Abstract

Diffraction of obliquely incident Gaussian beams on waveguide gratings with infinitely long grooves is described. The analysis is based on the angular spectrum of the beam and on the grating response derived from the four-wave coupled-mode theory, which simultaneously considers all four (approximately) synchronous waves. Results concerning the shapes of the emerging beams and their directions, the power content, and the possibility of beam splitting could differ significantly from those obtained by means of two-wave coupling. The analysis method is general and can be performed on many kinds of realistic beam shapes and applications.

© 1998 Optical Society of America

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  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]
  2. 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.
  3. A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultra-high 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.
  4. N. Izhaky, A. Hardy, “Four-wave coupled-mode theory of obliquely incident plane waves on waveguide diffraction gratings,” J. Opt. Soc. Am. A 14, 473–479 (1998).
    [CrossRef]
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  6. K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. 15, 632–637 (1979).
    [CrossRef]
  7. 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]
  8. A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991).
  9. W. Press, B. Flannery, S. Teudolsky, W. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U. Press, New York, 1986).
  10. R. S. Chu, T. Tamir, “Bragg diffraction of Gaussian beams by periodically modulated media,” J. Opt. Soc. Am. 66, 220–226 (1976).
    [CrossRef]
  11. J. Van Roey, P. E. Lagasse, “Coupled wave analysis of obliquely incident waves in thin film gratings,” Appl. Opt. 20, 423–429 (1981).
    [CrossRef] [PubMed]
  12. S. Zhang, T. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection gratings,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
    [CrossRef]
  13. M. R. Wang, “Analysis and observation of finite beam Bragg diffraction by a thick planar phase grating,” Appl. Opt. 35, 582–592 (1996).
    [CrossRef] [PubMed]

1998

N. Izhaky, A. Hardy, “Four-wave coupled-mode theory of obliquely incident plane waves on waveguide diffraction gratings,” J. Opt. Soc. Am. A 14, 473–479 (1998).
[CrossRef]

1996

1993

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]

1989

1988

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]

1981

1979

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

1976

Chu, R. S.

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, “Ultra-high 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, “Ultra-high 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.

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).

Goodman, J. W.

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

Hall, D. G.

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 coupled-mode theory of obliquely incident plane waves on waveguide diffraction gratings,” J. Opt. Soc. Am. A 14, 473–479 (1998).
[CrossRef]

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, “Ultra-high 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.

Izhaky, N.

N. Izhaky, A. Hardy, “Four-wave coupled-mode theory of obliquely incident plane waves on waveguide diffraction gratings,” J. Opt. Soc. Am. A 14, 473–479 (1998).
[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]

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultra-high 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.

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.

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).

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. 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. 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, “Ultra-high 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, “Ultra-high 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.

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. 15, 632–637 (1979).
[CrossRef]

Wang, M. R.

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, “Ultra-high 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).

Zhang, S.

Appl. Opt.

IEEE J. Quantum Electron.

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. 15, 632–637 (1979).
[CrossRef]

J. Lightwave Technol.

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.

J. Opt. Soc. Am. A

S. Zhang, T. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection gratings,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
[CrossRef]

N. Izhaky, A. Hardy, “Four-wave coupled-mode theory of obliquely incident plane waves on waveguide diffraction gratings,” J. Opt. Soc. Am. A 14, 473–479 (1998).
[CrossRef]

Other

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

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, “Ultra-high 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.

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

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

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Figures (11)

Fig. 1
Fig. 1

Schematic illustration of the waveguide diffraction grating with an obliquely incident (2-D) beam: (a) The waveguide diffraction grating. (b) The waveguide diffraction grating with an obliquely incident (2-D) beam. (c) AS of the incident (2-D) beam, coordinate systems, and notation. β(α) is the wave vector of one of the plane waves in the AS of the beam; β z = βcos(θ i ), β y = βsin(θ i ), θ i = α + θ0.

Fig. 2
Fig. 2

Incident Gaussian beam on the grating and its AS (W 0′ = 15 μm, d = 0, total power of 1 W): (a) Intensity distribution (in milliwatts per micrometer) in the space state, (1/2η)|f(y)|2. (b) AS magnitude, |F i )|.

Fig. 3
Fig. 3

Four-wave coupling (waveguide grating 1) for a TE+ incident plane wave |H i )|2: (a) Reflection coefficients: ℛ e - i ) for TE- and ℛ m - i ) for TM-. (b) Transmission coefficients: ℐ e + i ) for TE+ and ℐ m + i ) for TM+.

Fig. 4
Fig. 4

Four-wave coupling (waveguide grating 2) for a TE+ incident plane wave |H i )|2: (a) Reflection coefficient: ℛ e - i ) for TE-. (b) Reflection coefficient: ℛ m - i ) for TM-. (c) Transmission coefficient: ℐ e + i ) for TE+. (d) Transmission coefficient: ℐ m + i ) for TM+.

