Abstract

We have fabricated a two-dimensional photonic crystal that functions as a two-channel add/drop filter on azo-functionalized polymeric films using a fast, direct-writing method. By properly adjusting the resonance, this structure can be used as a multichannel add/drop filter in the plane of the film. Here, we were able to separate two channels at 1555.7 and 1570.7nm with attenuation of 13.1dB and FWHM of 6.7nm for each channel. The separated channels were directed towards the sides at an approximately 90° angle with respect to the input direction.

© 2009 Optical Society of America

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  1. J. H. Teng, J. R. Dong, S. J. Chua, B. S. Foo, M. Y. Lai, Y. J. Wang, S. S. Ang, and R. Yin, “Distributed Bragg reflector laser using buried SiO2 grating and self-aligned band gap tuning,” Appl. Phys. Lett. 90, 171107 (2007).
    [CrossRef]
  2. S. Rennon, L. Bach, and J. P. Reithmaier, “Complex couled distributed-feedback and Brag-reflector lasers for monolithic device integration based on focused-ion-beam technology,” IEEE J. Sel. Top. Quantum Electron. 7, 2 (2001).
  3. R. Kim, J. Zhang, O. Eknoya, H. F. Taylor, and T. L. Smith, “Narrowband Bragg reflectors in Ti:LiNbO3 optical waveguides,” Appl. Opt. 45, 4927-4932 (2006).
    [CrossRef] [PubMed]
  4. K. S. Feder, P. S. Westbrook, J. Ging, P. I. Rayes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photon. Technol. Lett. 15, 933 (2003).
    [CrossRef]
  5. E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Grating based electro-optic switch with azo nonlinear optical polymers,” Appl. Phys. Lett. 89, 171114 (2006).
    [CrossRef]
  6. H. Karimi-Alivijeh, M.-A. Baghban, G.-M. Parsanasab, E. Sarailou, A. Gharavi, and S. Javadpour, “Add/drop filter using in-plane slanted gratings in azo polymers,” Opt. Lett. 33, 2152-2154 (2008).
    [CrossRef]
  7. H. Karimi-Alavigeh, G.-M. Parsanasab, M.-A. Baghban, E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Fabrication of graded index waveguides in azo polymers using a direct writing technique,” Appl. Phys. Lett. 92, 041105 (2008).
    [CrossRef]
  8. A. Gharavi, H. Karimi-Alavijeh, E. Sarailou, M.-A. Baghban, and G.-M. Parsanasab, “Fabrication of 2-D photonic crystal structures using azo polymers,” 14th Conference on Optics and Photonics (Rafsanjan Vali-e-Asr University, 2008), pp. 170-173.
  9. P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azoaromatic polymers,” Appl. Phys. Lett. 60, 4-5 (1992).
    [CrossRef]
  10. D. Y. Kim, S. K. Tripathy, and L. Li. J Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166-1168(1995).
    [CrossRef]
  11. H. Saadeh, A. Gharavi, D. Yu, and L. Yu, “Polyimides with a diazo chromophore exhibiting high thermal stability and large electrooptic coefficients,” Macromolecules 30, 5403-5407(1997).
    [CrossRef]

2008 (2)

H. Karimi-Alivijeh, M.-A. Baghban, G.-M. Parsanasab, E. Sarailou, A. Gharavi, and S. Javadpour, “Add/drop filter using in-plane slanted gratings in azo polymers,” Opt. Lett. 33, 2152-2154 (2008).
[CrossRef]

H. Karimi-Alavigeh, G.-M. Parsanasab, M.-A. Baghban, E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Fabrication of graded index waveguides in azo polymers using a direct writing technique,” Appl. Phys. Lett. 92, 041105 (2008).
[CrossRef]

2007 (1)

J. H. Teng, J. R. Dong, S. J. Chua, B. S. Foo, M. Y. Lai, Y. J. Wang, S. S. Ang, and R. Yin, “Distributed Bragg reflector laser using buried SiO2 grating and self-aligned band gap tuning,” Appl. Phys. Lett. 90, 171107 (2007).
[CrossRef]

2006 (2)

R. Kim, J. Zhang, O. Eknoya, H. F. Taylor, and T. L. Smith, “Narrowband Bragg reflectors in Ti:LiNbO3 optical waveguides,” Appl. Opt. 45, 4927-4932 (2006).
[CrossRef] [PubMed]

E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Grating based electro-optic switch with azo nonlinear optical polymers,” Appl. Phys. Lett. 89, 171114 (2006).
[CrossRef]

2003 (1)

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Rayes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photon. Technol. Lett. 15, 933 (2003).
[CrossRef]

2001 (1)

S. Rennon, L. Bach, and J. P. Reithmaier, “Complex couled distributed-feedback and Brag-reflector lasers for monolithic device integration based on focused-ion-beam technology,” IEEE J. Sel. Top. Quantum Electron. 7, 2 (2001).

