Abstract

We propose a mode conversion method that enables transformation of the propagating mode from fundamental to higher-order modes by utilizing asymmetric graded index (A-GRIN) structures. Refractive index variations of two different asymmetric gradient profiles, i.e., exponential and Luneburg lens profiles, have been approximated by two-dimensional photonic crystals (PCs). The basic structure is composed of constant radii with different lattice sizes. The designed GRIN mode converters provide relatively high transmission efficiency in the spectral region of interest and achieve the transformation in compact configuration. Numerical approaches utilizing the finite-difference time-domain and plane wave expansion methods are used to analyze the mode conversion phenomenon of proposed GRIN PC media. Analytical formulation based on ray theory is outlined to explore both ray trajectories and the physical concept of a wavefront retardation mechanism.

© 2013 Optical Society of America

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  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
  2. H. Kurt and D. S. Citrin, “Annular photonic crystals,” Opt. Express 13, 10316–10326 (2005).
    [CrossRef]
  3. M. Turduev, I. Giden, and H. Kurt, “Modified annular photonic crystals with enhanced dispersion relations: polarization insensitive self-collimation and nanophotonic wire waveguide designs,” J. Opt. Soc. Am. B 29, 1589–1598 (2012).
    [CrossRef]
  4. I. H. Giden and H. Kurt, “Modified annular photonic crystals for enhanced band gap properties and iso-frequency contour engineering,” Appl. Opt. 51, 1287–1296 (2012).
    [CrossRef]
  5. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
    [CrossRef]
  6. H. Kurt and D. S. Citrin, “Photonic-crystal heterostructure waveguides,” IEEE J. Quantum Electron. 43, 78–84 (2007).
    [CrossRef]
  7. W. D. Zhou, J. Sabarinathan, P. Bhattacharya, B. Kochman, E. Berg, P. C. Yu, and S. Pang, “Characteristics of a photonic bandgap single defect microcavity electroluminescent device,” IEEE J. Quantum Electron. 37, 1153–1160 (2001).
    [CrossRef]
  8. M. Yano, F. Yamagishi, and T. Tsuda, “Optical MEMS for photonic switching-compact and stable optical cross connect switches for simple, fast and flexible wavelength applications in recent photonic networks,” IEEE J. Sel. Top. Quantum Electron. 11, 383–394 (2005).
    [CrossRef]
  9. M. Shah, J. D. Craw, and S. Wang, “Optical waveguide mode conversion experiments,” Appl. Phys. Lett. 20, 66–69 (1972).
    [CrossRef]
  10. F. Auracher, “Mode order converter,” Opt. Commun. 11, 187–190 (1974).
    [CrossRef]
  11. Y. Huang, G. Xu, and S. Ho, “An ultra compact optical mode order converter,” IEEE Photon. Technol. Lett. 18, 2281–2283 (2006).
  12. B. Lee and S. Shin, “Mode order converter in a multimode waveguide,” Opt. Lett. 28, 1660–1662 (2003).
    [CrossRef]
  13. M. Yang, H. Chen, K. Webb, S. Minin, S. Chuang, and G. Cueva, “Demonstration of mode conversion in an irregular waveguide,” Opt. Lett. 31, 383–385 (2006).
    [CrossRef]
  14. J. Castro, D. Geraghty, S. Honkanen, C. Greiner, D. Iazikov, and T. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide Bragg gratings,” Opt. Express 13, 4180–4184 (2005).
    [CrossRef]
  15. J. Castillo, J. Castro, R. Kostuk, and D. Geraghty, “Study of multichannel parallel anti-symmetric waveguide Bragg gratings for telecom applications,” IEEE Photon. Technol. Lett. 19, 85–87 (2007).
    [CrossRef]
  16. J. Kurz, J. Huang, X. Xie, T. Saida, and M. Fejer, “Mode multiplexing in optical frequency mixers,” Opt. Lett. 29, 551–553 (2004).
    [CrossRef]
  17. M. Pruessner, J. Khurgin, T. Stievater, W. Rabinovich, R. Bass, J. Boos, and V. Urick, “Demonstration of a mode-conversion cavity add-drop filter,” Opt. Lett. 36, 2230–2232 (2011).
    [CrossRef]
  18. J. Tan, M. Lu, A. Stein, and W. Jiang, “High-purity transmission of a slow light odd mode in a photonic crystal waveguide,” Opt. Lett. 37, 3189–3191 (2012).
    [CrossRef]
  19. V. Delaubert, M. Lassen, D. Pulford, H. Bachor, and C. Harb, “Spatial mode discrimination using second harmonic generation,” Opt. Express 15, 5815–5826 (2007).
    [CrossRef]
  20. B. Desiatov, I. Goykhman, and U. Levy, “Nanoscale mode selector in silicon waveguide for on chip nanofocusing applications,” Nano Lett. 9, 3381–3386 (2009).
    [CrossRef]
  21. J. Leuthold, J. Eckner, E. Gamper, P. A. Besse, and H. Melchior, “Multimode interference couplers for the conversion and combining of zero- and first-order modes,” J. Lightwave Technol. 16, 1228–1239 (1998).
    [CrossRef]
  22. M. Lu, B. K. Juluri, S.-C. S. Lin, B. Kiraly, T. Gao, and T. J. Huang, “Beam aperture modifier and beam deflector using gradient-index photonic crystals,” J. Appl. Phys. 108, 103505 (2010).
    [CrossRef]
  23. H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express 15, 1240–1253 (2007).
    [CrossRef]
  24. H. Kurt and D. S. Citrin, “A novel optical coupler design with graded-index photonic crystals,” IEEE Photon. Technol. Lett. 19, 1532–1534 (2007).
    [CrossRef]
  25. H. T. Chien, C. Lee, H. K. Chiu, K. C. Hsu, C. C. Chen, J. A. Ho, and C. Cho, “The comparison between the graded photonic crystal coupler and various coupler,” J. Lightwave Technol. 27, 2570–2574 (2009).
  26. B. Vasić and R. Gajić, “Self-focusing media using graded photonic crystals: focusing, Fourier transforming and imaging, directive emission and directional cloaking,” J. Appl. Phys. 110, 053103 (2011).
    [CrossRef]
  27. F. Gaufillet and É. Akmansoy, “Graded photonic crystals for graded index lens,” Opt. Commun. 285, 2638–2641 (2012).
    [CrossRef]
  28. C. Tan, T. Niemi, C. Peng, and M. Pessa, “Focusing effect of a graded index photonic crystal lens,” Opt. Commun. 284, 3140–3143 (2011).
    [CrossRef]
  29. H. W. Wang and L. W. Chen, “High transmission efficiency of arbitrary waveguide bends formed by graded index photonic crystals,” J. Opt. Soc. Am. B 28, 2098–2104 (2011).
    [CrossRef]
  30. E. Centeno, D. Cassagne, and J.-P. Albert, “Mirage and superbending effect in two-dimensional graded photonic crystals,” Phys. Rev. B 73, 235119 (2006).
    [CrossRef]
  31. B. Vasic, G. Isic, R. Gajic, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18, 20321–20333 (2010).
    [CrossRef]
  32. E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, and J.-M. Lourtioz, “Graded photonic crystals curve the flow of light: an experimental demonstration by the mirage effect,” Appl. Phys. Lett. 92, 133501 (2008).
    [CrossRef]
  33. H. Kurt, B. B. Oner, M. Turduev, and I. H. Giden, “Modified Maxwell fish-eye approach for efficient coupler design by graded photonic crystals,” Opt. Express 20, 22018–22033 (2012).
    [CrossRef]
  34. M. Wienold, A. Tahraoui, L. Schrottke, R. Sharma, X. Lü, K. Biermann, R. Hey, and H. Grahn, “Lateral distributed-feedback gratings for single-mode high-power terahertz quantum-cascade lasers,” Opt. Express 20, 11207–11217 (2012).
    [CrossRef]
  35. T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
    [CrossRef]
  36. S. Blaaberg, P. M. Petersen, and B. Tromborg, “Structure, stability, and spectra of lateral modes of a broad-area semiconductor laser,” IEEE J. Quantum Electron. 43, 959–973 (2007).
    [CrossRef]
  37. C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).
    [CrossRef]
  38. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
    [CrossRef]
  39. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  40. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
    [CrossRef]
  41. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refraction like behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696 (2000).
    [CrossRef]
  42. C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).
  43. B. B. Oner, M. Turduev, I. H. Giden, and H. Kurt, “Efficient mode converter design using asymmetric graded index photonic structures,” Opt. Lett. 38, 220–222 (2013).
    [CrossRef]
  44. S. Y. Huang and S. Y. Wang, “Ray optics of a planar waveguide with an exponential index profile,” J. Appl. Phys. 55, 647–651 (1984).
    [CrossRef]

