Abstract

It is often desired to functionally grade and/or spatially vary a periodic structure like a photonic crystal or metamaterial, yet no general method for doing this has been offered in the literature. A straightforward procedure is described here that allows many properties of the lattice to be spatially varied at the same time while producing a final lattice that is still smooth and continuous. Properties include unit cell orientation, lattice spacing, fill fraction, and more. This adds many degrees of freedom to a design such as spatially varying the orientation to exploit directional phenomena. The method is not a coordinate transformation technique so it can more easily produce complicated and arbitrary spatial variance. To demonstrate, the algorithm is used to synthesize a spatially variant self-collimating photonic crystal to flow a Gaussian beam around a 90° bend. The performance of the structure was confirmed through simulation and it showed virtually no scattering around the bend that would have arisen if the lattice had defects or discontinuities.

© 2012 OSA

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, Molding the Flow of Light (Princeton University Press, 1995).
  2. H. Benistry, V. Berger, J.-M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals, Towards Nanoscale Photonic Devices (Springer, 2005).
  3. S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Metamaterials, (CRC Press, 2009).
  4. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
    [CrossRef] [PubMed]
  5. D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express18(3), 3181–3186 (2010).
    [CrossRef] [PubMed]
  6. E. G. Johnson, M. K. Poutous, Z. A. Roth, P. Srinivasan, A. J. Pung, and Y. O. Yilmaz, “Advanced fabrication methods for 3D meta-optics,” Proc. SPIE7927, 792706, 792706-7 (2011).
    [CrossRef]
  7. Z. A. Roth, P. Srinivasan, M. K. Poutous, A. Pung, R. C. Rumpf, and E. G. Johnson, “Azimuthally varying guided mode resonance filters,” Micromachines3(1), 180–193 (2012).
    [CrossRef]
  8. 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(13), 133501 (2008).
    [CrossRef]
  9. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express18(19), 20321–20333 (2010).
    [CrossRef] [PubMed]
  10. R. C. Rumpf, “Design and optimization of nano-optical elements by coupling fabrication to optical behavior,” Ph.D. Dissertation, University of Central Florida (2006), pp. 171–183.
  11. L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett.27(11), 900–902 (2002).
    [CrossRef] [PubMed]
  12. S. C. Chapra and R. P. Canale, Numerical Methods for Engineers with Software and Programming Applications, 4th Ed., 820–856 (McGraw-Hill, 2002).
  13. B. Noble and J. W. Daniel, Applied Linear Algebra, 3rd ed. (Prentice Hall, 1988), pp. 66–73.
  14. Y. Sadd, Iterative Methods for Sparse Linear Systems, 2nd ed. (Yousef Sadd, 2000).
  15. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett.74(9), 1212–1214 (1999).
    [CrossRef]
  16. R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express13(18), 7076–7085 (2005).
    [CrossRef] [PubMed]
  17. J. Shin and S. Fan, “Conditions for self-collimation in three-dimensional photonic crystals,” Opt. Lett.30(18), 2397–2399 (2005).
    [CrossRef] [PubMed]
  18. Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett.96(17), 173902 (2006).
    [CrossRef] [PubMed]
  19. M. Born and E. Wolf, Principles of Optics, 6th Ed., 673–678 (Cambridge University Press, 1980).
  20. R. C. Rumpf, “Design and optimization of nano-optical elements by coupling fabrication to optical behavior,” Ph.D. Dissertation, University of Central Florida (2006), pp. 109–124.
  21. A. Taflove and S. C. Hagness, Computational Electrodynamics, the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  22. Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A Perfectly Matched Anisotropic Absorber for Use as an Absorbing Boundary Condition,” IEEE Trans. Antenn. Propag.43(12), 1460–1463 (1995).
    [CrossRef]

2012 (1)

Z. A. Roth, P. Srinivasan, M. K. Poutous, A. Pung, R. C. Rumpf, and E. G. Johnson, “Azimuthally varying guided mode resonance filters,” Micromachines3(1), 180–193 (2012).
[CrossRef]

2011 (1)

E. G. Johnson, M. K. Poutous, Z. A. Roth, P. Srinivasan, A. J. Pung, and Y. O. Yilmaz, “Advanced fabrication methods for 3D meta-optics,” Proc. SPIE7927, 792706, 792706-7 (2011).
[CrossRef]

2010 (2)

D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express18(3), 3181–3186 (2010).
[CrossRef] [PubMed]

