition between the two dynamical regime occurs at
F&11.
photorefractive crystal is a collection of uncorrelated optical domains, each one having a transverse
size limited by a correlation length l intrinsic of the crystal excitations. Then the medium gain will
have un upper cuto! at a transverse wavenumber 1/l and the ampli"cation of spatial details will be
e!ective only up to that frequency. This implies that, for F such that D"w /(F "l , a transition
from a boundary to a bulk dominated regime occurs. In the former regime, the separation between
phase singularities is given by Eq. (69), in the latter it is independent of F. Fig. 41 reports the
two regimes, and yields a value F K11 corresponding to l &170 lm, for w &600 lm and
¸"200 cm.
The reduction of the boundary in#uence is also marked by the reduction of the topological
charge imbalance. This is because a regular "eld should have a balance between topological
charges of di!erent sign, while any imbalance means that two phase singularities of opposite
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
57
sign created close to the boundary have been divided so that one has remained within the
boundary. Therefore, a boundary layer of area a ) 1D2 can be de"ned, which contains N &a/1D2
defects (N ;N when 1D2;a). Only within such layer a topological imbalance occurs. Other
independent evidences of such a dynamical transition are o!ered in Ref. [94].
5. Open problems and conclusions
We devote this conclusive section to a brief outline of three problem areas in optical patterns.
The "rst one (Section 5.1) deals with localized structures in devices made of nonlinear slices in
a feedback con"guration. How easy is to create and destroy small isolated spots of light intensity
over a uniform dark background (or the complementary, that is, dark spots on a bright background) furthermore being able to change their positions? This might be the clue for a new
information storage system, to be used in a next generation of computers.
The second problem (Section 5.2) is how costly, in terms of probes and e!ectors, is to control
chaotic patterns. For a discrete dynamical system, a single probe on the PoincareH section can signal
chaotic instabilities and suggest how to correct them by small perturbations [302]. It is not even
necessary to have access to the whole PoincareH section, but one single coordinate can be compared
with its value at a previous period, provided the correction is continuous [303]. An adaptive
version of these controls [304,381] provides higher order corrections. But in the case of space}time
chaos, there may be defects, or large discontinuity points, whereby any perturbative approach fails.
Furthermore, pattern control is a "eld problem, formally in"nite dimensional. Is there any suitable
tool to reduce the number of probes to a manageable one?
Finally, a new blossoming "eld is that of atom optics [305]. In this case, the propagating "eld is
represented by the atomic Schroedinger t(r, t) describing the space}time amplitude probability of
localizing the atom. In the case of an atomic beam pointing toward a direction z, we can extract
a main plane wave eIX\SR and account for the variations of the transverse (x, y) plane through the
Laplacian term of the Schroedinger equation. The structure of this equation in such a case includes
an eikonal operator. We should expect all phenomena of optics occurring in such a case, and a few
of them have already been observed [305]. Can we foresee a spontaneous patterning of a uniform
Schroedinger wavefront due to some interaction of Kerr type with a light "eld? Such a matter
would have been highly speculative a few years ago, because coherence constraints were di$cult to
satisfy with thermal beams having a De Broglie length much smaller than the interatomic
separation. After the beautiful evidence of Bose}Einstein condensation [17] and the make up of the
atom laser [18], these coherence problems have been overcome, thus the set up presented in
Section 5.3 may "nd a laboratory implementation within a short time.
5.1. Localized structures in feedback systems
Localization of single light (or dark) spot in feedback systems constitute a counterpart of soliton
formation in free propagation in nonlinear media. Though the existence of this kind of structures
and their relation with modulational instabilities in optical cavities "lled with a nonlinear medium
have been theoretically explored in the 80's in some pioneering works [197,306}314] full attention
to this matter is rather recent.
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 42. From Ref. [315]. The continuous-dotted line represents the homogeneous stationary solution of Eq. (70) in the
monostable regime (a"0.1, c"!0.001). This solution is stable in the continuous region, unstable with respect to
modulated perturbations in the dotted region. The circles indicate the minimum and maximum values of the numerically
evaluated pattern solutions. Localized structures exist in regions B, B.
Fig. 43. From Ref. [315]. Same as Fig. 41, but for c"0.025. The uniform stationary state is now bistable.
The localized structures (LS) that have been studied can be roughly classi"ed in two categories.
The "rst one occurs in systems displaying a spatial instability of the basic uniform state of a "nite
wavenumber, close to or in the bistability regime for the uniform state itself. The second one is
associated with bistability between uniform states, without the presence of bifurcations to delocalized patterned states.
As an example of the "rst type, consider pattern formation in a bistable cavity close to nascent
optical bistability [201,202,315]. The model consists of a cavity "lled with two-level atoms, with
an injected "eld. Assuming the validity of the mean "eld approximation, in the limit of weak
dispersion the cavity electric "eld evolves according to
4
RE
"4y#E(c!E)!4*
E! E .
