Abstract

In the first part of a two-part study on the equivalent-circuit representation of any given Fabry–Perot resonator (FPR) that supports, by nature, infinitely many resonance modes, the complex-variable pole-zero structure of its scattering coefficients is extensively analyzed in general terms through partial-fraction expansion based on a corollary to Mittag-Leffler’s theorem for meromorphic functions. By finding the right offset constant in the expansion from the theory, we present two sets of uniformly converging series of partial fractions for the two scattering coefficients. We compare quality of convergence between the two series sets and find that a set obtained by the fraction-reciprocated reflection coefficient for the FPR is relatively better than the other one, which is fortunate for the subsequent work in the second part.

© 2014 Optical Society of America

PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Ann. Chim. Phys. 16, 115–144 (1899).
  2. L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, Inc., 1979), Chap. 5, pp. 187–190, Sec. 2.
  3. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: II. Uniformly converging multimode equivalent-circuit models,” J. Opt. Soc. Am. A.31, 411–420 (2014).
  4. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984). Fig. 9.9.
  5. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
    [CrossRef]
  6. A. I. Markushevich, Theory of Functions of a Complex Variable (AMS Chelsea Publishing, 2005), Vol. II, Chap. II.10, pp. 299–304, Sec. 51.
  7. L. Euler, Introduction to the Analysis of Infinite, Book I (Springer-Verlag, 1989). Translated by J. D. Bantom in 1840 from L. Euler, Introduction in Analysin Infinitorum (1748).
  8. L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, 1979), Chap. 5, pp. 197, Sec. 2.3, Eq. (24).
  9. S. Lang, Complex Analysis, 4th ed. (Springer-Verlag, 1999), Chap. XIII, pp. 382.
  10. T. W. Gamelin, Complex Analysis (Springer-Verlag, 2001), pp. 356. Eq. (3.3).
  11. T. W. Gamelin, Complex Analysis (Springer-Verlag, 2001), pp. 351. Sec. XIII.2, Exer. 1.
  12. T. W. Gamelin, Complex Analysis (Springer Verlag, 2001), pp. 350. Sec. XIII.2, Example of Eq. (2.1).
  13. T. W. Gamelin, Complex Analysis (Springer-Verlag, 2001), pp. 118. Sec. IV.5.
  14. T. W. Gamelin, Complex Analysis (Springer Verlag, 2001), pp. 357. Sec. XIII.3, Exer. 12.
  15. L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, 1979), Chap. 5, pp. 208–212, Sec. 3.2.

1999 (1)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

1899 (1)

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Ann. Chim. Phys. 16, 115–144 (1899).

Ahlfors, L. V.

L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, Inc., 1979), Chap. 5, pp. 187–190, Sec. 2.

L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, 1979), Chap. 5, pp. 208–212, Sec. 3.2.

L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, 1979), Chap. 5, pp. 197, Sec. 2.3, Eq. (24).

Euler, L.

L. Euler, Introduction to the Analysis of Infinite, Book I (Springer-Verlag, 1989). Translated by J. D. Bantom in 1840 from L. Euler, Introduction in Analysin Infinitorum (1748).

Fabry, C.

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Ann. Chim. Phys. 16, 115–144 (1899).

Fan, S.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

Gamelin, T. W.

T. W. Gamelin, Complex Analysis (Springer-Verlag, 2001), pp. 351. Sec. XIII.2, Exer. 1.

T. W. Gamelin, Complex Analysis (Springer-Verlag, 2001), pp. 118. Sec. IV.5.

T. W. Gamelin, Complex Analysis (Springer-Verlag, 2001), pp. 356. Eq. (3.3).

T. W. Gamelin, Complex Analysis (Springer Verlag, 2001), pp. 350. Sec. XIII.2, Example of Eq. (2.1).

T. W. Gamelin, Complex Analysis (Springer Verlag, 2001), pp. 357. Sec. XIII.3, Exer. 12.

Haus, H. A.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984). Fig. 9.9.

Joannopoulos, J. D.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

Khan, M. J.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

Lang, S.

S. Lang, Complex Analysis, 4th ed. (Springer-Verlag, 1999), Chap. XIII, pp. 382.

Manolatou, C.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

Markushevich, A. I.

A. I. Markushevich, Theory of Functions of a Complex Variable (AMS Chelsea Publishing, 2005), Vol. II, Chap. II.10, pp. 299–304, Sec. 51.

Pérot, A.

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Ann. Chim. Phys. 16, 115–144 (1899).

Song, G. H.

G. H. Song, “Mathematical modeling of Fabry–Perot resonators: II. Uniformly converging multimode equivalent-circuit models,” J. Opt. Soc. Am. A.31, 411–420 (2014).

Villeneuve, P. R.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

Ann. Chim. Phys. (1)

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Ann. Chim. Phys. 16, 115–144 (1899).

IEEE J. Quantum Electron. (1)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

Other (13)

A. I. Markushevich, Theory of Functions of a Complex Variable (AMS Chelsea Publishing, 2005), Vol. II, Chap. II.10, pp. 299–304, Sec. 51.

L. Euler, Introduction to the Analysis of Infinite, Book I (Springer-Verlag, 1989). Translated by J. D. Bantom in 1840 from L. Euler, Introduction in Analysin Infinitorum (1748).

L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, 1979), Chap. 5, pp. 197, Sec. 2.3, Eq. (24).

S. Lang, Complex Analysis, 4th ed. (Springer-Verlag, 1999), Chap. XIII, pp. 382.

T. W. Gamelin, Complex Analysis (Springer-Verlag, 2001), pp. 356. Eq. (3.3).

T. W. Gamelin, Complex Analysis (Springer-Verlag, 2001), pp. 351. Sec. XIII.2, Exer. 1.

T. W. Gamelin, Complex Analysis (Springer Verlag, 2001), pp. 350. Sec. XIII.2, Example of Eq. (2.1).

T. W. Gamelin, Complex Analysis (Springer-Verlag, 2001), pp. 118. Sec. IV.5.

T. W. Gamelin, Complex Analysis (Springer Verlag, 2001), pp. 357. Sec. XIII.3, Exer. 12.

L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, 1979), Chap. 5, pp. 208–212, Sec. 3.2.

L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, Inc., 1979), Chap. 5, pp. 187–190, Sec. 2.

G. H. Song, “Mathematical modeling of Fabry–Perot resonators: II. Uniformly converging multimode equivalent-circuit models,” J. Opt. Soc. Am. A.31, 411–420 (2014).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984). Fig. 9.9.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Metrics