Abstract

A detailed account is given of the practical and theoretical aspects of grating or periodic couplers. Such couplers are used to launch light into thin film optical waveguides. The theoretical description stresses considerations of reciprocity. A Green's function formulation is used to calculate the radiated fields occurring when a guided wave is incident upon a grating coupler. Experiments were performed on grating couplers, formed in photoresist on top of thin film waveguides of 7059 glass deposited on glass of lower refractive index. Measurements of the coupling length and proportion of power in the various beams are presented as a function of the grating period for waveguides supporting one and many modes. Good agreement is found between the theoretical and experimental results. The methods used in the fabrication of the gratings are discussed in detail, and guidelines for the design of grating couplers are given.

© 1975 Optical Society of America

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References

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  1. M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
    [CrossRef]
  2. D. Dalgoutte, Opt. Commun. 8, 124 (1973).
    [CrossRef]
  3. J. H. Harris, R. K. Winn, D. G. Dalgoutte, Appl. Opt. 11, 2234 (1972).
    [CrossRef] [PubMed]
  4. J. H. Harris, R. Shubert, J. N. Polky, J. Opt. Soc. Am. 60, 1007 (1970).
    [CrossRef]
  5. R. Ulrich, J. Opt. Soc. Am. 60, 1337 (1970).
    [CrossRef]
  6. J. E. Midwinter, F. Zernike, Appl. Phys. Lett. 16, 198 (1970).
    [CrossRef]
  7. P. K. Tien, R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
    [CrossRef]
  8. R. Ulrich, J. Qpt. Soc. Am. 61, 1467 (1971).
  9. P. K. Tien, R. J. Martin, Appl. Phys. Lett. 18, 398 (1971).
    [CrossRef]
  10. C. Stanley, W. Duncan, J. A. McMurray, Appl. Phys. Lett. 24, 380 (1974).
    [CrossRef]
  11. S. T. Peng, T. Tamir, H. L. Bertoni, Electron Lett. 9, 150 (1973).
    [CrossRef]
  12. K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, IEEE J. Quantum Electron. QE-9, 1 (1973).
  13. L. L. Hope, Opt. Commun. 5, 179 (1972).
    [CrossRef]
  14. M. Neviere, R. Petit, M. Cadilhac, Opt. Commun. 8, 113 (1973).
    [CrossRef]
  15. See, for example, R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), p. 59.
  16. J. H. Harris, R. Shubert, IEEE Trans. Trans. Microwave Theory Tech. MTT-19, 269 (1971).
    [CrossRef]
  17. R. Shubert, J. H. Harris, J. Opt. Soc. Am. 61, 154 (1971).
    [CrossRef]
  18. R. Shubert, “Optical Waveguides and Applications to Integrated Optics,” Ph.D. Thesis, University of Washington, Seattle (1971).
  19. G. N. Jackson, Thin Solid Films 5, 209 (1970).
    [CrossRef]
  20. M. J. Beesley, J. G. Castledine, Appl. Opt. 9, 2720 (1970).
    [CrossRef] [PubMed]
  21. J. A. Aas, Appl. Opt. 11, 1579 (1972).
    [CrossRef] [PubMed]
  22. J. A. Ratcliffe, “Some Aspects of Diffraction Theory and Their Application to the Ionosphere,” in Rep. on Prog. in Physics XIX (1956), p. 188.
    [CrossRef]
  23. S. Fujiwara, Y. Iguchi, J. Opt. Soc. Am. 58, 361 (1968).
    [CrossRef]

1974 (1)

C. Stanley, W. Duncan, J. A. McMurray, Appl. Phys. Lett. 24, 380 (1974).
[CrossRef]

1973 (4)

S. T. Peng, T. Tamir, H. L. Bertoni, Electron Lett. 9, 150 (1973).
[CrossRef]

K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, IEEE J. Quantum Electron. QE-9, 1 (1973).

D. Dalgoutte, Opt. Commun. 8, 124 (1973).
[CrossRef]

M. Neviere, R. Petit, M. Cadilhac, Opt. Commun. 8, 113 (1973).
[CrossRef]

1972 (3)

1971 (4)

R. Ulrich, J. Qpt. Soc. Am. 61, 1467 (1971).

P. K. Tien, R. J. Martin, Appl. Phys. Lett. 18, 398 (1971).
[CrossRef]

J. H. Harris, R. Shubert, IEEE Trans. Trans. Microwave Theory Tech. MTT-19, 269 (1971).
[CrossRef]

R. Shubert, J. H. Harris, J. Opt. Soc. Am. 61, 154 (1971).
[CrossRef]

1970 (7)

1968 (1)

Aas, J. A.