3
Rt
(70)
Here, E and c are the deviations of the electric "eld and cooperation parameter (see Appendix A)
from their values at the critical point corresponding to nascent bistability, y is the injected "eld,
and * is the detuning between the atom and "eld frequencies. The input-output characteristics
y vs E describing the uniform solutions of (70) is monostable for c(0, bistable for c'0. In both
cases, subcritical bifurcations to patterned states exist for a given range of y values. These are due to
the instability mechanism described in Section 3.2, though an expansion close to the critical point
has led to a formally di!erent model equation. Figs. 42 and 43 display the uniform solution for
c(0, c'0 in the plane E!y, together with the maximum and minimum values of the "eld E in
the regime in which bifurcation to hexagons occur. In both cases, localized spots of the kind shown
in Fig. 44 can be created by addressing a narrow light pulse to speci"c spatial locations.
Similar patterns have been predicted in the simulation of a bistable cavity "lled with a Kerr
medium [194,316], and in systems with purely absorptive nonlinearity [317,318]. In other studies
the mean "eld approximation as well as the instantaneous response of the medium have been
relaxed, and still localization of patterns has been observed [319].
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
59
Fig. 44. From Ref. [315]. The two dimensional localized structures observed in simulation of the model described by
Eq. (70) for D"0.1, c"0.025, y"!0.0005.
Due to the potential interest of these kinds of structures as pixels in information processing
systems, investigations have been devoted to the existence of these spots in a Fabry-Perot cavity
"lled with semiconductors [320,321]. It has been shown that the localized structures are robust
with respect to the perturbations introduced by charge carriers di!usion and dynamics. Finally, the
existence of stable localized structures has been predicted also for cavities "lled with a medium
displaying optical nonlinearities [322].
In all the cases reported above, the shape of a localized structure looks as shown in Fig. 45. The
plot shows also the "eld distribution of a LS as compared to that of the delocalized hexagonal
pattern that can be excited for the same parameter values. In its central point, the LS "eld
distribution "ts nicely the hexagons distribution, thus stressing the role of the modulational
instability in the LS formation. Each spot can be interpreted as a single, elementary building block
of the underlying set of hexagons. The peak is connected to the low uniform state by a shallow
trough (see the oscillation at bottom left of Fig. 45).
It has been argued that these peripheral undulations have a central role in determining the
stability of the structure observed, introducing a sort of `pinninga that prevents the system to
switch uniformly to the upper patterned or lower uniform state [113,309,323,324].
From an experimental point of view, LS of this kind have been reported in Fabry-Perot cavities
containing a LC cell [325,326] and in the Kerr slice with feedback [327,250] experiment (Fig. 46).
The experimental observation agrees with the theoretical predictions for these systems [251,326].
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
Fig. 45. From Ref. [315]. Plot in the complex E plane of the stable localized structures observed in simulations of
a model of a cavity "lled with purely absorptive medium. Also plotted is the coexistent hexagonal solution. The localized
state solutions spiral out, and back to the plane wave "xed point (bottom left). The inset shows the spatial pro"le of the
localized structure.
Fig. 46. From Ref. [250]. Localized structures observed in a LCLV with di!ractive plus interferential feedback. (a) near
"eld image, (b) intensity pro"le.
A di!erent kind of LS occurs when a system displays bistability between uniform states, without
the presence of bifurcations to patterned states. This is the case of nonliner interferometers or lasers
in which the role of di!raction is negligible, due e.g. to the choice of a self-impinging geometry or to
the presence of strong di!usion [113,328}331]. A typical example of this class of structures arises in
a class A laser with saturable absorber, for which the electric "eld obeys the equation [330]:
RE
cE
(1!d)
"aE#b
E#
#
E.
Rt
1#"E" 1#e"E"
(71)
This equation supports stable LS like the one shown in Fig. 47. One immediately notices that the
lundulations at the periphery of the peak are absent in this case. The stability of these structures is
attributed to nonvariational e!ects leading to a coupling of the spot amplitude and phase, resulting
in a negative feedback for perturbations around these stationary solutions [330}332].
LS of this second kind have been observed in cavities with nonlinear gain and/or absorption
[333}336], using dyes or photorefractives as active media. It has been shown that structures of
various sizes can exist, depending both on the material di!usion and the level of pumping [336].
5.2. Control of patterns
Another research line which is currently attracting a lot of interest in the optical community is
the possibility of controlling desired or regular patterns within space}time chaotic regimes.
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
61
Fig. 47. From Ref. [330]. Two dimensional localized structure numerically found in a model of laser with saturable
absorber (Eq. (71)).
The idea of control of chaos was initially introduced in the case of non extended chaotic systems.
It consists in stabilizing a desired temporal dynamics within a chaotic regime by means of tiny,
judiciously chosen perturbations. Such perturbations can a!ect either a control parameter of the
system, or a state variable of the system. Since a chaotic dynamics can be seen as a continuous
motion in the phase space visiting closely an in"nite set of unstable periodic behaviors (the
so-called unstable periodic orbits or UPO) [337], then control of chaos allows one to stabilize each
UPO and then to use the same chaotic system to produce an in"nite set of periodic motions.
