Abstract

We propose the use of two-dimensional (2D) photonic crystals (PhCs) with engineered defects for the generation of an arbitrary-profile beam from a focused input beam. The cylindrical harmonics expansion of complex-source beams is derived and used to compute the scattered wave function of a 2D PhC via the multiple scattering method. The beam shaping problem is then solved using a genetic algorithm. We illustrate our procedure by generating different orders of Hermite–Gauss profiles, while maintaining reasonable losses and tolerance to variations in the input beam and the slab refractive index.

© 2012 Optical Society of America

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References

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  1. F. Dickey and S. Holswade, Laser Beam Shaping Applications (Taylor & Francis, 2005), Chap. 8.
  2. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
    [CrossRef]
  3. M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
    [CrossRef]
  4. N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012).
    [CrossRef]
  5. B. R. Brown and A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. 5, 967–969 (1966).
    [CrossRef]
  6. R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991).
    [CrossRef]
  7. H. Laabs and B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
    [CrossRef]
  8. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
    [CrossRef]
  9. O. Rasoga and D. Dragoman, “Engineered beam shaping effect in anisotropic photonic crystals,” Appl. Opt. 49, 2161–2167 (2010).
    [CrossRef]
  10. A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two-dimensional nonlinear beam shaping,” Opt. Lett. 37, 2136–2138 (2012).
    [CrossRef]
  11. H. Kurt, “Limited-diffraction light propagation with axicon-shape photonic crystals,” J. Opt. Soc. Am. B 26, 981–986 (2009).
    [CrossRef]
  12. H. Kurt and M. Turduev, “Generation of a two-dimensional limited-diffraction beam with self-healing ability by annular-type photonic crystals,” J. Opt. Soc. Am. B 29, 1245–1256(2012).
    [CrossRef]
  13. D. P. San-Roman-Alerigi, T. K. Ng, Y. Zhang, A. Ben Slimane, M. Alsunaidi, and B. S. Ooi, “Generation of J0-Bessel–Gauss beam by a heterogeneous refractive index map,” J. Opt. Soc. Am. A 29, 1252–1258 (2012).
    [CrossRef]
  14. L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultralow-loss 3 dB photonic crystal waveguide splitter,” Opt. Lett. 29, 1623–1625 (2004).
    [CrossRef]
  15. P. Pottier, S. Mastroiacovo, and R. M. De La Rue, “Power and polarization beam-splitters, mirrors, and integrated interferometers based on air-hole photonic crystals and lateral large index-contrast waveguides,” Opt. Express 14, 5617–5633(2006).
    [CrossRef]
  16. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
    [CrossRef]
  17. R. M. De La Rue and C. Seassal, “Photonic crystal devices: some basics and selected topics,” Laser Photon. Rev. 6, 564–597 (2012).
    [CrossRef]
  18. A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag. 40, 96–99 (1992).
    [CrossRef]
  19. S. Nojima, “Theoretical analysis of feedback mechanisms of two-dimensional finite-sized photonic-crystal lasers,” J. Appl. Phys. 98, 043102 (2005).
    [CrossRef]
  20. A. Vukovic, P. Sewell, and T. M. Benson, “Strategies for global optimization in photonics design,” J. Opt. Soc. Am. A 27, 2156–2168 (2010).
    [CrossRef]
  21. N. C. Evans and D. L. Shealy, “Design and optimization of an irradiance profile-shaping system with a genetic algorithm method,” Appl. Opt. 37, 5216–5221 (1998).
    [CrossRef]
  22. S. N. Sivanandam and S. N. Deepa, Introduction to Genetic Algorithms (Springer-Verlag, 2008).
  23. M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University, 2009).
  24. L. Felsen, “Complex rays,” Philips Res. Rep. 30, 187–195 (1975).
  25. E. Heyman and L. B. Felsen, “Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A 18, 1588–1611 (2001).
    [CrossRef]
  26. R. Mahillo-Isla, M. Gonzalez-Morales, and C. Dehesa-Martinez, “Regularization of complex beams,” in Proceedings of the 12th International Conference on Mathematical Methods in Electromagnetic Theory (IEEE, 2008), pp. 242–244.
  27. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).
  28. T. Oguzer, A. Altintas, and A. I. Nosich, “Accurate simulation of reflector antennas by the complex source-dual series approach,” IEEE Trans. Antennas Propag. 43, 793–801 (1995).
    [CrossRef]
  29. S. G. Johnson, and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
    [CrossRef]
  30. P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimization of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
    [CrossRef]
  31. M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002).
    [CrossRef]

