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Whispering gallery resonators� Dielectric spherical structures � Can sustain high-Q electromagnetic modes (WGM), which are

electromagnetic waves that circulate and are strongly confined within the sphere.

� The waves are totally internally reflected and focused by the surface. Due to minimal reflection losses these modes can reach unusuallyhigh quality factors.

� Whispering gallery modes have beenstudied at optical wavelengths in dielectric systems such as micrometer-sized liquid dropletsand glass spheres since the earlydays of lasers.

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WGM�s� WGM�s have attracted much attention due to potential

applications in photonics, quantum electrodynamics (QED), atom optics and telecommunication.

� Such applications in technological and scientific fields are e.g. the realization of microlasers, narrow filters, optical switching, ultrafine sensing, displacement measurements, high resolution spectroscopy, Raman sources and studies of nonlinear optical effects.

� For example in the case of microlasers, microspheresoffer large reduction in demanded pump power and fibre-coupling provides a convenient method to transport optical power.

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WGM�s� If the dielectric medium is microscopic, one can obtain a

very small mode volume and high finesse. � Utilization of small spheres increases the free spectral

range and reduces the number of exited modes.� In order to couple light in or out of the microsphere, it is

necessary to utilize overlapping of the evanescent field of the whispering gallery modes with the evanescent field of a phase-matched optical waveguide. Such coupling has been implemented by use of tapered optical fibers, side-polished optical fibers and prisms.

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Q values in WGM�s� Microsphere optical resonators with typical diameters

from 10 to 1000 micrometers have losses significantly lower than other optical resonators. Q-factors as high as 109 have been observed in spheres of low loss materials such as fused silica. Q-factor can substantially be limited by surface scattering and absorption in the bulk which limit the Q-factor to maximum 1011. Rayleigh scattering represents a lower limit for loss in liquid and glass spheres

� Moreover, very small spheres with diameters less than 30 µm suffer from large temperature effects causing the WGMs to experience instabilities and thermal shifting of the spectrum.

αλπ0

2 snQ < for fused silica, α=2 106 cm-1

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WGM�s� Whispering-gallery mode resonances

correspond to light that is trapped in circular orbits just within the surface of the structure. The modes are most strongly coupled along the equatorial plane and they can be thought to propagate along a zig-zag paths around the sphere.

� In the view of ray optics, the light is trapped inside the dielectric sphere by continuous total internal reflections at the curved boundary of surface. Whispering gallery modes occur at discrete frequencies that depend on index of refraction ns and radius r0 of the sphere.

� Each mode has a propagation constant βlparallel to the surface and in the direction of the zig-zag path. The propagation constant has a value:

� The projection of the propagation constant is βm and it has a value

θφ

r

equator

uφuθ

ur

mode path

βl

βm

0

)1(Rll

l+=β

0Rm

m =β

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WGM�s� Whispering gallery modes are characterized by two

polarization (transversal electric ie. TE-modes and transversal magnetic ie. TM-modes) and three mode numbers n, l and m which are the radial, angular and azimuthal mode numbers, respectively. The value of l is close to the number of wavelengths that fit into the optical length of the equator. The value l - m + 1 is equal to the number of field maxima in the polar direction, ie. perpendicular to the equatorial plane. Mode number n is equal to the number of field maxima in the direction along the radius of the sphere and 2l is the number of maxima in the azimuthal variation of the resonant field around the equator. The resonant wavelength is determined by the values of n and l.

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Lowest order WGM�s

Identification of whispering gallery modes, where radial mode number n equals to the number of field maxima in radial direction, l is the angular mode number and azimuthal mode number m equals to the number of field maxima in the equator plane. The fundamental mode is defined as n = 1 and m = l. The physical meaning of mode numbers is also seen.

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WGM�s

TE-mode with mode numbers n = 1 and m = l = 20.

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Dielectric spherical resonator

In this case, the electromagnetic field problems are solved in 3D spherical coordinates. When solving dynamic problems in spherical coordinates, spherical Bessel functions jn(x), nn(x), corresponding spherical Hankel functions hn

(1)(x), hn(2)(x), associated Legendre

functions Pnm(x), Qn

m(x) and exponential functions e±jmϕ are needed.

