Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Generalized formulation for performance degradations due to bending and edge scattering loss in microdisk resonators

J. E. Heebner, T. C. Bond, and J. S. Kallman

Open Access Open Access

Abstract

We present a generalized formulation for the treatment of both bending (whispering gallery) loss and scattering loss due to edge roughness in microdisk resonators. The results are applicable to microrings and related geometries. For thin disks with radii greater than the bend-loss limit, we find that the finesse limited by the scattering losses induced by edge roughness is independent of radii. While a strong lateral refractive index contrast is necessary to prevent bending losses, unless the radii are of the order of a few microns, lateral air-cladding is detrimental and only enhances scattering losses. The generalized formulation provides a framework for selecting the refractive index contrast that optimizes the finesse at a given radius.

©2007 Optical Society of America

1. Introduction

Photonic devices incorporating microresonators have become increasingly prevalent in recent years. Despite their enormous potential, their widespread applicability remains stifled due to challenges associated with their modeling and fabrication. Because these devices typically achieve performance enhancements through long effective path lengths, minimizing losses is crucial. Primary loss mechanisms in microdisk and microring resonators are generally due to intrinsic bending or whispering gallery radiation and scattering resulting from edge roughness due to imperfect fabrication. It is well known that bending losses increase with decreasing index contrast. Scattering losses resulting from edge roughness however, behave in the opposite manner, increasing with increasing index contrast. In general, modeling this tradeoff requires detailed, numerically-intensive finite element and/or finite difference time domain simulations or cumbersome approximations [1, 2, 3] to be accurate in the design phase. The designer does not have a tool early in the design phase to select the appropriate materials or geometry optimized to target a desired level of performance. In this paper, we implement and combine the results of two powerful analytic methods into a greatly simplified framework. These include solution of the whispering gallery dispersion relation for an infinite dielectric cylinder and a volume current analysis of edge roughness scattering [4, 5, 6, 7, 8, 9] For the design of simpler waveguide geometries, generalized formulations such as that of Kogelnik [10] have been and continue to be instrumental both in fostering intuition and in making early, high-level design choices. The following derivation proceeds in the same spirit.

2. Review of the optical whispering gallery dispersion relation

The Helmholtz equation in cylindrical coordinates applicable to the axial fields of whispering gallery modes (WGMs) is

(∂2∂z2+∂2∂r2+1r∂∂r+1r2∂2∂φ2+k2)Ψz(r,z,φ)=0.

The wavevector associated with circulating WGM field solutions of an infinite dielectric cylinder is directed primarily in the azimuthal direction. Wavevector contributions and field variation in the axial direction are generally small. In disks of finite height, the axial variations can be accounted for by using effective index methods. The time-independent modal field distribution for a WGM can thus be separated into a radially-dependent mode profile and an azimuthal phase dependence, Ψz(r)eimφ. Here, Ψz refers to the axial electric (TM) or magnetic (TE) modal field amplitude and m is the azimuthal quantization number. The Bessel equation for the radial field dependence becomes:

(∂2∂r2+1r∂∂r+k2−m2r2)Ψz(r)=0.

Field solutions consist of Bessel functions of the first kind, Jm(k̃ 1 r) inside the disk boundary, (r < R) and Hankel functions of the first kind, Hm (1)(k̃ 2 r) outside the disk boundary, (r > R). Here, a complex propagation constant and frequency k̃j = njω̃/c are introduced for reasons that will become apparent later. Applying the boundary condition that the axial field components be continuous across the interface gives the complete radial dependence:

Ψz,in(r,φ)=AmJm(k˜1r)ei(±mφ)
Ψz,out(r,φ)=AmJm(k˜1R)Hm(1)(k˜2R)Hm(1)(k˜2r)ei(±mφ).

The radial and azimuthal field components are easily derived from the axial field components by use of Maxwell’s equations for TM modes:

Hr=mZ0k˜0rEz
Hφ=iZ0k˜0∂∂rEz,

and TE modes:

Er=−mZ0n2k˜0rHz
Eφ=−iZ0n2k˜0∂∂rHz.

where Z 0 is the impedance of free space. Satisfying boundary conditions for both the electric and magnetic fields across the interface results in a dispersion relation that can be written respectively for TM and TE modes as:

k˜1Jm′(k˜1R)Jm(k˜1R)=k˜2Hm′(1)(k˜2R)Hm(1)(k˜2R)
Jm′(k˜1R)k˜1Jm(k˜1R)=Hm′(1)(k˜2R)k˜2Hm(1)(k˜2R)

Multiple radial solutions exist at discrete propagation constant values in the complex plane giving rise to a resonance map. In the absence of gain or extrinsic loss, solutions necessarily involve an imaginary component of the frequency, implying a continuous decay rate of energy confined within the resonator. The complex frequency and decay rate γ are related according to ω˜=ω−iγ2. An intrinsic quality factor equal to the characteristic number of optical cycles before confined energy is lost to the radiation continuum may be defined as Qi=ωγ. This dimensionless parameter generalizes the imaginary part of the solution space. In order to generalize the real part, a normalized radius is defined as X = n 12πR/λ. By specifying only two parameters: the index ratio, n = n 1/n 2 and azimuthal mode number m, one or more radial mode solutions exist; each is characterized by two parameters: X and Qi. Written in normalized form, the dispersion relation becomes:

nJm′[(1−i12Qi)X]Jm[(1−i12Qi)X]−Hm′(1)[(1−i12Qi)Xn]Hm(1)[(1−i12Qi)Xn]=0
Jm′[(1−i12Qi)X]Jm[(1−i12Qi)X]−nHm′(1)[(1−i12Qi)Xn]Hm(1)[(1−i12Qi)Xn]=0

In order to solve for the complex roots of these equations, a global optimization scheme can be used to minimize the absolute value of each equation over the complex map [11]. Using this method, generalized plots of intrinsic quality factor against normalized radius may be obtained for the WGMs of a dielectric cylinder. Figure 1 displays the bending limited finesse (ℱi = Qi/m) vs. normalized radius for a variety of azimuthal mode numbers and index ratios. These results will later be combined with the results of the normalized volume current formulation for edge scattering.

 figure: Fig. 1.

Fig. 1. Bending limited finesse of the lowest order radial TM and TE whispering-gallery modes of a dielectric cylinder of index n 1 in a medium of index n 2 plotted against normalized radius. The family of diagonal lines represents varying refractive index ratio (n 1/n 2). The family of nearly vertical lines corresponds to whispering gallery mode resonances, each characterized by an azimuthal mode number m. The normalized radius is nearly equal to m although differs slightly due to the fact that the mode does not peak at the disk edge and experiences a suppressed effective index due to imperfect edge confinement. The plots were obtained by numerically solving the dispersion relation for whispering-gallery modes.

Download Full Size | PDF

3. Review of the volume current method formulation

The volume current method is used to determine edge scattering losses associated with microdisk resonators in the whispering gallery mode regime. Only the fundamental (lowest order) whispering gallery mode characterized by a single radial lobe is considered. We assume a time dependence of exp(-iωt). A dielectric perturbation on the disk edge may be written as a spatially dependent permittivity distribution Δε(r′,z′,φ′). The dielectric perturbation introduces dipole currents which contribute to outwardly radiated (scattered) fields. The expressions for the perturbed current densities manifest themselves in the form of surface-parallel and surface-perpendicular contributions.

The surface-parallel current contribution is readily derived starting from the curl of the Maxwell equation:

∇×J=−∂∂t∇×D+∇×∇×H
=+iω∇×(εE)+∇×∇×H
=+iω∇ε×E+iωε∇×E+∇×∇×H
=+iω∇ε×E−ω2μ0εH+∇×∇×H
=+iω∇ε×E

By definition, the permittivity gradient is oriented normal to the interface; taking the line integral from just below to just above the interface eliminates the curl:

J∥=limε→0∫−ε+εduû×(∇×J)=limε→0∫−ε+εduiωû×(∇ε×E∥)=−iωΔεE∥

The surface-perpendicular current contribution is readily derived starting from the continuity relation in the absence of free charges. In combination with the expanded Maxwell equation for the divergence of the displacement vector:

∇·J=−∂ρ∂t=iωε0∇·E
∇·D=ε∇·E+∇ε·E=0
∇·J=−iωε0∇εε2·D

Again, the permittivity gradient is oriented normal to the interface; taking the integral eliminates the divergence:

J⊥=û∫du∇·J=iωε0∫du∇1εD⊥=iωε0Δ(ε−1)D⊥

The current density associated with boundary-continuous parallel electric and perpendicular displacement fields in the presence of the dielectric perturbation is thus given as:

J(r′,z′,φ′)=−iω[ΔεE∥(r′,z′)−ε0Δ(ε−1)D⊥(r′,z′)]eimφ′

The dielectric perturbation at the disk boundary can be written as a radial step variation at the interface between the two dissimilar refractive indices (n 1 for the core, and n 2 for the cladding),

Δεin=ε0(n22−n12)step[−ΔR(z′,φ′)]
Δεout=ε0(n12−n22)step[ΔR(z′,φ′)]
Δ(εin−1)=1ε0(1n22−1n12)step[−ΔR(z′,φ′)]
Δ(εout−1)=1ε0(1n12−1n22)step[ΔR(z′,φ′)]

The interpretation is that the field just inside the interface is perturbed by a lower permittivity when the radial variation is negative and the field just outside the interface is perturbed by a higher permittivity when the radial variation is positive.

