Abstract

Self-imaging phenomena in periodic dielectric waveguides has been predicted and investigated based on multimode interference effect by using the plane wave expansion method and the finite-difference time-domain method. Asymmetric and symmetric interferences were discussed and respective imaging positions were calculated. As examples of application, a demultiplexer and a filter with ultracompact and simple structures were designed and demonstrated theoretically for optical communication wavelengths.

© 2009 Optical Society of America

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  1. R. Ulrich and G. Ankele, "Self-imaging in homogeneous planar optical waveguides," Appl. Phys. Lett. 27, 337-339 (1975).
    [CrossRef]
  2. 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]
  3. J. B. Xiao, X. Liu, and X. H. Sun, "Design of an ultracompact MMI wavelength demultiplexer in slot waveguide structures," Opt. Express 15, 8300-8308 (2007).
    [CrossRef] [PubMed]
  4. J. K. Hong and S. S. Lee, "1 × 2 Wavelength multiplexer with high transmittances using extraneous self-imaging phenomenon," J. Lightwave Technol. 25, 1264-1268 (2007).
    [CrossRef]
  5. D. J. Y.  Feng, and T. S. Lay, "Compact multimode interference couplers with arbitrary power splitting ratio," Opt. Express 16, 7175-7180 (2008).
    [CrossRef] [PubMed]
  6. X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, "Compact variable optical attenuator based on multimode interference coupler," IEEE Photon. Technol. Lett. 17, 2361-2363 (2005).
    [CrossRef]
  7. S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, "Multimode interference photonic switches," J. Lightwave Technol. 20, 675-681 (2002).
    [CrossRef]
  8. F. Wang, J. Y. Yang, L. M. Chen, X. Q. Jiang, and M. H. Wang, "Optical switch based on multimode interference coupler," IEEE Photon. Technol. Lett. 18, 421-423 (2006).
    [CrossRef]
  9. H. J. Kim, I. Park, B. H. O, S. G. Park, E. H. Lee, and S. G. Lee, "Self-imaging phenomena in multi-mode photonic crystal line-defect waveguides: application to wavelength de-multiplexing," Opt. Express 12, 5625-5633 (2004).
    [CrossRef] [PubMed]
  10. Y. Zhang, Z. Li, and B. Li, "Multimode interference effect and self-imaging principle in two-dimensional silicon photonic crystal waveguides for terahertz waves," Opt. Express 14, 2679-2689 (2006).
    [CrossRef] [PubMed]
  11. D. Modotto, M. Conforti, A. Locatelli, and C. D. Angelis, "Imaging properties of multimode photonic crystal waveguides and waveguide arrays," J. Lightwave Technol. 25, 402-409 (2007).
    [CrossRef]
  12. L. W. Chung and S. L. Lee, "Multimode-interference-based broad-band demultiplexers with internal photonic crystals," Opt. Express 14, 4923-4927 (2006).
    [CrossRef] [PubMed]
  13. Z. Li, Y. Zhang, and B. Li, "Terahertz photonic crystal switch in silicon based on self-imaging principle," Opt. Express 14, 3887-3892 (2006)
    [CrossRef] [PubMed]
  14. S. Fan, N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J.D. Joannopoulos, "Guided and defect modes in periodic dielectric waveguides," J. Opt. Soc. Am. B 12, 1267-1272 (1995).
    [CrossRef]
  15. P. G. Luan and K. D. Chang, "Transmission characteristics of finite periodic dielectric waveguides," Opt. Express 14, 3263-3272 (2006).
    [CrossRef] [PubMed]
  16. P. G. Luan and K. D Chang, "Periodic dielectric waveguide beam splitter based on co-directional coupling," Opt. Express 15, 4536-4545 (2007).
    [CrossRef] [PubMed]
  17. D. S. Gao, R. Hao, and Z. P. Zhou, "Mach-Zehnder interferometer based on coupled dielectric pillars," Chin. Phys. Lett. 24, 3172-3174 (2007).
    [CrossRef]
  18. W. Huang, Y. Zhang, and B. Li, "Ultracompact wavelength and polarization splitters in periodic dielectric waveguides," Opt. Express 16, 1600-1609 (2008).
    [CrossRef] [PubMed]
  19. Y. Zhang, W. Huang, and B. Li, "Fabry-Pérot microcavities with controllable resonant wavelengths in periodic dielectric waveguides," Appl. Phys. Lett. 93, 0311101-3 (2008).
    [CrossRef]
  20. 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] [PubMed]
  21. A. Lavrinenko, P. I. Borel, L. H. Frandsen, M. Thorhauge, A. Harpøth, M. Kristensen, T. Niemi, and H. M. H. Chong, "Comprehensive FDTD modelling of photonic crystal waveguide components," Opt. Express 12, 234-248 (2004).
    [CrossRef] [PubMed]
  22. D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Nanopillars photonic crystal waveguides," Opt. Express 12, 617-622 (2004).
    [CrossRef] [PubMed]
  23. D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Numerical characterization of nanopillar photonic crystal waveguides and directional couplers," Opt. Quant. Electron. 37, 331-341 (2005).
    [CrossRef]
  24. J. Smajic, C. Hafner, and D. Erni, "On the design of photonic crystal multiplexers," Opt. Express 11, 566-571 (2003).
    [CrossRef] [PubMed]
  25. M. Y. Tekeste and J. M. Yarrison-Rice, "High efficiency photonic crystal based wavelength demultiplexer," Opt. Express 14, 7931-7942 (2006).
    [CrossRef] [PubMed]
  26. R. Costa, A. Melloni, and M. Martinelli, "Bandpass resonant filters in photonic-crystal waveguides," IEEE Photon. Technol. Lett. 15, 401-403 (2003).
    [CrossRef]

