Abstract

The diffraction directivity of parallelogrammic gratings with second-order pitch is examined for a plane wave normally incident upon a corrugated waveguide structure. The three diffracted components are assumed to be in the form of guided waves, which permits a self-consistent calculation. The efficiencies of diffraction into the horizontal components are obtained. Also, the dependence of efficiency on grating thickness, waveguide thickness, grating pitch, and angle of inclination is determined. The approach provides a useful simulation tool for optimizing the design parameters for waveguide couplers with an orthogonal source.

© 1998 Optical Society of America

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References

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  1. G. W. Taylor, C. Kwan, “Determination of diffraction efficiency for a second-order corrugated waveguide,” IEEE J. Quantum Electron. 33, 176–186 (1997).
    [CrossRef]
  2. G. W. Taylor, C. Kwan, “Diffraction into a corrugated waveguide from normally incident radiation,” IEEE J. Lightwave Technol. (to be published).
  3. M. Li, S. J. Sheard, “Waveguide couplers using parallelogrammic-shaped blazed gratings,” Opt. Commun. 109, 239–245 (1994).
    [CrossRef]
  4. I. I. Mokhun’, M. O. Sopin, “Geometric-optical approach to the problem of excitation of a corrugated optical waveguide at near-normal angles of incidence,” Quantum Electron. 26, 836–838 (1996).
    [CrossRef]
  5. W. Streifer, D. R. Scifres, R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. 12, 422–428 (1976).
    [CrossRef]
  6. M. Matsumoto, “Analysis of the blazing effect in second-order grating,” IEEE J. Quantum Electron. 28, 2016–2023 (1992).
    [CrossRef]
  7. L. B. Mashev, E. K. Popov, E. G. Loewen, “Optimization of the grating efficiency in grazing incidence,” Appl. Opt. 26, 4738–4741 (1987).
    [CrossRef] [PubMed]
  8. R. F. Kazarinov, C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
    [CrossRef]
  9. W.-H. Lee, W. Streifer, “Radiation loss calculations for corrugated dielectric waveguides. II. TM polarization,” J. Opt. Soc. Am. 69, 1671–1676 (1979).
    [CrossRef]

1997

G. W. Taylor, C. Kwan, “Determination of diffraction efficiency for a second-order corrugated waveguide,” IEEE J. Quantum Electron. 33, 176–186 (1997).
[CrossRef]

1996

I. I. Mokhun’, M. O. Sopin, “Geometric-optical approach to the problem of excitation of a corrugated optical waveguide at near-normal angles of incidence,” Quantum Electron. 26, 836–838 (1996).
[CrossRef]

1994

M. Li, S. J. Sheard, “Waveguide couplers using parallelogrammic-shaped blazed gratings,” Opt. Commun. 109, 239–245 (1994).
[CrossRef]

1992

M. Matsumoto, “Analysis of the blazing effect in second-order grating,” IEEE J. Quantum Electron. 28, 2016–2023 (1992).
[CrossRef]

1987

1985

R. F. Kazarinov, C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
[CrossRef]

1979

1976

W. Streifer, D. R. Scifres, R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. 12, 422–428 (1976).
[CrossRef]

Burnham, R.

W. Streifer, D. R. Scifres, R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. 12, 422–428 (1976).
[CrossRef]

Henry, C. H.

R. F. Kazarinov, C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
[CrossRef]

Kazarinov, R. F.

R. F. Kazarinov, C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
[CrossRef]

Kwan, C.

G. W. Taylor, C. Kwan, “Determination of diffraction efficiency for a second-order corrugated waveguide,” IEEE J. Quantum Electron. 33, 176–186 (1997).
[CrossRef]

G. W. Taylor, C. Kwan, “Diffraction into a corrugated waveguide from normally incident radiation,” IEEE J. Lightwave Technol. (to be published).

Lee, W.-H.

Li, M.

M. Li, S. J. Sheard, “Waveguide couplers using parallelogrammic-shaped blazed gratings,” Opt. Commun. 109, 239–245 (1994).
[CrossRef]

Loewen, E. G.

Mashev, L. B.

Matsumoto, M.

M. Matsumoto, “Analysis of the blazing effect in second-order grating,” IEEE J. Quantum Electron. 28, 2016–2023 (1992).
[CrossRef]

Mokhun’, I. I.

I. I. Mokhun’, M. O. Sopin, “Geometric-optical approach to the problem of excitation of a corrugated optical waveguide at near-normal angles of incidence,” Quantum Electron. 26, 836–838 (1996).
[CrossRef]

Popov, E. K.