4. Spectral density formulation for edge roughness

The mode is primarily affected by perturbations along the direction in which the propagation vector is dominant - here azimuthal. Moreover, etch processes tend to deliver uniform corrugations along height, z. This justifies a decomposition of the radial variation into a Fourier series expansion of corrugation harmonics of azimuthal quantization number, M

ΔR(z′,φ′)=∑M=−∞∞ΔRM(z′)e−iMφ′

The amplitude of each corrugation harmonic may be determined from an experimental measurement of the corrugation. Typically, Gaussian statistics apply as in Borselli [7] where the roughness can be characterized by two parameters: the rms value of the roughness,σ and its correlation length, Sc. Here, the correlation function for edge roughness, C(s) is related to the measured variation in radius, ΔR(s) with respect to arc length, s at the disk edge over a suitable measured total arc length S meas.

C(s)=1Smeas∫0Smeasds′ΔR(s′)ΔR(s′−s)

The value of the correlation function at zero is equal to the mean squared roughness,C(0) = σ 2. A Gaussian correlation function can be defined as:

C(s)=σ2e−π(sSc)2

When written in this form, the correlation length is within a factor of π4ln2=1.064 of a full width at half maximum (FWHM) definition. The result of further manipulations is also cleaner, hence the motivation. The spectral density is equal to the Fourier transform of the correlation function which is a Gaussian function of spatial frequency variable fs,

𝒞(fs)=σ2Sce−π(Scfs)2

If the entire circumference of a disk were to be mapped, a Fourier series representation with harmonics of integer azimuthal quantization numbers would emerge naturally. In practice, it is not often feasible to measure the entire circumference, thus the amplitude coefficients of the Fourier series expansion of the corrugation must be extrapolated from the limited data. To obtain those amplitudes, the spatial frequency variable is expressed as fs = M/2πR and the spectral density is integrated around each integer M.

ΔRM2=12πR∫M−12M+12dM𝒞(M2πR)=σ2Sc2πR∫M−12M+12dMe−π(Sc2πRM)2
≈σ2Sc2πRe−π(Sc2πRM)2

For most cases of interest, in comparison to the disk circumference, the correlation length is very small Sc/2πR < < 1 allowing the final (trapezoidal) approximation to the integral to hold.

5. Far field scattered power

Returning now to the electrodynamics of scattering, the vector potential in the far-field consists of the volume-integrated current density vector with a retardation phasor term to account for coherent interaction among the current density elements,

A=μ04πr∫dV′J(r′,z′,φ′)eikr′cosψ

Here, the volume integral is represented in cylindrical coordinates appropriate to the geometry of the circulating mode while the far-field scattering direction is represented in spherical coordinates (see Fig. 2). The angle cosine between the current density element and the observation point is expanded in spherical coordinates as

cosψ≈cos(θ)cos(θ′)+sin(θ)sin(θ′)cos(φ−φ′)

For most geometries in which the disk height is smaller than the radius d < < R the small polar angle approximations about θ= 90° can be made,

 figure: Fig. 2.

Fig. 2. The geometry used in the volume current method formulation for edge scattering losses in microresonators, here shown for a microdisk. The roughness perturbations on the disk edge are parameterized in cylindrical coordinates (r′,z′,φ′) while the scattered radiation is parameterized in spherical coordinates (r,θ,φ).

Download Full Size | PDF

sin(θ′)≈1
cos(θ′)≈z′r′.

Incorporating these approximations results in a volume integral written completely in cylindrical coordinates

A=μ04πr∫−d2+d2dz′∫0∞dr′∫02πr′dφ′J(r′,z′,φ′)eikz′cosθeikr′sin(θ)cos(φ−φ′)

Note that the integrated vector potential will be azimuthally independent due to the inherent symmetry of the geometry yet retain a polar dependence. In the far-field, the electric and magnetic fields and Poynting vector are expressed in terms of the vector potential as

EFF=iωr̂×(A×r̂)
HFF=−iωε2μ0(A×r̂)
SFF=EFF×HFF=ω22μ0c∣r̂×A∣2r̂

The power radiates as transverse electromagnetic waves into all angles of the far-field. The scattered power per unit solid angle thus consists of only polar and azimuthal contributions,

dPsdΩ=r2SFF·r̂=Z08λ2[∣Nθ∣2+∣Nφ∣2]

where the radiation vector N = 4πr A/μ 0 is introduced for convenience [9]. The total scattered power results from the solid angle integral:

Ps=∫dφ∫sinθdθdPsdΩ=2π∫0πsinθdθdPsdΩ

Finally, the loss per unit length attributed to scattering is directly related to the scattered power according to:

αs=12πRPsPg

where Pg is the power in the guided mode.

6. TM scattering losses

For TM WGM modes, the modal electric field is perpendicular to the plane of the disk (directed along the z-axis) and thus does not give rise to radiated fields polarized in the azimuthal direction. Thus, the radiation vector only possesses a polar component (the projection of the z-component),

Nθ=−sinθNz

resulting in

Nθ=iω∫−d2+d2dz′∫02πdφ′∫0∞r′dr′ΔεsinθEz(r′,z′)eimφ′eikz′cosθeikr′sin(θ)cos(φ−φ′)

The resulting scattered field retains the orthogonality of the corrugation harmonics; thus, the integral for the radiation vector can be treated separately for each harmonic and later summed incoherently. The modal field is continuous across the interface greatly simplifying the result. Incorporating the unit step perturbation as a limit in the integral results in:

NθM=iωε0sinθ∫−d2+d2dz′∫02πdφ′∫RR+ΔRM(z′)e−iMφ′r′dr′
(n12−n22)Ez(r′,z′)eikz′cosθeimφ′eikr′sin(θ)cos(φ−φ′)

Because the perturbation is small ΔRM < < R and localized to the disk edge surface all radial variables are replaced with the nominal disk radius and the integral is collapsed.

NθM=iωε0R(n12−n22)sinθ∫−d2+d2dz′ΔRM(z′)Ez(R,z′)eikz′cosθ
∫02πdφ′ei(m−M)φ′eikRsin(θ)cos(φ−φ′)

Implementing the identity:

∫02πdφ′ei(m−M)φ′eikRsin(θ)cos(φ−φ′)=2πim−MJm−M(kRsinθ)ei(m−M)φ

and retaining only the square modulus of the polar radiation vector component yields:

∣NθM∣2=(2πωε0R(n12−n22))2sin2θ∣Jm−M(kRsinθ)∣2
∣∫−d2+d2dz′ΔRM(z′)Ez(R,z′)eikz′cosθ∣2

Next the scattered power is calculated assuming that the field and corrugation are z-independent:

Ps=∑M=−∞∞2πR2k04(n12−n22)28Z0ΔRM2∣Ez(R)∣2
∫0πdθsin3θ∣Jm−M(kRsinθ)∣2∣∫−d2+d2dz′eikz′cosθ∣2

The calculation of the z integral is straightforward and results in a polar sinc pattern. Incorporating both this and the Gaussian correlation function, the expression for the scattering loss becomes,

αs=Rk04(n12−n22)2σ24Sc2πR1Pgλd∣Ez(R)∣22Z0∑M=−∞∞e−π(Sc2πRM)2
∫0πdθsin3θ∣Jm−M(kRsinθ)∣2dλsinc2(dcosθλ)

7. TE scattering losses

The edge scattering loss derivation for TE WGM modes is considerably more complicated. First, the modal electric field lies in the plane of the disk with both radial, Er and azimuthal, Eφ components. While the azimuthal component may be negligible for very low index contrast microresonators, in general it can be quite strong and cannot be neglected. Second, each of these components couples to both polar and azimuthal components of the radiated fields. Third, discontinuities in the planar electric field components exist unless the roughness is locally flat [12]. Fortunately, this approximation is valid for typical (shallow) roughness distributions where the corrugation depth is much smaller than the correlation length (σ < < Sc). For a treatment of deep perturbations, see Johnson [13].