Beesley, M. J.

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, Electron Lett. 9, 150 (1973).
[CrossRef]

Cadilhac, M.

M. Neviere, R. Petit, M. Cadilhac, Opt. Commun. 8, 113 (1973).
[CrossRef]

Castledine, J. G.

Chang, W. S. C.

K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, IEEE J. Quantum Electron. QE-9, 1 (1973).

Dakss, M. L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[CrossRef]

Dalgoutte, D.

D. Dalgoutte, Opt. Commun. 8, 124 (1973).
[CrossRef]

Dalgoutte, D. G.

Duncan, W.

C. Stanley, W. Duncan, J. A. McMurray, Appl. Phys. Lett. 24, 380 (1974).
[CrossRef]

Fujiwara, S.

Harrington, R. F.

See, for example, R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), p. 59.

Harris, J. H.

Heidrich, P. F.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[CrossRef]

Hope, L. L.

L. L. Hope, Opt. Commun. 5, 179 (1972).
[CrossRef]

Iguchi, Y.

Jackson, G. N.

G. N. Jackson, Thin Solid Films 5, 209 (1970).
[CrossRef]

Kuhn, L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[CrossRef]

Martin, R. J.

P. K. Tien, R. J. Martin, Appl. Phys. Lett. 18, 398 (1971).
[CrossRef]

McMurray, J. A.

C. Stanley, W. Duncan, J. A. McMurray, Appl. Phys. Lett. 24, 380 (1974).
[CrossRef]

Midwinter, J. E.

J. E. Midwinter, F. Zernike, Appl. Phys. Lett. 16, 198 (1970).
[CrossRef]

Neviere, M.

M. Neviere, R. Petit, M. Cadilhac, Opt. Commun. 8, 113 (1973).
[CrossRef]

Ogawa, K.

K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, IEEE J. Quantum Electron. QE-9, 1 (1973).

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, Electron Lett. 9, 150 (1973).
[CrossRef]

Petit, R.

M. Neviere, R. Petit, M. Cadilhac, Opt. Commun. 8, 113 (1973).
[CrossRef]

Polky, J. N.

Ratcliffe, J. A.

J. A. Ratcliffe, “Some Aspects of Diffraction Theory and Their Application to the Ionosphere,” in Rep. on Prog. in Physics XIX (1956), p. 188.
[CrossRef]

Rosenbaum, F. J.

K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, IEEE J. Quantum Electron. QE-9, 1 (1973).

Scott, B. A.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[CrossRef]

Shubert, R.

J. H. Harris, R. Shubert, IEEE Trans. Trans. Microwave Theory Tech. MTT-19, 269 (1971).
[CrossRef]

R. Shubert, J. H. Harris, J. Opt. Soc. Am. 61, 154 (1971).
[CrossRef]

J. H. Harris, R. Shubert, J. N. Polky, J. Opt. Soc. Am. 60, 1007 (1970).
[CrossRef]

R. Shubert, “Optical Waveguides and Applications to Integrated Optics,” Ph.D. Thesis, University of Washington, Seattle (1971).

Sopori, B. L.

K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, IEEE J. Quantum Electron. QE-9, 1 (1973).

Stanley, C.

C. Stanley, W. Duncan, J. A. McMurray, Appl. Phys. Lett. 24, 380 (1974).
[CrossRef]

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, Electron Lett. 9, 150 (1973).
[CrossRef]

Tien, P. K.

P. K. Tien, R. J. Martin, Appl. Phys. Lett. 18, 398 (1971).
[CrossRef]

P. K. Tien, R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
[CrossRef]

Ulrich, R.

Winn, R. K.

Zernike, F.

J. E. Midwinter, F. Zernike, Appl. Phys. Lett. 16, 198 (1970).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (4)

J. E. Midwinter, F. Zernike, Appl. Phys. Lett. 16, 198 (1970).
[CrossRef]

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
[CrossRef]

P. K. Tien, R. J. Martin, Appl. Phys. Lett. 18, 398 (1971).
[CrossRef]

C. Stanley, W. Duncan, J. A. McMurray, Appl. Phys. Lett. 24, 380 (1974).
[CrossRef]

Electron Lett. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, Electron Lett. 9, 150 (1973).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Ogawa, W. S. C. Chang, B. L. Sopori, F. J. Rosenbaum, IEEE J. Quantum Electron. QE-9, 1 (1973).

