Abstract

The self-imaging phenomena in multimode Bragg reflection waveguides (BRWs) have been predicted and investigated by using the plane-wave expansion method and the finite-difference time-domain method. A compact wavelength splitter based on self-imaging principles in BRWs is presented, and its transmission characteristics are investigated by using the finite-difference time-domain method. Calculated results indicate that, for the wavelength splitter without any waveguide bend optimizations, two optical waves with different wavelengths can be spatially separated, and corresponding transmittances are 95.6% and 90.1%, respectively. The simple and compact wavelength splitter is expected to be applied to highly dense photonic integrated circuits.

© 2012 Optical Society of America

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  9. J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Hingerl, “Arbitrary angle waveguiding applications of two-dimensional curvilinear-lattice photonic crystals,” Appl. Phys. Lett. 84, 4687–4689 (2004).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2011

2009

2008

J. Li and K. S. Chiang, “Light guidance in a photonic bandgap slab waveguide consisting of two different Bragg reflectors,” Opt. Commun. 281, 5797–5803 (2008).
[CrossRef]

B. Chen, T. Tang, Z. Wang, H. Chen, and Z. Liu, “Flexible optical waveguides based on the omnidirectional reflection of one-dimensional photonic crystals,” Appl. Phys. Lett. 93, 181107 (2008).
[CrossRef]

W. Huang, Y. Zhang, and B. Li, “Ultracompact wavelength and polarization splitters in periodic dielectric waveguide,” Opt. Express 16, 1600–1609 (2008).
[CrossRef]

2007

2006

B. Nistad, M. W. Haakestad, and J. Skaar, “Dispersion properties of planar Bragg waveguides,” Opt. Commun. 265, 153–160 (2006).
[CrossRef]

Y. Zhang and B. J. Li, “Photonic crystal-based bending waveguides for optical interconnections,” Opt. Express 14, 5723–5732 (2006).
[CrossRef]

T. Niemi, L. H. Frandsen, K. K. Hede, A. Harpoth, P. I. Borel, and M. Kristensen, “Wavelength-division demultiplexing using photonic crystal waveguide,” IEEE Photon. Technol. Lett. 18, 226–228 (2006).
[CrossRef]

2005

H.-Y. Sang, Z.-Y. Li, and B.-Y. Gu, “Propagation properties of planar Bragg waveguides studied by an analytical Bloch-mode method,” J. Appl. Phys. 98, 043114 (2005).
[CrossRef]

2004

J. Zimmermann, M. Kamp, A. Forchel, and R. März, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (2004).
[CrossRef]

H. Kim, I. Park, B. O. S. Park, E. Lee, and S. Lee, “Self-imaging phenomena in multi-mode photonic crystal line-defect waveguides: application to wavelength de-multiplexing,” Opt. Express 12, 5625–5633 (2004).
[CrossRef]

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Hingerl, “Arbitrary angle waveguiding applications of two-dimensional curvilinear-lattice photonic crystals,” Appl. Phys. Lett. 84, 4687–4689 (2004).
[CrossRef]

A. Mizrahi and L. Schächter, “Bragg reflection waveguides with a matching layer,” Opt. Express 12, 3156–3170(2004).
[CrossRef]

2002

2001

1999

1998

J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

1995

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
[CrossRef]

1990

G. Lenz and J. Salzman, “Bragg reflection waveguide composite structures,” IEEE J. Quantum Electron. 26, 519–531 (1990).
[CrossRef]

1989

S. R. A. Dods, “Bragg reflection waveguide,” J. Opt. Soc. Am. A 6, 1465–1476 (1989).
[CrossRef]

J. Salzman and G. Lenz, “The Bragg reflection waveguide directional coupler,” IEEE Photon. Technol. Lett. 1, 319–322(1989).
[CrossRef]

1976

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Abolghasem, P.

Allen, T.

Bijlani, B.

Borel, P. I.

