Abstract

Normal-incidence planar-optical waveguide-imbedded phase gratings of finite aperture width and length are analyzed with Svidzinskii’s [Sov. J. Quantum Electron. 10, 1103 (1980)] two-dimensional Bragg-diffraction theory. Svidzinskii’s characteristic-grating equations are adapted for the rectangular-grating case, and an overlap integral is used to extend the theory to account for the mode structure of the waveguide. The combined theory is used to optimize the throughput of a system composed of an input grating coupler, a waveguide, and an output grating coupler for both the highly multimode (thick-waveguide) and the few-mode (thin-waveguide) cases.

© 1995 Optical Society of America

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References

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  1. G. N. Lawrence, P. J. Cronkite, “Physical optics analysis of the focusing grating coupler,” Appl. Opt. 27, 672–678 (1988).
    [CrossRef] [PubMed]
  2. K. K. Svidzinskii, “Optical properties of specially shaped waveguide diffraction gratings,” Sov. J. Quantum Electron. 11, 1323–1327 (1981).
    [CrossRef]
  3. F. Lin, E. M. Strzelecki, T. Jannson, “Optical multiplanar VLSI interconnects based on multiplexed waveguide holograms,” Appl. Opt. 29, 1126–1133 (1990).
    [CrossRef] [PubMed]
  4. S. Tang, R. T. Chen, “1-to-42 optoelectronic interconnection for intra-multichip-module clock signal distribution,” Appl. Phys. Lett. 64, 2931–2933 (1994).
    [CrossRef]
  5. W. Driemeier, “Coupled-wave analysis of the Bragg effect waveguide coupler,” J. Mod. Opt. 38, 363–377 (1991).
    [CrossRef]
  6. F. Sauer, “Fabrication of diffractive–reflective optical interconnects for infrared operation based on total internal reflection,” Appl. Opt. 28, 386–388 (1989).
    [CrossRef] [PubMed]
  7. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech J. 48, 2909–2947 (1969).
  8. L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
    [CrossRef]
  9. L. Solymar, M. P. Jordan, “Finite beams in large volume holograms,” Microwaves Opt. Acoust. 1(3), 89–92 (1977).
    [CrossRef]
  10. P. St. J. Russell, L. Solymar, “The properties of holographic overlap gratings,” Opt. Acta 26, 329–347 (1979).
    [CrossRef]
  11. K. K. Svidzinskii, “Theory of Bragg diffraction by limited-aperture gratings in a planar optical waveguide,” Sov. J. Quantum Electron. 10, 1103–1109 (1980).
    [CrossRef]
  12. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, Mass., 1991).
  13. R. P. Kenan, “Theory of crossed-beam diffraction gratings,” IEEE J. Quantum Electron. QE-14, 924–930 (1978).
    [CrossRef]
  14. See Ref. 7; this coupling constant expression is valid only for Δn ≪ n2.
  15. See Ref. 13; Eq. (18), p. 925.
  16. See Ref. 12; Eq. (1.4–18), p. 24.

1994

S. Tang, R. T. Chen, “1-to-42 optoelectronic interconnection for intra-multichip-module clock signal distribution,” Appl. Phys. Lett. 64, 2931–2933 (1994).
[CrossRef]

1991

W. Driemeier, “Coupled-wave analysis of the Bragg effect waveguide coupler,” J. Mod. Opt. 38, 363–377 (1991).
[CrossRef]

1990

1989

1988

1981

K. K. Svidzinskii, “Optical properties of specially shaped waveguide diffraction gratings,” Sov. J. Quantum Electron. 11, 1323–1327 (1981).
[CrossRef]

1980

K. K. Svidzinskii, “Theory of Bragg diffraction by limited-aperture gratings in a planar optical waveguide,” Sov. J. Quantum Electron. 10, 1103–1109 (1980).
[CrossRef]

1979

P. St. J. Russell, L. Solymar, “The properties of holographic overlap gratings,” Opt. Acta 26, 329–347 (1979).
[CrossRef]

1978

R. P. Kenan, “Theory of crossed-beam diffraction gratings,” IEEE J. Quantum Electron. QE-14, 924–930 (1978).
[CrossRef]

1977

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

L. Solymar, M. P. Jordan, “Finite beams in large volume holograms,” Microwaves Opt. Acoust. 1(3), 89–92 (1977).
[CrossRef]

1969

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

Chen, R. T.

