Department of Physics

Whispering gallery modes

Seminar

Author: Matjaž Gomilšek

Mentor: asist. dr. Miha Ravnik

Ljubljana, November 2011

Abstract

Whispering gallery modes are specific resonances (or modes) of a wave field that are confined inside a given resonator

(cavity) with smooth edges due to continuous total internal reflection. The most interesting from a practical viewpoint are

electromagnetic whispering gallery modes, since they posses many unique properties, such as ultra-high Q-factors, low

mode volumes, small sizes of resonators supporting them and operation at optical and telecommunication frequencies of

light. This combined with the ease of fabrication and on-chip integration of devices using them, makes whispering gallery

modes ideally suited for a vast array of applications. In this seminar we will look at some key issues concerning whispering

gallery modes and resonators such as: wave theory of whispering gallery modes, resonator performance parameters,

resonator geometries, coupling of whispering gallery modes to and from resonators and at some practical applications of

whispering gallery modes.

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Contents

1. INTRODUCTION ........................................................................................................................................................... 2

2. WAVE THEORY OF WHISPERING GALLERY MODES ..................................................................................................... 3

2.1. General scalar wave theory ................................................................................................................................. 3 2.2. Scalar whispering gallery modes in a 2D cylinder (example) .............................................................................. 4

3. RESONATOR PARAMETERS ......................................................................................................................................... 5

3.1. Quality factor ....................................................................................................................................................... 5 3.1.1. Loss mechanisms .......................................................................................................................................... 6

3.2. Finesse ................................................................................................................................................................. 6 3.3. Mode volume....................................................................................................................................................... 7

4. RESONATOR GEOMETRIES AND FABRICATION OF RESONATORS .............................................................................. 7

4.1. Dielectric sphere resonator ................................................................................................................................. 7 4.1.1. Spectrum ....................................................................................................................................................... 8 4.1.2. Fabrication .................................................................................................................................................... 9

4.2. Dielectric cylinder, disk, ring and racetrack resonators ...................................................................................... 9 4.2.1. Fabrication .................................................................................................................................................. 10

4.3. Toroid microresonator....................................................................................................................................... 10 4.3.1. Fabrication .................................................................................................................................................. 11

4.4. Optical bottle microresonator ........................................................................................................................... 11 4.4.1. Spectrum ..................................................................................................................................................... 11 4.4.2. Fabrication .................................................................................................................................................. 12

5. RESONATOR COUPLING ............................................................................................................................................ 12

5.1. Free wave coupling ............................................................................................................................................ 12 5.2. Fluorescence coupling ....................................................................................................................................... 12 5.3. Evanescent coupling .......................................................................................................................................... 13

6. APPLICATIONS ........................................................................................................................................................... 14

6.1. Photonic filters .................................................................................................................................................. 14 6.2. Sensors ............................................................................................................................................................... 14 6.3. Ultralow-threshold lasers .................................................................................................................................. 15

7. CONCLUSION ............................................................................................................................................................. 15

8. REFERENCES .............................................................................................................................................................. 15

1. Introduction Whispering gallery modes or waves are specific resonances or (eigen-)modes of a wave field (e.g. sound waves,

electromagnetic waves, …) inside a given resonator (a cavity) with smooth edges. They correspond to waves circling around

the cavity, supported by continuous total internal reflection off the cavity surface, that meet the resonance condition (after

one roundtrip they return to the same point with the same phase (modulo ) and hence interfere constructively with

themselves, forming standing waves). These resonances depend greatly on the geometry of the resonator cavity.1

The term whispering gallery waves was first used by Lord Rayleigh in the 19th century to describe the phenomenon of the

whispering gallery located under the dome of St. Paul’s cathedral in London. It was known that a sound (a whisper) uttered

at one end of the dome could still be heard loudly at the opposite end of the dome, a large distance away from the source.

Lord Rayleigh described this phenomenon by noting that sound seemed to “stick” to the dome’s walls and propagate only

inside a narrow layer near the surface of the concave wall of the gallery (see Figure 1). While in free space sound intensity

decreases proportionally to the square of the distance from the source, in a whispering gallery, inside the narrow layer, the

sound intensity decreased only directly proportionally to the distance from the source, i.e. much slower than in free space,

and can therefore be heard loudly even a large distance away. Lord Rayleigh named these peculiar sound waves whispering

gallery waves.2

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Figure 1. A look at the dome (upper left) of St. Paul's cathedral from below

3 (left), a sketch of the whispering gallery (center)

2 and the sound

intensity profile showing the whispering gallery phenomenon that Lord Rayleigh studied (notice how sound seems to stick to the walls)4.

Then in the beginning of the 20th century it was realized that there existed electromagnetic waves in dielectric spheres that

had almost the same structure as acoustic whispering gallery waves. But it took until about the 1990s before they began to

be widely studied and applications of whispering gallery modes in optics started to appear2. Currently the field is very active

and interest in applications of whispering gallery modes and whispering gallery resonators is very high due to their unique

combination of high Q-factors, low mode volumes, small size, the ability to operate at optical (and telecommunication)

frequencies of light and the ease of fabrication and on-chip integration.

