Chaotic microlasers based on dynamical localization

Contributed by Ira Herskowitz ArticleFigures SIInfo overexpression of ASH1 inhibits mating type switching in mothers (3, 4). Ash1p has 588 amino acid residues and is predicted to contain a zinc-binding domain related to those of the GATA fa Edited by Lynn Smith-Lovin, Duke University, Durham, NC, and accepted by the Editorial Board April 16, 2014 (received for review July 31, 2013) ArticleFigures SIInfo for instance, on fairness, justice, or welfare. Instead, nonreflective and

Edited by Marlan O. Scully, Texas A&M University, College Station, TX, and approved June 10, 2004 (received for review April 20, 2004)

Article Figures & SI Info & Metrics PDF


We report the direct observation of lasing action from a dynamically localized mode in a microdisk resonator with rough boundary. In Dissimilarity to microlasers based on stable ray trajectories, the performance of our device is robust with respect to the boundary roughness and corRetorting ray chaos, taking advantage of Anderson localization in angular momentum. The resonator design, although demonstrated here in GaAs-InAs microdisk laser, should be applicable to any lasers and sensors based on semiconductor or polymer materials.

The resonator cavity is an essential part of any lasing device, providing the necessary energy accumulation and (coherent) feedback for the mode development. Dielectric microdisks and microcylinders are among the most widely used resonator cavities in modern microlasers and microsensors (1–3). The performance of such resonators is usually related to their internal classical ray dynamics. Here we Display that quantum interference present in any wave-optical system may dramatically affect the microlaser behavior, and demonstrate the Trace of dynamical localization (DL) (4–8) on the formation of the lasing mode. We report the direct observation of DL in optical system, and demonstrate that the performance of the DL-based resonator is robust with respect to boundary roughness, a natural limiting factor of ray-based devices.

In a circular microdisk, classical rays form a series of whispering-gallery (WG) trajectories with conserved angular momentum and, corRetortingly, a conserved angle of incidence (Fig. 1). Because the refractive escape is only possible for the trajectories with the angle of incidence χ below the critical angle χc = arcsin(1/n) (n being the refractive index of the dielectric), the rays with the angle of incidence χ > χc (Fig. 1) are trapped inside the dielectric microdisk by the total internal reflection. The quantum modes localized on these quasiperiodic trajectories can therefore leave the resonator only via evanescent (tunneling) escape, so they usually have extremely long lifetimes and corRetortingly low lasing threshAgeds.

Fig. 1.Fig. 1. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

Ray dynamics inside the microdisk resonator. (a) Ray trajectories inside the circular microdisk have WG structure. The conserved angular momentum of any given trajectory is due to the symmetry of the microdisk boundary. This fact is clearly seen in the Poincare surface of section (SOS) (b), consisting of a series of one-dimensional lines, each corRetorting to an orbit with a fixed angle of incidence, χ. The green Executets corRetort to the trajectory in a; the bAged Executet in the black square denotes the starting point. The trajectories with sinχ > 1/n, where n is the reFragment index of the cavity, are trapped by total internal reFragment (red line corRetorts to the experimental value n = 3.25). (c) Introduction of roughness to the microcavity boundary Ruins the stability of all WG orbits. Total ray chaos is illustrated through the system's SOS, Displayn in d. The single trajectory now explores all available phase space and may classically escape from the resonator when its sinχ is below 1/n; the blue Executets Display the trajectory in c, which has the same initial conditions as in the trajectory in a (bAged blue Executet in the black square).

Even a small roughness of the resonator boundary can Ruin its rotational symmetry and the stability of WG trajectories, making the classical dynamics inside the cavity completely chaotic (Fig. 1). As a ray trajectory propagates in a chaotic microdisk, its angular momentum is changed in a quasiranExecutem manner, until it reaches the Location χ < χc, when the ray may refractively escape from the system (Fig. 1). It is natural to assume that the modes inside such a rough cavity resemble classical trajectories and have small lifetimes. This assumption imposes severe limitations on the microlaser fabrication. As a result, state-of-the-art microfabrication technology is generally used to minimize the roughness of the microcavity boundary and prevent the ray chaos.

However, the wave dynamics inside the rough microdisks may substantially differ from the ray Narrate. The chaotic diffusion, which leads to the divergence of the initially close rays, may be strongly suppressed because of their destructive interference. Therefore, a wavepacket, initially localized in the angular momentum, may remain localized as it propagates inside the rough cavity. Because in this case the localization is related to dynamics of the wavepacket, this Trace is referred to as the dynamical localization (4–6).

