Abstract

A new bending-loss formula for dielectric waveguides with rectangular cross section is obtained. Some important differences are shown, compared with the existing formula obtained from a slab-waveguide model.

© 1978 Optical Society of America

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References

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  1. E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
    [Crossref]
  2. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
    [Crossref]
  3. S. Kawakami, M. Miyagi, and S. Nishida, “Bending losses of a dielectric slab optical waveguide with double or multiple claddings: theory,” Appl. Opt. 14, 2588–2597 (1975).
    [Crossref] [PubMed]
  4. M. Miyagi and G. L. Yip, “Field deformation and polarization change in a step-index optical fibre due to bending,” Opt. Quant. Electron. 8, 335–341 (1976).
    [Crossref]
  5. Y. Takuma, S. Kawakami, and S. Nishida, “Radiation and propagation along a uniformly curved slab waveguide,” Trans. of Inst. Electron. Commun. Eng. Japan. 60-C, 706–713 (1977).
  6. S. Kawakami, M. Miyagi, and Y. Suematsu, “Fundamentals of optical-fiber communication [V · finish],” J. of Inst. Electron. Eng. Commun. Eng. Japan 60, 1047–1056 (1977).
  7. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).
    [Crossref]
  8. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Product (Academic, New York1965), p. 963.
  9. L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Micro. Theory Tech. MTT-22, 718–727 (1974).
    [Crossref]
  10. E. F. Kuester and D. C. Chang, “Surface-wave radiation from curved dielectric slabs and fibers,” IEEE J. Quant. Electron. QE-11, 903–907 (1975).
    [Crossref]
  11. J. A. Arnaud, “Transverse coupling in fiber optics-Part III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).
    [Crossref]
  12. M. Miyagi and G. L. Yip, “Design theory of high-Q optical ring resonator with asymmetric three-layered dielectrics” (unpublished).

1977 (2)

Y. Takuma, S. Kawakami, and S. Nishida, “Radiation and propagation along a uniformly curved slab waveguide,” Trans. of Inst. Electron. Commun. Eng. Japan. 60-C, 706–713 (1977).

S. Kawakami, M. Miyagi, and Y. Suematsu, “Fundamentals of optical-fiber communication [V · finish],” J. of Inst. Electron. Eng. Commun. Eng. Japan 60, 1047–1056 (1977).

1976 (2)

D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).
[Crossref]

M. Miyagi and G. L. Yip, “Field deformation and polarization change in a step-index optical fibre due to bending,” Opt. Quant. Electron. 8, 335–341 (1976).
[Crossref]

1975 (2)

S. Kawakami, M. Miyagi, and S. Nishida, “Bending losses of a dielectric slab optical waveguide with double or multiple claddings: theory,” Appl. Opt. 14, 2588–2597 (1975).
[Crossref] [PubMed]

E. F. Kuester and D. C. Chang, “Surface-wave radiation from curved dielectric slabs and fibers,” IEEE J. Quant. Electron. QE-11, 903–907 (1975).
[Crossref]

1974 (2)

J. A. Arnaud, “Transverse coupling in fiber optics-Part III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).
[Crossref]

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

1969 (2)

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

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

Arnaud, J. A.

J. A. Arnaud, “Transverse coupling in fiber optics-Part III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).
[Crossref]

Chang, D. C.

E. F. Kuester and D. C. Chang, “Surface-wave radiation from curved dielectric slabs and fibers,” IEEE J. Quant. Electron. QE-11, 903–907 (1975).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Product (Academic, New York1965), p. 963.

Kawakami, S.

S. Kawakami, M. Miyagi, and Y. Suematsu, “Fundamentals of optical-fiber communication [V · finish],” J. of Inst. Electron. Eng. Commun. Eng. Japan 60, 1047–1056 (1977).

Y. Takuma, S. Kawakami, and S. Nishida, “Radiation and propagation along a uniformly curved slab waveguide,” Trans. of Inst. Electron. Commun. Eng. Japan. 60-C, 706–713 (1977).

S. Kawakami, M. Miyagi, and S. Nishida, “Bending losses of a dielectric slab optical waveguide with double or multiple claddings: theory,” Appl. Opt. 14, 2588–2597 (1975).
[Crossref] [PubMed]

Kuester, E. F.

E. F. Kuester and D. C. Chang, “Surface-wave radiation from curved dielectric slabs and fibers,” IEEE J. Quant. Electron. QE-11, 903–907 (1975).
[Crossref]

Lewin, L.

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

Marcatili, E. A. J.

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

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

Marcuse, D.

Miyagi, M.