2012 (6)

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[CrossRef]

N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012).
[CrossRef]

R. M. De La Rue and C. Seassal, “Photonic crystal devices: some basics and selected topics,” Laser Photon. Rev. 6, 564–597 (2012).
[CrossRef]

H. Kurt and M. Turduev, “Generation of a two-dimensional limited-diffraction beam with self-healing ability by annular-type photonic crystals,” J. Opt. Soc. Am. B 29, 1245–1256(2012).
[CrossRef]

A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two-dimensional nonlinear beam shaping,” Opt. Lett. 37, 2136–2138 (2012).
[CrossRef]

D. P. San-Roman-Alerigi, T. K. Ng, Y. Zhang, A. Ben Slimane, M. Alsunaidi, and B. S. Ooi, “Generation of J0-Bessel–Gauss beam by a heterogeneous refractive index map,” J. Opt. Soc. Am. A 29, 1252–1258 (2012).
[CrossRef]

2010 (2)

2009 (1)

2008 (1)

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

2006 (1)

2005 (2)

S. Nojima, “Theoretical analysis of feedback mechanisms of two-dimensional finite-sized photonic-crystal lasers,” J. Appl. Phys. 98, 043102 (2005).
[CrossRef]

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimization of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

2004 (1)

2003 (1)

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

2002 (1)

M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002).
[CrossRef]

2001 (2)

1998 (1)

1996 (1)

H. Laabs and B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[CrossRef]

1995 (1)

T. Oguzer, A. Altintas, and A. I. Nosich, “Accurate simulation of reflector antennas by the complex source-dual series approach,” IEEE Trans. Antennas Propag. 43, 793–801 (1995).
[CrossRef]

1992 (1)

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag. 40, 96–99 (1992).
[CrossRef]

1991 (1)

1975 (1)

L. Felsen, “Complex rays,” Philips Res. Rep. 30, 187–195 (1975).

1966 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Alsunaidi, M.

Altintas, A.

T. Oguzer, A. Altintas, and A. I. Nosich, “Accurate simulation of reflector antennas by the complex source-dual series approach,” IEEE Trans. Antennas Propag. 43, 793–801 (1995).
[CrossRef]

Andreasen, J.

N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012).
[CrossRef]

Arie, A.

Arnold, C. B.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[CrossRef]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

Bachelard, N.

N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012).
[CrossRef]

Baets, R.

Beckx, S.

Benson, T. M.

Bogaerts, W.

Borel, P. I.

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimization of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultralow-loss 3 dB photonic crystal waveguide splitter,” Opt. Lett. 29, 1623–1625 (2004).
[CrossRef]

Brown, B. R.

De La Rue, R. M.

Deepa, S. N.

S. N. Sivanandam and S. N. Deepa, Introduction to Genetic Algorithms (Springer-Verlag, 2008).

Dehesa-Martinez, C.

R. Mahillo-Isla, M. Gonzalez-Morales, and C. Dehesa-Martinez, “Regularization of complex beams,” in Proceedings of the 12th International Conference on Mathematical Methods in Electromagnetic Theory (IEEE, 2008), pp. 242–244.

Dickey, F.

F. Dickey and S. Holswade, Laser Beam Shaping Applications (Taylor & Francis, 2005), Chap. 8.

Dragoman, D.

Dumon, P.

Duocastella, M.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[CrossRef]

Elsherbeni, A. Z.

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag. 40, 96–99 (1992).
[CrossRef]

Evans, N. C.

Felsen, L.

L. Felsen, “Complex rays,” Philips Res. Rep. 30, 187–195 (1975).

Felsen, L. B.

