Abstract

A volume grating for outcoupling and line focusing of waveguided infrared light is designed and optimized. A local grating vector approach is used in combination with the rigorous coupled-wave analysis. By design, this volume grating coupler is holographically constructed on top of a waveguide by the interference of two coherent 364-nm ultraviolet waves formed with two aberration-optimized cylindrical lenses. This focusing coupler exhibits preferential-order coupling (92.9%) into the cover as well as very low focal intensity side lobes. This is accomplished through a chirped, slanted-fringe volume grating with a designed spatial variation in the attenuation coefficient (describing the outcoupling of the guided mode) along the length of the grating. This is achieved by a specific variation in the grating slant angle along the grating length. By design, the 1000-μm-length coupler focuses an 850-nm infrared guided wave to a line with an intensity FWHM of 3.32 μm and a 90% power width of 5.53 μm at a focal distance of 4 mm directly above the grating. Its performance is compared with that of a corresponding electron-beam-written surface-relief coupler design.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. Streibl, R. Volkel, J. Schwider, P. Habel, N. Lindlein, “Parallel optoelectronic interconnections with high packing density through a light-guiding plate using grating couplers and field lenses,” Opt. Commun. 99, 167–171 (1993).
    [CrossRef]
  2. S. H. Song, S. D. Jung, “Back-board optical signal interconnection module using focusing grating coupler arrays,” U.S. patent5,469,518 (21November1995).
  3. C. Zhao, R. Chen, “Performance consideration of three-dimensional optoelectronic interconnection for intra-multichip-module clock signal distribution,” Appl. Opt. 36, 2537–2544 (1997).
    [CrossRef] [PubMed]
  4. S. Ura, T. Suhara, H. Nishihara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913–918 (1986).
    [CrossRef]
  5. S. Nishiwaki, J. Asada, S. Uchida, “Optical head employing a concentric-circular focusing grating coupler,” Appl. Opt. 33, 1819–1827 (1994).
    [CrossRef] [PubMed]
  6. S. Nishiwaki, Y. Taketomi, S. Uchida, T. Tomita, J. Asada, “Optical head apparatus including a waveguide layer with concentric or spiral periodic structure,” U.S. patent5,200,939 (6April1993).
  7. H. Sunagawa, T. Suhara, H. Nishihara, “Optical pickup apparatus for detecting and correcting focusing and tracking errors in detected recorded signals,” U.S. patent5,153,860 (6October1992).
  8. N. Eriksson, M. Hagberg, A. Larsson, “Highly directional grating outcouplers with tailorable radiation characteristics,” IEEE J. Quantum Electron. 32, 1038–1047 (1996).
    [CrossRef]
  9. S. Kristjansson, M. Li, N. Eriksson, K. Killius, A. Larsson, “Circular grating coupled DBR laser with integrated focusing outcoupler,” IEEE Photon. Technol. Lett. 9, 416–418 (1997).
    [CrossRef]
  10. S. Ura, H. Sunagawa, T. Suhara, H. Nishihara, “Focusing grating couplers for polarization detection,” J. Lightwave Technol. 6, 1028–1033 (1988).
    [CrossRef]
  11. S. Ura, M. Shinohara, T. Suhara, H. Nishihara, “Integrated-optic grating-scale-displacement sensor using linearly focusing grating couplers,” IEEE Photon. Technol. Lett. 6, 239–241 (1994).
    [CrossRef]
  12. T. Suhara, T. Taniguchi, M. Uemukai, H. Nishihara, T. Hirata, S. Iio, “Monolithic integrated-optic position/displacement sensor using waveguide gratings and QW-DFB laser,” IEEE Photon. Technol. Lett. 7, 1195–1197 (1995).
    [CrossRef]
  13. T. Suhara, N. Nozaki, H. Nishihara, “An integrated acoustooptic printer head,” Proceedings of the Fourth European Conference on Integrated Optics, Glasgow, Scotland, Vol. 87, pp. 119–122 (1987).
  14. T. Suhara, K. Okada, T. Saso, H. Nishihara, “Focusing grating coupler in AlGaAs optical waveguide,” IEEE Photon. Technol. Lett. 4, 903–905 (1992).
    [CrossRef]
  15. D. Heitmann, C. Ortiz, “Calculation and experimental verification of two-dimensional focusing grating couplers,” IEEE J. Quantum Electron. QE-17, 1257–1263 (1981).
    [CrossRef]
  16. G. N. Lawrence, P. J. Cronkite, “Focusing grating coupler design method using holographic optical elements,” Appl. Opt. 27, 679–683 (1988).
    [CrossRef] [PubMed]
  17. S. Nishiwaki, J.-I. Asada, K. Ohshima, T. Kitagawa, “Fabrication of a concentric-circular focusing grating coupler by a conic-wave-front interference method and light-convergence experiments using the coupler,” Appl. Opt. 34, 7372–7382 (1995).
    [CrossRef] [PubMed]
  18. H. Sunagawa, T. Yamada, H. Miura, “Optical waveguide device,” U.S. patent5,436,991 (25July1995).
  19. I. A. Avrutsky, A. S. Svakhin, V. A. Sychugov, O. Parriaux, “High-efficiency single-order waveguide grating coupler,” Opt. Lett. 15, 1446–1448 (1990).
    [CrossRef] [PubMed]
  20. J. C. Brazas, L. Li, A. L. Mckeon, “High-efficiency input coupling into optical waveguides using gratings with double-surface corrugation,” Appl. Opt. 34, 604–609 (1995).
    [CrossRef] [PubMed]
  21. R. Waldhausl, E. B. Kley, P. Dannberg, A. Brauer, W. Karthe, “Grating couplers in planar polymer waveguides with beam shaping properties,” in Nanofabrication Technologies and Device Integration, W. Karthe, ed., Proc. SPIE2213, 122–132 (1994).
    [CrossRef]
  22. M. Li, S. Sheard, “Experimental study of waveguide grating couplers with parallelogramic tooth profiles,” Opt. Eng. 35, 3101–3106 (1996).
    [CrossRef]
  23. M. Hagberg, N. Eriksson, A. Larsson, “Investigation of high-efficiency surface-emitting lasers with blazed grating outcouplers,” IEEE J. Quantum Electron. 32, 1596–1605 (1996).
    [CrossRef]
  24. W. Driemeier, “Bragg-effect grating couplers integrated in multicomponent polymeric waveguides,” Opt. Lett. 15, 725–727 (1990).
    [CrossRef] [PubMed]
  25. M. L. Jones, R. P. Kenan, C. M. Verber, “Rectangular characteristic gratings for waveguide input and output coupling,” Appl. Opt. 34, 4149–4158 (1995).
    [CrossRef] [PubMed]
  26. Q. Huang, P. Ashley, “Holographic Bragg grating input–output couplers for polymer waveguides at 850-nm wavelength,” Appl. Opt. 36, 1198–1203 (1997).
    [CrossRef] [PubMed]
  27. T. Liao, S. Sheard, M. Li, J. Zhu, P. Prewett, “High-efficiency focusing waveguide grating coupler with parallelogramic groove profiles,” J. Lightwave Technol. 15, 1142–1148 (1997).
    [CrossRef]
  28. S. Sheard, T. Liao, G. Yang, P. Prewett, J. Zhu, “Focusing waveguide grating coupler using a diffractive doublet,” Appl. Opt. 36, 4349–4353 (1997).
    [CrossRef] [PubMed]
  29. W. Gambogi, W. Gerstadt, S. Mackara, A. Weber, “Holographic transmission elements using improved photopolymer films,” in Holographic Imaging and Materials, T. Jeong, ed., Proc. SPIE1555, 256–267 (1991).
  30. W. Gambogi, A. Weber, T. Trout, “Advances and applications of Dupont holographic photopolymer,” in Computer and Optically Generated Holographic Optics, S. Lee, ed., Proc. SPIE2043, 2–13 (1993).
  31. W. Driemeier, “Prepolymer-based waveguiding thin film for the holographic recording of dry-developing refractive-index gratings,” Opt. Commun. 76, 25–29 (1990).
    [CrossRef]
  32. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Formulation for the stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  33. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–938 (1985).
    [CrossRef]
  34. T. Touam, S. Najafi, “Symmetric profile beams from waveguides with asymmetric grating couplers,” Appl. Opt. 36, 2554–2558 (1997).
    [CrossRef] [PubMed]
  35. T. Tamir, S. T. Peng, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975).
    [CrossRef]
  36. M. Neviere, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5, pp. 123–157.
    [CrossRef]
  37. T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” IEEE J. Quantum Electron. 22, 544–550 (1986).
    [CrossRef]
  38. G. Hadjicostas, J. Butler, G. Evans, N. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
    [CrossRef]
  39. Zemax Optical Design Program User’s Guide, Focus Software, Inc., P.O. Box 18228, Tucson, Arizona, 85731.
  40. S. Solimeno, B. Crosignani, P. Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), Chap. 8.

