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Slow Scale Maxwell-Bloch Equations for Active Photonic Crystals

By Gandhi Alagappan
Progress In Electromagnetics Research B, Vol. 55, 169-194, 2013


We present a theory to describe the transient and steady state behaviors of the active modes of a photonic crystal with active constituents (active photonic crystal). Using a couple mode model, we showed that the full vectorial Maxwell-Bloch equations describing the physics of light matter interaction in the active photonic crystal can be written as system of integro-differential equations. Using the method of moments and the mean value theorem, we showed that the system of integro-differential equations can be transformed to a set of differential equations in slow time and slow spatial scales. The slow time (spatial) scale refers to a duration (distance) that is much longer than the optical time period (lattice constant of the photonic crystal). In the steady state, the slow scale equations reduce to a nonlinear matrix eigenvalue problem, from which the nonlinear Bloch modes can be obtained by an iterative method. For cases, where the coupling between the modes are negligible, we describe the transient behavior as an onedimensional problem in the spatial coordinate, and the steady behaviors are expressed using simple analytical expressions.


Gandhi Alagappan, "Slow Scale Maxwell-Bloch Equations for Active Photonic Crystals," Progress In Electromagnetics Research B, Vol. 55, 169-194, 2013.


    1. John, S., "Strong localization of photons in certain disordered dielectric superlattice," Phys. Rev. Lett., Vol. 5, No. 8, 2486, 1987.

    2. Joannopoulos, J. D., S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd Edition, Princeton University Press, 2008.

    3. Altug, H., D. Englund, and J. Vuckovic, "Ultrafast photonic crystal nanocavity laser," Nat. Phys., Vol. 2, 484, 2006.

    4. Matsubara, H., et al., "GaN photonic-crystal surface-emitting laser at blue-violet wavelength," Science, Vol. 319, 445, 2008.

    5. Park, H. G., et al., "Electrically driven single-cell photonic crystal laser," Science, Vol. 305, 1444, 2004.

    6. Painter, O., et al., "Two-dimensional photonic band-gap defect mode laser," Science, Vol. 28, No. 4, 1819, 1999.

    7. Vujic, D. and S. John, "Pulse reshaping in photonic crystal waveguides and microcavities with Kerr nonlinearity: Critical issues for all-optical switching," Phys. Rev. A, Vol. 72, 013807, 2005.

    8. Ellis, B., et al., "Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser," Nat. Photonics, Vol. 5, 297, 2011.

    9. Strauf, S., et al., "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett., Vol. 96, 127404, 2006.

    10. Asakawa, K., et al., "Photonic crystal and quantum dot technologies for all-optical switch and logic device," New J. Phys., Vol. 8, 208, 2006.

    11. Ma, X. and S. John, "Optical pulse dynamics for quantum-dot logic operations in a photonic-crystal waveguide," Phys. Rev. A, Vol. 84, 053848, 2011.

    12. Imada, M., et al., "Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure," Appl. Phys. Lett., Vol. 7, No. 5, 316, 1999.

    13. Meie, M., et al., "Laser action from two-dimensional distributed feedback in photonic crystal," Appl. Phys. Lett., Vol. 74, 7, 1999.

    14. Vurgaftman, I. and J. R. Meyer, "Photonic-crystal distributed-feedback quantum cascade lasers," IEEE J. Quant. Electron., Vol. 38, 592, 2002.

    15. Chassagneu, Y., et al., "Electrically pumped photonic-crystal terahertz lasers controlled by boundary condition," Nat., Vol. 45, No. 7, 174, 2009.

    16. Kim, M., et al., "Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared," Appl. Phys. Lett., Vol. 8, No. 8, 191105, 2006.

    17. Miyai, E., et al., "Lasers producing tailored beam," Nat., Vol. 44, No. 1, 946, 2006.

    18. Noda, S., et al., "Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design," Science, Vol. 29, No. 3, 1123, 2001.

    19. Ma, X. and S. John, "Optical pulse dynamics for quantum-dot logic operations in a photonic-crystal waveguide," Phys. Rev. A, Vol. 84, 053848, 2011.

    20. Bhattacharya, P., J. Sabarinathan, J. Topol'ancik, and S. Chakravarty, "Quantum dot photonic crystal light sources," Proceedings of the IEEE, Vol. 93, 1825, 2005.

    21. Topol'ancik, J., S. Chakravarty, P. Bhattacharya, and S. Chakrabarti, "Electrically injected quantum-dot photonic crystal microcavity light sources," Opt. Lett., Vol. 3, No. 1, 232, 2006.

    22. Shi, S. and D. W. Prather, "Lasing dynamics of a silicon photonic crystal microcavity," Opt. Express, Vol. 15, 10294, 2007.

    23. Makarova, M., et al., "Enhanced light emission in photonic crystal nanocavities with Erbium-doped silicon nanocrystal," Appl. Phys. Lett., Vol. 9, No. 2, 161107, 2008.

    24. Christiansen, M. B., et al., "Polymer photonic crystal dye lasers as optofluidic cell sensors," Opt. Express, Vol. 1, No. 7, 2722, 2009.