2008 (3)

2007 (5)

2006 (6)

2005 (2)

D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Numerical characterization of nanopillar photonic crystal waveguides and directional couplers," Opt. Quant. Electron. 37, 331-341 (2005).
[CrossRef]

X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, "Compact variable optical attenuator based on multimode interference coupler," IEEE Photon. Technol. Lett. 17, 2361-2363 (2005).
[CrossRef]

2004 (3)

2003 (2)

J. Smajic, C. Hafner, and D. Erni, "On the design of photonic crystal multiplexers," Opt. Express 11, 566-571 (2003).
[CrossRef] [PubMed]

R. Costa, A. Melloni, and M. Martinelli, "Bandpass resonant filters in photonic-crystal waveguides," IEEE Photon. Technol. Lett. 15, 401-403 (2003).
[CrossRef]

2002 (1)

2001 (1)

1995 (2)

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]

S. Fan, N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J.D. Joannopoulos, "Guided and defect modes in periodic dielectric waveguides," J. Opt. Soc. Am. B 12, 1267-1272 (1995).
[CrossRef]

1975 (1)

R. Ulrich and G. Ankele, "Self-imaging in homogeneous planar optical waveguides," Appl. Phys. Lett. 27, 337-339 (1975).
[CrossRef]

Angelis, C. D.

Ankele, G.

R. Ulrich and G. Ankele, "Self-imaging in homogeneous planar optical waveguides," Appl. Phys. Lett. 27, 337-339 (1975).
[CrossRef]

Borel, P. I.

Chang, K. D

Chang, K. D.

Chen, J. C.

Chen, L. M.

F. Wang, J. Y. Yang, L. M. Chen, X. Q. Jiang, and M. H. Wang, "Optical switch based on multimode interference coupler," IEEE Photon. Technol. Lett. 18, 421-423 (2006).
[CrossRef]

Chigrin, D. N.

D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Numerical characterization of nanopillar photonic crystal waveguides and directional couplers," Opt. Quant. Electron. 37, 331-341 (2005).
[CrossRef]

D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Nanopillars photonic crystal waveguides," Opt. Express 12, 617-622 (2004).
[CrossRef] [PubMed]

Chong, H. M. H.

Chung, L. W.

Conforti, M.

Costa, R.

R. Costa, A. Melloni, and M. Martinelli, "Bandpass resonant filters in photonic-crystal waveguides," IEEE Photon. Technol. Lett. 15, 401-403 (2003).
[CrossRef]

Devenyi, A.

Erni, D.

Fan, S.

Feng, D. J. Y.

Frandsen, L. H.

Gao, D. S.

D. S. Gao, R. Hao, and Z. P. Zhou, "Mach-Zehnder interferometer based on coupled dielectric pillars," Chin. Phys. Lett. 24, 3172-3174 (2007).
[CrossRef]

Hafner, C.

Hao, R.

D. S. Gao, R. Hao, and Z. P. Zhou, "Mach-Zehnder interferometer based on coupled dielectric pillars," Chin. Phys. Lett. 24, 3172-3174 (2007).
[CrossRef]

Harpøth, A.

Hong, J. K.