Frandsen, L. H.

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimization of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultralow-loss 3 dB photonic crystal waveguide splitter,” Opt. Lett. 29, 1623–1625 (2004).
[CrossRef]

Gigan, S.

N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012).
[CrossRef]

Gonzalez-Morales, M.

R. Mahillo-Isla, M. Gonzalez-Morales, and C. Dehesa-Martinez, “Regularization of complex beams,” in Proceedings of the 12th International Conference on Mathematical Methods in Electromagnetic Theory (IEEE, 2008), pp. 242–244.

Harpøth, A.

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimization of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultralow-loss 3 dB photonic crystal waveguide splitter,” Opt. Lett. 29, 1623–1625 (2004).
[CrossRef]

Herman, R. M.

Heyman, E.

Holswade, S.

F. Dickey and S. Holswade, Laser Beam Shaping Applications (Taylor & Francis, 2005), Chap. 8.

Joannopoulos, J. D.

Johnson, S. G.

Juwiler, I.

Kishk, A. A.

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag. 40, 96–99 (1992).
[CrossRef]

Kivshar, Y. S.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

Kristensen, M.

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimization of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultralow-loss 3 dB photonic crystal waveguide splitter,” Opt. Lett. 29, 1623–1625 (2004).
[CrossRef]

Kurt, H.

Laabs, H.

H. Laabs and B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[CrossRef]

Lohmann, A. W.

Mahillo-Isla, R.

R. Mahillo-Isla, M. Gonzalez-Morales, and C. Dehesa-Martinez, “Regularization of complex beams,” in Proceedings of the 12th International Conference on Mathematical Methods in Electromagnetic Theory (IEEE, 2008), pp. 242–244.

Mastroiacovo, S.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Ng, T. K.

Nojima, S.

S. Nojima, “Theoretical analysis of feedback mechanisms of two-dimensional finite-sized photonic-crystal lasers,” J. Appl. Phys. 98, 043102 (2005).
[CrossRef]

Nosich, A. I.

T. Oguzer, A. Altintas, and A. I. Nosich, “Accurate simulation of reflector antennas by the complex source-dual series approach,” IEEE Trans. Antennas Propag. 43, 793–801 (1995).
[CrossRef]

Oguzer, T.

T. Oguzer, A. Altintas, and A. I. Nosich, “Accurate simulation of reflector antennas by the complex source-dual series approach,” IEEE Trans. Antennas Propag. 43, 793–801 (1995).
[CrossRef]

Ooi, B. S.

Ozygus, B.

H. Laabs and B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[CrossRef]

Pottier, P.

Qiu, M.

M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002).
[CrossRef]

Rasoga, O.

San-Roman-Alerigi, D. P.

Seassal, C.

R. M. De La Rue and C. Seassal, “Photonic crystal devices: some basics and selected topics,” Laser Photon. Rev. 6, 564–597 (2012).
[CrossRef]

Sebbah, P.

N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012).
[CrossRef]

Sewell, P.

Shadrivov, I. V.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

Shapira, A.

Shealy, D. L.

Shiloh, R.

Sivanandam, S. N.

S. N. Sivanandam and S. N. Deepa, Introduction to Genetic Algorithms (Springer-Verlag, 2008).

Skorobogatiy, M.

M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University, 2009).

Slimane, A. Ben

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

Thorhauge, M.

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Turduev, M.

Vukovic, A.

Wiaux, V.

Wiggins, T. A.

Wouters, J.

Xing, P. F.

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimization of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

Yang, J.

M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University, 2009).

Zhang, Y.

Zhuang, Y. X.

