Abstract

We explore two-dimensional triangular lattice photonic crystals composed of air holes in a dielectric background which are subject to a graded-index distribution along the direction transverse to the propagation. The proper choice of the parameters such as the input beam width, gradient coefficient, and the operating frequency allow the realizations of the focusing (lens) and guiding (waveguide) effects upon which more complex optical devices such as couplers can be designed. Numerical results obtained by the finite-difference time-domain and planewave expansion methods validate the application of Gaussian optics within a range of parameters where close agreement between them are observed.

© 2007 Optical Society of America

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  4. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional Photonic band-gap defect mode Laser," Science 284, 1819-1821 (1999).
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  8. A. Mekis, J. C. Chen, I. Kurand, S. Fan, P. R. Villeneuve, and J. D. Joannopolous, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
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    [CrossRef]
  13. X. Yu and S. Fan, "Bends and splitters for self-collimated beams in photonic crystals," Appl. Phys. Lett. 83, 3251-3253 (2003).
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  14. D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. J. Schneider, and J. Murakowski, "Dispersion-based optical routing in photonic crystals," Opt. Lett. 29, 50-52 (2004).
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    [CrossRef]

2006 (5)

H. Benisty, J. M. Lourtioz, A. Chelnokov, S. Combrie, X. Checoury, "Recent advances toward optical devices in semiconductor-based photonic crystals," Proceedings of the IEEE 94, 997-1023 (2006).
[CrossRef]

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

E. Foca, H. Föll, J. Carstensen, V. V. Sergentu, I. M. Tiginyanu, F. Daschner, and R. Knöchel, "Strongly frequency dependent focusing efficiency of a concave lens based on two-dimensional photonic crystals," Appl. Phys. Lett. 88, 011102 (1-3) (2006).
[CrossRef]

F. S. Roux and I. De Leon, "Planar photonic crystal gradient index lens, simulated with a finite difference time domain method," Phys. Rev. B 74, 113103 (1-4) (2006).
[CrossRef]

M. Zickar, W. Noell, C. Marxer, and N. de Rooij, "MEMS compatible micro-GRIN lenses for fiber to chip coupling of light," Opt. Express 14, 4237-4249 (2006).
[CrossRef] [PubMed]

2005 (1)

E. Centeno and D. Cassagne, "Graded photonic crystals," Opt. Lett. 74, 2278-2280 (2005).
[CrossRef]

2004 (2)

D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystals waveguides," Appl. Phys. Lett. 85, 1101-1103 (2004).
[CrossRef]

D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. J. Schneider, and J. Murakowski, "Dispersion-based optical routing in photonic crystals," Opt. Lett. 29, 50-52 (2004).
[CrossRef] [PubMed]

2003 (4)

S. Guo and S. Albin, "Simple plane wave implementation for photonic crystal calculations," Opt. Express, vol.  11, 167-175 (2003).
[CrossRef]

M. R. Mackenzie and C. Y. Kwok, "Theoretical analysis of integrated collimating waveguide lens," J. Lightwave Technol. 21, 1046-1052 (2003).
[CrossRef]

X. Yu and S. Fan, "Bends and splitters for self-collimated beams in photonic crystals," Appl. Phys. Lett. 83, 3251-3253 (2003).
[CrossRef]

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537 (2003).
[CrossRef] [PubMed]

2000 (2)

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, "Design and fabrication of silicon photonic crystal optical waveguides," J. Lightwave Technol. 18, 1402-1411 (2000).
[CrossRef]

1999 (4)

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]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional Photonic band-gap defect mode Laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
[CrossRef]

1997 (1)

J. D. Joannoupoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

1996 (2)

P. Lalanne, "Effective medium theory applied to photonic crystals composed of cubic or square cylinders," Appl. Opt. 27, 5369-5380 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurand, S. Fan, P. R. Villeneuve, and J. D. Joannopolous, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

1965 (1)

1956 (1)

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP  2, 466-475 (1956).

