Unidirectional light emission from low-index polymer microlasers
M. Schermer, S. Bittner, G. Singh, C. Ulysse, M. Lebental, J. Wiersig
aa r X i v : . [ phy s i c s . op ti c s ] M a r Unidirectional light emission from low-index polymer microlasers
M. Schermer, S. Bittner, G. Singh, C. Ulysse, M. Lebental, ∗ and J. Wiersig † Institut f¨ur Theoretische Physik, Otto-von-Guericke-Universit¨at Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany Laboratoire de Photonique Quantique et Mol´eculaire, CNRS UMR 8537,Institut d’Alembert FR 3242, ´Ecole Normale Sup´erieure de Cachan,61 avenue du Pr´esident Wilson, F-94235 Cachan, France Laboratoire de Photonique et Nanostructures, CNRS UPR20, Route de Nozay, F-91460 Marcoussis, France (Dated: May 8, 2019)We report on experiments with deformed polymer microlasers that have a low refractive indexand exhibit unidirectional light emission. We demonstrate that the highly directional emission isdue to transport of light rays along the unstable manifold of the chaotic saddle in phase space.Experiments, ray-tracing simulations, and mode calculations show very good agreement.
The physics of optical microcavities is a topical re-search field for more than one decade [1]. Microcav-ities [2, 3] confine photons for a long time τ due tototal internal reflection at the boundary of the cavity.These so-called whispering-gallery modes (WGMs) havehigh quality factors Q = ωτ , where ω is the resonancefrequency. The in-plane light emission from an idealcircular-shaped microlaser is isotropic due to the rota-tional symmetry. To overcome this disadvantage micro-cavities with deformed boundaries have been fabricatedleading to significantly improved emission patterns [4–8]. Even unidirectional emission is possible, which hasbeen demonstrated for several shapes, e.g., the spiral [9–11], cavities with holes [12, 13], the lima¸con [14–20], thecircle with a point scatterer [21], and the notched el-lipse [22, 23].The ray dynamics inside a deformed microcavity is(partially) chaotic, i.e., neighboring ray trajectories devi-ate from each other exponentially fast. Because of this,deformed microdisks have attracted attention as mod-els for studying ray-wave correspondence in open sys-tems [5]. This is analog to the study of quantum-classicalcorrespondence in the field of quantum chaos [24]. Inopen chaotic systems the long-time behavior of trajec-tories is governed by the chaotic saddle (or chaotic re-peller for noninvertible dynamical systems) and its un-stable manifold [25, 26]. The chaotic saddle is the set ofpoints in phase space that never visits the leaky regionboth in forward and backward time evolution. The un-stable manifold of a chaotic saddle is the set of pointsthat converges to the saddle in backward time evolution.This unstable manifold therefore describes how trajecto-ries, after a transient time, escape from the open chaoticsystem. It has been shown both theoretically and exper-imentally that this kind of unstable manifold in chaoticmicrocavities determines the far-field pattern of all high- Q modes [7, 27]. As a consequence, the high- Q modes insuch a cavity have very similar far-field patterns. Thisuseful property has been termed universal far-field pat- ∗ [email protected] † [email protected] PSfrag replacements (a) (b) µ m 0 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ FIG. 1. (a) Top view of a shortegg microlaser with R = 80 µ m in real colors with an optical microscope. (b) Measuredfar-field intensity pattern of a polymer shortegg microlaser ina polar plot as a function of the azimuthal angle φ . tern [27]. Based on this concept it was predicted thatthe light emission from high- Q modes in a lima¸con cavitywith refractive index n between 2.7 and 3.9 is universaland unidirectional [14]. This was confirmed experimen-tally by a number of groups [15–20].Microcavities made from materials with low refractiveindex n ≤ . ≤ n ≤ .
8, its emission isstrongly concentrated in a single direction [see Fig. 1(b)],the far-field pattern is universal, and the quality factorsare reasonably high. Numerical simulations and experi-mental data show good agreement.The boundary of the shortegg cavity is given in polarcoordinates by ρ ( φ ) = R [1 + ε cos( φ ) + ε cos(2 φ ) + ε cos(3 φ )] (1)with mean radius R and deformation parameters ε =0 . ε = − . ε = − .
05. The ray dynamics in-side microcavities is best described in a two-dimensionalphase space representation, the so-called Poincar´e surfaceof section. Whenever the ray hits the cavity’s boundary,its position s in terms of the arclength coordinate alongthe circumference and its tangential momentum sin( χ )are recorded. Here, χ is the angle of incidence measuredfrom the surface normal. An angle χ < χ >
0) indi-cates clockwise (counterclockwise) propagation direction.When | χ | is larger than the critical angle for total inter-nal reflection, χ c = arcsin (1 /n ), the ray is completelyreflected. In the leaky region of phase space | χ | < χ c theray is only partially reflected according to Fresnel’s law.The ray dynamics inside the shortegg is chaotic exceptfor small islands of regular motion in the leaky region(not shown). Figure 2 depicts the unstable manifold ofthe chaotic saddle for the case of transverse electric (TE)polarization (i.e., electric field parallel to the plane of thecavity) with effective refractive index n = 1 . s/s max ≈ .