This seminar is organized as follows: first we introduce some basic theory behind wave phenomena and show an example of

whispering gallery modes as well as their derivation from wave theory for a 2D cylindrical (disk) resonator. Then we look at

resonator parameters (various quantities describing the spectrum of whispering gallery modes of a given resonator) and

some of the more common resonator geometries and describe their spectral properties as well as methods for the

fabrication of these resonators. Then we describe how to introduce coupling of whispering gallery modes in and out of the

resonator and finally describe some applications of whispering gallery modes. In most cases we restrict our attention to

whispering gallery modes of light.

2. Wave theory of whispering gallery modes In this chapter we introduce some general theory behind wave phenomena (the wave equation) and work through the

example of an idealized version of whispering gallery studied by Lord Rayleigh (a 2D cylindrical cavity). As we shall see the

specific waves we are interested in (the whispering gallery waves or modes) have a special structure which also allows for a

much simpler geometrical description of their general properties. Later in the seminar we use this description, alongside the

rigorous wave description, to interpret some of the properties of these modes.

2.1. General scalar wave theory

The basic equation describing scalar waves (e.g. acoustic waves or single components of the EM field) is the wave equation

for the (complex) wave field amplitude (here is the wave speed or phase velocity profile):

Here represents wave field sources. By inserting the separation of variables ansatz (i.e. an ansatz for a single

Fourier component): , (from we see that solutions with

dissipate in time (so called “leaky modes”) and that ; usually since otherwise

the solution dissipates very quickly)5 into the free wave equation ( ) we obtain the Helmholtz or amplitude

equation:

This is an eigenvalue equation for the operator with eigenvalues and eigenfunctions . Under specified

boundary conditions (defined depending on the physics of the problem) this equation can only be solved for a discrete set

of eigenfrequencies (resonant angular frequencies)5, which we choose to index with a (multi-)index (components

of are also called mode numbers).

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These special solutions (called the eigenmodes of the specific phase velocity profile ) are important for two reasons: 1)

they allow us to write the general solution of the free wave equation as a linear combination of eigenmodes, and 2) they

allow us to simply obtain the solution to the harmonically forced wave equation (angular frequency distinct from all

), like so:

1) Free: (coefficients arbitrary):

2) Forced: (here: ):

The expansion of with in 2) is justified since the eigenfunctions form a complete set. Since the

expansion coefficients do not depend on the angular frequency of the forcing we see that the energy contained

inside a specific eigenmode of the forced solution has a Lorentzian dependency on :

The energy is greatest when: (so by knowing the eigenfrequencies we also know the resonant

forcing angular frequencies), the height of the resonant peak is:

(a smaller

therefore means a stronger peak) and the width (full-width-at-half-maximum or FWHM) of the peak is:

(a smaller therefore means a narrower peak). We define the ratio

as the quality or Q-factor of

the eigenmode or resonance5 (more on this in the chapter on resonator parameters).

2.2. Scalar whispering gallery modes in a 2D cylinder (example)

Let us look at an example of scalar waves in an idealized whispering gallery (the one that Lord Rayleigh studied), namely a

2D cylindrical cavity of radius . Let us imagine that instead of a hard wall at the edge of the cavity only the wave speed

changes (i.e. equals inside the cavity and outside, and we call the refractive index of the cavity). We

will attempt to find the eigenmodes and eigenfrequencies (the spectrum) of such a cavity. First we observe that the jump in

results in two separate 2D Helmholtz equations (here we define the wavenumber and write ):

We will impose the following boundary conditions: 0.a) the function should be finite everywhere (no singularities),

0.b) it should describe only outgoing waves for (no waves traveling from towards the origin at , 1) (and

2)) (and ) should be continuous at the edge of the cavity at (for vector EM waves these two continuity

conditions 1) and 2) are slightly modified, depending on the polarization of the eigenmode, to satisfy Maxwell’s equations).5

From the boundary conditions 0.a) and 0.b), and applying the separation of variables in a cylindrical coordinate system

, we obtain (using Bessel and Hankel functions

of the first kind of order , where

for

(outgoing wave; exponentially decaying for )):

The boundary condition 1) gives us the ratio over and the boundary condition 2) gives us the characteristic equation

which determines the eigenfrequency (here written in terms of ):

Here the prime denotes ordinary differentiation, and due to linearity the actual values of and remain undetermined.

The characteristic equation is transcendental and can only be solved numerically. It has infinitely many complex roots

( , meaning that the eigenmodes are leaky) for each , which we number by in terms of increasing .

Our multi-index (mode numbers) to label the eigenmodes is: .

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Figure 2. Eigenmode profiles (real part) for m=4 (top) and m=15 (bottom) at N=1,2,3,4 (left to right). Green line (N=1) is the geometrical ray path.

Despite the cylindrical separation of v. we see a joint radial-angular dependence (a spiral) since: .

Figure 2 shows some eigenmode profiles. We see that the localization of the mode increases with (since

if , the mode is pushed towards the edge for large ) and decreases with (when increases the

mode can oscillate more times inside the cavity). We also see that modes that are more localized also leak out of the cavity

less (higher Q-factor). One of the principal features of whispering gallery modes is high localization and so we call modes

with and the whispering gallery modes for this cavity (modes with small are sometimes called higher

order whispering gallery modes). They indeed correspond to whispering gallery modes (waves) described by Lord Rayleigh.