Appl. Opt. (3)

Appl. Phys. Lett. (2)

M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

A. Z. Elsherbeni and A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,” IEEE Trans. Antennas Propag. 40, 96–99 (1992).
[CrossRef]

T. Oguzer, A. Altintas, and A. I. Nosich, “Accurate simulation of reflector antennas by the complex source-dual series approach,” IEEE Trans. Antennas Propag. 43, 793–801 (1995).
[CrossRef]

J. Appl. Phys. (1)

S. Nojima, “Theoretical analysis of feedback mechanisms of two-dimensional finite-sized photonic-crystal lasers,” J. Appl. Phys. 98, 043102 (2005).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (2)

Laser Photon. Rev. (2)

R. M. De La Rue and C. Seassal, “Photonic crystal devices: some basics and selected topics,” Laser Photon. Rev. 6, 564–597 (2012).
[CrossRef]

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[CrossRef]

Nat. Photonics (1)

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[CrossRef]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Opt. Commun. (1)

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimization of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (1)

H. Laabs and B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[CrossRef]

Opt. Lett. (2)

Philips Res. Rep. (1)

L. Felsen, “Complex rays,” Philips Res. Rep. 30, 187–195 (1975).

Phys. Rev. Lett. (1)

N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012).
[CrossRef]

Other (5)

F. Dickey and S. Holswade, Laser Beam Shaping Applications (Taylor & Francis, 2005), Chap. 8.

R. Mahillo-Isla, M. Gonzalez-Morales, and C. Dehesa-Martinez, “Regularization of complex beams,” in Proceedings of the 12th International Conference on Mathematical Methods in Electromagnetic Theory (IEEE, 2008), pp. 242–244.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

S. N. Sivanandam and S. N. Deepa, Introduction to Genetic Algorithms (Springer-Verlag, 2008).

M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University, 2009).

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Figures (6)

Fig. 1.
Fig. 1.

Basic photonic lattice configuration (Ns=104). To generate a desired beam profile, defects can be present or absent. We impose a vertical mirror symmetry, resulting in 256 possible configurations. The dotted line indicates the plane used for the computation of the desired beam profile.

Fig. 2.
Fig. 2.

Band structure for a square lattice of air holes of diameter D=0.6Λ in a dielectric medium with refractive index 2.76. The location of the partial bandgap is shaded. Eigenmodes were computed using the MIT Photonic-Bands software package [29].

Fig. 3.
Fig. 3.

Convergence of the standard GA used to find the configuration shown in Fig. 4. The fitness value reached is 1/I47.6.

Fig. 4.
Fig. 4.

Generation of order 1 Hermite–Gauss beam. (a) Optimized configuration and field profile (Ns=41). The target plane is indicated by a dashed line. (b) Comparison of computed irradiance along target plane and desired profile (arbitrary units). This design is characterized by I=0.021, η=0.705.

Fig. 5.
Fig. 5.

Generation of order 2 Hermite–Gauss beam. (a) Optimized configuration and field profile (Ns=28). The target plane is indicated by a dashed line. (b) Comparison of computed irradiance along target plane and desired profile (arbitrary units). This design is characterized by I=0.044, η=0.785.

Fig. 6.
Fig. 6.

Tolerance of PhC lattice configurations to (a) variations of the Rayleigh distance of the input beam and (b) group refractive index of the slab. The design values are indicated by a dotted line.

Equations (17)

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

[2+k2(x,y)]u(x,y)=0,
ui(x,y)=H0(1)(krs),
rs[(yy)2+(xx)2]1/2=[y2+(xixR)2]1/2.
ui(x,y)ug(x,y)exp(kxR+iπ/4),
ug(x,y)=2πk(xixR)exp{ik(x+12y2xixR)}.
ui(ρn,θn)=l=anl0Jl(kρn)eilθn.
ui(ρn,θn)=H0(1)(k|rnrsn|),
anl0=(1)lHl(krsn)eilμ,
rsn=(XnixR)2+Yn2,
cosμ=XnixRrsn.
anl0=ileikXn.
us(x,y)=nlbnlHl(1)(kρn)eilθn.
Tnnll=δnnδll(1δnn)ei(ll)ϕnnHll(1)(kRnn)snl,
I(x0)=||u(x0,y)|2|u¯(x0,y)|2|dy|u¯(x0,y)|2dy,
|u¯m(x0,y)|2=[Hm(ξ)]2exp(ξ2),
H1(ξ)=2ξ,H2(ξ)=4ξ22,
η=Sx(x0,y)dySx(xin,y)dy,

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