Scifres, D. R.

W. Streifer, D. R. Scifres, R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. 12, 422–428 (1976).
[CrossRef]

Sheard, S. J.

M. Li, S. J. Sheard, “Waveguide couplers using parallelogrammic-shaped blazed gratings,” Opt. Commun. 109, 239–245 (1994).
[CrossRef]

Sopin, M. O.

I. I. Mokhun’, M. O. Sopin, “Geometric-optical approach to the problem of excitation of a corrugated optical waveguide at near-normal angles of incidence,” Quantum Electron. 26, 836–838 (1996).
[CrossRef]

Streifer, W.

W.-H. Lee, W. Streifer, “Radiation loss calculations for corrugated dielectric waveguides. II. TM polarization,” J. Opt. Soc. Am. 69, 1671–1676 (1979).
[CrossRef]

W. Streifer, D. R. Scifres, R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. 12, 422–428 (1976).
[CrossRef]

Taylor, G. W.

G. W. Taylor, C. Kwan, “Determination of diffraction efficiency for a second-order corrugated waveguide,” IEEE J. Quantum Electron. 33, 176–186 (1997).
[CrossRef]

G. W. Taylor, C. Kwan, “Diffraction into a corrugated waveguide from normally incident radiation,” IEEE J. Lightwave Technol. (to be published).

Appl. Opt.

IEEE J. Quantum Electron.

R. F. Kazarinov, C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985).
[CrossRef]

W. Streifer, D. R. Scifres, R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. 12, 422–428 (1976).
[CrossRef]

M. Matsumoto, “Analysis of the blazing effect in second-order grating,” IEEE J. Quantum Electron. 28, 2016–2023 (1992).
[CrossRef]

G. W. Taylor, C. Kwan, “Determination of diffraction efficiency for a second-order corrugated waveguide,” IEEE J. Quantum Electron. 33, 176–186 (1997).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

M. Li, S. J. Sheard, “Waveguide couplers using parallelogrammic-shaped blazed gratings,” Opt. Commun. 109, 239–245 (1994).
[CrossRef]

Quantum Electron.

I. I. Mokhun’, M. O. Sopin, “Geometric-optical approach to the problem of excitation of a corrugated optical waveguide at near-normal angles of incidence,” Quantum Electron. 26, 836–838 (1996).
[CrossRef]

Other

G. W. Taylor, C. Kwan, “Diffraction into a corrugated waveguide from normally incident radiation,” IEEE J. Lightwave Technol. (to be published).

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Figures (6)

Fig. 1
Fig. 1

Schematic of the incident wave producing the transmitted wave, the reflected wave, and the diffracted wave resulting from a periodic grating.

Fig. 2
Fig. 2

Diffraction efficiency η versus grating thickness t at different blazed angles for a parallelogrammic grating. (a) Diffraction efficiency at an inclined angle 0° (solid curve) and at 26.6° (dashed curve); (b) diffraction efficiency at an inclined angle 0° (solid curve), at 26.6° (dashed curve), and at 35° (dotted curve); (c) diffraction efficiency at an inclined angle at 26.6° (dashed curve), at 35° (dotted curve), and at 42° (dotted–dashed curve).

Fig. 3
Fig. 3

Diffraction efficiency η versus normalized tooth width w/Λ for a parallelogrammic grating at several different blazed angles showing a peak in efficiency around 42° and a tooth width of approximately 0.38Λ.

Fig. 4
Fig. 4

Diffraction efficiency η versus waveguide thickness d for a parallelogrammic grating for several blaze angles, with other grating parameters remaining constant as indicated.

Fig. 5
Fig. 5

Diffraction efficiency η and diffraction constant 2α (representing 2α0 or 2α2 as noted in text, since these are the same) versus blaze angle θ for a parallelogrammic grating. The grating schematics shown below the θ axis indicate the angle of the blaze with respect to the vertical.

Fig. 6
Fig. 6

Diffraction efficiency η versus blaze angle θ on a log scale for a parallelogrammic grating. The grating schematics shown below the θ axis indicate the angle of the blaze with respect to the vertical.