S. Tang, R. T. Chen, “1-to-42 optoelectronic interconnection for intra-multichip-module clock signal distribution,” Appl. Phys. Lett. 64, 2931–2933 (1994).
[CrossRef]

Cronkite, P. J.

Driemeier, W.

W. Driemeier, “Coupled-wave analysis of the Bragg effect waveguide coupler,” J. Mod. Opt. 38, 363–377 (1991).
[CrossRef]

Jannson, T.

Jordan, M. P.

L. Solymar, M. P. Jordan, “Finite beams in large volume holograms,” Microwaves Opt. Acoust. 1(3), 89–92 (1977).
[CrossRef]

Kenan, R. P.

R. P. Kenan, “Theory of crossed-beam diffraction gratings,” IEEE J. Quantum Electron. QE-14, 924–930 (1978).
[CrossRef]

Kogelnik, H.

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

Lawrence, G. N.

Lin, F.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, Mass., 1991).

Russell, P. St. J.

P. St. J. Russell, L. Solymar, “The properties of holographic overlap gratings,” Opt. Acta 26, 329–347 (1979).
[CrossRef]

Sauer, F.

Solymar, L.

P. St. J. Russell, L. Solymar, “The properties of holographic overlap gratings,” Opt. Acta 26, 329–347 (1979).
[CrossRef]

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

L. Solymar, M. P. Jordan, “Finite beams in large volume holograms,” Microwaves Opt. Acoust. 1(3), 89–92 (1977).
[CrossRef]

Strzelecki, E. M.

Svidzinskii, K. K.

K. K. Svidzinskii, “Optical properties of specially shaped waveguide diffraction gratings,” Sov. J. Quantum Electron. 11, 1323–1327 (1981).
[CrossRef]

K. K. Svidzinskii, “Theory of Bragg diffraction by limited-aperture gratings in a planar optical waveguide,” Sov. J. Quantum Electron. 10, 1103–1109 (1980).
[CrossRef]

Tang, S.

S. Tang, R. T. Chen, “1-to-42 optoelectronic interconnection for intra-multichip-module clock signal distribution,” Appl. Phys. Lett. 64, 2931–2933 (1994).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

S. Tang, R. T. Chen, “1-to-42 optoelectronic interconnection for intra-multichip-module clock signal distribution,” Appl. Phys. Lett. 64, 2931–2933 (1994).
[CrossRef]

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

Bell Syst. Tech J.

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

IEEE J. Quantum Electron.

R. P. Kenan, “Theory of crossed-beam diffraction gratings,” IEEE J. Quantum Electron. QE-14, 924–930 (1978).
[CrossRef]

J. Mod. Opt.

W. Driemeier, “Coupled-wave analysis of the Bragg effect waveguide coupler,” J. Mod. Opt. 38, 363–377 (1991).
[CrossRef]

Microwaves Opt. Acoust.

L. Solymar, M. P. Jordan, “Finite beams in large volume holograms,” Microwaves Opt. Acoust. 1(3), 89–92 (1977).
[CrossRef]

Opt. Acta

P. St. J. Russell, L. Solymar, “The properties of holographic overlap gratings,” Opt. Acta 26, 329–347 (1979).
[CrossRef]

Sov. J. Quantum Electron.

K. K. Svidzinskii, “Theory of Bragg diffraction by limited-aperture gratings in a planar optical waveguide,” Sov. J. Quantum Electron. 10, 1103–1109 (1980).
[CrossRef]

K. K. Svidzinskii, “Optical properties of specially shaped waveguide diffraction gratings,” Sov. J. Quantum Electron. 11, 1323–1327 (1981).
[CrossRef]

Other

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, Mass., 1991).

See Ref. 7; this coupling constant expression is valid only for Δn ≪ n2.

See Ref. 13; Eq. (18), p. 925.

See Ref. 12; Eq. (1.4–18), p. 24.

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Figures (16)

Fig. 1
Fig. 1

Signal-distribution scheme in which the waveguide input/output coupler is attached to the back side of a multichip module substrate.

Fig. 2
Fig. 2

Input coupler for a waveguide with thickness d. The incident beam, F0(z), fills the width of the grating aperture, w, and the diffracted beam, G1(x), propagates in the plane of the waveguide. Δθ indicates the deviation from the Bragg angle.

Fig. 3
Fig. 3

Input-coupler/waveguide/output-coupler system with a path-folding grating pair. Incident light is diffracted by the input coupler and is guided toward the output coupler by a waveguide. Input and output beams are offset laterally and travel in opposite directions. Example incident-, diffracted-, and output-beam profiles are shown to illustrate a highly multimode waveguide system (compare with profiles in Fig. 4).

