Abstract

An approximate technique based on a sum rule is used to treat mode conversion at corner bends in dielectric waveguides. Matrix elements which describe the mode coupling are expressed as spatial integrals over electromagnetic field distributions for the guided modes. These matrix elements provide information on the magnitude, average propagation constant, and coherence of power propagating in radiation modes. Numerical results are obtained for single mode and multimode slab waveguides, and implications for the design of low-loss interconnections and mode converters for integrated optics are discussed.

© 1977 Optical Society of America

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References

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  1. See, for example, D. Marcuse, J. Opt. Soc. Am. 66, 216 (1976); D. Marcuse, J. Opt. Soc. Am. 66, 311 (1976); E. F. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975); A. W. Snyder, I. White, D. J. Mitchell, Electron. Lett. 11, 332 (1975); L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974); A. Gedeon, B. Carnstam, Opt. Quant. Electron 7, 456 (1975); D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971); E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).
    [CrossRef]
  2. W. B. Gardner, Bell Syst. Tech. J. 54, 457 (1975).
  3. T. Tamir, Ed., Integrated Optics (Springer Verlag, Berlin, 1975); H. Kogelnik, IEEE Trans. Microwave Theory Tech. MTT-23, 2 (1975); H. F. Taylor, A. Yariv, Proc. IEEE 62, 1044 (1974); S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).
    [CrossRef]
  4. J. S. Cook, W. L. Mammel, R. J. Grow, Bell Syst. Tech. J. 52, 1439 (1973).
  5. E. G. Neumann, H. D. Rudolph, Appl. Phys. 8, 107 (1975).
    [CrossRef]
  6. P. L. Chu, Electron. Lett. 10, 459 (1974).
    [CrossRef]
  7. H. F. Taylor, Appl. Opt. 13, 642 (1974).
    [CrossRef] [PubMed]
  8. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  9. A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-18, 383 (1970).
    [CrossRef]
  10. R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973), p. 78; H. A. Bethe, Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Academic, New York, 1957), p. 260; M. Mizushima, Quantum Mechanics of Atomic Spectra and Atomic Structure (Benjamin, New York, 1970), p. 89.
  11. D. Marcuse, Light Transmission Optics, (Van Nostrand Reinhold, New York, 1972), Chap. 8.

1976 (1)

1975 (2)

W. B. Gardner, Bell Syst. Tech. J. 54, 457 (1975).

E. G. Neumann, H. D. Rudolph, Appl. Phys. 8, 107 (1975).
[CrossRef]

1974 (2)

1973 (1)

J. S. Cook, W. L. Mammel, R. J. Grow, Bell Syst. Tech. J. 52, 1439 (1973).

1970 (1)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-18, 383 (1970).
[CrossRef]

Chu, P. L.

P. L. Chu, Electron. Lett. 10, 459 (1974).
[CrossRef]

Cook, J. S.

J. S. Cook, W. L. Mammel, R. J. Grow, Bell Syst. Tech. J. 52, 1439 (1973).

Gardner, W. B.

W. B. Gardner, Bell Syst. Tech. J. 54, 457 (1975).

Grow, R. J.

J. S. Cook, W. L. Mammel, R. J. Grow, Bell Syst. Tech. J. 52, 1439 (1973).

Loudon, R.

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973), p. 78; H. A. Bethe, Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Academic, New York, 1957), p. 260; M. Mizushima, Quantum Mechanics of Atomic Spectra and Atomic Structure (Benjamin, New York, 1970), p. 89.

Mammel, W. L.

J. S. Cook, W. L. Mammel, R. J. Grow, Bell Syst. Tech. J. 52, 1439 (1973).

Marcuse, D.

Neumann, E. G.

E. G. Neumann, H. D. Rudolph, Appl. Phys. 8, 107 (1975).
[CrossRef]

Rudolph, H. D.