The "rst method for the control of chaos is reported in Ref. [302], and it consists in a tiny
perturbation of a control parameter each time the chaotic trajectory intersects the PoincareH section
of the #ow. However, the time lapse for a natural passage of the #ow at the right point of the
PoincareH section suitable for control may be very large. To minimize such a waiting time,
a technique of targeting has been also introduced [338].
Another technique to constrain a nonlinear system x(t) to follow a prescribed goal dynamics u(t)
is based upon the addition to the equation of motion dx/dt"F(x) of a term U(t) choosen in such
a way that "x(t)!u(t)"P0 as tPR. Refs. [339,382,383] considers U(t)"(du/dt)!F(u(t)).
In other papers the e!ects of periodic [340,341,384] and stochastic [342] perturbations are
shown to produce outstanding changes in the dynamics, which however are quite di$cult to
predict and in general are not goal oriented.
A further method [303] has been proposed, based upon the continuous application of a delayed
feedback term in order to force the dynamical evolution of the system toward the desired periodic
dynamics whenever the system gets close to such a periodic behavior.
On the other hand, many experimental systems have been studied with the aim of establishing
control over chaos.
Experimental chaos control and higher order periodic orbit stabilization have been successfully
demonstrated for a thermal convection loop [343], a yttrium iron garnet oscillator [341], a diode
resonator [344], an optical multimode chaotic solid-state laser [345], a Belouzov}Zabotinsky
chemical reaction [346,385], a CO laser with modulation of losses [347]. In most cases stabiliz
ation of UPOs was achieved by the technique of occasional proportional feedback (OPF) introduced in Ref. [344].
Even though a discrete hyperchaotic (with more than one positive Liapunov exponent) dynamics
has been controlled [348] and targetted [349], extension of the above methods from a return map
for a discrete system to a continuous dynamics for an extended system was still an open problem.
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
A recent advancement is constituted by an adaptive technique, initially introduced for chaos
recognition [350,386], and then applied to chaos control [304], targeting [351], and synchronization [352], as well as to "ltering noise from a chaotic time series [353]. Such a technique has been
shown to be e!ective in controlling defects and space-like structures emerging in delayed dynamical systems [354], that is, large dimensional systems displaying many aspects in common with
space}time chaos.
Space}time chaos control would indeed consist in slightly perturbing an extended system in
order to stabilize some of the unstable patterns embedded in the turbulent regime. Some preliminary attempts have been done [355], by trying to extend the method of Ref. [303] to a space
extended system.
Another proposal [356] suggests to use a spatial modulation of the input pump "eld in an
optical pattern forming system to stabilize a series of unstable homogeneous solutions, such as
squares, hexagons and honeycombs.
A further method has been proposed, based on a spatial "lter with delayed feedback, and able to
stabilize and steer the weakly turbulent output of a spatially extended system [357]. The method
was shown to be e!ective in the case of the generalized complex Swift}Hohenberg equation
introduced in Refs. [21,358] as the generic model for pattern formation in the transverse section of
semiconductor lasers.
Many experimental implementations of the above methods are still in progress, however
a general question is in order, namely, extension of the control techniques from discrete dynamical
systems to patterns has been shown to be e!ective only when one forces an apriori known pattern,
as e.g. by a Fourier mask; then, is the number of independent controller (probes and e!ectors)
increasing with the size of the system (that is, with the aspect ratio) in a linear way or with a larger
power? In what circumstances does this increase of complication pay for, that is, what is the trade
o! for pattern control?
5.3. Patterns in atom optics
The dual version of the Young double-slit experiment [359] consists of a supercooled atomic
beam, with a de Broglie wavelength comparable with the optical wavelength, crossing the standing
wave of a laser beam detuned by D"u !u with respect to the atomic transition frequency
u so that a local dephasing d
(r) is induced on the t-function at each transverse coordinate
r depending on the local "eld intensity "E(r)"
l
d
"(n(r)!1) .
v
(72)
Here l is the size of the laser beam along the atomic path (interaction length), v is the speed of the
atomic beam,
m
n(r)K1!
;(r)
k
(73)
is the pseudo refraction index for the propagating wavefunction, where the energy shift ;(r) is given
in terms of the atomic-dipole moment k and the local intensity n" f (r)" (n being the average photon
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
63
Fig. 48. From Ref. [360]. Interaction between the wavefunction of an atomic beam and the laser standing wave
propagating along x and con"ned within a narrow range l along z. The forward-propagating wavefunction undergoes
a dephasing as it crosses the laser region. The transmitted wavefunction is further dephased along the free propagation
path, and it seems to the incident one in the laser volume, giving rise to a phase to amplitude conversion which modi"es
the laser intensity distribution and hence the induced dephasing on W. (a) Forward and backward beams in a counterpropagating con"guration; (b) feedback con"guration.
Fig. 49. From Ref. [360]. Two level atom (transition frequency u ) interacting with a detuned laser "eld (optical
frequency u ). r,(x, y) are the transverse coordinate and z the longitudinal one, l is the longitudinal size of the
interaction region. The laser "eld is a standing wave made of two counterpropagatig waves perpendicular to z and with
a pattern f (r) in the transverse direction.
number and f (r) the normalized "eld distribution) as
n
u
" f (r)"k .
