Abstract

The theory of periodic couplers is examined from the point of view of the reciprocity theorem and a modified Born scattering approach. Design equations for several coupler configurations are presented. Efficiency and aperture size calculations are compared with previously published data for grating and Bragg couplers.

© 1972 Optical Society of America

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References

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  1. J. H. Harris, R. Shubert, in Conference Abstracts, International Scientific Radio Union, Spring Meeting, Washington, D.C. (April1969), p. 71.
  2. P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
    [CrossRef]
  3. M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, Appl. Phys. Lett. 16, 523 (1970).
    [CrossRef]
  4. M. Kogelnik, T. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).
  5. J. H. Harris, R. Shubert, J. N. Polky, J. Opt. Soc. Am. 60, 1007 (1970).
    [CrossRef]
  6. P. K. Tien, R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
    [CrossRef]
  7. R. Ulrich, J. Opt. Soc. Am. 60, 1337 (1970).
    [CrossRef]
  8. J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 583 (1970).
    [CrossRef]
  9. J. H. Harris, R. Shubert, IEEE Trans. MTT 19, 269 (1971).
    [CrossRef]
  10. R. Ulrich, J. Opt. Soc. Am. 61, 1467 (1971).
    [CrossRef]
  11. T. Tamir, H. L. Bertoni, J. Opt. Soc. Am. 61, 1397 (1971).
    [CrossRef]
  12. D. S. Jones, The Theory of Electromagnetism (Pergamon, New York, 1964), pp. 63–65.
  13. R. Shubert, J. H. Harris, J. Opt. Soc. Am. 61, 154 (1971).
    [CrossRef]
  14. T. A. Shankoff, Appl. Opt. 7, 2101 (1968).
    [CrossRef] [PubMed]
  15. D. B. Ostrowsky, A. Jacques, Appl. Phys. Lett. 18, 556 (1971).
    [CrossRef]
  16. W. S. C. Chang, IEEE J. Quantum Electron. QE7, 167 (1971).
    [CrossRef]
  17. D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).
  18. Neglect of reflected guided waves in the broadside coupler (κ = κw, θ = 90°) is unwarranted and this special case must be excluded from the analysis of this paper. The reciprocity theorem provides anomalous results when the reflected wave is neglected.

1971

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

D. B. Ostrowsky, A. Jacques, Appl. Phys. Lett. 18, 556 (1971).
[CrossRef]

W. S. C. Chang, IEEE J. Quantum Electron. QE7, 167 (1971).
[CrossRef]

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

T. Tamir, H. L. Bertoni, J. Opt. Soc. Am. 61, 1397 (1971).
[CrossRef]

R. Ulrich, J. Opt. Soc. Am. 61, 1467 (1971).
[CrossRef]

1970

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

M. Kogelnik, T. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 583 (1970).
[CrossRef]

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

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

R. Ulrich, J. Opt. Soc. Am. 60, 1337 (1970).
[CrossRef]

1969

D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

1968

Bertoni, H. L.

Chang, W. S. C.

W. S. C. Chang, IEEE J. Quantum Electron. QE7, 167 (1971).
[CrossRef]

Dakss, M. L.

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

Harris, J. H.

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

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

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

J. H. Harris, R. Shubert, in Conference Abstracts, International Scientific Radio Union, Spring Meeting, Washington, D.C. (April1969), p. 71.

Heidrich, P. F.

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

Jacques, A.

D. B. Ostrowsky, A. Jacques, Appl. Phys. Lett. 18, 556 (1971).
[CrossRef]

Jones, D. S.

D. S. Jones, The Theory of Electromagnetism (Pergamon, New York, 1964), pp. 63–65.

Kogelnik, M.

M. Kogelnik, T. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

Kuhn, L.

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

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).

Martin, R. J.

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

Midwinter, J. E.

J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 583 (1970).
[CrossRef]

Ostrowsky, D. B.

D. B. Ostrowsky, A. Jacques, Appl. Phys. Lett. 18, 556 (1971).
[CrossRef]

Polky, J. N.

Scott, B. A.

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

Shankoff, T. A.

Shubert, R.

J. H. Harris, R. Shubert, IEEE Trans. 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]

J. H. Harris, R. Shubert, in Conference Abstracts, International Scientific Radio Union, Spring Meeting, Washington, D.C. (April1969), p. 71.

Sosnowski, T.

M. Kogelnik, T. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

Tamir, T.

Tien, P. K.

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

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

Ulrich, R.

Appl. Opt.

Appl. Phys. Lett.

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

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

D. B. Ostrowsky, A. Jacques, Appl. Phys. Lett. 18, 556 (1971).
[CrossRef]

Bell Syst. Tech. J.

D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).

M. Kogelnik, T. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

IEEE J. Quantum Electron.

