Anomalous patterns, symmetries and lasers


F.Papoff $^{\clubsuit}$ and G.D'Alessandro $^{\spadesuit}$
$^{\clubsuit}$ Dept of Physics, University of Manchester, Manchester M13 9PL
$^{\spadesuit}$ Department of Mathematics, University of Southampton, Southampton SO17 1BJ

Patterns and lasers

One usually assumes that a laser emits a single bright spot. However, if the beam is sufficiently wide the beam intensity may not be uniform. For example, bright spots or lines can arrange themselves in a regular pattern (Figure 1).

Symmetries of lasers and patterns

The symmetry of the pattern reflects the symmetry of the laser that produces it. A laser in a typical experiment (Figure 2) is a gas filled tube positioned between two spherical mirrors (optical cavity). Hence the laser is axially symmetric with respect to the line that joins the centre of the two mirrors (optical axis of the cavity). In other words, the geometrical symmetry of the laser is represented by the O₂ symmetry group, i.e. the group of rotations in the plane around a point plus the reflection with respect to an axis through the point.

Simple laser models

Most lasers have another distinctive feature apart from the axial symmetry that makes their modelling a relatively straightforward affair: the light in the optical cavity is linearly polarised. Light is an electric field of very high frequency (10¹⁵ Hertz. The frequency of FM radio is only 10⁸ Hertz). As such, it is described by a vector field whose two components lie in the plane orthogonal to the direction of propagation. However, most lasers contain optical elements that let through only one component of the electric field: these elements are called polarisers and the light that has gone through them has only one vector component and it is called ``linearly polarised''.

The great theoretical advantage of dealing with linearly polarised light is that the differential equations that describe it are scalar equations instead of the vector equations that would be needed to describe a vector (non-polarised) electric field.

``Regular'' and ``anomalous'' patterns

To summarise, a laser can be described by scalar equations that have O₂ symmetry. This statement allows us to use group bifurcation theory to predict the symmetries of the patterns that may appear generically as continuous changes from a perfectly symmetric solution (``regular'' patterns). First of all, they must have the symmetry of a subgroup of O<sub>2</sub>. Secondly, they must grow from instabilities of the fully symmetric solution that are seeded by a class of patterns that we will call ``natural'' patterns. More formally, the null space of the linearised operator splits naturally into subspaces that support different irreducible representations of O₂. In the case of a laser with O₂ symmetry these spaces are spanned by the functions $\{ \cos(n \phi), \sin(n \phi) \}$ with n integer and are fixed under the action of the group D_n, i.e. the subgroup of O₂ that represents the symmetries of a polygon with n sides. Regular laser patterns usually conserve the main features, like the appearance, of the ``natural'' patterns from which they have developed.

For example, we may expect to see quite often, when increasing the diameter of the laser beam, the transition from a pattern with only one spot to a pattern formed by six spots on a ring (see Figure 3). The symmetry of the six spot pattern is D₆ (the group of symmetries of a regular hexagon) and the six spot pattern grows from a ``natural'' pattern.

A bifurcation to a pattern that does not satisfy the two requirements of a ``regular'' pattern, i.e. to an ``anomalous'' pattern, should be observable only by choosing very carefully the values of at least two control parameters. It should therefore be exceedingly rare. For example, we do not expect to observe generically the bifurcation from one spot to a pattern with co-prime number of spots on concentric rings (see Figure 3).

However, these patterns are commonly observed in experiments, as shown in the photograph in Figure 4 taken from a paper by Green et al. appeared in Physical Review Letters in 1990.

Why are anomalous patterns observed?

The optical cavity of a laser usually contains apertures, metallic discs with holes, that are needed to make sure that the light emitted by the laser has the required stability and optical properties. The apertures are circular and so one could suppose that the laser with an intra-cavity aperture has O₂ symmetry, exactly as a laser without aperture. However, this is not true.

The approximation that the electric field is a scalar breaks down at the aperture, because a linearly polarised electric field cannot satisfy the boundary conditions at the edge of the aperture (see Figure 5). The physical interpretation of this phenomenon is that the incident light induces currents in the aperture (edge currents) that in turn generate an electric field that is no longer linearly polarised.

The polarising element in the cavity selects only one of the components and thus creates a preferred direction that breaks the O₂ symmetry, reducing it to the much simpler D₂ symmetry. This group is formed by the identity, the reflection with respect to the polariser, the reflection with respect to an axis orthogonal to the polariser and a rotation by $\pi$ with respect to the cavity axis.

