Abstract

The radiation characteristics of waveguide diffractive doublets consisting of double gratings located on two surfaces of waveguide cladding film are modeled based on a singular perturbation method. We determine the conditions under which the presence of the upper grating does not affect the radiation characteristics of the waveguide diffractive doublet as a whole. This allows independent performance of the upper grating, which may be replaced by a general diffractive optical element, and of the lower grating as a waveguide grating coupler. The results obtained provide an alternative method for determining the thickness of cladding film in the waveguide diffractive doublets for guided-wave manipulation.

© 1998 Optical Society of America

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References

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  1. I. A. Avrutsky, A. S. Savakhin, V. A. Sychugov, “Interference phenomena in waveguides with two corrugated boundaries,” J. Mod. Opt. 36, 1303–1320 (1989).
    [CrossRef]
  2. I. A. Avrutsky, A. S. Savakhin, V. A. Sychugov, “High-efficiency single-order waveguide grating coupler,” Opt. Lett. 15, 1446–1448 (1989).
    [CrossRef]
  3. A. Alphones, “Double grating coupler on a grounded dielectric slab waveguide,” Opt. Commun. 92, 35–39 (1992).
    [CrossRef]
  4. L. Li, “Analysis of planar waveguide grating couplers with double surface corrugations of identical period,” Opt. Commun. 114, 406–412 (1995).
    [CrossRef]
  5. 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]
  6. S. J. Sheard, T. Liao, G. Yang, P. R. Prewett, J. G. Zhu, “Focusing waveguide grating coupler using diffractive doublet,” Appl. Opt. 36, 4349–4353 (1997);also in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 360–363.
  7. T. Liao, S. J. Sheard, G. Yang, “An integrated waveguide diffractive doublet for guided-wave manipulation,” Appl. Opt. 36, 5476–5481 (1997).
    [CrossRef] [PubMed]
  8. T. Liao, S. J. Sheard, G. Yang, “Demonstration of guided-wave optical fan-out using waveguide diffractive doublet,” Opt. Commun. 137, 1–5 (1997).
    [CrossRef]
  9. M. T. Wlodarczyk, S. R. Seshadri, “Analysis of grating couplers for planar dielectric waveguides,” J. Appl. Phys. 58, 69–87 (1985).
    [CrossRef]
  10. S. R. Seshadri, “Coupling of guided modes in thin films with surface corrugation,” J. Appl. Phys. 63, R115–146 (1988).
    [CrossRef]

1997 (3)

1995 (2)

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]

L. Li, “Analysis of planar waveguide grating couplers with double surface corrugations of identical period,” Opt. Commun. 114, 406–412 (1995).
[CrossRef]

1992 (1)

A. Alphones, “Double grating coupler on a grounded dielectric slab waveguide,” Opt. Commun. 92, 35–39 (1992).
[CrossRef]

1989 (2)

I. A. Avrutsky, A. S. Savakhin, V. A. Sychugov, “Interference phenomena in waveguides with two corrugated boundaries,” J. Mod. Opt. 36, 1303–1320 (1989).
[CrossRef]

I. A. Avrutsky, A. S. Savakhin, V. A. Sychugov, “High-efficiency single-order waveguide grating coupler,” Opt. Lett. 15, 1446–1448 (1989).
[CrossRef]

1988 (1)

S. R. Seshadri, “Coupling of guided modes in thin films with surface corrugation,” J. Appl. Phys. 63, R115–146 (1988).
[CrossRef]

1985 (1)

M. T. Wlodarczyk, S. R. Seshadri, “Analysis of grating couplers for planar dielectric waveguides,” J. Appl. Phys. 58, 69–87 (1985).
[CrossRef]

Alphones, A.

A. Alphones, “Double grating coupler on a grounded dielectric slab waveguide,” Opt. Commun. 92, 35–39 (1992).
[CrossRef]

Avrutsky, I. A.

I. A. Avrutsky, A. S. Savakhin, V. A. Sychugov, “Interference phenomena in waveguides with two corrugated boundaries,” J. Mod. Opt. 36, 1303–1320 (1989).
[CrossRef]

I. A. Avrutsky, A. S. Savakhin, V. A. Sychugov, “High-efficiency single-order waveguide grating coupler,” Opt. Lett. 15, 1446–1448 (1989).
[CrossRef]

Brazas, J. C.

Li, L.

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]

L. Li, “Analysis of planar waveguide grating couplers with double surface corrugations of identical period,” Opt. Commun. 114, 406–412 (1995).
[CrossRef]

Liao, T.

Mckeon, A. L.

Prewett, P. R.

Savakhin, A. S.

I. A. Avrutsky, A. S. Savakhin, V. A. Sychugov, “High-efficiency single-order waveguide grating coupler,” Opt. Lett. 15, 1446–1448 (1989).
[CrossRef]

I. A. Avrutsky, A. S. Savakhin, V. A. Sychugov, “Interference phenomena in waveguides with two corrugated boundaries,” J. Mod. Opt. 36, 1303–1320 (1989).
[CrossRef]

Seshadri, S. R.

