Abstract

We develop a 3D vectorial description of microresonators of the microdisk and microring types based on the aperiodic Fourier modal method. Such a rigorous coupled-wave analysis allows us to evaluate accurately the resonant wavelengths, the quality factor, and the full profile of whispering-gallery modes. The results are compared with 2D (effective index) as well as 3D finite-difference time domain calculations.

© 2008 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  4. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621-623 (2002).
    [CrossRef] [PubMed]
  5. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711-713 (1999).
    [CrossRef]
  6. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925-928 (2003).
    [CrossRef] [PubMed]
  7. P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. I. Hopman, R. Costa, A. Melloni, I. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38, 731-759 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. V. A. Labay and J. Bornemann, “Matrix singular value decomposition for pole-free solutions of homogeneous matrix equations as applied to numerical modeling methods,” IEEE Microw. Guid. Wave Lett. 2, 49-51 (1992).
    [CrossRef]
  24. E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344-1351 (1992).
    [CrossRef]
  25. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguide structures,” J. Lightwave Technol. 12, 2080-2084 (1994).
    [CrossRef]
  26. O. Conradi, “Determining the resonator wavelength of VCSELs by Cauchy's integral formula,” Opt. Quantum Electron. 31, 1047-1058 (1999).
    [CrossRef]
  27. J. H. Mathews, “Nelder-Mead search for a minimum,” http://math.fullerton.edu/mathews/n2003/NelderMeadMod.html.
  28. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  29. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. 107, 6756-6769 (1997), implemented by S. G. Johnson, http://ab-initio.mit.edu/wiki/index.php/Harminv.
    [CrossRef]
  30. P. Bienstmann and R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523-540 (2002).
    [CrossRef]

2006 (3)

A. M. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3-14 (2006).
[CrossRef]

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. I. Hopman, R. Costa, A. Melloni, I. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38, 731-759 (2006).
[CrossRef]

A. Morand, Y. Zhang, B. Martin, K. P. Huy, D. Amans, P. Benech, J. Verbert, E. Hadji, and J. M. Fédéli, “Ultra-compact microdisk resonator filters on SOI substrate,” Opt. Express 14, 12814-12821 (2006).
[CrossRef] [PubMed]

2005 (4)

L. Prkna, M. Hubálek, and J. Ctyroký, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217-223 (2005).
[CrossRef]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Ctyroký, “Cylindrical integrated optical microresonators: modeling by 3-D vectorial coupled mode theory,” Opt. Commun. 256, 46-67 (2005).
[CrossRef]

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844-1849 (2005).
[CrossRef]

K. Phan-Huy, A. Morand, D. Amans, and P. Benech, “Analytical study of the whispering-gallery mode in two-dimensional microgear cavity using coupled-mode theory,” J. Opt. Soc. Am. B 22, 1793-1803 (2005).
[CrossRef]

2004 (1)

Ph. Lalanne, J. P. Hugonin, and J. S. Gerard, “Electromagnetic study of the quality factor of pillar microcavities in the small diameter limit,” Appl. Phys. Lett. 84, 4726-4728 (2004).
[CrossRef]

2003 (2)

K. J. Vahala, “Optical microcavities,” Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

2002 (2)

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621-623 (2002).
[CrossRef] [PubMed]

P. Bienstmann and R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523-540 (2002).
[CrossRef]

2000 (1)

1999 (2)

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711-713 (1999).
[CrossRef]

O. Conradi, “Determining the resonator wavelength of VCSELs by Cauchy's integral formula,” Opt. Quantum Electron. 31, 1047-1058 (1999).
[CrossRef]

1997 (2)

A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. 107, 6756-6769 (1997), implemented by S. G. Johnson, http://ab-initio.mit.edu/wiki/index.php/Harminv.
[CrossRef]

F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. 7, 371-373 (1997).
[CrossRef]

1996 (3)

1995 (1)

1994 (1)

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguide structures,” J. Lightwave Technol. 12, 2080-2084 (1994).
[CrossRef]

1993 (1)

A. S. Sudbø, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. 2, 211-233 (1993).
[CrossRef]

1992 (2)

V. A. Labay and J. Bornemann, “Matrix singular value decomposition for pole-free solutions of homogeneous matrix equations as applied to numerical modeling methods,” IEEE Microw. Guid. Wave Lett. 2, 49-51 (1992).
[CrossRef]

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344-1351 (1992).
[CrossRef]

Appl. Phys. Lett. (1)

Ph. Lalanne, J. P. Hugonin, and J. S. Gerard, “Electromagnetic study of the quality factor of pillar microcavities in the small diameter limit,” Appl. Phys. Lett. 84, 4726-4728 (2004).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