The modal field amplitudes projected in cartesian coordinates are

Ex=cosφ′Er−sinφ′Eφ
Ey=sinφ′Er+cosφ′Eφ

The radiation vector will consist of both polar θ and azimuthal φ components each arising from these modal field components.

Nθ=cosθ(cosφNx+sinφNy)
Nφ=−sinφNx+cosφNy

For convenience, field projection variables are defined:

Kθ(θ,φ,r′,z′,φ′)=cosθ{cos(φ−φ′)ε0Δε−1Dr+sin(φ−φ′)ΔεEφ}
Kφ(θ,φ,r′,z′,φ′)={−sin(φ−φ′)ε0Δε−1Dr+cos(φ−φ′)ΔεEφ}

resulting in a compact expression for the radiation vector components

Nθ=−iω∫−d2+d2dz′∫02πdφ′∫0∞r′dr′Kθ(θ,φ,r′,z′,φ′)eimφ′eikz′cosθeikr′sin(θ)cos(φ−φ′)
Nφ=−iω∫−d2+d2dz′∫02πdφ′∫0∞r′dr′Kφ(θ,φ,r′,z′,φ′)eimφ′eikz′cosθeikr′sin(θ)cos(φ−φ′)

The integral for the radiation vector is treated separately for each harmonic. Furthermore, for convenience, the radial and azimuthal field contributions can be treated separately and summed later. Incorporating the unit step perturbation as a limit in the integral results in:

Nθ,rM=−iωcosθ∫−d2+d2dz′∫02πdφ′∫RR+ΔRM(z′)e−iMφ′r′dr′(1n12−1n22)
[cos(φ−φ′)Dr(r′,z′)]eikz′cosθeimφ′eikr′sin(θ)cos(φ−φ′)
Nθ,φM=−iωε0cosθ∫−d2+d2dz′∫02πdφ′∫RR+ΔRM(z′)e−iMφ′r′dr′(n12−n22)
[sin(φ−φ′)Eφ(r′,z′)]eikz′cosθeimφ′eikr′sin(θ)cos(φ−φ′)
Nφ,rM=−iω∫−d2+d2dz′∫02πdφ′∫RR+ΔRM(z′)e−iMφ′r′dr′(1n12−1n22)
[−sin(φ−φ′)Dr(r′,z′)]eikz′cosθeimφ′eikr′sin(θ)cos(φ−φ′)
Nφ,φM=−iωε0∫−d2+d2dz′∫02πdφ′∫RR+ΔRM(z′)e−iMφ′r′dr′(n12−n22)
[cos(φ−φ′)Eφ(r′,z′)]eikz′cosθeimφ′eikr′sin(θ)cos(φ−φ′)

Because the perturbation is small ΔRM < < R and localized to the disk edge surface, all radial variables can be replaced with the disk radius to collapse the integral.

Nθ,rM=−iωR(1n12−1n22)cosθ∫−d2+d2dz′ΔRM(z′)Dr(R,z′)eikz′cosθ
∫02πdφ′[cos(φ−φ′)]ei(m−M)φ′eikRsin(θ)cos(φ−φ′)
Nθ,φM=−iωε0R(n12−n22)cosθ∫−d2+d2dz′ΔRM(z′)Eφ(R,z′)eikz′cosθ
∫02πdφ′[sin(φ−φ′)]ei(m−M)φ′eikRsin(θ)cos(φ−φ′)
Nφ,rM=−iωR(1n12−1n22)∫−d2+d2dz′ΔRM(z′)Dr(R,z′)eikz′cosθ
∫02πdφ′[−sin(φ−φ′)]ei(m−M)φ′eikRsin(θ)cos(φ−φ′)
Nφ,φM=−iωε0R(n12−n22)∫−d2+d2dz′ΔRM(z′)Eφ(R,z′)eikz′cosθ
∫02πdφ′[cos(φ−φ′)]ei(m−M)φ′eikRsin(θ)cos(φ−φ′)

Implementing the identity

∫02πdφ′e±i(φ−φ′)ei(m−M)φ′eikRsin(θ)cos(φ−φ′)=2πim−M±1Jm−M±1(kRsinθ)ei(m−M)φ

and retaining only the square modulus of the radiation vector components yields:

∣Nθ,rM∣2=(2πωR(1n12−1n22))2cos2θ∣Jm−M+1(kRsinθ)−Jm−M−1(kRsinθ)∣24
∣∫−d2+d2dz′ΔRM(z′)Dr(R,z′)eikz′cosθ∣2
∣Nθ,φM∣2=(2πωε0R(n12−n22))2cos2θ∣Jm−M+1(kRsinθ)+Jm−M−1(kRsinθ)∣24
∣∫−d2+d2dz′ΔRM(z′)Eφ(R,z′)eikz′cosθ∣2
∣Nφ,rM∣2=(2πωR(1n12−1n22))2∣Jm−M+1(kRsinθ)+Jm−M−1(kRsinθ)∣24
∣∫−d2+d2dz′ΔRM(z′)Dr(R,z′)eikz′cosθ∣2
∣Nφ,φM∣2=(2πωε0R(n12−n22))2∣Jm−M+1(kRsinθ)−Jm−M−1(kRsinθ)∣24
∣∫−d2+d2dz′ΔRM(z′)Eφ(R,z′)eikz′cosθ∣2

Assuming that the field and corrugation are z-independent, the scattered power is given as:

Ps=∑M=−∞∞2πR2k04(n12−n22)28Z0ΔRM2
∫0πdθ{[sinθcos2θ∣Jm−M+1(kRsinθ)−Jm−M−1(kRsinθ)∣24
+sinθ∣Jm−M+1(kRsinθ)+Jm−M−1(kRsinθ)∣24]∣Dr(R)∣2(n12n22ε0)2
+[sinθcos2θ∣Jm−M+1(kRsinθ)+Jm−M−1(kRsinθ)∣24
+sinθ∣Jm−M+1(kRsinθ)−Jm−M−1(kRsinθ)∣24]∣Eφ(R)∣2}
∣∫−d2+d2dz′eikz′cosθ∣2

The calculation of the z integral is straightforward and again, results in a polar sinc pattern. Incorporating both this and the Gaussian correlation function, the expression for the scattering loss results, here split into azimuthal and radial field contributions:

αs=Rk04(n12−n22)2σ24Sc2πR
{1Pgλd∣Dr(R)∣22Z0(n14n24ε02)∑M=−∞∞e−π(Sc2πRM)2∫0πdθ[sinθcos2θ∣Jm−M′(kRsinθ)∣2
+sinθ(m−M)2∣Jm−M(kRsinθ)∣2(kRsinθ)2]dλsinc2(dcosθλ)
+1Pgλd∣Eφ(R)∣22Z0∑M=−∞∞e−π(Sc2πRM)2∫0πdθ[sinθcos2θ(m−M)2∣Jm−M(kRsinθ)∣2(kRsinθ)2
+sinθ∣Jm−M′(kRsinθ)∣2]dλsinc2(dcosθλ)}

8. Normalized formulation for edge scattering losses

The expressions for the edge scattering loss scale with the inverse fourth power of the wavelength, typical of scattering processes. They also predict that the loss is strongly dependent on index contrast in proportion to at least the square of the permittivity difference. Finally, the expression is insensitive to disk height d when the height exceeds the wavelength as we will show later. The quality factor can be expressed in normalized units, from which useful limiting approximate forms can be derived. The quality factor is given by the radians per cycle divided by the fractional loss per cycle. Its association with the scattering loss is given by Qs -1 = αsR/m,

