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Quasi-normal-modes description of transmission properties for photonic bandgap structures
Author(s)
Language
English
Obiettivo Specifico
1.7. Osservazioni di alta e media atmosfera
Status
Published
JCR Journal
JCR Journal
Peer review journal
Yes
Title of the book
Issue/vol(year)
4/26 (2009)
Publisher
Henry M. Van Driel, University of Toronto
Pages (printed)
876-891
Issued date
March 31, 2009
Alternative Location
Last version
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-26-4-876
Abstract
We use the “quasi-normal-modes” (QNM) approach for discussing the transmission properties of double-side opened optical cavities: in particular, this approach is specified for one-dimensional (1D) “photonic bandgap” (PBG) structures. Moreover, we conjecture that the density of the modes is a dynamical variable that has the flexibility of varying with respect to the boundary conditions as well as the initial conditions; in fact, the electromagnetic
(e.m.) field generated by two monochromatic counterpropagating pump waves leads to interference effects inside a quarter-wave symmetric 1D-PBG structure. Finally, here, for the first time to the best of our knowledge, a large number of theoretical assumptions on QNM metrics for an open cavity, never discussed in literature, are proved, and a simple and direct method to calculate the QNM norm for a 1D-PBG structure
is reported.
(e.m.) field generated by two monochromatic counterpropagating pump waves leads to interference effects inside a quarter-wave symmetric 1D-PBG structure. Finally, here, for the first time to the best of our knowledge, a large number of theoretical assumptions on QNM metrics for an open cavity, never discussed in literature, are proved, and a simple and direct method to calculate the QNM norm for a 1D-PBG structure
is reported.
References
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4. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge”, Phys. Rev. Lett. 53, 2169–2172 (1984).
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“Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic
interactions”, Phys. Rev. E 60, 4891–4898 (1999).
7. J. E. Sipe, L. Poladian, and C. Martijn de Sterke, “Propagation through nonuniform grating structures”, J. Opt. Soc. Am. A 4, 1307–1320 (1994).
8. E. S. C. Ching, P. T. Leung, and K. Young, “Optical processes in microcavities—the role of quasi-normal modes”, in Optical Processes in Microcavities, R. K. Chang and A. J. Campillo, eds. (World Scientific,
1996), pp. 1–75.
9. P. T. Leung, S. Y. Liu, and K.Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities”, Phys. Rev. A 49, 3057–3067 (1994).
10. P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the
wave equation I: completeness of expansion”, J. Phys. A 30, 2139–2151 (1997).
11. P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the
wave equation II: linear space structure”, J. Phys. A 30, 2153–2162 (1997).
12. E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems”, Rev. Mod. Phys. 70,
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13. P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in open systems via a biorthogonal basis”, Phys. Rev. E 57, 6101–6104 (1998).
14. K. C. Ho, P. T. Leung, A. Maassen van den Brink, and K. Young, “Second quantization of open systems using
quasinormal modes”, Phys. Rev. E 58, 2965–2978 (1998).
15. A. Maassen van den Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open
wave systems”, J. Phys. A 34, 2607–2624 (2001).
16. B. J. Hoenders, “On the decomposition of the electromagnetic field into its natural modes,” in Proceedings of the 4th International Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf,
eds. (Plenum, 1978), pp. 221–233.
17. B. J. Hoenders and H. A. Ferwerda, “On a new proposal concerning the calculation of the derivatives of a function subject to errors”, Optik (Jena) 40, 14–17 (1974).
18. B. J. Hoenders, “Completeness of a set of modes connected with the electromagnetic field of a homogeneous sphere embedded in an infinite medium”, J. Math. Phys. 11, 1815–1832 (1978).
19. B. J. Hoenders, “On the completeness of the natural modes for quantum mechanical potential scattering”, J. Math. Phys. 20, 329–335 (1979).
20. M. Bertolotti, “Linear one-dimensional resonant cavities”, in Microresonators as Building Blocks for VLSI Photonics of AIP Conference Proceedings, F. Michelotti, A. Driessen, and M. Bertolotti, eds. (AIP, 2004), Vol. 709, pp. 19–47.
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and C. M. Bowden, “Quasi normal modes description of waves in 1D photonic crystals,” Proc. SPIE 5036, 392–401 (2003).
24. A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and
C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals”, Phys. Rev. E 68, 026614 (2003).
25. S. Severini, A. Settimi, C. Sibilia, M. Bertolotti, A. Napoli, and A. Messina, “Quasi-normal frequencies in open
cavities: an application to photonic crystals”, Acta Phys. Hung. B 23, 135–142 (2005).
26. S. Severini, A. Settimi, C. Sibilia, M. Bertolotti, A. Napoli, and A. Messina, “Quantum counterpropagation in open
optical cavities via the quasi-normal-mode approach”, Laser Phys. 16, 911–920 (2006).
27. M. Born and E. Wolf, Principles of Optics (Macmillan,1964).
28. G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Variable—Theory and Technique (McGraw-Hill, 1983).
29. E. N. Economou, Green’s Functions in Quantum Physics (Springer-Verlag, 1979).
30. A. Bachelot and A. Motet-Bachelot, “Les résonances d’un trou noir de Schwarzschild”, Ann. Inst. Henri Poincaré, Sect. A 59, 3–68 (1993).
31. D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential”, Am. J. Phys. 61, 1118–1124
(1993).
32. M. G. Rozman, P. Reineken, and R. Tehver, “Scattering by locally periodic one-dimensional potentials”, Phys. Lett. A
187, 127–131 (1994).
33. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite,
one-dimensional, photonic band-gap structures”, Phys. Rev. E 53, 4107–4121 (1996).