IEEE Trans. Trans. Microwave Theory Tech. MTT-19 (1)

J. H. Harris, R. Shubert, IEEE Trans. Trans. Microwave Theory Tech. MTT-19, 269 (1971).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Qpt. Soc. Am. (1)

R. Ulrich, J. Qpt. Soc. Am. 61, 1467 (1971).

Opt. Commun. (3)

D. Dalgoutte, Opt. Commun. 8, 124 (1973).
[CrossRef]

L. L. Hope, Opt. Commun. 5, 179 (1972).
[CrossRef]

M. Neviere, R. Petit, M. Cadilhac, Opt. Commun. 8, 113 (1973).
[CrossRef]

Thin Solid Films (1)

G. N. Jackson, Thin Solid Films 5, 209 (1970).
[CrossRef]

Other (3)

J. A. Ratcliffe, “Some Aspects of Diffraction Theory and Their Application to the Ionosphere,” in Rep. on Prog. in Physics XIX (1956), p. 188.
[CrossRef]

R. Shubert, “Optical Waveguides and Applications to Integrated Optics,” Ph.D. Thesis, University of Washington, Seattle (1971).

See, for example, R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), p. 59.

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Figures (22)

Fig. 1
Fig. 1

(a) Direct focusing into edge of waveguide; (b) prism coupler; (c) taper coupler.

Fig. 2
Fig. 2

(a) Grating coupler used as an input coupler; (b) wave vector diagram for input coupling; ka is the wave vector of the incident light, kw of the guided wave, and K = 2π/Λ, where Λ is the period of grating.

Fig. 3
Fig. 3

(a) The incident light (P0) is scattered (Ps), reflected (Pr), transmitted (Pt), and coupled out of the waveguide into two beams in the air (P1a, P2a) and two substrate beams (P1s, P2s). (b) Wave vector diagram for output coupling. The particular choice of K allows four output beams.

Fig. 4
Fig. 4

The Green's functions Gw and Gg are defined for the uniform multilayer waveguide structures shown.

Fig. 5
Fig. 5

Schematic of rf sputtering system for deposition of glass.

Fig. 6
Fig. 6

Interferometer used to produce gratings in photoresist.

Fig. 7
Fig. 7

Response of AZ-1350 photoresist to 476-nm laser light.

Fig. 8
Fig. 8

Refractive index of AZ-1350 photoresist as a function of wavelength. Point marked as a square is from manufacturer's data.

Fig. 9
Fig. 9

Experimental arrangement for measuring output coupling efficiency. Some of the incident laser beam is reflected to a photodiode to provide a reference signal. A prism coupler is used to launch light into the waveguide. The intensities of the output beams are measured. The hemispherical lens prevents multiple reflections.

Fig. 10
Fig. 10

Intensity of light as a function of distance along exit aperture (from edge of grating). The discontinuity in the line for the TE1 mode is due to a scattering center in the guide. The value of α is found from the slope of the lines.

Fig. 11
Fig. 11

Details of grating couplers (of different grating periods and depths) used in single mode experiments.

Fig. 12
Fig. 12

Equivalent structure used for computation of fq.

Fig. 13
Fig. 13

The mode index (kw/k0) for the composite waveguide structure as a function of the thickness of the photoresist film.

Fig. 14
Fig. 14

The amplitude of the axial magnetic field Hz as a function of transverse position in a composite waveguide of 0.78-μm glass with 0.39-μm photoresist film.

Fig. 15
Fig. 15

The variation of the decay rate α as a function of grating period Λ for a grating depth of 0.190 μm for the TE0 mode in the single mode structure. The curved ends to the lines represent the fact that the calculations are not valid close to the values of Λ at which Bragg reflection occurs or for which Wood's anomalies are allowed.

Fig. 16
Fig. 16

The variation of the decay rate α as a function of grating period for various values of the grating depth for the TE0 mode in the single mode waveguide structure. The gaps in the curves occur for grating periods that give rise to Bragg reflections or Wood's anomalies.

Fig. 17
Fig. 17

The variation of decay rate a as α function of grating period for the TE1 mode in the single mode waveguide structure.

