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Vietnam national university, Hanoi

College of Technology

Ho Duc Vinh

Mapping WGMs of

Erbium doped glass microsphere using

Near-field optical probe

Master thesis

Supervisor: Dr. Tran Thi Tam

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CONTENT

1. INTRODUCTION

2. CHAPTER I: MORPHOLOGY DEPENDENT RESONANCES

3. CHAPTER II: COUPLING MICROSPHERES WGMs BASED ON

NEAR-FIELD OPTICS

4. CHAPTER III: FABRICATION OF MICROSPHERE AND TAPER

FIBER

5. CHAPTER IV: EXPERIMENTS AND RESULTS

CONCLUSION

Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N5

chapter 1: Morphology Dependent Resonances

(MDRs-WGMs)

1.1. Dielectric Microsphere – A simple Model of WGMs:

Microspheres act as high Q resonators in optical regime. The curved surface

of a microshere leads to efficient confinement of light waves. The light waves

totally reflect at the surface and propagate along the circumference. If they round in

phase, resonant standing waves are produced near the surface. Such resonances are

called “”morphology dependent resonances (MDRs)”” because the resonance

frequencies strongly depend on the size parameter

λ

π ax 2= , (where a is the radius of

microstructure and λ is the light wavelength). Alternatively , the resonant modes

are often called “”Whispering Gallery Modes (WGMs)””. The WGMs are named

because of the similarity with acoustic waves traveling around the inside wall of a

gallery. Early this century, L.Rayleigh [46] first observed and analyzed the

“”whispers”” propagating around the dome of St.Catherine’s cathedral in England.

Optical processes associated with WGMs have been studied extensively in recent

years [45].

WGMs are characterized by three numbers, n, l and m, for both polarizations

corresponding to TE (transverse electric) and TM (transverse magnetic) modes. TE

and TM modes have no radial components of electric and magnetic fields,

respectively. These integers distinguish intensity distribution of the resonant mode

inside a microsphere (a simple model system of Micro resonators). The order

number n indicates the number of peaks in the radial intensity distribution inside the

sphere and the mode number l is the number of waves of resonant light along the

circumference of the sphere. The azimuthal mode number m describes azimuthal

spatial distribution of the mode. For the perfect sphere, modes of WGMs are

degenerate in respect to m.

In this section, firstly, it presents a simple model of WGMs in terms ray and

wave optics for a qualitative interpretation.

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Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N6

1.1.1 Ray and Wave Optics Approach:

The most intuitive picture describing the optical resonances of microsphere is

based upon ray and wave optics.

* Ray optics:

Consider a microsphere with radius a and a refractive index ( )n ω , and a ray

of light propagating inside, hitting the surface with angle of incidence inθ (Figure

1.1.a).

Figure 1.1. a/ Ray at glancing angle is totally reflected

b/ If optical path = integral number of wavelengths, a resonance is formed

If arcsin(1/ ( ))in c nθ θ ω> = , then total internal reflection occurs. Because of

spherical symmetry, all subsequent angles of incidence are the same, and the ray is

trapped. Leakage occurs only through diffractive effects, i.e., because of the

finiteness of λ/a , where λ is the wavelength in vacuum. The leakage is expected to

be exponentially small. This simple geometric picture leads to the concept of

resonances. For large microspheres ( a >> λ ), the trapped ray propagates close to

the surface, and traverses a distance aπ2≈ in one round trip [52]. If one round trip

exactly equals l wavelengths in the medium (l = integer), then a standing wave can

occur (Figure 1.1 b).This condition translates into

2

( )

a

n

λπ

ω

≈ l (1.1)

A dimensionless size parameter x is defined for this system

λ

π ax 2= (1.2)

cinc θθ >

Inphase

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Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N7

In terms of which the resonance condition is

( )

x

n ω

≈

l

(1.3)

Consider the ray in Figure 1.1.a as a photon. Its momentum is

[2 ( / ( ))]p k nπ λ ω= =h h (1.4)

where p is the momentum of photons, h is the Planck’s constant divided by π2 , and

k is the wave number. If this ray strikes the surface at near-glancing incidence

( 2πθ ≈in ), then the angular momentum, denoted as h l , is

2 ( / ( ))a p a nπ λ ω≈ =h hl (1.5)

which is identical to Equation 1.3. The point of this derivation is to identify the

integer l , originally introduced as the number of wavelengths in the circumference,

as the angular momentum in the usual sense.