L. Prkna, M. Hubálek, and J. Ctyroký, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217-223 (2005).
[CrossRef]

A. M. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3-14 (2006).
[CrossRef]

IEEE Microw. Guid. Wave Lett. (2)

F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. 7, 371-373 (1997).
[CrossRef]

V. A. Labay and J. Bornemann, “Matrix singular value decomposition for pole-free solutions of homogeneous matrix equations as applied to numerical modeling methods,” IEEE Microw. Guid. Wave Lett. 2, 49-51 (1992).
[CrossRef]

J. Chem. Phys. (1)

A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. 107, 6756-6769 (1997), implemented by S. G. Johnson, http://ab-initio.mit.edu/wiki/index.php/Harminv.
[CrossRef]

J. Lightwave Technol. (2)

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344-1351 (1992).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguide structures,” J. Lightwave Technol. 12, 2080-2084 (1994).
[CrossRef]

J. Opt. Soc. Am. A (5)

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

Nature (3)

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621-623 (2002).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

K. J. Vahala, “Optical microcavities,” Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Ctyroký, “Cylindrical integrated optical microresonators: modeling by 3-D vectorial coupled mode theory,” Opt. Commun. 256, 46-67 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Opt. Quantum Electron. (3)

P. Bienstman, S. Selleri, L. Rosa, H. P. Uranus, W. C. I. Hopman, R. Costa, A. Melloni, I. C. Andreani, J. P. Hugonin, P. Lalanne, D. Pinto, S. S. A. Obayya, M. Dems, and K. Panajotov, “Modelling leaky photonic wires: a mode solver comparison,” Opt. Quantum Electron. 38, 731-759 (2006).
[CrossRef]

P. Bienstmann and R. Baets, “Advanced boundary conditions for eigenmode expansion models,” Opt. Quantum Electron. 34, 523-540 (2002).
[CrossRef]

O. Conradi, “Determining the resonator wavelength of VCSELs by Cauchy's integral formula,” Opt. Quantum Electron. 31, 1047-1058 (1999).
[CrossRef]

Pure Appl. Opt. (1)

A. S. Sudbø, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. 2, 211-233 (1993).
[CrossRef]

Other (5)

J. H. Mathews, “Nelder-Mead search for a minimum,” http://math.fullerton.edu/mathews/n2003/NelderMeadMod.html.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

T. M. Benson, S. V. Boriskina, P. Sewell, A. Vukovic, S. C. Greedy, and A. I. Nosich, “Microcavities: an inspiration for advanced modelling techniques,” in Proceedings of the 7th International Conference Transparent Optical Networks (ICTON) (IEEE, 2005), pp. 272-275.
[CrossRef]

T. M. Benson, S. V. Boriskina, P. Sewell, A. Vukovic, S. C. Greedy, and A. I. Nosich, “Micro-optical resonators for microlasers and integrated optoelectronics,” in Frontiers in Planar Lightwave Circuit Technology: Design, Simulation and Fabrication, S.Janz, J.Ctyroký, and S.Tanez, eds. (Springer, 2006), pp. 39-70.
[CrossRef]

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

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Figures (4)

Fig. 1
Fig. 1

(a) Generic microresonator with z-axis cylindrical symmetry and its periodization on which the Fourier modal method relies; note that PMLs are also included in the periodic window Λ; (b) sketch of the radial layer decomposition; r s denotes the radial position of the generic interface between homogeneous layers.

Fig. 2
Fig. 2

Convergence plots for microdisk quasi-TE and -TM modes of order m = 7 : resonant wavelengths versus truncation order M for two different values of the window (period) Λ, and ratio Π Λ of physical to total window. Here the absorption figure of the PML is set to be γ = 0.5 j 0.5

Fig. 3
Fig. 3

Microring resonator: cross-sectional view of the absolute value of electric and magnetic (normalized by j ϵ 0 μ 0 ) field components of quasi-TE mode of azimuthal order m = 12 , with resonant wavelength λ = 1.1820 μ m and Q = 400 . The box represents the section of the ring, while the reported numbers are the maximum value of the relative quantities. Top: distribution of axial vertical components E z and H z ; center: azimuthal components E θ and H θ ; bottom: radial components E r and H r .

Fig. 4
Fig. 4

As in Fig. 3 for the microdisk, quasi-TM mode of order M = 9 , with resonant wavelength λ = 1.0358 μ m and Q = 215 . Top: distribution of axial vertical components E z and H z ; center: azimuthal components E θ and H θ ; bottom: radial components E r and H r .