1QsTM=4π3n12πRmλ(1−n22n12)2(σλn1)2Scλn11Pgd∣Ez(R)∣24kZ0∑M=−∞∞e−π(Scλn1λn12πRM)2
∫0πdθsin3θ∣Jm−M(kRsinθ)∣2dλsinc2(dcosθλ)
1QsTEr=4π3n12πRmλ(1−n22n12)2(σλn1)2Scλn11Pgd∣Dr(R)∣24kZ0(n14n24ε02)∑M=−∞∞e−π(Scλn1λn12πRM)2
∫0πdθ[sinθcos2θ∣Jm−M′(kRsinθ)∣2+sinθ(m−M)2∣Jm−M(kRsinθ)∣2(kRsinθ)2]
dλsinc2(dcosθλ)
1QsTEφ=4π3n12πRmλ(1−n22n12)2(σλn1)2Scλn11Pgd∣Eφ(R)∣24kZ0∑M=−∞∞e−π(Scλn1λn12πRM)2
∫0πdθ[sinθcos2θ(m−M)2∣Jm−M(kRsinθ)∣2(kRsinθ)2+sinθ∣Jm−M′(kRsinθ)∣2]
dλsinc2(dcosθλ)

These expressions can be written in a compact form by defining normalized units:

1Qsp=4π3Xm(1−1n2)2ξ2ℓcΓzpΓrp𝒢mp

where X, ξ, ℓc respectively are normalized quantities representing the radius X = n 12πR/λ, roughness ξ = n 1 σ/λ, and correlation length ∓c = n 1 Sc/λ. The superscript symbol p refers to the polarization state (TM, TEr, or TEφ). The quantity Γpz = Pc/Pg is the ratio of power vertically confined to total power guided or simply the vertical confinement factor. The usual confinement factor for a planar waveguide can be applied here when using the effective index method. This requires solution of the simple planar slab waveguide dispersion relation (kz vs. ω) as in the Kogelnik formulation [10]. The quantity Γpr is the edge confinement factor and is defined as the ratio of the intensity at the disk edge, (∣E(R)∣2/2Z 0) to the characteristic intensity of the mode, (2kPc/d). This requires solution of the simple infinite cylinder whispering gallery mode dispersion relation (X,Qi vs. m,n). The radial mode profile is normalized to a power per unit disk height of Pc/d by integrating the azimuthal component of the Poynting vector along the radial dimension from the disk center out to the radiation boundary (Rr = mλ/2πn 2). Finally, a geometric factor resulting from the sum/integral of the far-field scattering pattern into azimuthal and polar angles is defined, 𝒢mp=∑M=−∞∞e−π(ℓcXM)2𝒫m−MP where 𝒫m−Mp refers to the associated polar integral.

There are useful limits to consider: thick and thin cylinder. In the limit of a thick cylinder (d >> λ, but where d << R such that the small polar angle approximation, Eqn. 37 still holds), the scattered radiation is directed outward at θ = 90°. The following expressions result for the polar integrals where the derivation is assisted by making the change of variables d cos θ/λ ≡ τ′)

𝒫m−MTM=∫0πdθsin3θ∣Jm−M(kRsinθ)∣2dλsinc2(dcosθλ)
=∫−dλdλdτ′[1−(λdτ′)2]∣Jm−M(kR1−(λdτ′)2)∣2sinc2τ′
→d>>λ∫−∞∞dτ′∣Jm−M(kR)∣2sinc2τ′=∣Jm−M(kR)∣2
𝒫m−MTEr=∫0πdθ[sinθcos2θ∣Jm−M′(kRsinθ)∣2
+sinθ(m−M)2∣Jm−M(kRsinθ)∣2(kRsinθ)2]dλsinc2(dcosθλ)
=∫−dλdλdτ′[(λdτ′)2∣Jm−M′(kR1−(λdτ′)2)∣2
+(m−M)2∣Jm−M(kR1−(λdτ′)2)∣2(kR1−(λdτ′)2)2]sinc2τ′
→d>>λ∫−∞∞dτ′(m−M)2∣Jm−M(kR)∣2(kR)2sinc2τ′=(m−M)2∣Jm−M(kR)∣2(kR)2
𝒫m−MTEφ=∫0πdθ[sinθcos2θ(m−M)2∣Jm−M(kRsinθ)∣2(kRsinθ)2
+sinθ∣Jm−M′(kRsinθ)∣2]dλsinc2(dcosθλ)
=∫−dλ+dλdτ′[(λdτ′)2(m−M)2∣Jm−M(kR1−(λdτ′)2)∣2(kR1−(λdτ′)2)2
+∣J′m−M(kR1−(λdτ′)2)∣2]sinc2τ′
→d>>λ∫−∞∞dτ′∣Jm−M′(kR)∣2sinc2τ′=∣Jm−M′(kR)∣2

In the limit of a thin cylinder (d << λ), the following expressions result for the polar integrals:

𝒫m−MTM→d<<λdλ∫0πdθsin3θ∣Jm−M(kRsinθ)∣2
𝒫m−MTEr→d<<λdλ∫0πdθ[sin⁡θcos⁡2θ∣Jm−M′(kRsin⁡θ)∣2+sin⁡θ(m−M)2∣Jm−M(kRsinθ)∣2(kRsinθ)2
𝒫m−MTEφ→d<<λdλ∫0πdθ[sin⁡θcos⁡2θ(m−M)2∣Jm−M(kRsin⁡θ)∣2(kRsin⁡θ)2+sin⁡θ∣Jm−M′(kRsinθ)∣2]

The polar integrals for scattering as a function of corrugation order are plotted for comparison in Fig. 3. These display characteristics similar to those in Rabiei [9]. The scattering as a function of corrugation order can be expressed as scattering as a function of local azimuthal coordinate, Δφ through a modified grating equation,

Δφ=arccos[n(1−Mm)]

If the correlation length is small such that the width of the spectral density of the roughness distribution (Xλ/n 1 Sc) is much wider than that of the polar integral (2X/n), then the summation term is greatly simplified. For typical fabrication processes, the relation (Sc << λ/2n 2) is generally the case.

𝒢mp→Sc<<λ2n2∑M=−∞∞𝒫m−Mp
 figure: Fig. 3.

Fig. 3. Distribution of the polar integral terms, Pm-M versus corrugation order M for a) TM, b) TE radial, and c) TE azimuthal. Here, the azimuthal order for the mode is m=50, and hence the center for the distribution where the mode is scattered directly radially outward is M = m=50. The index ratio (for this example n = 3) restricts the participating corrugation orders from the full 0 < M < 2m because of Snell’s law or phase matching conditions. For λ = 1.55μm, the resonant radii for TM and TE at m = 50 are 4.605 and 4.686μm respectively. The total sums Gm are shown for each component in the thick and thin limits. The thick/thin cylinder limits are denoted by thick/thin linewidths respectively. The thin curves have been normalized by factoring out the normalized thickness δ = d/λ parameter. The scattering distributions are also plotted as an angular distribution on the right. d) The field solution associated with a whispering gallery mode parameterized by m = 50 interacting with a periodic sidewall corrugation parameterized by M = 40. The angular deviation associated with the scattered wave is indicated by Δφ which is related to m and M by Eqn. 86

Download Full Size | PDF

 figure: Fig. 4.

Fig. 4. Variation in the edge confinement factor associated with the electric field amplidudes of a) TM, b) TE radial, c) TE azimuthal, and d) TE net as a function of normalized radius for varying index contrasts (n =1.25,1.35,1.5,1.7,2.0,2.5,3.0,3.5). Note that (a) and (d) approach the approximate form 1/X for high index contrasts.

Download Full Size | PDF

This allows the further reduction of the 𝒢mp parameter to simple limiting values:

𝒢mTM,TEr,TEφ→d>>λTHICK1,12,12
𝒢mTM,TEr,TEφ→d<<λTHIN43δ,43δ,43δ

Here, a normalized thickness has been defined as δ = d/λ. Prior work [8] neglected cross-term radiation vectors Nθ,ϕ and Nϕ,r resulting in geometric factors of (4/3,2/3,2). The corrected set of geometric factors incorporates the cross-terms and a neglected factor of 1/2:(43,23+22,2+232)=(43,43,43).

 figure: Fig. 5.

Fig. 5. Exact solution (solid line) for the finesse limited by bending and edge scattering losses for disk refractive indices of n 1=1.5, 3.0, n 2=1, TM polarization, λ=1.55 microns, d=300nm, σ=1 nm, and Sc=75nm. Note in the asymptotic limit, the validity of the edge scattering limited finesse approximation (dashed line) for thin microresonator disks.