We can also give a geometrical interpretation to these modes, which will also serve to identify whispering gallery modes in

general resonators: they are the wave analog to a ray reflecting times off of the edge of the cavity at a grazing angle by

total internal reflection, before circling the cavity one full time and returning to the starting point with the same phase

(modulo ) it had at the beginning (forming a standing wave in the wave picture). Whispering gallery modes of a general

resonator are therefore naturally highly localized near the edge of the cavity, and are due to the effect of total internal

reflection also much better confined inside the cavity (higher Q-factor) than (most) other eigenmodes of the cavity.

3. Resonator parameters

Since the spectrum (eigenfrequencies) depends crucially on the geometry of the cavity (on the phase velocity profile ,

where we can even model bulk medium losses by introducing a complex instead of ) it is appropriate to

talk about resonator parameters which are quantities that quantify different aspects of the resonator’s (cavity’s) spectrum.

We will assume that the resonator is in “vacuum” or “air”, so that for large .

3.1. Quality factor

One of the most important quantities that describe the performance of any resonator (on resonance) is the quality factor or

Q-factor. It can be defined as6:

where is the angular frequency and is the frequency of the resonance, is the cavity ring down lifetime i.e.

the time required for the field intensity to decay by a factor of and is the linewidth or the “uncertainty” of

the frequency of the resonance (full-width-at-half-maximum or FWHM of the resonance peak) in angular frequency.

In many applications large Q-factors are needed. We say that resonators with Q-factors from about up to have a

high Q-factor and those with Q-factors above an ultra-high Q-factor7.

The above expressions for the quality factor can be interpreted in the following ways. On the one hand the Q-factor

measures the characteristic time for the natural (exponential) decay of the energy stored inside the resonator in terms of

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the number of full field oscillations (times ). This means that for a higher Q-factor the time that energy is stored inside

the resonator is proportionally longer.

On the other hand it can be interpreted as the total energy of the circulating modes inside the resonator at equilibrium

divided by the amount of energy that has to be pumped to the resonator in the time of one full field oscillation (times )

so as to maintain the equilibrium. This means that for a higher Q-factor the total field intensity (i.e. the total energy stored

inside the resonator) of the circulating modes is proportionally higher at the same pumping power. Ultra-high Q-factors of

some (whispering gallery) resonators can thus enable the circulating intensities to achieve extremely high values (even

beyond all known nonlinearities of the resonator medium) even at moderate pumping powers (in the range of milliwatts)

and thus simplify the study and use of extreme nonlinear optical effects.

Lastly the Q-factor measures the frequency of the resonant peak in terms of the number of FWHM linewidths of that

resonance peak. This becomes important (especially in connection with the finesse of the resonator, as defined below) in

interferometric applications of the resonator and, more generally, in the consideration of the tuning characteristics of the

resonator (how much we can change the resonance spectrum of a certain resonator by varying external parameter).

As hinted at in the second chapter we can also express the quality factor of the circulating mode from the complex

propagation constant (wave number)

(here

is the phase constant, is wavelength in

vacuum and is the intensity attenuation coefficient due to various (bulk and surface) cavity loss mechanisms) as5:

3.1.1. Loss mechanisms

The intrinsic quality factor (for example of a spherical resonator) gets contributions due to losses from many processes

and can be written as8:

Here describes intrinsic material absorption, surface absorption losses (due to surface coatings or adsorbed

material, for example due to adsorbed water (common with silica resonators) or due to other contaminants),

describes scattering losses (intrinsic and inherent to the surface of the cavity, such as imperfections in the form of surface-

roughness) and describes bending loss (or whispering gallery or tunnel or radiation loss).

For example, the absorption and (mainly Rayleigh) scattering losses for silica (SiO2) at (the vacuum

wavelength of minimum loss) set an upper bound for an absorption limited Q-factor of8

, where is

the refractive index of silica and is the intensity attenuation coefficient.9

Bending losses (the losses in the example in chapter 2.2.) arise from the fact, that total internal reflection at a curved

interface is never complete and always results in a transmitted wave on the lower refractive index side, corresponding to

loss of energy in the case of a whispering gallery mode (this is because the local phase velocity of the evanescent waves at a

curved interface would otherwise exceed the wave speed , which is not possible). The resulting quality factor

exhibits a very strong dependence on the resonator radius for a fixed resonance wavelength (for a sphere

,

where is the radius of the sphere).8

When a cavity is coupled to an external mode (e.g. to a prism or a waveguide mode as described in chapter 5.) the total (or

loaded) quality factor gets an additional contribution due to losses to the external mode (such losses to external

modes can be desired behavior in practice, since they provide a way to couple the resonator to other optical structures):

3.2. Finesse

Finesse is another performance parameter of a resonator which is very important (larger finesses are usually desirable). It is

defined as the ratio of the free spectral range

(the spacing between resonant frequencies) and the FWHM

linewidth of the resonances6:

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While the free spectral range tells us the distance between the resonant peaks in absolute terms, the finesse of a resonator

tells us the distance between the resonant peaks in terms of the number of FWHM linewidths that go in between two

consecutive resonant peaks. Since the resonant frequencies of most resonators are (at least) approximately equidistant a

resonator interferometer can only distinguish between frequencies that lie inside an interval the length of one free spectral

range (this is also why we call a resonator that can be tuned over one free spectral range a “fully tunable” resonator).