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express18(19), 20321–20333 (2010).
[CrossRef] [PubMed]

2008 (1)

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(13), 133501 (2008).
[CrossRef]

2006 (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett.96(17), 173902 (2006).
[CrossRef] [PubMed]

2005 (2)

R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express13(18), 7076–7085 (2005).
[CrossRef] [PubMed]

J. Shin and S. Fan, “Conditions for self-collimation in three-dimensional photonic crystals,” Opt. Lett.30(18), 2397–2399 (2005).
[CrossRef] [PubMed]

2002 (1)

L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett.27(11), 900–902 (2002).
[CrossRef] [PubMed]

1999 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett.74(9), 1212–1214 (1999).
[CrossRef]

1995 (1)

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A Perfectly Matched Anisotropic Absorber for Use as an Absorbing Boundary Condition,” IEEE Trans. Antenn. Propag.43(12), 1460–1463 (1995).
[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(13), 133501 (2008).
[CrossRef]

Cai, L. Z.

L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett.27(11), 900–902 (2002).
[CrossRef] [PubMed]

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(13), 133501 (2008).
[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(13), 133501 (2008).
[CrossRef]

Etrich, C.

R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express13(18), 7076–7085 (2005).
[CrossRef] [PubMed]

Fan, S.

J. Shin and S. Fan, “Conditions for self-collimation in three-dimensional photonic crystals,” Opt. Lett.30(18), 2397–2399 (2005).
[CrossRef] [PubMed]

Gabrielli, L. H.

D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express18(3), 3181–3186 (2010).
[CrossRef] [PubMed]

Gajic, R.

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express18(19), 20321–20333 (2010).
[CrossRef] [PubMed]

Hingerl, K.

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express18(19), 20321–20333 (2010).
[CrossRef] [PubMed]

Iliew, R.

R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express13(18), 7076–7085 (2005).
[CrossRef] [PubMed]

Isic, G.

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express18(19), 20321–20333 (2010).
[CrossRef] [PubMed]

Johnson, E. G.

Z. A. Roth, P. Srinivasan, M. K. Poutous, A. Pung, R. C. Rumpf, and E. G. Johnson, “Azimuthally varying guided mode resonance filters,” Micromachines3(1), 180–193 (2012).
[CrossRef]

E. G. Johnson, M. K. Poutous, Z. A. Roth, P. Srinivasan, A. J. Pung, and Y. O. Yilmaz, “Advanced fabrication methods for 3D meta-optics,” Proc. SPIE7927, 792706, 792706-7 (2011).
[CrossRef]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett.74(9), 1212–1214 (1999).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett.74(9), 1212–1214 (1999).
[CrossRef]

Kingsland, D. M.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A Perfectly Matched Anisotropic Absorber for Use as an Absorbing Boundary Condition,” IEEE Trans. Antenn. Propag.43(12), 1460–1463 (1995).
[CrossRef]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett.74(9), 1212–1214 (1999).
[CrossRef]

Lederer, F.

R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express13(18), 7076–7085 (2005).
[CrossRef] [PubMed]

Lee, J.-F.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A Perfectly Matched Anisotropic Absorber for Use as an Absorbing Boundary Condition,” IEEE Trans. Antenn. Propag.43(12), 1460–1463 (1995).
[CrossRef]

Lee, R.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A Perfectly Matched Anisotropic Absorber for Use as an Absorbing Boundary Condition,” IEEE Trans. Antenn. Propag.43(12), 1460–1463 (1995).
[CrossRef]

Lipson, M.

D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express18(3), 3181–3186 (2010).
[CrossRef] [PubMed]

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(13), 133501 (2008).
[CrossRef]

Lu, Z.

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett.96(17), 173902 (2006).
[CrossRef] [PubMed]

Murakowski, J. A.

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett.96(17), 173902 (2006).
[CrossRef] [PubMed]

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett.74(9), 1212–1214 (1999).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Poitras, C. B.

D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express18(3), 3181–3186 (2010).
[CrossRef] [PubMed]

Poutous, M. K.

Z. A. Roth, P. Srinivasan, M. K. Poutous, A. Pung, R. C. Rumpf, and E. G. Johnson, “Azimuthally varying guided mode resonance filters,” Micromachines3(1), 180–193 (2012).
[CrossRef]

E. G. Johnson, M. K. Poutous, Z. A. Roth, P. Srinivasan, A. J. Pung, and Y. O. Yilmaz, “Advanced fabrication methods for 3D meta-optics,” Proc. SPIE7927, 792706, 792706-7 (2011).
[CrossRef]

Prather, D. W.