1997 (6)

1996 (3)

N. Eriksson, M. Hagberg, A. Larsson, “Highly directional grating outcouplers with tailorable radiation characteristics,” IEEE J. Quantum Electron. 32, 1038–1047 (1996).
[CrossRef]

M. Li, S. Sheard, “Experimental study of waveguide grating couplers with parallelogramic tooth profiles,” Opt. Eng. 35, 3101–3106 (1996).
[CrossRef]

M. Hagberg, N. Eriksson, A. Larsson, “Investigation of high-efficiency surface-emitting lasers with blazed grating outcouplers,” IEEE J. Quantum Electron. 32, 1596–1605 (1996).
[CrossRef]

1995 (5)

1994 (2)

S. Ura, M. Shinohara, T. Suhara, H. Nishihara, “Integrated-optic grating-scale-displacement sensor using linearly focusing grating couplers,” IEEE Photon. Technol. Lett. 6, 239–241 (1994).
[CrossRef]

S. Nishiwaki, J. Asada, S. Uchida, “Optical head employing a concentric-circular focusing grating coupler,” Appl. Opt. 33, 1819–1827 (1994).
[CrossRef] [PubMed]

1993 (1)

N. Streibl, R. Volkel, J. Schwider, P. Habel, N. Lindlein, “Parallel optoelectronic interconnections with high packing density through a light-guiding plate using grating couplers and field lenses,” Opt. Commun. 99, 167–171 (1993).
[CrossRef]

1992 (1)

T. Suhara, K. Okada, T. Saso, H. Nishihara, “Focusing grating coupler in AlGaAs optical waveguide,” IEEE Photon. Technol. Lett. 4, 903–905 (1992).
[CrossRef]

1990 (4)

I. A. Avrutsky, A. S. Svakhin, V. A. Sychugov, O. Parriaux, “High-efficiency single-order waveguide grating coupler,” Opt. Lett. 15, 1446–1448 (1990).
[CrossRef] [PubMed]

W. Driemeier, “Bragg-effect grating couplers integrated in multicomponent polymeric waveguides,” Opt. Lett. 15, 725–727 (1990).
[CrossRef] [PubMed]

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

W. Driemeier, “Prepolymer-based waveguiding thin film for the holographic recording of dry-developing refractive-index gratings,” Opt. Commun. 76, 25–29 (1990).
[CrossRef]

1988 (2)

G. N. Lawrence, P. J. Cronkite, “Focusing grating coupler design method using holographic optical elements,” Appl. Opt. 27, 679–683 (1988).
[CrossRef] [PubMed]

S. Ura, H. Sunagawa, T. Suhara, H. Nishihara, “Focusing grating couplers for polarization detection,” J. Lightwave Technol. 6, 1028–1033 (1988).
[CrossRef]

1986 (2)

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913–918 (1986).
[CrossRef]

T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” IEEE J. Quantum Electron. 22, 544–550 (1986).
[CrossRef]

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–938 (1985).
[CrossRef]

1981 (1)

D. Heitmann, C. Ortiz, “Calculation and experimental verification of two-dimensional focusing grating couplers,” IEEE J. Quantum Electron. QE-17, 1257–1263 (1981).
[CrossRef]

1975 (1)

T. Tamir, S. T. Peng, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975).
[CrossRef]

Amantea, R.