Fig. 4
Fig. 4

Input-coupler/waveguide/output-coupler system with a path-shifting grating pair. The output beam is shifted laterally with respect to the input beam, but it travels in the same direction after exiting the system. Example incident-, diffracted-, and output-beam profiles are shown to illustrate a highly multimode waveguide system (compare with profiles in Fig. 3).

Fig. 5
Fig. 5

Input coupler showing the input and the output angular-deviation parameters Δθ1 and Δθ2 (the relationship between the two parameters is shown in Fig. 7).

Fig. 6
Fig. 6

Phase diagram showing the relationship between the grating vector, K, and the input and the output wave vectors, kF and kG.

Fig. 7
Fig. 7

Angular relationship between the input and the output angular-deviation parameters, Δθ1 and Δθ2, for the coupler shown in Fig. 5.

Fig. 8
Fig. 8

Magnitude of the output wave vector, kG, as a function of the input-angle deviation, Δθ1 (see Fig. 6).

Fig. 9
Fig. 9

Input-coupler diffraction-efficiency curves as functions of the (internal) incidence-angle deviation from the Bragg angle for a 40-μm (thick waveguide) × 1-mm grating with cover, waveguide (film), and substrate bulk refractive indices 1.0, 1.527, and 1.514, respectively, at a wavelength of λ = 632.9 nm. Each curve represents a different value of Δn. The incident electric-field distribution, F0(z), is uniform. As expected, the diffraction efficiency increases as Δn increases.

Fig. 10
Fig. 10

Total system throughput as a function of the (external) input-angle deviation from the Bragg angle for a path-folding (Fig. 3) pair of gratings 40 μm × 1 mm with cover, waveguide (film), and substrate bulk refractive indices 1.0, 1.527, and 1.514, respectively, at a wavelength of λ = 632.9 nm. Each curve represents a different value of Δn. Diamonds are used to highlight the optimum Δn curve. A moderate value of Δn results in the highest on-Bragg throughput (compare with Fig. 9).

Fig. 11
Fig. 11

Total system throughput as a function of the (external) input-angle deviation from the Bragg angle for a path-shifting (Fig. 4) pair of gratings 40 μm × 1 mm with cover, waveguide (film), and substrate bulk refractive indices 1.0, 1.527, and 1.514, respectively, at a wavelength of λ = 632.9 nm. Each curve represents a different value of Δn. Diamonds are used to highlight the moderate Δn curve. A high value of Δn results in the highest on-Bragg throughput for this geometry (compare with Fig. 10).

Fig. 12
Fig. 12

On-Bragg throughput as a function of Δn for the path-folding (solid curve) and the path-shifting (dashed curve) systems as described in Figs. 10 and 11, respectively. The modal and the attenuation effects of the waveguide are neglected.

Fig. 13
Fig. 13

Diffraction-efficiency family of curves as a function of the internal-incidence-angle deviation from the Bragg angle for a 1 μm (thin waveguide) × 1 mm input coupler with cover, waveguide, and substrate bulk refractive indices 1.0, 1.6, and 1.5, respectively, at a wavelength of λ = 632.8 nm. Each curve represents a different value of Δn. The incident electric-field distribution, F0(z), is uniform, and, as expected, the diffraction efficiency increases as Δn increases.

Fig. 14
Fig. 14

Graph showing the power that is coupled into guided modes divided by the total input diffracted power in the two-mode coupler (as described in Fig. 12). For values of Δn less than 0.009, over 80% of the diffracted light is coupled into guided modes.

Fig. 15
Fig. 15

Total two-mode (thin-waveguide) system throughput as a function of the external incidence angle for both the path-folding and the path-shifting grating pair. Each curve represents a different value of Δn. A moderate Δn value of 0.006 (denoted by the diamonds) results in the optimum system throughput at the Bragg angle.

Fig. 16
Fig. 16

On-Bragg two-mode system throughput as a function of the waveguide length zl (see Fig. 3). This graph illustrates one of the drawbacks of the few-mode system. A variation in this particular waveguide of 5 μm can result in a 15% throughput variation.