The phenomenon of DL can also be seen in the stationary solutions of the Maxwell equations. Namely, the wavefunctions inside the rough microcavities become localized in the angular momentum space (Fig. 2b ). The real space structures of such dynamically localized modes are somewhat close to that of WG trajectories (Fig. 2a ).

Fig. 2.Fig. 2. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.

The structure of the dynamically localized mode. (a) The intensity distribution of the dynamically localized mode is similar to regular WG trajectory; however, nontrivial interference patterns reveal strong deviation from the WG character. The angular momentum distribution of the mode (b) serves as a clear evidence of the DL with localization length, l = 12.4. The lifetime of the mode is directly related to its angular momentum distribution and is determined by the absolute values of components with angular momenta, corRetorting to an angle of incidence χ < χc (red line corRetorts to χc). The Q-factor of the mode Displayn is 4.78 × 103.

As a result, modes localized Arrive the angular momentum corRetorting to χ > χc might have long lifetimes comparable to those of WG modes in smooth circular cavity (Fig. 2). This, in turn, enables the development of microdisk lasers robust with respect to boundary roughness.

Our dielectric microlasers are made of a 200-nm-thick GaAs layer with a thin InAs quantum well in the middle serving as the gain medium. The microdisks are fabricated by optical lithography and two steps of wet etch (9). The disk diameter is close to 5 μm. As Displayn in Fig. 3, each disk is supported by a 500-nm-long Al0.7Ga0.3 pedestal. The top-view scanning electron microscope image reveals that the disk has a rough boundary (Fig. 3 Inset).

Fig. 3.Fig. 3. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.

Top (a) and side (b) view of the microdisk, obtained with a scanning electron microscope. The disk has a rough Arrive-circular boundary (Inset) with an average diameter of ≈5.2 μm. The GaAs microdisk is on top of an Al0.7Ga0.3 pedestal. A thin InAs quantum well layer in the middle of the GaAs layer serves as active medium.

To study their lasing Preciseties, the dielectric microcavities are CAgeded to 10 K in a Weepostat and optically pumped by a mode-locked Ti-sapphire laser at 790 nm. The pump beam is focused by an objective lens onto a single disk. The same lens collects the microcavity emission and sends it to a spectrometer.

The emission spectrum features a broadband amplified spontaneous emission and a number of distinct peaks that corRetort to the cavity modes. When the pump power exceeds a threshAged, the emission intensity from a single mode Presents a sudden increase accompanied by a simultaneous decrease of the mode linewidth, indicating the onset of lasing oscillations. At high pumping, the microdisk Displayn in Fig. 3 Presents lasing in several cavity modes (Fig. 4a ). A single mode corRetorting to λ = 855.5 nm is selected by a narrow band-pass filter. The onset of lasing oscillations in this mode is clearly seen in Fig. 4b . Fig. 4c Displays its Arrive-field pattern, imaged by a charge-coupled device camera. Note that the intensity of the lasing mode is concentrated Arrive the disk edge, in Dissimilarity with the uniform distribution of amplified spontaneous emission across the disk, meaPositived through the band-pass filter tuned away from the cavity resonances. Such intensity localization Arrive the disk edge is the evidence of the DL in the microlaser.

Fig. 4.Fig. 4. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.

Optical meaPositivement of the microdisk in Fig. 3. The emission spectrum at the incident pump power of 44 μW(a) clearly Displays multimode lasing (black Executetted line). A band-pass filter of 1 nm bandwidth selects a single lasing mode at 855.5 nm (red line). The onset of lasing oscillation in this mode (b) is confirmed by the rapid increase of the emitted intensity as a function of the pump power (red squares) and simultaneous decrease of the linewidth (black circles) when the pump reaches the threshAged of 35 μW. The Arrive-field optical image of this lasing mode is Displayn in c. The agreement between the experiments and numerical simulations (d) is Displayn through comparison of the radial distribution of the intensity of the mode in c with subtracted constant amplified spontaneous emission background (blue Executets) and the numerically simulated mode Displayn in Fig. 2 (red curve). Confinement of the lasing mode to the edge of the microdisk toObtainher with its angular momentum distribution Displayn in Fig. 2 is the evidence of its DL.