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

Asada, J.

S. Nishiwaki, J. Asada, S. Uchida, “Optical head employing a concentric-circular focusing grating coupler,” Appl. Opt. 33, 1819–1827 (1994).
[CrossRef] [PubMed]

S. Nishiwaki, Y. Taketomi, S. Uchida, T. Tomita, J. Asada, “Optical head apparatus including a waveguide layer with concentric or spiral periodic structure,” U.S. patent5,200,939 (6April1993).

Asada, J.-I.

Ashley, P.

Avrutsky, I. A.

Bertoni, H. L.

T. Tamir, S. T. Peng, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975).
[CrossRef]

Brauer, A.

R. Waldhausl, E. B. Kley, P. Dannberg, A. Brauer, W. Karthe, “Grating couplers in planar polymer waveguides with beam shaping properties,” in Nanofabrication Technologies and Device Integration, W. Karthe, ed., Proc. SPIE2213, 122–132 (1994).
[CrossRef]

Brazas, J. C.

Butler, J.

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

Carlson, N.

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

Chen, R.

Cronkite, P. J.

Crosignani, B.

S. Solimeno, B. Crosignani, P. Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), Chap. 8.

Dannberg, P.

R. Waldhausl, E. B. Kley, P. Dannberg, A. Brauer, W. Karthe, “Grating couplers in planar polymer waveguides with beam shaping properties,” in Nanofabrication Technologies and Device Integration, W. Karthe, ed., Proc. SPIE2213, 122–132 (1994).
[CrossRef]

Driemeier, W.

W. Driemeier, “Bragg-effect grating couplers integrated in multicomponent polymeric waveguides,” Opt. Lett. 15, 725–727 (1990).
[CrossRef] [PubMed]

W. Driemeier, “Prepolymer-based waveguiding thin film for the holographic recording of dry-developing refractive-index gratings,” Opt. Commun. 76, 25–29 (1990).
[CrossRef]

Eriksson, N.

S. Kristjansson, M. Li, N. Eriksson, K. Killius, A. Larsson, “Circular grating coupled DBR laser with integrated focusing outcoupler,” IEEE Photon. Technol. Lett. 9, 416–418 (1997).
[CrossRef]

N. Eriksson, M. Hagberg, A. Larsson, “Highly directional grating outcouplers with tailorable radiation characteristics,” IEEE J. Quantum Electron. 32, 1038–1047 (1996).
[CrossRef]

M. Hagberg, N. Eriksson, A. Larsson, “Investigation of high-efficiency surface-emitting lasers with blazed grating outcouplers,” IEEE J. Quantum Electron. 32, 1596–1605 (1996).
[CrossRef]

Evans, G.

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

Gambogi, W.

W. Gambogi, W. Gerstadt, S. Mackara, A. Weber, “Holographic transmission elements using improved photopolymer films,” in Holographic Imaging and Materials, T. Jeong, ed., Proc. SPIE1555, 256–267 (1991).

W. Gambogi, A. Weber, T. Trout, “Advances and applications of Dupont holographic photopolymer,” in Computer and Optically Generated Holographic Optics, S. Lee, ed., Proc. SPIE2043, 2–13 (1993).

Gaylord, T. K.

Gerstadt, W.

W. Gambogi, W. Gerstadt, S. Mackara, A. Weber, “Holographic transmission elements using improved photopolymer films,” in Holographic Imaging and Materials, T. Jeong, ed., Proc. SPIE1555, 256–267 (1991).

Grann, E. B.

Habel, P.

N. Streibl, R. Volkel, J. Schwider, P. Habel, N. Lindlein, “Parallel optoelectronic interconnections with high packing density through a light-guiding plate using grating couplers and field lenses,” Opt. Commun. 99, 167–171 (1993).
[CrossRef]

Hadjicostas, G.

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

Hagberg, M.

M. Hagberg, N. Eriksson, A. Larsson, “Investigation of high-efficiency surface-emitting lasers with blazed grating outcouplers,” IEEE J. Quantum Electron. 32, 1596–1605 (1996).
[CrossRef]

N. Eriksson, M. Hagberg, A. Larsson, “Highly directional grating outcouplers with tailorable radiation characteristics,” IEEE J. Quantum Electron. 32, 1038–1047 (1996).
[CrossRef]

Heitmann, D.

D. Heitmann, C. Ortiz, “Calculation and experimental verification of two-dimensional focusing grating couplers,” IEEE J. Quantum Electron. QE-17, 1257–1263 (1981).
[CrossRef]

Hirata, T.

T. Suhara, T. Taniguchi, M. Uemukai, H. Nishihara, T. Hirata, S. Iio, “Monolithic integrated-optic position/displacement sensor using waveguide gratings and QW-DFB laser,” IEEE Photon. Technol. Lett. 7, 1195–1197 (1995).
[CrossRef]

Huang, Q.

Iio, S.

