Mapping wgms of erbium doped glass microsphere using near-field optical probe

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H F-XC A N GE H F-XC A N GE c u-tr a c k N y bu to k lic Vietnam national university, Hanoi College of Technology .d o Ho Duc Vinh Mapping WGMs of Erbium doped glass microsphere using Near-field optical probe Master thesis Supervisor: Dr. Tran Thi Tam 1 o .c m C m w o .d o w w w w w C lic k to bu y N O W ! PD O W ! PD c u-tr a c k .c 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 H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c 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 x = 2π a , (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. Ho Duc Vinh 5 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c 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 θ in (Figure 1.1.a). Inphase θ inc > θ c Figure 1.1. 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.e., 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 1.1 b).This condition translates into 2π a ≈ l λ n(ω ) (1.1) A dimensionless size parameter x is defined for this system x= Ho Duc Vinh 2π a λ 6 (1.2) K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c In terms of which the resonance condition is x≈ l n(ω ) (1.3) Consider the ray in Figure 1.1.a as a photon. 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 1.3. 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) π m = l .cos( − θ ) 2 (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 ± 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. Ho Duc Vinh 7 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c - 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.e., the system is spherically symmetric. The transverse electric (TE) modes are characterized by E ( r ) = Φ lm ( a ) X lm (θ , Φ ) where X lm =  l ( l + 1)  −1/ 2 (1.8) LYlm is the vector spherical harmonic and L = a × i∇ . The waves are then described by a scalar equation [19] l ( l − 1)  d 2Φ  2 + ω ε ( a ) − Φ = 0 da  a2  (1.9) 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 E (r ) = 1 ∇ × φ lm ( a ) X lm  ε ( a) (1.11) and is again reducible to a scalar equation [19] d 1  d Φ   2 l ( l + 1)  Φ = 0   + ω − da ε ( a )  da   ε (a) a2  (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. Ho Duc Vinh 8 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c 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 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. Ho Duc Vinh 9 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c 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 )  = n 2 (ω ) jl (n(ω ) x) hl(1) ( x) ' where x is the size parameter, x = ' (1.13) 2π a , 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)] ' jl (n(ω ) x)  xhl(1) ( x)  =  (1) hl ( x ) ' (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 Ho Duc Vinh 10 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c 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. 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: jl ( x) [ n (ω ) x jl ( n(ω ) x ) ] − n 2 (ω ) jl ( n(ω ) x) [ x jl ( x ) ] ' al = ' hl(2) ( x) [ n(ω ) x jl ( n(ω ) x) ] − n 2 (ω ) jl ( n(ω ) x)  xhl( 2) ( x)  ' ' (1.15) Similarly, bl coefficients are associated with the TMn,l WGMs as specified by equation 1.16, where hl(2) ( x) are the spherical Hankel functions of the second type [6]. jl ( x ) [ n(ω ) x jl ( n(ω ) x ) ] − jl ( n(ω ) x) [ x jl ( x ) ] ' bl = ' hl(2) ( x) [ n(ω ) x jl ( n(ω ) x) ] − jl ( n(ω ) x)  xhl(2) ( x )  ' ' (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]. Ho Duc Vinh 11 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c For a fixed radius, the WGMs have l values that are bound by x < l < n(ω ) x [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: Mode Density (a.u.) ω n, l = xn , l (1.17) a µε ∆λ λ1/ 2 Wavelength Figure 1.3. WGM mode spacing ∆λ and the WGM linewidth λ1/ 2 Based on the Lorenz-Mie theory, the separation between the adjacent peak wavelengths of the same mode order (n) WGMs with subsequent mode numbers Ho Duc Vinh 12 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c ( 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 λ1/ 2 , see Fig. 1.3 [57] ∆λ = λ 2 arctan( n2 (ω ) − 1) (1.18) 2π a n 2 (ω ) − 1 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]: n(ω ) xn ,l = v + 2 −1 3 α n v1 3 − P 3 2 −1 3 P(n 2 (ω ) − 2 P 2 3) + ( 2−2 3 )α n2 v −1 3 − αn v−2 3 + O(v−3 3 ) 3 ρ 10 ρ (1.19) where P = n(ω ) for TE modes, P = 1/ n(ω ) for TM modes, ρ 2 = n 2 (ω ) − 1 , α n are the roots of the Airy function, and O(ν −i 3 v = l + 1/ 2 , ) are the ith fractional forms of the Airy function . 1.2.2. WGM Separation: The separation between resonances ∆xn ,l is more useful than the absolute mode positions to determine the approximate sphere size and approximating mode numbers. Asymptotic analysis gives: n(ω )∆xn,l = 1 + 2−1 3 2−2 3 2 −4 3 α n v −2 3 − αn v 3 10  22 3 P ( n 2 (ω ) − 2 P 2 3) 2 −1 3  −5 3 −2 + −  α n v + O (v ) 4/3 3 9 ρ   (1.20) Although equation 1.20 is more accurate, a simple approximation to ∆x is given by: Ho Duc Vinh 13 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c 1/ 2 tan −1  (n (ω ) x / l )2 − 1 ∆x = 1/ 2 l (n(ω ) x / l ) 2 − 1 ∆x = for x − l ? 