The great-circle orbit of the rays need not be confined to the x-y plane (e.g.,

the equatorial plane). If the orbit is inclined at an angle θ with respect to the z-axis,

the z-component of the angular momentum of the mode is (see Figure. 1.2)

.cos( )

2

m π θ= −l (1.6)

For a perfect sphere, all of the m modes are degenerate (with 2 l +1

degeneracy). The degeneracy is partially lifted when the cavity is axisymmetrically

(along the z-axis) deformed from sphericity. For such distortions the integer values

for m are , ( 1),…0,± ± −l l where the degeneracy remains, because the resonance

modes are independent of the circulation direction (clockwise or counterclockwise)

[49]. Highly accurate measurements of the clockwise and counterclockwise

circulating m-mode frequencies reveal a splitting due to internal backscattering, that

couples the two counter propagating modes [47].

Geometrical interpretation of light interaction with a microsphere has several

limitations:

– It cannot explain escape of light from a WGM (for perfect spheres), and

hence the characteristic leakage rates cannot be calculated.

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Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N8

– Geometric optics provides no possibility for incident light to couple into a

WGM.

– The polarization of light is not taken into account.

– The radial character of the optical modes cannot be determined by

geometrical optics [7].

* Wave optics:

The proper description of the system should reply on Maxwell’s equations,

which, for a definite frequency ω and in units where C = 1, is

( ) ( ) 02 =−×∇×∇ ErE εω (1.7)

Here we assume that the dielectric constant ε depends only on the radius a,

i.e., the system is spherically symmetric. The transverse electric (TE) modes are

characterized by

( ) ( ) ( ),m mE r a X θ= Φ Φl l (1.8)

where ( ) 1/ 21m mX LY

−

= + l ll l is the vector spherical harmonic and L a i= × ∇ . The

waves are then described by a scalar equation [19]

( ) ( )

2

2

2

1

0d a

da a

ω ε

− Φ

+ − Φ =

l l

(1.9)

where the scalar function Φ is related to the radial function of the field as

( )ma aφΦ = l (1.10)

similarly, the transverse magnetic (TM) modes are characterized by

( ) ( ) ( )

1

m mE r a Xa

φ

ε

= ∇× l l (1.11)

and is again reducible to a scalar equation [19]

( )

( )

( )

2

2

11 0d d

da a da a a

ω

ε ε

+Φ + − Φ =

l l

(1.12)

where in this case the scalar function is again given by (1.10). Hence, the radial

character of the optical modes could be determined by wave optics.

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Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N9

1.1.2 Lorenz-Mie Theory:

A complete description of the interaction of light with a dielectric is given by

electromagnetic theory which is solved basically in wave optics above. The spherical

geometry suggests expanding the fields in terms of vector spherical harmonics.

Characteristic equations for the WGMs are derived by requiring continuity of the

tangential components of both the electric and magnetic fields at the boundary of

the dielectric sphere and the surrounding medium. Internal intensity distributions

are determined by expanding the incident wave (plane-wave of focused beam),

internal field, and external field, all in terms of vector spherical harmonics and again

imposing appropriate boundary conditions.

Figure 1.2: The resonant light wave propagates along the great circle whose normal

direction is inclined at an angle 2π θ− with respect to the z-axis.

The WGMs of a microsphere are analyzed by the localization principle and

the Generalized Lorenz-Mie Theory (GLMT) [36, 34, 51]. Therefore, each WGM is

characterized by a mode order n , a mode number l and an azimuthal mode m,

which are described above and are summarized here:

+ The radial mode order n indicates the number of maxima in the internal

electric field distribution in the radial direction.

+ The mode number l gives the number of maxima between 0o and 180o

degrees in the angular distribution of the energy of the WGM.