Finesse thus tells us the effective resolution of the resonator as an interferometer (roughly, the number of different

frequencies that it can distinguish).

Finesse is also connected to the quality factor, explicitly finesse is equal to the quality factor times the fraction of the

spectrum (up to the resonant frequency) covered by one free spectral range of the resonator. Finesse over the quality

factor is thus always less than or equal to one. Just as the quality factor, finesse also changes when coupling is introduced to

the resonator.

3.3. Mode volume

The mode volume of a field eigenmode in a resonator is defined as the ratio of the total energy stored inside the resonator

in that mode and the maximum energy density of that mode8:

For the electromagnetic field:

, where is the total electric permittivity and

the total magnetic permeability. Mode volume is related to the free spectral range (lower mode volume means

higher FSR) and also to lasing characteristics of a resonator (containing a lasing medium) since stimulated as well as

spontaneous emission rates are inversely proportional to mode volume and the lasing threshold (minimum pump power for

laser operation) is directly proportional to the mode volume10. Small mode volumes are usually desirable.

4. Resonator geometries and fabrication of resonators There exist many different resonators that support whispering gallery modes (WGMs). Here we focus on resonators that

support optical (or at least close to optical) frequency modes of the electromagnetic field. Because they are much smaller

than conventional resonators for light they are sometimes called microresonators. We list some of them along with their

specific characteristics and fabrication methods.

Common to all whispering gallery resonators is that they typically enable very high (or ultra-high) Q-factors and finesses at

very low mode volumes, and can be much more monolithic in design, compared to “usual” (i.e. non-whispering-gallery, for

example Fabry-Pérot) resonators, since they require no external mirrors to confine light.

4.1. Dielectric sphere resonator

One of the simplest resonator supporting whispering gallery modes is a (dielectric) sphere with a refractive index higher

than the surrounding material. From a geometrical optics perspective, light that travels close to the edge of the sphere is

continuously reflected back inside the sphere by total internal reflection at the cavity-air interface and can never leave the

sphere (it is trapped inside it). If the circulating beam of light returns to the same point with the same phase it interferes

constructively with itself and resonant standing waves form (a resonance). It is usual to orient our coordinate system so that

the beam of light is circulating around the sphere in the plane (azimuthally, around the sphere’s equator). The spherical

surface also serves to focus the light in the polar (vertical) direction (because of the polar curvature of the sphere light

travels the same effective optical path as if it were zig-zag-ing around the equator instead of just going around in a straight

line; see Figure 3 – this phenomenon can be understood as an extra Gouy phase shift from the polar confinement of light, in

analogy with the Gouy phase shift (an effective lengthening of the optical path length evident from the phase of the beam,

compared with the expected beam phase computed from the actual travelled distance) of ordinary Gaussian beams as they

travel through their focal points).11

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Figure 3. Fabrication of a silica microsphere resonator by electric arc heating. High temperature silica glows as an opaque molten globe, the fiber stem is visible underneath the globe

12 (left). A stationary fluorescence pattern representing the intensity of a combination of whispering gallery

modes in a 300m fused silica microsphere13

(center) and the geometrical optics approximation to whispering gallery mode propagation11

(right), where the equatorial (circular) path is the actual WGM ray path and the zig-zag shows the effective optical path length that the WGM experiences.

For a full characterization of the resonant behavior of the spherical resonator a wave-optics approach needs to be used,

rather than the geometrical description. Considering light as waves gives two major corrections to our understanding.

Firstly, light is not actually bouncing off the edge of the sphere but is in fact smoothly guided along the edge of the sphere

(this changes the effective optical path), and secondly, total internal reflection of a wave at a curved interface is never

complete which means there are bending losses associated with the curved spherical surface and light slowly leaks out of

the sphere (setting a bound on the maximum attainable Q-factor regardless of the material from which the sphere is made).

4.1.1. Spectrum

Eigenmodes of the electromagnetic field inside a dielectric sphere of radius placed in vacuum are described in spherical

coordinates by (using adjoint Legendre polynomials , spherical Bessel and spherical Hankel functions)2:

where is the angular frequency of the resonance, is the speed of light in vacuum, is the relative electric permittivity

and the relative magnetic permeability of the sphere. Here , and are the radial, polar and azimuthal mode numbers

respectively which, along with the polarization (transverse electric (TE) if the electric field is parallel to the surface of the

sphere, or transverse magnetic (TM) if the magnetic field is parallel to the surface of the sphere (the electric field is then

mostly radial))8, determine the eigenmode uniquely (the mode multi-index is ). The ratio and are

fixed by suitable boundary conditions at the sphere’s surface (exactly as in 2.2.). The resonant frequency depends only on

the indices and and on the polarization, but not on , thus the degeneracy of the modes is (this degeneracy is

lifted if the sphere is deformed into a spheroid).2

Here equals the number of field maxima along the radial direction inside the sphere (as in 2.2.), equals

roughly the number of wavelengths that can fit into the optical length of the equator, and determines the

sense (clockwise or counterclockwise) of circulation of the wave around the equator and its wave number in this direction.

equals the number of field maxima in the polar direction (i.e. perpendicularly to the equatorial plane). For small

indices the mode fills almost the entire volume of the sphere, whereas for large indices (roughly, greater than ) the

mode is highly localized near the surface of the sphere (see Figure 4).2

Figure 4. The radial and polar dependence of field intensity for eigenmodes of a spherical resonator, demonstrating the meaning of mode numbers (left) and a photograph of individual excited whispering gallery modes made visible by implanted Er

3+ ions via up-conversion luminescence (right).

