Abstract

The mode characteristics of a three-dimensional (3D) microdisk with a vertical refractive index distribution of n23.4n2 are investigated by the S-matrix method and 3D finite-difference time-domain (FDTD) technique. For the microdisk with a thickness of 0.2μm and a radius of 1μm, the mode wavelengths and quality factors for the HE7,1 mode obtained by 3D FDTD simulation and the S-matrix method are in good agreement as n2 increases from 1.0 to 2.6. But the Q factor obtained by the 3D FDTD rapidly decreases from 1.12×104 to 379 as n2 increases from 2.65 to 2.8 owing to the vertical radiation losses, which cannot be predicted by the proposed S-matrix method. The comparisons also show that quality factors obtained from the analytical solution of two-dimensional microdisks under the effective index approximation are five to seven times smaller than those of the 3D FDTD as n2=1 and R=1μm.

© 2006 Optical Society of America

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  1. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).
    [Crossref]
  2. M. K. Chin, D. Y. Chu, and S.-T. Ho, "Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes," J. Appl. Phys. 75, 3302-3307 (1994).
    [Crossref]
  3. N. C. Frateschi and A. F. J. Levi, "The spectrum of microdisk lasers," J. Appl. Phys. 80, 644-653 (1996).
    [Crossref]
  4. B.-J. Li and P.-L. Liu, "Numerical analysis of the whispering gallery modes by the finite-difference time-domain method," IEEE J. Quantum Electron. 32, 1583-1587(1996).
    [Crossref]
  5. R. P. Wang and M.-M. Dumitreschu, "Optical modes in semiconductor microdisk lasers," IEEE J. Quantum Electron. 34, 1933-1937 (1998).
    [Crossref]
  6. F. Laeri and J. U. Nöckel, "Nanoporous compound materials for optical applications—microlasers and microresonators," in Handbook of Advanced Electronic and Photonic Materials, H.S.Nalwa, ed. (Academic, 2001), Vol. 6, pp. 58-90.
  7. M. Hentschel and K. Richter, "Quantum chaos in optical systems: the annular billiard," Phys. Rev. E 66, 056207 (2002).
    [Crossref]
  8. A. I. Rahachou and I. V. Zozoulenko, "Effects of boundary roughness on a Q factor of whispering-gallery-mode lasing microdisk cavities," J. Appl. Phys. 94, 7929-7931 (2003).
    [Crossref]
  9. A. I. Rahachou and I. V. Zozoulenko, "Scattering matrix approach to the resonant states and Q values of microdisk lasing cavities," Appl. Opt. 43, 1761-1772 (2004).
    [Crossref] [PubMed]
  10. D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, 1982), pp. 290-294.
  11. W. H. Guo, W. J. Li, and Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
    [Crossref]
  12. K. Wörhoff, P. V. Lambeck, and A. Driessen, "Design, tolerance analysis, and fabrication of silicon oxynitride based planar optical waveguides for communication devices," J. Lightwave Technol. 17, 1401-1407 (1999).
    [Crossref]
  13. D. J. W. Klunder, E. Krioukov, F. S. Tan, T. Van Der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, "Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology," Appl. Phys. B 73, 603-608 (2001).
    [Crossref]

2004 (1)

2003 (1)

A. I. Rahachou and I. V. Zozoulenko, "Effects of boundary roughness on a Q factor of whispering-gallery-mode lasing microdisk cavities," J. Appl. Phys. 94, 7929-7931 (2003).
[Crossref]

2002 (1)

M. Hentschel and K. Richter, "Quantum chaos in optical systems: the annular billiard," Phys. Rev. E 66, 056207 (2002).
[Crossref]

2001 (2)

W. H. Guo, W. J. Li, and Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[Crossref]

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. Van Der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, "Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology," Appl. Phys. B 73, 603-608 (2001).
[Crossref]

1999 (1)

1998 (1)

R. P. Wang and M.-M. Dumitreschu, "Optical modes in semiconductor microdisk lasers," IEEE J. Quantum Electron. 34, 1933-1937 (1998).
[Crossref]

1996 (2)

N. C. Frateschi and A. F. J. Levi, "The spectrum of microdisk lasers," J. Appl. Phys. 80, 644-653 (1996).
[Crossref]

B.-J. Li and P.-L. Liu, "Numerical analysis of the whispering gallery modes by the finite-difference time-domain method," IEEE J. Quantum Electron. 32, 1583-1587(1996).
[Crossref]

1994 (1)

M. K. Chin, D. Y. Chu, and S.-T. Ho, "Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes," J. Appl. Phys. 75, 3302-3307 (1994).
[Crossref]

1992 (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).
[Crossref]

Bulthuis, H. F.