T. Niemi, L. H. Frandsen, K. K. Hede, A. Harpoth, P. I. Borel, and M. Kristensen, “Wavelength-division demultiplexing using photonic crystal waveguide,” IEEE Photon. Technol. Lett. 18, 226–228 (2006).
[CrossRef]

Busch, K.

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Hingerl, “Arbitrary angle waveguiding applications of two-dimensional curvilinear-lattice photonic crystals,” Appl. Phys. Lett. 84, 4687–4689 (2004).
[CrossRef]

Chen, B.

B. Chen, T. Tong, and H. Chen, “Study on a compact flexible photonic crystal waveguide and its bends,” Opt. Express 17, 5033–5038 (2009).
[CrossRef]

B. Chen, T. Tang, Z. Wang, H. Chen, and Z. Liu, “Flexible optical waveguides based on the omnidirectional reflection of one-dimensional photonic crystals,” Appl. Phys. Lett. 93, 181107 (2008).
[CrossRef]

Chen, C.

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. 17, 2039–2041 (1999).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef]

Chen, H.

B. Chen, T. Tong, and H. Chen, “Study on a compact flexible photonic crystal waveguide and its bends,” Opt. Express 17, 5033–5038 (2009).
[CrossRef]

B. Chen, T. Tang, Z. Wang, H. Chen, and Z. Liu, “Flexible optical waveguides based on the omnidirectional reflection of one-dimensional photonic crystals,” Appl. Phys. Lett. 93, 181107 (2008).
[CrossRef]

Chiang, K. S.

Clement, T. J.

DeCorby, R. G.

Dods, S. R. A.

Epp, E.

Fan, S.

Fink, Y.

Forchel, A.

J. Zimmermann, M. Kamp, A. Forchel, and R. März, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (2004).
[CrossRef]

Frandsen, L. H.

T. Niemi, L. H. Frandsen, K. K. Hede, A. Harpoth, P. I. Borel, and M. Kristensen, “Wavelength-division demultiplexing using photonic crystal waveguide,” IEEE Photon. Technol. Lett. 18, 226–228 (2006).
[CrossRef]

Gu, B.-Y.

H.-Y. Sang, Z.-Y. Li, and B.-Y. Gu, “Propagation properties of planar Bragg waveguides studied by an analytical Bloch-mode method,” J. Appl. Phys. 98, 043114 (2005).
[CrossRef]

Haakestad, M. W.

B. Nistad, M. W. Haakestad, and J. Skaar, “Dispersion properties of planar Bragg waveguides,” Opt. Commun. 265, 153–160 (2006).
[CrossRef]

Hagmann, F.

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Hingerl, “Arbitrary angle waveguiding applications of two-dimensional curvilinear-lattice photonic crystals,” Appl. Phys. Lett. 84, 4687–4689 (2004).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time Domain Method, 2nd ed. (Artech House, 2000).

Harpoth, A.

T. Niemi, L. H. Frandsen, K. K. Hede, A. Harpoth, P. I. Borel, and M. Kristensen, “Wavelength-division demultiplexing using photonic crystal waveguide,” IEEE Photon. Technol. Lett. 18, 226–228 (2006).
[CrossRef]

Haus, H. A.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

Hede, K. K.

T. Niemi, L. H. Frandsen, K. K. Hede, A. Harpoth, P. I. Borel, and M. Kristensen, “Wavelength-division demultiplexing using photonic crystal waveguide,” IEEE Photon. Technol. Lett. 18, 226–228 (2006).
[CrossRef]

Helmy, A. S.

Hingerl, K.

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Hingerl, “Arbitrary angle waveguiding applications of two-dimensional curvilinear-lattice photonic crystals,” Appl. Phys. Lett. 84, 4687–4689 (2004).
[CrossRef]

Huang, W.

Hwangbo, C. K.

Jiao, Y.

Joannopoulos, J. D.

Johnson, S. G.

Kamp, M.

J. Zimmermann, M. Kamp, A. Forchel, and R. März, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (2004).
[CrossRef]

Kim, H.

