Abstract

We present a method to combine two periodic lattice wave fields to generate a complex dual-lattice wave field which could be employed for microfabrication of corresponding two-dimensional dual-lattice structures. Since the addition of two periodic lattice wave fields is coherent in nature, the resultant dual-lattice structure is highly dependent on the relative phase difference between constituent wave fields. We show that it is possible to have control over the dual-lattice pattern by precisely controlling this relative phase difference. This control is enabled by making use of digitally addressable phase-only spatial light modulator (SLM). We provide the computational method for calculation of the corresponding phase mask to be displayed on the SLM and also verify the results experimentally by employing a simple 4f Fourier filter-based geometry. The method is completely scalable and reconfigurable in terms of the choice of periodic lattice wave fields and has the potential to form gradient phase masks which could be useful for fabrication of graded-index optical components.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).
  2. A. Mekis, J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
    [CrossRef]
  3. O. Painter, R. Lee, A. Scherer, A. Yariv, J. O’Brien, P. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
    [CrossRef]
  4. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
    [CrossRef]
  5. 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, 1212–1214 (1999).
    [CrossRef]
  6. C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104(R) (2002).
    [CrossRef]
  7. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
    [CrossRef]
  8. M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
    [CrossRef]
  9. Y. A. Vlasov, X.-Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bandgap crystals,” Nature 414, 289–293 (2001).
    [CrossRef]
  10. K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
    [CrossRef]
  11. M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
    [CrossRef]
  12. J. L. Stay, G. M. Burrow, and T. Gaylord, “Three-beam interference lithography methodology,” Rev. Sci. Instrum. 82, 023115 (2011).
    [CrossRef]
  13. J. Xavier, P. Rose, B. Terhalle, J. Joseph, and C. Denz, “Three-dimensional optically induced reconfigurable photorefractive nonlinear photonic lattices,” Opt. Lett. 34, 2625–2627 (2009).
    [CrossRef]
  14. J. Xavier and J. Joseph, “Tunable complex photonic chiral lattices by reconfigurable optical phase engineering,” Opt. Lett. 36, 403–405 (2011).
    [CrossRef]
  15. M. Kumar and J. Joseph, “Embedding a nondiffracting defect site in helical lattice wave-field by optical phase engineering,” Appl. Opt. 52, 5653–5658 (2013).
    [CrossRef]
  16. J. Lutkenhaus, D. George, M. Moazzezi, U. Philipose, and Y. Lin, “Digitally tunable holographic lithography using a spatial light modulator as a programmable phase mask,” Opt. Express 21, 26227–26235 (2013).
    [CrossRef]
  17. A. Kelberer, M. Boguslawski, P. Rose, and C. Denz, “Embedding defect sites into hexagonal nondiffracting wave fields,” Opt. Lett. 37, 5009–5011 (2012).
    [CrossRef]
  18. V. Arrizón, D. Sánchez-de-la-Llave, and G. Méndez, “Holographic generation of a class of nondiffracting fields with optimum efficiency,” Opt. Lett. 37, 2154–2156 (2012).
    [CrossRef]
  19. R. C. Rumpf and J. Pazos, “Synthesis of spatially variant lattices,” Opt. Express 20, 15263–15274 (2012).
    [CrossRef]

2013 (2)

2012 (3)

2011 (2)

J. L. Stay, G. M. Burrow, and T. Gaylord, “Three-beam interference lithography methodology,” Rev. Sci. Instrum. 82, 023115 (2011).
[CrossRef]

J. Xavier and J. Joseph, “Tunable complex photonic chiral lattices by reconfigurable optical phase engineering,” Opt. Lett. 36, 403–405 (2011).
[CrossRef]

2009 (1)

2008 (1)

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

2004 (1)

M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
[CrossRef]

2002 (1)

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

2001 (1)

Y. A. Vlasov, X.-Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bandgap crystals,” Nature 414, 289–293 (2001).
[CrossRef]

2000 (1)

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef]

1999 (2)

O. Painter, R. Lee, A. Scherer, A. Yariv, J. O’Brien, P. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[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, 1212–1214 (1999).
[CrossRef]

1998 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

1996 (1)

A. Mekis, J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

1994 (1)

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Arrizón, V.

Baba, T.

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

Biswas, R.

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Bo, X.-Z.

Y. A. Vlasov, X.-Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bandgap crystals,” Nature 414, 289–293 (2001).
[CrossRef]

Boguslawski, M.

Burrow, G. M.

J. L. Stay, G. M. Burrow, and T. Gaylord, “Three-beam interference lithography methodology,” Rev. Sci. Instrum. 82, 023115 (2011).
[CrossRef]

Busch, K.

M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
[CrossRef]

Campbell, M.

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef]

Chan, C.

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Chen, J.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Dapkus, P.

O. Painter, R. Lee, A. Scherer, A. Yariv, J. O’Brien, P. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef]

Denning, R.

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef]

Denz, C.

Deubel, M.

M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
[CrossRef]

Fan, S.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Gaylord, T.

J. L. Stay, G. M. Burrow, and T. Gaylord, “Three-beam interference lithography methodology,” Rev. Sci. Instrum. 82, 023115 (2011).
[CrossRef]

George, D.

Harrison, M.

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef]

Ho, K.

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Joannopoulos, J.

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

A. Mekis, J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Johnson, S. G.

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Joseph, J.

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, 1212–1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[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, 1212–1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kelberer, A.

Kim, I.