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. Van Der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, "Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology," Appl. Phys. B 73, 603-608 (2001).
[Crossref]

Chin, M. K.

M. K. Chin, D. Y. Chu, and S.-T. Ho, "Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes," J. Appl. Phys. 75, 3302-3307 (1994).
[Crossref]

Chu, D. Y.

M. K. Chin, D. Y. Chu, and S.-T. Ho, "Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes," J. Appl. Phys. 75, 3302-3307 (1994).
[Crossref]

Driessen, A.

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. Van Der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, "Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology," Appl. Phys. B 73, 603-608 (2001).
[Crossref]

K. Wörhoff, P. V. Lambeck, and A. Driessen, "Design, tolerance analysis, and fabrication of silicon oxynitride based planar optical waveguides for communication devices," J. Lightwave Technol. 17, 1401-1407 (1999).
[Crossref]

Dumitreschu, M.-M.

R. P. Wang and M.-M. Dumitreschu, "Optical modes in semiconductor microdisk lasers," IEEE J. Quantum Electron. 34, 1933-1937 (1998).
[Crossref]

Frateschi, N. C.

N. C. Frateschi and A. F. J. Levi, "The spectrum of microdisk lasers," J. Appl. Phys. 80, 644-653 (1996).
[Crossref]

Guo, W. H.

W. H. Guo, W. J. Li, and Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[Crossref]

Hentschel, M.

M. Hentschel and K. Richter, "Quantum chaos in optical systems: the annular billiard," Phys. Rev. E 66, 056207 (2002).
[Crossref]

Ho, S.-T.

M. K. Chin, D. Y. Chu, and S.-T. Ho, "Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes," J. Appl. Phys. 75, 3302-3307 (1994).
[Crossref]

Hoekstra, H. J. W. M.

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. Van Der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, "Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology," Appl. Phys. B 73, 603-608 (2001).
[Crossref]

Huang, Y. Z.

W. H. Guo, W. J. Li, and Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[Crossref]

Klunder, D. J. W.

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. Van Der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, "Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology," Appl. Phys. B 73, 603-608 (2001).
[Crossref]

Krioukov, E.

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. Van Der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, "Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology," Appl. Phys. B 73, 603-608 (2001).
[Crossref]

Laeri, F.

F. Laeri and J. U. Nöckel, "Nanoporous compound materials for optical applications—microlasers and microresonators," in Handbook of Advanced Electronic and Photonic Materials, H.S.Nalwa, ed. (Academic, 2001), Vol. 6, pp. 58-90.

Lambeck, P. V.

Levi, A. F. J.

N. C. Frateschi and A. F. J. Levi, "The spectrum of microdisk lasers," J. Appl. Phys. 80, 644-653 (1996).
[Crossref]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).
[Crossref]

Li, B.-J.

B.-J. Li and P.-L. Liu, "Numerical analysis of the whispering gallery modes by the finite-difference time-domain method," IEEE J. Quantum Electron. 32, 1583-1587(1996).
[Crossref]

Li, W. J.

W. H. Guo, W. J. Li, and Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[Crossref]

Liu, P.-L.

B.-J. Li and P.-L. Liu, "Numerical analysis of the whispering gallery modes by the finite-difference time-domain method," IEEE J. Quantum Electron. 32, 1583-1587(1996).
[Crossref]

Logan, R. A.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).
[Crossref]

Marcuse, D.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, 1982), pp. 290-294.

McCall, S. L.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).
[Crossref]

Nöckel, J. U.

F. Laeri and J. U. Nöckel, "Nanoporous compound materials for optical applications—microlasers and microresonators," in Handbook of Advanced Electronic and Photonic Materials, H.S.Nalwa, ed. (Academic, 2001), Vol. 6, pp. 58-90.

Otto, C.

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. Van Der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, "Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology," Appl. Phys. B 73, 603-608 (2001).
[Crossref]

Pearton, S. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).
[Crossref]

Rahachou, A. I.