Kim, S.

Koshiba, M.

Kristensen, M.

T. Niemi, L. H. Frandsen, K. K. Hede, A. Harpoth, P. I. Borel, and M. Kristensen, “Wavelength-division demultiplexing using photonic crystal waveguide,” IEEE Photon. Technol. Lett. 18, 226–228 (2006).
[CrossRef]

Lee, E.

Lee, S.

Lenz, G.

G. Lenz and J. Salzman, “Bragg reflection waveguide composite structures,” IEEE J. Quantum Electron. 26, 519–531 (1990).
[CrossRef]

J. Salzman and G. Lenz, “The Bragg reflection waveguide directional coupler,” IEEE Photon. Technol. Lett. 1, 319–322(1989).
[CrossRef]

Li, B.

Li, B. J.

Li, J.

Li, Z.-Y.

H.-Y. Sang, Z.-Y. Li, and B.-Y. Gu, “Propagation properties of planar Bragg waveguides studied by an analytical Bloch-mode method,” J. Appl. Phys. 98, 043114 (2005).
[CrossRef]

Liu, V.

Liu, Z.

B. Chen, T. Tang, Z. Wang, H. Chen, and Z. Liu, “Flexible optical waveguides based on the omnidirectional reflection of one-dimensional photonic crystals,” Appl. Phys. Lett. 93, 181107 (2008).
[CrossRef]

März, R.

J. Zimmermann, M. Kamp, A. Forchel, and R. März, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (2004).
[CrossRef]

McMullin, J. N.

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef]

Miller, D. A. B.

Mingaleev, S. F.

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Hingerl, “Arbitrary angle waveguiding applications of two-dimensional curvilinear-lattice photonic crystals,” Appl. Phys. Lett. 84, 4687–4689 (2004).
[CrossRef]

Mizrahi, A.

Nguyen, H. T.

Niemi, T.

T. Niemi, L. H. Frandsen, K. K. Hede, A. Harpoth, P. I. Borel, and M. Kristensen, “Wavelength-division demultiplexing using photonic crystal waveguide,” IEEE Photon. Technol. Lett. 18, 226–228 (2006).
[CrossRef]

Nistad, B.

B. Nistad, M. W. Haakestad, and J. Skaar, “Dispersion properties of planar Bragg waveguides,” Opt. Commun. 265, 153–160 (2006).
[CrossRef]

Pai, M. M.

Park, B. O. S.

Park, I.

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
[CrossRef]

Ponnampalam, N.

Ripin, D. J.

Salzman, J.

G. Lenz and J. Salzman, “Bragg reflection waveguide composite structures,” IEEE J. Quantum Electron. 26, 519–531 (1990).
[CrossRef]

J. Salzman and G. Lenz, “The Bragg reflection waveguide directional coupler,” IEEE Photon. Technol. Lett. 1, 319–322(1989).
[CrossRef]

Sang, H.-Y.

H.-Y. Sang, Z.-Y. Li, and B.-Y. Gu, “Propagation properties of planar Bragg waveguides studied by an analytical Bloch-mode method,” J. Appl. Phys. 98, 043114 (2005).
[CrossRef]

Schächter, L.

Skaar, J.

B. Nistad, M. W. Haakestad, and J. Skaar, “Dispersion properties of planar Bragg waveguides,” Opt. Commun. 265, 153–160 (2006).
[CrossRef]

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time Domain Method, 2nd ed. (Artech House, 2000).

Tang, T.

B. Chen, T. Tang, Z. Wang, H. Chen, and Z. Liu, “Flexible optical waveguides based on the omnidirectional reflection of one-dimensional photonic crystals,” Appl. Phys. Lett. 93, 181107 (2008).
[CrossRef]

Thomas, E. L.

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. 17, 2039–2041 (1999).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef]

Tong, T.

Villeneuve, P. R.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

Wang, Z.