S. R. Seshadri, “Coupling of guided modes in thin films with surface corrugation,” J. Appl. Phys. 63, R115–146 (1988).
[CrossRef]

M. T. Wlodarczyk, S. R. Seshadri, “Analysis of grating couplers for planar dielectric waveguides,” J. Appl. Phys. 58, 69–87 (1985).
[CrossRef]

Sheard, S. J.

Sychugov, V. A.

I. A. Avrutsky, A. S. Savakhin, V. A. Sychugov, “High-efficiency single-order waveguide grating coupler,” Opt. Lett. 15, 1446–1448 (1989).
[CrossRef]

I. A. Avrutsky, A. S. Savakhin, V. A. Sychugov, “Interference phenomena in waveguides with two corrugated boundaries,” J. Mod. Opt. 36, 1303–1320 (1989).
[CrossRef]

Wlodarczyk, M. T.

M. T. Wlodarczyk, S. R. Seshadri, “Analysis of grating couplers for planar dielectric waveguides,” J. Appl. Phys. 58, 69–87 (1985).
[CrossRef]

Yang, G.

Zhu, J. G.

Appl. Opt. (3)

J. Appl. Phys. (2)

M. T. Wlodarczyk, S. R. Seshadri, “Analysis of grating couplers for planar dielectric waveguides,” J. Appl. Phys. 58, 69–87 (1985).
[CrossRef]

S. R. Seshadri, “Coupling of guided modes in thin films with surface corrugation,” J. Appl. Phys. 63, R115–146 (1988).
[CrossRef]

J. Mod. Opt. (1)

I. A. Avrutsky, A. S. Savakhin, V. A. Sychugov, “Interference phenomena in waveguides with two corrugated boundaries,” J. Mod. Opt. 36, 1303–1320 (1989).
[CrossRef]

Opt. Commun. (3)

A. Alphones, “Double grating coupler on a grounded dielectric slab waveguide,” Opt. Commun. 92, 35–39 (1992).
[CrossRef]

L. Li, “Analysis of planar waveguide grating couplers with double surface corrugations of identical period,” Opt. Commun. 114, 406–412 (1995).
[CrossRef]

T. Liao, S. J. Sheard, G. Yang, “Demonstration of guided-wave optical fan-out using waveguide diffractive doublet,” Opt. Commun. 137, 1–5 (1997).
[CrossRef]

Opt. Lett. (1)

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of a waveguide diffractive doublet. The waveguide parameters are n 1 = 1.000, n 2 = 1.460, n 3 = 1.492, n 4 = 1.460; t f = 2.0 μm, λ = 1.3 μm, and Λ = 0.80 μm.

Fig. 2
Fig. 2

Radiation characteristics versus cladding film thickness d for 2σ1 = 0.10 μm, 2σ2 = 0.075 μm. Solid curve, Φ = 0.25π; dashed curve, Φ = 0.50π; dot–dash curve, Φ = 0.75π. (a) Radiation decay factor; (b) radiation directionality.

Fig. 3
Fig. 3

Radiation characteristics versus thickness of cladding film d for fixed 2σ1 = 0.10 μm, Φ = 0.25π. Solid curve, 2σ2 = 0.05; dashed curve, 2σ2 = 0.075 μm; dot–dash curve, 2σ2 = 0.10 μm; (a) Radiation decay factor; (b) radiation directionality.

Fig. 4
Fig. 4

Radiation characteristics versus thickness of cladding film d for fixed σ2 = 0.10 μm, Φ = 0. Solid curve, 2σ1 = 0; dashed curve, 2σ1 = 0.05 μm; dot–dash curve, 2σ1 = 0.10 μm; (a) radiation decay factor; (b) radiation directionality.

Fig. 5
Fig. 5

Effects of the index of the cladding film on radiation decay factor for fixed 2σ2 = 0.10 μm, Φ = 0. Solid curve, 2σ1 = 0.10 μm; dashed curve, 2σ1 = 0.0 μm. (a) n c = 1.460 and (b) n c = 1.400.

Equations (37)

