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

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

2005 (3)

2004 (1)

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]

2002 (2)

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]

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]

2001 (1)

1997 (1)

1996 (1)

1991 (1)

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

1983 (1)

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]

1974 (1)

1971 (1)

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

Absil, P. P.

Barclay, P. E.

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]

Borselli, M.

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]

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]

Boyd, R. W.

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]

Chu, S. T.

Deri, R. J.

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

Fink, Y.

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]

Haus, H. A.

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]

Heebner, J. E.

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]

Ho, P. T.

Hryniewicz, J. V.

Ibanescu, M.

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]

Jacobs, S.

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]

Joannopoulos, J. D.

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]

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]

Johnson, S. G.

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]

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]

Johnson, T. J.

Kapon, E.

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

Karalis, A.

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]

Kogelnik, H.

Kuznetsov, M.

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]

Laine, J. P.

Little, B. E.

Painter, O.

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]

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]

Park, Q.

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]

Povinelli, M. L.

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]

Rabiei, P.

Ramaswamy, V.

Skorobogatiy, M. A.

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]

Soljacic, M.

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]

Srinivasan, K.

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]

Tien, P. K.

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

Van, V.

Weisberg, O.

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]

Appl Phys B. (1)

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]

Appl Phys Lett. (1)

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]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

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

IEEE. J. Quantum Electron. (1)

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]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B. (1)

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]

Opt. Express (1)

Opt. Lett. (2)

Phys Rev E. (1)

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]

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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.

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).

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

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.

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.

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.

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.

Equations (133)

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( 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 ( ± )
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 ( ± ) .
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 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
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 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 = sin θ d P s d Ω = 2 π 0 π sin θ 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 e i ( m M ) φ′ e ikR sin ( θ ) cos ( φ φ′ )
0 2 π 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 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 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 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 = 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 = 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 = 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 = 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 = 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 = 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 = 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 = 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 { [ 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 [ 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 [ 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 ( 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 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 c Γ z p Γ r p 𝒢 m p
𝒫 m M TM = 0 π 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 J m M ( kR ) 2 sinc 2 τ = J m M ( kR ) 2
𝒫 m M TE r = 0 π [ 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 [ ( 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 ( 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 [ 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 [ ( 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 J m M ( kR ) 2 sinc 2 τ = J m M ( kR ) 2
𝒫 m M TM d < < λ d λ 0 π sin 3 θ J m M ( kR sin θ ) 2
𝒫 m M TEr d < < λ d λ 0 π [ 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 [ 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 c
1 𝓕 s TE = 2 π 3 ( 1 1 n 2 ) 2 ξ 2 c
1 𝓕 s TM TE = 16 π 3 3 ( 1 1 n 2 ) 2 ξ 2 c Γ z δ

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