(74)
;(r)" D 2e <
A?T
Here, starting from Eqs. (72)}(74), the novel mechanism to be considered is the following (see
Fig. 48):
1. the dephased W-"eld is propagated along a closed loop (this is feasible, since total re#ection
atomic mirrors are nowadays available) and hence di!raction provides a phase-to-amplitude
conversion, whereby d
(r) induces a local modi"cation of "W(r)".
2. The atomic probability density "W" modi"es the stationary "eld pattern f (r).
Thus, beyond a threshold controlled by the density of the atomic beam and by the frequency
position, the uniform transverse phase of the W wavefront spontaneously destabilizes toward
a pattern. The symmetry of the pattern can be studied in terms of the symmetry changes induced on
the laser "eld.
In fact, the energy shift ;(r) appears as a spatially inhomogeneous optical potential [359] to be
introduced into the Schroedinger equation for the wavefunction W(r, z, t). In the case of the forward
beam (see Fig. 49) we can write the wavefunction as
t"W(r, z, t)eSR\IX ,
(75)
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
where
k
u"
2m
(76)
is the dispersion relation for a free particle. It is important not to confuse the plane-wave frequency
u, corresponding to the translational degrees of freedom of the atom, with the frequency u corre
sponding to the internal degree of freedom (see Fig. 49 and Eq. (74)). The two level optical
transition has been treated separately using the formalism of Ref. [359].
We consider the potential ; localized over a z interval so small that the main z dependence in
Eq. (75) is included in the exponential factor.
The evolution equation for the wavefunction is
z
i
R t" t#;trect ,
R
l
2m
(77)
where rect (z/l)"1 in the z interval of width l, where the laser is active and vanishes identically
elsewhere.
#R, where is the transverse Laplacian and taking for W a slow
Expanding "
X
,
NCPN
z dependence, so that
RW
RW
;k ,
Rz
Rz
(78)
Eq. (77) reduces to
R
i
z
R
#v #i W"! ;Wrect .
Rz
2m ,
l
Rt
(79)
Eq. (79) is formally equal to the equation describing transverse pattern formation in an optical
beam propagating along z and with di!ractive e!ects in the transverse direction. Therefore, it
should be expected the possibility of occurrence of all phenomena described for patterns in optical
"elds.
By inspection, Eq. (79) is a legitimate Schroedinger equation, without ad hoc approximations, as
e.g., Hartree}Fock type, corresponding to consider ; proportional to "W".
A detailed analysis of Eq. (79) is contained in Ref. [360].
Ref. [360] establishes the longitudinal and transverse coherence constraints which the atomic
W has to satisfy in order to assure successful interference between the input and feedback
wavefronts. These conditions are very restrictive for thermal atomic beams, where the de Broglie
wavelength is comparable with the optical wavelength. They have become feasible nowadays that
Bose}Einstein condensates have a coherence length of macroscopic size. Based on these considerations, renewed interest [361] has arised to the problem of longitudinal patterning in atomic
beams, originally considered in Ref. [362].
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
65
Acknowledgements
F.T.A. and P.L.R. acknowledge "nancial support from the EEC Contract no. FMRXCT960010.
S.B. acknowledges "nancial support from the EEC Contract no. ERBFMBICT983466. Work
partly supported also by the Coordinated Project `Nonlinear dynamics in optical systemsa of the
Italian CNR, and by the 1998 Italy}Spain Integrated Action.
Appendix A. A reminder of nonlinear optics
We recall some general notions of nonlinear optics, which can be found in many standard
textbooks. We refer, e.g., to the recent book by Boyd [363].
A.1. Nonlinear susceptibility
The dipole moment per unit volume, or polarization P(t), of a material system depends upon the
strength of the applied optical "eld. Taking for simplicity scalar relations, the suitable generalization of the linear relation
P(t)"e sE(t)
of conventional optics is (in SI units)
(80)
P(t)"e (sE(t)#sE(t)#sE(t)#2) .
(81)
More generally, s would be a second-rank tensor, s a third-rank tensor, etc.
As well known, s is a nondimensional quantity, of the order of unity for condensed matter;
s (m/V) and s (m/V) are of the order respectively of the reciprocal of the atomic "eld E and
of its square. If we call a K0.5;10\ m the Bohr radius of the hydrogen atom, the order of
magnitude of E is E &(1/4pe ) (e/a)&5;10 V/m, e"1.6;10\C being the electron charge.
Therefore
s
s& &2;10\ m/V
E
and similarly
s
s& &4;10\ (m/V) .
