Abstract

Radiation losses in arbitrarily curved integrated-optic waveguides are computed numerically by means of the beam-propagation method. From the results obtained, design rules for bends and S-shaped curves connecting two straight waveguides are derived.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
    [Crossref]
  2. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972);“Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).
  3. M. Miyagi and S. Nishida, “Bending losses of dielectric rectangular waveguides for integrated optics,” J. Opt. Soc. Am. 68, 316–319 (1978).
    [Crossref]
  4. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66, 311–320 (1976).
    [Crossref]
  5. E. F. Kuester, “An alternative expression for the curvature loss of a dielectric waveguide and its application to the rectangular dielectric channel,” Radio Sci. 12, 573–578 (1977).
    [Crossref]
  6. A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975).
    [Crossref]
  7. L. Lewin, D. Chang, and E. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus, London, 1977).
  8. L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
    [Crossref]
  9. C. Winkler, J. Love, and A. Ghatak, “Loss calculations in bent multimode optical waveguides,” Opt. Quantum Electron. 11, 173–183 (1979).
    [Crossref]
  10. M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
    [Crossref]
  11. J. Van Roey, J. van der Donk, and P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
    [Crossref]
  12. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [Crossref]
  13. J. van der Donk, “The beam propagation method in integrated optics,” Ph.D. Thesis (University of Gent, Gent, Belgium, 1982).
  14. R. Baets and P. E. Lagasse, “Calculation of radiation loss in integrated-optics tapers and Y-junctions,” Appl. Opt. 21, 1972–1978 (1982).
    [Crossref] [PubMed]
  15. W. Gambling, M. Matsumura, and C. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979);K. Petermann, “Fundamental mode microbending loss in graded-index and W fibers,” Opt. Quantum Electron. 9, 167–175 (1977)
    [Crossref]

1982 (1)

1981 (1)

1979 (2)

C. Winkler, J. Love, and A. Ghatak, “Loss calculations in bent multimode optical waveguides,” Opt. Quantum Electron. 11, 173–183 (1979).
[Crossref]

W. Gambling, M. Matsumura, and C. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979);K. Petermann, “Fundamental mode microbending loss in graded-index and W fibers,” Opt. Quantum Electron. 9, 167–175 (1977)
[Crossref]

1978 (1)

1977 (1)

E. F. Kuester, “An alternative expression for the curvature loss of a dielectric waveguide and its application to the rectangular dielectric channel,” Radio Sci. 12, 573–578 (1977).
[Crossref]

1976 (2)

D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66, 311–320 (1976).
[Crossref]

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

1975 (2)

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[Crossref]

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975).
[Crossref]

1974 (1)

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[Crossref]

1969 (1)

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
[Crossref]

Baets, R.

Chang, D.

L. Lewin, D. Chang, and E. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus, London, 1977).

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Gambling, W.

W. Gambling, M. Matsumura, and C. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979);K. Petermann, “Fundamental mode microbending loss in graded-index and W fibers,” Opt. Quantum Electron. 9, 167–175 (1977)
[Crossref]

Ghatak, A.

C. Winkler, J. Love, and A. Ghatak, “Loss calculations in bent multimode optical waveguides,” Opt. Quantum Electron. 11, 173–183 (1979).
[Crossref]

Harris, J.

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[Crossref]

Heiblum, M.

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[Crossref]

Kuester, E.

L. Lewin, D. Chang, and E. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus, London, 1977).

Kuester, E. F.

E. F. Kuester, “An alternative expression for the curvature loss of a dielectric waveguide and its application to the rectangular dielectric channel,” Radio Sci. 12, 573–578 (1977).
[Crossref]

Lagasse, P. E.

Lewin, L.

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[Crossref]

L. Lewin, D. Chang, and E. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus, London, 1977).

Love, J.

C. Winkler, J. Love, and A. Ghatak, “Loss calculations in bent multimode optical waveguides,” Opt. Quantum Electron. 11, 173–183 (1979).
[Crossref]

Marcatili, E. A. J.

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
[Crossref]

Marcuse, D.

D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66, 311–320 (1976).
[Crossref]

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972);“Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).

Matsumura, M.

W. Gambling, M. Matsumura, and C. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979);K. Petermann, “Fundamental mode microbending loss in graded-index and W fibers,” Opt. Quantum Electron. 9, 167–175 (1977)
[Crossref]

Mitchell, D. J.

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975).
[Crossref]

Miyagi, M.

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Nishida, S.

Ragdale, C.

W. Gambling, M. Matsumura, and C. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979);K. Petermann, “Fundamental mode microbending loss in graded-index and W fibers,” Opt. Quantum Electron. 9, 167–175 (1977)
[Crossref]

Snyder, A. W.

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975).
[Crossref]

van der Donk, J.

J. Van Roey, J. van der Donk, and P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
[Crossref]

J. van der Donk, “The beam propagation method in integrated optics,” Ph.D. Thesis (University of Gent, Gent, Belgium, 1982).

Van Roey, J.

White, I.

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975).
[Crossref]

Winkler, C.

C. Winkler, J. Love, and A. Ghatak, “Loss calculations in bent multimode optical waveguides,” Opt. Quantum Electron. 11, 173–183 (1979).
[Crossref]

Appl. Opt. (1)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
[Crossref]

Electron. Lett. (1)

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[Crossref]

J. Opt. Soc. Am. (3)

Opt. Quantum Electron. (2)

W. Gambling, M. Matsumura, and C. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979);K. Petermann, “Fundamental mode microbending loss in graded-index and W fibers,” Opt. Quantum Electron. 9, 167–175 (1977)
[Crossref]

C. Winkler, J. Love, and A. Ghatak, “Loss calculations in bent multimode optical waveguides,” Opt. Quantum Electron. 11, 173–183 (1979).
[Crossref]

Radio Sci. (1)

E. F. Kuester, “An alternative expression for the curvature loss of a dielectric waveguide and its application to the rectangular dielectric channel,” Radio Sci. 12, 573–578 (1977).
[Crossref]

Other (3)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972);“Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).