A. I. Rahachou and I. V. Zozoulenko, "Scattering matrix approach to the resonant states and Q values of microdisk lasing cavities," Appl. Opt. 43, 1761-1772 (2004).
[Crossref] [PubMed]

A. I. Rahachou and I. V. Zozoulenko, "Effects of boundary roughness on a Q factor of whispering-gallery-mode lasing microdisk cavities," J. Appl. Phys. 94, 7929-7931 (2003).
[Crossref]

Richter, K.

M. Hentschel and K. Richter, "Quantum chaos in optical systems: the annular billiard," Phys. Rev. E 66, 056207 (2002).
[Crossref]

Sengo, G.

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. Van Der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, "Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology," Appl. Phys. B 73, 603-608 (2001).
[Crossref]

Slusher, R. E.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).
[Crossref]

Tan, F. S.

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. Van Der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, "Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology," Appl. Phys. B 73, 603-608 (2001).
[Crossref]

Van Der Veen, T.

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. Van Der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, "Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology," Appl. Phys. B 73, 603-608 (2001).
[Crossref]

Wang, R. P.

R. P. Wang and M.-M. Dumitreschu, "Optical modes in semiconductor microdisk lasers," IEEE J. Quantum Electron. 34, 1933-1937 (1998).
[Crossref]

Wörhoff, K.

Zozoulenko, I. V.

A. I. Rahachou and I. V. Zozoulenko, "Scattering matrix approach to the resonant states and Q values of microdisk lasing cavities," Appl. Opt. 43, 1761-1772 (2004).
[Crossref] [PubMed]

A. I. Rahachou and I. V. Zozoulenko, "Effects of boundary roughness on a Q factor of whispering-gallery-mode lasing microdisk cavities," J. Appl. Phys. 94, 7929-7931 (2003).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. Van Der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, "Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology," Appl. Phys. B 73, 603-608 (2001).
[Crossref]

Appl. Phys. Lett. (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).
[Crossref]

IEEE J. Quantum Electron. (2)

B.-J. Li and P.-L. Liu, "Numerical analysis of the whispering gallery modes by the finite-difference time-domain method," IEEE J. Quantum Electron. 32, 1583-1587(1996).
[Crossref]

R. P. Wang and M.-M. Dumitreschu, "Optical modes in semiconductor microdisk lasers," IEEE J. Quantum Electron. 34, 1933-1937 (1998).
[Crossref]

IEEE Microw. Wirel. Compon. Lett. (1)

W. H. Guo, W. J. Li, and Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[Crossref]

J. Appl. Phys. (3)

M. K. Chin, D. Y. Chu, and S.-T. Ho, "Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes," J. Appl. Phys. 75, 3302-3307 (1994).
[Crossref]

N. C. Frateschi and A. F. J. Levi, "The spectrum of microdisk lasers," J. Appl. Phys. 80, 644-653 (1996).
[Crossref]

A. I. Rahachou and I. V. Zozoulenko, "Effects of boundary roughness on a Q factor of whispering-gallery-mode lasing microdisk cavities," J. Appl. Phys. 94, 7929-7931 (2003).
[Crossref]

J. Lightwave Technol. (1)

Phys. Rev. E (1)

M. Hentschel and K. Richter, "Quantum chaos in optical systems: the annular billiard," Phys. Rev. E 66, 056207 (2002).
[Crossref]

Other (2)

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, 1982), pp. 290-294.

F. Laeri and J. U. Nöckel, "Nanoporous compound materials for optical applications—microlasers and microresonators," in Handbook of Advanced Electronic and Photonic Materials, H.S.Nalwa, ed. (Academic, 2001), Vol. 6, pp. 58-90.

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic diagram of a microdisk waveguide structure. (b) Microdisk structure and the boundaries used in 3D FDTD simulation (side view).

Fig. 2
Fig. 2

Mode wavelength and Q factor of the HE 7 , 1 mode, calculated by the S-matrix approach, versus (a) M L at L = 4 μ m and (b) L at M L = 6 μ m 1 .

Fig. 3
Fig. 3

Mode wavelength versus the refractive index of the cladding layers for the (a) HE 7 , 1 mode in a microdisk with R = 1 μ m and (b) HE 17 , 1 mode in a microdisk with R = 2 μ m .

Fig. 4
Fig. 4

Mode Q factors versus the refractive index of the cladding layers for the (a) HE 7 , 1 mode in a microdisk with R = 1 μ m and (b) HE 17 , 1 mode in a microdisk with R = 2 μ m .