E
(82)
(83)
A.2. The two level approximation
In the case of light resonant with an atomic transition between a ground level "g2 and an excited
level "e2, the contributions from other energy levels are negligible at all orders of perturbation. The
problem of the quantum atoms interacting with a classical electromagnetic "eld has a simple
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
solution which is an oscillatory energy exchange between the atoms and the "eld, at the Rabi
frequency
kE
,
X"
(84)
where k,1e"ed"g2 is the matrix element of the atomic dipole operator ed between upper and lower
states. For allowed transitions, d is of the order of a Bohr radius, thus k&10\ C m and
X&10 s\ for E"1 V/cm. This coherent Rabi oscillation is eventually cut o! by coupling the
atoms to the environment, which acts as a thermal bath, providing two damping rates: the "rst,
denoted c , due to decay of any coherent in-phase superposition of the atomic wavefunctions of
,
upper and lower states, due e.g. to elastic collisions, and the second, denoted c corresponding to
,
the decay of the energy stored in the atomic medium, due, e.g., to spontaneous emission. In dilute
gases, the two rates are equal; in condensed matter the phase decay is much faster than the energy
decay (elastic collisions occur more frequently than spontaneous emission processes). Numerically,
for the D transition in Na vapours at the pressure of a few millibar, c &c &10 s\; for dye
,
,
molecules in liquid solution c &10 s\ whereas c &10 s\; for a partly forbidden transition,
,
,
where the spontaneous lifetime is very long as for the Nd> ions in a glass matrix, c &10 s\
,
whereas c &10 s\.
,
The e!ect of the damping rates is to quench the coherent Rabi oscillation. Rather than a whole
sinusoidal waveform in time, whenever (c,c <X we have just a piece of straight line which is
,
equivalent to a "rst order perturbation theory, as if the atom was a harmonic oscillator rather than
a two-level system. This will provide the standard linear relation (80), where now s can be
evaluated by Fermi golden rule. We present as numerical example the calculation for the D line of
Na atoms at a density o of 10 m\ corresponding to a pressure of a few millibar. Fermi golden
rule gives the transition rate w per unit time in terms of the Rabi frequency and of the resonant
density in frequency (dn/du)"1/c of the "nal states (c"10 s\ being the width of the D line)
2po
X (s\ m\) .
(85)
w"
c
As the energy exchange rate w
u is equated to the standard electromagnetic rate uPE"
ue sE, we obtain
2pok
.
(86)
s"
c
e
Numerically, we evaluate s&10, whence a dramatic enhancement with respect to the standard
s of a nonresonant dilute gas (s&10\).
As the ratio X/c c becomes important, one should introduce the correction
, ,
e sE
e sE
P"
" ,
(87)
1#X/(c c ) 1#E/E
, ,
where E is the E value for which the Rabi frequency kE /
is equal to the mean damping rate
(c c .
, ,
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
67
Fig. 50. (a) Resonant two level interaction. Upper: absorption; middle: emission; lower: emission with "rst nonlinear
correction. (b) Nonresonant two level interaction (scattering). On the left: energy levels with photon exchanges (wavy
lines). "g2 represents the ground state, "e2 the excited state. On the right: matter propagators (solid lines) and light
propagators (wavy lines). In case (b) the lifetime of the excited level is very short (of the order of the reciprocal of the o! set
frequency /*=) so that the interaction diagram can be simpli"ed to the photon exchange.
An expansion of the denominator of Eq. (87) will provide a generalized P(E) where only odd
terms (s, s, s, etc.) are present. Thus, in the resonant interaction E replaces what in general
would be E . For the sake of numerical evaluation, in the case of Na vapours
(88)
E "
(c c "10 V/m ,
I , ,
thus s&1/E "10\ (m/V), rather than 4;10\ as in the nonresonant case. This shows
again the dramatic enhancement of nonlinear e!ects in resonant media.
As we move away from resonance (Fig. 50b) the strength of the interaction is reduced by the
square ratio of the Rabi frequency to the o!-set frequency *=/
. Thus, the scattering process
depicted in Fig. 50b corresponds to a s reduced with respect to the resonant value s by
X
.
(89)
s "s
*=/
The reduction factor corresponds to the passage from the one-vertex diagram of Fig. 50a (1st order
perturbation theory) to the two-vertex diagram of Fig. 50b (2nd order perturbation theory). For the
above example, take X"10 s\ and *=/
"10 s\; thus s &0.1.
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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
The nonlinear relation (87) is the basis of optical bistability. If we shine an external "eld E on
a Fabry-Perot cavity "lled with a medium, the transmission properties of the cavity will be
provided by the boundary conditions combined with the medium refractive index. In the case of
tuning between the input "eld frequency and the cavity resonance, the input intensity I yields an
in-cavity intensity I through the Airy's equation [363]
¹I
,
I "
(1!R(1!al))
(90)
where ¹ and R are the transmission and re#ection of the mirrors, l is the length of the intra-cavity
absorbing medium and a its absorption per unit length. This equation can be simpli"ed introducing the so-called cooperation number c"Ral/(1!R). The equation can be rewritten as
I
1
.
I "
¹ (1#c)
(91)
If we now account for the nonlinear polarization, the input-output relation E vs. E can have two
branches, depending on the s value, that is, on the density of resonant atoms "lling the cavity.