35 andone related by symmetry at s/s max ≈ .
65. However, incontrast to the lima¸con cavity these two overlap regionsare rather elongated and it is therefore not clear a priori how this can lead to unidirectional emission. In fact,we have optimized the boundary of the cavity such thatthe arms of the unstable manifold follow precisely thecurve of points that are emitted to the far-field angle 0 ◦ (red curve in Fig. 2). Due to this fact, a highly directedemission can be expected.The far-field intensity pattern resulting from the ray-tracing simulations is shown as dashed black curve inFig. 3. As predicted by the structure of the unstablemanifold, the emission is strongly peaked around the far-field angle φ = 0 ◦ with considerably smaller side peaksat ± ◦ and ± ◦ . The beam divergence is as smallas ± ◦ , which is much smaller than the beam divergencefrom the lima¸con cavity [14] and comparable to the beamdivergence from the notched ellipse [23] in the high-indexregime. The small oscillations of the far-field intensitypattern in Fig. 3 are due to the nontrivial fine struc-ture of the unstable manifold which is not visible on thescale shown in Fig. 2. Due to the small value of n theemission for TM polarization is similar with additionalsmall side peaks at ± ◦ (not shown). Extensive numer-ical investigations reveal that the unidirectional emissionpersists for 1 . ≤ n ≤ . PSfrag replacements . . . . . s/s max -1 . . . . s i n ( χ ) FIG. 2. Unstable manifold (blue points) of the chaotic saddlein the shortegg cavity for TE polarization. The red horizontallines mark the border of the leaky region | χ | < χ c . The solidred curve is the set of points that leave the cavity towards thefar-field angle 0 ◦ . Inset: typical trajectory from the unstablemanifold. PSfrag replacements -150 -100 -50 0 50 100 150Far-field angle φ (deg)0100200300400500 I n t e n s i t y ( a r b . un i t s ) FIG. 3. Far-field intensity pattern from ray-tracing simu-lations (dashed black curve) and mode calculations (solidred curve) for TE polarization. Left inset: near-field inten-sity pattern of the corresponding WGM with radial modenumber 1 and dimensionless frequency Re(Ω) ≈ .
41, and Q ≈
25 800. Right inset: near-field intensity pattern ofa whispering-gallery-like mode with radial mode number 3,Re(Ω) ≈ .
28, and Q ≈ spect to small variations of the deformation parameters ε , ε , and ε . However, the directionality is considerablyspoiled for ε = 0.Numerical simulations of Maxwell’s equations in theshortegg geometry are performed within the effective re-fractive index approximation in two dimensions (see, e.g.,Ref. [32]). Using the boundary element method [33] wedetermine the spatial mode pattern ψ ( x, y ) correspond-ing to the z -component of the magnetic (TE) and electric(TM) field as well as the complex resonant frequencies ω = ck where c is the speed of light in vacuum and k isthe wave number. The real part is the conventional fre-quency whereas the imaginary part determines the life-time τ = − / [2 Im ω ] of a given mode. For convenience,the dimensionless complex frequency Ω = ωR/c = kR isused.In the considered frequency regime the computed high- Q modes are WGMs (two examples for n = 1 . Q s are formedin such a strongly deformed cavity with chaotic ray dy-namics can be explained by the existence of partial bar-riers in phase space [34]. Moreover, we verified that theHusimi functions [35] (not shown) of the high- Q modes inthe leaky region are well localized on the unstable mani-fold. This presents another indication that the emissiondirectionality in the shortegg is due to the unstable man-ifold of the chaotic saddle.Shortegg cavities were experimentally investigated us-ing organic microlasers. The microlasers were fabricatedfrom a 700 nm thick layer of PMMA doped with 5 wt%of the laser dye DCM . The dye-doped polymer withbulk refractive index 1 .
54 was spin-coated on a Si waferwith a 2 µ m buffer layer of SiO to avoid leakage of thecavity modes into the Si wafer. The cavities were cre-ated from the polymer layer by electron-beam lithogra-phy which ensured nanometric precision [38]. They canbe considered as two-dimensional systems with effectiverefractive index n = 1 .