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Thus, if we choose we get no oscillations inside the sphere (also, modes with lower are better confined in the

sphere and leak out more slowly (higher Q-factor) than those with larger )14 and for large indices we get modes that are

localized very close to the surface of the sphere. If we choose we also get no oscillations of amplitude in the polar

direction, and since , such a mode is, for large indices , highly localized also in the polar direction. These

properties are very reminiscent of those ascribed by Lord Rayleigh to his whispering gallery waves and that is why we call

modes with and whispering gallery modes of the sphere2.

For whispering gallery modes the spectrum of a spherical resonator can be approximated as (Taylor expansion in terms of

obtained by approximating Bessel and Hankel functions by suitably rescaled Airy functions at the cavity edge)2:

where

,

and we used , where is the first root of the Airy function .8 The

spectrum is nearly equidistant, except for a factor (which is, for large , almost negligible). The spectra of TE

and TM modes are the same up to a constant shift (up to corrections in the term). Since and

(if ) it follows that:

always, for a fixed frequency

, since .8

4.1.2. Fabrication

Microsphere resonators are usually formed by means of surface tension and have been demonstrated as spheres made of

materials in liquid, amorphous and crystalline forms. The simplest, and also the earliest demonstrated optical

microresonator, is a micron-sized liquid droplet with a near perfect spherical surface due to surface tension6. Practical use

of droplets as whispering gallery resonators is hindered because they slowly evaporate and because they are more difficult

to manipulate than solid-state resonators.6

Nevertheless liquid resonators have proven to be useful in spectroscopy, fluorescence and lasing in dyes. Recently liquid

crystal droplets have been shown to be as much as two orders of magnitude more tunable than any solid-state resonator

(tuning with an external electric field of more than one spectral range at moderate voltages) at a high Q-factor of ,

possibly opening the way for new kinds of sensors and lasers.15

The first solid-state microsphere resonator was demonstrated in fused silica (SiO2)6. If the tip of a silica optical fiber is

melted by a flame or an electric arc, the melted silica forms a smooth sphere to minimize its surface energy (see Figure 3). If

the flame or electric arc is then removed the melted silica solidifies in a microsphere, with its radius controlled by adjusting

the size of the fiber tip. Reproducible size and shape of the microsphere has been demonstrated with sphere diameters

between and and Q-factors in the order of . Fused silica microspheres are very sensitive to external

contaminants, such as water and -OH absorption, so care must be taken to ensure an inert environment for the resonators.6

Currently, spherical resonators hold the record for the highest measured Q-factor for a whispering gallery resonator. A

quality factor of (and finesse of ) has been measured at in fused silica7,16 and

(and finesse of ) at for crystalline CaF2.17

4.2. Dielectric cylinder, disk, ring and racetrack resonators

Dielectric cylinders and disks (cylinders with small height) also support whispering gallery modes. Cylinders are in many

ways similar to spheres in terms of the whispering gallery modes they support (light circulates around the cylinder in an

analogous way to light circling around the equator of the sphere), but there is one fundamental difference. While in a

sphere the modes are stable under perturbations away from equatorial propagation, the same is not true for dielectric

cylinders. While for a sphere the polar curvature of the sphere surface confines and focuses light in the polar direction, in a

dielectric cylinder a light beam, which is sufficiently perturbed from propagating around the equator, can escape through

the top or bottom of the cylinder and thus leave the resonator.

Because of this added mode instability due to the lack of focusing in the polar direction, and the fact that it is harder to

manufacture disks with comparably low surface roughness as that of a spherical resonator, the Q-factors of cylinder and

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disk resonators are usually much lower than those of spheres (typically for disks). But because of their planar

geometry their fabrication is easier, faster and more controllable, and they are much more easily integrated in an integrated

optical network or on a chip, while at the same time taking up much less space than microspheres of comparable radii and

having smaller mode volumes. This makes them very useful for practical applications.6

Variants of disk resonators are ring resonators (disk resonators with a circular hole in the middle) and racetrack resonators

(ring resonators elongated in one planar direction) (see Figure 5). Since whispering gallery modes are already highly

localized at cavity-air interface ring and racetrack resonators have almost the same whispering gallery modal structure as

disk resonators, while higher order radial modes are much better suppressed. An additional advantage of ring and racetrack

resonators is that they allow for many times smaller mode volumes (than microspheres or disk resonators), at only a

fraction of the volume of the dielectric material.