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Spherical Bessel functions� Spherical Bessel functions describe standing waves in the

radial direction of spherical coordinates. Spherical Bessel functions are in close connection with ordinary Bessel functions of sylindrical coordinates. Spherical Bessel (and Hankel) functions bl(x) = jl(x), nl(x), hl

(1)(x) and hl(2)(x) can

be calculated as

,

where Bl+1/2 is the corresponding Bessel function in sylindrical coordinates. Spherical Hankel function hl

(2)(kr) = jl (kr) – jnl (kr) is essential because it describes spherical waves propagating outside the dielectric sphere and in the radial direction away from the center of the sphere.

)(2

)(21 krB

krkrb

ll += π

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Manufacturing of spherical microresonator and tapered fiber

� Melting a pure silica fiber can result a nearly spherical silica drop at the end of the fiber. A common method to fabricate microspheres is to melt the tip of a fiber with a high power (e.g. CO2) laser, a fusion splicer or an oxygen-hydrogen torch. Surface tension creates a spherical volume with usually rather low deformation (~2%). During the cooling, the fiber can be put into a hermetic chamber in order to protect it from dust andcondensates.

� In many cases, microspheres are optically coupled by the use of tapered optical fiber with diameter of typically few micrometers. Tapered fibersare formed by heating and stretching a standard telecommunication single-mode fiber. The core vanishes and the situation is similar to the glass rod with an air cladding. The optical field from the fiber tunnels into the microsphere through evanescent coupling. The coupling strength and resulting Q-factor depends considerably on the refractive index of the microsphere, sphere dimensions, tapered fibre dimensions and relative positions of the components, the difference in propagation constants between fiber and sphere and on the spherical mode orders.

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Of all resonant geometries micron-sized spherical dielectric resonators are unparalleled in terms of their ability to store and confine energy for long periods of time within meso-scale volumes. In the sphere light orbits near the surface where long confinement times (high-Q) effectively wraps a large interaction distance into a tiny volume. This characteristic makes them uniquely suited for studies of nonlinear coupling of light with matter. The electromagnetic modes of these micro-spheres can be efficiently excited with a tapered optical fiber.Research in this area has been twofold. On the one hand we explore applications in optical telecommunication using these meso-scale structures . On a more fundamental level we investigate the physics which can be observed in these microstructures; Nonlinear effects like stimulated Raman Scattering, Stimulated Four Wave Mixing and Brillouin Scattering may be observed at extremely low input power levels and furthermore the conditions under which they can be observed are altered due to the confinement of the resonator. Additionally, micro-spheres are a candidate to investigate cavity QED effects, such as gain enhancement of nonlinear effects. Research Areas:- Micro-resonators for Optical Communication- Nonlinear Optics in High-Q microresonators- All Optical Logic

Green upconversion in a Erbium doped Microsphere

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Microsphere coupled to two optical fibers (left)constitutes an add-drop filter. An optical signal sent toward the sphere along one fiber (green arrow, upper right) is added to the many signals traveling along the other fiber (blue arrows). This device can also be used to extract (or �drop�) a signal (red arrow, upper left) that was originally traveling through a fiber along with many others.

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Light emitted from donor molecules within a droplet shows a distinct asymmetry (a), because the laser beam that excites them is focused at the left (b). Light emitted from the acceptor molecules reveals a ringlike pattern (c), which demonstrates that they are excited only near the surface of the droplet. Most of the light emitted by donor molecules escapes before intercepting an acceptor. Only photons in a resonant mode, which circulate just beneath the surface (d), remain within the droplet long enough to have an appreciable chance of colliding with acceptor molecules. Many ringlike orbits are possible, but they all come together at two antipodal points (e), corresponding to the two diametrically opposed bright spots in the acceptor image (c). (Images a and c courtesy of the author.)

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Microsphere placed within one wavelength of the core of an optical fiber readily absorbs photons when the frequency of light passing through the fiber matches a resonant mode of the sphere (top). At resonance, photons enter the sphere and eventually leak out in all directions, causing distinct peaks in the intensity of light seen coming from the sphere as a function of the wavelength of laser excitation (bottom).

