Abstract

A novel hybrid diffraction method is introduced to simulate the diffraction and imaging of a planar-integrated concave grating that has total internal reflection (TIR) facets. The Kirchhoff–Huygens diffraction formula is adopted to simulate the propagation of the lightwave field in the free-propagation region, and a rigorous coupled-wave analysis is used to calculate the polarization-dependent diffraction by the grating. The hybrid diffraction method can be used to analyze accurately the imaging properties as well as the polarization-dependent diffraction characteristics of a concave grating. The dependence of several merit parameters of a concave grating with TIR facets on its basic geometric parameters is studied. Compared with one with metallic echelle facets, a concave grating with TIR facets shows a much lower polarization-dependent loss. Since more performance specifications can be considered in the design of a concave grating than with the conventional scalar method, design error can be reduced greatly with the present hybrid diffraction method.

© 2004 Optical Society of America

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References

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  1. M. K. Smit, C. van Dam, “Phasar-based WDM-devices: principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. 2, 236–250 (1996).
    [CrossRef]
  2. C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stall, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
    [CrossRef]
  3. J.-J. He, E. S. Koteles, B. Lamontagne, L. Erickson, A. Delâge, M. Davies, “Integrated polarisation compensator for WDM waveguide demultiplexers,” IEEE Photon. Technol. Lett. 11, 224–226 (1999).
    [CrossRef]
  4. S. Y. Sadov, K. A. McGreer, “Polarization dependence of diffraction gratings that have total internal reflection facets,” J. Opt. Soc. Am. A 17, 1590–1594 (2002).
    [CrossRef]
  5. M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photon. Technol. Lett. 11, 84–86 (1999).
    [CrossRef]
  6. M. C. Hutley, Diffraction Gratings (Academic, London, 1982).
  7. A. C. McGreer, “Diffraction from concave gratings in planar waveguides,” IEEE Photon. Technol. Lett. 7, 324–326 (1995).
    [CrossRef]
  8. J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. S. Koteles, “Sources of crosstalk in grating based monolithic integrated wavelength demultiplexers,” in Application of Photonic Technology 3: Closing the Gap between Theory, Development and Application, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 593–598 (1998).
  9. K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
    [CrossRef]
  10. S.-D. Wu, E. N. Glytsis, “Finite-number-of-periods holographic gratings with finite-width incident beams: analysis using the finite-difference frequency-domain method,” J. Opt. Soc. Am. A 19, 2018–2029 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
    [CrossRef]
  14. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  15. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  16. P. Lalanne, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  17. E. Popov, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. L. Shen, S. He, “Analysis for the convergence problem of the plane-wave expansion method for photonic crystals,” J. Opt. Soc. Am. A 19, 1021–1024 (2002).
    [CrossRef]
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  26. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980), p. 179.
  27. Z. Shi, S. He, “A three-focal-point method for the optimal design of a flat-top planar waveguide demultiplexer,” IEEE J. Sel. Top. Quantum Electron. 8, 1179–1185 (2002).
    [CrossRef]
  28. Z. Shi, J.-J. He, S. He, “An analytic method for designing passband flattened DWDM demultiplexers using spatial phase modulation,” J. Lightwave Technol. 21, 2314–2321 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  31. T. Ikegami, “Reflectivity of mode at facet on oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
    [CrossRef]

2003

2002

2000

D. Chowdhury, “Design of low-loss and polarization-insensitive reflection grating-based planar demultiplexers,” IEEE J. Sel. Top. Quantum Electron. 6, 233–239 (2000).
[CrossRef]

E. Popov, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

1999

J.-J. He, E. S. Koteles, B. Lamontagne, L. Erickson, A. Delâge, M. Davies, “Integrated polarisation compensator for WDM waveguide demultiplexers,” IEEE Photon. Technol. Lett. 11, 224–226 (1999).
[CrossRef]

M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photon. Technol. Lett. 11, 84–86 (1999).
[CrossRef]

1998

1997

1996

1995

1993

W. J. Tsay, D. M. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microwave Guid. Wave Lett. 3, 250–252 (1993).
[CrossRef]

1992

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stall, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

R. Marz, C. Cremer, “On the theory of planar spectrographs,” J. Lightwave Technol. 10, 2017–2022 (1992).
[CrossRef]

1986

1972

T. Ikegami, “Reflectivity of mode at facet on oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1965).

Chowdhury, D.

D. Chowdhury, “Design of low-loss and polarization-insensitive reflection grating-based planar demultiplexers,” IEEE J. Sel. Top. Quantum Electron. 6, 233–239 (2000).
[CrossRef]

Costa, R.