Equations (28)

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Λ = 2 λ / 2 n ,
G 1 ( x ) = j κ exp [ j ( 2 π n Δ θ λ ) ( x + w ) ] 0 w F 0 ( z ) × exp [ j ( 2 π n Δ θ λ z ) J 0 [ 2 κ ( x z ) 1 / 2 ] d z ,
F R 1 ( z R ) = F R 0 ( z R ) κ exp [ ( j 2 π n Δ θ λ ) z R ] × z R w F R 0 ( ξ ) exp [ ( j 2 π n Δ θ λ ) ξ ] × ( d ξ z R ) 1 / 2 J 1 { 2 κ [ d ( ξ z R ) ] 1 / 2 } d ξ,
κ = π Δ n / λ .
η = 0 d | G 1 ( x ) | 2 d x 0 w | F 0 ( z ) | 2 d z ,
G 1 ( x ) = j ( w / x ) 1 / 2 J 1 [ 2 κ ( x w ) 1 / 2 ] .
η = 1 J 0 2 [ 2 κ ( w d ) 1 / 2 ] J 1 2 [ 2 κ ( w d ) 1 / 2 ] .
G 2 ( z 2 ) = j κ exp [ j ( 2 π n Δ θ 2 λ ) ( z 2 + d ) ] × 0 d F 2 ( x 2 ) exp ( j 2 π n Δ θ 2 λ x 2 ) × J 0 [ 2 κ ( x 2 z 2 ) 1 / 2 ] d x 2
F 2 ( x 2 ) = G 1 ( d x 1 ) = j κ exp [ j ( 2 π n Δ θ 1 λ ) ( w 1 + d x 1 ) ] × 0 w 1 F 0 ( z 1 ) exp ( j 2 π n Δ θ 1 λ z 1 ) × J 0 { 2 κ [ z 1 ( d x 1 ) 1 / 2 ] } d z 1
F 2 ( x 2 ) = G 1 ( x ) = j κ exp [ j ( 2 π n Δ θ 1 λ ) ( w 1 + x ) ] × 0 w 1 F 0 ( z 1 ) exp ( j 2 π n Δ θ 1 λ z 1 ) × J 0 [ 2 κ ( z 1 x ) 1 / 2 ] d z 1
η T = 0 w 2 | G 2 ( z 2 ) | 2 d z 2 0 w 1 | F 0 ( z 1 ) | 2 d z 1 .
| k F | exp ( j Δ θ 1 ) + | K | exp ( j 3 π 4 ) = | k G | exp [ j ( π 2 + Δ θ 2 ) ] .
Δ θ 2 0 . 1 ( Δ θ 1 ) 2 ;
Δ θ out 0 . 001 ( Δ θ 1 ) 4 .
E ( x , z ) = m α m E m ( x ) exp ( j β m z ) ,
α m = 0 d G 1 ( x ) E m ( x ) d x 0 d | E m ( x ) | 2 d x .
G 1 ( x ) = m α m E m ( x ) + β α R ( β ) E R ( β ) d β ,
G 1 ( x ) = m α m E m ( x ) + ρ α R ( ρ ) E R ( ρ , x ) d ρ ,
x G 1 ( x ) E R * ( ρ ˜ , x ) d x = ρ α R ( ρ ) x E R * ( ρ ˜ , x ) E R ( ρ , x ) d x d ρ .
β* 2 ωμ 0 E ( ρ ) E * ( ρ ˜ ) d x = β* | β | P δ ( ρ ρ ˜ ) ,
α R ( ρ ) = | β | 0 d G 1 ( x ) E R ( ρ , x ) d x 2 P ωμ 0 ,
R ( x , z ) = ρ α R ( ρ ) E R ( ρ , x ) exp ( j ρ z ) d ρ .
W ( x , z ) = E ( x , z ) + R ( x , z ) .
0 d | G 1 ( x ) | 2 d x = 0 d | E ( x , w ) | 2 d x + 0 d | R ( x , w ) | 2 d x
η T = 1 w 1 0 w 2 | G 2 ( z ) | 2 d z ,
G 2 ( z ) = κ 0 d ( w 1 / x ) 1 / 2 J 1 [ 2 κ ( x w 1 ) 1 / 2 ] J 0 [ 2 κ ( x z ) 1 / 2 ] d x
G 2 ( z ) = κ 0 d [ w 1 / ( d x ) ] 1 / 2 J 1 { 2 κ [ ( d x ) w 1 ] 1 / 2 } × J 0 [ 2 κ ( x z ) 1 / 2 ] d x
η g = x | E ( x , w 1 ) | 2 d x 0 d | G 1 ( x ) | 2 d x ,

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