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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 8 — Apr. 16, 2007
  • pp: 4452–4473
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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  »View Author Affiliations


Optics Express, Vol. 15, Issue 8, pp. 4452-4473 (2007)
http://dx.doi.org/10.1364/OE.15.004452


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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

2. Review of the optical whispering gallery dispersion relation

(2z2+2r2+1rr+1r22φ2+k2)Ψz(r,z,φ)=0.
(1)

(2r2+1rr+k2m2r2)Ψz(r)=0.
(2)

Field solutions consist of Bessel functions of the first kind, Jm( 1 r) inside the disk boundary, (r < R) and Hankel functions of the first kind, Hm (1)( 2 r) outside the disk boundary, (r > R). Here, a complex propagation constant and frequency j = 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(±)
(3)
Ψz,out(r,φ)=AmJm(k˜1R)Hm(1)(k˜2R)Hm(1)(k˜2r)ei(±).
(4)

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
(5)
Hφ=iZ0k˜0rEz,
(6)

and TE modes:

Er=mZ0n2k˜0rHz
(7)
Eφ=iZ0n2k˜0rHz.
(8)

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)
(9)
Jm(k˜1R)k˜1Jm(k˜1R)=Hm(1)(k˜2R)k˜2Hm(1)(k˜2R)
(10)

nJm[(1i12Qi)X]Jm[(1i12Qi)X]Hm(1)[(1i12Qi)Xn]Hm(1)[(1i12Qi)Xn]=0
(11)
Jm[(1i12Qi)X]Jm[(1i12Qi)X]nHm(1)[(1i12Qi)Xn]Hm(1)[(1i12Qi)Xn]=0
(12)

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

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]

]. 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.

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.

3. Review of the volume current method formulation

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

×J=t×D+××H
(13)
=+iω×(εE)+××H
(14)
=+iωε×E+iωε×E+××H
(15)
=+iωε×Eω2μ0εH+××H
(16)
=+iωε×E
(17)

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=limε0ε+εduû×(×J)=limε0ε+εduû×(ε×E)=iωΔεE
(18)

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=ρt=iωε0·E
(19)
·D=ε·E+ε·E=0
(20)
·J=iωε0εε2·D
(21)

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

J=ûdu·J=iωε0du1εD=iωε0Δ(ε1)D
(22)

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(ε1)D(r,z)]eimφ
(23)

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(n22n12)step[ΔR(z,φ)]
(24)
Δεout=ε0(n12n22)step[ΔR(z,φ)]
(25)
Δ(εin1)=1ε0(1n221n12)step[ΔR(z,φ)]
(26)
Δ(εout1)=1ε0(1n121n22)step[ΔR(z,φ)]
(27)

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)ei
(28)

C(s)=1Smeas0SmeasdsΔR(s)ΔR(ss)
(29)

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)=σ2eπ(sSc)2
(30)

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)=σ2Sceπ(Scfs)2
(31)

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=12πRM12M+12dM𝒞(M2πR)=σ2Sc2πRM12M+12dMeπ(Sc2πRM)2
(32)
σ2Sc2πReπ(Sc2πRM)2
(33)

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=μ04πrdVJ(r,z,φ)eikrcosψ
(34)

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

cosψcos(θ)cos(θ)+sin(θ)sin(θ)cos(φφ)
(35)

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,

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).
sin(θ)1
(36)
cos(θ)zr.
(37)

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

A=μ04πrd2+d2dz0dr02πrJ(r,z,φ)eikzcosθeikrsin(θ)cos(φφ)
(38)

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)
(39)
HFF=iωε2μ0(A×r̂)
(40)
SFF=EFF×HFF=ω22μ0cr̂×A2r̂
(41)

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[Nθ2+Nφ2]
(42)

where the radiation vector N = 4πr A/μ 0 is introduced for convenience [9

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]

]. The total scattered power results from the solid angle integral:

Ps=sinθdPsdΩ=2π0πsinθdPsdΩ
(43)

αs=12πRPsPg
(44)

where Pg is the power in the guided mode.

6. TM scattering losses

Nθ=sinθNz
(45)

resulting in

Nθ=iωd2+d2dz02πdφ0rdrΔεsinθEz(r,z)eimφeikzcosθeikrsin(θ)cos(φφ)
(46)

NθM=iωε0sinθd2+d2dz02πdφRR+ΔRM(z)eiMφrdr
(n12n22)Ez(r,z)eikzcosθeimφeikrsin(θ)cos(φφ)
(47)

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(n12n22)sinθd2+d2dzΔRM(z′)Ez(R,z′)eikzcosθ
02πei(mM)φ′eikRsin(θ)cos(φφ′)
(48)

Implementing the identity:

02πei(mM)φ′eikRsin(θ)cos(φφ′)=2πimMJmM(kRsinθ)ei(mM)φ
(49)

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

NθM2=(2πωε0R(n12n22))2sin2θJmM(kRsinθ)2
d2+d2dz′ΔRM(z′)Ez(R,z′)eikzcosθ2
(50)