J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 583 (1970).
[CrossRef]

W. S. C. Chang, IEEE J. Quantum Electron. QE7, 167 (1971).
[CrossRef]

IEEE Trans. MTT

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

J. Opt. Soc. Am.

Other

J. H. Harris, R. Shubert, in Conference Abstracts, International Scientific Radio Union, Spring Meeting, Washington, D.C. (April1969), p. 71.

D. S. Jones, The Theory of Electromagnetism (Pergamon, New York, 1964), pp. 63–65.

Neglect of reflected guided waves in the broadside coupler (κ = κw, θ = 90°) is unwarranted and this special case must be excluded from the analysis of this paper. The reciprocity theorem provides anomalous results when the reflected wave is neglected.

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Figures (5)

Fig. 1
Fig. 1

Radiation of beams from a periodic structure serving as an output coupler. P0, power incident on coupler from the waveguide; Pna and Pns, radiated collimated beams in air and substrate, respectively; Ps, nondirective scattered fields; Pr, and Pi, reflected and transmitted power not coupled out of the waveguide; Pi, input beam for coupling; Pc, power coupled into the waveguide.

Fig. 2
Fig. 2

Geometric representation of a thick sinusoidal grating of thickness tg(x) = t3(1 − cosκx)/2. x1, x2, x3, x4 are points at which the electric field is reduced from its value at the surface by the factor 1/e, i.e., tg(x1) = tg(x4) = 1/|u3w|, tg(x2) = tg(x4) = 1/|u4w|. The effective thickness of the guide (Appendix B) is t′ = t + (1/|u3w| + 1/|u4w|).

Fig. 3
Fig. 3

Angular location of beams radiated by a guided wave of average propagation constant kw in the presence of a periodic structure of period Λ(κ = 2π/Λ). The angular location of the beams is described by angle θns,a, where n is the order of the beam and s,a refers to either the substrate or air. Note that cosθns,a = (kw)/ks,a.

Fig. 4
Fig. 4

Configurations for couplers: I. Thick sinusoidal periodic grating of low index3 with grating thickness tg(x) = a(1 − cosκx), tg(x1) = tg(x4) = 1/|u3|, tg(x2) = tg(x3) = 1/|u4|. II. Thick sinusoidal periodic grating of low index with reflector on the grating. III. Bragg coupler4 with refractive index n3(x,z) = n ¯ 3 + Δncos(κxx + κxz). IV. Thin square wave grating15 of length Λ/2. V. Thin square wave grating with reflector in the substrate.

Fig. 5
Fig. 5

Approximate beamwidth (√2/α) for maximum coupling efficiency vs refractive index (ns) and thickness (t3), of a sinusoidal grating (Fig. 4I). The beamwidth and grating thickness are expressed in microns. n1 = 1.52 μ, n2 = 1.73 μ, n4 = 1.0 μ, nw = 1.695 μ, and tfilm = 0.76 μ.

Tables (1)

Tables Icon

Table I Equations for Determining Aperture Size of Periodic Couplers Illustrated in Fig. 4a

Equations (44)