Indeed, from Eq. (87) the absorption factor is intensity dependent as
a
,
a"
1#I /I
(92)
where I J"E ". The relation (91) between I and I can be rewritten as
c
I "¹I 1#
,
1#2I /I
(93)
where c is the cooperation number corresponding to the unperturbed a . It is easy to show that
the output}input relation I vs. I is monotonic for c (5 (monostable) and two-branched
(bistable) for c '5.
A.3. The s and s nonlinear optics
In the case of a two-level medium, we have seen that the nonresonant interaction can be
represented by a simpli"ed one-vertex diagram, displaying only the light propagators. Indeed, the
lifetime of the excited state /*= as given by Heisemberg principle is very short, and we can
consider its role as that of a catalyst which eventually leaves the medium in the unperturbed
"g2 state.
In the case of s processes (Fig. 51a) we must conserve the energy of the incident and outgoing
quanta. On the left we present a di!erence frequency generation or down-conversion process, on
the right a sum frequency generation or up-conversion process.
If the impinging and outgoing "elds are all light "elds, we speak of parametric processes. In
particular for X"u/2 we have the degenerate parametric down-conversion (left) and the second
harmonic generation (right). But X may be the quantum of a wave"eld, which is di!erent from
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
69
Fig. 51. Interaction diagrams for (a) s or three-quanta and (b) s or four-quanta processes.
Table 2
From Ref. [363]. Typical strength and response time of optical nonlinearities depending on the physical mechanisms
lying at the basis of the nonlinear response
Mechanism
n
(cm/W)
s
(esu)
Response time
(s)
Electronic polarization
Molecular orientation
Electrostriction
Saturated atomic absorption
Thermal e!ects
Photorefractive e!ect
10\
10\
10\
10\
10\
(large)
10\
10\
10\
10\
10\
(large)
10\
10\
10\
10\
10\
(intensity-dependent)
The photorefractive e!ect often leads to a very stron nonlinear response. These responses usually cannot be described in
terms of a s (or an n ) nonlinear susceptibility, because the nonlinear polarization does not depend on the applied "eld
strength in the same manner as the other mechanisms listed.
a light "eld. If it is a molecular vibration, we will speak of a Raman process, if it is a sound wave, we
will speak of a Brillouin process.
In the left case, the energy of an incident photon is splitted into a lower frequency photon plus
a quantum of a material excitation (Stokes process). In the right case, the outgoing photon energy is
increased by the material quantum (anti-Stokes process).
The s nonlinearity (Fig. 51b) gives rise to an intensity dependent refractive index. Indeed,
neglecting for the time being the tensor nature of the sQ, the induced polarization can be written as
P
"(s#s"E")E ,
e
corresponding to a refractive index
(94)
n"n #n I ,
(95)
where IJ"E" is the light intensity. Such a change in the refractive index is called optical Kerr
e!ect. Some of the physical processes that can produce a Kerr e!ect are listed in Table 2, together
with the characteristic time scale for the nonlinear response to develop.
70
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
A.4. The photorefractive (PR) ewect
The photorefractive e!ect is the change in refractive index of a medium resulting from the
optically induced redistribution of electrons and holes.
Suppose we shine on a PR crystal two input optical "elds of equal frequency and amplitudes
A <A (Fig. 52). The corresponding intensity grating (horizontal lines in the "gure) will shift
electrons from donor atoms, inducing a space charge wave. This is equivalent to a ripple of "eld
amplitude [363]
A AH
E ,
E "!i
"A ""A " (96)
where E is an inner "eld, function of the material properties. The dephasing between the
impinging "elds and the ripple "eld E acts as a nonlinear transfer of energy from the pump to the
signal wave. The two intensities I and I are respectively ampli"ed and attenuated as
dI
II
"$C ,
dz
I #I
(97)
where C depends upon the PR properties. As it appears from this skeleton description, the PR
behavior is in general more complex than a Kerr medium, and it is approximated by a s process
only in special domains of operations.
We can generalize the two beam coupling to a four wave mixing, as shown in the con"gurations
of the passive phase-conjugate mirrors (Fig. 53) and the double phase-conjugate mirrors (Fig. 54).
This latter con"guration has the remarkable property that one of the output waves can be an
ampli"ed phase-conjugate wave, even though the two input waves A and A are mutually
incoherent, so that no gratings are formed by their interference. The nonlinear interaction leads to
the generation of the output waves A (phase-conjugate of A ) and A (phase-conjugate of A );
however A is incoherent with A and A with A , whence the creation of the grating shown in the
"gure.
Fig. 52. From Ref. [363]. Typical geometry for studying two-bam coupling in a photorefractive crystal.
Fig. 53. From Ref. [363]. Geometry of the linear passive phase-conjugate mirror. Only the A wave is applied externally;
this wave excites the oscillation of the waves A and A , which act as pump waves for the four-wave mixing process that
generates the conjugate wave A .
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
71
Fig. 54. From Ref. [363]. Geometry of the double phase-conjugate mirror. Waves A and A are applied externally and
need not to be phase-coherent. The generated wave A is the phase conjugate of A , and the generated wave A is the
phase conjugate of A .