+ Each mode WGM of the microsphere also has an azimuthal angular

dependence from 0o and 360o, which is define with an azimuthal mode number m.

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Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N10

However, for sphere, WGMs differing only in azimuthal mode number have

identical resonance frequencies.

The characteristic eigenvalue equations for the natural resonant frequencies

of dielectric microsphere have been solved in homogeneous surroundings. WGMs

correspond to solutions of these characteristic equations of the electromagnetic

fields in the presence of a sphere. The characteristic equations are obtained by

expanding the fields in vector spherical harmonics and then matching the tangential

components of the electric and magnetic fields at the surface of the sphere. No

incident field is assumed in deriving the characteristic equations [17].

For modes having no radial component of the magnetic field (transverse

magnetic or TM modes) the characteristic equation is,

[ ]

” (1)

2 (1)

( )( ) ( ( ) )

( ) ( ( ) ) ( )

xh xn j n x

n j n x h x

ω ω

ω ω

=

ll

l l

(1.13)

where x is the size parameter,

λ

π ax 2= , a is the radius, λ is the wavelength, and

( )n ω is the ratio of the refractive index of dielectric microsphere to that of the

surrounding medium.

The characteristic equation for modes having no radial component of the

electric field (transverse electric or TE modes) is:

[ ]

” (1)

(1)

( )( ) ( ( ) )

( ( ) ) ( )

xh xn x j n x

j n x h x

ω ω

ω

=

ll

l l

(1.14)

The characteristic equations are independent of the incident field. In equation

1.13 and equation 1.14, jl(x) and hl

(1)(x) are the spherical Bessel and the Hankel

functions of the first kind, respectively. The prime (‘) denotes differentiation with

respect to the argument. The transcendental equation is satisfied only by a discrete

set of characteristic values of the size parameter, xn,l , corresponding to the radial nth

root for each angular l.

The elastically scattered field can be written as an expansion of vector

spherical wave functions with TE coefficients (al) and TM coefficients (bl) for a

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Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N11

plane wave incident on a dielectric microsphere. The scattered field becomes infinite

at complex frequencies ( , )nω l corresponding to the complex size parameters x(n,l) , at

which, al and bl become infinite.

Fig. 2.3: Three light waves; the linearly polarized incident plane wave, the spherical

wave inside the sphere and the spherical wave scattered by the sphere.

al coefficients are associated with TEn,l WGMs specified by:

[ ] [ ]

[ ]

‘ ‘2

”(2) 2 (2)

( ) ( ) ( ( ) ) ( ) ( ( ) ) ( )

( ) ( ) ( ( ) ) ( ) ( ( ) ) ( )

j x n x j n x n j n x x j x

a

h x n x j n x n j n x xh x

ω ω ω ω

ω ω ω ω

−

=

−

l l l l

l

l l l l

(1.15)

Similarly, bl coefficients are associated with the TMn,l WGMs as specified by

equation 1.16, where (2) ( )h xl are the spherical Hankel functions of the second type

[6].

[ ] [ ]

[ ]

‘ ‘

”(2) (2)

( ) ( ) ( ( ) ) ( ( ) ) ( )

( ) ( ) ( ( ) ) ( ( ) ) ( )

j x n x j n x j n x x j x

b

h x n x j n x j n x xh x

ω ω ω

ω ω ω

−

=

−

l l l l

l

l l l l

(1.16)

The WGMs of the microsphere occur at the zeros of the denominators (or

poles) of al and bl coefficients. These complex poles occur at discrete values of the

complex size parameter x. The modes are radiative for real frequencies, and hence

the modes are virtual when the resonance frequencies are complex.

+ The real part of the pole frequency is close to real resonance frequency

[19].

+ The imaginary part of the pole frequency determines the linewidth of the

resonance [37].

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Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N12

For a fixed radius, the WGMs have l values that are bound by ( )x n xω< <l

[28] (see equation 1.3), where the upper limit is the maximum number of

wavelengths that fit inside the circumference. The radial electric field distribution of

the lowest order modes (nth) shows a peak just inside the surface. The higher the

mode order becomes, the more the mode distribution goes to inner region [30].