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett.96(17), 173902 (2006).
[CrossRef] [PubMed]

Pung, A.

Z. A. Roth, P. Srinivasan, M. K. Poutous, A. Pung, R. C. Rumpf, and E. G. Johnson, “Azimuthally varying guided mode resonance filters,” Micromachines3(1), 180–193 (2012).
[CrossRef]

Pung, A. J.

E. G. Johnson, M. K. Poutous, Z. A. Roth, P. Srinivasan, A. J. Pung, and Y. O. Yilmaz, “Advanced fabrication methods for 3D meta-optics,” Proc. SPIE7927, 792706, 792706-7 (2011).
[CrossRef]

Roth, Z. A.

Z. A. Roth, P. Srinivasan, M. K. Poutous, A. Pung, R. C. Rumpf, and E. G. Johnson, “Azimuthally varying guided mode resonance filters,” Micromachines3(1), 180–193 (2012).
[CrossRef]

E. G. Johnson, M. K. Poutous, Z. A. Roth, P. Srinivasan, A. J. Pung, and Y. O. Yilmaz, “Advanced fabrication methods for 3D meta-optics,” Proc. SPIE7927, 792706, 792706-7 (2011).
[CrossRef]

Rumpf, R. C.

Z. A. Roth, P. Srinivasan, M. K. Poutous, A. Pung, R. C. Rumpf, and E. G. Johnson, “Azimuthally varying guided mode resonance filters,” Micromachines3(1), 180–193 (2012).
[CrossRef]

Sacks, Z. S.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A Perfectly Matched Anisotropic Absorber for Use as an Absorbing Boundary Condition,” IEEE Trans. Antenn. Propag.43(12), 1460–1463 (1995).
[CrossRef]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett.74(9), 1212–1214 (1999).
[CrossRef]

Schneider, G. J.

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett.96(17), 173902 (2006).
[CrossRef] [PubMed]

Schuetz, C. A.

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett.96(17), 173902 (2006).
[CrossRef] [PubMed]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Shi, S.

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett.96(17), 173902 (2006).
[CrossRef] [PubMed]

Shin, J.

J. Shin and S. Fan, “Conditions for self-collimation in three-dimensional photonic crystals,” Opt. Lett.30(18), 2397–2399 (2005).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Spadoti, D. H.

D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express18(3), 3181–3186 (2010).
[CrossRef] [PubMed]

Srinivasan, P.

Z. A. Roth, P. Srinivasan, M. K. Poutous, A. Pung, R. C. Rumpf, and E. G. Johnson, “Azimuthally varying guided mode resonance filters,” Micromachines3(1), 180–193 (2012).
[CrossRef]

E. G. Johnson, M. K. Poutous, Z. A. Roth, P. Srinivasan, A. J. Pung, and Y. O. Yilmaz, “Advanced fabrication methods for 3D meta-optics,” Proc. SPIE7927, 792706, 792706-7 (2011).
[CrossRef]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett.74(9), 1212–1214 (1999).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett.74(9), 1212–1214 (1999).
[CrossRef]

Vasic, B.

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express18(19), 20321–20333 (2010).
[CrossRef] [PubMed]

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(13), 133501 (2008).
[CrossRef]

Wang, Y. R.

L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett.27(11), 900–902 (2002).
[CrossRef] [PubMed]

Yang, X. L.

L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett.27(11), 900–902 (2002).
[CrossRef] [PubMed]

Yilmaz, Y. O.

E. G. Johnson, M. K. Poutous, Z. A. Roth, P. Srinivasan, A. J. Pung, and Y. O. Yilmaz, “Advanced fabrication methods for 3D meta-optics,” Proc. SPIE7927, 792706, 792706-7 (2011).
[CrossRef]

Appl. Phys. Lett. (2)

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(13), 133501 (2008).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett.74(9), 1212–1214 (1999).
[CrossRef]

IEEE Trans. Antenn. Propag. (1)

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A Perfectly Matched Anisotropic Absorber for Use as an Absorbing Boundary Condition,” IEEE Trans. Antenn. Propag.43(12), 1460–1463 (1995).
[CrossRef]

Micromachines (1)