Equations (42)

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n g 2 x ,   z = n ¯ 2 x + q = - ,   q 0   B q x exp - i 2 π qz Λ ,
B q x = n 2 2 - n 1 2 i 2 π q exp - i 2 π qw 2 x Λ - exp - i 2 π qw 1 x Λ ,
w 1 x = - w 2 - ax ,   w 2 x = w 2 - ax
B q x = n 2 2 - n 1 2 i 2 π q exp - i π qw Λ exp i 2 π q Λ   ax - exp i π qw Λ exp i 2 π q Λ   ax = - n 2 2 - n 1 2 π q sin π qw Λ exp i 2 π q Λ   ax .
n g 2 x ,   z ,   a = n ¯ 2 + q = -   q 0 n 2 2 - n 1 2 π q sin π qw Λ × exp i 2 π q ax + z Λ ,
n ¯ 2 = wn 1 2 + Λ - w n 2 2 / Λ ,   0 < x < t ,
n 2 x ,   z ,   a = n 1 2 x < 0 n g 2 x ,   z ,   a 0 < x < t n 2 2 t < x < d n 3 2 d < x .
2 E 0 ˜ x 2 - β 0 2 E 0 ˜ + k 0 2 n ¯ 2 E 0 ˜ + 2 i β 0 E 0 ˜ z + k 0 2 B 1 E 1 ˜ + B 2 E 2 ˜ = 0 ,   m = 0 ,
2 E 1 ˜ x 2 - β 1 2 E 1 ˜ + k 0 2 n ¯ 2 E 1 ˜ + 2 i β 1 E 1 ˜ z + k 0 2 B - 1 E 0 ˜ + B 1 E 2 ˜ = 0 ,   m = 1 ,
2 E 2 ˜ x 2 - β 2 2 E 2 ˜ + k 0 2 n ¯ 2 E 2 ˜ + 2 i β 2 E 2 ˜ z + k 0 2 B - 1 E 1 ˜ + B - 2 E 0 ˜ = 0 ,   m = 2 .
B 1 x = - n 2 2 - n 1 2 π sin π w Λ exp i 2 π a Λ   x ,
B - 1 x = - n 2 2 - n 1 2 π sin π w Λ exp - i 2 π a Λ   x ,
B 2 x = - n 2 2 - n 1 2 2 π sin 2 π w Λ exp i 4 π a Λ   x ,
B - 2 x = - n 2 2 - n 1 2 2 π sin 2 π w Λ exp - i 4 π a Λ   x .
β 0 = - β 2 ,
E i ˜ = A i x exp ± α i z exp - i β i z ,   i = 0 ,   1 ,   2 ,
A i x = A yi   exp δ i x x 0 A yi   cos   κ i x + B yi   sin   κ i x 0 x d ,   i = 0 ,   1 ,   2 A yi   cos   κ i d + B yi   sin   κ i d exp - γ i x - d d x ,
κ i = n 2 k 0   sin   θ bi , δ i = n 2 2 - n 1 2 k 0 2 - n 2 2 k 0 2 sin 2   θ bi 1 / 2 , γ i = n 2 2 - n 3 2 k 0 2 - n 2 2 k 0 2 sin 2   θ bi 1 / 2 ,
E 1 x x ,   z = C 1   exp ik 1 x , k 1 = k 0 2 n 1 2 - β 1 2 1 / 2 = k 0 n 1 ,
E gx x ,   z = C 2   exp ik g x + C 3   exp - ik g x + E 1 gp , k g = k 0 2 n ¯ 2 - β 1 2 1 / 2 = k 0 n ¯ ,
E 2 x x ,   z = C 4   exp ik 2 d - x + C 5   exp - ik 2 d - x , k 2 = k 0 2 n 2 2 - β 1 2 1 / 2 = k 0 n 2 ,
E 3 x x ,   z = C 6   exp - ik 3 x - d , k 3 = k 0 2 n 3 2 - β 1 2 1 / 2 = k 0 n 3 ,
E 1 gp = k 0 2 exp ik g x     B - 1 E 0 ˜ x ,   z + B 1 E 2 ˜ x ,   z exp - ik g x - 2 ik g d x - k 0 2 exp - ik g x     B - 1 E 0 ˜ x ,   z + B 1 E 2 ˜ x ,   z exp ik g x - 2 ik g d x .
d d z 1 2 η f   A 0 2 exp + 2 α 0 z d + 1 2 η f   A 2 2 exp + 2 α 2 z d = 1 2 η 1   C 1 C 1 * + 1 2 η 3   C 6 C 6 * .
A i = A yi κ 2 + δ 2 2 κ 2 ,   i = 0 ,   2 ,
