Fractal(?) Lasers
Lasers have been with us for over forty years now. They have become an indispensable tool for research and technology, and their physics is well understood. Nevertheless, every so often new basic aspects are still discovered. One example of this is the recent discovery by my colleague Han Woerdman together with student Gerwin Karman (who then was his graduate student, and who is now at the Philips Research Laboratory) of fractal laser modes.
Usually, lasers are constructed in such a way as to operate at one single transverse mode. This regime may be the most obvious one when lasers are used as a tool in experiments and applications, but even in some single mode regimes lasers sometimes do exhibit dynamical behavior which is of intrinsic interest. For example, lasers have been studied as an experimental realization of low-dimensional nonlinear dynamics, associated with the fact that the laser equations map onto the Lorenz equations. Furthermore, for strong detuning multiple transverse modes can become active. One can then enter a regime where patterns in the transverse direction can emerge spontaneously. Such transverse patterns have been studied as an example of non-equilibrium pattern formation, and the laser systems are linked to other pattern forming systems by the complex Ginzburg-Landau equation, which emerges as the universal amplitude equation near threshold for the instability.
The interesting twist to the ongoing laser story added recently by Karman and Woerdman is the possibility of obtaining transverse intensity profiles with fractal scaling properties. Their focus in this case was not on the dynamics; rather, they showed that in an unstable cavity laser, depending on the mirror shape (triangular, octagonal, etc.), the two-dimensional transverse intensity profiles show self-similar structure reminiscent of fractals like the Sierpinsky gasket.
Indeed, Karman and Woerdman found numerically that the eigenmodes of unstable cavity lasers possess fractal scaling, and using the box-counting method they estimate the fractal dimension D to be about 1.6-1.7 in one dimension. For a two-dimensional systems with a circular aperture, a similar study produced a value of D of about 1.3. The idea that the mode profiles of unstable cavity mode lasers has got a lot of attention in various journals.
The fractal scaling properties of unstable cavity lasers can be approached from various angles. Courtial and Padgett have termed the effect the "monitor outside the monitor effect", the inverse of the wellknown patterns that one can get when a camera is focussed on a monitor which shows the output of the camera. Then, one sees an infinite series of monitors which are getting smaller and smaller. But here, for the fractal laser setup, the image is blown up in every round trip, hence the name "monitor outside the monitor effect". This is an important ingredient of the unstable cavity lasers, but the diffraction effects are also important. New et al. have also shown that the static Fourier spectrum of the intensity profiles shows a lot of structure, which can be understood in terms of contributions from the edge waves of the mirror.
Sir Michael Berry, Kees Storm (graduate student) and I have recently also studied the problem for the one-dimensional case. The basis of our analysis is the observation that in the large Fresnel number limit (the Fresnel number is large if the wave number of the light is sufficiently small in comparison with the dimensions of the laser) one can write a very accurate asymptotic expression for the intensity profile of laser. This is possible because in this limit, multiple diffractions of the waves from the edge of the mirror, become negligible. We then are able to study the fractal properties explicitly. It turns out that the laser profiles are never true fractal profiles in the strict sense of the word, in that they are not characterized by a single scaling exponent. Rather, one can think of the mode profiles as having a scale-dependent fractal exponent D(k). However, for scales k much less than a crossover scale kc the local fractal dimension D(k) is, to a very good approximation, equal to 2. On these scales, the mode profiles then resemble fractal curves which, because their fractal dimension is so large, almost fill the plane!
Our analysis builds on and simplifies earlier work in the seventies on the mode profiles in unstable cavity mode lasers, and unifies the earlier approaches, in that the essential ingredients and observations of the earlier work are recovered from it. It also illustrates quite nicely that when one naievely looks a the power spectra, the fractal dimension appears to be 3/2, but that the appropriate fractal dimension is over large ranges of the spectrum, essentially 2.
References on the subject:
G. P. Karman and J. P. Woerdman, Fractal structure ofeigenmodesl of unstable-cavity lasers, Opt. Lett. 23 1909-1911 (1998).
G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, Fractal modes in unstable resonators, Nature 402, 138 (1999).
J. Courtial and M. J. Padgett, Monitor outside a monitor effect and self-similar fractal structures in the eigenmodes of unstable optical resonators, Phys. Rev. Lett. 85, 5320.
G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, Diffractive origin of fractal resonator modes, Optics Commun. 193, 261 (2001).
M. V. Berry, C. Storm and W. van Saarloos, Fractal(?) Lasers, submitted to Optics Commun., June 2001.
Wim van Saarloos
June 10, 2001
[Pattern formation] [Wim van Saarloos] [Instituut-Lorentz]