June 13, 2003

We report the surprising observation of highly directional light emission from nearly spherical fused silica optical resonators, in which most of the phase space is filled with non-chaotic regular trajectories. The observed emission pattern, along with a theoretical analysis, shows that non-perturbative phase space structures in the internal ray dynamics profoundly affect tunneling leakage of the whispering gallery modes. |

Electromagnetic fields in a uniform dielectric sphere can be calculated in much the same
way as quantum mechanical wave functions in a spherically symmetric potential. The sphere
exhibits whispering-gallery (WG) modes, which are long-lived resonances with electromagnetic
fields confined near the surface [1]. For small deviations from the spherical shape,
ε≡(r_{max} - r_{min})/(r_{max} + r_{min}) « 1 (where r_{max}, r_{min} are the maximal and minimal radii),
perturbative treatments [2] are routinely employed to infer deformation parameters from the
splitting of azimuthal degeneracy of the WG modes [3]. Certain strongly non-spherical
resonators, on the other hand, can be analyzed with methods from quantum chaos such
as random-matrix theory or periodic-orbit expansions [4]. Many generic resonator
shapes, however, fall into a transition regime in which none of these known methods
apply globally. When entering this regime from the perturbative side, calculations
may encounter singularities and undefined limits [5]. Experimental studies in this
regime can thus provide insight into how nature resolves the competition between
perturbative and non-perturbative physics, here with the resonator shape as a control
parameter.

In this paper we present studies of far-field emission patterns and resonance lifetimes of
deformed fused-silica microspheres at various degrees of deformation. Highly directional
emission from WG modes with Q-factors (proportional to the modal lifetime) near 10^{8} is
observed from microspheres with ε ≈ 1% and with the size parameter kR = 2πR/λ ≈ 785 ( λ ≈ 800nm,
R ≈ 100μm). These results are completely unexpected in the ray optics limit or in the earlier
perturbative wave treatment with ε as a small parameter. The observed emission pattern
illustrates how the non-perturbative phase space structure in the internal ray dynamics can
profoundly affect tunneling leakage of the WG modes. Directional emission patterns have
previously been observed only in more strongly deformed resonators where a significant
fraction of the internal rays shows chaotic motion [6] and consequently the Q-factors are much
smaller than reported here.

Deformed microspheres were formed by melting together two spheres of similar sizes. The
individual spheres were fabricated by melting an optical fiber tip with a focused CO_{2} laser
beam. The spheres were brought into contact and heated until surface tension produced a
completely convex surface. By carefully controlling the temperature of the glass using the CO_{2}
laser, it was possible to repeatedly reduce the degree of deformation. For reference, we
define the elongated axis of the resulting prolate spheroid, which also connects the
centers of the two original spheres, to be the x-axis. The z-axis is defined by the
remaining fiber stem, which breaks the rotational symmetry about the x-axis, making
the deformed microsphere completely non-axisymmetric. Images of these deformed
microspheres taken from three orthogonal directions have been shown in an earlier study
[7].

To investigate emission properties and resonance lifetimes of the deformed microspheres, we
use frustrated total internal reflection at the surface of a prism with refractive index n = 1.7 to
launch individual traveling-wave WG modes near the x-y plane of the spheroid (Fig. 1a inset).
The measurement was performed with a tunable diode laser (New Focus). In this approach,
the initial angle of incidence χ_{0} inside the microsphere with which the WG modes were
launched can be controlled by adjusting the angle of incidence ψ inside the prism.

WG mode spectra were obtained by measuring the far-field emission intensity as a function of the excitation wavelength. Since the stem holding the resonator is a strong leakage pathway, the observed long-lived WG modes exhibit minimal amount of scattering near the stem region and thus must have internal field patterns that do not overlap with the stem region. This is made possible by the fact that the resonator is slightly flattened in the z direction (along the stem axis). Experiments as well as ray simulations show that this deformation stabilizes the rays in the vicinity of the x-y plane. Our directionality measurements correspondingly were performed in this plane. By scanning the detector while keeping the resonator fixed, we recorded WG mode spectra at various angles, φ (from the x-axis), from which we constructed the far-field emission pattern at a given microsphere deformation ε. The microsphere was then reheated to reduce the degree of deformation and the far-field emission pattern was measured again. Figure 1 shows the evolution of the measured far-field emission pattern of a microsphere as its deformation is reduced. The excitation beam is s-polarized, and the detection scheme is polarization-insensitive. The far-field emission patterns are independent of the polarization of the excitation laser beam.