Tables (4)

Tables Icon

Table 1 Resonant Wavelength and Quality Factor of Quasi-TE Modes with Azimuthal Order m of a Microring Resonator a

Tables Icon

Table 2 As in Table 1 for Quasi-TM Modes of a Microring Resonator

Tables Icon

Table 3 As in Table 1 for Quasi-TE Modes of a Microdisk Resonator

Tables Icon

Table 4 As in Table 1 for Quasi-TM Modes of a Microdisk Resonator

Equations (33)

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ϵ r = n ϵ n e j n K z , μ r = n μ n e j n K z ,
E = n = M M ( S r n r 0 + S θ n θ 0 + S z n z 0 ) e j n K z e j m θ ,
H = j ϵ 0 μ 0 n = M M ( U r n r 0 + U θ n θ 0 + U z n z 0 ) e j n K z e j m θ ,
2 E + k 0 2 ϵ r E = ( ϵ r E ϵ r ) ,
2 H + k 0 2 ϵ r H = j ω ϵ 0 ϵ r × E .
2 E z + k 0 2 ϵ r E z = d d z ( 1 ϵ r d ϵ r d z E z ) = d d z ( 1 ϵ r d d z ( ϵ r E z ) ) + d 2 E z d z 2 ,
2 H z + k 0 2 ϵ r H z = 0 .
d 2 S z n d r 2 + 1 r d S z n d r m 2 r 2 S z n n K p [ ϵ ] n , p 1 p K l [ 1 ϵ ] p , l 1 S z l + k 0 2 p [ 1 ϵ ] n , p 1 S z p = 0 ,
d 2 U z n d r 2 + 1 r d U z n d r m 2 r 2 U z n ( n K ) 2 U z n + k 0 2 p ϵ n p U z p = 0 ,
S z n = i w n i [ a i J m ( k 0 λ i A r ) + d i H m ( 2 ) ( k 0 λ i A r ) ] ,
U z n = i v n i [ α i J m ( k 0 λ i B r ) + δ i H m ( 2 ) ( k 0 λ i B r ) ] ,
1 k 0 d p d r = U ( r ) p ,
[ S z U z ] = [ W 0 0 V ] p ( r ) .
p ( r ) = [ p e ( r ) p h ( r ) ] = { [ J 0 e ( r ) a 0 J 0 h ( r ) α 0 ] = J 0 ( r ) A 0 for r [ 0 , r 1 ] [ J s e ( r ) a s + H s e ( r ) d s J s h ( r ) α s + H s h ( r ) δ s ] = J s ( r ) A s + H s ( r ) D s for r [ r s , r s + 1 ] [ H f e ( r ) d f H f h ( r ) δ f ] = H f ( r ) D f for r [ r f 1 , r f ] ]
C s e ( r ) = C m ( k 0 N s e r ) , C ¯ s e ( r ) = d C m d ξ ξ = k 0 N s e r ,
C s h ( r ) = C m ( k 0 N s h r ) , C ¯ s h ( r ) = d C m d ξ ξ = k 0 N s h r ,
C s ( r ) = C m ( k 0 N s r ) , C ¯ s ( r ) = d C m d ξ ξ = k 0 N s r ,
U s R = G s H s [ U s L F s ] 1 E s ,
U s L = H s [ G s U s R ] 1 E s + F s ,
E s = N s [ H ( r s ) J ¯ ( r s ) H ¯ ( r s ) J ( r s ) ] X s 1 ,
F s = N s [ H ¯ ( r s ) J ( r s + 1 ) H ( r s + 1 ) J ¯ ( r s ) ] X s 1 ,
G s = N s [ H ( r s ) J ¯ ( r s + 1 ) H ¯ ( r s + 1 ) J ( r s ) ] X s 1 ,
H s = N s [ H ( r s + 1 ) J ¯ ( r s + 1 ) H ¯ ( r s + 1 ) J ( r s + 1 ) ] X s 1 ,
X s = H ( r s ) J ( r s + 1 ) H ( r s + 1 ) J ( r s ) .
C U θ = m r k 0 E K z E ¯ U z + 1 k 0 E ¯ 1 d S z d r = E K r E ¯ U z + 1 k 0 E ¯ 1 d S z d r ,
B S θ = m r k 0 K z S z + 1 k 0 d U z d r = K r S z + 1 k 0 d U z d r ,
Q = Re { ω c } 2 Im { ω c } ,
Q = Re { λ c } 2 Im { λ c } .
U 1 = U ( r 1 ) = N 0 J ¯ m ( k 0 N 0 r 1 ) J m ( k 0 N 0 r 1 ) ,
U f = U ( r f ) = N f H ¯ m ( 2 ) ( k 0 N f r f ) H m ( 2 ) ( k 0 N f r f ) ,
( U s L U s L + ) p ( r s + 1 ) = 0 .
λ f ( λ ) = min k σ k ( A ) , k = 1 , , n ,
λ c = arg min λ C , Re { λ } > 0 , Im { λ } < 0 f .

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