2013

2012

2011

C. Tan, T. Niemi, C. Peng, and M. Pessa, “Focusing effect of a graded index photonic crystal lens,” Opt. Commun. 284, 3140–3143 (2011).
[CrossRef]

B. Vasić and R. Gajić, “Self-focusing media using graded photonic crystals: focusing, Fourier transforming and imaging, directive emission and directional cloaking,” J. Appl. Phys. 110, 053103 (2011).
[CrossRef]

M. Pruessner, J. Khurgin, T. Stievater, W. Rabinovich, R. Bass, J. Boos, and V. Urick, “Demonstration of a mode-conversion cavity add-drop filter,” Opt. Lett. 36, 2230–2232 (2011).
[CrossRef]

H. W. Wang and L. W. Chen, “High transmission efficiency of arbitrary waveguide bends formed by graded index photonic crystals,” J. Opt. Soc. Am. B 28, 2098–2104 (2011).
[CrossRef]

2010

M. Lu, B. K. Juluri, S.-C. S. Lin, B. Kiraly, T. Gao, and T. J. Huang, “Beam aperture modifier and beam deflector using gradient-index photonic crystals,” J. Appl. Phys. 108, 103505 (2010).
[CrossRef]

F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

B. Vasic, G. Isic, R. Gajic, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18, 20321–20333 (2010).
[CrossRef]

2009

H. T. Chien, C. Lee, H. K. Chiu, K. C. Hsu, C. C. Chen, J. A. Ho, and C. Cho, “The comparison between the graded photonic crystal coupler and various coupler,” J. Lightwave Technol. 27, 2570–2574 (2009).

