Abstract

The resonant modes of rectangular two-dimensional optical resonators were analyzed exactly. Based on the characteristics of the Bessel function, the resonant frequencies of the rectangular microcavities are expressed in a simple way. In addition, a simple rule to judge when the finite length of a rectangular resonator can be considered infinite is given in realistic applications. The solution that is presented should be useful in possible applications of the rectangular resonators as filters for dense wavelength-division multiplexing.

© 2006 Optical Society of America

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  1. D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, and R. C. Tiberio, Opt. Lett. 22, 1244 (1997).
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  2. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
    [CrossRef]
  3. B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
    [CrossRef]
  4. A. W. Poon, F. Courvoisier, and R. K. Chang, Opt. Lett. 26, 632 (2001).
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  5. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
    [CrossRef]
  6. Y.-Z. Huang, Q. Chen, W.-H. Guo, and L.-J. Yu, IEEE Photon. Technol. Lett. 17, 2589 (2005).
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  7. M.-Q. Wang, Y.-Z. Huang, Q. Chen, and Z.-P. Cai, IEEE J. Quantum Electron. 42, 146 (2006).
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  8. Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989), pp. 345-455.
  9. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).
  10. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962).
  11. W. R. Smythe, Static and Dynamic Electricity, 3rd ed. (McGraw-Hill, 1968).
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    [CrossRef]
  13. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed. (Academic, 1980).

2006

M.-Q. Wang, Y.-Z. Huang, Q. Chen, and Z.-P. Cai, IEEE J. Quantum Electron. 42, 146 (2006).
[CrossRef]

2005

Y.-Z. Huang, Q. Chen, W.-H. Guo, and L.-J. Yu, IEEE Photon. Technol. Lett. 17, 2589 (2005).
[CrossRef]

2003

W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, IEEE J. Quantum Electron. 39, 1106 (2003).
[CrossRef]

2001

2000

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
[CrossRef]

1999

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

1998

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

1997

Cai, Z.-P.

M.-Q. Wang, Y.-Z. Huang, Q. Chen, and Z.-P. Cai, IEEE J. Quantum Electron. 42, 146 (2006).
[CrossRef]

Chang, R. K.

Chen, Q.

M.-Q. Wang, Y.-Z. Huang, Q. Chen, and Z.-P. Cai, IEEE J. Quantum Electron. 42, 146 (2006).
[CrossRef]

Y.-Z. Huang, Q. Chen, W.-H. Guo, and L.-J. Yu, IEEE Photon. Technol. Lett. 17, 2589 (2005).
[CrossRef]

Chu, S. T.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Courvoisier, F.

Fan, S.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed. (Academic, 1980).

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Guo, D. R.

Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989), pp. 345-455.

Guo, W. H.

W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, IEEE J. Quantum Electron. 39, 1106 (2003).
[CrossRef]

Guo, W.-H.

Y.-Z. Huang, Q. Chen, W.-H. Guo, and L.-J. Yu, IEEE Photon. Technol. Lett. 17, 2589 (2005).
[CrossRef]

Hagness, S. C.

Haus, H. A.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Ho, S. T.

Huang, Y. Z.

W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, IEEE J. Quantum Electron. 39, 1106 (2003).
[CrossRef]

Huang, Y.-Z.

M.-Q. Wang, Y.-Z. Huang, Q. Chen, and Z.-P. Cai, IEEE J. Quantum Electron. 42, 146 (2006).
[CrossRef]

Y.-Z. Huang, Q. Chen, W.-H. Guo, and L.-J. Yu, IEEE Photon. Technol. Lett. 17, 2589 (2005).
[CrossRef]

Ippen, E.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Joannopoulos, J. D.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

Khan, M. J.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

Kimerling, L. C.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Kokubun, Y.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
[CrossRef]

Little, B. E.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Lu, Q. Y.

W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, IEEE J. Quantum Electron. 39, 1106 (2003).
[CrossRef]

Manolatou, C.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

Pan, W.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
[CrossRef]

Panofsky, W. K. H.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962).

Phillips, M.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962).

Poon, A. W.

Rafizadeh, D.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed. (Academic, 1980).

Smythe, W. R.

W. R. Smythe, Static and Dynamic Electricity, 3rd ed. (McGraw-Hill, 1968).

Stair, K. A.

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Taflove, A.

Thoen, E. R.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

Tiberio, R. C.

Villeneuve, P. R.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

Wang, M.-Q.

M.-Q. Wang, Y.-Z. Huang, Q. Chen, and Z.-P. Cai, IEEE J. Quantum Electron. 42, 146 (2006).
[CrossRef]

Wang, Z. X.

Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989), pp. 345-455.

Yu, L. J.

W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, IEEE J. Quantum Electron. 39, 1106 (2003).
[CrossRef]

Yu, L.-J.

Y.-Z. Huang, Q. Chen, W.-H. Guo, and L.-J. Yu, IEEE Photon. Technol. Lett. 17, 2589 (2005).
[CrossRef]

Zhang, J. P.

IEEE J. Quantum Electron.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, IEEE J. Quantum Electron. 35, 1322 (1999).
[CrossRef]

M.-Q. Wang, Y.-Z. Huang, Q. Chen, and Z.-P. Cai, IEEE J. Quantum Electron. 42, 146 (2006).
[CrossRef]

W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, IEEE J. Quantum Electron. 39, 1106 (2003).
[CrossRef]

IEEE Photon. Technol. Lett.

Y.-Z. Huang, Q. Chen, W.-H. Guo, and L.-J. Yu, IEEE Photon. Technol. Lett. 17, 2589 (2005).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. Ippen, L. C. Kimerling, and W. Greene, IEEE Photon. Technol. Lett. 10, 549 (1998).
[CrossRef]

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, IEEE Photon. Technol. Lett. 12, 323 (2000).
[CrossRef]

Opt. Lett.

Other

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed. (Academic, 1980).

Z. X. Wang and D. R. Guo, Special Functions (World Scientific, 1989), pp. 345-455.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962).

W. R. Smythe, Static and Dynamic Electricity, 3rd ed. (McGraw-Hill, 1968).

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Figures (3)

Fig. 1
Fig. 1

Comparison of the resonant frequencies for a square microcavity obtained by the exact solution in the text and an approximate solution.[12]

Fig. 2
Fig. 2

Resonant modes for m = n = 0 : (a) k a = 3.83170597020751 , (b) k a = 13.32369193631422 , (c) k a = 19.61585851046824 .

Fig. 3
Fig. 3

Resonant modes for n = 4 and m = 2 : (a) k a = 13.17037085601612 , (b) k a = 19.51291278248821 .

Equations (18)

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2 φ ( r ) + k 2 φ ( r ) = 0 ,
φ ( r ) = n = A n J n ( k r ) cos ( n θ ) ,
φ ( x , y ) = n = A n J n ( k x 2 + y 2 ) cos [ n tan 1 ( y x ) ] ,
φ x = 0 , a = 0 , φ x x = 0 , a = 0 , φ y = 0 , b = 0 , φ y y = 0 , b = 0 .
J n ( x 2 + y 2 ) cos [ n tan 1 ( y x ) ] = m = J n m ( x ) J m ( y ) cos ( m π 2 ) , = m = J n m ( y ) J m ( x ) cos [ ( m n ) π 2 ] .
J m ( 0 ) = { 1 m = 0 0 m 0 } , J m ( 0 ) = { 1 2 m = 1 1 2 m = 1 0 others } ,
φ n , m = A n { J n m ( k x ) + J n m [ k ( a x ) ] J n m ( k a ) J n m ( 0 ) } { J m ( k y ) + J m [ k ( b y ) ] J m ( k b ) J m ( 0 ) } cos m π 2 cos n π 2 ,
S φ n , m φ s , t d x d y = δ n s , m t ,
J n m ( k a ) = 0 ,
J m ( k b ) = 0 .
J m ( k a ) = 0 .
k s a = π [ 2 + ( s 2 ϕ π ) 2 2 ] 1 2 ,
ϕ 2 tan 1 ( 1 n eff n eff 2 2 ) + π ,
( ρ + m + n ) x x ρ 1 J m ( x ) J n ( x ) d x + ( ρ m n 2 ) x x ρ 1 J m + 1 ( x ) J n + 1 ( x ) d x = x ρ { J m ( x ) J n ( x ) + J m + 1 ( x ) J n + 1 ( x ) } ,
0 a J m ( x ) J m + 1 ( x ) d x = s = 0 [ J m + s + 1 ( a ) ] 2 ,
0 a J m ( x ) J n ( x ) d x = ( 1 ) m m 2 ( 1 ) n n 2 s = 0 a m + n + 2 s + 1 { J m + s ( a ) J n + s ( a ) + J m + s + 1 ( a ) J n + s + 1 ( a ) } .
0 a J m ( x ) J n ( a x ) d x = s = 0 ( 1 ) s 2 J m + n + 2 s + 1 ( a ) ,
0 a J m ( x ) d x = 2 ( 1 ) m s = 0 J m + 2 s + 1 ( a ) .

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