Fig. 5
Fig. 5

Two-wave coupling (waveguide grating 2): (a) TE- power reflectivity coefficient ℛ e - i ) and the TE+ power transmission coefficient ℐ e + i ) as functions of the plane-wave incident angle TE+ - TE-. (b) TM- power reflectivity coefficient ℛ m - i ) and the TE+ power transmission coefficient ℐ e + i ) as functions of the plane-wave incident angle TE+ - TM-.

Fig. 6
Fig. 6

Grating AS response magnitude (waveguide 1) |Gout)| for four-wave coupling: (a) Reflected TE-out = θ i ) beam, z = 0. (b) Reflected TM-out) beam, z = 0. (c) Transmitted TE+out = θ i ) beam, z = L. (d) Transmitted TM+out) beam, z = L.

Fig. 7
Fig. 7

Grating AS response magnitude (waveguide 2) |Gout)| for four-wave coupling: (a) Reflected TE-out = θ i ) beam, z = 0. (b) Reflected TM-out) beam, z = 0. (c) Transmitted TE+out = θ i ) beam, z = L. (d) Transmitted TM+out) beam, z = L.

Fig. 8
Fig. 8

Grating AS response magnitude (waveguide 2) |Gout)| for two-wave coupling: (a) Reflected TE-out = θ i ) beam, z = 0, TE+ - TE-. (b) Reflected TM-out) beam, z = 0, TE+ - TM-. (c) Transmitted TE+out = θ i ) beam, z = L, TE+ - TE-. (d) Transmitted TE+out = θ i ) beam, z = L, TE+ - TM-.

Fig. 9
Fig. 9

Grating response (waveguide 1) intensity distribution (in milliwatts per micrometer) in the spatial space for four-wave coupling: (a) Reflected TE- beam, z = 0, power of 0.332 W. (b) Reflected TM- beam, z = 0, power of 0.041 W. (c) Transmitted TE+ beam, z = L, power of 0.564 W. (d) Transmitted TM+ beam, z = L, power of 0.063 W.

Fig. 10
Fig. 10

Grating response (waveguide 2) intensity distribution (in milliwatts per micrometer) in the spatial space for four-wave coupling: (a) Reflected TE- beam, z = 0, power of 0.624 W. (b) Reflected TM- beam, z = 0, power of 0.1615 W. (c) Transmitted TE+ beam, z = L, power of 0.1745 W. (d) Transmitted TM+ beam, z = L, power of 0.04 W.

Fig. 11
Fig. 11

Grating response (waveguide 2) intensity distribution (in milliwatts per micrometer) in the spatial space for two-wave coupling: (a) Reflected TE- beam, z = 0, power of 0.725 W, TE+ - TE-. (b) Reflected TM- beam, z = 0, power of 0.22 W, TE+ - TM-. (c) Transmitted TE+ beam, z = L, power of 0.275 W. TE+ - TE-. (d) Transmitted TE+ beam, z = L, power of 0.78 W, TE+ - TM-.

Tables (1)

Tables Icon

Table 1 Waveguides 1 and 2 and Their Gratings (First Order g = 1)

Equations (19)

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E inc y ,   z = E inc 0 W 0 W z exp - y 2 w 2 z exp - i ky 2 2 R z × exp - i k z - d - tan - 1 z - d z 0 ,
W z = W 0 1 + z - d z 0 2 1 / 2 , R z = z - d + z 0 2 z - d ,
y z = cos   θ 0 - sin   θ 0 sin   θ 0 cos   θ 0 y z .
F θ i = -   f y exp - i β   sin   θ i   y d y .
H θ i S 0 R θ i ;   0 R L R θ i ;   0 exp i δ ee L U 0 R θ i ;   0 T L R θ i ;   0 exp i δ mm L ,     TE + z = 0 TE - z = 0 TE + z = 0 TE + z = L TE + z = 0 TM - z = 0 TE + z = 0 TM + z = L ,
G θ i = F θ i H θ i .
IFT G θ i = 1 2 π -   G θ i exp i β y y d β y = β 2 π - π / 2 π / 2 cos   θ i G θ i exp i β   sin   θ i y d θ i ,
P grz P tot = β W 0 2 π - π / 2 π / 2 - θ 0 cos α exp - W 0 β   sin α 2 2 d α 0.99 ,
L y 3 L   tan λ π W 0 + θ 0 .
β i cos   θ i + β d cos   θ d = qG = q   2 π Λ ,
β i sin   θ i = β d sin   θ d ,
2 δ ab = β a cos   θ a + β b cos   θ b - q   2 π Λ .
θ m = sin - 1 β e / β m sin θ e .
δ ee = β e cos   θ e - q π Λ ,
δ mm = β m cos   θ m - q π Λ ,
δ em = 1 2 β e cos   θ e + β m cos   θ m - q 2 π Λ = δ ee + δ mm 2 .
d d z R z S z T z U z = M R z S z T z U z ,
M - i δ ee κ ee 0 κ em κ ee * i δ ee κ me * 0 0 κ me - i δ mm κ mm κ em * 0 κ mm * i δ mm ,
R 0 = A i = 1 , S L = T 0 = U L = 0 ,

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