34. J. P. Dowling, “Parity, time-reversal and group delay for inhomogeneous dielectric slabs: Application to pulse propagation in finite, one-dimensional, photonic bandgap
structures”, in Proceedings of IEEE-Optoelctronics (IEEE, 1998), Vol. 145, pp. 420–435.
35. A. Sopahaluwakan, “Characterization and simulation of localized states in periodic structures”, Ph.D. dissertation (University of Twente, 2006).
2. J. Maddox, “Photonic band-gaps bite the dust”, Nature 348, 481–481 (1990).
3. S. John, “Strong localization of photons in certain disordered dielectric superlattices”, Phys. Rev. Lett. 58, 2486–2489 (1987).
4. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge”, Phys. Rev. Lett. 53, 2169–2172 (1984).
5. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
6. M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov,
“Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic
interactions”, Phys. Rev. E 60, 4891–4898 (1999).
7. J. E. Sipe, L. Poladian, and C. Martijn de Sterke, “Propagation through nonuniform grating structures”, J. Opt. Soc. Am. A 4, 1307–1320 (1994).
8. E. S. C. Ching, P. T. Leung, and K. Young, “Optical processes in microcavities—the role of quasi-normal modes”, in Optical Processes in Microcavities, R. K. Chang and A. J. Campillo, eds. (World Scientific,
1996), pp. 1–75.
9. P. T. Leung, S. Y. Liu, and K.Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities”, Phys. Rev. A 49, 3057–3067 (1994).
10. P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the
wave equation I: completeness of expansion”, J. Phys. A 30, 2139–2151 (1997).
11. P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the
wave equation II: linear space structure”, J. Phys. A 30, 2153–2162 (1997).
12. E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems”, Rev. Mod. Phys. 70,
1545–1554 (1998).
13. P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in open systems via a biorthogonal basis”, Phys. Rev. E 57, 6101–6104 (1998).
14. K. C. Ho, P. T. Leung, A. Maassen van den Brink, and K. Young, “Second quantization of open systems using
quasinormal modes”, Phys. Rev. E 58, 2965–2978 (1998).
15. A. Maassen van den Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open
wave systems”, J. Phys. A 34, 2607–2624 (2001).
16. B. J. Hoenders, “On the decomposition of the electromagnetic field into its natural modes,” in Proceedings of the 4th International Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf,
eds. (Plenum, 1978), pp. 221–233.
17. B. J. Hoenders and H. A. Ferwerda, “On a new proposal concerning the calculation of the derivatives of a function subject to errors”, Optik (Jena) 40, 14–17 (1974).
18. B. J. Hoenders, “Completeness of a set of modes connected with the electromagnetic field of a homogeneous sphere embedded in an infinite medium”, J. Math. Phys. 11, 1815–1832 (1978).
19. B. J. Hoenders, “On the completeness of the natural modes for quantum mechanical potential scattering”, J. Math. Phys. 20, 329–335 (1979).
20. M. Bertolotti, “Linear one-dimensional resonant cavities”, in Microresonators as Building Blocks for VLSI Photonics of AIP Conference Proceedings, F. Michelotti, A. Driessen, and M. Bertolotti, eds. (AIP, 2004), Vol. 709, pp. 19–47.
21. B. J. Hoenders and M. Bertolotti, “The (quasi)normal natural mode description of the scattering process by dispersive photonic crystals”, Proc. SPIE 6182,
61821F–6182G (2006).
22. M. Maksimovic, M. Hammer, and E. van Groesen, “Field representation for optical defect resonances in multilayer microcavities using quasi-normal modes”, Opt. Lett. 281, 1401–1411 (2008).
23. S. Severini, A. Settimi, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. J. Bloemer,
and C. M. Bowden, “Quasi normal modes description of waves in 1D photonic crystals,” Proc. SPIE 5036, 392–401 (2003).
24. A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and
C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals”, Phys. Rev. E 68, 026614 (2003).
25. S. Severini, A. Settimi, C. Sibilia, M. Bertolotti, A. Napoli, and A. Messina, “Quasi-normal frequencies in open
cavities: an application to photonic crystals”, Acta Phys. Hung. B 23, 135–142 (2005).
26. S. Severini, A. Settimi, C. Sibilia, M. Bertolotti, A. Napoli, and A. Messina, “Quantum counterpropagation in open
optical cavities via the quasi-normal-mode approach”, Laser Phys. 16, 911–920 (2006).
27. M. Born and E. Wolf, Principles of Optics (Macmillan,1964).
28. G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Variable—Theory and Technique (McGraw-Hill, 1983).
29. E. N. Economou, Green’s Functions in Quantum Physics (Springer-Verlag, 1979).
30. A. Bachelot and A. Motet-Bachelot, “Les résonances d’un trou noir de Schwarzschild”, Ann. Inst. Henri Poincaré, Sect. A 59, 3–68 (1993).
31. D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential”, Am. J. Phys. 61, 1118–1124
(1993).
32. M. G. Rozman, P. Reineken, and R. Tehver, “Scattering by locally periodic one-dimensional potentials”, Phys. Lett. A
187, 127–131 (1994).
33. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite,
one-dimensional, photonic band-gap structures”, Phys. Rev. E 53, 4107–4121 (1996).
34. J. P. Dowling, “Parity, time-reversal and group delay for inhomogeneous dielectric slabs: Application to pulse propagation in finite, one-dimensional, photonic bandgap
structures”, in Proceedings of IEEE-Optoelctronics (IEEE, 1998), Vol. 145, pp. 420–435.
35. A. Sopahaluwakan, “Characterization and simulation of localized states in periodic structures”, Ph.D. dissertation (University of Twente, 2006).
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