For larger size parameters the first order resonances become narrow while the

higher order resonances heighten and become dominant [8].

The first peaks observed in the spectra are the first-order resonances. The

second order resonances begin to appear when the size parameter increases due to

decreasing the linewidths. As the size parameter increases further, the linewidths of

the first and second order resonances decrease further and third-order resonances

begin to appear.

The natural resonance frequencies associated with the TEn,l and TMn,l modes

are given by equation 1.17, where µ is the permeability and ε permittivity of the

surrounding lossless medium [23]. Thus, equation 1.17 definitions the complex

frequencies at which a dielectric sphere will resonate in one of its natural modes are:

,

,

n

n

x

a

ω

µε

= ll (1.17)

Figure 1.3. WGM mode spacing λ∆ and the WGM linewidth 2/1λ

Based on the Lorenz-Mie theory, the separation between the adjacent peak

wavelengths of the same mode order (n) WGMs with subsequent mode numbers

λ∆

2/1λ

Wavelength

M

od

e

D

en

si

ty

(

a.

u.

)

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Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N13

( l ), mode spacing ( λ∆ ), is approximately given by Eq. 1.18. The full width at half

maximum (FWHM) of the resonance or the resonance linewidth are denoted by 2/1λ ,

see Fig. 1.3 [57]

2 2

2

arctan( ( ) 1)

2 ( ) 1

n

a n

λ ω

λ

π ω

−

∆ =

−

(1.18)

1.2. Characteristics of Dielectric Microsphere:

1.2.1. WGM Position:

For spheres with large x, several expressions are derived to determine the

spectral location, separation, and width of WGMs. The positions of WGMs are

approximated by [26, 10]:

1 3 2 2

1 3 1 3 2 3 2 1 3 2 3 3 3

, 3

3 2 ( ( ) 2 3)( ) 2 ( 2 ) ( )

10n n n n

P P n Pn x v v v v O vωω α α α

ρ ρ

−

− − − − −−= + − + − +l

(1.19)

where ( )P n ω= for TE modes, 1/ ( )P n ω= for TM modes, 1/ 2v = +l ,

2 2 ( ) 1nρ ω= − , nα are the roots of the Airy function, and 3( )

i

O ν

−

are the ith fractional

forms of the Airy function .

1.2.2. WGM Separation:

The separation between resonances ,nx∆ l is more useful than the absolute

mode positions to determine the approximate sphere size and approximating mode

numbers. Asymptotic analysis gives:

1 3 2 3

2 3 2 4 3

,

2 2( ) 1

3 10n n n

n x v vω α α

− −

− −∆ = + −l

2 22 3 1 3

5 3 2

4 / 3

( ( ) 2 3)2 2 ( )

3 9 n

P n P v O vω α

ρ

−

− − −+ − +

(1.20)

Although equation 1.20 is more accurate, a simple approximation to x∆ is

given by:

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Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N14

1/ 21 2

1/ 22

tan ( ( ) / ) 1

( ( ) / ) 1

n x

x

n x

ω

ω

− − ∆ =

−

l

l l

for 1/ 2x − ?l (1.21a)

1tanx ρ

ρ

−

∆ = for x/l =1 (1.21b)

1.2.3. WGM Density:

An approximation to the mode density of high-Q WGMs, which is defined as

the number of resonance modes per frequency or size-parameter interval, is [50]:

WGM Density

( )2 1tanxρ ρ ρ

π

−−

= (1.22)

Equation 1.22 implies that the mode density increases rapidly as the refractive index

increases.

1.2.4. Spatial Distribution of WGMs:

Spatial characteristics of WGMs are described in terms of M and N at

resonant size parameters satisfying the characteristic equations. Since TE modes are

defined as the electric field having no radial components, these modes are

represented by the vector functions M. Similarly, since TM modes are defined as the

magnetic field having no radial components, these modes are represented by the

vector functions N. The corresponding electric fields are represented by the vector

functions N because the rotation of M is proportional to N.

Using the vector wave functions, the internal electric fields of a sphere are

expanded as a sum of electric fields (TE modes and one of TM modes). The spatial

distributions of the electric field of TE and TM modes of WGMs are obtained

by

2

TEE and

2

TME at the size parameter satisfying characteristic equations for a

given l .