    25. Smith, C. L. C., et al., "Enhanced transduction of photonic crystal dye lasers for gas sensing via swelling polymer film," Opt. Lett., Vol. 3, No. 6, 1392, 2011.

    26. Ziolkowski, R. W., J. M. Arnold, and D. M. Gogny, "Ultrafast pulse interactions with two-level atom," Phys. Rev. A, Vol. 52, 3082, 1995.

    27. Bermel, P., E. Lidorikis, Y. Fink, and J. D. Joannopoulos, "Active materials embedded in photonic crystals and coupled to electromagnetic radiation," Phys. Rev. B, Vol. 7, No. 3, 165125, 2006.

    28. Chua, S. L., Y. Chong, A. D. Stone, M. Soljacic, and J. B. Abad, "Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances," Opt. Express, Vol. 19, 1539, 2011.

    29. Milonni, P. W. and J. H. Eberly, Laser Physics, 2nd edition, Wiley, 2010.

    30. Yariv, A., Optical Electronics in Modern Communications, 5th edition, Oxford University Press, 1997.

    31. Sargent, M., M. O. Scully, and W. E. Lamb, Laser Physics, Addison-Wesley, Reading, Mass., 1977.

    32. Florescu, L., K. Busch, and S. John, "Semiclassical theory of lasing in photonic crystal," J. Opt. Soc. Am. B, Vol. 1, No. 9, 2215, 2002.

    33. Kogelnik, H. and C. V. Shank, "Couple-wave theory of distributed feedback laser," J. Appl. Phys., Vol. 43, 2327, 1972.

    34. Sakai, K., E. Miyai, and S. Noda, "Two-dimensional coupled wave theory for square-lattice photonic-crystal lasers with TM-polarization," Opt. Express, Vol. 1, No. 5, 3981, 2007.

    35. Sakai, K., E. Miyai, and S. Noda, "Coupled-wave model for squarelattice two-dimensional photonic crystal with transverse-electric-like mode," Appl. Phys. Lett., Vol. 89, 021101, 2006.

    36. Sakai, K., E. Miyai, and S. Noda, "Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization," IEEE J. Quant. Electron., Vol. 46, 788, 2010.

    37. Kaso, A. and S. John, "Nonlinear Bloch waves in resonantly doped photonic crystal," Physical Review E, Vol. 74, 046611, 2006.

    38. Kaso, A. and S. John, "Nonlinear Bloch waves in metallic photonic band-gap filament," Physical Review A, Vol. 76, 053838, 2007.

    39. Meleshko, S. V., Y. N. Grigoriev, N. H. Ibragimov, and V. F. Kovalev, Symmetries of Integro-differential Equations: With Applications in Mechanics and Plasma Physics, Springer, 2010.

    40. Meleshko, S. V., Methods for Constructing Exact Solutions of Partial Differential Equations: Mathematical and Analytical Techniques with Applications to Engineerin, Chap. 2, Springer, 2005.

    41. Bowden, C. M. and G. P. Agrawal, "Generalized Bloch-Maxwell formulation for semiconductor laser," Opt. Commun, Vol. 100, 147, 1993.

    42. Agrawal, G. P. and C. M. Bowden, "Concept of linewidth enhancement factor in semiconductor lasers: Its usefulness and limitation," IEEE Phot. Tech. Lett., Vol. 5, 640, 1993.

    43. Thomas, G. B., Calculus and Analytic Geometry, 9th Edition, Addison Wesley, 1995.

    44. Harrison, P., Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures, 3rd edition, Wiley, 2010.

    45. Sakoda, K., "Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices," Phys. Rev. B, Vol. 5, No. 2, 7982, 1995.

    46. Painter, O. and K. Srinivasan, "Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis," Phys. Rev. B, Vol. 6, No. 8, 035110, 2003.

    47. Lopez-Tejeira, F., T. Ochiai, K. Sakoda, and J. Sanchez-Dehesa, "Symmetry characterization of eigenstates in opal-based photonic crystals," Phys. Rev. B, Vol. 65, 195110, 2002.

    48. Dmitriev, V., "2D magnetic photonic crystals with square lattice-group theoretical standpoint," Progress In Electromagnetics Research, Vol. 58, 71, 2006.

    49. Alagappan, G. and X. W. Sun, "Symmetries of the eigenstates in an anisotropic photonic crystal," Phys. Rev. B, Vol. 7, No. 7, 195117, 2008.

    50. Sakoda, K., Optical Properties of Photonic Crystals, Springer, Berlin, 2001.

    51. Cornwell, J. F., Group Theory in Physics: An Introduction, Academic Press, San Diego, 1997.

    52. Gohberg, I., P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, London, 1982.

    53. Thyagarajan, K. and A. Ghatak, Lasers: Fundamentals and Application, 2nd edition, Springer, 2010.

    54. Martijn de Sterke, C. and J. E. Sipe, "Envelope-function approach for the electrodynamics of nonlinear periodic structure," Phys. Rev. A, Vol. 38, 514-5165, 1988.

    55. Alagappan, G., S. John, and E. P. Li, "Macroscopic response in active nonlinearphotonic crystal," Opt. Lett., Vol. 15, 3514, 2013.