Vol. 55

Front:[PDF file] Back:[PDF file]
Latest Volume
All Volumes
All Issues
2013-09-27

Slow Scale Maxwell-Bloch Equations for Active Photonic Crystals

By Gandhi Alagappan
Progress In Electromagnetics Research B, Vol. 55, 169-194, 2013
doi:10.2528/PIERB13081201

Abstract

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.

Citation


Gandhi Alagappan, "Slow Scale Maxwell-Bloch Equations for Active Photonic Crystals," Progress In Electromagnetics Research B, Vol. 55, 169-194, 2013.
doi:10.2528/PIERB13081201
http://jpier.org/PIERB/pier.php?paper=13081201

References


    1. John, S., "Strong localization of photons in certain disordered dielectric superlattice," Phys. Rev. Lett., Vol. 5, No. 8, 2486, 1987.
    doi:10.1103/PhysRevLett.58.2486

    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.
    doi:10.1038/nphys343

    4. Matsubara, H., et al., "GaN photonic-crystal surface-emitting laser at blue-violet wavelength," Science, Vol. 319, 445, 2008.
    doi:10.1126/science.1150413

    5. Park, H. G., et al., "Electrically driven single-cell photonic crystal laser," Science, Vol. 305, 1444, 2004.
    doi:10.1126/science.1100968

    6. Painter, O., et al., "Two-dimensional photonic band-gap defect mode laser," Science, Vol. 28, No. 4, 1819, 1999.
    doi:10.1126/science.284.5421.1819

    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.
    doi:10.1103/PhysRevA.72.013807

    8. Ellis, B., et al., "Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser," Nat. Photonics, Vol. 5, 297, 2011.
    doi:10.1038/nphoton.2011.51

    9. Strauf, S., et al., "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett., Vol. 96, 127404, 2006.
    doi:10.1103/PhysRevLett.96.127404

    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.
    doi:10.1088/1367-2630/8/9/208

    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.
    doi:10.1103/PhysRevA.84.053848

    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.
    doi:10.1063/1.124361

    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.
    doi:10.1109/JQE.2002.1005409

    15. Chassagneu, Y., et al., "Electrically pumped photonic-crystal terahertz lasers controlled by boundary condition," Nat., Vol. 45, No. 7, 174, 2009.
    doi:10.1038/nature07636

    16. Kim, M., et al., "Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared," Appl. Phys. Lett., Vol. 8, No. 8, 191105, 2006.
    doi:10.1063/1.2203234

    17. Miyai, E., et al., "Lasers producing tailored beam," Nat., Vol. 44, No. 1, 946, 2006.
    doi:10.1038/441946a

    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.
    doi:10.1126/science.1061738

    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.
    doi:10.1103/PhysRevA.84.053848

    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.
    doi:10.1109/JPROC.2005.853555

    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.
    doi:10.1364/OL.31.000232

    22. Shi, S. and D. W. Prather, "Lasing dynamics of a silicon photonic crystal microcavity," Opt. Express, Vol. 15, 10294, 2007.
    doi:10.1364/OE.15.010294

    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.
    doi:10.1063/1.2916711

    24. Christiansen, M. B., et al., "Polymer photonic crystal dye lasers as optofluidic cell sensors," Opt. Express, Vol. 1, No. 7, 2722, 2009.
    doi:10.1364/OE.17.002722

    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.
    doi:10.1364/OL.36.001392

    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.
    doi:10.1103/PhysRevA.52.3082

    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.
    doi:10.1103/PhysRevB.73.165125

    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.
    doi:10.1364/OE.19.001539

    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.
    doi:10.1364/JOSAB.19.002215

    33. Kogelnik, H. and C. V. Shank, "Couple-wave theory of distributed feedback laser," J. Appl. Phys., Vol. 43, 2327, 1972.
    doi:10.1063/1.1661499

    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.
    doi:10.1364/OE.15.003981

    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.
    doi:10.1063/1.2220057

    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.
    doi:10.1109/JQE.2009.2037597

    37. Kaso, A. and S. John, "Nonlinear Bloch waves in resonantly doped photonic crystal," Physical Review E, Vol. 74, 046611, 2006.
    doi:10.1103/PhysRevE.74.046611

    38. Kaso, A. and S. John, "Nonlinear Bloch waves in metallic photonic band-gap filament," Physical Review A, Vol. 76, 053838, 2007.
    doi:10.1103/PhysRevA.76.053838

    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.
    doi:10.1016/0030-4018(93)90571-L

    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.
    doi:10.1109/68.219695

    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.
    doi:10.1103/PhysRevB.52.7982

    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.
    doi:10.1103/PhysRevB.68.035110

    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.
    doi:10.1103/PhysRevB.65.195110

    48. Dmitriev, V., "2D magnetic photonic crystals with square lattice-group theoretical standpoint," Progress In Electromagnetics Research, Vol. 58, 71, 2006.
    doi:10.2528/PIER05061701

    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.
    doi:10.1103/PhysRevB.77.195117

    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.
    doi:10.1364/OL.38.003514