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2 x 2 + 2 z 2 + k 0 2 n 2 x ,   z E y x ,   z = 0 ,
i ω μ 0 H x x ,   z = - z   E y x ,   z ,
i ω μ 0 H z x ,   z = - x   E y x ,   z .
E y x ,   z = E y 0 x ,   z + δ E y 1 x ,   z .
/ z = / z 0 + / z 1 ,
δ 0   :   2 x 2 + 2 z 0 2 + k 0 2 n 2 x ,   z E y 0 x ,   z = 0 ,
δ 1   :   2 x 2 + 2 z 1 2 + k 0 2 n 2 x ,   z E y 1 x ,   z = - 2   2 z 0 z 1   E y 0 x ,   z .
E y x ,   z is continuous ,
H z x ,   z + dx dz   H x x ,   z is continuous .
δ 0   :   E y 0 x ,   z and   E y 0 x ,   z / x   are continuous ,
δ 1   :   E y 1 x ,   z + x z - x 0 ) E y 0 x ,   z / x   is continuous ,
δ 1   :   E y 1 x ,   z / x + x z - x 0 2 E y 0 x ,   z / x 2 - dx z dz 0   E y 0 x ,   z / z 0 is also continuous .
x 1 z = σ 1 sin Kz x 2 z = d + σ 2 sin Kz + Φ for   upper   grating for   lower   grating ,
E y x ,   z = m = - +   E ym x exp i β m z
β m = β 0 + 2 π m / Λ ,
E y x ,   z = C - 1 exp ik 1 x exp i β - 1 z , B - 1 exp ik 2 x + B - 1   exp - ik 2 x exp i β - 1 z , S - 1 cos k 3 t - x + ik 4 k 3 sin k 3 t - x exp i β - 1 z , S - 1 exp 1 - ik 4 x - t exp i β - 1 z , x < 0 0 < x < d . d < x < t x > t .
k j 2 = k 0 2 n j 2 - β - 1 2 ,   j = 1 ,   2 ,   3 ,   and   4 ,
| C m | 2 = 1 M A 1 2 G 1 2 + A 2 2 G 2 2 + 2 G 1 G 2 A 3 cos   Φ + A 4 sin   Φ ,
| S m | 2 = 1 M k 2 2 G 1 2 + k 1 2 sin 2 ϱ   + k 2 2 cos 2 ϱ G 2 2 + 2 G 1 G 2 k 2 2 cos ϱ cos   Φ + k 2 k 1 sin ϱ sin   Φ ,
G 1 1 2 E 1 k 0 2 σ 1 n 2 2 - n 1 2 , G 2 1 2 E 2 k 0 2 σ 2 n 3 2 - n 2 2 ,
A 1 2 = k 2 2 cos 2 ϱ   + k 4 2 sin 2 ϱ cos 2 ϱ 0 + k 2 k 3 k 4 2 k 3 2 - 1 sin 2 ϱ + k 2 2 k 4 2 k 3 2 cos 2 ϱ + k 3 2 sin 2 ϱ sin 2 ϱ 0 , A 2 2 = k 2 2 cos 2 ϱ 0 + k 2 2 k 4 2 k 3 2 + k 3 2 sin 2 ϱ 0 , A 3 = k 2 2 cos 2 ϱ 0 + k 2 2 k 4 2 k 3 2 + k 3 2 sin 2 ϱ 0 cos ϱ + k 2 k 3 k 4 2 k 3 2 - 1 sin ϱ , A 4 = - k 2 k 4 sin ϱ .
M = 1 2 k 2 2 + k 1 2 k 2 2 + k 4 2 cos 2 ϱ 0 + k 2 2 k 4 2 k 3 2 + k 3 2 sin 2 ϱ 0 - 1 2 k 2 2 - k 1 2 × k 2 2 - k 4 2 cos 2 ϱ 0 + k 2 2 k 4 2 k 3 2 - k 3 2 × sin 0 ϱ 0 cos 2 ϱ - 1 2   k 2 k 3 k 2 2 - k 1 2 × k 4 2 k 3 2 - 1 sin 2 ϱ 0 sin 2 ϱ + 2 k 1 k 2 2 k 4 ,
P r = 1 / 2 ω μ 0 | C - 1 | 2 Re k 0 2 n 1 2 - β - 1 2 1 / 2 + | S - 1 | 2 Re k 0 2 n 4 2 - β - 1 2 1 / 2 ,
α c = 1 / 2 ω μ 0 | C - 1 | 2 Re k 0 2 n 1 2 - β - 1 2 1 / 2 ,
α s = 1 / 2 ω μ 0 | S - 1 | 2 Re k 0 2 n 4 2 - β - 1 2 1 / 2 .
α = α c + α s ,
| C - 1 | 2 Φ = 0 ,     | C - 1 | 2 G 1 = 0 ,
tan   Φ = A 4 A 3 ,
σ 1 σ 2 = - γ s n 3 2 - n 2 2 E y 0 d A 1 2 n 2 2 - n 1 2 E y 0 0 ,
γ s 2 = A 3 2 + A 4 2 .
| C - 1 | 2 = 1 M A 2 2 - γ s 2 A 1 2 G 2 2 .
tan   Φ = k 1 k 2 tan   ϱ ,
σ 1 σ 2 = - γ c n 3 2 - n 2 2 E y 0 d k 2 n 2 2 - n 1 2 E y 0 0 ,
γ c 2 = k 1 2 sin 2 ϱ + k 2 2 cos 2 ϱ .
| C ¯ - 1 | 2 = 1 M   A 2 2 G 2 2 ,
| S ¯ - 1 | 2 = 1 M k 1 2 sin 2 ϱ + k 2 2 cos 2 ϱ G 2 2 .
α d = α ¯ d or   η c d = η c ¯ d .

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