S. Kawakami, M. Miyagi, and Y. Suematsu, “Fundamentals of optical-fiber communication [V · finish],” J. of Inst. Electron. Eng. Commun. Eng. Japan 60, 1047–1056 (1977).

M. Miyagi and G. L. Yip, “Field deformation and polarization change in a step-index optical fibre due to bending,” Opt. Quant. Electron. 8, 335–341 (1976).
[Crossref]

S. Kawakami, M. Miyagi, and S. Nishida, “Bending losses of a dielectric slab optical waveguide with double or multiple claddings: theory,” Appl. Opt. 14, 2588–2597 (1975).
[Crossref] [PubMed]

M. Miyagi and G. L. Yip, “Design theory of high-Q optical ring resonator with asymmetric three-layered dielectrics” (unpublished).

Nishida, S.

Y. Takuma, S. Kawakami, and S. Nishida, “Radiation and propagation along a uniformly curved slab waveguide,” Trans. of Inst. Electron. Commun. Eng. Japan. 60-C, 706–713 (1977).

S. Kawakami, M. Miyagi, and S. Nishida, “Bending losses of a dielectric slab optical waveguide with double or multiple claddings: theory,” Appl. Opt. 14, 2588–2597 (1975).
[Crossref] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Product (Academic, New York1965), p. 963.

Suematsu, Y.

S. Kawakami, M. Miyagi, and Y. Suematsu, “Fundamentals of optical-fiber communication [V · finish],” J. of Inst. Electron. Eng. Commun. Eng. Japan 60, 1047–1056 (1977).

Takuma, Y.

Y. Takuma, S. Kawakami, and S. Nishida, “Radiation and propagation along a uniformly curved slab waveguide,” Trans. of Inst. Electron. Commun. Eng. Japan. 60-C, 706–713 (1977).

Yip, G. L.

M. Miyagi and G. L. Yip, “Field deformation and polarization change in a step-index optical fibre due to bending,” Opt. Quant. Electron. 8, 335–341 (1976).
[Crossref]

M. Miyagi and G. L. Yip, “Design theory of high-Q optical ring resonator with asymmetric three-layered dielectrics” (unpublished).

Appl. Opt. (1)

Bell Syst. Tech. J. (3)

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

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

J. A. Arnaud, “Transverse coupling in fiber optics-Part III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).
[Crossref]

IEEE J. Quant. Electron. (1)

E. F. Kuester and D. C. Chang, “Surface-wave radiation from curved dielectric slabs and fibers,” IEEE J. Quant. Electron. QE-11, 903–907 (1975).
[Crossref]

IEEE Trans. Micro. Theory Tech. (1)

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

J. of Inst. Electron. Eng. Commun. Eng. Japan (1)

S. Kawakami, M. Miyagi, and Y. Suematsu, “Fundamentals of optical-fiber communication [V · finish],” J. of Inst. Electron. Eng. Commun. Eng. Japan 60, 1047–1056 (1977).

J. Opt. Soc. Am. (1)

Opt. Quant. Electron. (1)

M. Miyagi and G. L. Yip, “Field deformation and polarization change in a step-index optical fibre due to bending,” Opt. Quant. Electron. 8, 335–341 (1976).
[Crossref]

Trans. of Inst. Electron. Commun. Eng. Japan. (1)

Y. Takuma, S. Kawakami, and S. Nishida, “Radiation and propagation along a uniformly curved slab waveguide,” Trans. of Inst. Electron. Commun. Eng. Japan. 60-C, 706–713 (1977).

Other (2)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Product (Academic, New York1965), p. 963.

M. Miyagi and G. L. Yip, “Design theory of high-Q optical ring resonator with asymmetric three-layered dielectrics” (unpublished).

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Figures (1)

FIG. 1
FIG. 1

Geometry of a rectangular waveguide and a coordinate system erected (a), and the refractive-index distribution in the (x,y) plane (b).