Fig. 5
Fig. 5

Normalized real part of E r varies in the z direction at r = 0.92 μ m in a microdisk with R = 1 μ m and refractive index of the cladding layer n 2 = 1.0 , 2.5, 2.7, 3.0, and 3.17.

Tables (2)

Tables Icon

Table 1 Mode Wavelengths and Q Factors Obtained by 3D FDTD Simulation, S Matrix, and 2D Solution under EI Approximation with Vertical Refractive Index Distribution of 1.0 3.4 1.0

Tables Icon

Table 2 Mode Wavelengths and Q Factors Obtained by 3D FDTD Simulation, S Matrix, and 2D Solution under EI Approximation with Vertical Refractive Index Distribution of 1.0 2.65 1.45

Equations (24)

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

E z = Ag ( z ) J v ( κ r ) J v 1 ( κ R ) ,
H z = B f ( z ) J v ( γ r ) J v 1 ( γ R ) ,
E ϕ = A i v κ 2 r g ( z ) J v ( κ r ) J v 1 ( κ R ) + B i k γ f ( z ) J v ( γ r ) J v 1 ( γ R ) ,
H ϕ = A i k κ h ( z ) J v ( κ r ) J v 1 ( κ R ) + B i v γ 2 r f ( z ) J v ( γ r ) J v 1 ( γ R ) ,
d d z 1 ε ( z ) d ε ( z ) g ( z ) d z + k 2 ε ( z ) g ( z ) = κ 2 g ( z ) ,
d 2 f ( z ) d z 2 + k 2 ε ( z ) f ( z ) = γ 2 f ( z ) .
m = exp ( i p m z ) L , p m = 2 m π L ,
f ( z ) = m = + f m m , f m = m f ( z ) ,
g ( z ) = m = + g m m , g m = m g ( z ) .
e z , m = Ag m J v ( κ r ) J v 1 ( κ R ) ,
h z , m = B f m J v ( γ r ) J v 1 ( γ R ) ,
e ϕ , m = A i v κ 2 r J v ( κ r ) J v 1 ( κ R ) ( i p m g m ) + B i k γ J v ( γ r ) J v 1 ( γ R ) f m ,
h ϕ , m = A i k κ J v ( κ r ) J v 1 ( κ R ) s m + B i v γ 2 r J v ( γ r ) J v 1 ( γ R ) ( i p m f m ) ,
s m = n = M M m ε ( z ) n g n .
e z , m = C m in H v ( 1 ) ( q m r ) H v ( 1 ) ( q m R ) + C m out H v ( 2 ) ( q m r ) H v ( 2 ) ( q m R ) ,
h z , m = D m in H v ( 1 ) ( q m r ) H v ( 1 ) ( q m R ) + D m out H v ( 2 ) ( q m r ) H v ( 2 ) ( q m R ) ,
e ϕ , m = v p m q m 2 r [ C m in H v ( 1 ) ( q m r ) H v ( 1 ) ( q m R ) + C m out H v ( 2 ) ( q m r ) H v ( 2 ) ( q m R ) ] + i k q m [ D m in H v ( 1 ) ( q m r ) H v ( 1 ) ( q m R ) + D m out H v ( 2 ) ( q m r ) H v ( 2 ) ( q m R ) ] ,
h ϕ , m = i k q m [ C m in H v ( 1 ) ( q m r ) H v ( 1 ) ( q m R ) + C m out H v ( 2 ) ( q m r ) H v ( 2 ) ( q m R ) ] + v p m q m 2 r [ D m in H v ( 1 ) ( q m r ) H v ( 1 ) ( q m R ) + D m out H v ( 2 ) ( q m r ) H v ( 2 ) ( q m R ) ] ,
( C m out D m out ) = [ O 11 O 12 O 21 O 22 ] ( C m in D m in ) ,
S ( k ) = [ O 11 ( i , j ) O 12 ( i , j ) O 21 ( i , j ) O 22 ( i , j ) ] ,
τ ( k ) = 2 π d θ d k .
λ = 2 π k center , Q = k center Δ k .
P ( t ) = exp [ ( t t 0 ) 2 t w 2 ] cos ( 2 π f t )
J v ( n eff k R ) H v ( 2 ) ( k R ) = 1 n eff J v ( n eff k R ) H v ( 2 ) ( k R ) ,

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