4.2.1. Fabrication

Disk, circuit and ring resonators can be fabricated by one of the three basic processes: deep ultraviolet (DUV) lithography

which has high throughput, is compatible with CMOS, but produces some surface roughness due to a feature resolution of

only , electron beam lithography (EBL) which has high feature resolution of , has less proximity effects for

closely packed structures than DUV lithography (useful in designing coupling), but is much slower/has lower throughput

than DUV lithography, or nanoimprinting lithography (NIL) which has both high throughput and high feature resolution.6

Deep ultraviolet lithography uses UV light at the wavelength of or for etching the substrate to define the

resonator structure, while electron beam lithography uses accelerated electrons for etching. Nanoimprinting lithography

first requires the resonator structure to be fabricated using either DUV lithography or EBL, then a polymer is molded around

the structure and solidified to create a solid mold. This mold can then be used as a resonator directly, or it can be used to

make a second, harder, replica (for example, one made of silica (SiO2)).

Figure 5. Fabricated disk

18, ring

6 and racetrack

6 resonators (from left to right).

All three techniques can produce resonator structures of roughly the same intrinsic Q-factor of and total (loaded)

Q-factor of . DUV lithography has been shown to be able to produce resonator with intrinsic finesse of and

loaded finesse of .6

4.3. Toroid microresonator

A toroid microresonator (microcavity) is made from a dielectric material in the shape of a solid toroid along the inside of

which light can circulate by continuously bouncing off the toroid-air interface by total internal reflection (see Figure 6). It

supports whispering-gallery modes that are very similar to those of the disk, ring and race-track microresonators, and

because of its planarity shares most of the advantages of using such resonators over spherical ones. The fabrication of

toroid microresonator is slightly more involved (coupling also) than that of disk resonators, but is still easily chip integrable.

Figure 6. A toroid microcavity fabricated from a thermal oxide (left) and an array of such toroid microcavities (right).

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The biggest advantage of toroid microresonators over disk resonators is that they can achieve ultra-high Q-factors on the

order of , even (comparable to microsphere resonators due to the role of surface tension in their fabrication

giving them a very smooth surface, and many orders of magnitude larger than Q-factors of disk resonators)7,19, while sharing

most of the advantages of easy fabrication and integration of disk resonators.

4.3.1. Fabrication

The manufacturing process for a fabricating a microtoroid is illustrated on Figure 7, below. First a circular silica (SiO2) disk is

defined by dry etching, then the some of the silicon (Si) below the disk is removed by isotropic etching using XeF2 gas to

provide vertical confinement of light (the remaining silicon acts as a post that supports the disk) and finally the silica is

melted by irradiating the structure with a CO2 laser. Via surface tension the melted silica then forms a smooth toroidal

surface at the edges of the disk (the inner parts of the disk do not reshape as much since they quickly transfer their heat

through the silicon post, which has higher heat conductivity than silica).6

Figure 7. Illustration of the steps in the fabrication process of a microtoroid and a photograph of a finished microtoroid.

6

Microtoroids have been fabricated by this process with principal diameters between and , torus

thicknesses of to and Q-factors on the order of (comparable to microsphere resonators, due to the

smoothness of the surface formed by surface tension during fabrication). The fabrication process allows for much easier

control of the size of the fabricated microtoroid, than fabrication processes for producing microspheres, while the planar

geometry of the microtoroid allows for much easier integration into optical circuits than is possible for microspheres.6

4.4. Optical bottle microresonator

An optical bottle resonator is a type of microresonator made from an optical fiber (a long cylindrical dielectric fiber made

out of silica or plastic) with a bulge in the middle (the thickness in the “bottle” part of the fiber is slightly increased with

respect to the thickness of the surrounding fiber). Light, which circulates along the circumference of the fiber,

perpendicularly to the symmetry axis of the optical fiber, is radially confined by continuous total internal reflection (just as

in a disk or cylinder resonator), but (in contrast to a disk resonator) additional axial confinement is achieved by the gradually

changing thickness of the optical fiber (similarly to polar confinement in spherical resonators, and in contrast with uniform

optical fibers in which light is unconfined (only) in the axial direction to enable guiding of light down the fiber). 20

Figure 8. Optical bottle resonator geometry (left) and a false-color micrograph of a fluorescing resonator doped with erbium ions.

20

4.4.1. Spectrum

Typically the optical fiber’s thickness profile around the bottle is approximately parabolic in the axial direction :

, where is the maximum radius of the bottle and is the axial curvature of the resonator.

This thickness profile introduces an effective linear harmonic oscillator (LHO)-like potential in the axial direction and thus

complete confinement of light in the resonator is achieved. This holds in the adiabatic (or Born-Oppenheimer)

approximation (

).20

Eigenmodes of the electromagnetic field can be written in cylindrical coordinates inside the fiber as (using Bessel

functions of the first kind and LHO eigenfunctions ):

12

Here is the azimuthal and is the axial mode number (the LHO quantum number, which determines the number of

nodes of the axial intensity distribution). They, along with the polarization (TE or TM), determine the mode uniquely (the

mode multi-index is ).20

We can imagine light bouncing back and forth inside the LHO in the axial direction (see Figure 9), forming a standing wave

when the resonant condition is met. On resonance light forms a caustic (a region of significantly increased intensity) at the

turning points for classical motion inside a LHO. There we can imagine light “bouncing back”, as if hitting a mirror (like

in a Fabry-Pérot interferometer).20

Figure 9. Comparison of the optical bottle resonator (center left) with a Farby-Pérot resonator (left).