B. Chen, T. Tang, Z. Wang, H. Chen, and Z. Liu, “Flexible optical waveguides based on the omnidirectional reflection of one-dimensional photonic crystals,” Appl. Phys. Lett. 93, 181107 (2008).
[CrossRef]

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef]

J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Yariv, A.

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Yeh, P.

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Zarbakhsh, J.

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Hingerl, “Arbitrary angle waveguiding applications of two-dimensional curvilinear-lattice photonic crystals,” Appl. Phys. Lett. 84, 4687–4689 (2004).
[CrossRef]

Zeng, S.

Zhang, Y.

Zimmermann, J.

J. Zimmermann, M. Kamp, A. Forchel, and R. März, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (2004).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

B. Chen, T. Tang, Z. Wang, H. Chen, and Z. Liu, “Flexible optical waveguides based on the omnidirectional reflection of one-dimensional photonic crystals,” Appl. Phys. Lett. 93, 181107 (2008).
[CrossRef]

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Hingerl, “Arbitrary angle waveguiding applications of two-dimensional curvilinear-lattice photonic crystals,” Appl. Phys. Lett. 84, 4687–4689 (2004).
[CrossRef]

IEEE J. Quantum Electron.

G. Lenz and J. Salzman, “Bragg reflection waveguide composite structures,” IEEE J. Quantum Electron. 26, 519–531 (1990).
[CrossRef]

IEEE Photon. Technol. Lett.

T. Niemi, L. H. Frandsen, K. K. Hede, A. Harpoth, P. I. Borel, and M. Kristensen, “Wavelength-division demultiplexing using photonic crystal waveguide,” IEEE Photon. Technol. Lett. 18, 226–228 (2006).
[CrossRef]

J. Salzman and G. Lenz, “The Bragg reflection waveguide directional coupler,” IEEE Photon. Technol. Lett. 1, 319–322(1989).
[CrossRef]

J. Appl. Phys.

H.-Y. Sang, Z.-Y. Li, and B.-Y. Gu, “Propagation properties of planar Bragg waveguides studied by an analytical Bloch-mode method,” J. Appl. Phys. 98, 043114 (2005).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

J. Li and K. S. Chiang, “Light guidance in a photonic bandgap slab waveguide consisting of two different Bragg reflectors,” Opt. Commun. 281, 5797–5803 (2008).
[CrossRef]

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

B. Nistad, M. W. Haakestad, and J. Skaar, “Dispersion properties of planar Bragg waveguides,” Opt. Commun. 265, 153–160 (2006).
[CrossRef]

J. Zimmermann, M. Kamp, A. Forchel, and R. März, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230, 387–392 (2004).
[CrossRef]

Opt. Express

W. Huang, Y. Zhang, and B. Li, “Ultracompact wavelength and polarization splitters in periodic dielectric waveguide,” Opt. Express 16, 1600–1609 (2008).
[CrossRef]

H. Kim, I. Park, B. O. S. Park, E. Lee, and S. Lee, “Self-imaging phenomena in multi-mode photonic crystal line-defect waveguides: application to wavelength de-multiplexing,” Opt. Express 12, 5625–5633 (2004).
[CrossRef]

S. Zeng, Y. Zhang, and B. Li, “Self-imaging in periodic dielectric waveguides,” Opt. Express 17, 365–378 (2009).
[CrossRef]

Y. Zhang and B. J. Li, “Photonic crystal-based bending waveguides for optical interconnections,” Opt. Express 14, 5723–5732 (2006).
[CrossRef]

R. G. DeCorby, N. Ponnampalam, H. T. Nguyen, M. M. Pai, and T. J. Clement, “Guided self-assembly of integrated hollow Bragg waveguides,” Opt. Express 15, 3902–3915 (2007).
[CrossRef]

R. G. DeCorby, N. Ponnampalam, E. Epp, T. Allen, and J. N. McMullin, “Chip-scale spectrometry based on tapered hollow Bragg waveguides,” Opt. Express 17, 16632–16645 (2009).
[CrossRef]

A. Mizrahi and L. Schächter, “Bragg reflection waveguides with a matching layer,” Opt. Express 12, 3156–3170(2004).
[CrossRef]

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]

B. Chen, T. Tong, and H. Chen, “Study on a compact flexible photonic crystal waveguide and its bends,” Opt. Express 17, 5033–5038 (2009).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
[CrossRef]

Science

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef]

Other

Optiwave, “Optical FDTD,” http://www.optiwave.com/products/fdtd_overview.html .