Cremer, C.

R. Marz, C. Cremer, “On the theory of planar spectrographs,” J. Lightwave Technol. 10, 2017–2022 (1992).
[CrossRef]

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stall, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

Davies, M.

J.-J. He, E. S. Koteles, B. Lamontagne, L. Erickson, A. Delâge, M. Davies, “Integrated polarisation compensator for WDM waveguide demultiplexers,” IEEE Photon. Technol. Lett. 11, 224–226 (1999).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delâge, L. Erickson, M. Davies, E. S. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide rowland circle grating in InGaAsP/InP,” J. Lightwave Technol. 16, 631–638 (1998).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. S. Koteles, “Sources of crosstalk in grating based monolithic integrated wavelength demultiplexers,” in Application of Photonic Technology 3: Closing the Gap between Theory, Development and Application, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 593–598 (1998).

Delage, A.

J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. S. Koteles, “Sources of crosstalk in grating based monolithic integrated wavelength demultiplexers,” in Application of Photonic Technology 3: Closing the Gap between Theory, Development and Application, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 593–598 (1998).

Delâge, A.

J.-J. He, E. S. Koteles, B. Lamontagne, L. Erickson, A. Delâge, M. Davies, “Integrated polarisation compensator for WDM waveguide demultiplexers,” IEEE Photon. Technol. Lett. 11, 224–226 (1999).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delâge, L. Erickson, M. Davies, E. S. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide rowland circle grating in InGaAsP/InP,” J. Lightwave Technol. 16, 631–638 (1998).
[CrossRef]

Ebbinghaus, G.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stall, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

Emeis, N.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stall, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

Erickson, L.

J.-J. He, E. S. Koteles, B. Lamontagne, L. Erickson, A. Delâge, M. Davies, “Integrated polarisation compensator for WDM waveguide demultiplexers,” IEEE Photon. Technol. Lett. 11, 224–226 (1999).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delâge, L. Erickson, M. Davies, E. S. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide rowland circle grating in InGaAsP/InP,” J. Lightwave Technol. 16, 631–638 (1998).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. S. Koteles, “Sources of crosstalk in grating based monolithic integrated wavelength demultiplexers,” in Application of Photonic Technology 3: Closing the Gap between Theory, Development and Application, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 593–598 (1998).

Gaylord, T. K.

Glytsis, E. N.

Grann, E. B.

Haus, H.

H. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984), p. 48.

He, J.-J.

Z. Shi, J.-J. He, S. He, “An analytic method for designing passband flattened DWDM demultiplexers using spatial phase modulation,” J. Lightwave Technol. 21, 2314–2321 (2003).
[CrossRef]

J.-J. He, E. S. Koteles, B. Lamontagne, L. Erickson, A. Delâge, M. Davies, “Integrated polarisation compensator for WDM waveguide demultiplexers,” IEEE Photon. Technol. Lett. 11, 224–226 (1999).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delâge, L. Erickson, M. Davies, E. S. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide rowland circle grating in InGaAsP/InP,” J. Lightwave Technol. 16, 631–638 (1998).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. S. Koteles, “Sources of crosstalk in grating based monolithic integrated wavelength demultiplexers,” in Application of Photonic Technology 3: Closing the Gap between Theory, Development and Application, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 593–598 (1998).

He, S.

Heise, G.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stall, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

Hirayama, K.

Hunsperger, R. G.

R. G. Hunsperger, Integrated Optics: Theory and Technology, 2nd ed. (Springer-Verlag, Berlin, 1984), p. 89.

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, London, 1982).

Ikegami, T.

T. Ikegami, “Reflectivity of mode at facet on oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
[CrossRef]

Koteles, E. S.

J.-J. He, E. S. Koteles, B. Lamontagne, L. Erickson, A. Delâge, M. Davies, “Integrated polarisation compensator for WDM waveguide demultiplexers,” IEEE Photon. Technol. Lett. 11, 224–226 (1999).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delâge, L. Erickson, M. Davies, E. S. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide rowland circle grating in InGaAsP/InP,” J. Lightwave Technol. 16, 631–638 (1998).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. S. Koteles, “Sources of crosstalk in grating based monolithic integrated wavelength demultiplexers,” in Application of Photonic Technology 3: Closing the Gap between Theory, Development and Application, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 593–598 (1998).

Lalanne, P.

Lamontagne, B.