Fig. 18
Fig. 18

The proportion of power in the first order output beam into the substrate as a function of grating period Λ for the TE0 mode in the single mode waveguide.

Fig. 19
Fig. 19

Details of grating couplers (of different grating periods and depths) used in the multimode experiments.

Fig. 20
Fig. 20

The variation of the decay rate α as a function of the grating modulation depth for various modes in the multimode waveguide.

Fig. 21
Fig. 21

The proportion of power in the first order output beam into the substrate as a function of grating period for the TE0 mode in the multimode waveguide.

Fig. 22
Fig. 22

Wave vector diagram for the reverse grating coupler.

Tables (2)

Tables Icon

Table I Relative Output Powers in Various Orders for Single Mode Waveguide Structure

Tables Icon

Table II Relative Output Powers in Various Orders for the Multimode Waveguide Structure

Equations (51)

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k w n K = k a sin θ n a or k s sin θ n s ,
P 0 = P t + P r + P s + n ( P n s + P n a ) .
E y = ( 2 ω μ 0 P 0 k w ) 1 / 2 f ( x ) exp [ j ( ω t k w z ) ] ,
E y = ( 2 ω μ 0 P n s k s cos θ n s ) 1 / 2 g ( z ) exp [ j ( k s sin θ n s ) z ] ,
[ g ( z ) ] 2 d z = 1 .
E y i = ( 2 ω μ 0 P i k s cos θ i ) 1 / 2 h ( z ) exp [ j ( k s sin θ i ) z ] .
θ i = θ n s or θ n a .
( E d × H r E r × H d ) · d S = 0 ,
P c / P i = [ g ( z ) h ( z ) d z ] 2 P n s P 0 or [ g ( z ) h ( z ) d z ] 2 P n a P 0 .
E y = [ 2 ω μ P 0 k w ( z ) ] 1 / 2 f ( x , z ) exp [ j 0 z k w ( z ) d z ] .
k ¯ w = 1 m Λ 0 m Λ k w ( z ) d z
exp [ j 0 z k w ( z ) d z ] = exp { j k ¯ w z [ n = n = A n exp ( jnKz ) ] }
A n = 1 Λ 0 Λ [ exp ( j { 0 z [ k w ( z ) k ¯ w ] d z + nKz } ) ] d z .
exp [ 0 z α ( z ) d z ]
E y = ( 2 ω μ 0 P 0 k w j α ) 1 / 2 f ( x , z ) exp { 0 z [ j k w ( z ) + α ( z ) ] d z } .
[ 2 x 2 + 2 z 2 + k 2 ( x , z ) ] · ( E y + E y ) = 0 ,
[ 2 x 2 + 2 z 2 + k 2 ( x , z ) ] E y = [ 2 x 2 + 2 z 2 + k 2 ( x , z ) ] · E y = Q ,
Q ( 2 ω μ P 0 k w ) 1 / 2 [ 2 f z 2 2 j k w f z ] exp ( j 0 z k w d z ) p ( z ) ,
p ( z ) = exp ( 0 z α d z ) .
E y = E n exp [ j ( k ¯ w n K ¯ ) z ] · p ( z ) , Q = q n exp [ j ( k ¯ w n K ) z ] · p ( z ) .
[ 2 x 2 + k 2 ( x , z ) ( k ¯ w n K ) 2 ] E n = q n .
q n = 1 Λ 0 Λ ( 2 ω μ P 0 k w ) 1 / 2 · ( 2 f z 2 2 j k w f k w ) · exp [ j 0 z ( k w k ¯ w + n K ) d z ] · d z .
[ 2 x 2 + k 2 ( x , z ) ( k ¯ w n K ) 2 ] · G ( x , x ) = δ ( x x ) .
E n ( x ) = q n ( x ) G ( x , x ) d x = 1 Λ G ( x , x ) × ( 0 Λ ( 2 ω μ P 0 k ¯ w ) 1 / 2 [ 2 f ( x , z ) z 2 2 j k w f ( x , z ) d z ] · { exp [ j 0 z ( k w k ¯ w ) d z jnK z ] } d z ) d x .