B. Desiatov, I. Goykhman, and U. Levy, “Nanoscale mode selector in silicon waveguide for on chip nanofocusing applications,” Nano Lett. 9, 3381–3386 (2009).
[CrossRef]

2008

E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, and J.-M. Lourtioz, “Graded photonic crystals curve the flow of light: an experimental demonstration by the mirage effect,” Appl. Phys. Lett. 92, 133501 (2008).
[CrossRef]

2007

T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
[CrossRef]

S. Blaaberg, P. M. Petersen, and B. Tromborg, “Structure, stability, and spectra of lateral modes of a broad-area semiconductor laser,” IEEE J. Quantum Electron. 43, 959–973 (2007).
[CrossRef]

H. Kurt and D. S. Citrin, “A novel optical coupler design with graded-index photonic crystals,” IEEE Photon. Technol. Lett. 19, 1532–1534 (2007).
[CrossRef]

H. Kurt and D. S. Citrin, “Photonic-crystal heterostructure waveguides,” IEEE J. Quantum Electron. 43, 78–84 (2007).
[CrossRef]

J. Castillo, J. Castro, R. Kostuk, and D. Geraghty, “Study of multichannel parallel anti-symmetric waveguide Bragg gratings for telecom applications,” IEEE Photon. Technol. Lett. 19, 85–87 (2007).
[CrossRef]

H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express 15, 1240–1253 (2007).
[CrossRef]

V. Delaubert, M. Lassen, D. Pulford, H. Bachor, and C. Harb, “Spatial mode discrimination using second harmonic generation,” Opt. Express 15, 5815–5826 (2007).
[CrossRef]

2006

M. Yang, H. Chen, K. Webb, S. Minin, S. Chuang, and G. Cueva, “Demonstration of mode conversion in an irregular waveguide,” Opt. Lett. 31, 383–385 (2006).
[CrossRef]

Y. Huang, G. Xu, and S. Ho, “An ultra compact optical mode order converter,” IEEE Photon. Technol. Lett. 18, 2281–2283 (2006).

E. Centeno, D. Cassagne, and J.-P. Albert, “Mirage and superbending effect in two-dimensional graded photonic crystals,” Phys. Rev. B 73, 235119 (2006).
[CrossRef]

2005

M. Yano, F. Yamagishi, and T. Tsuda, “Optical MEMS for photonic switching-compact and stable optical cross connect switches for simple, fast and flexible wavelength applications in recent photonic networks,” IEEE J. Sel. Top. Quantum Electron. 11, 383–394 (2005).
[CrossRef]

J. Castro, D. Geraghty, S. Honkanen, C. Greiner, D. Iazikov, and T. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide Bragg gratings,” Opt. Express 13, 4180–4184 (2005).
[CrossRef]

H. Kurt and D. S. Citrin, “Annular photonic crystals,” Opt. Express 13, 10316–10326 (2005).
[CrossRef]

2004

2003

2001

S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef]

W. D. Zhou, J. Sabarinathan, P. Bhattacharya, B. Kochman, E. Berg, P. C. Yu, and S. Pang, “Characteristics of a photonic bandgap single defect microcavity electroluminescent device,” IEEE J. Quantum Electron. 37, 1153–1160 (2001).
[CrossRef]

2000

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refraction like behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696 (2000).
[CrossRef]

1998

1994

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).
[CrossRef]

1984

S. Y. Huang and S. Y. Wang, “Ray optics of a planar waveguide with an exponential index profile,” J. Appl. Phys. 55, 647–651 (1984).
[CrossRef]

1974

F. Auracher, “Mode order converter,” Opt. Commun. 11, 187–190 (1974).
[CrossRef]

1972

M. Shah, J. D. Craw, and S. Wang, “Optical waveguide mode conversion experiments,” Appl. Phys. Lett. 20, 66–69 (1972).
[CrossRef]

Akmansoy, E.

E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, and J.-M. Lourtioz, “Graded photonic crystals curve the flow of light: an experimental demonstration by the mirage effect,” Appl. Phys. Lett. 92, 133501 (2008).
[CrossRef]

Albert, J.-P.

E. Centeno, D. Cassagne, and J.-P. Albert, “Mirage and superbending effect in two-dimensional graded photonic crystals,” Phys. Rev. B 73, 235119 (2006).
[CrossRef]

Auracher, F.

F. Auracher, “Mode order converter,” Opt. Commun. 11, 187–190 (1974).
[CrossRef]

Bachor, H.

Bao, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

Bass, R.

Berg, E.

W. D. Zhou, J. Sabarinathan, P. Bhattacharya, B. Kochman, E. Berg, P. C. Yu, and S. Pang, “Characteristics of a photonic bandgap single defect microcavity electroluminescent device,” IEEE J. Quantum Electron. 37, 1153–1160 (2001).
[CrossRef]

Bermel, P.