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� Chemicals of interest could be detected at low concentrations in small samples.� NASA's Jet Propulsion Laboratory, Pasadena, California� Miniature high-resolution optical-absorption sensors for detecting trace amounts of chemical species of interest in gas

and liquid samples are undergoing development. The transducer in a sensor of this type is a fiber-optic-coupled optical resonator in the form of a transparent microsphere, (or a microcavity equivalent to a microsphere as described below).

� The principle of operation of these sensors is an updated version of that of conventional optical cavity-ringdown spectroscopy (CRDS), wherein resonators in the form of long and bulky Fabry-Perot cavities are used in order to obtain enough effective optical-path length to enable the resolution of small attenuation associated with trace concentrations of analytes. In addition to bulky apparatuses, conventional CRDS requires large samples to fill the Fabry-Perot cavities. In contrast, a microsphere or microcavity sensor of the type under development is designed to be immersed in a sample, which can be small because the microsphere is small. (Alternatively, the sample can be contained in a small cavity as described below.)

� The use of transparent microspheres as optical resonators has been reported in a number of prior Tech Brief articles. To recapitulate: In a transparent microsphere, resonance is achieved through glancing-incidence total internal reflection in one or more "whispering-gallery" modes, in which light propagates in equatorial planes near the surface, with integer numbers of wavelengths along nominally closed circumferential trajectories. In the absence of external influences, and assuming that the microsphere is made of a low-loss material, the high degree of confinement of light in whispering-gallery modes results in a high resonance quality factor (high Q).

� Suppose that the microsphere is illuminated by laser light at its resonance wavelength and is immersed in a sample liquid or gas that (1) has an index of refraction less than that of the microsphere material and (2) contains a highly diluted chemical species of interest that absorbs light at the resonance wavelength. In that case, the Q of the resonator is diminished through absorption by molecules of that species in the evanescent field of the whispering-gallery modes. Because of the smallness of microspheres (typical diameters from tens to hundreds of optical wavelengths), the smallness of the effective volumes of the evanescent fields (typically 10-9 cm3 or less), and the low level of optical losses intrinsic to microspheres themselves, it is possible to detect very small amounts of optically absorbing chemical species through decreases in Q; calculations have shown that in some cases, it should be possible to detect amounts as small as single atoms or molecules.

� The left side of the figure depicts a typical setup for a microsphere sensor immersed in a sample fluid that has an index of refraction less than that of the microsphere. If the index of refraction of the sample fluid exceeds that of a material that could be used to construct a microsphere, then one must use a setup like that shown on the right side of the figure: The sample is contained in a microspherical cavity in a capillary cell made from a transparent material that has an index of refraction less than that of the sample fluid. In this setup, the portion of the sample in the microspherical cavity serves as a the whispering-gallery-mode resonator, and coupling between the optical fiber and the microsphere is effected by use of a prism attached to a thin wall that acts as a tunneling (in quantum-mechanical analogy) gap for photons.

� The decrease in Q (and thus the amount of the chemical species of interest) can be determined either by measurement of the decrease in the cavity-ringdown time or, if the spectral purity of the laser is adequate, by traditional measurement of transmission bandwidth. Bandwidth measurement is ordinarily used when Q ranges from Å105 to Å108; cavity-ringdown measurement is more convenient for Q ≥108 (typically corresponding to ringdown time ≥30 ns). While the precision with which the absolute value of Q can be determined is usually no better than a few percent, variations in Q can be measured with greater precision. In state-of-the-art CRDS as performed with Fabry-Perot cavities, it is possible to resolve ringdown times to fractional variations as small as about 2 × 10-3 at data-acquisition rates of about 1 kHz. Hence, it is possible to obtain absorption spectra with satisfactory signal-to-noise ratios even though the losses added by the chemical species of interest may be only small fractions of the intrinsic optical losses of the resonators themselves.

�

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�A Microspherical Optical Resonatoris operated in the presence of a sample fluid that contains an optically absorbing species. The concentration of the species is determined from its effect on the Q of the resonator.

