Abstract

The problem of oblique incidence of plane waves and Gaussian beams on finite-aperture gratings (the number of grooves and their length and depth are all finite) in slab waveguides is analyzed by means of a four-wave two-dimensional coupled-mode theory (2D-CMT). This model considers the finite aperture of the gratings and the correct simultaneous interaction among all four relevant waves (TE+, TE-, TM+, and TM-) by means of Bragg diffraction at oblique incidence. The grating’s geometry and boundary conditions are properly accounted for, and the problem is solved numerically by a finite-difference method. Near-field and far-field distributions, as well as reflection and transmission (power) coefficients (as functions of the plane-wave incidence angle), are calculated. The model is compared with the degenerate case of two-wave coupling that considers interaction only between pairs (e.g., TE+TE-), and significant differences may be observed. Compatibility and differences between the 2D-CMT and the one-dimensional CMT (grooves with infinite length) are also presented, in addition to the influence of the beam width and the groove length on the emerging waves. The analysis is general and can be performed on many kinds of realistic beams, grating shapes, and applications.

© 1999 Optical Society of America

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References

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    [CrossRef]
  6. S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CTuC2, pp. 77–78.
  7. A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultrahigh brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper 7.
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    [CrossRef]
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    [CrossRef]
  13. N. Ramanujam, J. J. Burke, L. Li, “Guided wave deflectors using gratings with slowly-varying groove depth for beam shaping,” in Guided Wave Optoelectronics, T. Tamir, ed. (Plenum, New York, 1995), pp. 321–332.
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1998 (1)

1997 (1)

G. Weitman, A. Hardy, “Reduction of coupling coefficients for distributed Bragg reflection in corrugated narrow-rib waveguides,” IEE Proc.: Optoelectron. 144, 101–103 (1997).

1996 (1)

1993 (1)

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

1991 (1)

D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
[CrossRef]

1989 (1)

1988 (1)

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

1984 (1)

A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved optical planar waveguides,” Appl. Sci. Res. 41, 271–274 (1984).
[CrossRef]

1981 (2)

1980 (2)

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. 70, 300–304 (1980).
[CrossRef]

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

1979 (1)

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

1978 (1)

1977 (1)

L. Solimar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
[CrossRef]

1976 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory of thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

1967 (1)

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

Ames, W. F.

W. F. Ames, Numerical Methods for Partial Differential Equations, 3rd ed. (Academic, New York, 1992).

Burke, J. J.

N. Ramanujam, J. J. Burke, L. Li, “Guided wave deflectors using gratings with slowly-varying groove depth for beam shaping,” in Guided Wave Optoelectronics, T. Tamir, ed. (Plenum, New York, 1995), pp. 321–332.

Chester, C. R.

C. R. Chester, Techniques in Partial Differential Equations (McGraw-Hill, New York, 1971).

Chu, R. S.

Cook, B. D.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II.

DeMars, S. D.

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CTuC2, pp. 77–78.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultrahigh brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper 7.

Dzurko, K. M.

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CTuC2, pp. 77–78.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultrahigh brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper 7.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Flannery, B.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in fortran: The Art of Scientific Computing (Cambridge U. Press, New York, 1986).

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. 70, 300–304 (1980).
[CrossRef]

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hall, D. G.

D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
[CrossRef]

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

Hardy, A.

N. Izhaky, A. Hardy, “Four wave CMT of obliquely incident plane waves on waveguide diffraction gratings,” J. Opt. Soc. Am. A 15, 473–479 (1998); “Oblique incidence of Gaussian beams on waveguide gratings with the four-wave CMT,” Appl. Opt. 37, 5806–5815 (1998).
[CrossRef]

G. Weitman, A. Hardy, “Reduction of coupling coefficients for distributed Bragg reflection in corrugated narrow-rib waveguides,” IEE Proc.: Optoelectron. 144, 101–103 (1997).

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CTuC2, pp. 77–78.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultrahigh brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper 7.

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II.

Itzykson, C.

C. Itzykson, J. B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).

Izhaky, N.

Klein, W. R.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory of thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).

Korpel, A.

A. Korpel, “Acousto-optics—a review of fundamentals,” Proc. IEEE 69, 48–53 (1981).
[CrossRef]

Lagasse, P. E.

Lang, R. J.

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CTuC2, pp. 77–78.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultrahigh brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper 7.

Li, L.

N. Ramanujam, J. J. Burke, L. Li, “Guided wave deflectors using gratings with slowly-varying groove depth for beam shaping,” in Guided Wave Optoelectronics, T. Tamir, ed. (Plenum, New York, 1995), pp. 321–332.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. 70, 300–304 (1980).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1991).