J.-J. He, E. S. Koteles, B. Lamontagne, L. Erickson, A. Delâge, M. Davies, “Integrated polarisation compensator for WDM waveguide demultiplexers,” IEEE Photon. Technol. Lett. 11, 224–226 (1999).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delâge, L. Erickson, M. Davies, E. S. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide rowland circle grating in InGaAsP/InP,” J. Lightwave Technol. 16, 631–638 (1998).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. S. Koteles, “Sources of crosstalk in grating based monolithic integrated wavelength demultiplexers,” in Application of Photonic Technology 3: Closing the Gap between Theory, Development and Application, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 593–598 (1998).

Li, L.

Lierstuen, L. O.

Martinelli, M.

Marz, R.

R. Marz, C. Cremer, “On the theory of planar spectrographs,” J. Lightwave Technol. 10, 2017–2022 (1992).
[CrossRef]

McGreer, A. C.

A. C. McGreer, “Diffraction from concave gratings in planar waveguides,” IEEE Photon. Technol. Lett. 7, 324–326 (1995).
[CrossRef]

McGreer, K. A.

Melloni, A.

Moharam, M. G.

Monguzzi, P.

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, New York, 2000), p. 252.

Pommet, D. A.

Popov, E.

Pozar, D. M.

W. J. Tsay, D. M. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microwave Guid. Wave Lett. 3, 250–252 (1993).
[CrossRef]

Sadov, S. Y.

Schier, M.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stall, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

Shen, L.

Shi, Z.

Z. Shi, J.-J. He, S. He, “An analytic method for designing passband flattened DWDM demultiplexers using spatial phase modulation,” J. Lightwave Technol. 21, 2314–2321 (2003).
[CrossRef]

Z. Shi, S. He, “A three-focal-point method for the optimal design of a flat-top planar waveguide demultiplexer,” IEEE J. Sel. Top. Quantum Electron. 8, 1179–1185 (2002).
[CrossRef]

Smit, M. K.

M. K. Smit, C. van Dam, “Phasar-based WDM-devices: principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. 2, 236–250 (1996).
[CrossRef]

Smith, M. S. D.

M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photon. Technol. Lett. 11, 84–86 (1999).
[CrossRef]

Stall, L.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stall, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

Sudbo, A. Sv.

Tsay, W. J.

W. J. Tsay, D. M. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microwave Guid. Wave Lett. 3, 250–252 (1993).
[CrossRef]

van Dam, C.

M. K. Smit, C. van Dam, “Phasar-based WDM-devices: principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. 2, 236–250 (1996).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1965).

Wu, S.-D.

Appl. Opt.

IEEE J. Quantum Electron.

T. Ikegami, “Reflectivity of mode at facet on oscillation mode in double-heterostructure injection lasers,” IEEE J. Quantum Electron. QE-8, 470–476 (1972).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

D. Chowdhury, “Design of low-loss and polarization-insensitive reflection grating-based planar demultiplexers,” IEEE J. Sel. Top. Quantum Electron. 6, 233–239 (2000).
[CrossRef]

Z. Shi, S. He, “A three-focal-point method for the optimal design of a flat-top planar waveguide demultiplexer,” IEEE J. Sel. Top. Quantum Electron. 8, 1179–1185 (2002).
[CrossRef]

M. K. Smit, C. van Dam, “Phasar-based WDM-devices: principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. 2, 236–250 (1996).
[CrossRef]

IEEE Microwave Guid. Wave Lett.

W. J. Tsay, D. M. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microwave Guid. Wave Lett. 3, 250–252 (1993).
[CrossRef]

IEEE Photon. Technol. Lett.

C. Cremer, N. Emeis, M. Schier, G. Heise, G. Ebbinghaus, L. Stall, “Grating spectrograph integrated with photodiode array in InGaAsP/InGaAs/InP,” IEEE Photon. Technol. Lett. 4, 108–110 (1992).
[CrossRef]

J.-J. He, E. S. Koteles, B. Lamontagne, L. Erickson, A. Delâge, M. Davies, “Integrated polarisation compensator for WDM waveguide demultiplexers,” IEEE Photon. Technol. Lett. 11, 224–226 (1999).
[CrossRef]

M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photon. Technol. Lett. 11, 84–86 (1999).
[CrossRef]

A. C. McGreer, “Diffraction from concave gratings in planar waveguides,” IEEE Photon. Technol. Lett. 7, 324–326 (1995).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

A. Melloni, P. Monguzzi, R. Costa, M. Martinelli, “Design of curved waveguides: the matched bend,” J. Opt. Soc. Am. A 20, 130–137 (2003).
[CrossRef]

K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
[CrossRef]

S.-D. Wu, E. N. Glytsis, “Finite-number-of-periods holographic gratings with finite-width incident beams: analysis using the finite-difference frequency-domain method,” J. Opt. Soc. Am. A 19, 2018–2029 (2002).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
[CrossRef]