T. Suhara, T. Taniguchi, M. Uemukai, H. Nishihara, T. Hirata, S. Iio, “Monolithic integrated-optic position/displacement sensor using waveguide gratings and QW-DFB laser,” IEEE Photon. Technol. Lett. 7, 1195–1197 (1995).
[CrossRef]

Jones, M. L.

Jung, S. D.

S. H. Song, S. D. Jung, “Back-board optical signal interconnection module using focusing grating coupler arrays,” U.S. patent5,469,518 (21November1995).

Karthe, W.

R. Waldhausl, E. B. Kley, P. Dannberg, A. Brauer, W. Karthe, “Grating couplers in planar polymer waveguides with beam shaping properties,” in Nanofabrication Technologies and Device Integration, W. Karthe, ed., Proc. SPIE2213, 122–132 (1994).
[CrossRef]

Kenan, R. P.

Killius, K.

S. Kristjansson, M. Li, N. Eriksson, K. Killius, A. Larsson, “Circular grating coupled DBR laser with integrated focusing outcoupler,” IEEE Photon. Technol. Lett. 9, 416–418 (1997).
[CrossRef]

Kitagawa, T.

Kley, E. B.

R. Waldhausl, E. B. Kley, P. Dannberg, A. Brauer, W. Karthe, “Grating couplers in planar polymer waveguides with beam shaping properties,” in Nanofabrication Technologies and Device Integration, W. Karthe, ed., Proc. SPIE2213, 122–132 (1994).
[CrossRef]

Koyama, J.

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913–918 (1986).
[CrossRef]

Kristjansson, S.

S. Kristjansson, M. Li, N. Eriksson, K. Killius, A. Larsson, “Circular grating coupled DBR laser with integrated focusing outcoupler,” IEEE Photon. Technol. Lett. 9, 416–418 (1997).
[CrossRef]

Larsson, A.

S. Kristjansson, M. Li, N. Eriksson, K. Killius, A. Larsson, “Circular grating coupled DBR laser with integrated focusing outcoupler,” IEEE Photon. Technol. Lett. 9, 416–418 (1997).
[CrossRef]

N. Eriksson, M. Hagberg, A. Larsson, “Highly directional grating outcouplers with tailorable radiation characteristics,” IEEE J. Quantum Electron. 32, 1038–1047 (1996).
[CrossRef]

M. Hagberg, N. Eriksson, A. Larsson, “Investigation of high-efficiency surface-emitting lasers with blazed grating outcouplers,” IEEE J. Quantum Electron. 32, 1596–1605 (1996).
[CrossRef]

Lawrence, G. N.

Li, L.

Li, M.

T. Liao, S. Sheard, M. Li, J. Zhu, P. Prewett, “High-efficiency focusing waveguide grating coupler with parallelogramic groove profiles,” J. Lightwave Technol. 15, 1142–1148 (1997).
[CrossRef]

S. Kristjansson, M. Li, N. Eriksson, K. Killius, A. Larsson, “Circular grating coupled DBR laser with integrated focusing outcoupler,” IEEE Photon. Technol. Lett. 9, 416–418 (1997).
[CrossRef]

M. Li, S. Sheard, “Experimental study of waveguide grating couplers with parallelogramic tooth profiles,” Opt. Eng. 35, 3101–3106 (1996).
[CrossRef]

Liao, T.

T. Liao, S. Sheard, M. Li, J. Zhu, P. Prewett, “High-efficiency focusing waveguide grating coupler with parallelogramic groove profiles,” J. Lightwave Technol. 15, 1142–1148 (1997).
[CrossRef]

S. Sheard, T. Liao, G. Yang, P. Prewett, J. Zhu, “Focusing waveguide grating coupler using a diffractive doublet,” Appl. Opt. 36, 4349–4353 (1997).
[CrossRef] [PubMed]

Lindlein, N.

N. Streibl, R. Volkel, J. Schwider, P. Habel, N. Lindlein, “Parallel optoelectronic interconnections with high packing density through a light-guiding plate using grating couplers and field lenses,” Opt. Commun. 99, 167–171 (1993).
[CrossRef]

Mackara, S.

W. Gambogi, W. Gerstadt, S. Mackara, A. Weber, “Holographic transmission elements using improved photopolymer films,” in Holographic Imaging and Materials, T. Jeong, ed., Proc. SPIE1555, 256–267 (1991).

Mckeon, A. L.

Miura, H.

H. Sunagawa, T. Yamada, H. Miura, “Optical waveguide device,” U.S. patent5,436,991 (25July1995).

Moharam, M. G.

Najafi, S.

Neviere, M.

M. Neviere, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5, pp. 123–157.
[CrossRef]

Nishihara, H.

T. Suhara, T. Taniguchi, M. Uemukai, H. Nishihara, T. Hirata, S. Iio, “Monolithic integrated-optic position/displacement sensor using waveguide gratings and QW-DFB laser,” IEEE Photon. Technol. Lett. 7, 1195–1197 (1995).
[CrossRef]

S. Ura, M. Shinohara, T. Suhara, H. Nishihara, “Integrated-optic grating-scale-displacement sensor using linearly focusing grating couplers,” IEEE Photon. Technol. Lett. 6, 239–241 (1994).
[CrossRef]

T. Suhara, K. Okada, T. Saso, H. Nishihara, “Focusing grating coupler in AlGaAs optical waveguide,” IEEE Photon. Technol. Lett. 4, 903–905 (1992).
[CrossRef]

S. Ura, H. Sunagawa, T. Suhara, H. Nishihara, “Focusing grating couplers for polarization detection,” J. Lightwave Technol. 6, 1028–1033 (1988).
[CrossRef]

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913–918 (1986).
[CrossRef]

T. Suhara, N. Nozaki, H. Nishihara, “An integrated acoustooptic printer head,” Proceedings of the Fourth European Conference on Integrated Optics, Glasgow, Scotland, Vol. 87, pp. 119–122 (1987).