E gx x ,   z = D 2 F 2 exp ik g x + D 3 F 3 exp - ik g x + E 0 gp E 2 gp ,
E 0 gp = k 0 2 exp ik g x ×   B 1 E 1 ˜ x ,   z + B 2 E 2 ˜ x ,   z exp - ik g x - 2 ik g d x - k 0 2 exp - ik g x ×   B 1 E 1 ˜ x ,   z + B 2 E 2 ˜ x ,   z exp ik g x - 2 ik g d x ,
E 2 gp = k 0 2 exp ik g x ×   B - 1 E 1 ˜ x ,   z + B - 2 E 0 ˜ x ,   z exp - ik g x - 2 ik g d x - k 0 2 exp - ik g x ×   B - 1 E 1 ˜ x ,   z + B - 2 E 0 ˜ x ,   z exp ik g x - 2 ik g d x ,
E 1 x x ,   z = D 1 F 1 exp ik 1 x ,   x 0 ,
E 2 x x ,   z = D 4 F 4 exp ik 2 d - x + D 5 F 5 × exp - ik 2 d - x ,   t x d ,
E 3 x x ,   z = D 6 F 6 exp - ik 3 x - d ,   d x ,
D 1 A y 0 ,   A y 1 ,   A y 2 = A y 0 ,
F 1 A y 0 ,   A y 1 ,   A y 2 = A y 2
η L = α 0 d   A y 0 A y 0 * A y 1 A y 1 * ,   η R = α 2 d   A y 2 A y 2 * A y 1 A y 1 * .
d d x ν 0 d H m ˜ d x + k 0 2 - β 1 2 ν 0 H m ˜ = q m q = 0 2 β q β m B ˆ m - q H m ˜ - d d x   B ˆ m - q d H m ˜ d x ,
E 1 gp = k 0 2 n 2 2 - n 1 2 2 π ik g   A y 0   sin π w Λ exp i   2 π a Λ   x × i 2 π a Λ - k g - δ 1 κ 1 2 - 2 π a Λ - k g 2 + - i 2 π a Λ + k g + δ 1 κ 1 2 - 2 π a Λ + k g 2 cos   κ 1 x + κ 1 + i 2 π a Λ - k g δ 1 κ 1 κ 1 2 - 2 π a Λ - k g 2 + - κ 1 - i 2 π a Λ + k g δ 1 κ 1 κ 1 2 - 2 π a Λ + k g 2 sin   κ 1 x + k 0 2 n 2 2 - n 1 2 2 π ik g   A y 2   sin π w Λ exp - i   2 π a Λ   x × - i 2 π a Λ + k g - δ 1 κ 1 2 - 2 π a Λ + k g 2 + - i - 2 π a Λ + k g + δ 1 κ 1 2 - - 2 π a Λ + k g 2 cos   κ 1 x + κ 1 - i 2 π a Λ + k g δ 1 κ 1 κ 1 2 - 2 π a Λ + k g 2 + - κ 1 - i - 2 π a Λ + k g δ 1 κ 1 κ 1 2 - - 2 π a Λ + k g 2 sin   κ 1 x ,
E 0 gp = k 0 2 n 2 2 - n 1 2 2 π ik g   A y 1   sin π w Λ exp i   2 π a Λ   x × i 2 π a Λ - k g - δ 0 κ 0 2 - 2 π a Λ - k g 2 + - i 2 π a Λ + k g + δ 0 κ 0 2 - 2 π a Λ + k g 2 cos   κ 0 x + κ 0 + i 2 π a Λ - k g δ 0 κ 0 κ 0 2 - 2 π a Λ - k g 2 + - κ 0 - i 2 π a Λ + k g δ 0 κ 0 κ 0 2 - 2 π a Λ + k g 2 sin   κ 0 x + k 0 2 n 2 2 - n 1 2 4 π ik g   A y 2   sin 2 π w Λ exp i   4 π a Λ   x × i 4 π a Λ - k g - δ 0 κ 0 2 - 4 π a Λ - k g 2 + - i 4 π a Λ + k g + δ 0 κ 0 2 - 4 π a Λ + k g 2 cos   κ 0 x + κ 0 + i 4 π a Λ - k g δ 0 κ 0 κ 0 2 - 4 π a Λ - k g 2 + - κ 0 - i 4 π a Λ + k g δ 0 κ 0 κ 0 2 - 4 π a Λ + k g 2 sin   κ 0 x ,