1/ 2 tan −1 ρ for x/l =1 ρ (1.21a) (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 = x ρ 2 ( ρ − tan −1 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 2 by ETE and ETM 2 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 Ho Duc Vinh 14 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c 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). Ho Duc Vinh 15 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c A B 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 θ = sin −1 ( m / 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 Ho Duc Vinh 16 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c the maximum intensity for each m is inclined at an angle θ = sin −1 ( m / l ) . The maximum intensity peak agrees with the ray optics picture of an m-mode circulating in a confined orbit inclined at θ = sin −1 ( m / l ) and with its normal inclined at an angle θ = cos −1 (m / 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 θ = sin −1 (m / l ) . The m = 1 mode is confined near the pole region. The m = 15 mode is located near θ = sin −1 (15 / 30) = 30o and the m = 30 mode is near the equatorial plane ( θ = 90o ). 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: Q = Re( xn,l ) 2 Im( xn,l ) = ω n,l = ω n,l τ ∆ω n,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: E (t ) = E0 exp(−iω 0t − ω0 t) 2Q (1.24) The decay term leads to a broadening of the resonance linewidth, giving a Lorentzian lineshape for the energy distribution | E (ω ) | 2 ∝ Ho Duc Vinh 1 (ω − ω 0 ) + (ω 0 2Q ) 2 2 17 (1.25) K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c Figure 1.7. The angle average intensity as a function of the normalized radius r/a for TE WGMs with l=60, and n=1, 2, 3. The refractive index of microsphere is 1.4 When resonant standing waves grow inside a sphere, the spherical particle acts as a high Q resonator. A fraction of the resonant light wave leaks due to the diffractive effect and the quality factor Q of the resonator is limited by the diffractive losses. The electric fields of the WGMs extend beyond the particle boundary as evanescent waves. The lowest order WGM has the maximum of the internal distribution at the region nearest the surface of the sphere, and has the shortest penetration depth toward the outer region of the sphere. For a given mode number l, the n=1 modes have the highest Q (smallest ∆x), with a peak intensity located closest to the surface and the evanescent wave penetrating shortest into the surrounding medium. As n increases, the Q value decreases, the peak intensity moves away from the surface, and the evanescent wave penetration into the surrounding medium increases. For a fixed radial mode order n, modes with higher angular momentum or higher mode number l have higher Q values. Figure 1.7 shows the angle averaged intensity as a function of the normalized radius r/a for TE WGMs with l = 60 and n = 1, 2 and 3. The refractive index of the sphere is 1.4. 2 This result is obtained by computing ETE integrated over the total solid angle [42]. Ho Duc Vinh 18 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c Figure 1.8: The resonance curves for the same WGMs and the sphere as in Fig.1.7 as a function of the size parameter, where each x0 is centered. TE60,1 TE60,2 TE60,3 Resonance size parameter 47.491 51.677 55.218 Quality factor 9.4 x 106 3.9 x 104 1.6 x 103 Table.1: The resonance size parameters and the quality factors of TE MDRs with l = 60 and n = 1, 2 and 3. Figure 1.8 shows the resonance curves for the same WGMs and sphere as in Figure 1.7 as a function of the size parameter, where each resonant size parameter xo is centered. The resonance curve of the lowest order WGM is extremely narrow compared with the higher order WGMs. The quality factor Q of the WGM can be also defined as: Q= x0 ∆x (1.26) where ∆x is the full width at the half maximum of the resonance curve. The resonance size parameters and the quality factors of these modes are summarized in Table.1. The lowest order WGM with the same mode number has the highest quality factor and is therefore most strongly confined inside the sphere. Ho Duc Vinh 19 K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c Figure 1.9: the resonance curves for the first order TE MDRs with l = 30, 45 and 60 as a function of the size parameter x 0 is also centered. The refractive index of the sphere is 1.4. Table.2: The resonance size parameters and the quality factors of the first order TE WGMs with l = 30, 45 and 60. TE30,1 TE45,1 TE60,1 Resonance size parameter 24.969 36.299 47.491 Quality factor 2.3 x 103 1.3 x 105 9.4 x 106 Figure 1.9 shows the resonance curves for the first order TE WGMs with l = 30, 45 and 60 as a function of the size parameter, where x 0 is also centered. The refractive index of the sphere is 1.4. Q increases as the mode number is increased for a fixed n. The resonance curve of TE60,1 WGM is also extremely narrow compared with the lower mode WGMs. The resonance size parameters and the quality factors of these modes are summarized in Table.2. On the other hand, one can describe the performance of a resonator element in terms of its capacity to store energy. The quality factor (Q-factor) determines how long a photon can be stored inside a WGM [18]. Therefore the quality (Q) of a resonance is governed by the losses associated with it. The observed resonator quality is the geometric sum of the qualities of each mechanism. 