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, 235119-235119 (2006).
[CrossRef]

Albin, S.

Arriaga, J.

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

Asano, T.

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537 (2003).
[CrossRef] [PubMed]

Baba, T.

D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystals waveguides," Appl. Phys. Lett. 85, 1101-1103 (2004).
[CrossRef]

Bayindir, M.

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

Benisty, H.

H. Benisty, J. M. Lourtioz, A. Chelnokov, S. Combrie, X. Checoury, "Recent advances toward optical devices in semiconductor-based photonic crystals," Proceedings of the IEEE 94, 997-1023 (2006).
[CrossRef]

Carstensen, J.

E. Foca, H. Föll, J. Carstensen, V. V. Sergentu, I. M. Tiginyanu, F. Daschner, and R. Knöchel, "Strongly frequency dependent focusing efficiency of a concave lens based on two-dimensional photonic crystals," Appl. Phys. Lett. 88, 011102 (1-3) (2006).
[CrossRef]

Cassagne, D.

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

E. Centeno and D. Cassagne, "Graded photonic crystals," Opt. Lett. 74, 2278-2280 (2005).
[CrossRef]

Centeno, E.

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

E. Centeno and D. Cassagne, "Graded photonic crystals," Opt. Lett. 74, 2278-2280 (2005).
[CrossRef]

Checoury, X.

H. Benisty, J. M. Lourtioz, A. Chelnokov, S. Combrie, X. Checoury, "Recent advances toward optical devices in semiconductor-based photonic crystals," Proceedings of the IEEE 94, 997-1023 (2006).
[CrossRef]

Chelnokov, A.

H. Benisty, J. M. Lourtioz, A. Chelnokov, S. Combrie, X. Checoury, "Recent advances toward optical devices in semiconductor-based photonic crystals," Proceedings of the IEEE 94, 997-1023 (2006).
[CrossRef]

Chen, C.

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurand, S. Fan, P. R. Villeneuve, and J. D. Joannopolous, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Combrie, S.

H. Benisty, J. M. Lourtioz, A. Chelnokov, S. Combrie, X. Checoury, "Recent advances toward optical devices in semiconductor-based photonic crystals," Proceedings of the IEEE 94, 997-1023 (2006).
[CrossRef]

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional Photonic band-gap defect mode Laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Daschner, F.

E. Foca, H. Föll, J. Carstensen, V. V. Sergentu, I. M. Tiginyanu, F. Daschner, and R. Knöchel, "Strongly frequency dependent focusing efficiency of a concave lens based on two-dimensional photonic crystals," Appl. Phys. Lett. 88, 011102 (1-3) (2006).
[CrossRef]

De Leon, I.

F. S. Roux and I. De Leon, "Planar photonic crystal gradient index lens, simulated with a finite difference time domain method," Phys. Rev. B 74, 113103 (1-4) (2006).
[CrossRef]

de Rooij, N.

Doll, T.

Fan, S.

X. Yu and S. Fan, "Bends and splitters for self-collimated beams in photonic crystals," Appl. Phys. Lett. 83, 3251-3253 (2003).
[CrossRef]

J. D. Joannoupoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurand, S. Fan, P. R. Villeneuve, and J. D. Joannopolous, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Foca, E.

E. Foca, H. Föll, J. Carstensen, V. V. Sergentu, I. M. Tiginyanu, F. Daschner, and R. Knöchel, "Strongly frequency dependent focusing efficiency of a concave lens based on two-dimensional photonic crystals," Appl. Phys. Lett. 88, 011102 (1-3) (2006).
[CrossRef]

Föll, H.

E. Foca, H. Föll, J. Carstensen, V. V. Sergentu, I. M. Tiginyanu, F. Daschner, and R. Knöchel, "Strongly frequency dependent focusing efficiency of a concave lens based on two-dimensional photonic crystals," Appl. Phys. Lett. 88, 011102 (1-3) (2006).
[CrossRef]

Guo, S.