Z. A. Roth, P. Srinivasan, M. K. Poutous, A. Pung, R. C. Rumpf, and E. G. Johnson, “Azimuthally varying guided mode resonance filters,” Micromachines3(1), 180–193 (2012).
[CrossRef]

Opt. Express (3)

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express18(19), 20321–20333 (2010).
[CrossRef] [PubMed]

D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express18(3), 3181–3186 (2010).
[CrossRef] [PubMed]

R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express13(18), 7076–7085 (2005).
[CrossRef] [PubMed]

Opt. Lett. (2)

J. Shin and S. Fan, “Conditions for self-collimation in three-dimensional photonic crystals,” Opt. Lett.30(18), 2397–2399 (2005).
[CrossRef] [PubMed]

L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett.27(11), 900–902 (2002).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett.96(17), 173902 (2006).
[CrossRef] [PubMed]

Proc. SPIE (1)

E. G. Johnson, M. K. Poutous, Z. A. Roth, P. Srinivasan, A. J. Pung, and Y. O. Yilmaz, “Advanced fabrication methods for 3D meta-optics,” Proc. SPIE7927, 792706, 792706-7 (2011).
[CrossRef]

Science (1)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Other (10)

R. C. Rumpf, “Design and optimization of nano-optical elements by coupling fabrication to optical behavior,” Ph.D. Dissertation, University of Central Florida (2006), pp. 171–183.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, Molding the Flow of Light (Princeton University Press, 1995).

H. Benistry, V. Berger, J.-M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals, Towards Nanoscale Photonic Devices (Springer, 2005).

S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Metamaterials, (CRC Press, 2009).

M. Born and E. Wolf, Principles of Optics, 6th Ed., 673–678 (Cambridge University Press, 1980).

R. C. Rumpf, “Design and optimization of nano-optical elements by coupling fabrication to optical behavior,” Ph.D. Dissertation, University of Central Florida (2006), pp. 109–124.

A. Taflove and S. C. Hagness, Computational Electrodynamics, the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

S. C. Chapra and R. P. Canale, Numerical Methods for Engineers with Software and Programming Applications, 4th Ed., 820–856 (McGraw-Hill, 2002).

B. Noble and J. W. Daniel, Applied Linear Algebra, 3rd ed. (Prentice Hall, 1988), pp. 66–73.

Y. Sadd, Iterative Methods for Sparse Linear Systems, 2nd ed. (Yousef Sadd, 2000).

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Figures (10)

Fig. 1
Fig. 1

2D and 3D unit cells along with a subset of their spatial harmonics.

Fig. 2
Fig. 2

Common formulations for the grating vectors for cubic symmetries.

Fig. 3
Fig. 3

Spatially variant functions and their effects on the lattice.

Fig. 4
Fig. 4

Construction of the spatially variant K function

Fig. 5
Fig. 5

Comparison of approaches for constructing the spatially variant 1D grating from the spatially variant K function.

Fig. 6
Fig. 6

Using the threshold technique to control the fill fraction of a lattice.

Fig. 7
Fig. 7

Spatially varying multiple parameters to compensate for lattice distortion.

Fig. 8
Fig. 8

Example of a 3D spatially variant lattice

Fig. 9
Fig. 9

Self-collimating photonic crystal unit cell and its isofrequency contours.

Fig. 10
Fig. 10

FDTD simulation of a Gaussian beam turning a 90° bend inside a spatially variant self-collimating photonic crystal (Media 1).

Equations (16)

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ε uc ( s )= m=1 M a m exp( j K m s )
K m = 2πp Λ x x ^ + 2πq Λ y y ^ + 2πr Λ z z ^
ε m ( s ) a m exp[ j K m ( s ) s ]
ϕ m ( s )= K m ( s )
[ D x D y D z ] Φ m =[ K x,m K y,m K z,m ].
Φ m = ( A ) 1 b m ,
A = A T A           b m = A T b m ,
A=[ D x D y D z ]           b m =[ K x,m K y,m K z,m ].
ε m ( s )= a m exp[ j ϕ m ( s ) ]
ε ( s )=Re[ m=1 M ε m ( s ) ]
ε( s )={ ε 1       ε ( s )γ( s ) ε 2       ε ( s )>γ( s )
γcos( πf ).
Δ coarse Λ/ N coarse .
Δ fine L min N fine            L min = 2π max[ K m ] .
v g ( k )= k ω( k )
K 1 = 2π a x ^            K 2 = 2π a y ^ .

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