50 (see Ref. [32] for details of theeffective refractive index calculation). The cavities werecompletely supported by the silica layer[38]. An opti-cal microscope image of a shortegg cavity is shown inFig. 1(a). The results presented here were measured witha cavity with radius R = 80 µ m [Re(Ω) ≃ ǫ , as for theshortegg and ǫ = 0. The microlasers were pumped by apulsed, frequency-doubled Nd:YAG laser (532 nm, 0 . . · cm − and is about three times lowerthan that of the quadrupolar microlaser. This evidencesthat the particular shape of the shortegg leads to an en-hancement of the quality factors compared to other cav- Poly(methyl methacrylate) (PMMA A6 resist by Microchem) PSfrag replacements (a) I n t e n s i t y ( c o un t s ) I n t e n s i t y ( c o un t s ) (b) µ m)0 . . F o u r i e rtr a n s f o r m ( a r b . un i t s ) (c) FIG. 4. (a) Measured lasing spectrum in the direction φ = 0 ◦ for a shortegg microlaser with R = 80 µ m just above the laserthreshold. (b) Normalized Fourier transform of the spectrumin (a). The vertical dotted lines indicate the expected opticallengths of the diameter, the stable four bounce inscribed orbit( ℓ (4b)geo ≃ . R = 462 . µ m), and the perimeter. (c) Idem(a) far above threshold. ities with similar shapes. The lasing emission was TEpolarized.A spectrum recorded for φ = 0 ◦ just above threshold isshown in Fig. 4(a) and features a sequence of equidistantlasing modes. Lasing modes with large Re(Ω) are oftenrelated to specific sets of trajectories. If this is the case,then the optical length of these trajectories is inverselyproportional to the free spectral range, ℓ opt = 2 π/ ∆ k ,and can be determined from the Fourier transform ofthe spectrum [32]. The Fourier transform of the spec-trum in Fig. 4(a) is presented in Fig. 4(b). Its first twosignificant peaks are at ℓ opt = 755 . µ m and 785 . µ m,and further peaks are found at multiples of these lengths.They correspond to a geometrical length of 455 µ m and473 µ m, respectively, for a group refractive index of n g = 1 .
66. These lengths are significantly longer thanthe length of the diameter, ℓ (diam)geo ≃ . R = 313 µ m,and hence evidence that the lasing modes are of thewhispering gallery type. On the other hand they are PSfrag replacements (a) (b)
FIG. 5. (a) Photograph of a lasing shortegg cavity. The lasingemission appears in red (real color). The green pump beamis not visible. The direction φ = 0 ◦ is turned towards thecamera. (b) Sketch of the cavity geometry and the lasingemission corresponding to (a). also significantly shorter than the length of the perime-ter, ℓ (per)geo ≃ . R = 509 µ m, which indicates thatthe observed modes are WGMs with higher radial ex-citation. In contrast, the spectrum of the quadrupolarmicrolaser (not shown) corresponds to the optical lengthof the diameter orbit. The shortegg cavity exhibits alarger number of resonances for higher pump intensities[see Fig. 4(c)], but the Fourier transform features onlypeaks at the same optical lengths as in Fig. 4(b).The measured far-field intensity pattern in Fig. 1(b)shows the maximal intensity of each spectrum as a func-tion of the azimuthal angle. It was recorded for a pumpintensity two times higher than the threshold. No sig-nificant effect of bleaching was observed during the mea-surement. The quadrupolar microlaser exhibits a verybroad far-field intensity pattern without sharp emissionlobes (not shown). On the contrary, the far-field intensitypattern of the shortegg micro-lasers exhibits one domi-nant emission lobe at φ = 0 ◦ with a divergence of about ± ◦ , in very good agreement with the theoretical predic-tions. Four much smaller lobes at ± ◦ and ± ◦ arealso observed. This is consistent with the numerical sim- ulations even though the lasing modes probably have ahigher radial excitation than the modes shown in Fig. 3,which once again shows the universality of the emissionpatterns.A photograph of the lasing cavity made by a CMOSsensor camera with a high-magnification zoom lens is pre-sented in Fig. 5(a). The observation direction was φ = 0 ◦ and the camera had an inclination angle of 10 ◦ with re-spect to the plane of the cavity. The photograph showsthat the red lasing emission towards φ = 0 ◦ originatesfrom two small regions of the cavity boundary around φ = ± ◦ . Figure 5(b) shows a sketch of the cavitywhere the emitting parts of the cavity are indicated inred. Their positions agree very well with the calculatednear-field intensity patterns shown in the insets of Fig. 3.We have demonstrated both theoretically and exper-imentally that low-index polymer microlasers with theso-called shortegg shape exhibit lasing modes with highlydirectional emission. The lasing modes are based on theunstable manifold of the chaotic saddle which leads toa universal (i.e., mode-independent) far-field pattern ofthe high-Q modes. The theoretical predictions of near-and far-field intensity patterns showed excellent agree-ment with the experimental findings.Fruitful discussions with J. Zyss, J.-B. Shim andM. Kraft are acknowledged. M. S. thanks A. Ebersp¨acherfor providing his computer code package. S. 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