Download Full Size | PDF

As a final useful approximation, the edge confinement factor may be approximated in the high index contrast limit as Γr ≈ 1/X. This simple inverse radius dependence is found for both TM and TE when the radial and azimuthal contributions are summed. It is dominated by the azimuthal component in the high index contrast limit validating approximations made in Little [5]. For a breakdown of these contributions, see Fig. 4. Because the edge confinement is inversely related to the normalized disk radius, it directly cancels the increased circumferential path length per round trip. Ultimately this leads to a edge scattering limited finesse (ℱs = Qs/m) that is independent of radius. Incorporating all these approximations results in simple forms for the edge scattering limited finesse. For thick, vertically extended (d > λ) microresonators, the expressions for TM and TE differ slightly:

 figure: Fig. 6.

Fig. 6. Finesse limited by bending and edge scattering losses, for both TM and TE polarization, λ=1.55 microns, d=300nm, n 2=1, σ=1, 10 nm, Sc=75nm. The index ratios are n = 1.25,1.35,1.5,1.7,2.0,2.5,3.0,3.5. Note the clamping of finesse with increasing normalized radius in the edge scattering limited regime.

Download Full Size | PDF

1𝓕sTE=4π3(1−1n2)2ξ2ℓc
1𝓕sTE=2π3(1−1n2)2ξ2ℓc

However, for thin, thumbtack-like (d < λ) microresonators, they are equivalent:

1𝓕sTMTE=16π33(1−1n2)2ξ2ℓcΓzδ

The dependence of the fractional loss per round trip (inverse of finesse) on the index contrast, roughness, and edge confinement is consistent with prior models of scattering losses in straight waveguides [1, 2] and microracetracks [3].

In practice it is often found that TE operation results in higher losses [5, 8]. This has been attributed to a variety of interpretations such as a higher field strength at the sidewall and arguments involving incomplete derivations of geometric factors. The derivation presented in this paper shows that neither is the case. Rather, it is likely that other mechanisms are responsible for the discrepancy such as the lower vertical confinement factors associated with TM operation. The validity of this approximation for a thin disk is shown by plotting the bending and edge scattering limited finesse as a function of normalized radius in Fig. 5. The field strengths at the disk edge and bending loss contributions were obtained by numerical solution of the dispersion relation for whispering-gallery modes of a cylinder. Figure 6 displays the TM and TE finesse curves for two different values of roughness across a variety of refractive index ratios. Note that at small radii, the advantage of high indices to combat bending loss is apparent as higher indices leaded to higher finesse. For larger radii however, where bending loss is insignificant, the ordering of the curves interchanges where edge scattering limited operation becomes prevalent. Figure 7 displays the tradeoff from a different perspective, plotting the finesse versus refractive index ratio for a few radii (parameterized by m). With increasing refractive index, bending losses decrease while edge scattering losses increase. The tradeoff is thus clear: if index ratio is a design variable, one desires enough to negate the effects of bending loss - but only just enough - as any more leads to increased edge scattering loss in a practical device.

 figure: Fig. 7.

Fig. 7. The tradeoff between edge scattering and bending loss as a function of index contrast. There exists an optimum index contrast whose value increases as the resonator is made smaller (lower azimuthal number m). Specific choices made for this plot are: TM polarization, λ=1.55 microns, d=300nm, n 2=1, σ=3 nm, Sc=75nm.

Download Full Size | PDF

9. Conclusion

In conclusion, we have presented a generalized formulation incorporating both bending and edge scattering loss limited finesse (and quality factor) in microdisk resonators. The results are also applicable to microrings, microracetracks, and microtoroids. In this formulation, bend loss limited operation is completely specified by a refractive index ratio and an azimuthal quantization number (or alternatively, a normalized radius). The formulation predicts an edge scattering limited finesse that is independent of disk radius and polarization in the thin disk regime. It is completely specified by the same index ratio, along with a product of two normalized roughness parameters, the normalized disk height and the vertical confinement factor. Effectively, only three bundled parameters are thus required to completely specify these primary loss mechanisms for all lowest order whispering gallery modes. We expect that the generalized expressions should be widely applicable both in predicting the impact of fabrication imperfections and/or in selecting the lateral index contrast for optimizing the performance of microresonators.

We would like to thank Ellen Chang for her contributions to this work. This work was supported by the Laboratory for Physical Sciences, College Park, MD.

References and links

1. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. 10, 23952419, (1971). [CrossRef]  

2. R. J. Deri and E. Kapon, “Low-Loss III-V Semiconductor Optical Waveguides.” IEEE J. Quantum Electron. , 27, 626–640, (1991). [CrossRef]  

3. V. Van, P. P. Absil, J. V. Hryniewicz, and P. T. Ho, “Propagation loss in single-mode GaAs-AlGaAs microring resonators: measurement and model,” J. Lightwave Technol. 19, 1734–1739, (2001). [CrossRef]  

4. M. Kuznetsov and H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE. J. Quantum Electron. QE–19, 1505–1514, (1983). [CrossRef]  

5. B. E. Little and S. T. Chu, “Estimating surface-roughness loss and output coupling in microdisk resonators,” Opt. Lett. 21, 1390–1392, (1996). [CrossRef]   [PubMed]  

6. B. E. Little, J. P. Laine, and S. T. Chu, “Surface-roughness-induced contradirectional coupling in ring and disk resonators,” Opt. Lett. 22, 4–6, (1997). [CrossRef]   [PubMed]  

7. M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl Phys Lett. 85, 3693–3695, (2004). [CrossRef]  

8. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13, 1515–1530, (2005). [CrossRef]   [PubMed]  

9. P. Rabiei, “Calculation of Losses in Micro-Ring Resonators With Arbitrary Refractive Index or Shape Profile and Its Applications,” J. Lightwave Technol. 23, 1295–1301, (2005). [CrossRef]  

10. H. Kogelnik and V. Ramaswamy, “Scaling rules for thin-film optical waveguides,” Appl. Opt. 13, pp. 1857–1862, (1974). [CrossRef]   [PubMed]  

11. J. E. Heebner, R. W. Boyd, and Q. Park, “SCISSOR Solitons & other propagation effects in microresonator modified waveguides,” J. Opt. Soc. Am. B. 19, 722–731, (2002). [CrossRef]  

12. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Pertubation theory for Maxwell’s equations with shifting material boundaries,” Phys Rev E. 65, 066611, (2002). [CrossRef]  

13. S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl Phys B. 81, 283–293, (2005). [CrossRef]  

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)
Fig. 1.
Fig. 1. Bending limited finesse of the lowest order radial TM and TE whispering-gallery modes of a dielectric cylinder of index n 1 in a medium of index n 2 plotted against normalized radius. The family of diagonal lines represents varying refractive index ratio (n 1/n 2). The family of nearly vertical lines corresponds to whispering gallery mode resonances, each characterized by an azimuthal mode number m. The normalized radius is nearly equal to m although differs slightly due to the fact that the mode does not peak at the disk edge and experiences a suppressed effective index due to imperfect edge confinement. The plots were obtained by numerically solving the dispersion relation for whispering-gallery modes.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. The geometry used in the volume current method formulation for edge scattering losses in microresonators, here shown for a microdisk. The roughness perturbations on the disk edge are parameterized in cylindrical coordinates (r′,z′,φ′) while the scattered radiation is parameterized in spherical coordinates (r,θ,φ).

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. Distribution of the polar integral terms, P m-M versus corrugation order M for a) TM, b) TE radial, and c) TE azimuthal. Here, the azimuthal order for the mode is m=50, and hence the center for the distribution where the mode is scattered directly radially outward is M = m=50. The index ratio (for this example n = 3) restricts the participating corrugation orders from the full 0 < M < 2m because of Snell’s law or phase matching conditions. For λ = 1.55μm, the resonant radii for TM and TE at m = 50 are 4.605 and 4.686μm respectively. The total sums G m are shown for each component in the thick and thin limits. The thick/thin cylinder limits are denoted by thick/thin linewidths respectively. The thin curves have been normalized by factoring out the normalized thickness δ = d/λ parameter. The scattering distributions are also plotted as an angular distribution on the right. d) The field solution associated with a whispering gallery mode parameterized by m = 50 interacting with a periodic sidewall corrugation parameterized by M = 40. The angular deviation associated with the scattered wave is indicated by Δφ which is related to m and M by Eqn. 86

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Variation in the edge confinement factor associated with the electric field amplidudes of a) TM, b) TE radial, c) TE azimuthal, and d) TE net as a function of normalized radius for varying index contrasts (n =1.25,1.35,1.5,1.7,2.0,2.5,3.0,3.5). Note that (a) and (d) approach the approximate form 1/X for high index contrasts.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. Exact solution (solid line) for the finesse limited by bending and edge scattering losses for disk refractive indices of n 1=1.5, 3.0, n 2=1, TM polarization, λ=1.55 microns, d=300nm, σ=1 nm, and Sc =75nm. Note in the asymptotic limit, the validity of the edge scattering limited finesse approximation (dashed line) for thin microresonator disks.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. Finesse limited by bending and edge scattering losses, for both TM and TE polarization, λ=1.55 microns, d=300nm, n 2=1, σ=1, 10 nm, Sc =75nm. The index ratios are n = 1.25,1.35,1.5,1.7,2.0,2.5,3.0,3.5. Note the clamping of finesse with increasing normalized radius in the edge scattering limited regime.