Huang, W.

W. Huang, Y. Zhang, and B. Li, "Ultracompact wavelength and polarization splitters in periodic dielectric waveguides," Opt. Express 16, 1600-1609 (2008).
[CrossRef] [PubMed]

Y. Zhang, W. Huang, and B. Li, "Fabry-Pérot microcavities with controllable resonant wavelengths in periodic dielectric waveguides," Appl. Phys. Lett. 93, 0311101-3 (2008).
[CrossRef]

Inayoshi, H.

Ishikawa, S.

X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, "Compact variable optical attenuator based on multimode interference coupler," IEEE Photon. Technol. Lett. 17, 2361-2363 (2005).
[CrossRef]

Jiang, X. Q.

F. Wang, J. Y. Yang, L. M. Chen, X. Q. Jiang, and M. H. Wang, "Optical switch based on multimode interference coupler," IEEE Photon. Technol. Lett. 18, 421-423 (2006).
[CrossRef]

X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, "Compact variable optical attenuator based on multimode interference coupler," IEEE Photon. Technol. Lett. 17, 2361-2363 (2005).
[CrossRef]

Joannopoulos, J. D.

Joannopoulos, J.D.

Johnson, S. G.

Kim, H. J.

Kristensen, M.

Lavrinenko, A.

Lavrinenko, A. V.

D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Numerical characterization of nanopillar photonic crystal waveguides and directional couplers," Opt. Quant. Electron. 37, 331-341 (2005).
[CrossRef]

D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Nanopillars photonic crystal waveguides," Opt. Express 12, 617-622 (2004).
[CrossRef] [PubMed]

Lay, T. S.

Lee, S. L.

Lee, S. S.

Li, B.

Li, X.

X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, "Compact variable optical attenuator based on multimode interference coupler," IEEE Photon. Technol. Lett. 17, 2361-2363 (2005).
[CrossRef]

Li, Z.

Liu, X.

Locatelli, A.

Luan, P. G.

Martinelli, M.

R. Costa, A. Melloni, and M. Martinelli, "Bandpass resonant filters in photonic-crystal waveguides," IEEE Photon. Technol. Lett. 15, 401-403 (2003).
[CrossRef]

Meade, R. D.

Melloni, A.

R. Costa, A. Melloni, and M. Martinelli, "Bandpass resonant filters in photonic-crystal waveguides," IEEE Photon. Technol. Lett. 15, 401-403 (2003).
[CrossRef]

Modotto, D.

Morishima, G.

Nagai, S.

Niemi, T.

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]

Smajic, J.

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]

Sotomayor Torres, C. M.

D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Numerical characterization of nanopillar photonic crystal waveguides and directional couplers," Opt. Quant. Electron. 37, 331-341 (2005).
[CrossRef]

D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Nanopillars photonic crystal waveguides," Opt. Express 12, 617-622 (2004).
[CrossRef] [PubMed]

Sun, X. H.

Tekeste, M. Y.

Thorhauge, M.

Ulrich, R.

R. Ulrich and G. Ankele, "Self-imaging in homogeneous planar optical waveguides," Appl. Phys. Lett. 27, 337-339 (1975).
[CrossRef]

Utaka, K.

Wang, F.

F. Wang, J. Y. Yang, L. M. Chen, X. Q. Jiang, and M. H. Wang, "Optical switch based on multimode interference coupler," IEEE Photon. Technol. Lett. 18, 421-423 (2006).
[CrossRef]

Wang, M. H.

F. Wang, J. Y. Yang, L. M. Chen, X. Q. Jiang, and M. H. Wang, "Optical switch based on multimode interference coupler," IEEE Photon. Technol. Lett. 18, 421-423 (2006).
[CrossRef]

X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, "Compact variable optical attenuator based on multimode interference coupler," IEEE Photon. Technol. Lett. 17, 2361-2363 (2005).
[CrossRef]

Winn, N.

Wu, Y. Y.

X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, "Compact variable optical attenuator based on multimode interference coupler," IEEE Photon. Technol. Lett. 17, 2361-2363 (2005).
[CrossRef]

Xiao, J. B.

Yang, J. Y.

F. Wang, J. Y. Yang, L. M. Chen, X. Q. Jiang, and M. H. Wang, "Optical switch based on multimode interference coupler," IEEE Photon. Technol. Lett. 18, 421-423 (2006).
[CrossRef]

X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, "Compact variable optical attenuator based on multimode interference coupler," IEEE Photon. Technol. Lett. 17, 2361-2363 (2005).
[CrossRef]

Yarrison-Rice, J. M.