Appendix B. Rescaling the Maxwell}Bloch equations to account for detuning
and a large aspect ratio
For a single mode, the Maxwell}Bloch equations have been reported in Section 1.3.
The stationary solutions are
c k
DM " , ,
g
EM "E
D
!1 ,
DM
(98)
where E"c c /(4g) is called the saturation intensity and
, ,
g
PM " DM EM .
(99)
c
,
Let us now replace the inversion D with N"D !D (o!set with respect to the pump value).
Furthermore, we introduce dimensionless variables
E
D
D
P
e" , d" , r" , p"
,
(100)
E
DM
DM
DM E g/c
,
and rescale the time with respect to the polarization decay time c . The equations then are
,
rewritten as [13,19]
e "p(p!e), p "!p#(r!N)e, NQ "!bN#pe .
(101)
If furthermore we introduce a normalized detuning X"(u !u )/c between the cavity line
,
u and the atomic line u , as well as a di!ractive operator, we obtain Eqs. (11), but now "eld and
polarization are complex quantities.
Appendix C. Multiple scale analysis of the bifurcation problem for the non lasing solution
of the Maxwell}Bloch equations
Expanding all relevant variables of the Maxwell}Bloch equations in powers of a smallness
parameter e, by writing
X"eX , (E, P, N)"(E , P , N )#e(E , P , N )#e(E , P , N )#2
(102)
72
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
and deriving the spatial scaling from the breadth of the band of the unstable modes above
threshold
X"(r!1)x,
>"(r!1)y ,
(103)
one can assume r"1#e. It results X"(ex, >"(ey. As for the time scales, Refs. [21,22]
consider ¹ "et and ¹ "et as two di!erent secular time scales. Plugging all those expressions
into the Maxwell}Bloch equations, and identifying the coe$cients of each power of e, one gets, at
order zero
0"!pE #pP ,
P "(1!N )E , bN "(EHP #E PH) ,
(104)
which gives E "P "N "0. At order one, the relations are
0"!pE #pP ,
P "E ,
bN "0 ,
(105)
implying N "0 and E "P "t, t being a complex variable. Finally, at order two, one gets the
following equations:
RE
!ia
E "!pE #pP ,
R¹
RP
#P #iX P "E ,
R¹
bN "(EHP #E PH) ,
(106)
which yield, in terms of the new variable t
Rt
pE !pP "!
#ia
t,
R¹
Rt
!E #P "!
!iX t,
R¹
bN ""t" .
(107)
In order for the "rst two equations of (107) to be compatible, one must require a condition on t,
which in fact corresponds to the dynamical equation sought for. At the actual order, this condition
reads
Rt
"ia
t!iX pt .
(p#1)
R¹
(108)
Finally, choosing P "!(R/R¹ #iX )t"![i/(1#p)](a
#X )t and N ""t", the next
@
order (order three) yields
RE
RE
# !ia
E "!pE #pP ,
R¹
R¹
RP
RP
# #P #iX P "E #E !N E !N E ,
R¹
R¹
RN
RN
1
# #bN " (EHP #EHP #E PH#E PH) ,
2 R¹
R¹
(109)
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
73
which can be rewritten as
Rt
pE !pP "!
,
R¹
R
1
Rt
(110)
! !
#iX P #t! "t"t ,
!E #P "!
R¹
b
R¹
R
ia
1
"t"#
(t
tH!tH
t) .
bN "! !
2(1#p)
b R¹
The new solvability condition represents the behavior of t as a function of ¹ . If one uses the
relation
R
R
#iX P "
#iX
R¹
R¹
one gets
Rt
1
!
!iX t "
(X #a
)t ,
R¹
(1#p) p
p
Rt
"pt! "t"t!
(X #a
)t .
(p#1)
b
(1#p) R¹
(111)
(112)
The "nal equation for t is obtained by writing Rt/Rt"e Rt/R¹ #e Rt/R¹ . Reintroducing the
original variables x"X/(e, y">/(e, X"eX , r!1"e and rede"ning et"t, the "nal equa
tion reads
(p#1)
Rt
p
p
"p(r!1)t!
(X#a
)t#ia
t!iXpt! "t"t .
Rt
(1#p)
b
(113)
Appendix D. Symmetries and normal form equations
We analyze the normal form equations arising for symmetry requirements in the case of three
transverse modes: a central one, with complex amplitude z , and two higher order ones (rotating
and counter rotating along an azimuthal coordinate h) with respective complex amplitudes z and
z and angular momenta $1. The cavity "eld can be expressed as
(114)
E"f (r)(z eF#z e\F)eSR#f (r)z eSR ,
where f and f are the space distributions of the modes. The optical frequencies u and u are in
general di!erent. The slow time dependence due to the dynamics is accounted for in the amplitudes
z (t) (i"0, 1, 2).
G
The zero intensity situation is described by z "z "z "0, the central mode by z "z "0
and an azimuthal standing wave by z "0, z "z .