Beginning with a strongly deformed microsphere, the emission was observed to have a
strong peak at φ= 45^{o} in the far field [see Fig. 1 (a)]. Since clockwise traveling waves were
excited, as indicated by the white arrow in the sphere image in Fig. 1 (a), this far-field
emission direction corresponds to light escaping tangential from the region at φ= -45^{o} on the
surface of the microsphere, shown by the solid arrow in the sphere image. A much smaller
peak (5% of the large peak height) in the far-field emission pattern was observed at
φ= 135^{o}, corresponding to light escaping from the region at φ= 45^{o} on the microsphere
surface (dotted arrow). As the deformation was reduced, the far-field emission peak at
φ= 135^{o} grew to about one quarter of the φ= 45^{o}-peak height in Fig. 1 (b). In Fig.
1 (c), the two peaks reached nearly equal intensity. The bright emission spots at
φ= ± 45^{o} on the resonator surface can be viewed directly with a CCD camera. For
reference, we call the pattern in Fig. 1 (a) asymmetric and the pattern in Fig. 1 (c)
symmetric (around φ= 90^{o}). Figures 1 (e)-(g) also show that as the deformation was
reduced, the Q-factor of the relevant WG modes increased by nearly four orders of
magnitude.

The measurements discussed thus far were performed with input condition χ_{0} ≈ 90^{o} (sinχ _{
0} ≈ 1).
Figure 1 (d) shows the far-field emission pattern from the same microsphere as Fig. 1 (c) but
with light launched at sinχ _{0} ≈ 0.8. The emission pattern is nearly identical to Fig 1 (c) where
sinχ_{0} ≈ 1, although the corresponding Q-factor is nearly two orders of magnitude smaller [Figs. 1
(g) and (h)].

The observed emission patterns in Fig. 1 (a)-(d) disagree with the intuitive expectation that
WG modes in an oval resonator should preferentially emit tangential to the points of highest
curvature, into the far-field direction φ= ± 90^{o}. This is what one obtains when modeling our
spheroids as triaxial ellipsoids, whose internal ray dynamics exhibits no chaos [8],
independently of the axis ratios.

The observed emission pattern of the most deformed microsphere, Fig. 1 (a), is well
explained by a ray model. The peak at φ= 45^{o} can be attributed to an effect known as
dynamical eclipsing [6]. In the ray model, a WG mode corresponds to light rays trapped close
to the perimeter of the dielectric resonator by total internal reflection, which prevents light
escape unless a critical angle χ_{c}≡ arcsin 1/n = 43.6^{o} is reached, where n = 1.45 is the refractive
index of the fused silica. At small but finite ε, any oval can be approximated by the first terms
of a multipole expansion, which after proper choice of origin is quadrupolar. In this limit, a
stable 4-bounce orbit shaped like a diamond forms with its sharp vertices at the
highest-curvature points of the resonator with an angle of incidence χ_{4} ≈ 45^{o} [inset to
Fig. 1 (d)]. In the phase space describing the possible types of ray motion [4], this
creates “islands of stability”: rays launched near the diamond orbit will retain similar
reflection points and angles even if they do not close onto themselves after four
bounces. Chaotic rays, which exponentially diverge from trajectories with closely
neighboring initial conditions, cannot penetrate the diamond-shaped 4-bounce islands. At
the deformation used in Fig. 1 (a), most of the phase space supporting the WG
mode in question is chaotic. However, since χ_{4} ≈ χ_{c} chaotic WG rays are prevented from
refractively escaping at the points of highest curvature. Instead, as trajectories flow
around the islands in one direction, refractive escape occurs near φ= -45^{o} on the
resonator surface. From this circulation around the islands, asymmetric far-field emission
patterns result as a hallmark of refractive escape. This is what we observe in Fig. 1
(a).