F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Besse, P. A.

Bhattacharya, P.

W. D. Zhou, J. Sabarinathan, P. Bhattacharya, B. Kochman, E. Berg, P. C. Yu, and S. Pang, “Characteristics of a photonic bandgap single defect microcavity electroluminescent device,” IEEE J. Quantum Electron. 37, 1153–1160 (2001).
[CrossRef]

Biermann, K.

Blaaberg, S.

S. Blaaberg, P. M. Petersen, and B. Tromborg, “Structure, stability, and spectra of lateral modes of a broad-area semiconductor laser,” IEEE J. Quantum Electron. 43, 959–973 (2007).
[CrossRef]

Boos, J.

Cassagne, D.

E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, and J.-M. Lourtioz, “Graded photonic crystals curve the flow of light: an experimental demonstration by the mirage effect,” Appl. Phys. Lett. 92, 133501 (2008).
[CrossRef]

E. Centeno, D. Cassagne, and J.-P. Albert, “Mirage and superbending effect in two-dimensional graded photonic crystals,” Phys. Rev. B 73, 235119 (2006).
[CrossRef]

Castillo, J.

J. Castillo, J. Castro, R. Kostuk, and D. Geraghty, “Study of multichannel parallel anti-symmetric waveguide Bragg gratings for telecom applications,” IEEE Photon. Technol. Lett. 19, 85–87 (2007).
[CrossRef]

Castro, J.

J. Castillo, J. Castro, R. Kostuk, and D. Geraghty, “Study of multichannel parallel anti-symmetric waveguide Bragg gratings for telecom applications,” IEEE Photon. Technol. Lett. 19, 85–87 (2007).
[CrossRef]

J. Castro, D. Geraghty, S. Honkanen, C. Greiner, D. Iazikov, and T. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide Bragg gratings,” Opt. Express 13, 4180–4184 (2005).
[CrossRef]

Centeno, E.

E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, and J.-M. Lourtioz, “Graded photonic crystals curve the flow of light: an experimental demonstration by the mirage effect,” Appl. Phys. Lett. 92, 133501 (2008).
[CrossRef]

E. Centeno, D. Cassagne, and J.-P. Albert, “Mirage and superbending effect in two-dimensional graded photonic crystals,” Phys. Rev. B 73, 235119 (2006).
[CrossRef]

Chen, C. C.

Chen, H.

Chen, L. W.

Chien, H. T.

Chiu, H. K.

Cho, C.

Chuang, S.

Citrin, D. S.

H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express 15, 1240–1253 (2007).
[CrossRef]

H. Kurt and D. S. Citrin, “A novel optical coupler design with graded-index photonic crystals,” IEEE Photon. Technol. Lett. 19, 1532–1534 (2007).
[CrossRef]

H. Kurt and D. S. Citrin, “Photonic-crystal heterostructure waveguides,” IEEE J. Quantum Electron. 43, 78–84 (2007).
[CrossRef]

H. Kurt and D. S. Citrin, “Annular photonic crystals,” Opt. Express 13, 10316–10326 (2005).
[CrossRef]

Craw, J. D.

M. Shah, J. D. Craw, and S. Wang, “Optical waveguide mode conversion experiments,” Appl. Phys. Lett. 20, 66–69 (1972).
[CrossRef]

Cueva, G.

Czernecki, R.

T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
[CrossRef]

Delaubert, V.

Desiatov, B.

B. Desiatov, I. Goykhman, and U. Levy, “Nanoscale mode selector in silicon waveguide for on chip nanofocusing applications,” Nano Lett. 9, 3381–3386 (2009).
[CrossRef]

DiGiovanni, D. J.

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).
[CrossRef]

É. Akmansoy,

F. Gaufillet and É. Akmansoy, “Graded photonic crystals for graded index lens,” Opt. Commun. 285, 2638–2641 (2012).
[CrossRef]

Eckner, J.

Fan, S.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

Fejer, M.

Franssen, G.

T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
[CrossRef]

Gajic, R.

B. Vasić and R. Gajić, “Self-focusing media using graded photonic crystals: focusing, Fourier transforming and imaging, directive emission and directional cloaking,” J. Appl. Phys. 110, 053103 (2011).
[CrossRef]

B. Vasic, G. Isic, R. Gajic, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18, 20321–20333 (2010).
[CrossRef]

Gamper, E.

Gao, T.

M. Lu, B. K. Juluri, S.-C. S. Lin, B. Kiraly, T. Gao, and T. J. Huang, “Beam aperture modifier and beam deflector using gradient-index photonic crystals,” J. Appl. Phys. 108, 103505 (2010).
[CrossRef]

Gaufillet, F.

F. Gaufillet and É. Akmansoy, “Graded photonic crystals for graded index lens,” Opt. Commun. 285, 2638–2641 (2012).
[CrossRef]

Geraghty, D.