Figure 1.4 shows the internal intensity distributions in the equatorial plane of

a sphere with index of refraction ratio n(ω) = 1,4 for (A) TE30,1, (B) TE30,2, and

(C)TE30,3 modes, where the subscript denote angular mode and the order numbers.

The resonant size parameter is shown in the upper side of each figure. Here the

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Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N15

distributions are obtained by adding WGMs rounding along the +Φ (m = 30) and –

Φ (m = -30) directions. Remarkably, the number of peaks in the angular distribution

is identical as the mode number l multiplied by a factor 2 (l from 0o to 180o), while

the number of peaks in the radial intensity is the mode order n.

Figure 1.4. The internal intensity distributions in the equatorial plane for

(A)TE30,1, (B)TE30,2 and (C)TE30,3 modes of a sphere with n(ω) = 1,4. The resonant

size parameters are shown in the upper side of each figure.

As the mode order increases, number of peaks in the internal intensity profile

increases, corresponding to the mode order, and the highest peak is located at the

most inner side in the radial direction. An illustration of the angle-averaged radial

intensity distribution for the same mode number with n=1,2,3 is shown in

figure.1.5.(A).

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Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N16

Fig 1.5. (A) Typical illustration of angle-averaged intensity distribution profile

along the radial direction for WGMs n=1,2,3 with same l.

(B) ) Typical illustration of internal-intensity distribution as

a function of θ for TE WGMs with m = 1, l/2, and l.

Figure1.6. The internal intensity distribution as a function of θ for TE WGMs with

l=30, and m=1, 15, and 30. The maximum intensity of each m-mode

is located near 1sin ( / )mθ −= l

The dependence of the internal intensity distribution on the azimuthal mode

number m is depicted in figure 1.5(B), in which the angular internal intensity

distribution is a function of θ. Three WGMs for m=1, l/2 and l are illustrated as the

angle θ varies from 0o to 90o. These WGMs have the same resonance frequency, but

A B

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Chapter 1 Morphology Dependent Resonances

Ho Duc Vinh K10N17

the maximum intensity for each m is inclined at an angle 1sin ( / )mθ −= l . The

maximum intensity peak agrees with the ray optics picture of an m-mode circulating

in a confined orbit inclined at 1sin ( / )mθ −= l and with its normal inclined at an angle

1cos ( / )mθ −= l .

Figure 1.6. shows the angular distribution of three TE WGMs with l = 30 and

m = 1, 15 and 30 as a function of θ varied form 0 to 90 degrees. The maximum

intensity of each m mode is located near 1sin ( / )mθ −= l . The m = 1 mode is

confined near the pole region. The m = 15 mode is located near

1sin (15/ 30) 30oθ −= = and the m = 30 mode is near the equatorial plane ( 90oθ = ).

These results are consistent with the qualitative interpretation mentioned in the

previous subsection although the spatial distributions shown in this figure have

somewhat broader structure.

1.2.5. Resonator Quality of Microsphere WGMs:

Based on the theory of electromagnetic fields, the quality factor-Q of a

resonance is defined as:

, ,

,

, ,

Re( )

2 Im( )

n n

n

n n

x

Q

x

ω

ω τ

ω

= = =

∆

l l

l

l l

(1.23)

whereτ is the life time of a wave on a WGM. In a perfectly smooth homogeneous

lossless sphere the Q values are limited by diffractive leakage losses and can be as

high as 1010. In reality, volume inhomogeneities, surface roughness, and absorption

restrict the maximum Q values to be less than 1010. Local or global shape

deformations and nonlinear effects can further reduce the maximum Q value.

For frequencies near a WGM, the electric field inside the cavity varies as:

)

2

exp()( 000 tQ

tiEtE ωω −−= (1.24)

The decay term leads to a broadening of the resonance linewidth, giving a

Lorentzian lineshape for the energy distribution

2

0

2

0

2

)2()(

1|)(|

Q

E

ωωω

ω

+−

∝ (1.25)

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