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Surface Gratings for Optical Coupling With Microspheres

Far-field coupling offers advantages over near-field coupling. NASA's Jet Propulsion Laboratory, Pasadena, CaliforniaA diffraction grating consisting of a periodic gradient in the index of refraction of a thin surface layer has been shown to be effective as a means of far-field coupling of monochromatic light into or out of the "whispering-gallery" electromagnetic modes of a transparent microsphere. This far-field coupling can be an alternative to the near-field (evanescent-wave) coupling afforded by prism- and fiber-optic couplers described in the immediately preceding article. Far-field coupling is preferable to near-field coupling in applications in which there are requirements for undisturbed access to the entire surfaces of microspheres. Examples of such applications include (1) a proposed atomic cavity in which cold atoms would orbit in a toroidal trap around a microsphere and (2) a photonic quantum logic gate based on coupling between a high-Q (where Q is the resonance quality factor) microsphere and trapped individual resonant ions.

.

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In preparation for experiments to demonstrate this concept, fused silica microspheres with a diameter of about 180 µm were fabricated, then coated with layers of molten germanium-doped glass powder 3 to 5 µm thick. The purpose served by the germanium doping was to increase the photosensitivity of the surface layers for the grating-fabrication step described next. An index-of-refraction grating was formed in the surface layer of each microsphere by exposing the layer to ultraviolet light (wavelength = 244 nm) from a frequency-doubled argon laser. The laser beam power was 40 mW, the exposure time was 5 to 10 minutes, and the expected index modulation was (1 to 3) × 10-4. The spatial period and length of the grating were Å2 µm and Å15 µm, respectively. The spatial period was chosen to provide first-order phase matching between a whispering-gallery mode of the microsphere and a free-space beam oriented at Å45° to the surface of the microsphere.

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Figure 1 schematically depicts the experimental setup used to demonstrate the grating-based coupling scheme. Laser light at a wavelength of Å1,550 nm was coupled into the whispering-gallery modes of a microsphere by a standard prism coupler, then coupled out of the microsphere by the grating. The laser was gradually tuned over a frequency range that included some whispering-gallery-mode resonances. The resulting measurements (see Figure 2) showed that at the resonances, some light was depleted from the input beam and there were corresponding increases in the amount of light emitted from the microsphere through the surface grating.

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From the measurement data, the maximum grating coupling efficiency was calculated to be 14 percent. The grating loaded the resonance sufficiently to decrease the Q of the microsphere to a value in the range of (0.2 to 2) × 106.[The initial Q (without the grating) was 1.2 × 108.] Higher Q could be obtained by reducing the strength of the grating. Efficiency of coupling could be increased by optimizing the exposure to ultraviolet light, improving the grating profile, and minimizing scattering losses. Parasitic coupling to low-Q higher-order modes in the microsphere could be prevented by decreasing the diameter of the microsphere

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WGM�s� Main motivation: very high Q factors� Physical consequences:

� ultranarrow linewidths� high energy densities and intensities� ultrafine sensing

� Experiments/Applications:� optical switching & bistability� measuring displacements� high-resolution spectroscopy� quantum non-demolition measurements� spectral hole burning memory� ultrafine environmental sensing (phase or reflection response)� laser frequency locking and stabilization

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WGM�s

Coupling

� in order not to disturb the high Q mode⇒evanescent wave light input coupling

Couplers

prisms

tapered fiber

polished half-block coupler

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Calculation of the modes

� Characteristic equations: resonant modes vs. wavelenght

� Intrinsic losses

0

0

//)1(

RmRll

m

l

=+=

ββ

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Classical theory of fields:� all the sources are inside the sphere� em fields of a sphere are vector fields� if direction of polarization is constant� solution to the Helmholtz equation is separable� => TE and TM modes� => same thing as representing the fields with the

help of 2 vector potentials

=> Spherical Bessel functions for the radial dep.=> Spherical Harmonics for the angular dep.

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Coupling...

� dielectric sphere is an open cavity with tunneling out of the it

=> eigenvalues complex to satisfy the radiation condition

� only near field contributes to the coupling

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Interpretation of modes� n,l, m characterize the modes� l-m+1 = # of field maxima in polar direction

» (between 2 poles)

� n = # of field maxima in the radial direction� propagation constants

� different values of m mean that modes travel in zig-zag-paths with different inclinations with respect to the equitorial plane

� when m=0, inclination is 90º� when m=l �fundamental mode�, inclination angle in

smallest ~

lmR

mm

≤

=0

βml ββ ,

radl

1≈