L. Lewin, D. Chang, and E. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus, London, 1977).

J. van der Donk, “The beam propagation method in integrated optics,” Ph.D. Thesis (University of Gent, Gent, Belgium, 1982).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Curved waveguide: (a) three-dimensional configuration, (b) equivalent two-dimensional slab.

Fig. 2
Fig. 2

Insertion loss and amplitude of fundamental bend mode along a uniformly curved guide.

Fig. 3
Fig. 3

Pure bending loss of uniformly curved step-index slab as a function of V and R.

Fig. 4
Fig. 4

Transition loss of uniformly curved step-index slab as a function of V and R.

Fig. 5
Fig. 5

Arbitrarily curved waveguide.

Fig. 6
Fig. 6

S-curved guide. L = 10 mm, W = 7.5 mm. (a) 1/R as a function of arc length for four different geometries defined by Eqs. (19)(22), (b) insertion loss as a function of s.

Fig. 7
Fig. 7

Insertion loss of S-curved guides as a function of W for the four geometries defined by Eqs. (19)(22). (a) L = 1.5 mm, (b) L = 10 mm.

Fig. 8
Fig. 8

Waveguide with a change of direction over an angle 2θf.

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

w = u + j υ = R ln p + R R ,
n ( u , υ ) = e u / R n [ x ( u , υ ) , z ( u , υ ) ] .
u = R ln ( 1 + x R ) x x 2 2 R x
n ( u , 0 ) = n ( x , 0 ) ( 1 + x R ) .
x s 2 E y ( x ̂ , s ) + k 0 2 n 2 ( x ̂ , s ) [ 1 + x ̂ R ( s ) ] 2 E y ( x ̂ , s ) = 0 .
2 ψ i r 2 + 1 r ψ i r + 1 r 2 2 ψ i θ 2 + k 0 2 n 2 ( r ) ψ i = 0 .
lim r r ( ψ i r + j k n ψ i ) = 0 .
2 ϕ j r 2 + 1 r ϕ j r + 1 r 2 + 2 ϕ j θ 2 + k 0 2 n * 2 ( r ) ϕ j = 0
lim r r ( ϕ j r j k n * ϕ j ) = 0 .
0 ψ i ϕ j * r d r = K i δ i j ,
0 G i G j r d r = K i δ i j , | K i | = 1 .
ψ ( r , θ ) = a 0 ψ 0 ( r , θ ) + other terms ,
a 0 = 1 K 0 0 ψ ( r , θ ) r ϕ 0 * ( r , θ ) d r
| a 0 | exp ( α 0 R θ ) = | 0 ψ ( r , θ ) G 0 ( r ) r d r | .
R = R λ s 2 d 3 ,
log α 0 R d / λ s log C n log R
α 0 R d / λ s C R n .
bend α 0 ( R ) d s ,
z ( θ ) = 0 θ R ( θ ) cos θ d θ , x ( θ ) = 0 θ R ( θ ) sin θ d θ .
x ( z ) = W 2 ( cos + π z L 1 ) ,
x ( z ) = W 2 π sin 2 π L z W L z ,
± L 2 4 W ( 1 + W 2 L 2 ) ,
the suboptimal curve R 0 ( θ ) mentioned above .
0 θ f R ( θ ) cos θ d θ = L 2
0 θ f R ( θ ) sin θ d θ = W 2 .
R 0 ( θ ) = k [ sin ( θ f 0 θ ) ] 1 / n + 1 ,
R 0 ( θ ) = k cos ( θ f θ ) 1 / n + 1 , θ [ 0 , θ f ] ,
k = L 0 θ f cos ( θ ) n / n + 1 d θ .
R 0 ( θ ) = R 0 ( θ ) + δ R ( θ ) , 0 θ θ f 0 = R 0 ( θ f 0 ) + δ R ( θ ) , θ θ θ f 0 + δ θ f 0 , α 0 ( R ) = α 0 ( R 0 ) + d α 0 d R 0 δ R .
L ( R ) = L ( R 0 ) + 0 θ f 0 [ α 0 ( R 0 ) + d α 0 d R 0 R 0 ] δ R d θ + α 0 [ R 0 ( θ f 0 ) ] R 0 ( θ f 0 ) δ θ f 0 + higher-order terms .
α 0 [ R 0 ( θ f 0 ) ] R 0 ( θ f 0 ) = 0 ,
0 θ f 0 [ α 0 ( R 0 ) + d α 0 d R 0 R 0 ] δ R d θ = 0
0 θ f 0 δ R sin θ d θ = 0 , 0 θ f 0 δ R cos θ d θ = 0 .
α 0 ( R 0 ) + d α 0 d R 0 R 0 = a sin θ + b cos θ .
R 0 ( θ ) = k [ sin ( θ f 0 θ ) ] 1 / n + 1 ,
0 θ f o sin θ [ sin ( θ f 0 θ ) ] 1 / n + 1 d θ 0 θ f o sin θ [ sin ( θ f 0 θ ) ] 1 / n + 1 d θ = W L ,
k = 1 2 n n + 1 L cos θ f 0 + W sin θ f 0 ( sin θ f 0 ) n / n + 1 .