Bowtie Laser

High Power Directional Emission from Lasers with Chaotic Resonators

This page contains additional information concerning the June 5, 1998, cover story in Science. Our research into chaos and its wave manifestations in dielectric optical microcavities is further presented on the picture pages 1-3 , and on the movie page, see to the left. More background information on microlasers and ray chaos can be found in the hitchhiker's guide to dielectric cavities.

Here is what it boils down to (cum grano salis):

poster figure

Click here to see a news release on this work that the Max Planck Society has posted on EurekAlert!.

All images on this page (plus the wave and Husimi plots in the Science article, and including all the color plots now being found in various news releases) have been created by me with computer programs which I wrote mainly at Yale University, but then significantly improved at the Max Planck Institute for Physics of Complex Systems. The exact solutions of the wave equations for deformed dielectric resonators, of which the above pictures are an example, are calculated using the Max Planck Institute's DEC Alpha computers.

Here is the idea:

The point of our work is to improve upon the low output power and missing directionality in the lasing emission from resonators with a circular cross section, an example of which is shown here: undeformed laser cross section

Although high intensity (red) can build up in the resonator, there is no preferred escape direction, and in fact leakage is very small because it can only take place by quantum-mechanical tunneling through the barrier made up of the jump in refractive index between inside and outside region. The inside is a semiconductor with refractive index n > 1, the outside is air with n=1. The waves are squeezed toward the interface (indicated by the black circle) - this can be visualized as the result of a centrifugal force acting on them as they rotate around. This picture assumes n=2 in the circular cylinder. In the new Bell Labs experiment, an important role is played by the larger value of n=3.3, because larger n means better confinement of the light. That would make it even more difficult for any rays to leak out if they were following a similar whispering-gallery pattern. On the other hand, better confinement is also the prerequisite for getting lasing out of the bowtie mode, see below. This is why the bowtie takes over and does all the work soon after it is born in a bifurcation (i.e. a chaos-induced instability tantamount to the tearing-apart of a more well-behaved, regular ray pattern existing at smaller deformations).

To download the manuscript of the Science article, please choose High power directional emission from lasers with chaotic resonators.

The published article has a small error in it, which was corrected in a Letter by Claire Gmachl [Science 282, p.47 (1998)]: The deformation of the resonator shape at which the bowtie mode ceases to be a stable periodic orbit was given as eps= 0.23 (for a definition of eps, see below). The correct upper limit for the deformation parameter to create a stable bowtie mode is eps =0.18, as was pointed out to us by Anthony E. Siegman. None of our findings are are affected by this error.

Also available online is an article that was written about our work by Volker Drach for "Physik in unserer Zeit" 5/98 (in German).

Here are two other papers on the subject:

Semiclassical theory of the emission properties of wave-chaotic resonant cavities
Authors: Evgenii E. Narimanov, Gregor Hackenbroich, Philippe Jacquod, A. Douglas Stone
We develop a perturbation theory for the lifetime and emission intensity for isolated resonances in asymmetric resonant cavities. The inverse lifetime $\Gamma$ and the emission intensity $I(\theta)$ in the open system are expressed in terms of matrix elements of operators evaluated with eigenmodes of the closed resonator. These matrix elements are calculated in a semiclassical approximation which allows us to represent $\Gamma$ and $I(\theta)$ as sums over the contributions of rays which escape the resonator by refraction.

Signature of Dynamical Localization in the Lifetime Distribution of Wave-Chaotic Dielectric Resonators
Authors: Oleg A. Starykh, Philippe R. J. Jacquod, Evgenii E. Narimanov , A. Douglas Stone
We consider the effect of dynamical localization on the lifetimes of the resonances in open wave-chaotic dielectric cavities. We show that dynamical localization leads to a log-normal distribution of the resonance lifetimes which scales with the localization length in excellent agreement with the results of numerical calculations for open rough microcavities.

The theoretical perspective: Technical details and figures

Please note that the material below is largely unpublished work.

We are considering the "flatty" shape. In actual units , we write the cavity shape as

r(phi)=R * Sqrt[1 + 2*eps* Cos(2 phi)].

Here, R has the dimension of length, e.g. in microns. The experimental shape has a fixed short diameter of 50 microns, i.e. at phi = pi/2 one has to set
R Sqrt[1 - 2 eps] = 25 microns.

The emission wavelength is approximately 5.2 microns, and is related to the size parameter by
kR = 2 Pi R/lambda = 2 Pi * 25 microns / (Sqrt[1 - 2 eps] * 5.2 microns).

This implies at eps = 0.06
kR = 32.2,

and at eps = 0.15

Ray-wave duality in oval resonators

The spectrum of this shape at eps=0.15 is shown below. Note that I am showing a spectral interval at about twice the frequency as the experiment. The states with the experimentally relevant frequencies are treated in the paper, and will be discussed further below. The fact that the results are essentially the same for the much higher-energy states shown here just serves to show that frequency does not matter much when we look at phenomena that have a clssical origin. This is the case for the emission directionality of these states, as well as for the resonance widths ! This latter fact is explained in our Nature paper, where it is argued that the linewidths on the wavenumber axis will be the same for all resonances that are semiclassically quantized on the same portion of the classical phase space, provided that the dominant escape mechanism is a classical process such as phase space diffusion.
flat shape spectrum

Most of the resonances are too broad to support lasing. What is the origin of the sharper features ?