20 A micrograph of q=1,2,3,4 modes (right).

21

The spectrum of an optical bottle microresonator is given by the wave number inside the bottle as20:

where

is the effective LHO energy spacing, is the refractive index of the optical fiber and is the

wavelength of light in vacuum. Here denotes the radius of the bottle at the caustic. For higher and higher

this radius shrinks and so the axial position of the caustic increases with increasing mode numbers.

4.4.2. Fabrication

Some advantages of optical bottle resonators include ease of manufacture (optical fibers are easy to make, and the

thickness of a fiber can easily be modified by heating and stretching the fiber) and high tunability (mechanically stretching

the fiber changes its thickness and also the resonant frequencies of the bottle; another option is electrical thermo-optic

tuning), while also maintaining ultra-high quality factors of typical of spherical and toroid resonators. A fully tunable

optical bottle microresonator has also been demonstrated.20

5. Resonator coupling

For practical application of whispering gallery mode resonators efficient and controllable coupling of light to and from the

resonator is crucial. There are many ways in which we can couple light in and out of a WGM resonator, but we can divide

them into three categories: free wave coupling, fluorescent coupling and evanescent coupling.

There are two important parameters describing coupling performance: efficiency (the ratio of power actually transferred to

the resonator and total input power) and ideality (the ratio of the power transferred to the desired excitation mode and the

total power transferred to the resonator). It is desirable that both parameters be as close to as possible.22

5.1. Free wave coupling

By illuminating the resonator with light from the outside we can excite whispering gallery modes inside the resonator, just

as already excited whispering gallery modes can also leak out of the resonator into free space. This form of coupling is

usually extremely inefficient, since it is based on radiative exchange between radiating modes of the resonator and free

space, which is extremely small for ultra-high-Q resonators (efficiency is much less than for a sphere of radius

and illumination wavelength )23. Efficient coupling of light in WGM resonators must thus be realized in some other way.22

5.2. Fluorescence coupling

A variant on free space coupling is fluorescent coupling. Here we dope the resonator with a fluorescent substance (a dye or

quantum dots), which emits a broad spectrum of light when excited by short wavelength light. To excite the resonator we

illuminate the doped whispering gallery resonator with short wavelength light (which more easily penetrates into the

13

resonator than longer wavelength light), the fluorescent substance in the resonator starts emitting a broad spectrum of

longer wavelength light, which then gets trapped inside the resonator. Here lower Q-factor modes quickly leak out of the

resonator and the only highest Q-factor modes (precisely the whispering gallery modes) remain. By this process we get

semi-efficient in-coupling with virtually zero ideality (all the available modes get excited), and out-coupling is the same as in

free wave coupling, but the coupling scheme itself is quite simple.

5.3. Evanescent coupling

This form of coupling occurs when we bring a structure possessing an optical evanescent field in the vicinity of the

resonator, so that the evanescent fields of the resonator and the coupler overlap significantly. This then enables the

tunneling of the wave field from the coupler (prism, tapered fiber, …) to the resonator and back (in complete analogy with

quantum tunneling in quantum mechanics) and can be much more efficient than free wave coupling.

One of the earliest evanescent coupling methods is the use of a prism coupler22, where the evanescent field of a laser beam

which undergoes total internal reflection in a dielectric prism, overlaps with the evanescent field of a WGM resonator (see

Figure 10). Coupling efficiencies of were achieved with a prism coupler22, but such coupling has low ideality, since it

is hard to control which whispering gallery modes get excited. A variation on the prism coupler is a side-polished fiber

coupler, which offers better compatibility with fiber optics but only limited efficiency.

Figure 10. Illustration of prism and side-polished fiber coupling.

2

The best known way to couple light with whispering gallery modes is through the use of a tapered optical fiber (Figure 11).22

This is a (single mode) optical fiber that is, through a process of heating and stretching, significantly thinned in one part

(possibly even below the wavelength of the light). Light which travels inside an optical fiber has an evanescent field that falls

off exponentially with radial displacement and thus normally most of the optical power is concentrated on the inside of the

fiber. But in a tapered optical fiber, where the thinness of the fiber would force light to focus to a diameter close to or less

that its wavelength, the evanescent field of the light can stretch out so far, that most of the optical power of the guided

beam is actually outside the fiber, in its evanescent field. When this evanescent field is brought to overlap with the

evanescent field of the resonator highly efficient resonator-waveguide coupling can occur.

Figure 11. Illustration of tapered fiber coupling (left)

2, an example of a sphere (center)

12 and a microtoroid (right)

8 coupled to a tapered fiber.

For efficient coupling between a tapered fiber and a resonator an additional criticality condition must also be met. The

condition of critical coupling is a fundamental property of resonator-waveguide coupling and refers to the condition in

which internal resonator loss and waveguide coupling loss must be equal for a matched waveguide-resonator system, at

which point the resulting transmission at the output of the waveguide goes to zero on resonance and so all the energy

pumped inside the waveguide gets transferred to the resonator. In the context of critical coupling the coupled fiber taper-

resonator system is effectively a single-mode coupler and thus the ideality of the coupling is extremely high. With fiber

tapers coupling efficiencies of up to were achieved for coupling of fused silica resonators.22