The change of symmetry has two consequences: both the bifurcation structure and the ``natural'' patterns of a laser with aperture may be completely different from those of a laser without aperture. In particular, the ``natural'' patterns, i.e. the seeds from which the observed patterns grow, split into two families, one with an even number of spots and one with an odd number of spots. An example of a ``natural'' pattern that belongs to the odd family is shown in Figure 6.

These two changes are such that the patterns that would be considered ``anomalous'' in a cavity without aperture are now perfectly ``regular'' and should thus be readily observable.

Numerical simulations

The results stated until now are based on symmetric bifurcation theory. The beauty of this mathematical technique is that with very little knowledge of the model's equations and their solutions it is possible to state the symmetry of the patterns that may be observed. However, the drawback is that it provides very little information on what patterns can actually be observed. Therefore, to complement the prediction of group bifurcation theory we have developed numerical codes to integrate the partial differential equations of a laser with an intra-cavity waveguide. The optical cavity of this laser contains a metallic pipe (waveguide), that we assume perfectly conducting. This laser is conceptually identical to the laser with an aperture, but it is considerably easier to handle numerically.

We have written a code to find the ``natural'' patterns of the system and another one to integrate the laser equations. The end result is that the ``anomalous'' patterns observed in the experiment by Green et al. are indeed readily observable (see Figure 7).

Future work

There is still a lot of work to do on this subject. The following are two topics that we plan to tackle in the near future.

• We have not yet built the bifurcation structure of the laser with intra-cavity waveguide. We know some of the patterns that appear when the laser is switched on, but we need to construct a scheme to classify them.

• The group of Prof Pierre Glorieux at the University of Lille has performed experiments in an optical system very similar to those discussed here. In order to fit better the theory to the experiment we need to develop more accurate models of their laser. In particular, we must try to tackle the thorny problem of the intra-cavity aperture. This problem is difficult in two ways: first of all is not at all trivial to express in a computationally and analytically efficient way the electric field after an aperture. Second, we believe that the study of the bifurcations of a laser with aperture is probably better approached by considering the laser as an O₂ system with a D₂ perturbation, rather than as a straightforward D₂ system, like the waveguide laser. In this case, however, we expect that the group theoretical analysis of the possible bifurcations and patterns becomes considerably more complicated.

References

The content of this poster is a brief and rough summary of two papers published in Physical Review Letters and Physical Review A:

F.Papoff et al. - Diffraction induced polarization in optical pattern formation, Phys. Rev. Lett. 82, 2087-2091 (1999)

F.Papoff et al. - Effects of polarization and non-paraxial approximation on pattern formation, Phys. Rev. A 60, 648-662 (1999)

General references to bifurcation theory are

J.Guckenheimer and P.Holmes, Nonlinear Oscillations, Dynamical systems and Bifurcations of Vector Fields, Springer-Verlag (1993)

P.Glendinning, Stability, instability and chaos, Cambridge University Press (1995)

An excellent introduction to the group theory approach to bifurcations is

J.D.Crawford and E.Knobloch, Symmetry and symmetry breaking bifurcations in fluid dynamics, Annual Review of Fluid Mechanics, 23, 341 (1991)

The ``holy grail'' of symmetries and bifurcations (that should be approached only by the direct descendants of Sir Percival) are the two books

M.Golubitsky and D.G.Schaeffer, Singularities and groups in bifurcation theory - Vol I, Springer-Verlag (1985)

M.Golubitsky, I.Stewart and D.G.Schaeffer, Singularities and groups in bifurcation theory - Vol II, Springer-Verlag (1988)

Contact points

There are a few people in this department interested in groups and/or bifurcations. If you would like to know more about this topic please contact them:

Name			E-mail			Office
Dr David Chillingworth			homepage			7011
Dr Giampaolo D'Alessandro			homepage			6005
Prof James Vickers			homepage			2015

This page is the summary of two papers:

• F.Papoff, G.D'Alessandro, W.J.Firth and G.-L. Oppo, Diffraction-induced polarisation effects in optical pattern formation, Phys. Rev. Lett. 82(10), 2087-2091 (1999).
• F. Papoff, G. D'Alessandro and G.-L. Oppo, Combined effects of polarization and non-parazial propagation on pattern formation, Phys. Rev. A 60, 648-662 (1999).

This web page was produced in year 2000. Some details may have changed since then.