1997 (1)

H. Saadeh, A. Gharavi, D. Yu, and L. Yu, “Polyimides with a diazo chromophore exhibiting high thermal stability and large electrooptic coefficients,” Macromolecules 30, 5403-5407(1997).
[CrossRef]

1995 (1)

D. Y. Kim, S. K. Tripathy, and L. Li. J Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166-1168(1995).
[CrossRef]

1992 (1)

P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azoaromatic polymers,” Appl. Phys. Lett. 60, 4-5 (1992).
[CrossRef]

Ang, S. S.

J. H. Teng, J. R. Dong, S. J. Chua, B. S. Foo, M. Y. Lai, Y. J. Wang, S. S. Ang, and R. Yin, “Distributed Bragg reflector laser using buried SiO2 grating and self-aligned band gap tuning,” Appl. Phys. Lett. 90, 171107 (2007).
[CrossRef]

Bach, L.

S. Rennon, L. Bach, and J. P. Reithmaier, “Complex couled distributed-feedback and Brag-reflector lasers for monolithic device integration based on focused-ion-beam technology,” IEEE J. Sel. Top. Quantum Electron. 7, 2 (2001).

Baghban, M.-A.

H. Karimi-Alivijeh, M.-A. Baghban, G.-M. Parsanasab, E. Sarailou, A. Gharavi, and S. Javadpour, “Add/drop filter using in-plane slanted gratings in azo polymers,” Opt. Lett. 33, 2152-2154 (2008).
[CrossRef]

H. Karimi-Alavigeh, G.-M. Parsanasab, M.-A. Baghban, E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Fabrication of graded index waveguides in azo polymers using a direct writing technique,” Appl. Phys. Lett. 92, 041105 (2008).
[CrossRef]

A. Gharavi, H. Karimi-Alavijeh, E. Sarailou, M.-A. Baghban, and G.-M. Parsanasab, “Fabrication of 2-D photonic crystal structures using azo polymers,” 14th Conference on Optics and Photonics (Rafsanjan Vali-e-Asr University, 2008), pp. 170-173.

Carver, G. E.

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Rayes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photon. Technol. Lett. 15, 933 (2003).
[CrossRef]

Chua, S. J.

J. H. Teng, J. R. Dong, S. J. Chua, B. S. Foo, M. Y. Lai, Y. J. Wang, S. S. Ang, and R. Yin, “Distributed Bragg reflector laser using buried SiO2 grating and self-aligned band gap tuning,” Appl. Phys. Lett. 90, 171107 (2007).
[CrossRef]

Dong, J. R.

J. H. Teng, J. R. Dong, S. J. Chua, B. S. Foo, M. Y. Lai, Y. J. Wang, S. S. Ang, and R. Yin, “Distributed Bragg reflector laser using buried SiO2 grating and self-aligned band gap tuning,” Appl. Phys. Lett. 90, 171107 (2007).
[CrossRef]

Eknoya, O.

Feder, K. S.

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Rayes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photon. Technol. Lett. 15, 933 (2003).
[CrossRef]

Foo, B. S.

J. H. Teng, J. R. Dong, S. J. Chua, B. S. Foo, M. Y. Lai, Y. J. Wang, S. S. Ang, and R. Yin, “Distributed Bragg reflector laser using buried SiO2 grating and self-aligned band gap tuning,” Appl. Phys. Lett. 90, 171107 (2007).
[CrossRef]

Gharavi, A.