This ray mechanism was originally proposed and tested for effectively 2D systems where
islands of stability and chaotic regions are mutually exclusive. In our 3D spheroids, the same
diamond-shaped stable orbit exists in or near the x-y plane, and refractively escaping rays in
its vicinity have lifetimes that correspond to ≤ 10 round trips in the resonator, translating to a
Q-factor near 10^{4}. During this time, a given ray can be considered as moving in a
cross-sectional plane that may be inclined and slowly precessing around the z-axis. The
variation of cross-sectional shape, Δ ε, probed by such rays causes no significant differences in the
size of the 4-bounce islands [9, 10]. Therefore, chaotic rays in the 3D resonator behave as in a
corresponding fixed 2D resonator during a relatively short time scale preceding
an escape event, and in particular avoid islands of stability, leading to dynamical
eclipsing.

However, refractive escape ceases to be the dominant leakage mechanism at the very small deformation used in Figs. 1 (c) and (d), since in this case chaotic diffusion is no longer effective and the stable diamond orbit itself also becomes fully confined by total internal reflection. In addition, ray simulations neglecting tunneling escape cannot produce the symmetric emission patterns observed in Figs. 1 (c) and (d) [11]. The symmetric emission in Figs. 1 (c, d) reveals the importance of tunneling as a correction to the ray picture, as discussed below.

Tunneling becomes the only decay mechanism in an ideal sphere where each WG mode in
the plane of excitation corresponds to a unique azimuthal angular momentum number m with
respect to the z-axis, which is semiclassically related to the angle of incidence by
sinχ_{0} = m/(nkR), using the fact that the modes of interest remain close to the x-y plane and
hence have total angular momentum l ≈ m. At kR » 1, the tunneling escape rate is negligible
when sinχ _{0} ≈ 1 but accelerates exponentially as sinχ _{0} approaches sinχ _{c}. In a deformed microsphere,
the angle of incidence χ varies as function of φ. χ(φ) is a well-defined function of φ provided it covers
an angular momentum range where chaos is neglibible [12]. A circulating ray with varying
χ(φ) >χ _{c} will then escape with exponentially strong selectivity near the minima of
χ(φ).

The emission locations and directions observed in the experiment indicate that the minima
of χ(φ) should lie near the corners of the unstable rectangular 4-bounce orbit [inset to Fig. 1 (d)].
We determined from ray tracing that this only occurs near but above χ_{c}. Thus, the symmetric
far-field patterns in Figs. 1 (c) and (d) are a tunneling probe for the minima of
the ray-optical χ(φ) in the vicinity of the critical angle χ_{c}. To further corroborate this
interpretation, recall that the emission patterns as shown in Figs. 1 (c) and (d)
are insensitive to the initial angle of incidence, χ_{0}, as they should be if the detected
light originates near sinχ ≈ sinχ _{c}. The question remains how the light coupled into the
resonator is able to reduce its angular momentum from a high value corresponding to
sinχ_{0} to the much lower sinχ _{c}. Our observation shows that such a dynamical process
is present in the spheroid, but does not unambiguously reveal its mechanism. A
possible mechanism is Arnol’d diffusion, a phase-space transport process that crucially
depends on the fact that the resonators are in fact three-dimensional [7], with no axial
symmetry. While this exceedingly slow process can lead to very long lifetime, we defer a
discussion of these truly 3D effects to a future publication [10], and focus instead on the
phase space regions that are near sinχ _{c} and are responsible for the observed emission
patterns.

Figures 2 (a) and (b) plot the numerical results of the intensity patterns of two WG modes under traveling-wave excitation [12, 13]. The calculation assumes s-polarized light, uses size parameters near kR ≈ 113 [14], and models the shape felt by modes in the x-y plane as the cross section of a quadrupolar cylinder. This 2D model is justified by the 3D ray simultions discussed earlier: the emission patterns probe ray motion near the critical angle on relatively short time scales over which deviations from a planar dynamics are negligible.