Look at the classical phase space portrait (surface of section, or SOS) for eps=0.15,
sos e03

The total-internal-reflection condition requires sin(chi) to be above sin(chi_c)=0.3, which is satisfied approximately for the 4 small islands seen above - but not for the large "bouncing-ball " islands. The small islands correspond to the bowtie orbit,
Click below to investigate the question:
Is the bowtie orbit trapped by total internal reflection ?
In the spectral interval I plotted earlier, at kR=69.9 and gamma=0.047, there is a resonance that does indeed show the bowtie orbit (eps=0.15):
flat shape bowtie wave function

Its directionality is shown here (eps=0.15).
In all directionality plots, the top panel is the near field and the bottom shows the far field intensity.
flat shape bowtie directionality

As we can already see from the wave function, the ribbon making the bowtie is actually broader than could be supported by the island itself. To show where in phase space this state is located, look at the Husimi plot.

A Husimi plot is a representation of the wave field as a probability distribution in the classical phase space. It is obtained by forming the overlap between the wave function and a minimum-uncertainty wave packet centered at the given classical coordinates.

Here I show the Husimi plot superimposed with the corresponding SOS:
flat shape bowtie Husimi function

The rays are sticking to the neighborhood of the bowtie- and to a lesser extent also to the mainbouncing-ball island.

In making this plot, I did not use a logarithmic scale, but took a square root of the function to enhance smaller values close to 0.
To enhance small Husimi weight even more, here I raise the function to the power 1/4, and plot only the stable islands, not the chaos, so as not to obscure the Husimi plot:

flat shape bowtie Husimi function

Let us first check if such a state retains its bowtie character when eps is lowered. After all, there are clearly many anticrossings, and especially at smaller eps below eps=0.133 we cannot trust this to be the correct state. Here is the wave function for eps = 0.134:

It becomes more difficult to distinguish this from a pure bouncing-ball orbit.
The directionality looks like this:

It also becomes more similar to that of the bouncing ball orbits existing at the same deformation. Of course, so does the resonance width, which increases.

At the smaller deformation of eps=0.06, the spectrum of the flatty shape looks quite different from the one at eps=0.15. Its more regular appearance coincides with the more regular classical phase in which the bowtie orbit has ceased to exist:

The dominant mode spacing is that of the bouncing balls, approximately given by pi/n.
Here is the SOS:

The bouncing-ball resonance at kR=68 has the following wave function:

The directionality is shown here:

Clearly, this is a much less directional emission. The envelope shows broad peaks, but its minima are not very deep, i.e. they don't go to zero, only to 1/2 of the maximum envelope.

The experimentally relevant wavenumbers

The wavenumbers used so far are twice those actually present in the experiment. This shows the power of classical arguments, allowing us to infer features of the wave solutions that are valid universally in different spectral ranges, by comparing to the results at the experimental wavenumbers.

This will be done now, keeping the deformation at eps=0.06 for now.

Here is the corresponding spectrum, showing a rather similar dominant mode spacing as before.

Whispering-gallery states

At k=33.61, there is a resonance of width gamma=7*10^(-4) which shows up in the spectrum as a tiny spike (consisting of one data point, which hence is invisible in the above image). Here is its wavefunction:

The directionality is still rather broad, although the envelope has intervals where it vanishes.

To see where in phase space this function is localized, consider the Husimi plot:

The Husimi density beautifully follows the chaotic layer roughly up to tangency to the critical line for escape. The 6 islands corresponding to the triangular orbits are clearly avoided. Also note the 8 islands, the lowest of which is centered at phi=0, sinchi=0.25 in the SOS. The latter causes partial dynamical eclipsing (as opposed to a total eclipse), which explains why the directionality shows the combination of tangential escape and split peaks in the far field. Here is the shape of the orbit responsible for the eclipsing:

Another WG mode (the one used in the Science article) is at kR=31.3, gamma=0.033, Q=955:
WG mode gamma=0.033

The Husimi plot is shown here:
WG mode gamma=0.033

The directionality is shown here:
WG mode gamma=0.033

Bowtie state

A bowtie state at deformation eps=0.15 and wavenumber kR=35.83 and gamma=0.043 (i.e. Q=833) is shown here. It is different from the one used in the Science article:
bowtie mode gamma=0.04

The unsmoothed directionality is shown here:
bowtie mode gamma=0.04

Here is the Husimi plot.
bowtie mode

Interestingly, the state is not localized on top of the bowtie islands, but instead surrounds it almost like a halo. The question is: is this due to dynamical localization or does the ray simply escape before having a chance to diffuse far away from the "sticky" region near the islands ?

Related information is found on the following web pages:

This page © Copyright Jens Uwe Nöckel, 06/2003

Last modified: Mon Dec 27 18:13:27 PST 2010