Halevi, P.

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

Joannopolous, J. D.

A. Mekis, J. C. Chen, I. Kurand, S. Fan, P. R. Villeneuve, and J. D. Joannopolous, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Joannoupoulos, J. D.

J. D. Joannoupoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[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, 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, 1212-1214 (1999).
[CrossRef]

Kim, I.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional Photonic band-gap defect mode Laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Knöchel, R.

E. Foca, H. Föll, J. Carstensen, V. V. Sergentu, I. M. Tiginyanu, F. Daschner, and R. Knöchel, "Strongly frequency dependent focusing efficiency of a concave lens based on two-dimensional photonic crystals," Appl. Phys. Lett. 88, 011102 (1-3) (2006).
[CrossRef]

Kogelnik, H.

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]

Krokhin, A. A.

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

Kurand, I.

A. Mekis, J. C. Chen, I. Kurand, S. Fan, P. R. Villeneuve, and J. D. Joannopolous, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Kwok, C. Y.

Lalanne, P.

P. Lalanne, "Effective medium theory applied to photonic crystals composed of cubic or square cylinders," Appl. Opt. 27, 5369-5380 (1996).
[CrossRef]

Lee, R. K.

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
[CrossRef]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional Photonic band-gap defect mode Laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Loncar, M.

Lourtioz, J. M.

H. Benisty, J. M. Lourtioz, A. Chelnokov, S. Combrie, X. Checoury, "Recent advances toward optical devices in semiconductor-based photonic crystals," Proceedings of the IEEE 94, 997-1023 (2006).
[CrossRef]

Mackenzie, M. R.

Marxer, C.

Mekis, A.

A. Mekis, J. C. Chen, I. Kurand, S. Fan, P. R. Villeneuve, and J. D. Joannopolous, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Mori, D.

D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystals waveguides," Appl. Phys. Lett. 85, 1101-1103 (2004).
[CrossRef]

Murakowski, J.

Noda, S.

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537 (2003).
[CrossRef] [PubMed]

Noell, W.

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]

O'Brien, J. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional Photonic band-gap defect mode Laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Ozbay, E.

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

Painter, O.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional Photonic band-gap defect mode Laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Prather, D. W.

Pustai, D. M.

Roux, F. S.

F. S. Roux and I. De Leon, "Planar photonic crystal gradient index lens, simulated with a finite difference time domain method," Phys. Rev. B 74, 113103 (1-4) (2006).
[CrossRef]

Rytov, S. M.

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP  2, 466-475 (1956).

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]

Scherer, A.

Schneider, G. J.

Sergentu, V. V.

E. Foca, H. Föll, J. Carstensen, V. V. Sergentu, I. M. Tiginyanu, F. Daschner, and R. Knöchel, "Strongly frequency dependent focusing efficiency of a concave lens based on two-dimensional photonic crystals," Appl. Phys. Lett. 88, 011102 (1-3) (2006).
[CrossRef]

Sharkawy, A.

Shi, S.

Song, B. S.

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537 (2003).
[CrossRef] [PubMed]

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]

Temelkuran, B.

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

Tiginyanu, I. M.

E. Foca, H. Föll, J. Carstensen, V. V. Sergentu, I. M. Tiginyanu, F. Daschner, and R. Knöchel, "Strongly frequency dependent focusing efficiency of a concave lens based on two-dimensional photonic crystals," Appl. Phys. Lett. 88, 011102 (1-3) (2006).
[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, 1212-1214 (1999).
[CrossRef]

Venkataraman, S.

Villeneuve, P. R.

J. D. Joannoupoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurand, S. Fan, P. R. Villeneuve, and J. D. Joannopolous, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Vuckovic, J.

Xu, Y.

Yariv, A.

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

GRIN PC structure: (a) 2D triangular lattice PCs with air holes in dielectric background and the desired refractive index variation along the transverse direction. The two different ways of the realization of such an index variation either (b) by varying the radii of holes or (c) by changing the refractive indices of holes.