H. Sunagawa, T. Suhara, H. Nishihara, “Optical pickup apparatus for detecting and correcting focusing and tracking errors in detected recorded signals,” U.S. patent5,153,860 (6October1992).

Nishiwaki, S.

Nozaki, N.

T. Suhara, N. Nozaki, H. Nishihara, “An integrated acoustooptic printer head,” Proceedings of the Fourth European Conference on Integrated Optics, Glasgow, Scotland, Vol. 87, pp. 119–122 (1987).

Ohshima, K.

Okada, K.

T. Suhara, K. Okada, T. Saso, H. Nishihara, “Focusing grating coupler in AlGaAs optical waveguide,” IEEE Photon. Technol. Lett. 4, 903–905 (1992).
[CrossRef]

Ortiz, C.

D. Heitmann, C. Ortiz, “Calculation and experimental verification of two-dimensional focusing grating couplers,” IEEE J. Quantum Electron. QE-17, 1257–1263 (1981).
[CrossRef]

Parriaux, O.

Peng, S. T.

T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” IEEE J. Quantum Electron. 22, 544–550 (1986).
[CrossRef]

T. Tamir, S. T. Peng, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975).
[CrossRef]

Pommet, D. A.

Porto, P.

S. Solimeno, B. Crosignani, P. Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), Chap. 8.

Prewett, P.

T. Liao, S. Sheard, M. Li, J. Zhu, P. Prewett, “High-efficiency focusing waveguide grating coupler with parallelogramic groove profiles,” J. Lightwave Technol. 15, 1142–1148 (1997).
[CrossRef]

S. Sheard, T. Liao, G. Yang, P. Prewett, J. Zhu, “Focusing waveguide grating coupler using a diffractive doublet,” Appl. Opt. 36, 4349–4353 (1997).
[CrossRef] [PubMed]

Saso, T.

T. Suhara, K. Okada, T. Saso, H. Nishihara, “Focusing grating coupler in AlGaAs optical waveguide,” IEEE Photon. Technol. Lett. 4, 903–905 (1992).
[CrossRef]

Schwider, J.

N. Streibl, R. Volkel, J. Schwider, P. Habel, N. Lindlein, “Parallel optoelectronic interconnections with high packing density through a light-guiding plate using grating couplers and field lenses,” Opt. Commun. 99, 167–171 (1993).
[CrossRef]

Sheard, S.

T. Liao, S. Sheard, M. Li, J. Zhu, P. Prewett, “High-efficiency focusing waveguide grating coupler with parallelogramic groove profiles,” J. Lightwave Technol. 15, 1142–1148 (1997).
[CrossRef]

S. Sheard, T. Liao, G. Yang, P. Prewett, J. Zhu, “Focusing waveguide grating coupler using a diffractive doublet,” Appl. Opt. 36, 4349–4353 (1997).
[CrossRef] [PubMed]

M. Li, S. Sheard, “Experimental study of waveguide grating couplers with parallelogramic tooth profiles,” Opt. Eng. 35, 3101–3106 (1996).
[CrossRef]

Shinohara, M.

S. Ura, M. Shinohara, T. Suhara, H. Nishihara, “Integrated-optic grating-scale-displacement sensor using linearly focusing grating couplers,” IEEE Photon. Technol. Lett. 6, 239–241 (1994).
[CrossRef]

Solimeno, S.

S. Solimeno, B. Crosignani, P. Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), Chap. 8.

Song, S. H.

S. H. Song, S. D. Jung, “Back-board optical signal interconnection module using focusing grating coupler arrays,” U.S. patent5,469,518 (21November1995).

Streibl, N.

N. Streibl, R. Volkel, J. Schwider, P. Habel, N. Lindlein, “Parallel optoelectronic interconnections with high packing density through a light-guiding plate using grating couplers and field lenses,” Opt. Commun. 99, 167–171 (1993).
[CrossRef]

Suhara, T.

T. Suhara, T. Taniguchi, M. Uemukai, H. Nishihara, T. Hirata, S. Iio, “Monolithic integrated-optic position/displacement sensor using waveguide gratings and QW-DFB laser,” IEEE Photon. Technol. Lett. 7, 1195–1197 (1995).
[CrossRef]

S. Ura, M. Shinohara, T. Suhara, H. Nishihara, “Integrated-optic grating-scale-displacement sensor using linearly focusing grating couplers,” IEEE Photon. Technol. Lett. 6, 239–241 (1994).
[CrossRef]

T. Suhara, K. Okada, T. Saso, H. Nishihara, “Focusing grating coupler in AlGaAs optical waveguide,” IEEE Photon. Technol. Lett. 4, 903–905 (1992).
[CrossRef]

S. Ura, H. Sunagawa, T. Suhara, H. Nishihara, “Focusing grating couplers for polarization detection,” J. Lightwave Technol. 6, 1028–1033 (1988).
[CrossRef]

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913–918 (1986).
[CrossRef]

T. Suhara, N. Nozaki, H. Nishihara, “An integrated acoustooptic printer head,” Proceedings of the Fourth European Conference on Integrated Optics, Glasgow, Scotland, Vol. 87, pp. 119–122 (1987).

H. Sunagawa, T. Suhara, H. Nishihara, “Optical pickup apparatus for detecting and correcting focusing and tracking errors in detected recorded signals,” U.S. patent5,153,860 (6October1992).

Sunagawa, H.

S. Ura, H. Sunagawa, T. Suhara, H. Nishihara, “Focusing grating couplers for polarization detection,” J. Lightwave Technol. 6, 1028–1033 (1988).
[CrossRef]

H. Sunagawa, T. Suhara, H. Nishihara, “Optical pickup apparatus for detecting and correcting focusing and tracking errors in detected recorded signals,” U.S. patent5,153,860 (6October1992).