E n s = { 2 ω μ 0 P n s [ k s 2 ( k ¯ w n K ) 2 ] 1 / 2 } 1 / 2 · ( 2 α ) 1 / 2 · exp ( α z ) · exp [ j ( k w n K ) z ] .
α P n s P 0 = { [ k s 2 ( k ¯ w n K ) 2 ] 1 / 2 2 k ¯ w } ( 1 Λ G < ( x , x ) { 0 Λ ( 2 f z 2 2 j k w f z ) · exp [ j 0 z ( k w k ¯ w ) d z jnK z ] d z } d x ) 2 .
P 0 = n ( P n s + P n a )
exp [ 0 z α ( z ) d z ]
[ g ( z ) h ( z ) d z ] 2
G M ( x , x ) = 1 Λ 0 Λ G ( x , x ) d z .
E n = ( 2 ω μ P 0 k w ) 1 / 2 m A m [ 2 k ¯ w K ( n m ) K 2 ( n m ) 2 ] · 1 Λ 0 Λ G M ( x , x ) f ( x , z ) exp [ j ( n m ) K z ] d z d x .
E n = ( 2 ω μ P 0 k ¯ w ) 1 / 2 m A m 2 k ¯ w ( n m ) K 2 ( n m ) 2 2 n K k ¯ w n 2 K 2 · 1 Λ 0 Λ { f ( x , z ) [ G g ( x , x ) + G w ( x , x ) 2 k 2 ( x , z ) G g ( x , x ) 2 k g 2 ( x ) G w ( x , x ) 2 k w 2 ( x ) ] · exp [ j ( n m ) K z ] } d z d x .
I = f ( x , z ) [ G w + G g 2 k 2 ( x , z ) G w 2 k w 2 ( x ) G g 2 k g 2 ( x ) ] ,
I d x = t 2 t 2 + t g ( z ) f ( x , z ) G w 2 ( k 3 2 k 4 2 ) d x + t 2 + t g ( z ) t 2 + t 3 f ( x , z ) G g 2 ( k 4 2 k 3 2 ) d x .
I d x = ¼ f q t g ( k 3 2 k 4 2 ) [ ( 1 + cos K z ) G g q ( 1 cos K z ) G w q ] ,
0 Λ d z I d x = t g 8 ( k 3 2 k 4 2 ) f q ( G w q + G g q ) , for n m = ± 1 . = 0 , n m ± 1
A n = j n J n ( Δ β / 2 K ) ,
α P n s = { [ k 2 2 ( k ¯ w n K ) 2 ] 1 / 2 2 k ¯ w } · [ t g 8 ( k 3 2 k 4 2 ) f q ( G w q < + G g q < ) ] 2 , n = 1 0 n 1 .
I n / I = [ J n ( ϕ / 2 ) ] 2 { n [ J n ( ϕ / 2 ) ] 2 } ; ϕ = 2 π λ 0 ( n g 1 ) d ,
[ d d x 2 + k 2 ( x , z ) ( k w n K ) 2 ] E y = 0 .
E y = 1 j [ a i · exp ( j u i x ) + a i + · exp ( j u i x ) ] in region i ,
[ D 1 , N + 1 ] [ A ] = 0 ,
G | x + G | x = 0 ; G x | x + G x | x = 1 .
G i = ( 1 / j ) [ b i · exp ( j u i x ) + b i + · exp ( j u i x ) ]
b p ± = L p ± / D 1 , N + 1 * for x < x , b p ± = M p ± / D 1 , N + 1 * for x > x ,
D 1 , N + 1 * = 2 u q D 1 , N .
b p = M p / D 1 , N + 1 * = [ i = p 1 q + 1 ( 2 u i ) ] [ D 1 , q exp ( j u q x ) exp ( j u q x ) D 1 , q ] / D 1 , p , p > q = 1 2 u p [ ( D 1 , p exp ( j u p x ) / D 1 , p ) exp ( j u p x ) ] , p = q ,
G > ( x , x ) = ( b 4 / j ) exp ( j u 4 x ) .
b 1 + = L 1 + / D 1 , N + 1 * = [ i = 2 q 1 ( 2 u i ) ] [ exp ( j u q x ) D q , N + 1 ʹ exp ( j u q x ) D q , N + 1 ] / D 1 , N + 1 , q < N + 1 = [ i = 2 q 1 ( 2 u i ) ] [ exp ( j u q x ) ] / D 1 , q q = N + 1
G < ( x , x ) = ( b 1 + / j ) exp ( j u 1 x ) .
D i , j = D i , j ( u j ) and D i , j ʹ = D i , j ( u i ) .

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