Abstract

The problem of a guided wave reflected by a thin-film grating is considered. A solution method is proposed based on a coupled-beam theory and a boundary-value iteration. The method is applied to different configurations. A focusing wavelength demultiplexer is analyzed, which could be used with stripe waveguides as input and output channels.

© 1982 Optical Society of America

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References

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  1. G. Hatakoshi and S. Tanaka, “Grating lenses for integrated optics,” Opt. Lett. 2, 142–144 (1978).
    [CrossRef] [PubMed]
  2. P. K. Tien, “Method of forming novel curved-line gratings and their use as reflectors and resonators in integrated optics,” Opt. Lett. 1, 64–66 (1977).
    [CrossRef] [PubMed]
  3. J. P. Hugonin and R. Petit, “Theoretical and numerical study of a locally deformed stratified medium,” J. Opt. Soc. Am. 71, 664–674 (1981).
    [CrossRef]
  4. J. Van Roey and P. E. Lagasse, “Coupled wave analysis of obliquely incident waves in thin film gratings,” Appl. Opt. 20, 423–429 (1981).
    [CrossRef] [PubMed]
  5. G. Hatakoshi and S. Tanaka, “Coupled-mode theory for a plane and a cylindrical wave in optical waveguide Bragg grating lenses,” J. Opt. Soc. Am. 71, 40–48 (1981).
    [CrossRef]
  6. G. Hatakoshi and S. Tanaka, “Coupling of two cylindrical guided waves in optical waveguide Bragg lenses,” J. Opt. Soc. Am. 71, 121–123 (1981).
    [CrossRef]
  7. J. Van Roey, J. van der Donk, and P. E. Lagasse, “Beam propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
    [CrossRef]
  8. K. Wagatsuma, H. Sakaki, and 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]
  9. G. Mur, “A differential-equation method for the computation of the electromagnetic scattering by an inhomogeneity in a cylindrical waveguide,” J. Eng. Math. 12, 49–67 (1978).
    [CrossRef]
  10. D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Algebra (Freeman, San Francisco, 1963).
  11. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]

1981 (5)

1979 (1)

K. Wagatsuma, H. Sakaki, and 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 (2)

G. Mur, “A differential-equation method for the computation of the electromagnetic scattering by an inhomogeneity in a cylindrical waveguide,” J. Eng. Math. 12, 49–67 (1978).
[CrossRef]

G. Hatakoshi and S. Tanaka, “Grating lenses for integrated optics,” Opt. Lett. 2, 142–144 (1978).
[CrossRef] [PubMed]

1977 (1)

1969 (1)

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

Faddeev, D. K.

D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Algebra (Freeman, San Francisco, 1963).

Faddeeva, V. N.

D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Algebra (Freeman, San Francisco, 1963).

Hatakoshi, G.

Hugonin, J. P.

Kogelnik, H.

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

Lagasse, P. E.

Mur, G.

G. Mur, “A differential-equation method for the computation of the electromagnetic scattering by an inhomogeneity in a cylindrical waveguide,” J. Eng. Math. 12, 49–67 (1978).
[CrossRef]

Petit, R.

Saito, S.

K. Wagatsuma, H. Sakaki, and 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, and 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]

Tanaka, S.

Tien, P. K.

van der Donk, J.

Van Roey, J.

Wagatsuma, K.

K. Wagatsuma, H. Sakaki, and 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]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

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

IEEE J. Quantum Electron. (1)

K. Wagatsuma, H. Sakaki, and 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]

J. Eng. Math. (1)

G. Mur, “A differential-equation method for the computation of the electromagnetic scattering by an inhomogeneity in a cylindrical waveguide,” J. Eng. Math. 12, 49–67 (1978).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Lett. (2)

Other (1)

D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Algebra (Freeman, San Francisco, 1963).

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Figures (11)

Fig. 1
Fig. 1

Thin-film waveguide with a grating acting as a reflector.

Fig. 2
Fig. 2

Propagation through a grating.

Fig. 3
Fig. 3

Reflecting grating and its boundary conditions.

Fig. 4
Fig. 4

Oblique incidence of a plane wave on an infinitely wide grating.

Fig. 5
Fig. 5

Gaussian beam incident upon a square grating: 1, incident beam; 2, reflected beam; 3, transmitted beam. Distance d is 50 μm.

Fig. 6
Fig. 6

Reflected power as a function of e−2 width of a Gaussian beam.

Fig. 7
Fig. 7

Focusing of a diverging beam by a curved grating: 1, incident beam; 2, reflected beam in its focus; 3, transmitted (diverged) beam. Distance d is 500 μm.

Fig. 8
Fig. 8

Amplitude of the focal fields: (a) = 12 m−1, (b) = 12.2 m−1, (c) = 12.25 m−1. The amplitude of the incident field has a maximum equal to 1. The position is shown relative to the position of the incident beam.

Fig. 9
Fig. 9

Focusing by a curved and tilted grating. 1, incident beam; 2, reflected field in its focus; 3, transmitted (diverged) beam. Distance d is 500 μm.

Fig. 10
Fig. 10

Transverse shift of the focused field. Guided wavelength: 594 to 606 nm (from right to left with 2-nm steps). The position is shown relative to the position of the incident beam.

Fig. 11
Fig. 11

Focused field for guided wavelengths 601.5 nm (left) and 598.5 nm (right). The position is shown relative to the position of the incident beam.

Equations (27)

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x y 2 a + β a 2 a = - 2 j β a κ a b e - j U b , x y 2 b + β b 2 b = - 2 j β b κ b a e j U a ,
κ b a = - κ a b * ,
a corrected = a + κ a b cos α a e - j U b Δ y , b corrected = b + κ b a cos α b e j U a Δ y ,
a = incident field , b = 0.
b 1 ( 3 ) = Ū ¯ a 1 + V ¯ ¯ b 1 ( 1 ) .
V ¯ ¯ b 1 + c = 0 ,
R = b 1 ( 3 ) = V ¯ ¯ b 1 + c
b 1 = b 1 + h ( V ¯ ¯ ) R ,
R = g ( V ¯ ¯ ) R ,
g ( λ ) < 1             if λ is an eigenvalue of V ¯ ¯ .
g ( t ) = cos [ p arccos ( A t + B ) ] ,
= 1 ch ( p argch B ) .
A λ min + B = - 1 ,
A λ max + B = 1.
A λ max + B = cos π 2 p .
z 1 = b 1 - 1 γ 1 ( V ¯ ¯ b 1 + c ) z i = z i - 1 - 1 γ 1 ( V ¯ ¯ z i - 1 + c )             i = 2 , , p .
γ i = 1 A [ cos ( 2 i - 1 ) π 2 p - B ] .
d F d y = κ cos α B e j μ y , d B d y = κ cos α F e - j μ y ,
λ min = 1 , λ max = ch 2 κ L cos α .
z i = z i - 1 - 1 γ i r i - 1 ,
V ¯ ¯ z i - 1 + c .
E - 1 = [ ch ( p argch λ max + cos π 2 p λ max - 1 ) ] 1 / p .
lim p E = λ max - 1 λ max + 1 .
p - ln ln λ max + 1 λ max - 1 .
p > - ln ln 1 + T 1 - T ,
p > - ln / 2 T .
12 { [ ( x - x i ) 2 + ( y - y i ) 2 ] 1 / 2 + [ ( x - x f ) 2 + ( y - y f ) 2 ] 1 / 2 } ,