H. Sunagawa, T. Yamada, H. Miura, “Optical waveguide device,” U.S. patent5,436,991 (25July1995).

Svakhin, A. S.

Sychugov, V. A.

Taketomi, Y.

S. Nishiwaki, Y. Taketomi, S. Uchida, T. Tomita, J. Asada, “Optical head apparatus including a waveguide layer with concentric or spiral periodic structure,” U.S. patent5,200,939 (6April1993).

Tamir, T.

T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” IEEE J. Quantum Electron. 22, 544–550 (1986).
[CrossRef]

T. Tamir, S. T. Peng, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975).
[CrossRef]

Taniguchi, T.

T. Suhara, T. Taniguchi, M. Uemukai, H. Nishihara, T. Hirata, S. Iio, “Monolithic integrated-optic position/displacement sensor using waveguide gratings and QW-DFB laser,” IEEE Photon. Technol. Lett. 7, 1195–1197 (1995).
[CrossRef]

Tomita, T.

S. Nishiwaki, Y. Taketomi, S. Uchida, T. Tomita, J. Asada, “Optical head apparatus including a waveguide layer with concentric or spiral periodic structure,” U.S. patent5,200,939 (6April1993).

Touam, T.

Trout, T.

W. Gambogi, A. Weber, T. Trout, “Advances and applications of Dupont holographic photopolymer,” in Computer and Optically Generated Holographic Optics, S. Lee, ed., Proc. SPIE2043, 2–13 (1993).

Uchida, S.

S. Nishiwaki, J. Asada, S. Uchida, “Optical head employing a concentric-circular focusing grating coupler,” Appl. Opt. 33, 1819–1827 (1994).
[CrossRef] [PubMed]

S. Nishiwaki, Y. Taketomi, S. Uchida, T. Tomita, J. Asada, “Optical head apparatus including a waveguide layer with concentric or spiral periodic structure,” U.S. patent5,200,939 (6April1993).

Uemukai, M.

T. Suhara, T. Taniguchi, M. Uemukai, H. Nishihara, T. Hirata, S. Iio, “Monolithic integrated-optic position/displacement sensor using waveguide gratings and QW-DFB laser,” IEEE Photon. Technol. Lett. 7, 1195–1197 (1995).
[CrossRef]

Ura, S.

S. Ura, M. Shinohara, T. Suhara, H. Nishihara, “Integrated-optic grating-scale-displacement sensor using linearly focusing grating couplers,” IEEE Photon. Technol. Lett. 6, 239–241 (1994).
[CrossRef]

S. Ura, H. Sunagawa, T. Suhara, H. Nishihara, “Focusing grating couplers for polarization detection,” J. Lightwave Technol. 6, 1028–1033 (1988).
[CrossRef]

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913–918 (1986).
[CrossRef]

Verber, C. M.

Volkel, R.

N. Streibl, R. Volkel, J. Schwider, P. Habel, N. Lindlein, “Parallel optoelectronic interconnections with high packing density through a light-guiding plate using grating couplers and field lenses,” Opt. Commun. 99, 167–171 (1993).
[CrossRef]

Waldhausl, R.

R. Waldhausl, E. B. Kley, P. Dannberg, A. Brauer, W. Karthe, “Grating couplers in planar polymer waveguides with beam shaping properties,” in Nanofabrication Technologies and Device Integration, W. Karthe, ed., Proc. SPIE2213, 122–132 (1994).
[CrossRef]

Weber, A.

W. Gambogi, W. Gerstadt, S. Mackara, A. Weber, “Holographic transmission elements using improved photopolymer films,” in Holographic Imaging and Materials, T. Jeong, ed., Proc. SPIE1555, 256–267 (1991).

W. Gambogi, A. Weber, T. Trout, “Advances and applications of Dupont holographic photopolymer,” in Computer and Optically Generated Holographic Optics, S. Lee, ed., Proc. SPIE2043, 2–13 (1993).

Yamada, T.

H. Sunagawa, T. Yamada, H. Miura, “Optical waveguide device,” U.S. patent5,436,991 (25July1995).

Yang, G.

Zhao, C.

Zhu, J.

S. Sheard, T. Liao, G. Yang, P. Prewett, J. Zhu, “Focusing waveguide grating coupler using a diffractive doublet,” Appl. Opt. 36, 4349–4353 (1997).
[CrossRef] [PubMed]

T. Liao, S. Sheard, M. Li, J. Zhu, P. Prewett, “High-efficiency focusing waveguide grating coupler with parallelogramic groove profiles,” J. Lightwave Technol. 15, 1142–1148 (1997).
[CrossRef]

Appl. Opt. (9)

C. Zhao, R. Chen, “Performance consideration of three-dimensional optoelectronic interconnection for intra-multichip-module clock signal distribution,” Appl. Opt. 36, 2537–2544 (1997).
[CrossRef] [PubMed]

S. Nishiwaki, J. Asada, S. Uchida, “Optical head employing a concentric-circular focusing grating coupler,” Appl. Opt. 33, 1819–1827 (1994).
[CrossRef] [PubMed]

G. N. Lawrence, P. J. Cronkite, “Focusing grating coupler design method using holographic optical elements,” Appl. Opt. 27, 679–683 (1988).
[CrossRef] [PubMed]

S. Nishiwaki, J.-I. Asada, K. Ohshima, T. Kitagawa, “Fabrication of a concentric-circular focusing grating coupler by a conic-wave-front interference method and light-convergence experiments using the coupler,” Appl. Opt. 34, 7372–7382 (1995).
[CrossRef] [PubMed]