H. Karimi-Alivijeh, M.-A. Baghban, G.-M. Parsanasab, E. Sarailou, A. Gharavi, and S. Javadpour, “Add/drop filter using in-plane slanted gratings in azo polymers,” Opt. Lett. 33, 2152-2154 (2008).
[CrossRef]

H. Karimi-Alavigeh, G.-M. Parsanasab, M.-A. Baghban, E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Fabrication of graded index waveguides in azo polymers using a direct writing technique,” Appl. Phys. Lett. 92, 041105 (2008).
[CrossRef]

E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Grating based electro-optic switch with azo nonlinear optical polymers,” Appl. Phys. Lett. 89, 171114 (2006).
[CrossRef]

H. Saadeh, A. Gharavi, D. Yu, and L. Yu, “Polyimides with a diazo chromophore exhibiting high thermal stability and large electrooptic coefficients,” Macromolecules 30, 5403-5407(1997).
[CrossRef]

A. Gharavi, H. Karimi-Alavijeh, E. Sarailou, M.-A. Baghban, and G.-M. Parsanasab, “Fabrication of 2-D photonic crystal structures using azo polymers,” 14th Conference on Optics and Photonics (Rafsanjan Vali-e-Asr University, 2008), pp. 170-173.

Ging, J.

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Rayes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photon. Technol. Lett. 15, 933 (2003).
[CrossRef]

Gosselin, J.

P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azoaromatic polymers,” Appl. Phys. Lett. 60, 4-5 (1992).
[CrossRef]

Javadpour, S.

H. Karimi-Alavigeh, G.-M. Parsanasab, M.-A. Baghban, E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Fabrication of graded index waveguides in azo polymers using a direct writing technique,” Appl. Phys. Lett. 92, 041105 (2008).
[CrossRef]

H. Karimi-Alivijeh, M.-A. Baghban, G.-M. Parsanasab, E. Sarailou, A. Gharavi, and S. Javadpour, “Add/drop filter using in-plane slanted gratings in azo polymers,” Opt. Lett. 33, 2152-2154 (2008).
[CrossRef]

E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Grating based electro-optic switch with azo nonlinear optical polymers,” Appl. Phys. Lett. 89, 171114 (2006).
[CrossRef]

Karimi-Alavigeh, H.

H. Karimi-Alavigeh, G.-M. Parsanasab, M.-A. Baghban, E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Fabrication of graded index waveguides in azo polymers using a direct writing technique,” Appl. Phys. Lett. 92, 041105 (2008).
[CrossRef]

Karimi-Alavijeh, H.

A. Gharavi, H. Karimi-Alavijeh, E. Sarailou, M.-A. Baghban, and G.-M. Parsanasab, “Fabrication of 2-D photonic crystal structures using azo polymers,” 14th Conference on Optics and Photonics (Rafsanjan Vali-e-Asr University, 2008), pp. 170-173.

Karimi-Alivijeh, H.

Kim, D. Y.

D. Y. Kim, S. K. Tripathy, and L. Li. J Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166-1168(1995).
[CrossRef]

Kim, R.

Kumar, L. Li. J

D. Y. Kim, S. K. Tripathy, and L. Li. J Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166-1168(1995).
[CrossRef]

Lai, M. Y.

J. H. Teng, J. R. Dong, S. J. Chua, B. S. Foo, M. Y. Lai, Y. J. Wang, S. S. Ang, and R. Yin, “Distributed Bragg reflector laser using buried SiO2 grating and self-aligned band gap tuning,” Appl. Phys. Lett. 90, 171107 (2007).
[CrossRef]

Natansohn, A.

P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azoaromatic polymers,” Appl. Phys. Lett. 60, 4-5 (1992).
[CrossRef]

Parsanasab, G.-M.

H. Karimi-Alavigeh, G.-M. Parsanasab, M.-A. Baghban, E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Fabrication of graded index waveguides in azo polymers using a direct writing technique,” Appl. Phys. Lett. 92, 041105 (2008).
[CrossRef]

H. Karimi-Alivijeh, M.-A. Baghban, G.-M. Parsanasab, E. Sarailou, A. Gharavi, and S. Javadpour, “Add/drop filter using in-plane slanted gratings in azo polymers,” Opt. Lett. 33, 2152-2154 (2008).
[CrossRef]

A. Gharavi, H. Karimi-Alavijeh, E. Sarailou, M.-A. Baghban, and G.-M. Parsanasab, “Fabrication of 2-D photonic crystal structures using azo polymers,” 14th Conference on Optics and Photonics (Rafsanjan Vali-e-Asr University, 2008), pp. 170-173.

Rayes, P. I.

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Rayes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photon. Technol. Lett. 15, 933 (2003).
[CrossRef]

Reithmaier, J. P.