Zhang, Y.

Zhou, H. F.

X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, "Compact variable optical attenuator based on multimode interference coupler," IEEE Photon. Technol. Lett. 17, 2361-2363 (2005).
[CrossRef]

Zhou, Z. P.

D. S. Gao, R. Hao, and Z. P. Zhou, "Mach-Zehnder interferometer based on coupled dielectric pillars," Chin. Phys. Lett. 24, 3172-3174 (2007).
[CrossRef]

Appl. Phys. Lett. (2)

R. Ulrich and G. Ankele, "Self-imaging in homogeneous planar optical waveguides," Appl. Phys. Lett. 27, 337-339 (1975).
[CrossRef]

Y. Zhang, W. Huang, and B. Li, "Fabry-Pérot microcavities with controllable resonant wavelengths in periodic dielectric waveguides," Appl. Phys. Lett. 93, 0311101-3 (2008).
[CrossRef]

Chin. Phys. Lett. (1)

D. S. Gao, R. Hao, and Z. P. Zhou, "Mach-Zehnder interferometer based on coupled dielectric pillars," Chin. Phys. Lett. 24, 3172-3174 (2007).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

R. Costa, A. Melloni, and M. Martinelli, "Bandpass resonant filters in photonic-crystal waveguides," IEEE Photon. Technol. Lett. 15, 401-403 (2003).
[CrossRef]

X. Q. Jiang, X. Li, H. F. Zhou, J. Y. Yang, M. H. Wang, Y. Y. Wu, and S. Ishikawa, "Compact variable optical attenuator based on multimode interference coupler," IEEE Photon. Technol. Lett. 17, 2361-2363 (2005).
[CrossRef]

F. Wang, J. Y. Yang, L. M. Chen, X. Q. Jiang, and M. H. Wang, "Optical switch based on multimode interference coupler," IEEE Photon. Technol. Lett. 18, 421-423 (2006).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. B (1)

Opt. Express (14)

P. G. Luan and K. D. Chang, "Transmission characteristics of finite periodic dielectric waveguides," Opt. Express 14, 3263-3272 (2006).
[CrossRef] [PubMed]

P. G. Luan and K. D Chang, "Periodic dielectric waveguide beam splitter based on co-directional coupling," Opt. Express 15, 4536-4545 (2007).
[CrossRef] [PubMed]

L. W. Chung and S. L. Lee, "Multimode-interference-based broad-band demultiplexers with internal photonic crystals," Opt. Express 14, 4923-4927 (2006).
[CrossRef] [PubMed]

Z. Li, Y. Zhang, and B. Li, "Terahertz photonic crystal switch in silicon based on self-imaging principle," Opt. Express 14, 3887-3892 (2006)
[CrossRef] [PubMed]

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] [PubMed]

A. Lavrinenko, P. I. Borel, L. H. Frandsen, M. Thorhauge, A. Harpøth, M. Kristensen, T. Niemi, and H. M. H. Chong, "Comprehensive FDTD modelling of photonic crystal waveguide components," Opt. Express 12, 234-248 (2004).
[CrossRef] [PubMed]

D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Nanopillars photonic crystal waveguides," Opt. Express 12, 617-622 (2004).
[CrossRef] [PubMed]

D. J. Y.  Feng, and T. S. Lay, "Compact multimode interference couplers with arbitrary power splitting ratio," Opt. Express 16, 7175-7180 (2008).
[CrossRef] [PubMed]

J. B. Xiao, X. Liu, and X. H. Sun, "Design of an ultracompact MMI wavelength demultiplexer in slot waveguide structures," Opt. Express 15, 8300-8308 (2007).
[CrossRef] [PubMed]

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

Y. Zhang, Z. Li, and B. Li, "Multimode interference effect and self-imaging principle in two-dimensional silicon photonic crystal waveguides for terahertz waves," Opt. Express 14, 2679-2689 (2006).
[CrossRef] [PubMed]

W. Huang, Y. Zhang, and B. Li, "Ultracompact wavelength and polarization splitters in periodic dielectric waveguides," Opt. Express 16, 1600-1609 (2008).
[CrossRef] [PubMed]

J. Smajic, C. Hafner, and D. Erni, "On the design of photonic crystal multiplexers," Opt. Express 11, 566-571 (2003).
[CrossRef] [PubMed]

M. Y. Tekeste and J. M. Yarrison-Rice, "High efficiency photonic crystal based wavelength demultiplexer," Opt. Express 14, 7931-7942 (2006).
[CrossRef] [PubMed]

Opt. Quant. Electron. (1)

D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, "Numerical characterization of nanopillar photonic crystal waveguides and directional couplers," Opt. Quant. Electron. 37, 331-341 (2005).
[CrossRef]

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Figures (12)

Fig. 1.
Fig. 1.