J. C. Brazas, L. Li, A. L. Mckeon, “High-efficiency input coupling into optical waveguides using gratings with double-surface corrugation,” Appl. Opt. 34, 604–609 (1995).
[CrossRef] [PubMed]

M. L. Jones, R. P. Kenan, C. M. Verber, “Rectangular characteristic gratings for waveguide input and output coupling,” Appl. Opt. 34, 4149–4158 (1995).
[CrossRef] [PubMed]

Q. Huang, P. Ashley, “Holographic Bragg grating input–output couplers for polymer waveguides at 850-nm wavelength,” Appl. Opt. 36, 1198–1203 (1997).
[CrossRef] [PubMed]

S. Sheard, T. Liao, G. Yang, P. Prewett, J. Zhu, “Focusing waveguide grating coupler using a diffractive doublet,” Appl. Opt. 36, 4349–4353 (1997).
[CrossRef] [PubMed]

T. Touam, S. Najafi, “Symmetric profile beams from waveguides with asymmetric grating couplers,” Appl. Opt. 36, 2554–2558 (1997).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (5)

T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” IEEE J. Quantum Electron. 22, 544–550 (1986).
[CrossRef]

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

M. Hagberg, N. Eriksson, A. Larsson, “Investigation of high-efficiency surface-emitting lasers with blazed grating outcouplers,” IEEE J. Quantum Electron. 32, 1596–1605 (1996).
[CrossRef]

D. Heitmann, C. Ortiz, “Calculation and experimental verification of two-dimensional focusing grating couplers,” IEEE J. Quantum Electron. QE-17, 1257–1263 (1981).
[CrossRef]

N. Eriksson, M. Hagberg, A. Larsson, “Highly directional grating outcouplers with tailorable radiation characteristics,” IEEE J. Quantum Electron. 32, 1038–1047 (1996).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

S. Kristjansson, M. Li, N. Eriksson, K. Killius, A. Larsson, “Circular grating coupled DBR laser with integrated focusing outcoupler,” IEEE Photon. Technol. Lett. 9, 416–418 (1997).
[CrossRef]

T. Suhara, K. Okada, T. Saso, H. Nishihara, “Focusing grating coupler in AlGaAs optical waveguide,” IEEE Photon. Technol. Lett. 4, 903–905 (1992).
[CrossRef]

S. Ura, M. Shinohara, T. Suhara, H. Nishihara, “Integrated-optic grating-scale-displacement sensor using linearly focusing grating couplers,” IEEE Photon. Technol. Lett. 6, 239–241 (1994).
[CrossRef]

T. Suhara, T. Taniguchi, M. Uemukai, H. Nishihara, T. Hirata, S. Iio, “Monolithic integrated-optic position/displacement sensor using waveguide gratings and QW-DFB laser,” IEEE Photon. Technol. Lett. 7, 1195–1197 (1995).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

T. Tamir, S. T. Peng, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975).
[CrossRef]

J. Lightwave Technol. (3)

T. Liao, S. Sheard, M. Li, J. Zhu, P. Prewett, “High-efficiency focusing waveguide grating coupler with parallelogramic groove profiles,” J. Lightwave Technol. 15, 1142–1148 (1997).
[CrossRef]

S. Ura, H. Sunagawa, T. Suhara, H. Nishihara, “Focusing grating couplers for polarization detection,” J. Lightwave Technol. 6, 1028–1033 (1988).
[CrossRef]

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “An integrated-optic disk pickup device,” J. Lightwave Technol. 4, 913–918 (1986).
[CrossRef]

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

Opt. Commun. (2)

N. Streibl, R. Volkel, J. Schwider, P. Habel, N. Lindlein, “Parallel optoelectronic interconnections with high packing density through a light-guiding plate using grating couplers and field lenses,” Opt. Commun. 99, 167–171 (1993).
[CrossRef]

W. Driemeier, “Prepolymer-based waveguiding thin film for the holographic recording of dry-developing refractive-index gratings,” Opt. Commun. 76, 25–29 (1990).
[CrossRef]

Opt. Eng. (1)

M. Li, S. Sheard, “Experimental study of waveguide grating couplers with parallelogramic tooth profiles,” Opt. Eng. 35, 3101–3106 (1996).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–938 (1985).
[CrossRef]

Other (11)

M. Neviere, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5, pp. 123–157.
[CrossRef]

Zemax Optical Design Program User’s Guide, Focus Software, Inc., P.O. Box 18228, Tucson, Arizona, 85731.

S. Solimeno, B. Crosignani, P. Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), Chap. 8.

W. Gambogi, W. Gerstadt, S. Mackara, A. Weber, “Holographic transmission elements using improved photopolymer films,” in Holographic Imaging and Materials, T. Jeong, ed., Proc. SPIE1555, 256–267 (1991).

W. Gambogi, A. Weber, T. Trout, “Advances and applications of Dupont holographic photopolymer,” in Computer and Optically Generated Holographic Optics, S. Lee, ed., Proc. SPIE2043, 2–13 (1993).

S. H. Song, S. D. Jung, “Back-board optical signal interconnection module using focusing grating coupler arrays,” U.S. patent5,469,518 (21November1995).

S. Nishiwaki, Y. Taketomi, S. Uchida, T. Tomita, J. Asada, “Optical head apparatus including a waveguide layer with concentric or spiral periodic structure,” U.S. patent5,200,939 (6April1993).

H. Sunagawa, T. Suhara, H. Nishihara, “Optical pickup apparatus for detecting and correcting focusing and tracking errors in detected recorded signals,” U.S. patent5,153,860 (6October1992).

T. Suhara, N. Nozaki, H. Nishihara, “An integrated acoustooptic printer head,” Proceedings of the Fourth European Conference on Integrated Optics, Glasgow, Scotland, Vol. 87, pp. 119–122 (1987).

