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 1 LUAN MOI download : skknchat@gmail. 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 TIEU LUAN MOI download : skknchat@gmail.com Chapter 1 Morphology Dependent Resonances chapter 1: Morphology Dependent Resonances (MDRs-WGMs) 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 2π a frequencies strongly depend on the size parameter x = , (where a is the radius of λ 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. 5 Ho Duc Vinh K10N LUAN MOI download : skknchat@gmail.com Chapter 1 Morphology Dependent Resonances 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 θ in (Figure 1.
Inphase θ inc > θ c Figure 1. a/ Ray at glancing angle is totally reflected b/ If optical path = integral number of wavelengths, a resonance is formed If θ in > θ c = arcsin(1/ n(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., because of the finiteness of a / λ , 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 ≈ 2π a 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.This condition translates into λ 2π a ≈ l (1.1) n(ω ) A dimensionless size parameter x is defined for this system 2π a x= (1.2) λ 6 Ho Duc Vinh K10N LUAN MOI download : skknchat@gmail.com Chapter 1 Morphology Dependent Resonances In terms of which the resonance condition is l x≈ (1.3) n(ω ) Consider the ray in Figure 1.
Its momentum is p = h k = h [2π (λ / n(ω ))] (1.4) where p is the momentum of photons, h is the Planck’s constant divided by 2π , and k is the wave number. If this ray strikes the surface at near-glancing incidence ( θ in ≈ π 2 ), then the angular momentum, denoted as h l , is h l ≈ a p = a 2π h (λ / n(ω )) (1.5) which is identical to Equation 1. 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., 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.6) 2 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 ± l , ± (l − 1),.0, 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. 7 Ho Duc Vinh K10N LUAN MOI download : skknchat@gmail.com Chapter 1 Morphology Dependent Resonances - 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 ∇ × (∇ × E ) − ω 2ε (r )E = 0 (1.7) Here we assume that the dielectric constant ε depends only on the radius a, i., the system is spherically symmetric. The transverse electric (TE) modes are characterized by E ( r ) = Φ lm ( a ) X lm (θ , Φ ) (1.8) −1/ 2 where X lm = l ( l + 1) LYlm is the vector spherical harmonic and L = a × i∇. The waves are then described by a scalar equation [19] d 2Φ 2 l ( l − 1) + ω ε ( a ) − Φ = 0 (1.9) da a2 where the scalar function Φ is related to the radial function of the field as Φ = aφ lm ( a ) (1.10) similarly, the transverse magnetic (TM) modes are characterized by 1 E (r ) = ∇ × φ lm ( a ) X lm (1.12) da ε ( a ) da ε (a) a2 where in this case the scalar function is again given by (1. Hence, the radial character of the optical modes could be determined by wave optics.
8 Ho Duc Vinh K10N LUAN MOI download : skknchat@gmail.com Chapter 1 Morphology Dependent Resonances 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.2: The resonant light wave propagates along the great circle whose normal direction is inclined at an angle π 2 − 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.
9 Ho Duc Vinh K10N LUAN MOI download : skknchat@gmail.com Chapter 1 Morphology Dependent Resonances 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, [ n(ω ) jl (n(ω ) x)] ' ' xhl(1) ( x ) = (1.13) n 2 (ω ) jl (n(ω ) x) hl(1) ( x) 2π a where x is the size parameter, x = , a is the radius, λ is the wavelength, and λ 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: [ n(ω ) x jl (n(ω ) x)] ' ' xhl(1) ( x) = (1) (1.14) jl (n(ω ) x) hl ( x ) The characteristic equations are independent of the incident field.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 10 Ho Duc Vinh K10N LUAN MOI download : skknchat@gmail.com Chapter 1 Morphology Dependent Resonances plane wave incident on a dielectric microsphere. The scattered field becomes infinite at complex frequencies ω ( n , l ) corresponding to the complex size parameters x(n,l) , at which, al and bl become infinite.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: jl ( x) [ n (ω ) x jl ( n(ω ) x ) ] − n 2 (ω ) jl ( n(ω ) x) [ x jl ( x ) ] ' ' al = (1.15) hl(2) ( x) [ n(ω ) x jl ( n(ω ) x) ] − n 2 (ω ) jl ( n(ω ) x) xhl( 2) ( x) ' ' Similarly, bl coefficients are associated with the TMn,l WGMs as specified by equation 1.