Fig. 2.
Fig. 2.

GRIN PC structure: 2D triangular lattice photonic crystals with air holes in dielectric background. The right and the left plots show the refractive index variation along the dotted section.

Fig. 3.
Fig. 3.

The steady-state magnetic field variation for the (a) self-collimated and (b) converging cases for the input beam at frequencies a/λ = 0.12 and a/λ = 0.24, respectively.

Fig. 4.
Fig. 4.

Equi-frequency curves of the three lowest bands. The refractive indices of air holes are varied (εa = 1.0 and εa = 4.0) keeping the background refractive indices the same. The dotted circles and hexagonal shapes highlight the frequency curves of a/λ = 0.12 and 0.16 for the first band and a/λ = 0.24 for the second band.

Fig. 5.
Fig. 5.

The beam profiles across the y section for the (a) self-collimation at three different distances and (b) focusing cases (solid line is for the input beam and dotted line is at the focal plane).

Fig. 6.
Fig. 6.

(a)–6(c) Three different input beam widths (values are listed in Table I) and the steady-state field plot for the focusing cases at frequency below the band-gap (a/λ = 0.16) and (d–f) above the band-gap (a/λ = 0.24).

Fig. 7.
Fig. 7.

(a). Beam amplitude plots of Figs. 5 (a)–5(c) along the middle section of the GRIN PC and (b) for the Figs. 5 (d)–5(f). Solid, dashed and dotted lines correspond to different input beam width in increasing order.

Fig. 8.
Fig. 8.

The beam profiles at the input and output (focal) planes at frequency below the band-gap (a–c) and above the band-gap (e–f).

Fig. 9.
Fig. 9.

The steady-state field map for frequencies (a) inside band-gap and (b) at high frequencies.

Fig. 10.
Fig. 10.

The steady-state field map for the (a) self-collimation (a/λ = 0.24) and (b) focusing cases (a/λ = 0.16) for a long propagation distances.

Tables (2)

Tables Icon

Table I. FWHM of the beam profile at the input and focal planes for two different frequencies.

Tables Icon

Table II. Comparison of spot-size width values obtained by FDTD and Gaussian optics for two different frequencies.

Equations (15)

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1 q ( x ) = 1 R ( x ) j λ 0 π ω 2 ( x ) n 0 ,
[ r 2 r 2 ] = [ A B C D ] [ r 1 r 1 ] ,
[ A B C D ] = [ cos α d sin α d α α sin α d cos α d ] .
q out = cos ( α d ) q in + sin ( α d ) α α sin ( α d ) q in + cos ( α d ) .
q in = i π n 0 w 0 2 λ 0 = i q i ,
q out = i cos ( α x ) q i + sin ( α x ) α i α sin ( α x ) q i + cos ( α x ) .
Re ( q out ) = Re ( A q in + B C q in + C ) = 0 .
w 0 = ( λ π n 0 α ) 1 2 .
q out ( x ) = i π n x w x 2 λ 0 ,
q i cos 2 ( α x ) + α 2 sin 2 ( α x ) q i 2 = π n x w x 2 λ 0 ,
w x = n 0 w 0 2 n x [ cos 2 ( α x ) + α 2 sin 2 ( α x ) π n 0 w 0 2 λ 0 ] .
H z ( x , y ) = H 0 w 0 w ( x ) exp ( i k y 2 2 q ( x ) ) = H 0 w 0 w ( x ) exp [ i k y 2 2 ( 1 R ( x ) i λ 0 π w 2 ( x ) n 0 ) ]
= H 0 w 0 w ( x ) exp ( i π n 0 y 2 λ 0 R ( x ) ) exp ( y 2 w 2 ( x ) )
f = 1 C = 1 n x α sin ( α d ) .
n eff 2 = f n 1 2 + ( 1 f ) n 2 2 + 1 3 [ a λ π f ( 1 f ) ( n 1 2 n 2 2 ) ] 2 ,

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