The emission patterns in Figs. 2 (a) and (b) agree qualitatively with the asymmetric and
the symmetric emission patterns shown in Figs. 1 (b) and (d), respectively. To understand the
physical mechanism for the emission in Fig. 2 (b), we plot in Fig. 2 (c) the distribution of
angular momentum numbers for the WG mode in (b). As shown in Fig. 2 (c), there is
negligible overlap with the window for refractive escape below the critical angle χ_{c}, which in
this plot translates to a critical angular momentum m_{c} = kR ≈ 113. This indicates that the
emission from this mode is due to tunneling escape. By comparison, both refractive and
tunneling escape contribute to the emission in Fig. 2 (a), with the refractive escape playing a
dominant role.

The peak splitting of width Δm ≈ 4 in Fig. 2 (c) is a straightforward consequence of the oval
deformation: as the wave circulates around the resonator with varying radius between r_{min}
and r_{max}, its angular momentum oscillates but has high probabitity of being near its extrema,
given by the extrema of m ≈ r(φ)nkR sinχ(φ) over the surface. The fact that the minima of χ(φ) do not
occur at φ= 0^{o}, 180^{o} on the surface is due to the 4-bounce island structure in the vicinity of
sinχ = sin 45^{o}.

For comparison, Fig. 2 (c) also shows the distribution of angular momentum number of an
ultra-high Q WG mode with emission directionality as expected for the ellipse [see Fig. 2 (d)].
The mode is confined at a high m, corresponding to sinχ _{0} ≈ 0.91. Note that here the tunneling
loss is negligible compared with other loss mechanisms such as scattering or absorption loss of
the material that limits the actual Q-factor of fused silica microspheres to of order 10^{9} [15].
The two modes in Fig. 2 (c) have practically no overlap in angular momentum space. The
qualitative difference in the far-field patterns of these two modes further confirms
that the mode shown in Fig. 1 (c), which was excited at sinχ _{0} ≈ 1, is not confined to
sinχ_{0} ≈ 1.

In essence, our experiment exploits the peculiar feature that for fused silica the critical angle lies near the phase-space islands corresponding to the stable 4-bounce orbit. These islands are the dominant structure in the WG region and persist even at small ε where most of the phase space is filled with non-chaotic, “regular” trajectories. The island formation is a non-perturbative consequence of the breakdown of conservation laws [8], which qualitatively distinguishes the generic quadrupolar shape from an ideal ellipse even though the two shapes differ only to second order in ε. The emission characteristics of a silica spheroid in the seemingly trivial small-ε regime will continue to be strongly affected by ray patterns that wrap around the perimeter in approximately four bounces.

Whether any non-perturbative structure in the ray dynamics can be resolved by the wave
field, depends on the size parameter kR [16]: a directionality measurement will be
able to distinguish the peaks at φ= 90^{o} ± 45^{o} if the conjugate angular momentum m
satisfies the uncertainty relation Δφ Δm ≥ 1/2 with Δφ ≈ π/4. This implies Δm > 2/π ≈ 1, which in our
spheroids translates to a fluctuating angle of incidence of Δsinχ =Δ m/(nkR) ≈ 10^{-3}. Even
the least deformed resonator (ε ≈ 1%) exceeded this estimated resolution threshold
significantly.

It is truly remarkable that these peculiar properties, for which chaos plays no dominant role, can be exploited to engineer WG modes that can feature both high-Q and highly directional emission. This makes WG resonators with small deformation highly promising for a variety of applications [17], such as microlasers, nonlinear optics, and quantum information processing.

This work was supported in part by NSF under grants No. DMR9733230 and No. DMR0201784.

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[11] Even at small deformation, the separatrix region we are probing is created by the homoclinic tangle of the unstable, rectangle-shaped periodic orbit shown dashed in the inset to Fig. 1 (d), and this tangle exhibits an asymmetry which will produce asymmetric emission whenever the escape is predominantly ray-based [10].

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[14] We have performed the same calculations at wave numbers kR ≈ 70, and close to the current limits of our numerical technique at kR ≈ 200. All these results are consistent with the ones presented in Fig. 2, showing that the phase-space structure causing the observed emission is already fully resolved in Fig. 2. Extrapolating to the experiment (Fig. 1) at kR ≈ 785, we expect no further changes within the angular resolution of the detector apparatus.

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