O. Painter, R. Lee, A. Scherer, A. Yariv, J. O’Brien, P. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[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, 1212–1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kumar, M.

Kurland, I.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Lee, R.

O. Painter, R. Lee, A. Scherer, A. Yariv, J. O’Brien, P. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef]

Lin, Y.

Luo, C.

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

Lutkenhaus, J.

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Mekis, A.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Méndez, G.

Moazzezi, M.

Norris, D. J.

Y. A. Vlasov, X.-Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bandgap crystals,” Nature 414, 289–293 (2001).
[CrossRef]

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, 1212–1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

O’Brien, J.

O. Painter, R. Lee, A. Scherer, A. Yariv, J. O’Brien, P. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef]

Painter, O.

O. Painter, R. Lee, A. Scherer, A. Yariv, J. O’Brien, P. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef]

Pazos, J.

Pendry, J.

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

Pereira, S.

M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
[CrossRef]

Philipose, U.

Rose, P.

Rumpf, R. C.

Sánchez-de-la-Llave, D.

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, 1212–1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Scherer, A.

O. Painter, R. Lee, A. Scherer, A. Yariv, J. O’Brien, P. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef]

Sharp, D.

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef]

Sigalas, M.

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Soukoulis, C.

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Soukoulis, C. M.

M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
[CrossRef]

Stay, J. L.

J. L. Stay, G. M. Burrow, and T. Gaylord, “Three-beam interference lithography methodology,” Rev. Sci. Instrum. 82, 023115 (2011).
[CrossRef]

Sturm, J. C.

Y. A. Vlasov, X.-Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bandgap crystals,” Nature 414, 289–293 (2001).
[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, 1212–1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Terhalle, B.

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, 1212–1214 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Turberfield, A.

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef]

Villeneuve, P. R.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Vlasov, Y. A.

Y. A. Vlasov, X.-Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bandgap crystals,” Nature 414, 289–293 (2001).
[CrossRef]

Von Freymann, G.

M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
[CrossRef]

Wegener, M.

M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Xavier, J.

Yariv, A.

O. Painter, R. Lee, A. Scherer, A. Yariv, J. O’Brien, P. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (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, 1212–1214 (1999).
[CrossRef]

Nat. Mater. (1)

M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
[CrossRef]

Nat. Photonics (1)

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

Nature (2)

Y. A. Vlasov, X.-Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bandgap crystals,” Nature 414, 289–293 (2001).
[CrossRef]

M. Campbell, D. Sharp, M. Harrison, R. Denning, and A. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. B (2)

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

A. Mekis, J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Rev. Sci. Instrum. (1)

J. L. Stay, G. M. Burrow, and T. Gaylord, “Three-beam interference lithography methodology,” Rev. Sci. Instrum. 82, 023115 (2011).
[CrossRef]

Science (1)

O. Painter, R. Lee, A. Scherer, A. Yariv, J. O’Brien, P. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef]

Solid State Commun. (1)

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Other (1)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Schematic diagram for the experimental setup. MO, microscope objective; SLM, spatial light modulator (phase only).

Fig. 2.
Fig. 2.

Multiple beam interference patterns. Fourier space picture representing (a) two-beam arrangement, (b) three-beam arrangement, and (c) sum of both two-beam and beam-beam arrangements where one beam is in common. (d)–(f) Numerically calculated interference pattern due to the beam arrangements shown in (a)–(c), respectively.

Fig. 3.
Fig. 3.

Complex wave field visualization for (a) two-beam interference, (b) three-beam interference, and (c) coherent sum of two-beam and three-beam interference. The color represents the phase distribution of the complex field according to the color bars associated with each figure while brightness varies linearly with amplitude. (d) Extracted phase profile from the complex field shown in (c). Insets in (a)–(d) show the numerically obtained FT intensity profile of corresponding complex/phase profiles. (e) Enlarged view of FT of phase profile of (d). Spots are numbered from 1 to 6 where spot numbers 1, 3, 4, and 5 are responsible for forming the dual-lattice wave field.

Fig. 4.
Fig. 4.

(Left to right) Dual-lattice wave-field phase profile (first column), experimentally obtained FT intensity profile (second column), numerically simulated interference intensity profile (third column), and experimentally recorded interference intensity profile (fourth column). Rows (a) to (e) represent the results for different phase differences between the coherent sum of two-beam and three-beam wave fields: 0, π/4, π/2, 3π/4, and π, respectively.

Fig. 5.
Fig. 5.

Experimentally recorded dual-lattice pattern in a positive photoresist. (a) Image obtained by optical microscope and (b) SEM profile with corresponding length scale bar.

Fig. 6.
Fig. 6.

Synthesis of a gradient phase mask. Complex wave fields due to (a) two-beam and (b) three-beam interference. (c) Spatially varying phase profile which varies linearly in the radial direction from value π/2π. (d) Resultant complex wave field due to superposition of the wave fields shown in (a) and (b), where the initial phase offset between fields is given by (c). (e) Extracted phase profile from the resultant complex field (d). (f) Intensity profile obtained from the complex wave field in (d).

Equations (5)

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

Edual-lattice=E2beam+E3beam,
Enbeam=i=1nEieiki·r.
I(r)=i=1n|Ei|2+i=1nj=1j1nEiEj*·exp[i(kikj).r+iψij],
ki=k*[cos(qiπ)*sinθi,sin(qiπ)*sinθi,cosθi],
Edual-lattice=E2beam+E3beam×eiφ,

Metrics