S. Rennon, L. Bach, and J. P. Reithmaier, “Complex couled distributed-feedback and Brag-reflector lasers for monolithic device integration based on focused-ion-beam technology,” IEEE J. Sel. Top. Quantum Electron. 7, 2 (2001).

Rennon, S.

S. Rennon, L. Bach, and J. P. Reithmaier, “Complex couled distributed-feedback and Brag-reflector lasers for monolithic device integration based on focused-ion-beam technology,” IEEE J. Sel. Top. Quantum Electron. 7, 2 (2001).

Rochon, P.

P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azoaromatic polymers,” Appl. Phys. Lett. 60, 4-5 (1992).
[CrossRef]

Saadeh, H.

H. Saadeh, A. Gharavi, D. Yu, and L. Yu, “Polyimides with a diazo chromophore exhibiting high thermal stability and large electrooptic coefficients,” Macromolecules 30, 5403-5407(1997).
[CrossRef]

Sarailou, E.

H. Karimi-Alivijeh, M.-A. Baghban, G.-M. Parsanasab, E. Sarailou, A. Gharavi, and S. Javadpour, “Add/drop filter using in-plane slanted gratings in azo polymers,” Opt. Lett. 33, 2152-2154 (2008).
[CrossRef]

H. Karimi-Alavigeh, G.-M. Parsanasab, M.-A. Baghban, E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Fabrication of graded index waveguides in azo polymers using a direct writing technique,” Appl. Phys. Lett. 92, 041105 (2008).
[CrossRef]

E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Grating based electro-optic switch with azo nonlinear optical polymers,” Appl. Phys. Lett. 89, 171114 (2006).
[CrossRef]

A. Gharavi, H. Karimi-Alavijeh, E. Sarailou, M.-A. Baghban, and G.-M. Parsanasab, “Fabrication of 2-D photonic crystal structures using azo polymers,” 14th Conference on Optics and Photonics (Rafsanjan Vali-e-Asr University, 2008), pp. 170-173.

Shkunov, V.

H. Karimi-Alavigeh, G.-M. Parsanasab, M.-A. Baghban, E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Fabrication of graded index waveguides in azo polymers using a direct writing technique,” Appl. Phys. Lett. 92, 041105 (2008).
[CrossRef]

E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Grating based electro-optic switch with azo nonlinear optical polymers,” Appl. Phys. Lett. 89, 171114 (2006).
[CrossRef]

Smith, T. L.

Taylor, H. F.

Teng, J. H.

J. H. Teng, J. R. Dong, S. J. Chua, B. S. Foo, M. Y. Lai, Y. J. Wang, S. S. Ang, and R. Yin, “Distributed Bragg reflector laser using buried SiO2 grating and self-aligned band gap tuning,” Appl. Phys. Lett. 90, 171107 (2007).
[CrossRef]

Tripathy, S. K.

D. Y. Kim, S. K. Tripathy, and L. Li. J Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166-1168(1995).
[CrossRef]

Wang, Y. J.

J. H. Teng, J. R. Dong, S. J. Chua, B. S. Foo, M. Y. Lai, Y. J. Wang, S. S. Ang, and R. Yin, “Distributed Bragg reflector laser using buried SiO2 grating and self-aligned band gap tuning,” Appl. Phys. Lett. 90, 171107 (2007).
[CrossRef]

Westbrook, P. S.

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Rayes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photon. Technol. Lett. 15, 933 (2003).
[CrossRef]

Xie, S.

P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azoaromatic polymers,” Appl. Phys. Lett. 60, 4-5 (1992).
[CrossRef]

Yin, R.

J. H. Teng, J. R. Dong, S. J. Chua, B. S. Foo, M. Y. Lai, Y. J. Wang, S. S. Ang, and R. Yin, “Distributed Bragg reflector laser using buried SiO2 grating and self-aligned band gap tuning,” Appl. Phys. Lett. 90, 171107 (2007).
[CrossRef]

Yu, D.

H. Saadeh, A. Gharavi, D. Yu, and L. Yu, “Polyimides with a diazo chromophore exhibiting high thermal stability and large electrooptic coefficients,” Macromolecules 30, 5403-5407(1997).
[CrossRef]

Yu, L.

H. Saadeh, A. Gharavi, D. Yu, and L. Yu, “Polyimides with a diazo chromophore exhibiting high thermal stability and large electrooptic coefficients,” Macromolecules 30, 5403-5407(1997).
[CrossRef]

Zhang, J.