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. The tradeoff between edge scattering and bending loss as a function of index contrast. There exists an optimum index contrast whose value increases as the resonator is made smaller (lower azimuthal number m). Specific choices made for this plot are: TM polarization, λ=1.55 microns, d=300nm, n 2=1, σ=3 nm, Sc =75nm.

View in Article | Download Full Size | PDF

Equations (133)

Equations on this page are rendered with MathJax. Learn more.

( ∂ 2 ∂ z 2 + ∂ 2 ∂ r 2 + 1 r ∂ ∂ r + 1 r 2 ∂ 2 ∂ φ 2 + k 2 ) Ψ z ( r , z , φ ) = 0 .
( ∂ 2 ∂ r 2 + 1 r ∂ ∂ r + k 2 − m 2 r 2 ) Ψ z ( r ) = 0 .
Ψ z , in ( r , φ ) = A m J m ( k ˜ 1 r ) e i ( ± mφ )
Ψ z , out ( r , φ ) = A m J m ( k ˜ 1 R ) H m ( 1 ) ( k ˜ 2 R ) H m ( 1 ) ( k ˜ 2 r ) e i ( ± mφ ) .
H r = m Z 0 k ˜ 0 r E z
H φ = i Z 0 k ˜ 0 ∂ ∂ r E z ,
E r = − m Z 0 n 2 k ˜ 0 r H z
E φ = − i Z 0 n 2 k ˜ 0 ∂ ∂ r H z .
k ˜ 1 J m ′ ( k ˜ 1 R ) J m ( k ˜ 1 R ) = k ˜ 2 H m ′ ( 1 ) ( k ˜ 2 R ) H m ( 1 ) ( k ˜ 2 R )
J m ′ ( k ˜ 1 R ) k ˜ 1 J m ( k ˜ 1 R ) = H m ′ ( 1 ) ( k ˜ 2 R ) k ˜ 2 H m ( 1 ) ( k ˜ 2 R )
n J m ′ [ ( 1 − i 1 2 Q i ) X ] J m [ ( 1 − i 1 2 Q i ) X ] − H m ′ ( 1 ) [ ( 1 − i 1 2 Q i ) X n ] H m ( 1 ) [ ( 1 − i 1 2 Q i ) X n ] = 0
J m ′ [ ( 1 − i 1 2 Q i ) X ] J m [ ( 1 − i 1 2 Q i ) X ] − n H m ′ ( 1 ) [ ( 1 − i 1 2 Q i ) X n ] H m ( 1 ) [ ( 1 − i 1 2 Q i ) X n ] = 0
∇ × J = − ∂ ∂ t ∇ × D + ∇ × ∇ × H
= + i ω ∇ × ( ε E ) + ∇ × ∇ × H
= + i ω ∇ ε × E + iωε ∇ × E + ∇ × ∇ × H
= + i ω ∇ ε × E − ω 2 μ 0 ε H + ∇ × ∇ × H
= + i ω ∇ ε × E
J ∥ = lim ε → 0 ∫ − ε + ε du u ̂ × ( ∇ × J ) = lim ε → 0 ∫ − ε + ε du iω u ̂ × ( ∇ ε × E ∥ ) = − i ω Δ ε E ∥
∇ · J = − ∂ ρ ∂ t = i ω ε 0 ∇ · E
∇ · D = ε ∇ · E + ∇ ε · E = 0
∇ · J = − i ωε 0 ∇ ε ε 2 · D
J ⊥ = u ̂ ∫ du ∇ · J = i ωε 0 ∫ du ∇ 1 ε D ⊥ = i ω ε 0 Δ ( ε − 1 ) D ⊥
J ( r ′ , z ′ , φ ′ ) = − i ω [ Δε E ∥ ( r ′ , z ′ ) − ε 0 Δ ( ε − 1 ) D ⊥ ( r ′ , z ′ ) ] e imφ ′
Δ ε in = ε 0 ( n 2 2 − n 1 2 ) step [ − Δ R ( z ′ , φ ′ ) ]
Δ ε out = ε 0 ( n 1 2 − n 2 2 ) step [ Δ R ( z ′ , φ ′ ) ]
Δ ( ε in − 1 ) = 1 ε 0 ( 1 n 2 2 − 1 n 1 2 ) step [ − Δ R ( z ′ , φ ′ ) ]
Δ ( ε out − 1 ) = 1 ε 0 ( 1 n 1 2 − 1 n 2 2 ) step [ Δ R ( z ′ , φ ′ ) ]
Δ R ( z ′ , φ ′ ) = ∑ M = − ∞ ∞ Δ R M ( z ′ ) e − i Mφ ′
C ( s ) = 1 S meas ∫ 0 S meas d s ′ Δ R ( s ′ ) Δ R ( s ′ − s )
C ( s ) = σ 2 e − π ( s S c ) 2
𝒞 ( f s ) = σ 2 S c e − π ( S c f s ) 2
Δ R M 2 = 1 2 πR ∫ M − 1 2 M + 1 2 dM𝒞 ( M 2 πR ) = σ 2 S c 2 πR ∫ M − 1 2 M + 1 2 dM e − π ( S c 2 πR M ) 2
≈ σ 2 S c 2 πR e − π ( S c 2 πR M ) 2
A = μ 0 4 πr ∫ d V ′ J ( r ′ , z ′ , φ ′ ) e i kr ′ cos ψ
cos ψ ≈ cos ( θ ) cos ( θ ′ ) + sin ( θ ) sin ( θ ′ ) cos ( φ − φ ′ )
sin ( θ ′ ) ≈ 1
cos ( θ ′ ) ≈ z ′ r ′ .
A = μ 0 4 πr ∫ − d 2 + d 2 d z ′ ∫ 0 ∞ d r ′ ∫ 0 2 π r ′ dφ ′ J ( r ′ , z ′ , φ ′ ) e i kz ′ cos θ e i kr ′ sin ( θ ) cos ( φ − φ ′ )
E FF = i ω r ̂ × ( A × r ̂ )
H FF = − i ω ε 2 μ 0 ( A × r ̂ )
S FF = E FF × H FF = ω 2 2 μ 0 c ∣ r ̂ × A ∣ 2 r ̂
d P s d Ω = r 2 S FF · r ̂ = Z 0 8 λ 2 [ ∣ N θ ∣ 2 + ∣ N φ ∣ 2 ]
P s = ∫ dφ ∫ sin θ dθ d P s d Ω = 2 π ∫ 0 π sin θ dθ d P s d Ω
α s = 1 2 πR P s P g
N θ = − sin θ N z
N θ = i ω ∫ − d 2 + d 2 d z ′ ∫ 0 2 π d φ ′ ∫ 0 ∞ r ′ d r ′ Δ ε sin θ E z ( r ′ , z ′ ) e imφ ′ e i kz ′ cos θ e ik r ′ sin ( θ ) cos ( φ − φ ′ )
N θ M = i ω ε 0 sin θ ∫ − d 2 + d 2 d z ′ ∫ 0 2 π d φ ′ ∫ R R + Δ R M ( z ′ ) e − iMφ ′ r ′ d r ′
( n 1 2 − n 2 2 ) E z ( r ′ , z ′ ) e i kz ′ cos θ e imφ ′ e ik r ′ sin ( θ ) cos ( φ − φ ′ )
N θ M = i ωε 0 R ( n 1 2 − n 2 2 ) sin θ ∫ − d 2 + d 2 dz ′ Δ R M ( z′ ) E z ( R , z′ ) e ikz ′ cos θ
∫ 0 2 π dφ ′ e i ( m − M ) φ′ e ikR sin ( θ ) cos ( φ − φ′ )