1 Qobserved Ho Duc Vinh = 20 1 1 + Q0 Qcoupling (1.27) K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N N O W ! PD O W ! PD .c Alternatively, the observed spectral width is the sum of the widths of all the different loss mechanisms. The internal losses (Q0) are composed of absorption losses (Qabs), diffraction leakage losses (Qr) and scattering losses (Qs) .The external loss is due to coupling (Qcoupling). 1 1 1 1 = + + Q0 Qabs Qr Qs (1.28) In an optical microsphere WGM resonator, energy storage may be thought of as the retention of individual light rays that have been inserted into the cavity [15]. The value of the quality factor roughly equals the number of times a given ray can be expected to travel around the sphere before succumbing to a loss process. In silica microspheres, internal loss effects include scattering from surface irregularities, absorption due to molecular resonances, Rayleigh scattering. Surface scattering is extremely low, since extremely smooth surfaces can be fabricated. Therefore, absorption and Rayleigh scattering dominate the losses [30]. 1.2.6. Mode volume of microsphere WGMs: In many applications, not only temporal confinement of light (i.e. the Qfactor), but also the spatial extension to which the light is confined is an important performance parameter. Several definitions of mode volume can be encountered in literature, and are discussed in this section. The most common definition of mode volume is related to the definition of the energy density of the optical mode. It is defined as the equivalent volume, the mode occupies if the energy density was distributed homogeneously throughout the mode volume, at the peak value: ϖ e (r ) + ϖ m (r ) = 1 1 BB εE E + 2 2µ (1.29) r VMode ∫ (ϖ ( r ) + ϖ ( r ) ) dV = ∫ ε ( r ) E ( r ) d r = max (ϖ ( r ) + ϖ ( r ) ) max ε ( r ) Er ( r ) ) ( e 2 3 m (1.30) 2 e m The mode volumes using these formulas can be well approximated by:  1.02 D 11 / 6 (λ / n )7 / 6 Vm , sphere ≅  11 / 6 (λ / n ) 7 / 6 1.08D Ho Duc Vinh 21 TE TM (1.31) K10N y bu to k .d o m o o c u-tr a c k lic to k lic C m w w w .d o C bu y Morphology Dependent Resonances Chapter 1 w w w c u-tr a c k .c H F-XC A N GE H F-XC A N GE N .c Chapter 2: Coupling Microsphere WGMs based on Near-field Optics 2.1. Introduction of Near-field optics Near-field optics has developed very rapidly from around the middles 1980s after preliminary trials in the microwave frequency region, as proposed as early as 1928. At the early stages of this development, most technical efforts were devoted to realizing super-high-resolution optical microscopy beyond the diffraction limit. However, the possibility of exploiting the optical near-field, phenomenon of quasistatic electromagnetic interaction at subwavelength distances between nanometric particles has opened new ways to nanometric optical science and technology, and many applications in the field of nanometric fabrication and manipulation have been proposed and implemented. For optical telecommunication system, near-field optics can demonstrate lots of photonic phenomena such as quantum electrodynamics (QED), CQED… And one of spectacular examples is near-field interaction of microcavities and tip guide. It is my purpose to use a simple and practical theory so that we can understand easily the fundamental physics of the near-field in three dimensions and to obtain a general expression for each field component which will serve as a guide to more complicated cases. This part will show that the analytic forms of the near-field components around a microsphere produced by an incident plane wave can be obtained and that the effect of near-field can be evaluated in some applications. Comparison of our theory with an experimental result reported by other authors shows good agreement. It will also verify that the localization area of the near field is proportional to the size of the microsphere, and that the field momentum is locally modified by the interference between the near field and the incident field and that the modulation amount is dependent on the size of the sphere, instead of the wavelength of the light. Also, It could be seen the relationship of Evanescent-field with Near-field optics and the scope of near-field optics in the modern optical telecommunication depicted hereinafter. Ho Duc Vinh 22 K10N y bu to k .d o m o o c u-tr a c k lic w w w .d o Coupling Microsphere WGMs based on near-field optics m C lic k to Chapter 2 w w w C bu y N O W ! PD O W ! PD c u-tr a c k .c
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