Appl. Opt. (1)

Appl. Phys. Lett. (5)

J. H. Teng, J. R. Dong, S. J. Chua, B. S. Foo, M. Y. Lai, Y. J. Wang, S. S. Ang, and R. Yin, “Distributed Bragg reflector laser using buried SiO2 grating and self-aligned band gap tuning,” Appl. Phys. Lett. 90, 171107 (2007).
[CrossRef]

E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Grating based electro-optic switch with azo nonlinear optical polymers,” Appl. Phys. Lett. 89, 171114 (2006).
[CrossRef]

H. Karimi-Alavigeh, G.-M. Parsanasab, M.-A. Baghban, E. Sarailou, A. Gharavi, S. Javadpour, and V. Shkunov, “Fabrication of graded index waveguides in azo polymers using a direct writing technique,” Appl. Phys. Lett. 92, 041105 (2008).
[CrossRef]

P. Rochon, J. Gosselin, A. Natansohn, and S. Xie, “Optically induced and erased birefringence and dichroism in azoaromatic polymers,” Appl. Phys. Lett. 60, 4-5 (1992).
[CrossRef]

D. Y. Kim, S. K. Tripathy, and L. Li. J Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166-1168(1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

S. Rennon, L. Bach, and J. P. Reithmaier, “Complex couled distributed-feedback and Brag-reflector lasers for monolithic device integration based on focused-ion-beam technology,” IEEE J. Sel. Top. Quantum Electron. 7, 2 (2001).

IEEE Photon. Technol. Lett. (1)

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Rayes, and G. E. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photon. Technol. Lett. 15, 933 (2003).
[CrossRef]

Macromolecules (1)

H. Saadeh, A. Gharavi, D. Yu, and L. Yu, “Polyimides with a diazo chromophore exhibiting high thermal stability and large electrooptic coefficients,” Macromolecules 30, 5403-5407(1997).
[CrossRef]

Opt. Lett. (1)

Other (1)

A. Gharavi, H. Karimi-Alavijeh, E. Sarailou, M.-A. Baghban, and G.-M. Parsanasab, “Fabrication of 2-D photonic crystal structures using azo polymers,” 14th Conference on Optics and Photonics (Rafsanjan Vali-e-Asr University, 2008), pp. 170-173.

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Figures (6)

Fig. 1
Fig. 1

(a) Chemical structure and (b) absorption spectrum of the polymeric material used in this experiment.

Fig. 2
Fig. 2

Experimental setup used to form 2D photonic crystals. M is mirror; BS is beam splitter; SLED is the super-luminescent light emitting diode; and OSA is optical spectrum analyzer.

Fig. 3
Fig. 3

(a) Output signal exiting straight through the PC; the inset shows the spectrum of the input signal; (b) dropped signals reflected by the 2D PC; λ 1 is reflected to one side and λ 2 is reflected to the other side of the PC in opposite directions; and (c) geometry of the input/output signals.

Fig. 4
Fig. 4

Guided beam in the polymeric film and dropped beam reflected by the PC, captured by an IR camera. The schematic of Fig. 3c is relevant to this photo.

Fig. 5
Fig. 5

(a) Photograph of an actual fabricated 2D structure, and (b) diffraction pattern obtained exposing the PC to a red laser. The inset shows the angles of rotation, α 1 and α 2 .

Fig. 6
Fig. 6

Intensity modulation of the captured photograph of written photonic crystal (gray lines) and sinusoidal curves fitted to them (dark lines); in the (a) x direction and (b) y direction; (c) refractive index modulation pattern of written structures, simulated by a computer program.

Equations (9)

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

b = h tan θ sin 45 ° + cos 45 ° tan θ .
I total = 2 | E 0 | 2 ( 1 + cos ( 2 k x sin θ ) ) ,
Λ = λ w 2 sin θ ,
Λ = λ R 2 n cos α ,
Δ n x Δ n m ( 1 + cos ( 2 k x sin θ ) ) ,
Δ n y Δ n m ( 1 + cos ( 2 k y sin θ ) ) ,
Δ n ( x , y ) = Δ n x + Δ n y = Δ n m [ 2 + cos ( 2 k x sin θ ) + cos ( 2 k y sin θ ) 4 ] ,
n ( x , y ) = n 0 + Δ n ( x , y ) .
n ( x , y ) = n 0 + Δ n m ( 2 + cos ( 2 π x / Λ ) + cos ( 2 π y / Λ ) 4 ) .

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