(a) A single row of PD rods in air (single PDW). (b) Four parallel rows of PD rods in air (multimode PDW). (c) Band structure for TM mode in the single PDW. The orange line is light line, the shaded region is for extended modes, and the solid curve below the light line is for guided mode. (d) Band structure for TM mode in the multimode PDW. The solid curves below the light line are guided modes for the 0th to 3rd band modes. The insets denote the supercells used for calculations.

Fig. 2.
Fig. 2.

(a) A single row of PD rods in air (single PDW). (b) Five parallel rows of PD rods in air (multimode PDW). (c) Band structure for TM mode in single PDW. The orange line is light line, the shaded region represents extended modes, and the solid curve below the light line is guided mode. (d) Band structure for TM mode in multimode PDW. The solid curves below the light line are five guided modes (0th to 4th modes). The insets denote the supercells used for calculations.

Fig. 3.
Fig. 3.

(a) An asymmetric multimode PDW structure (model-I). The length and width of the multimode region are 65a and 3a, respectively. (b) Scheme of the imaged optical field distribution in the multimode region. Single images are reproduced at x = Lm and at x = Ld .

Fig. 4.
Fig. 4.

The simulated steady-state electric field distributions and Poynting vector (x-component) distributions in the asymmetric multimode PDW structure (model-I) showed in Fig. 3(a). (a) Steady-state electric field distribution at 0.132(a/λ), (b) Poynting vector distribution at 0.132(a/λ). (c) Steady-state electric field distribution at 0.156(a/λ), (d) Poynting vector distribution at 0.156(a/λ).

Fig. 5.
Fig. 5.

(a) A symmetric multimode PDW structure (model-II). Length and width of the multimode region are 50a and 6a, respectively. (b) Scheme of the imaged optical field distribution in the multimode region. Single images are reproduced at x = Ls and two-fold images were reproduced at x = Lf .

Fig. 6.
Fig. 6.

The simulated steady-state electric field distributions and Poynting vector (x-component) distributions in the symmetric multimode PDW structure (model-II) showed in Fig. 5(a). (a) Steady-state electric field distribution at 0.119(a/λ), (b) Poynting vector distribution at 0.119(a/λ). (c) Steady-state electric field distribution at 0.140(a/λ), (d) Poynting vector distribution at 0.140(a/λ).

Fig. 7.
Fig. 7.

Schematic diagram of the designed wavelength demultiplexer in PDWs.

Fig. 8.
Fig. 8.

The simulated steady-state electric field distributions in the wavelength demultiplexer, (a) for λ 1 = 1550 nm, and (b) for λ 2 = 1310 nm.

Fig. 9.
Fig. 9.

The calculated normalized powers at the outputs of the wavelength demultiplexer. The solid lines are for the multimode region with L = 49a while the dashed lines are for L = 50a.

Fig. 10.
Fig. 10.

Schematic diagram of the designed wavelength filter in PDWs.

Fig. 11.
Fig. 11.

The simulated steady-state electric field distributions in the wavelength filter, (a) for λ 1 = 1550 nm, and (b) for λ 2 = 1310 nm.

Fig. 12.
Fig. 12.

The calculated normalized power at the output of the wavelength filter as a function of wavelength from 1260 to 1600 nm.

Tables (4)

Tables Icon

Table 1. Parameters and calculated results for average value of Ld at 0.132(a/λ)

Tables Icon

Table 2. Parameters and calculated results for average value of Lm at 0.156(a/λ)

Tables Icon

Table 3. Parameters and calculated results for average value of L s1 at 0.119(a/λ)

Tables Icon

Table 4. Parameters and calculated results for average value of L s1 at 0.140(a/λ)

Equations (5)

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

βnLm=knπ withkn={1,3,5,7fornodd2,4,6,8forneven
βnLd=knπ withkn=2,4,6,8……
βnLs=knπ withkn={1,3,5,7formirroredimage2,4,6,8fordirectimage
Lf1=Ls1/2
Lfk=(k1/2) Ls1 , k=1or2

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