∫ 0 2 π dφ ′ e i ( m − M ) φ′ e ikR sin ( θ ) cos ( φ − φ′ ) = 2 πi m − M J m − M ( kR sin θ ) e i ( m − M ) φ
∣ N θ M ∣ 2 = ( 2 πωε 0 R ( n 1 2 − n 2 2 ) ) 2 sin 2 θ ∣ J m − M ( kR sin θ ) ∣ 2
∣ ∫ − d 2 + d 2 dz′ Δ R M ( z′ ) E z ( R , z′ ) e ikz ′ cos θ ∣ 2
P s = ∑ M = − ∞ ∞ 2 πR 2 k 0 4 ( n 1 2 − n 2 2 ) 2 8 Z 0 Δ R M 2 ∣ E z ( R ) ∣ 2
∫ 0 π d θ sin 3 θ ∣ J m − M ( k R sin θ ) ∣ 2 ∣ ∫ − d 2 + d 2 d z ′ e ikz ′ cos θ ∣ 2
α s = R k 0 4 ( n 1 2 − n 2 2 ) 2 σ 2 4 S c 2 πR 1 P g λd ∣ E z ( R ) ∣ 2 2 Z 0 ∑ M = − ∞ ∞ e − π ( S c 2 πR M ) 2
∫ 0 π dθ sin 3 θ ∣ J m − M ( kR sin θ ) ∣ 2 d λ sinc 2 ( d cos θ λ )
E x = cos φ′ E r − sin φ′ E φ
E y = sin φ′ E r + cos φ′ E φ
N θ = cos θ ( cos φ N x + sin φ N y )
N φ = − sin φ N x + cos φ N y
K θ ( θ , φ , r′ , z′ , φ′ ) = cos θ { cos ( φ − φ′ ) ε 0 Δ ε − 1 D r + sin ( φ − φ′ ) Δ ε E φ }
K φ ( θ , φ , r′ , z′ , φ′ ) = { − sin ( φ − φ′ ) ε 0 Δ ε − 1 D r + cos ( φ − φ′ ) Δ ε E φ }
N θ = − i ω ∫ − d 2 + d 2 dz′ ∫ 0 2 π dφ ′ ∫ 0 ∞ r′dr′ K θ ( θ , φ , r′ , z′ , φ′ ) e imφ′ e ikz ′ cos θ e ikr ′ sin ( θ ) cos ( φ − φ′ )
N φ = − i ω ∫ − d 2 + d 2 dz′ ∫ 0 2 π dφ ′ ∫ 0 ∞ r′dr′ K φ ( θ , φ , r′ , z′ , φ′ ) e imφ′ e ikz ′ cos θ e ikr ′ sin ( θ ) cos ( φ − φ′ )
N θ , r M = − iω cos θ ∫ − d 2 + d 2 dz′ ∫ 0 2 π dφ′ ∫ R R + Δ R M ( z′ ) e − i Mφ′ r′dr′ ( 1 n 1 2 − 1 n 2 2 )
[ cos ( φ − φ′ ) D r ( r′ , z′ ) ] e ikz′ cos θ e imφ′ e ikr′ sin ( θ ) cos ( φ − φ′ )
N θ , φ M = − iω ε 0 cos θ ∫ − d 2 + d 2 dz′ ∫ 0 2 π dφ′ ∫ R R + Δ R M ( z′ ) e − i Mφ′ r′dr′ ( n 1 2 − n 2 2 )
[ sin ( φ − φ′ ) E φ ( r′ , z′ ) ] e ikz′ cos θ e imφ′ e ikr′ sin ( θ ) cos ( φ − φ′ )
N φ , r M = − iω ∫ − d 2 + d 2 dz′ ∫ 0 2 π dφ′ ∫ R R + Δ R M ( z′ ) e − i Mφ′ r′dr′ ( 1 n 1 2 − 1 n 2 2 )
[ − sin ( φ − φ′ ) D r ( r′ , z′ ) ] e ikz′ cos θ e imφ′ e ikr′ sin ( θ ) cos ( φ − φ′ )
N φ , φ M = − iω ε 0 ∫ − d 2 + d 2 dz′ ∫ 0 2 π dφ′ ∫ R R + Δ R M ( z′ ) e − i Mφ′ r′dr′ ( n 1 2 − n 2 2 )
[ cos ( φ − φ′ ) E φ ( r′ , z′ ) ] e ikz′ cos θ e imφ′ e ikr′ sin ( θ ) cos ( φ − φ′ )
N θ , r M = − iω R ( 1 n 1 2 − 1 n 2 2 ) cos θ ∫ − d 2 + d 2 dz ′ Δ R M ( z′ ) D r ( R , z′ ) e ikz ′ cos θ
∫ 0 2 π dφ′ [ cos ( φ − φ′ ) ] e i ( m − M ) φ ′ e ikR sin ( θ ) cos ( φ − φ′ )
N θ , φ M = − iω ε 0 R ( n 1 2 − n 2 2 ) cos θ ∫ − d 2 + d 2 dz ′ Δ R M ( z′ ) E φ ( R , z′ ) e ikz ′ cos θ
∫ 0 2 π dφ′ [ sin ( φ − φ′ ) ] e i ( m − M ) φ′ e ikR sin ( θ ) cos ( φ − φ′ )
N φ , r M = − iω R ( 1 n 1 2 − 1 n 2 2 ) ∫ − d 2 + d 2 dz ′ Δ R M ( z′ ) D r ( R , z′ ) e ikz ′ cos θ
∫ 0 2 π dφ′ [ − sin ( φ − φ′ ) ] e i ( m − M ) φ′ e ikR sin ( θ ) cos ( φ − φ′ )
N φ , φ M = − iω ε 0 R ( n 1 2 − n 2 2 ) ∫ − d 2 + d 2 dz ′ Δ R M ( z′ ) E φ ( R , z′ ) e ikz ′ cos θ
∫ 0 2 π dφ′ [ cos ( φ − φ′ ) ] e i ( m − M ) φ′ e ikR sin ( θ ) cos ( φ − φ′ )
∫ 0 2 π d φ ′ e ± i ( φ − φ′ ) e i ( m − M ) φ′ e ikR sin ( θ ) cos ( φ − φ′ ) = 2 πi m − M ± 1 J m − M ± 1 ( kR sin θ ) e i ( m − M ) φ
∣ N θ , r M ∣ 2 = ( 2 πωR ( 1 n 1 2 − 1 n 2 2 ) ) 2 cos 2 θ ∣ J m − M + 1 ( kR sin θ ) − J m − M − 1 ( kR sin θ ) ∣ 2 4
∣ ∫ − d 2 + d 2 dz ′ Δ R M ( z′ ) D r ( R , z′ ) e ikz′ cos θ ∣ 2
∣ N θ , φ M ∣ 2 = ( 2 πω ε 0 R ( n 1 2 − n 2 2 ) ) 2 cos 2 θ ∣ J m − M + 1 ( kR sin θ ) + J m − M − 1 ( kR sin θ ) ∣ 2 4
∣ ∫ − d 2 + d 2 dz ′ Δ R M ( z ′ ) E φ ( R , z ′ ) e ikz ′ cos θ ∣ 2
∣ N φ , r M ∣ 2 = ( 2 πωR ( 1 n 1 2 − 1 n 2 2 ) ) 2 ∣ J m − M + 1 ( kR sin θ ) + J m − M − 1 ( kR sin θ ) ∣ 2 4
∣ ∫ − d 2 + d 2 dz ′ Δ R M ( z ′ ) D r ( R , z ′ ) e ikz ′ cos θ ∣ 2
∣ N φ , φ M ∣ 2 = ( 2 πω ε 0 R ( n 1 2 − n 2 2 ) ) 2 ∣ J m − M + 1 ( kR sin θ ) − J m − M − 1 ( kR sin θ ) ∣ 2 4
∣ ∫ − d 2 + d 2 dz ′ Δ R M ( z ′ ) E φ ( R , z ′ ) e ikz ′ cos θ ∣ 2
P s = ∑ M = − ∞ ∞ 2 πR 2 k 0 4 ( n 1 2 − n 2 2 ) 2 8 Z 0 Δ R M 2