Moharam, M. G.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Morton, K. W.

R. D. Richtmyer, K. W. Morton, Difference Methods for Initial-Value Problems, 2nd ed. (Wiley, New York, 1967).

Oliner, A. A.

A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved optical planar waveguides,” Appl. Sci. Res. 41, 271–274 (1984).
[CrossRef]

Peng, S. T.

A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved optical planar waveguides,” Appl. Sci. Res. 41, 271–274 (1984).
[CrossRef]

Press, W.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in fortran: The Art of Scientific Computing (Cambridge U. Press, New York, 1986).

Ramanujam, N.

N. Ramanujam, J. J. Burke, L. Li, “Guided wave deflectors using gratings with slowly-varying groove depth for beam shaping,” in Guided Wave Optoelectronics, T. Tamir, ed. (Plenum, New York, 1995), pp. 321–332.

Richtmyer, R. D.

R. D. Richtmyer, K. W. Morton, Difference Methods for Initial-Value Problems, 2nd ed. (Wiley, New York, 1967).

Saito, S.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

Sakaki, H.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

Schoenfelder, A.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultrahigh brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper 7.

Scifres, D. R.

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CTuC2, pp. 77–78.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultrahigh brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper 7.

Solimar, L.

L. Solimar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
[CrossRef]

Tamir, T.

Teudolsky, S.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in fortran: The Art of Scientific Computing (Cambridge U. Press, New York, 1986).

Van Roey, J.

Vetterling, W.

W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in fortran: The Art of Scientific Computing (Cambridge U. Press, New York, 1986).

Waarts, R. G.

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

Wagatsuma, K.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

Wang, M. R.

Weitman, G.

G. Weitman, A. Hardy, “Reduction of coupling coefficients for distributed Bragg reflection in corrugated narrow-rib waveguides,” IEE Proc.: Optoelectron. 144, 101–103 (1997).

Welch, D. F.

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CTuC2, pp. 77–78.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultrahigh brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper 7.

Weller-Brophy, L. A.

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991).

Young, L.

Zhang, S.

Zuber, J. B.

C. Itzykson, J. B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

L. Solimar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
[CrossRef]

Appl. Sci. Res. (1)

A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved optical planar waveguides,” Appl. Sci. Res. 41, 271–274 (1984).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory of thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

IEE Proc.: Optoelectron. (1)

G. Weitman, A. Hardy, “Reduction of coupling coefficients for distributed Bragg reflection in corrugated narrow-rib waveguides,” IEE Proc.: Optoelectron. 144, 101–103 (1997).

IEEE J. Quantum Electron. (2)

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632–637 (1979).
[CrossRef]

IEEE Trans. Sonics Ultrason. (1)

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

J. Lightwave Technol. (1)

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

Proc. IEEE (1)

A. Korpel, “Acousto-optics—a review of fundamentals,” Proc. IEEE 69, 48–53 (1981).
[CrossRef]

Prog. Opt. (1)

D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
[CrossRef]

Other (14)

N. Ramanujam, J. J. Burke, L. Li, “Guided wave deflectors using gratings with slowly-varying groove depth for beam shaping,” in Guided Wave Optoelectronics, T. Tamir, ed. (Plenum, New York, 1995), pp. 321–332.

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper CTuC2, pp. 77–78.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultrahigh brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper 7.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1991).

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991).

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II.

C. R. Chester, Techniques in Partial Differential Equations (McGraw-Hill, New York, 1971).

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Figures (13)

Fig. 1
Fig. 1

Schematic illustration of the slab waveguide with its sinusoidal rectangular (finite-aperture) surface-relief grating at oblique incidence.

Fig. 2
Fig. 2

Near-field coordinate systems for the TE+ wave [ge+(z, y)] and the TE- wave [ge-(z, y)]. The grating coordinate system is (z, y).

Fig. 3
Fig. 3

Schematic illustration of the waveguide diffraction grating with an obliquely incident two-dimensional (2D) beam.

Fig. 4
Fig. 4

2D coupling (waveguide grating II): absolute values of the near fields for TE+ plane-wave incidence at θi=θe=60° (d=50 µm, Δz=4 µm, Δy=2 µm). The four-wave (4W) interactions are denoted by solid curves, and the two-wave (2W) interactions are denoted by dashed curves (TE–TE) and dotted curves (TE–TM). (a) TE+: |ge+(y)|, power=0.653 W (4W), 0.75 W (TE–TE), 0.91 W (TE–TM); (b) TE-: |ge-(y)|, power=0.242 W (4W), 0.25 W (TE–TE); (c) TM+: |gm+(y˜)|, power=0.015 W (4W); (d) TM-: |gm-(y˜)|, power=0.08 W (4W), 0.09 W (TE–TM).