Ps=M=2πR2k04(n12n22)28Z0ΔRM2Ez(R)2
0πdθsin3θJmM(kRsinθ)2d2+d2dzeikzcosθ2
(51)

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(n12n22)2σ24Sc2πR1PgλdEz(R)22Z0M=eπ(Sc2πRM)2
0πsin3θJmM(kRsinθ)2dλsinc2(dcosθλ)
(52)

7. TE scattering losses

Ex=cosφ′Ersinφ′Eφ
(53)
Ey=sinφ′Er+cosφ′Eφ
(54)

Nθ=cosθ(cosφNx+sinφNy)
(55)
Nφ=sinφNx+cosφNy
(56)

For convenience, field projection variables are defined:

Kθ(θ,φ,r′,z′,φ′)=cosθ{cos(φφ′)ε0Δε1Dr+sin(φφ′)ΔεEφ}
(57)
Kφ(θ,φ,r′,z′,φ′)={sin(φφ′)ε0Δε1Dr+cos(φφ′)ΔεEφ}
(58)

resulting in a compact expression for the radiation vector components

Nθ=iωd2+d2dz′02π0r′dr′Kθ(θ,φ,r′,z′,φ′)eimφ′eikzcosθeikrsin(θ)cos(φφ′)
(59)
Nφ=iωd2+d2dz′02π0r′dr′Kφ(θ,φ,r′,z′,φ′)eimφ′eikzcosθeikrsin(θ)cos(φφ′)
(60)

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=cosθd2+d2dz′02πdφ′RR+ΔRM(z′)eiMφ′r′dr′(1n121n22)
[cos(φφ′)Dr(r′,z′)]eikz′cosθeimφ′eikr′sin(θ)cos(φφ′)
(61)
Nθ,φM=ε0cosθd2+d2dz′02πdφ′RR+ΔRM(z′)eiMφ′r′dr′(n12n22)
[sin(φφ′)Eφ(r′,z′)]eikz′cosθeimφ′eikr′sin(θ)cos(φφ′)
(62)
Nφ,rM=d2+d2dz′02πdφ′RR+ΔRM(z′)eiMφ′r′dr′(1n121n22)
[sin(φφ′)Dr(r′,z′)]eikz′cosθeimφ′eikr′sin(θ)cos(φφ′)
(63)
Nφ,φM=ε0d2+d2dz′02πdφ′RR+ΔRM(z′)eiMφ′r′dr′(n12n22)
[cos(φφ′)Eφ(r′,z′)]eikz′cosθeimφ′eikr′sin(θ)cos(φφ′)
(64)

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=R(1n121n22)cosθd2+d2dzΔRM(z′)Dr(R,z′)eikzcosθ
02πdφ′[cos(φφ′)]ei(mM)φeikRsin(θ)cos(φφ′)
(65)
Nθ,φM=ε0R(n12n22)cosθd2+d2dzΔRM(z′)Eφ(R,z′)eikzcosθ
02πdφ′[sin(φφ′)]ei(mM)φ′eikRsin(θ)cos(φφ′)
(66)
Nφ,rM=R(1n121n22)d2+d2dzΔRM(z′)Dr(R,z′)eikzcosθ
02πdφ′[sin(φφ′)]ei(mM)φ′eikRsin(θ)cos(φφ′)
(67)
Nφ,φM=ε0R(n12n22)d2+d2dzΔRM(z′)Eφ(R,z′)eikzcosθ
02πdφ′[cos(φφ′)]ei(mM)φ′eikRsin(θ)cos(φφ′)
(68)

Implementing the identity

02πdφe±i(φφ′)ei(mM)φ′eikRsin(θ)cos(φφ′)=2πimM±1JmM±1(kRsinθ)ei(mM)φ
(69)

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

Nθ,rM2=(2πωR(1n121n22))2cos2θJmM+1(kRsinθ)JmM1(kRsinθ)24
d2+d2dzΔRM(z′)Dr(R,z′)eikz′cosθ2
(70)
Nθ,φM2=(2πωε0R(n12n22))2cos2θJmM+1(kRsinθ)+JmM1(kRsinθ)24
d2+d2dzΔRM(z)Eφ(R,z)eikzcosθ2
(71)
Nφ,rM2=(2πωR(1n121n22))2JmM+1(kRsinθ)+JmM1(kRsinθ)24
d2+d2dzΔRM(z)Dr(R,z)eikzcosθ2
(72)
Nφ,φM2=(2πωε0R(n12n22))2JmM+1(kRsinθ)JmM1(kRsinθ)24
d2+d2dzΔRM(z)Eφ(R,z)eikzcosθ2
(73)

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

Ps=M=2πR2k04(n12n22)28Z0ΔRM2
0π{[sinθcos2θJmM+1(kRsinθ)JmM1(kRsinθ)24
+sinθJmM+1(kRsinθ)+JmM1(kRsinθ)24]Dr(R)2(n12n22ε0)2
+[sinθcos2θJmM+1(kRsinθ)+JmM1(kRsinθ)24
+sinθJmM+1(kRsinθ)JmM1(kRsinθ)24]Eφ(R)2}
d2+d2dzeikz′cosθ2
(74)