J. Castillo, J. Castro, R. Kostuk, and D. Geraghty, “Study of multichannel parallel anti-symmetric waveguide Bragg gratings for telecom applications,” IEEE Photon. Technol. Lett. 19, 85–87 (2007).
[CrossRef]

J. Castro, D. Geraghty, S. Honkanen, C. Greiner, D. Iazikov, and T. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide Bragg gratings,” Opt. Express 13, 4180–4184 (2005).
[CrossRef]

Giden, I.

Giden, I. H.

Gomez-Reino, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

Goykhman, I.

B. Desiatov, I. Goykhman, and U. Levy, “Nanoscale mode selector in silicon waveguide for on chip nanofocusing applications,” Nano Lett. 9, 3381–3386 (2009).
[CrossRef]

Grahn, H.

Greiner, C.

Grzegory, I.

T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Harb, C.

Hey, R.

Hingerl, K.

Ho, J. A.

Ho, S.

Y. Huang, G. Xu, and S. Ho, “An ultra compact optical mode order converter,” IEEE Photon. Technol. Lett. 18, 2281–2283 (2006).

Honkanen, S.

Hsu, K. C.

Huang, J.

Huang, S. Y.

S. Y. Huang and S. Y. Wang, “Ray optics of a planar waveguide with an exponential index profile,” J. Appl. Phys. 55, 647–651 (1984).
[CrossRef]

Huang, T. J.

M. Lu, B. K. Juluri, S.-C. S. Lin, B. Kiraly, T. Gao, and T. J. Huang, “Beam aperture modifier and beam deflector using gradient-index photonic crystals,” J. Appl. Phys. 108, 103505 (2010).
[CrossRef]

Huang, Y.

Y. Huang, G. Xu, and S. Ho, “An ultra compact optical mode order converter,” IEEE Photon. Technol. Lett. 18, 2281–2283 (2006).

Iazikov, D.

Ibanescu, M.

F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Isic, G.

Jiang, W.

Joannopoulos, J.

Joannopoulos, J. D.

F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

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

Johnson, S.

Johnson, S. G.

F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

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

Juluri, B. K.

M. Lu, B. K. Juluri, S.-C. S. Lin, B. Kiraly, T. Gao, and T. J. Huang, “Beam aperture modifier and beam deflector using gradient-index photonic crystals,” J. Appl. Phys. 108, 103505 (2010).
[CrossRef]

Khurgin, J.

Kiraly, B.

M. Lu, B. K. Juluri, S.-C. S. Lin, B. Kiraly, T. Gao, and T. J. Huang, “Beam aperture modifier and beam deflector using gradient-index photonic crystals,” J. Appl. Phys. 108, 103505 (2010).
[CrossRef]

Kochman, B.

W. D. Zhou, J. Sabarinathan, P. Bhattacharya, B. Kochman, E. Berg, P. C. Yu, and S. Pang, “Characteristics of a photonic bandgap single defect microcavity electroluminescent device,” IEEE J. Quantum Electron. 37, 1153–1160 (2001).
[CrossRef]

Kostuk, R.

J. Castillo, J. Castro, R. Kostuk, and D. Geraghty, “Study of multichannel parallel anti-symmetric waveguide Bragg gratings for telecom applications,” IEEE Photon. Technol. Lett. 19, 85–87 (2007).
[CrossRef]

Kurt, H.

Kurz, J.

Lassen, M.

Lauterbach, C.

T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
[CrossRef]

Lee, B.

Lee, C.

Leszczynski, M.

T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
[CrossRef]

Leuthold, J.

Levy, U.

B. Desiatov, I. Goykhman, and U. Levy, “Nanoscale mode selector in silicon waveguide for on chip nanofocusing applications,” Nano Lett. 9, 3381–3386 (2009).
[CrossRef]

Lin, S.-C. S.

M. Lu, B. K. Juluri, S.-C. S. Lin, B. Kiraly, T. Gao, and T. J. Huang, “Beam aperture modifier and beam deflector using gradient-index photonic crystals,” J. Appl. Phys. 108, 103505 (2010).
[CrossRef]

Lourtioz, J.-M.

E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, and J.-M. Lourtioz, “Graded photonic crystals curve the flow of light: an experimental demonstration by the mirage effect,” Appl. Phys. Lett. 92, 133501 (2008).
[CrossRef]

Lu, M.

J. Tan, M. Lu, A. Stein, and W. Jiang, “High-purity transmission of a slow light odd mode in a photonic crystal waveguide,” Opt. Lett. 37, 3189–3191 (2012).
[CrossRef]

M. Lu, B. K. Juluri, S.-C. S. Lin, B. Kiraly, T. Gao, and T. J. Huang, “Beam aperture modifier and beam deflector using gradient-index photonic crystals,” J. Appl. Phys. 108, 103505 (2010).
[CrossRef]

Lü, X.

Meade, R. D.

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

Melchior, H.

Minin, S.

Mossberg, T.

Niemi, T.

C. Tan, T. Niemi, C. Peng, and M. Pessa, “Focusing effect of a graded index photonic crystal lens,” Opt. Commun. 284, 3140–3143 (2011).
[CrossRef]

Notomi, M.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refraction like behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696 (2000).
[CrossRef]

Oner, B. B.

Oskooi, F.

F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Pang, S.