Additional benefits of using fiber tapers placed alongside resonators as couplers are that they allow simple focusing and

alignment of the input beam as well as collection of the output beam. Since fiber tapers are basically just modified optical

fibers their use as couplers also allows for very easy integration of coupled resonator-waveguide systems with other fiber

optical systems.22

14

6. Applications

Resonators based on whispering gallery modes have a vast array of practical applications. Some of them include24:

interferometry, spectroscopy and fluorescence studies (because of their high Q-factors and finesses), metrology and light

storage devices (because of long photon storage times), as filters in optical fiber telecommunications, study of nonlinear

optical effects and generation of (optical) frequency combs (because of very high circulating intensities at moderate

pumping powers), study of nonclassical light and cavity quantum electrodynamics or cQED effects (where ultra-high Q-

factors are needed), biosensing and other sensing applications (detection of analytes down to a single molecule, measuring

changes in pressure, temperature or motion by measuring changes in the Q-factor and resonant frequencies), ultralow-

threshold microlasers (small mode volumes and high Q-factors enable thresholds below ) and cavities for single atom

lasers (with zero threshold), tunable coupled resonator optical waveguides (CROWs) for photonic integrated circuits and

even optical trapping using the whispering gallery carousel trap (light gradients and momentum transfer cause particles to

rotate around the equator of a whispering gallery resonator greatly ( ) aiding biosensing by actively attracting

particles to the resonator, instead of relying on diffusion).

6.1. Photonic filters

Use of photonic filters based on optical whispering gallery resonators is among the most developed applications of

whispering gallery resonators. The intent is to use them for processing signals in optical communications, where ring

resonators with Q-factors are adequate. The most common designs (see Figure 12) are a whispering gallery

resonator coupled either to a single fiber taper bus (a drop filter) or to two buses (an add/drop filter).24

Figure 12. Comparison of a drop (left) and add/drop (right) WGM filter designs with their standing wave analogs.

6

Both filters are useful for wavelength division multiplexing (where we send many different wavelength signals down a single

optical fiber), since they only filter out those signal that match the resonant frequencies of the resonator and leave other

signals in the fiber unchanged. In an add/drop filter a signal with a resonant wavelength may also be added to the stream of

signals in the optical fiber.

6.2. Sensors

Since the evanescent field of whispering gallery modes protrudes outside the resonator volume such modes are affected by

the environment in which the resonator is placed. The environment affects both the resonant frequencies as well as the

quality factors of whispering gallery modes. Because whispering gallery resonators can have extremely large Q-factors

(small linewidth) a shift in their resonant frequencies is easily measured. This means that they act as very sensitive sensors

which can be affected by things such as the pressure, temperature and chemical (or bio-) composition of their surroundings

(we can also treat the surface of the resonator so that it binds only specific molecules, to improve selectivity).

Bellow a scheme for measuring the presence of viruses (with single virus resolution) that bind to the surface of a glass

microsphere is presented (Figure 13). Coupling is achieved by a tapered optical fiber, and a tunable near-IR laser sweeps

across different wavelengths to determine the resonance frequencies of the resonator (on resonance transmission of the

laser beam through the optical fiber drops), which directly depend on the number of bound viruses.25

Figure 13. Measuring the presence of the influenza A virus using a whispering gallery resonator.

25

15

6.3. Ultralow-threshold lasers

By doping the whispering gallery resonator with a lasing medium (for example a dye, quantum dots or nanocrystals) we can

achieve laser operation in such resonators (in droplets or in solid-state resonators). The advantages of using whispering

gallery resonators as laser resonators are their (potentially ultra-)high Q-factors and very low mode volumes which directly

translate into ultralow lasing thresholds (under one microwatt optical pump power and even lower). Because of their small

size and easy integration on-chip they are very promising for lasing applications, and since we can couple many such

resonators to one optical fiber we can even achieve multi-wavelength lasing.24

An example of an ultralow threshold laser is a laser with lasing threshold26 at with a quality factor at

a lasing wavelength of that has been achieved with a silica microsphere functionalized with doped Nd3+:Gd2O3 .

Another example is sub-nanowatt threshold lasing at the temperature of with GaInP microdisk and microring

resonators (diameters - and quality factors ) with embedded InP quantum dots27.

7. Conclusion In this seminar we looked at some key issues surrounding the use of whispering gallery modes and resonators, such as:

wave theory of whispering gallery modes, resonator performance parameters, typical resonator geometries, coupling of

whispering gallery modes to and from resonators and finally at a few practical applications of whispering gallery modes.

These applications are vast and numerous and interest in these unique wave modes is very high, so it is certain that the field

of whispering gallery modes and their use in photonics will continue its growth in the future, as even more innovative uses

and applications are conceived of.

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Research and Applications 156, 29-59 (2010). 15. Humar, M., Ravnik, M., Pajk, S. & Muševic, I. Electrically tunable liquid crystal optical microresonators. Nat Photon 3, 595-600 (2009). 16. Gorodetsky, M.L., Savchenkov, A.A. & Ilchenko, V.S. Ultimate Q of optical microsphere resonators. Opt. Lett. 21, 453-455 (1996). 17. Savchenkov, A.A., Matsko, A.B., Ilchenko, V.S. & Maleki, L. Optical resonators with ten million finesse. Opt. Express 15, 6768-6773 (2007). 18. Srinivasan, K., Borselli, M., Painter, O., Stintz, A. & Krishna, S. Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with

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