E. G. Neumann, H. D. Rudolph, Appl. Phys. 8, 107 (1975).
[CrossRef]

Snyder, A. W.

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-18, 383 (1970).
[CrossRef]

Taylor, H. F.

Appl. Opt. (1)

Appl. Phys. (1)

E. G. Neumann, H. D. Rudolph, Appl. Phys. 8, 107 (1975).
[CrossRef]

Bell Syst. Tech. J. (2)

W. B. Gardner, Bell Syst. Tech. J. 54, 457 (1975).

J. S. Cook, W. L. Mammel, R. J. Grow, Bell Syst. Tech. J. 52, 1439 (1973).

Electron. Lett. (1)

P. L. Chu, Electron. Lett. 10, 459 (1974).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MTT-18, 383 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (4)

T. Tamir, Ed., Integrated Optics (Springer Verlag, Berlin, 1975); H. Kogelnik, IEEE Trans. Microwave Theory Tech. MTT-23, 2 (1975); H. F. Taylor, A. Yariv, Proc. IEEE 62, 1044 (1974); S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).
[CrossRef]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973), p. 78; H. A. Bethe, Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Academic, New York, 1957), p. 260; M. Mizushima, Quantum Mechanics of Atomic Spectra and Atomic Structure (Benjamin, New York, 1970), p. 89.

D. Marcuse, Light Transmission Optics, (Van Nostrand Reinhold, New York, 1972), Chap. 8.

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Figures (6)

Fig. 1
Fig. 1

Local coordinate axes for a corner bend in a dielectric waveguide. A rotation of the axes takes place at an interface plane defined by the bend.

Fig. 2
Fig. 2

Dependence of power transferred from the fundamental mode at a bend on the guiding strength h.

Fig. 3
Fig. 3

Dependence of propagation constants for the first two guided modes and of the mean propagation constant for power coupled into radiation modes at a corner bend β ¯ r on the guiding strength h.

Fig. 4
Fig. 4

Dependence of the rms deviation from the mean of power coupled into radiation modes at a corner bend on the guiding strength h.

Fig. 5
Fig. 5

Coupling lengths for transfer of power from |0〉 to |1〉 and from |0〉 to radiation modes, L01 and L0r, and the radiation mode coherence length lc plotted as a function of guiding strength h.

Fig. 6
Fig. 6

Power loss at a succession of two identical corner bends, each of angle θ/2, plotted as a function of the separation between the bends L for the case that h = 1. The solid curve was plotted by calculating matrix elements involving guided- and radiation-mode fields; the dashed curve was plotted from Eq. (29). Parameters used for the dashed curve were λσr/2π(ncn0) = 0.19, (n0 − λ β ¯ r /2π)/(ncn0) = 0.10, s = −0.70, b = 0.47λ/(ncn0).

Equations (47)