W. D. Zhou, J. Sabarinathan, P. Bhattacharya, B. Kochman, E. Berg, P. C. Yu, and S. Pang, “Characteristics of a photonic bandgap single defect microcavity electroluminescent device,” IEEE J. Quantum Electron. 37, 1153–1160 (2001).
[CrossRef]

Peng, C.

C. Tan, T. Niemi, C. Peng, and M. Pessa, “Focusing effect of a graded index photonic crystal lens,” Opt. Commun. 284, 3140–3143 (2011).
[CrossRef]

Perez, M. V.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

Perlin, P.

T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
[CrossRef]

Pessa, M.

C. Tan, T. Niemi, C. Peng, and M. Pessa, “Focusing effect of a graded index photonic crystal lens,” Opt. Commun. 284, 3140–3143 (2011).
[CrossRef]

Petersen, P. M.

S. Blaaberg, P. M. Petersen, and B. Tromborg, “Structure, stability, and spectra of lateral modes of a broad-area semiconductor laser,” IEEE J. Quantum Electron. 43, 959–973 (2007).
[CrossRef]

Poole, C. D.

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).
[CrossRef]

Pruessner, M.

Pulford, D.

Rabinovich, W.

Roundy, D.

F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Sabarinathan, J.

W. D. Zhou, J. Sabarinathan, P. Bhattacharya, B. Kochman, E. Berg, P. C. Yu, and S. Pang, “Characteristics of a photonic bandgap single defect microcavity electroluminescent device,” IEEE J. Quantum Electron. 37, 1153–1160 (2001).
[CrossRef]

Saida, T.

Schrottke, L.

Schwarz, U. T.

T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
[CrossRef]

Shah, M.

M. Shah, J. D. Craw, and S. Wang, “Optical waveguide mode conversion experiments,” Appl. Phys. Lett. 20, 66–69 (1972).
[CrossRef]

Sharma, R.

Shin, S.

Skierbiszewski, C.

T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
[CrossRef]

Stein, A.

Stievater, T.

Suski, T.

T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
[CrossRef]

Swietlik, T.

T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Tahraoui, A.

Tan, C.

C. Tan, T. Niemi, C. Peng, and M. Pessa, “Focusing effect of a graded index photonic crystal lens,” Opt. Commun. 284, 3140–3143 (2011).
[CrossRef]

Tan, J.

Tromborg, B.

S. Blaaberg, P. M. Petersen, and B. Tromborg, “Structure, stability, and spectra of lateral modes of a broad-area semiconductor laser,” IEEE J. Quantum Electron. 43, 959–973 (2007).
[CrossRef]

Tsuda, T.

M. Yano, F. Yamagishi, and T. Tsuda, “Optical MEMS for photonic switching-compact and stable optical cross connect switches for simple, fast and flexible wavelength applications in recent photonic networks,” IEEE J. Sel. Top. Quantum Electron. 11, 383–394 (2005).
[CrossRef]

Turduev, M.

Urick, V.

Vasic, B.

B. Vasić and R. Gajić, “Self-focusing media using graded photonic crystals: focusing, Fourier transforming and imaging, directive emission and directional cloaking,” J. Appl. Phys. 110, 053103 (2011).
[CrossRef]

B. Vasic, G. Isic, R. Gajic, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18, 20321–20333 (2010).
[CrossRef]

Vengsarkar, A. M.

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).
[CrossRef]

Villeneuve, P. R.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

Vynck, K.

E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, and J.-M. Lourtioz, “Graded photonic crystals curve the flow of light: an experimental demonstration by the mirage effect,” Appl. Phys. Lett. 92, 133501 (2008).
[CrossRef]

Wang, H. W.

Wang, S.

M. Shah, J. D. Craw, and S. Wang, “Optical waveguide mode conversion experiments,” Appl. Phys. Lett. 20, 66–69 (1972).
[CrossRef]

Wang, S. Y.

S. Y. Huang and S. Y. Wang, “Ray optics of a planar waveguide with an exponential index profile,” J. Appl. Phys. 55, 647–651 (1984).
[CrossRef]

Webb, K.

Wienold, M.

Wiesenfeld, J. M.

C. D. Poole, J. M. Wiesenfeld, D. J. DiGiovanni, and A. M. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).
[CrossRef]

Winn, J. N.

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

Xie, X.

Xu, G.

Y. Huang, G. Xu, and S. Ho, “An ultra compact optical mode order converter,” IEEE Photon. Technol. Lett. 18, 2281–2283 (2006).

Yamagishi, F.

M. Yano, F. Yamagishi, and T. Tsuda, “Optical MEMS for photonic switching-compact and stable optical cross connect switches for simple, fast and flexible wavelength applications in recent photonic networks,” IEEE J. Sel. Top. Quantum Electron. 11, 383–394 (2005).
[CrossRef]

Yang, M.

Yano, M.

M. Yano, F. Yamagishi, and T. Tsuda, “Optical MEMS for photonic switching-compact and stable optical cross connect switches for simple, fast and flexible wavelength applications in recent photonic networks,” IEEE J. Sel. Top. Quantum Electron. 11, 383–394 (2005).
[CrossRef]

Yu, P. C.

