## Frequency-domain formulation of photonic crystals using sources and gain |

Optics Express, Vol. 21, Issue 2, pp. 1972-1985 (2013)

http://dx.doi.org/10.1364/OE.21.001972

Enhanced HTML Acrobat PDF (2629 KB)

### Abstract

We present a formulation to analyze photonic periodic structures from viewpoints of sources and gain. The approach is based on a generalized eigenvalue problem and mode expansions of sources which sustain optical fields with phase boundary conditions. Using this scheme, we calculate power spectra, dispersion relations, and quality factors of Bloch modes in one-dimensional periodic structures consisting of dielectrics or metals. We also compare the results calculated from this scheme with those from the complex-frequency method. The outcomes of these two approaches generally agree well and only deviate slightly in the regime of low quality factors.

© 2013 OSA

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(260.2030) Physical optics : Dispersion

(260.2110) Physical optics : Electromagnetic optics

(260.5740) Physical optics : Resonance

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: November 15, 2012

Revised Manuscript: January 11, 2013

Manuscript Accepted: January 11, 2013

Published: January 17, 2013

**Citation**

Po-Jui Chiang and Shu-Wei Chang, "Frequency-domain formulation of photonic crystals using sources and gain," Opt. Express **21**, 1972-1985 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-1972

Sort: Year | Journal | Reset

### References

- L. Brillouin, Wave Propagation in Periodic Structures (Dover, 1953).
- E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett.58, 2059–2062 (1987). [CrossRef] [PubMed]
- S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett.58, 2486–2489 (1987). [CrossRef] [PubMed]
- J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008), 2nd ed.
- P. Russell, “Photonic crystal fibers,” Science299, 358–362 (2003). [CrossRef] [PubMed]
- O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284, 1819–1821 (1999). [CrossRef] [PubMed]
- S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001). [CrossRef] [PubMed]
- H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science305, 1444–1447 (2004). [CrossRef] [PubMed]
- F. Lemarchand, A. Sentenac, and H. Giovannini, “Increasing the angular tolerance of resonant grating filters with doubly periodic structures,” Opt. Lett.23, 1149–1151 (1998). [CrossRef]
- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
- M. I. Stockman, “Electromagnetic theory of SERS,” in Surface-Enhanced Raman Scattering, Topics in Applied Physics, K. Kneipp, M. Moskovits, and H. Kneipp, Eds. (Springer-Verlag, 2006). 47–65. [CrossRef]
- S. I. Bozhevolnyi and V. M. Shalaev, “Nanophotonics with surface plasmons–part I,” Photonics Spectra, Jan. Issue, 58–66 (2006).
- V. M. Shalaev and S. I. Bozhevolnyi, “Nanophotonics with surface plasmons–part II,” Photonics Spectra, Feb. Issue, 66–73 (2006).
- W. Cai, R. Sainidou, J. Xu, A. Polman, and F. J. G. de Abajo, “Efficient generation of propagating plasmons by electron beams,” Nano Lett.9, 1176–1181 (2009). [CrossRef] [PubMed]
- A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science308, 670–672 (2005). [CrossRef] [PubMed]
- C. W. Cheng, M. N. Abbas, M. H. Shih, and Y. C. Chang, “Characterization of the surface plasmon polariton band gap in an Ag/SiO2/Ag T-shaped periodical structure,” Opt. Express19, 23698–23705 (2011). [CrossRef] [PubMed]
- W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996). [CrossRef]
- A. Kocabas, S. S. Senlik, and A. Aydinli, “Plasmonic band gap cavities on biharmonic gratings,” Phys. Rev. B77, 195130 (2008).
- S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett.86, 3008–3011 (2001). [CrossRef] [PubMed]