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P 0 = ( P s + P t + P r ) + Σ ( P n s + P n a ) .
E v = ( 2 ω μ P 0 / k w ) 1 2 f ( z ) exp i k w x             H z = ( k w / ω μ ) E y ,
E y n s = ( 2 ω μ P n s / k s sin θ ) 1 2 g ( x ) exp i ( k s cos θ ) x , H x n s = ( k s sin θ / ω μ ) E y n s ,
E y i = ( 2 ω μ P i / k s sin θ ) 1 2 h ( x ) exp - i ( k s cos θ ) x , H x i = - ( k s sin θ / ω μ ) E y i ,
E y c = ( 2 ω μ P c / k w ) 1 2 f ( z ) exp ( - i k w x )             H z c = - ( k w / ω μ ) E y c .
( E × H r - E r × H ) · d S = 0 ,
P c / P i = [ - g ( x ) h ( x ) d x ] 2 P n s / P 0 .
E y = ( 2 ω μ P 0 / k w ) 1 2 f ( z ) exp i 0 x k w ( x ) d x .
k ¯ w = ( 1 / m Λ ) 0 m Λ k w ( x ) d x ,
exp i 0 x k w ( x ) d x = ( exp i k ¯ w x ) ( - A n exp - i n κ x ) ,
A n = ( 1 / Λ ) m Λ ( m + 1 ) / Λ exp i ( 0 x ( k w - k ¯ w ) d x + n κ x ) d x .
k ¯ w - n κ < k s ,
k ¯ w - n κ < k 0 ,
E y = ( P 0 2 ω μ k w + i α ) 1 2 f ( z ) exp [ 0 x ( i k w - α ) d x ] .
E y n s z 1 = { P n s 2 ω μ / [ k s 2 - ( k ¯ w - n κ ) 2 ] 1 2 } 1 2 × ( 2 α ) 1 2 exp [ - 0 x α d x + i ( k ¯ w - n κ ) x ] .
g ( x ) = ( 2 α ) 1 2 exp [ - 0 x α d x ] .
α = n ( α P n s / p 0 + α P n a / P 0 ) .
( G ) top guide = 1 / i ( u 2 + u 3 ) .
( E i ) = u 2 ω [ ( k 2 - k 3 2 ) t / 2 ] - 1 2 ( 2 ω μ P 0 / k ¯ w ) 1 2 .
( k 0 2 / 4 ) ( n max 2 - n 3 2 ) ( E i G d ) ~ [ ( 2 α ) ( 2 ω μ P n s ) / u 1 ] 1 2 .
α ~ 2.46 ( n 1 / n w ) [ ( n max 2 - n 3 2 ) / ( n 2 2 - n 3 2 ) ] 2 × ( n 2 2 - n w 2 ) ( d 2 / t λ 2 ) sin θ ,
A n = i n J n ( Δ n w Λ / 2 λ ) ,
A 0 ~ 1             A n ~ ( Δ n w Λ / 4 λ ) / n !             n 0
[ 2 + k 2 ( z , x ) ] E y = - [ 2 + k 2 ( z , x ) ] E y = - Q ,
E y = ( 2 ω μ P 0 ) 1 2 [ ( k w + i α ) - 1 2 f ( z ) ] × [ exp i 0 x ( k w + i α ) d x ] = ( 2 ω μ P 0 ) 1 2 [ Ψ ] { Φ } ,
Q = ( 2 ω μ P 0 ) 1 2 [ Ψ x x + 2 i ( k w + i α ) 1 2 f x + ( α 2 - 2 i α k w ) Ψ ] Φ .
Q ~ [ ( 2 ω μ P 0 / k w ) 1 2 ( f x x + 2 i k w f x ) exp i 0 x k w d x ] p ( x ) ,
p ( x ) = exp - 0 x α d x .
Q ~ [ Σ q n exp i ( k ¯ w - n κ ) x ] p ( x ) .
E y = [ Σ E n exp i ( k ¯ w - n κ ) x ] p ( x )
[ ( d 2 / d z 2 ) + k 2 ( z , x ) - ( k ¯ w - n κ ) 2 ] E n ~ q n .
[ ( d 2 / d z 2 ) + k 2 ( z , x ) - ( k ¯ w - n κ ) 2 ] G = δ ( z - z ) ,
E n = - q n G d z = 1 Λ 0 Λ d x - d z G ( 2 ω μ P 0 / k ¯ w ) 1 2 ( f x x + 2 i k ¯ w f x ) × exp [ i 0 x ( k w - k ¯ w ) d x + i n K x ] .
E n ~ ( 2 ω μ P 0 / k ¯ w ) 1 2 n A m [ 2 k ¯ w κ ( n - m ) - κ 2 ( n - m ) 2 ] × 1 Λ 0 Λ - G B f exp [ i ( n - m ) κ x ] d x d z .
[ ( d 2 / d z 2 ) + k 2 ( z , x ) - k w 2 ] f = 0
E n s , a ~ ( 2 ω μ P 0 / k ¯ w ) 1 2 n A m [ 2 k ¯ w κ ( n - m ) - κ 2 ( n - m ) 2 ] ( 2 κ w κ n - κ 2 n 2 ) × 1 Λ 0 Λ G B s , a f [ k 2 ( z , x ) - k ¯ 2 ( z ) ] exp [ i ( n - m ) κ x ] d x d z ,
( G B { [ ( d 2 / d z 2 ) + k 2 ( z , x ) - k ¯ w 2 ] f } - f { [ ( d 2 / d z 2 ) + k ¯ 2 ( z ) - ( k ¯ w - n κ ) 2 ] G B } ) d z = 0.
G M = 1 2 ( G g + G a ) ,
E n s , a ~ ( 2 ω μ P 0 / k ¯ w ) 1 2 1 Λ m A m [ 2 k ¯ w κ ( n - m ) - κ 2 ( n - m ) 2 ] × G M s , a f exp i ( n - m ) κ x d x d z .
P n s , a / P 0 = ( u 1 , 4 / 2 α k ¯ w ) [ E n / ( 2 ω μ P 0 / k ¯ w ) 1 2 ] 2 ,
f = F / ( - F 2 d z ) 1 2 ,
F = { [ u 2 w / ( k 2 2 - k 3 2 ) 1 2 ] exp i u 3 w ( z - t ) z > t , cos ( μ 2 w z - tan - 1 u 1 w / i u 2 w ) 0 < z < t , [ u 2 w / ( k 2 2 - k 1 2 ) 1 2 ] exp - i u 1 w z z < 0.
- F 2 d z = 1 2 [ t + i ( 1 / u 1 w + 1 / u 3 w ) ] t / 2.
F = { F a 0 < x < x 1 , x 4 < x < Λ , ( F σ - F a ) ( x - x 1 , 4 / x 2 , 3 x 1 , 4 ) + F a x 1 , 3 < x < x 2 , 4 , F σ x 2 < x < x 3 .

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