W. D. Zhou, J. Sabarinathan, P. Bhattacharya, B. Kochman, E. Berg, P. C. Yu, and S. Pang, “Characteristics of a photonic bandgap single defect microcavity electroluminescent device,” IEEE J. Quantum Electron. 37, 1153–1160 (2001).
[CrossRef]

Zhou, W. D.

W. D. Zhou, J. Sabarinathan, P. Bhattacharya, B. Kochman, E. Berg, P. C. Yu, and S. Pang, “Characteristics of a photonic bandgap single defect microcavity electroluminescent device,” IEEE J. Quantum Electron. 37, 1153–1160 (2001).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

M. Shah, J. D. Craw, and S. Wang, “Optical waveguide mode conversion experiments,” Appl. Phys. Lett. 20, 66–69 (1972).
[CrossRef]

E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, and J.-M. Lourtioz, “Graded photonic crystals curve the flow of light: an experimental demonstration by the mirage effect,” Appl. Phys. Lett. 92, 133501 (2008).
[CrossRef]

Comput. Phys. Commun.

F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

IEEE J. Quantum Electron.

S. Blaaberg, P. M. Petersen, and B. Tromborg, “Structure, stability, and spectra of lateral modes of a broad-area semiconductor laser,” IEEE J. Quantum Electron. 43, 959–973 (2007).
[CrossRef]

H. Kurt and D. S. Citrin, “Photonic-crystal heterostructure waveguides,” IEEE J. Quantum Electron. 43, 78–84 (2007).
[CrossRef]

W. D. Zhou, J. Sabarinathan, P. Bhattacharya, B. Kochman, E. Berg, P. C. Yu, and S. Pang, “Characteristics of a photonic bandgap single defect microcavity electroluminescent device,” IEEE J. Quantum Electron. 37, 1153–1160 (2001).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

M. Yano, F. Yamagishi, and T. Tsuda, “Optical MEMS for photonic switching-compact and stable optical cross connect switches for simple, fast and flexible wavelength applications in recent photonic networks,” IEEE J. Sel. Top. Quantum Electron. 11, 383–394 (2005).
[CrossRef]

IEEE Photon. Technol. Lett.

Y. Huang, G. Xu, and S. Ho, “An ultra compact optical mode order converter,” IEEE Photon. Technol. Lett. 18, 2281–2283 (2006).

J. Castillo, J. Castro, R. Kostuk, and D. Geraghty, “Study of multichannel parallel anti-symmetric waveguide Bragg gratings for telecom applications,” IEEE Photon. Technol. Lett. 19, 85–87 (2007).
[CrossRef]

H. Kurt and D. S. Citrin, “A novel optical coupler design with graded-index photonic crystals,” IEEE Photon. Technol. Lett. 19, 1532–1534 (2007).
[CrossRef]

J. Appl. Phys.

T. Swietlik, G. Franssen, R. Czernecki, M. Leszczynski, C. Skierbiszewski, I. Grzegory, T. Suski, P. Perlin, C. Lauterbach, and U. T. Schwarz, “Mode dynamics of high power (InAl)GaN based laser diodes grown on bulk GaN substrate,” J. Appl. Phys. 101, 083109 (2007).
[CrossRef]

M. Lu, B. K. Juluri, S.-C. S. Lin, B. Kiraly, T. Gao, and T. J. Huang, “Beam aperture modifier and beam deflector using gradient-index photonic crystals,” J. Appl. Phys. 108, 103505 (2010).
[CrossRef]

B. Vasić and R. Gajić, “Self-focusing media using graded photonic crystals: focusing, Fourier transforming and imaging, directive emission and directional cloaking,” J. Appl. Phys. 110, 053103 (2011).
[CrossRef]

S. Y. Huang and S. Y. Wang, “Ray optics of a planar waveguide with an exponential index profile,” J. Appl. Phys. 55, 647–651 (1984).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Nano Lett.

B. Desiatov, I. Goykhman, and U. Levy, “Nanoscale mode selector in silicon waveguide for on chip nanofocusing applications,” Nano Lett. 9, 3381–3386 (2009).
[CrossRef]

Opt. Commun.

F. Auracher, “Mode order converter,” Opt. Commun. 11, 187–190 (1974).
[CrossRef]

F. Gaufillet and É. Akmansoy, “Graded photonic crystals for graded index lens,” Opt. Commun. 285, 2638–2641 (2012).
[CrossRef]

C. Tan, T. Niemi, C. Peng, and M. Pessa, “Focusing effect of a graded index photonic crystal lens,” Opt. Commun. 284, 3140–3143 (2011).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refraction like behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696 (2000).
[CrossRef]

E. Centeno, D. Cassagne, and J.-P. Albert, “Mirage and superbending effect in two-dimensional graded photonic crystals,” Phys. Rev. B 73, 235119 (2006).
[CrossRef]

Other

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

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Supplementary Material (4)

» Media 1: MPG (2228 KB)     
» Media 2: MPG (2666 KB)     
» Media 3: MPG (2126 KB)     
» Media 4: MPG (2114 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

Schematic representation of mode converting structure with corresponding refractive index variation plots. In homogeneous and asymmetric GRIN media are shown on the left and the transverse profiles of the exponential and Luneburg lens refractive index profiles are presented on the right.