M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

P. Lalanne, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

E. Popov, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

L. Shen, S. He, “Analysis for the convergence problem of the plane-wave expansion method for photonic crystals,” J. Opt. Soc. Am. A 19, 1021–1024 (2002).
[CrossRef]

S. Y. Sadov, K. A. McGreer, “Polarization dependence of diffraction gratings that have total internal reflection facets,” J. Opt. Soc. Am. A 17, 1590–1594 (2002).
[CrossRef]

Other

J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. S. Koteles, “Sources of crosstalk in grating based monolithic integrated wavelength demultiplexers,” in Application of Photonic Technology 3: Closing the Gap between Theory, Development and Application, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 593–598 (1998).

M. C. Hutley, Diffraction Gratings (Academic, London, 1982).

R. G. Hunsperger, Integrated Optics: Theory and Technology, 2nd ed. (Springer-Verlag, Berlin, 1984), p. 89.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, New York, 2000), p. 252.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1965).

H. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984), p. 48.

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980), p. 179.

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of a concave grating in a Rowland mount.

Fig. 2
Fig. 2

Schematic diagram of the hybrid diffraction method.

Fig. 3
Fig. 3

Geometry of a straight TIR-blazed grating analyzed with the RCWA.

Fig. 4
Fig. 4

Retrodiffraction efficiency as neff increases for (a) TIR grating, (b) metallic echelle grating, (c) dielectric echelle grating.

Fig. 5
Fig. 5

The retrodiffraction efficiency of a TIR grating as αg increases.

Fig. 6
Fig. 6

PDL as αg increases for (a) TIR grating, (b) metallic echelle grating.

Fig. 7
Fig. 7

Diffraction efficiencies at different orders as αg increases (with mb=12) for (a) TIR grating, (b) metallic echelle grating.

Fig. 8
Fig. 8

Maximal allowable diffraction order as the relative wavelength deviation (|Δλ|max/Δλc) increases.

Fig. 9
Fig. 9

(a) Geometric parameters of the grating tooth as θd varies. (b) Phase difference and the relative magnitude difference of the retrodiffraction coefficient (with respect to that at the grating pole) as θd varies. (c) Spectral response of the designed concave-grating demultiplexer at the central channel in TE case calculated by the hybrid diffraction method with two different treatments in step 3.

Fig. 10
Fig. 10

Retrodiffraction efficiency of the designed concave-grating demultiplexer with αg=20° and mb=12 as wavelength increases.

Fig. 11
Fig. 11

Image fields and spectral responses at the -20th, 0th, and 20th channels of a designed concave-TIR-grating demultiplexer with αg=20° and mb=12.

Tables (1)

Tables Icon

Table 1 Performance Statistics for a Designed Concave-TIR-Grating Demultiplexer a

Equations (16)

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

sin αin+sin αdiff,0=mbλ0neffΛ,
u/λ |rout,c-r0|mb/(neffΛ cos αdiff,0),
|rout,c-r0| (u/λ)λ0cos αdiff,0(sin αin+sin αdiff,0).
αg(αin+αdiff,0)/2αinαdiff,0.
|rl-og| =2rRc
|rl-rin,c|+|rl-rout,c|-|r0-rin,c|
+|r0-rout,c| =lmbλ0/neff,
Eicp(ric)=12neffpλ1/2inputEinp(rin)|rin-ric| (1+cos θd)×exp(-jneffp k0|rin-ric|)dl,
2z2 [Hly]=l-1-1(Kl-1K-I)[Hly],
lm,p=Λ-1-Λ/2Λ/2l(x)exp-j2π (m-p)xΛdx
l-1m,p=Λ-1-Λ/2Λ/2[1/l(x)]exp-j2π (m-p)xΛdx,
HI,diffy=mRmTMexp[-j(kmxx-kmI,zz)],
Ediff,hp(ric)=mEdiffp,m(rref,h)×exp[-jkhm(ric-rref,h)]=mEicp(rref,h)Rm,hp×exp[-jkhm(ric-rref,h)],
Eoutp(rout)=12neffλ1/2hmEic(rref,h)Rm,hpexp[-jkhm(ric-rref,h)](cos αim+cos αd)(|ric-rout|)1/2exp(-jk|ric-rout|)dl,
Ip(λ)=imageplaneEoutp(rout)E0p*(rout)dl2inputplaneEinp(rin)Einp*(rin)dlimageplaneE0p(rout)E0p*(rout)dl,
emp(λ)em,maxpsin[mπ(λ-λmaxp)/λ]mπ(λ-λmaxp)/λ2,

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