- F. Wu, D. Han, X. Hu, X. Liu, and J. Zi, “Complete surface plasmon-polariton band gap and gap-governed waveguiding, bending and splitting,” J. Phys.: Condens. Matter21, 185010 (2009).
- V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B55, 7427–7444 (1997). [CrossRef]
- S. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express11, 167–175 (2003). [CrossRef] [PubMed]
- C. C. Chang, R. L. Chern, C. C. Chang, and R. R. Hwang, “Interfacial operator approach to computing modes of surface plasmon polaritons for periodic structures,” Phys. Rev. B72, 205112 (2005). [CrossRef]
- R. L. Chern, C. C. Chang, and C. C. Chang, “Analysis of surface plasmon modes and band structures for plasmonic crystals in one and two dimensions,” Phys. Rev. E73, 036605 (2006). [CrossRef]
- E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B65, 155120 (2002). [CrossRef]
- A. G. Hanif, T. Arima, and T. Uno, “Finite-difference frequency-domain algorithm for band-diagram calculation of 2-D photonic crystals composed of Debye-type dispersive materials,” IEEE Antennas Wireless Propag. Lett.11, 41–44 (2012). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Arctech House, 2005), 3rd ed.
- P. G. Petropoulos, “Phase error control for FD-TD methods of second and fourth order accuracy,” IEEE Trans. Antennas Propag.42, 859–862 (1994). [CrossRef]
- Y. Cao, Z. Hou, and Y. Liu, “Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals,” Solid State Commun.132, 539–543 (2004). [CrossRef]
- Y. Tanaka, Y. Tomoyasu, and S. Tamura, “Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch,” Phys. Rev. B62, 7387–7392 (2000). [CrossRef]
- T. Søndergaard and B. Tromborg, “General theory for spontaneous emission in active dielectric microstructures: example of a fiber amplifier,” Phys. Rev. A64, 033812 (2001). [CrossRef]
- Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B81, 125431 (2010). [CrossRef]
- V. Siahpoush, T. Søndergaard, and J. Jung, “Green’s function approach to investigate the excitation of surface plasmon polaritons in a nanometer-thin metal film,” Phys. Rev. B85, 075305 (2012). [CrossRef]
- A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys.22, 17–23 (1977).
- E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron.36, 213–221 (2004). [CrossRef]
- E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-d modeling with accurate 2-d analysis,” IEEE J. Sel. Top. Quantum Electron.11, 1135–1142 (2005). [CrossRef]
- A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron.39, 1253–1272 (2007). [CrossRef]
- S. W. Chang, “Full frequency-domain approach to reciprocal microlasers and nanolasers-perspective from Lorentz reciprocity,” Opt. Express19, 21116–21134 (2011). [CrossRef] [PubMed]
- S. W. Chang, “Confinement factors and modal volumes of micro- and nanocavities invariant to integration regions,” IEEE J. Sel. Top. Quantum. Electron.18, 1771–1780 (2012). [CrossRef]
- E. Merzbacher, Quantum Mechanics (Wiley and Sons, New York, 1998), 3rd ed.
- H. A. Lorentz, “The theorem of poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Verh. K. Akad. Wet. Amsterdam, Afd. Natuurkd.4, 176–187 (1896).
- J. R. Carson, “A generalization of reciprocal theorem,” Bell System Technical Journal3, 393–399 (1924).
- C. A. Balanis, Advanced Engineering Electromagnetics (Wiley and Sons, 1989).
- F. Olyslager, “Properties of and generalized full-wave transmission line models for hybrid (bi)(an)isotropic waveguides,” IEEE Trans. Microw. Theory Techn.44, 2064–2075 (1996). [CrossRef]
- D. Pissoort and F. Olyslager, “Study of eigenmodes in periodic waveguides using the lorentz reciprocity theorem,” IEEE Trans. Microw. Theory Techn.52, 542–553 (2004). [CrossRef]
- A. D. Yaghjian, “Bidirectionality of reciprocal, lossy or lossless, uniform or periodic waveguides,” IEEE Microw. Wireless Compon. Lett.17, 480–482 (2007). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.