∫ 0 π dθ { [ sin θ cos 2 θ ∣ J m − M + 1 ( kR sin θ ) − J m − M − 1 ( kR sin θ ) ∣ 2 4
+ sin θ ∣ J m − M + 1 ( kR sin θ ) + J m − M − 1 ( kR sin θ ) ∣ 2 4 ] ∣ D r ( R ) ∣ 2 ( n 1 2 n 2 2 ε 0 ) 2
+ [ sin θ cos 2 θ ∣ J m − M + 1 ( kR sin θ ) + J m − M − 1 ( kR sin θ ) ∣ 2 4
+ sin θ ∣ J m − M + 1 ( kR sin θ ) − J m − M − 1 ( kR sin θ ) ∣ 2 4 ] ∣ E φ ( R ) ∣ 2 }
∣ ∫ − d 2 + d 2 dz ′ e ikz′ cos θ ∣ 2
α s = R k 0 4 ( n 1 2 − n 2 2 ) 2 σ 2 4 S c 2 πR
{ 1 P g λd ∣ D r ( R ) ∣ 2 2 Z 0 ( n 1 4 n 2 4 ε 0 2 ) ∑ M = − ∞ ∞ e − π ( S c 2 πR M ) 2 ∫ 0 π dθ [ sin θ cos 2 θ ∣ J m − M ′ ( kR sin θ ) ∣ 2
+ sin θ ( m − M ) 2 ∣ J m − M ( kR sin θ ) ∣ 2 ( kR sin θ ) 2 ] d λ sinc 2 ( d cos θ λ )
+ 1 P g λd ∣ E φ ( R ) ∣ 2 2 Z 0 ∑ M = − ∞ ∞ e − π ( S c 2 πR M ) 2 ∫ 0 π dθ [ sin θ cos 2 θ ( m − M ) 2 ∣ J m − M ( kR sin θ ) ∣ 2 ( kR sin θ ) 2
+ sin θ ∣ J m − M ′ ( kR sin θ ) ∣ 2 ] d λ sinc 2 ( d cos θ λ ) }
1 Q s TM = 4 π 3 n 1 2 πR mλ ( 1 − n 2 2 n 1 2 ) 2 ( σ λ n 1 ) 2 S c λ n 1 1 P g d ∣ E z ( R ) ∣ 2 4 k Z 0 ∑ M = − ∞ ∞ e − π ( S c λ n 1 λ n 1 2 πR M ) 2
∫ 0 π dθ sin 3 θ ∣ J m − M ( kR sin θ ) ∣ 2 d λ sinc 2 ( d cos θ λ )
1 Q s TE r = 4 π 3 n 1 2 π R m λ ( 1 − n 2 2 n 1 2 ) 2 ( σ λ n 1 ) 2 S c λ n 1 1 P g d ∣ D r ( R ) ∣ 2 4 k Z 0 ( n 1 4 n 2 4 ε 0 2 ) ∑ M = − ∞ ∞ e − π ( S c λ n 1 λ n 1 2 π R M ) 2
∫ 0 π d θ [ sin θ cos 2 θ ∣ J m − M ′ ( k R sin θ ) ∣ 2 + sin θ ( m − M ) 2 ∣ J m − M ( k R sin θ ) ∣ 2 ( k R sin θ ) 2 ]
d λ sinc 2 ( d cos θ λ )
1 Q s TE φ = 4 π 3 n 1 2 π R m λ ( 1 − n 2 2 n 1 2 ) 2 ( σ λ n 1 ) 2 S c λ n 1 1 P g d ∣ E φ ( R ) ∣ 2 4 k Z 0 ∑ M = − ∞ ∞ e − π ( S c λ n 1 λ n 1 2 π R M ) 2
∫ 0 π d θ [ sin θ cos 2 θ ( m − M ) 2 ∣ J m − M ( k R sin θ ) ∣ 2 ( k R sin θ ) 2 + sin θ ∣ J m − M ′ ( k R sin θ ) ∣ 2 ]
d λ sinc 2 ( d cos θ λ )
1 Q s p = 4 π 3 X m ( 1 − 1 n 2 ) 2 ξ 2 ℓ c Γ z p Γ r p 𝒢 m p
𝒫 m − M TM = ∫ 0 π dθ sin 3 θ ∣ J m − M ( kR sin θ ) ∣ 2 d λ sin c 2 ( d cos θ λ )
= ∫ − d λ d λ dτ′ [ 1 − ( λ d τ ′ ) 2 ] ∣ J m − M ( kR 1 − ( λ d τ ′ ) 2 ) ∣ 2 sinc 2 τ ′
→ d > > λ ∫ − ∞ ∞ dτ ′ ∣ J m − M ( kR ) ∣ 2 sinc 2 τ ′ = ∣ J m − M ( kR ) ∣ 2
𝒫 m − M TE r = ∫ 0 π dθ [ sin θ cos 2 θ ∣ J m − M ′ ( kR sin θ ) ∣ 2
+ sin θ ( m − M ) 2 ∣ J m − M ( kR sin θ ) ∣ 2 ( kR sin θ ) 2 ] d λ sinc 2 ( d cos θ λ )
= ∫ − d λ d λ dτ ′ [ ( λ d τ ′ ) 2 ∣ J m − M ′ ( kR 1 − ( λ d τ ′ ) 2 ) ∣ 2
+ ( m − M ) 2 ∣ J m − M ( k R 1 − ( λ d τ ′ ) 2 ) ∣ 2 ( k R 1 − ( λ d τ ′ ) 2 ) 2 ] sinc 2 τ ′
→ d > > λ ∫ − ∞ ∞ dτ ′ ( m − M ) 2 ∣ J m − M ( kR ) ∣ 2 ( kR ) 2 sinc 2 τ ′ = ( m − M ) 2 ∣ J m − M ( kR ) ∣ 2 ( kR ) 2
𝒫 m − M TE φ = ∫ 0 π dθ [ sin θ cos 2 θ ( m − M ) 2 ∣ J m − M ( kR sin θ ) ∣ 2 ( kR sin θ ) 2
+ sin θ ∣ J m − M ′ ( kR sin θ ) ∣ 2 ] d λ sinc 2 ( d cos θ λ )
= ∫ − d λ + d λ dτ ′ [ ( λ d τ ′ ) 2 ( m − M ) 2 ∣ J m − M ( kR 1 − ( λ d τ ′ ) 2 ) ∣ 2 ( kR 1 − ( λ d τ ′ ) 2 ) 2
+ ∣ J ′ m − M ( kR 1 − ( λ d τ ′ ) 2 ) ∣ 2 ] sinc 2 τ ′
→ d > > λ ∫ − ∞ ∞ dτ ′ ∣ J m − M ′ ( kR ) ∣ 2 sinc 2 τ ′ = ∣ J m − M ′ ( kR ) ∣ 2
𝒫 m − M TM → d < < λ d λ ∫ 0 π dθ sin 3 θ ∣ J m − M ( kR sin θ ) ∣ 2
𝒫 m − M TEr → d < < λ d λ ∫ 0 π dθ [ sin ⁡ θ cos ⁡ 2 θ ∣ J m − M ′ ( kR sin ⁡ θ ) ∣ 2 + sin ⁡ θ ( m − M ) 2 ∣ J m − M ( kR sin θ ) ∣ 2 ( kR sin θ ) 2
𝒫 m − M TE φ → d < < λ d λ ∫ 0 π dθ [ sin ⁡ θ cos ⁡ 2 θ ( m − M ) 2 ∣ J m − M ( kR sin ⁡ θ ) ∣ 2 ( kR sin ⁡ θ ) 2 + sin ⁡ θ ∣ J m − M ′ ( kR sin θ ) ∣ 2 ]
Δ φ = arccos [ n ( 1 − M m ) ]
𝒢 m p → S c < < λ 2 n 2 ∑ M = − ∞ ∞ 𝒫 m − M p
𝒢 m TM , TE r , TE φ → d >> λ THICK 1 , 1 2 , 1 2
𝒢 m TM , TE r , TE φ → d << λ THIN 4 3 δ , 4 3 δ , 4 3 δ
1 𝓕 s TE = 4 π 3 ( 1 − 1 n 2 ) 2 ξ 2 ℓ c
1 𝓕 s TE = 2 π 3 ( 1 − 1 n 2 ) 2 ξ 2 ℓ c
1 𝓕 s TM TE = 16 π 3 3 ( 1 − 1 n 2 ) 2 ξ 2 ℓ c Γ z δ
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use