The cylindrical geometry of the cavity imposes the following constraints on mode amplitudes
[96]
H : (z , z , z )P(eFz , e\Fz , z ) ,
K : (z , z , z )P(z , z , z ) ,
(115)
74
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
(H being the rotation operation, and K the re#ection operator around a privileged plane). If one
considers the modes as born from Hopf bifurcations, then there is an additional time symmetry
B : (z , z , z )P(e@z , e@z , e@z ) .
(116)
The normal form for the nonlinear interaction among the three modes, assuming it to be invariant
under the above symmetries is [97,98] (dots denote time derivatives)
z "j z #(a("z "#"z ")#b"z ")z ,
z "j z #(c"z "#d"z "#e"z ")z #ez ,
z "j z #(d"z "#c"z "#e"z ")z #ez ,
(117)
(j , j , a, b, c, d, e being complex coe$cients and e"o ePC being a symmetry breaking parameter).
C
The parameter e is reminiscent of the breaking of the cylindrical symmetry induced by the pumping
procedure which privileges a de"ned plane, thus breaking the rotational invariance.
Putting z "o ePG, and operating a change in the variables (o "A cos(a/2), o "A sin(a/2) and
G
G
d"u !u ) Eqs. (117) are rewritten as
AQ "(jP #(cP!(1/2)(cP!dP)sin(a))A)A#(o sin(a)cos(d)cos(u )#ePo)A ,
C
C
a"!(cP!dP)sin(2a)A#2o (cos(u )cos(d)cos(a)#sin(u )sin(d)) ,
C
C
C
a
a
sin(u !d)!tg
sin(u #d) ,
dQ "!(cG!dG)Acos(a)#o cotan
C
C
C
2
2
o "(jP #aPA#bPo)o ,
u "jG #aGA#bGo ,
a
a
#dG sin
u "jG #A cG cos
2
2
(118)
(119)
#eGo .
The solutions of Eqs. (118) and (119) reproduce the experimental behavior for certain parameter
values that it is possible to derive explicitly. First of all, it is to notice that Eqs. (118) constitute
a closed four dimensional system.
The laboratory experiment shows that any initial condition close to a central mode evolves in
time toward a zero intensity state. Therefore, in the phase-space of the solutions of Eqs. (118) and
(119), the zero con"guration will be stable in the o direction. A "rst condition for the correspond
ence with the experiments is thus: jP (0 and bP'0. The fact that jP (0 is supported by the linear
stability analysis contained in Ref. [364].
Furthermore, if o "0, there are "xed points at a"p/2, d"0, p, de"ned by
A"!2(jP $o cos(u ))/(cP#dP) .
C
C
(120)
These solutions are standing waves, and they are part of the `skeletona of pure modes in the
dynamical model.
Stability of these solutions in the (a, d) directions can be inspected looking at the eigenvalues of
the Jacobian of the "rst three Eqs. (118). The experimentally observed dynamics again shows that
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
75
states close to zero are followed by a standing wave. This implies that the zero con"guration must
be unstable in the o "o "o direction, and this occurs if !2jP G2o cos(u )'0. If the
C
C
eigenvalues of the Jacobian have negative real parts, the two standing waves will be stable in the
a and d directions also. The time average of the intensity for these states can be calculated as
1EEH2, with E given by
E"eP\B o(eF\B #e\F\B ) f (r)eSR .
(121)
The pattern looks like a set of two bright spots either parallel (d"0) or perpendicular (d"p) to the
privileged plane (the plane with respect to whom the symmetry is broken).
The conditions for switching to the central mode is now derived. This can be obtained by
imposing the standing wave to be unstable in the o and o directions, in order that initial states
close to them (with small components of o ) evolve towards a central mode con"guration.
In the subspace (o "o , d"0), the dynamics is ruled by
o "(jP #aPA#bPo)o ,
AQ "(jP #o cos(u )#(cP#dP)A#ePo)A .
C
C
(122)
In the plane (o , A), three "xed points are present (O, S= and C), de"ned by (0, 0),
(0, (!2(jP #o cos(u ))/(cP#dP)) and ((!jP /bP, 0) respectively. If the following conditions are
C
C
satis"ed:
aP
(jP #o cos(u ))'0 ,
jP !2
C
C
cP#dP eP
! jP #(jP #o cos(u ))(0 ,
C
C
bP (123)
then S= is unstable in the o direction, whereas C is stable in the A direction. Moreover, since
bP'0, also C is unstable in the o direction. In order for both conditions to be satis"ed at the same
time, D,aPeP!bP(cP#dP)/2 must be negative. This implies that the (o , A) plane includes a fourth
"xed point P de"ned by
1 cP#dP
jP !aP(jP #o cos(u )) ,
o"
C
C
D
2
1
A" (!ePjP #bP (jP #o cos(u ))) .
C
C
D
P undergoes a Hopf bifurcation if
o"!
cP#dP
A .
2bP
(124)
(125)
A periodic orbit which emerges from P and which gets close to the other "xed points, will
su!er a critical slowing down that gives rise to the PA phenomenon among the O, S= and
C con"gurations.
76
F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83
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