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time Domain Method, 2nd ed. (Artech House, 2000).

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Figures (6)

Fig. 1.
Fig. 1.

(a) TE-mode dispersion curves of the BRW with the width of defect channel h 0 = 3.75 a . A schematic drawing of the BRW is shown in the inset. The geometry is uniform in the Y direction, and the light is confined in and along the defect channel (the + Z direction). n 1 = 1.6 , n 2 = 4.6 , n 0 = 1 , h 1 = 0.75 a , and h 2 = 0.25 a , where a denotes the lattice constant of the Bragg stack (the one-dimensional photonic crystal). (b) The TE-mode dispersion curves of the BRW with h 0 = 7.75 a . The other parameters are the same as those above.

Fig. 2.
Fig. 2.

Self-imaging phenomenon in the multimode BRW with h 02 = 7.75 a , which is asymmetrically connected with a single-mode BRW with h 01 = 3.75 a . The other parameters are the same as Fig. 1. The positions for a mirrored single image (symmetrical image about x = 0 ) and for a direct single image are z = L m and L d .

Fig. 3.
Fig. 3.

Self-imaging phenomena in the multimode BRW with h 02 = 7.75 a shown in Fig. 2. The time-averaged Poynting vector P z distribution (a) at the normalized frequency 0.2495 [ 2 π c / a ] and (b) at the normalized frequency 0.1931 [ 2 π c / a ] .

Fig. 4.
Fig. 4.

Schematic drawing of the wavelength splitter with [ h 01 , h 02 , L , R ] = [ 3.75 a , 9.75 a , 37.5 a , 9 a ] .

Fig. 5.
Fig. 5.

Transmission characteristics of the wavelength splitter shown in Fig. 4.

Fig. 6.
Fig. 6.

Steady-state electric field E y distribution for the wavelength splitter shown in Fig. 4 (a) at the normalized frequency 0.2495 [ 2 π c / a ] and (b) at the normalized 0.1931 [ 2 π c / a ] .

Tables (3)

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Table 1. Calculated L d at the Frequency 0.2495 [ 2 π c / a ]

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Table 2. Calculated L m at the Frequency 0.1931 [ 2 π c / a ]

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Table 3. Transmission Characteristics of the Wavelength Splitter Shown in Fig. 4

Equations (6)

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ψ ( z , x ) = n = 0 p 1 c n φ n ( x ) e j β n z ,
ψ ( L m , x ) = n = 0 p 1 c n φ n ( x ) e j β n L m = c 0 φ 0 ( x ) e j β 0 L m + c 2 φ 2 ( x ) e j β 2 L m + c 4 φ 4 ( x ) e j β 4 L m + + c 1 φ 1 ( x ) e j β 1 L m + c 3 φ 3 ( x ) e j β 3 L m + c 5 φ 5 ( x ) e j β 5 L m + = ψ ( 0 , x ) ,
ψ ( 0 , x ) = n = 0 p 1 c n φ n ( x ) = c 0 φ 0 ( x ) + c 2 φ 2 ( x ) + c 4 φ 4 ( x ) + c 1 φ 1 ( x ) c 3 φ 3 ( x ) c 5 φ 5 ( x ) .
ψ n ( x ) = { ψ n ( x ) n = 0 , 2 , 4 , ψ n ( x ) n = 1 , 3 , 5 , .
β n L m = k n π with k n = { 1 , 3 , 5 , 7 for odd modes 2 , 4 , 6 , 8 for even modes .
β n L d = k n π with k n = 2 , 4 , 6 , 8 .

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