E 2 gp = k 0 2 n 2 2 - n 1 2 2 π ik g   A y 1   sin π w Λ exp - i   2 π a Λ   x × i - 2 π a Λ - k g - δ 2 κ 2 2 - 2 π a Λ + k g 2 + - i - 2 π a Λ + k g + δ 2 κ 2 2 - - 2 π a Λ + k g 2 cos   κ 2 x + κ 2 - i 2 π a Λ + k g δ 2 κ 2 κ 2 2 - 2 π a Λ + k g 2 + - κ 2 - i - 2 π a Λ + k g δ 2 κ 2 κ 2 2 - - 2 π a Λ + k g 2 sin   κ 2 x + k 0 2 n 2 2 - n 1 2 4 π ik g   A y 0   sin 2 π w Λ exp - i   4 π a Λ   x × i - 4 π a Λ - k g - δ 2 κ 2 2 - 4 π a Λ + k g 2 + - i - 4 π a Λ + k g + δ 2 κ 2 2 - - 4 π a Λ + k g 2 cos   κ 2 x + κ 2 - i 4 π a Λ + k g δ 2 κ 2 κ 2 2 - 4 π a Λ + k g 2 + - κ 2 - i - 4 π a Λ + k g δ 2 κ 2 κ 2 2 - - 4 π a Λ + k g 2 sin   κ 2 x ,
C 1 = k 2 + k 3 k 2 - k 3   A - B + k 2 + k 3 k 2 - k 3   C - D + 1 2 1 + k 1 k g A + 1 2 1 - k 1 k g B ,
C 6 = 1 2 c 1 A + B + C + D + k 1 k g A + C - B - D - A + C - B + D + ,
A = 1 2 1 - k g k 2 exp - i k 2 d - t - k g t ,   B = 1 2 1 + k g k 2 exp - i k 2 d - t + k g t , C = 1 2 1 + k g k 2 exp i k 2 d - t - k g t ,   D = 1 2 1 - k g k 2 exp i k 2 d - t + k g t , = 1 2 E 1 gp 0 + i k g   E 1 gp 0 ,   = 1 2 E 1 gp 0 - i k g   E 1 gp 0 , = i E 1 gp t sin   k 2 d - t - 1 k 2   E 1 gp t cos   k 2 d - t ,   = E 1 gp t cos   k 2 d - t + 1 k 2   E 1 gp t sin   k 2 d - t .
E 1 gp x = i   2 π a Λ k 0 2 n 2 2 - n 1 2 2 π ik g   A y 0   sin π w Λ exp i   2 π a Λ   x × i 2 π a Λ - k g - δ 1 κ 1 2 - 2 π a Λ - k g 2 + - i 2 π a Λ + k g + δ 1 κ 1 2 - 2 π a Λ + k g 2 cos   κ 1 x + κ 1 + i 2 π a Λ - k g δ 1 κ 1 κ 1 2 - 2 π a Λ - k g 2 + - κ 1 - i 2 π a Λ + k g δ 1 κ 1 κ 1 2 - 2 π a Λ + k g 2 sin   κ 1 x - i   2 π a Λ k 0 2 n 2 2 - n 1 2 2 π ik g   A y 2   sin π w Λ × exp - i   2 π a Λ   x × - i 2 π a Λ + k g - δ 1 κ 1 2 - 2 π a Λ + k g 2 + - i - 2 π a Λ + k g + δ 1 κ 1 2 - - 2 π a Λ + k g 2 cos   κ 1 x + κ 1 - i 2 π a Λ + k g δ 1 κ 1 κ 1 2 - 2 π a Λ + k g 2 + - κ 1 - i - 2 π a Λ + k g δ 1 κ 1 κ 1 2 - - 2 π a Λ + k g 2 sin   κ 1 x + k 0 2 n 2 2 - n 1 2 2 π ik g   A y 0   sin π w Λ × exp i   2 π a Λ   x κ 1 × - i 2 π a Λ - k g - δ 1 κ 1 2 - 2 π a Λ - k g 2 + - i 2 π a Λ + k g + δ 1 κ 1 2 - 2 π a Λ + k g 2 sin   κ 1 x + κ 1 + i 2 π a Λ - k g δ 1 κ 1 κ 1 2 - 2 π a Λ - k g 2 + - κ 1 - i 2 π a Λ + k g δ 1 κ 1 κ 1 2 - 2 π a Λ + k g 2 cos   κ 1 x + k 0 2 n 2 2 - n 1 2 2 π ik g   A y 2   sin π w Λ × exp - i   2 π a Λ   x κ 1 × - - i 2 π a Λ + k g - δ 1 κ 1 2 - 2 π a Λ + k g 2 + - i - 2 π a Λ + k g + δ 1 κ 1 2 - - 2 π a Λ + k g 2 cos   κ + κ 1 - i 2 π a Λ + k g δ 1 κ 1 κ 1 2 - 2 π a Λ + k g 2 + - κ 1 - i - 2 π a Λ + k g δ 1 κ 1 κ 1 2 - - 2 π a Λ + k g 2 sin   κ 1 x .

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