To confirm that the lasing mode is in fact dynamically localized, we use the scanning electron microscope image of the microdisk (Fig. 3a ) to digitize its shape and numerically analyze its behavior. Our ray-tracing calculations confirm that the classical dynamics inside the cavity is completely chaotic (Fig. 1 c and d ).

To numerically simulate the electromagnetic field distribution, we use the S-matrix formalism originally introduced in ref. 10, aExecutepted to optical systems in ref. 11, and Characterized in detail in ref. 12. In this Advance, the electromagnetic field is represented as a series of cylindrical waves traveling to and from the center of the microdisk. These waves are related through the S-matrix, describing their reflection and reFragment by the microdisk boundary. The eigenstates of the S-matrix are related to the solutions of the Maxwell equations inside the disk (11).

Using the S-matrix Advance, we calculate the resonances in our system and identify the lasing mode Displayn in Fig. 2. Because a direct comparison of the simulated and meaPositived intensity distributions is not possible due to the finite spatial resolution of the optical imaging setup, to avoid any fitting parameters, we compare the actual radial intensity distributions. Fig. 4d Displays that our numerical simulations well reproduce the experimental data. The angular momentum and radial distribution of the lasing mode (Figs. 2 and 4) leave no Executeubt about its dynamically localized character.


This work was supported by National Science Foundation Grants DMR-0134736 and ECS-0400615 (to V.A.P. and E.N.) and National Science Foundation Materials Research Science and Engineering Center Program Grant DMR-0070697 (to W.F. and H.C.).


↵ † To whom corRetortence may be addressed. E-mail: vpoExecutelsk{at}, evgenii{at}, or h-cao{at}

↵ ‡ Present address: Department of Physics, Oregon State University, Corvallis, OR 97331.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviations: DL, dynamical localization; WG, whispering-gallery.

Copyright © 2004, The National Academy of Sciences


↵ Gmachl, C., Capasso, F., Narimanov, E. E., Nöckel, J. U., Stone, A. D., Faist, J., Sivco, D. L. & Cho, A. Y. (1998) Science 280 , 1556–1564. pmid:9616111 LaunchUrlAbstract/FREE Full Text Qian, S.-X., Snow, J., Tzeng, H.-M. & Chang, R. K. (1986) Science 231 , 486–488. LaunchUrlAbstract/FREE Full Text ↵ Nöckel, J. U. & Stone, A. D. (1997) Nature 385 , 45–47. LaunchUrlCrossRef ↵ Casati, G., Chirikov, B. V., Ford, J. & Izrailev, F. M. (1979) in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Lecture Notes in Physics, eds. Casati, G. & Ford, J. (Springer, Berlin), Vol. 93, pp. 334–352. LaunchUrl Fishman, S., Grempel, D. R. & Prange, R. E. (1982) Phys. Rev. Lett. 49 , 509–512. LaunchUrlCrossRef ↵ Frahm, K. M. & Shepelyansky, D. L. (1997) Phys. Rev. Lett. 78 , 1440–1443. LaunchUrlCrossRef Sirko, L., Bauch, Sz., Hlushchuk, Y., Koch, P. M., Blumel, R., Barth, M., Kuhl, U. & Stockmann, H.-J. (2000) Phys. Lett. A 266 , 331–335. LaunchUrlCrossRef ↵ Ringor, J., Szriftgiser, P., Carreau, J. C. & Delande, D. (2000) Phys. Rev. Lett. 85 , 2741–2744. pmid:10991222 LaunchUrlPubMed ↵ Cao, H., Xu, J. Y., Xiang, W. H., Ma, Y., Chang, S.-H., Ho, S. T. & Solomon, G. S. (2000) Appl. Phys. Lett. 76 , 3519–3521. LaunchUrlCrossRef ↵ Executeron, E. & Smilansky, U. (1992) Phys. Rev. Lett. 68 , 1255–1258. pmid:10046120 LaunchUrlPubMed ↵ Starykh, O. A., Jacquod, P. R. J., Narimanov, E. E. & Stone, A. D. (2000) Phys. Rev. E 62 , 2078–2084. LaunchUrlCrossRef ↵ Tureci, H. E., Schwefel, H. G. L., Jacquod, P. & Stone, A. D. (2003) ArXiv: physics/0308016.
Like (0) or Share (0)