Fig. 5
Fig. 5

2D coupling (waveguide grating II): normalized far-field intensity patterns, obtained from the near fields given in Fig. 4. The 4W interaction is denoted by solid curves, and the 2W interactions are denoted by dashed curves (TE–TE) and dotted curves (TE–TM). (a) |Ge+(θ)|2, (b) |Ge-(θ)|2, (c) |Gm+(θ)|2, (d) |Gm-(θ)|2.

Fig. 6
Fig. 6

4W-2D coupling (waveguide grating II) for TE+ plane-wave incidence (d=50 µm, Δz=Δy=0.4 µm). The maximum error in total power conservation (because of numerical calculations) is 0.31%. (a) TE- power reflectivity coefficient Re-(θi) and TE+ power transmission coefficient Te+(θi) as functions of the plane-wave incidence angle, (b) TM- power reflectivity coefficient Rm-(θi) and TM+ power transmission coefficient Tm+(θi) as functions of the plane-wave incidence angle.

Fig. 7
Fig. 7

Influence of the grating’s width (waveguide grating I) on the TE- wave (as an example) for TE+ plane-wave incidence at θi=60° (by the 4W-2D coupling): (a) d=500 µm, Δz=Δy=10 µm, |Ae-(z, y)|; (b) d=10 µm, Δz=4 µm, Δy=0.4 µm, |Ae-(z, y)|; (c) near field from (a), TE- power=0.512 W; (d) near field from (b), TE- power=0.035 W.

Fig. 8
Fig. 8

Compatibility of the 4W-2D coupling with the 4W-1D-coupling for TE+ plane-wave incidence at θi=60° (d=600 µm, Δz=Δy=2 µm) in terms of near-field intensity |ge+(y)|2 in waveguide grating I.

Fig. 9
Fig. 9

4W-2D coupling (waveguide grating II): absolute values of the interacting fields inside the grating for TE+ Gaussian beam incidence (W0=30 µm, total power=1 W) at θ0=60° (d=50 µm, Δz=4 µm, Δy=2 µm). (a) TE+ wave, |Ae+(z, y)|; (b) TE- wave, |Ae-(z, y)|; (c) TM+ wave, |Am+(z, y)|; (d) TM- wave, Am-(z, y)|.

Fig. 10
Fig. 10

4W-2D coupling (waveguide grating II): absolute values of the near fields for TE+ Gaussian beam incidence. The solid curves are related to the waves in Fig. 9 (W0=30 µm). The dashed curves are obtained under similar conditions, except that W0=2 µm (Δz=Δy=0.2 µm). (a) TE+: |ge+(y)|, power=0.430 W (solid curve), 0.640 W (dashed curve); (b) TE-: |ge-(y)|, power=0.344 W (solid curve), 0.085 W (dashed curve); (c) TM+: |gm+(y˜)|, power=0.016 W (solid curve), 0.012 W (dashed curve); (d) TM-: |gm-(y˜)|, power=0.208 W (solid curve), 0.270 W (dashed curve).

Fig. 11
Fig. 11

4W-2D coupling (waveguide grating II): normalized far-field intensity patterns obtained from the near fields given in Fig. 10 (solid curves refer to W0=30 µm, and dashed curves refer to W0=2 µm). (a) |Ge+(θ)|2, (b) |Ge-(θ)|2, (c) |Gm+(θ)|2, (d) |Gm-(θ)|2.

Fig. 12
Fig. 12

Parallelogram grating structure and its 2W characteristic coordinate system.

Fig. 13
Fig. 13

AS of the incident (2D) beam, coordinate systems, and notation. β(α) is the wave vector of one of the plane waves in the AS of the beam. βz=β cos α, βy=β sin α, and θi=α+θ0.

Tables (1)

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Table 1 Waveguides (I, II) and Their Gratings (First Order, q=1)

Equations (47)