Fig. 2.
Fig. 2.

(a) Dispersion curves corresponding to the first band are shown. (b) Group index (ng) dependency of each dispersion bands is shown. The index variation at a/λ=0.10 is also given in the same plot as an inset.

Fig. 3.
Fig. 3.

Schematic view of proposed GRIN PC structures where (a) and (b) demonstrate GRIN structure with the Luneburg lens index profile and its index distribution plots [continuous and discrete versions (stair-step plots)], respectively. Similarly, (c) the exponential index profile GRIN PC structure is represented and (d) the related index variation plot is shown.

Fig. 4.
Fig. 4.

Calculated transmission efficiencies of A-GRIN PC structures with exponential and Luneburg lens index distributions.

Fig. 5.
Fig. 5.

Plots (a) and (b) demonstrate the collections of EFCs of different lateral sizes (rectangular cells are depicted as an inset) of rectangular PC cells. The corresponding Δy ranges from 0.44a to 1.26a and vice versa regarding the OA (upper and lower part of the structure), respectively. The arrows, which are normal to corresponding EFCs in the same plots, represent light flow direction. (c) The corresponding light propagation directions within each cell are superimposed with the help of two different types of arrows (solid and dashed). The medium has an exponential index profile.

Fig. 6.
Fig. 6.

(a) Ray propagation in a GRIN medium with n(y) index profile and (b) ray paths and regarding geometrical wavefronts in an asymmetric GRIN structure with an exponential index profile.

Fig. 7.
Fig. 7.

Instantaneous electric fields of (a) an incident even mode source without structure and converted even to odd mode source utilizing asymmetric GRIN PC structures with (b) an exponential (Media 1) and (c) Luneburg lens (Media 2) index profiles. In plots the OA indicates the optical axis and the signs “+” and “−” represent the mismatch in the phase fronts of the decomposed propagating beam. The radial phase profiles of (d) an ideal incident even mode source and configurations with (e) exponential and (f) Luneburg lens profiles. Phase profiles are extracted over predefined radial sections.

Fig. 8.
Fig. 8.

Instantaneous electric fields of an (a) ideal even mode source without any structure, (b) an exponential index A-GRIN PC (Media 3), and (c) Luneburg lens index A-GRIN PC (Media 4); (d), (e), and (f), respectively, present the corresponding phase profiles of the three cases.

Fig. 9.
Fig. 9.

Ray propagation in a GRIN medium with n(y) index profile.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

nL-lens(y)={bδ(y/ymax)2y0bδ((y+ymax)/ymax)2y<0,
nexp(y)={n0exp(αy)y0n0exp(α(y+ymax))y<0,
dds[ndrds]=n
|∇⃗S|2=n2,
y(x)=1αln(u0αcos(αx)+u˙0sin(αx)Aα),
y˙(x)=(u0αsin(αx)+u˙0cos(αx)u0αcos(αx)+u˙0sin(αx)),
OPL=x0xLdx=1I0x0xn(s)2ds,
Δφ=kn(y)ds,
nexp(y)=n0exp(αy)y0.
ds=(dx)2+(dy)2;dxds=11+(y˙)2.
I0=n(y)dxds=n(y)1+(y˙)2=const.
x=y0yI0n(y)2I02dy=y0yI0n(y)1(I0n(y))2dy.
u=I0n(y)=I0n0eαy=Aeαy;A=I0n0.
x=y0yAeαy1(Aeαy)2dy=|y(x)u(x)|=1αu0u11u2du=1α(sin1(u)sin1(u0)),
αx+sin1(u0)=sin1(u).
sin(τ)=u(x).
u˙(x)=cos(τ)τ˙(x)=cos(τ)(αx+sin1(u0))=cos(τ)α.
u˙(x)=αcos(αx+sin1(u0))=α(cos(αx)cos(sin1(u0))u0sin(αx)).
u(x)=u˙(x)dx=cos(sin1(u0))sin(αx)+u0cos(αx).
τ0=τ(x=0)=sin1(u0);u˙0=αcos(τ0);cos(τ0)=u˙0α.
u(x)=u˙0αsin(αx)+u0cos(αx),
u˙(x)=u˙0cos(αx)αu0sin(αx).
(u(x)u˙(x))=(u0cos(αx)+u˙0αsin(αx)αu0sin(αx)+u˙0cos(αx))=(HfHaH˙fH˙a)(u0u˙0),
Hf=cos(αx);Ha=sin(αx)α;H˙f=αsin(αx);H˙a=cos(αx).
y(x)=1αln(u(x)A)=1αln(u˙0sin(αx)+u0αcos(αx)Aα),
y˙(x)=u˙(x)αu(x)=1αH˙fu0+H˙au˙0Hfu0+Hau˙0=(αu0sin(αx)+u˙0cos(αx)αu0cos(αx)+u˙0sin(αx)).

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