R. Waldhausl, E. B. Kley, P. Dannberg, A. Brauer, W. Karthe, “Grating couplers in planar polymer waveguides with beam shaping properties,” in Nanofabrication Technologies and Device Integration, W. Karthe, ed., Proc. SPIE2213, 122–132 (1994).
[CrossRef]

H. Sunagawa, T. Yamada, H. Miura, “Optical waveguide device,” U.S. patent5,436,991 (25July1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Geometry of a focusing grating coupler. The grating has a length L and focuses the guided wave to a focal line at (x f , z f ). The grating thickness is t g and the waveguide thickness is t w . Both (a) a surface-relief and (b) a volume focusing coupler are shown. The focal line is above the center of the grating at x f = L/2. A representative diffracted ray with propagation angle θ is also shown.

Fig. 2
Fig. 2

Representative focusing coupler segment with a localized grating vector K(x 0), slant angle ϕ(x 0), grating period Λ(x 0) = 2π/|K(x 0)|, diffracted wave vector k c1(x 0), input guided power P in(x 0 - Δx/2), diffracted power ΔP c1(x 0), and output guided power P in(x 0 + Δx/2). The focal plane z = z f is also shown.

Fig. 3
Fig. 3

Attenuation coefficient α versus the normal component of the grating vector K z for a modulation of Δε = 0.06 (Δn = 0.0194) (dashed curve) and Δε = 0.1 (Δn = 0.0324) (solid curve) with K x = -12.3 μm-1 corresponding to the ideal K x value at the end of the grating (x = L). The K z corresponding to the maximum attenuation coefficient K z max) is also labeled. These calculations are performed with the following parameters: n c = 1.0, n g = 1.543, n w = 1.55, n s = 1.51, t g = 10.0 μm, and t w = 1.6 μm.

Fig. 4
Fig. 4

Ultraviolet recording configuration used to produce the volume grating focusing coupler. The chief ray and a ray passing through x 0 are shown for the two aberration-optimized cylindrical lenses. The following quantities are labeled for both lenses: the angles between the chief rays and the grating surface, θ m ; the focal lines for the ideal focusing cylindrical waves (x mf i , z mf i ); the surfaces of the lenses, S m1 and S m2; the lens centers (x m , z m ); and the localized wave vectors, k m , where m = 1, 2 corresponds to the respective lenses. The following quantities are labeled for lens 2 (corresponding quantities exist for lens 1 but are not labeled): the distance between the lens center and the prism surface measured along the chief ray, D 2; the incident angle, γ2; and the surface crossing point, x 2″. The center thickness t 1 and lens refractive index are labeled on lens 2 (corresponding quantities exist for lens 1 but are not labeled).

Fig. 5
Fig. 5

Flow chart illustrating the procedure for optimizing the cylindrical lenses for the ultraviolet interferometric recording.

Fig. 6
Fig. 6

Scale drawing of the ultraviolet recording configuration needed to produce the volume grating focusing coupler.

Fig. 7
Fig. 7

(a) Desired surface grating vector K x d (x) as a function of position for the surface-relief and volume couplers and the normal grating vector K z (x) for the volume coupler. (b) The corresponding grating period Λ and slant angle ϕ for the volume coupler. (c) The deviation from the desired surface grating vector ΔK x (x) as a function of position.

Fig. 8
Fig. 8

Attenuation coefficient α versus the longitudinal position x for the two volume cases labeled as Δε = 0.1 (Δn = 0.0324) and Δε = 0.06 (Δn = 0.0194) and the surface-relief focusing coupler. The oscillation around x = 500 μm is due to the i = 2 diffracted order in the substrate being at cutoff for the coupler segment corresponding to x = 500 μm.

Fig. 9
Fig. 9

Intensity profile at the grating surface normalized by the initial guided power I c1/P 0 with the corresponding coupling efficiency CE labeled.

Fig. 10
Fig. 10

Ray trace for the volume focusing grating coupler with the focal line width d labeled.

Fig. 11
Fig. 11

Intensity profile I f at the focal plane (z = 4 mm) normalized by the guided power at the beginning of the coupler P 0.

Tables (3)

Tables Icon

Table 1 Parameter Values of the Surface-Relief and Volume Grating Focusing Couplers

Tables Icon

Table 2 Parameters of Optimized Recording Configuration Comprised of the Prism and the Two Cylindrical Lenses

Tables Icon

Table 3 Numerical Data for the Various Types of Couplers

Equations (13)

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

K x d x = n c k 0 x f - x x f - x 2 + z f 2 1 / 2 - Nk 0 ,
M k ˜ x 0 V = 0 ,
Δ P c 1 x 0 = P in x 0 - Δ x / 2 η c 1 x 0 × 1 - exp - 2 α x 0 Δ x .
I c 1 x 0 = P 0 η c 1 x 0 2 α x 0 exp - 2   0 x 0   α τ d τ .
CE = 1 P 0 0 L   I c 1 x 0 d x 0 .
k 1 L - k 2 L = K x L x ˆ + K z α max , k mx 2 L + k mz 2 L = 2 π n g / λ uv 2 m = 1 ,   2 ,
k 1 L = - 23.4 x ˆ - 12.6   μ m - 1 , k 2 L = - 11.1 x ˆ - 24.2   μ m - 1 ,
k 1 x d x = K x d x + k 2 x i x ,
γ 1 d x = cos - 1 k 1 x d x / k 0 - θ 1 .
k 1 x x = k 0 cos θ 1 + γ 1 x .
k 2 x d x = k 1 x x - K x d x .
K x x = k 1 x x - k 2 x x ,
Δ K x x = K x x - K x d x .

Metrics