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ρ=πλΛ2κ10,
βi cos θi+βd cos θd=qG=q 2πΛ,
βi sin θi=βd sin θd,
2δab=βa cos θa+βb cos θb-q 2πΛ,
θm=sin-1[(βe/βm)sin θe].
δee=βe cos θe-qπΛ,
δmm=βm cos θm-qπΛ,
δem=12βe cos θe+βm cos θm-q2πΛ=δee+δmm2.
E(r)ae+(z, y)Ete(+)(x)+ae-(z, y)Ete(-)(x)+am+(z, y)Etm(+)(x)+am-(z, y)Etm(-)(x),
aj±(z,y)Aj±(z,y)exp[-i(βjy sin θj±βjzcos θj)],
βe cos θe2ωμ-|E0(x)|2 dx=1,TEmodes,
βm cos θm2ωε0-[|H0(x)|2/n2(x)]dx=1,TMmodes,
zcos(θe)Ae+(z, y)cos(θe)Ae-(z, y)cos(θm)Am+(z, y)cos(θm)Am-(z, y)+ysin(θe)Ae+(z, y)-sin(θe)Ae-(z, y)sin(θm)Am+(z, y)-sin(θm)Am-(z, y)
=QAe+(z, y)Ae-(z, y)Am+(z, y)Am-(z, y),
Q=0κee* exp(-i2δeez)0κem* exp(-i2δemz)κee exp(i2δeez)0κme exp(i2δemz)00κme* exp(-i2δemz)0κmm* exp(-i2δmmz)κem exp(i2δemz)0κmm exp(i2δmmz)0
Ae+(0zL, y=0)=Ae+(z=0, 0yd)=Ai=1,
Ae-(0zL, y=0)=Ae-(z=L, 0yd)=0,
Am+(0zL, y=0)=Am+(z=0, 0yd)=0,
Am-(0zL, y=0)=Am-(z=L, 0yd)=0,
ge+(z=z0, y)=Ae+[z=(-y+d cos θe)/sin θe, y=d]exp[-iβe(L cos θe+d sin θe)],y0yd cos θeAe+[z=L, y=(y+L sin θe)/cos θe]exp[-iβe(L cos θe+d sin θe)],-L sin θeyy0,
z0=L cos θe+d sin θe,
y0=-L sin θe+d cos θe.
Gm-(θ)=cos(θ)0L sin θm+d cos m×gm-(z˜=d sin θm,y˜)exp[-iβm sin(θ)y˜]dy˜.
Einc(z, y)=Einc0W0W(z)exp-y2w2(z)exp-i ky22R(z)×exp-ik(z-s)-tan-1z-sz0,
cos(θi) A+(z, y)z+sin(θi) A+(z, y)y
=κ exp(i2δz)A-(z, y),
cos(θd) A-(z, y)z-sin(θd) A-(z, y)y
=κ*exp(-i2δz)A+(z, y),
zycos θisin θi-cos θdsin θdξη,
A+(ξ, η)ξ=κ exp[i2δz(ξ, η)]A-(ξ, η),
A-(ξ, η)η=-κ*exp[-i2δz(ξ, η)]A+(ξ, η);
2A+(ξ, η)ξη+|κ|2A+(ξ, η)+i2δ cos θd
×A+(ξ, η)ξ=0,
2A-(ξ, η)ηξ+|κ|2A-(ξ, η)+i2δ cos θi
×A-(ξ, η)η=0.
A+(ξ, η)A¯+(ξ, η)exp(-i2δη cos θd),
A-(ξ, η)A¯-(ξ, η)exp(-i2δξ cos θi)
2A¯+(ξ, η)ξη+|κ|2A¯+(ξ, η)=0,
2A¯-(ξ, η)ηξ+|κ|2A¯-(ξ, η)=0.
A-(ξ0, η0)=-κ* exp(-i2δξ0 cos θi)×0η0 exp(i2δη cos θd)×J0(2|κ|ξ0(η0-η))dη,
A+(ξ0, η0)=1-|κ|exp(-i2δη0 cos θd)×0η0 exp(i2δη cos θd)×ξ0η0-η1/2J1(2|κ|ξ0(η0-η))dη,
Ae+(z, y+Δy)=1-ΔyΔzcot θeAe+(z, y)+ΔyΔzcot(θe)Ae+(z-Δz, y)+Δysin θe[κee exp(i2δeez)Ae-(z, y)+κem exp(i2δemz)Am-(z, y)],
Ae-(z, y+Δy)=1-ΔyΔzcot θeAe-(z, y)+ΔyΔzcot(θe)Ae-(z+Δz, y)-Δysin θe[κee* exp(-i2δeez)Ae+(z, y)+κme* exp(-i2δemz)Am+(z, y)],
Am+(z, y+Δy)=1-ΔyΔzcot θmAm+(z, y)+ΔyΔzcot(θm)Am+(z-Δz, y)+Δysin θm[κme exp(i2δemz)Ae-(z, y)+κmm exp(i2δmmz)Am-(z, y)],
Am-(z, y+Δy)=1-ΔyΔzcot θmAm-(z, y)+ΔyΔzcot(θm)Am-(z+Δz, y)-Δysin θm[κem* exp(-i2δemz)Ae+(z, y)+κmm* exp(-i2δmmz)Am+(z, y)].
zy=cos θ0sin θ0-sin θ0cos θ0zy
F(α)=-f(y)exp(-iyβ sin α)dy,

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