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(n12n22)2σ24Sc2πR
{1PgλdDr(R)22Z0(n14n24ε02)M=eπ(Sc2πRM)20π[sinθcos2θJmM(kRsinθ)2
+sinθ(mM)2JmM(kRsinθ)2(kRsinθ)2]dλsinc2(dcosθλ)
+1PgλdEφ(R)22Z0M=eπ(Sc2πRM)20π[sinθcos2θ(mM)2JmM(kRsinθ)2(kRsinθ)2
+sinθJmM(kRsinθ)2]dλsinc2(dcosθλ)}
(75)

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=4π3n12πR(1n22n12)2(σλn1)2Scλn11PgdEz(R)24kZ0M=eπ(Scλn1λn12πRM)2
0πsin3θJmM(kRsinθ)2dλsinc2(dcosθλ)
(76)
1QsTEr=4π3n12πRmλ(1n22n12)2(σλn1)2Scλn11PgdDr(R)24kZ0(n14n24ε02)M=eπ(Scλn1λn12πRM)2
0πdθ[sinθcos2θJmM(kRsinθ)2+sinθ(mM)2JmM(kRsinθ)2(kRsinθ)2]
dλsinc2(dcosθλ)
(77)
1QsTEφ=4π3n12πRmλ(1n22n12)2(σλn1)2Scλn11PgdEφ(R)24kZ0M=eπ(Scλn1λn12πRM)2
0πdθ[sinθcos2θ(mM)2JmM(kRsinθ)2(kRsinθ)2+sinθJmM(kRsinθ)2]
dλsinc2(dcosθλ)
(78)

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

1Qsp=4π3Xm(11n2)2ξ2cΓzpΓrp𝒢mp
(79)

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/λτ′)

𝒫mMTM=0πsin3θJmM(kRsinθ)2dλsinc2(dcosθλ)
=dλdλdτ′[1(λdτ)2]JmM(kR1(λdτ)2)2sinc2τ
d>>λJmM(kR)2sinc2τ=JmM(kR)2
(80)
𝒫mMTEr=0π[sinθcos2θJmM(kRsinθ)2
+sinθ(mM)2JmM(kRsinθ)2(kRsinθ)2]dλsinc2(dcosθλ)
=dλdλ[(λdτ)2JmM(kR1(λdτ)2)2
+(mM)2JmM(kR1(λdτ)2)2(kR1(λdτ)2)2]sinc2τ
d>>λ(mM)2JmM(kR)2(kR)2sinc2τ=(mM)2JmM(kR)2(kR)2
(81)
𝒫mMTEφ=0π[sinθcos2θ(mM)2JmM(kRsinθ)2(kRsinθ)2
+sinθJmM(kRsinθ)2]dλsinc2(dcosθλ)
=dλ+dλ[(λdτ)2(mM)2JmM(kR1(λdτ)2)2(kR1(λdτ)2)2
+JmM(kR1(λdτ)2)2]sinc2τ
d>>λJmM(kR)2sinc2τ=JmM(kR)2
(82)

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

𝒫mMTMd<<λdλ0πsin3θJmM(kRsinθ)2
(83)
𝒫mMTErd<<λdλ0π[sinθcos2θJmM(kRsinθ)2+sinθ(mM)2JmM(kRsinθ)2(kRsinθ)2
(84)
𝒫mMTEφd<<λdλ0π[sinθcos2θ(mM)2JmM(kRsinθ)2(kRsinθ)2+sinθJmM(kRsinθ)2]
(85)

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

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]

]. 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(1Mm)]
(86)

If the correlation length is small such that the width of the spectral density of the roughness distribution (/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.

𝒢mpSc<<λ2n2M=𝒫mMp
(87)
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
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.

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

𝒢mTM,TEr,TEφd>>λTHICK1,12,12
(88)
𝒢mTM,TEr,TEφd<<λTHIN43δ,43δ,43δ
(89)

Here, a normalized thickness has been defined as δ = d/λ. Prior work [8

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]

] 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).

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.

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.
1𝓕sTE=4π3(11n2)2ξ2c
(90)
1𝓕sTE=2π3(11n2)2ξ2c
(91)

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

1𝓕sTMTE=16π33(11n2)2ξ2cΓzδ
(92)

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

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

, 2

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

] and microracetracks [3

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]

].

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.

9. Conclusion

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]

OCIS Codes
(130.2790) Integrated optics : Guided waves
(130.3120) Integrated optics : Integrated optics devices
(230.5750) Optical devices : Resonators
(230.7370) Optical devices : Waveguides

ToC Category:
Integrated Optics

History
Original Manuscript: January 29, 2007
Revised Manuscript: March 8, 2007
Manuscript Accepted: March 9, 2007
Published: April 3, 2007

Citation
John E. Heebner, Tiziana C. Bond, and Jeff S. Kallman, "Generalized formulation for performance degradations due to bending and edge scattering loss in microdisk resonators," Opt. Express 15, 4452-4473 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4452


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References

  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, 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]

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