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U ( x , y , z ) = u ( x , y ) exp ( i ω t - i β z ) ,
Q u = β 2 u .
Q = 2 x 2 + 2 y 2 + n 2 k 0 2 ,
γ F γ = - - u γ * ( F u γ ) d x d y
f ( x , y ) = m a m m + β a β β .
β a β β
ρ β a β β ) d β ,
U i = c 0 0 exp ( i ω t - i β 0 z ) , U t = m a m m exp ( i ω t - i β m z ) + β a β β exp ( i ω t - i β z ) ,
a γ = c 0 γ G 0 ,
G = exp [ i ( β 0 + β γ ) ( tan θ / 2 ) x ] .
G = exp ( i x )
P γ = ( n 0 k 0 / 2 ω μ 0 ) a γ * a γ ,
i S T k = j i S j j T k ,
P t = γ = m , β 0 G * γ γ G 0 .
P g = m 0 G * m m G 0 ,
P r = 1 - P g .
β ¯ r = β a β * a β β / P r , σ r = [ β a β * a β ( β - β ¯ r ) 2 / P r ) ] 1 / 2 .
( Q ) N β = β 2 N β
β 2 N ¯ = γ = m , β a γ * a γ β γ 2 N .
β 2 N ¯ = γ = m , β 0 G * ( Q ) N γ γ G 0 ,
β 2 N ¯ = 0 G * ( Q ) N G 0 .
Q G = G ( Q - 2 + 2 i d d x )
G * Q = ( Q + 2 + 2 i d d x ) G * .
β 2 ¯ = β 0 2 - 2 ,
β 4 ¯ = ( β 0 2 - 2 ) 2 - 4 2 0 | d d x 2 | 0 .
β r 2 N ¯ = ( β 2 N ¯ - m P m β m 2 N ) ( 1 - P g ) - 1 .
β r 2 ¯ = n 0 2 k 0 2 + 2 n 0 k 0 ( Δ β ) r ¯ + ( Δ β ) r 2 ¯ ,
β r 4 ¯ = n 0 4 k 0 4 + 4 n 0 3 k 0 3 ( Δ β ) r ¯ + 6 n 0 2 k 0 2 ( Δ β ) r 2 ¯
β r ¯ = n 0 k 0 + Δ β r ¯ ,
σ r = { [ ( Δ β ) r 2 ¯ ] - [ ( Δ β ) r ¯ ] 2 } 1 / 2 .
Δ P 0 / P 0 = - γ = m , β 0 - i n 0 k 0 θ x γ γ i n 0 k 0 θ x 0 .
Δ P 0 / P 0 = - ( n 0 k 0 θ ) 2 0 x 2 0 .
Δ P 1 / P 0 = ( n 0 k 0 θ ) 2 0 x 1 2 .
Δ P r = - ( Δ P 0 + Δ P 1 ) .
Δ P 0 / P 0 = - 2 ( n 0 k 0 θ ) 2 a 0 π / 2 α x 2 cos 2 α x d x , Δ P 1 / P 0 = 2 ( n 0 k 0 θ ) 2 a 2 ( 0 π / 2 α x cos α x sin 2 α x d x ) 2 .
Δ P q / P 0 = 2 K q ( 2 n 0 k 0 θ a / π ) 2 ,
Δ P q / P 0 = K q ( n 0 n c - n 0 ) ( h θ ) 2 .
L 01 = π / ( β 0 - β 1 ) , L 0 r = π / ( β 0 - β ¯ r ) .
a 0 = c 0 ( 1 - α 2 / 2 ) , a 1 = i α c 0 ,
a 0 = c 0 ( 1 - α 2 / 2 ) exp ( - i β 0 L 01 ) , a 1 = i α c 0 exp ( - i β 1 L 01 ) = - i α c 0 exp ( - i β 0 L 01 ) .
a 1 = a 1 + i α a 0 = - i α 3 c 0 exp ( - i β 0 L 01 ) / 2.
a 1 2 a 1 2 .
Δ P 1 ( 2 ) = Δ P 1 ( 1 ) 1 + exp [ i ( β 0 - β 1 ) L ] 2 / 4 = Δ P 1 ( 1 ) [ 1 + cos ( β 0 - β 1 ) L ] / 2.
1 Δ P r ( 1 ) d P r ( 1 ) d β = b s + 1 s ! ( - Δ β ) s exp ( b Δ β ) ,             Δ β < 0 = 0.             Δ β > 0 ,
b = ( 2 π n 0 / λ - β ¯ r ) / σ r 2 , s = ( 2 π n 0 / λ - β ¯ r ) 2 / σ r 2 - 1.
Δ P r ( 2 ) = d P r ( 1 ) d β [ 1 + cos ( β 0 - β ) L ] d β / 2 .
Δ P r ( 2 ) = Δ P r ( 1 ) { 1 + [ b / ( b 2 + L 2 ) 1 / 2 ] s + 1 × cos [ ( β 0 - n 0 k 0 ) L + ( s + 1 ) ϕ ] } / 2

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