Equations (29)

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E = - f ( μ ) H ν ( 2 ) { [ ( a n 0 k 0 ) 2 - μ 2 ] 1 / 2 r } × exp ( - j μ x - j β z ) d μ .
P r = 2 ω μ 0 R - a n 0 k 0 a n 0 k 0 f ( μ ) 2 d μ .
E x = { cos ( u x H x ) cos ( u T y + φ ) ,             x < H , y < T u x v x exp ( - w x H ( x - H ) ) cos ( u T y + φ ) ,             x > H , y < T u v ˆ cos ( u x H x ) exp ( - w 2 T ( y - Y ) ) ,             x < H , y > T u v cos ( u x H x ) exp ( w 1 T ( y + T ) ) ,             x < H , y < T
w x = u x tan u x ,
u x 2 + w x 2 = ( 1 - c 2 ) ( n 0 k 0 H ) 2 v x 2 ,
2 u = tan - 1 ( w 1 / u ) + tan - 1 ( w 2 / u ) + ( q - 1 ) π ,
u 2 + w 1 2 = ( 1 - b 2 ) ( n 0 k 0 T ) 2 v 2
u 2 + w 2 2 = ( 1 - a 2 ) ( n 0 k 0 T ) 2 v ˆ 2 .
P g = n 0 k 0 2 ω μ 0 [ 1 + 1 2 ( 1 + w x v x 2 ) ( 1 w 1 + 1 w 2 ) + 1 w x + u x 2 2 w x v x 2 ( w 1 v 2 + w 2 v ˆ 2 ) ] H T n 0 k 0 2 ω μ 0 [ 1 + 1 2 ( 1 + w x v x 2 ) ( 1 w 1 + 1 w 2 ) + 1 w x ] H T .
f ( μ ) = f 1 ( μ ) f 2 ( μ ) ,
f 1 ( μ ) = u π v ˆ / H ν ( 2 ) { [ ( a n 0 k 0 ) 2 - μ 2 ] 1 / 2 ( R + T ) } ,
f 2 ( μ ) = H [ 1 2 ( sin ( μ H - u x ) μ H - u x + sin ( μ H + u x ) μ H + u x ) + cos u x ( μ H ) 2 + u x 2 ( w x cos μ H - μ H sin μ H ) ] .
P r = n 0 k 0 H 2 ω μ 0 v x 2 u 2 u x 2 w x 2 v ˆ 2 ( w 2 c π T R ) 1 / 2 × exp ( - w 2 3 3 v 3 c 3 + 2 w 2 c )
c = [ 1 - 1 - c 2 1 - b 2 ( u x v v x w 2 ) 2 ] 1 / 2 ,
= 2 n 0 k 0 ( 1 - b 2 ) 3 / 2 R .
P r = 1 2 ω μ 0 u 2 w 2 v ˆ 2 ( H T ) exp ( - w 2 3 3 v 3 + 2 w 2 ) .
e - 1 = f 1 ( μ 1 ) / f 1 ( 0 ) 2
( 1 - b 2 ) 1 / 2 α R = { ( 1 - b 2 1 - c 2 ) 1 / 2 ( w 2 c 2 π v 3 ) 1 / 2 [ 1 + 1 2 ( 1 + w x v x 2 ) ( 1 w 1 + 1 w 2 ) + 1 w x ] - 1 × u 2 v x 3 u x 2 w x 2 v ˆ 2 1 / 2 exp ( - w 2 3 3 v 3 c 3 + 2 w 2 c ) , v x v x s u 2 w 2 4 v 2 v ˆ 2 [ 1 + 1 2 ( 1 w 1 + 1 w 2 ) ] - 1 exp ( - w 2 3 3 v 3 + 2 w 2 ) ,             v x v x s ,
v x s = ( 1 - c 2 1 - b 2 ) 1 / 2 ( w 2 2 v ) 1 / 2 ( u x + π ) .
E x = { cos ( u x H x ) J ν ( ( n 0 k 0 ) 2 - ( u x / H ) 2 r ) ; x < H , r < R u x v x exp [ - w x H ( x - H ) ] J ν ( ( n 0 k 0 ) 2 - ( u x / H ) 2 r ) ; x > H , r < R J ν ( ( n 0 k 0 ) 2 - ( u x / H ) 2 R ) cos ( u x H x ) H ν ( 2 ) ( ( a n 0 k 0 ) 2 - ( u x / H ) 2 r ) / H ν ( 2 ) ( ( a n 0 k 0 ) 2 - ( u x / H ) 2 R ) ; x < H , r > R
d d R log J ν ( ( n 0 k 0 ) 2 - ( u x / H ) 2 R ) = d d R log H ν ( 2 ) ( ( a n 0 k 0 ) 2 - ( u x / H ) 2 R ) .
P g = 1 2 π ω μ 0 ( 1 + 1 w x ) [ 3 ( 4 q - 1 ) π 4 n 0 k 0 R ] 1 / 3 H .
f 1 ( μ ) = 1 π J ν ( ( n 0 k 0 ) 2 - ( u x / H ) 2 R ) / H ν ( 2 ) × ( ( a n 0 k 0 ) 2 - μ 2 R ) .
( 1 - b 2 ) 1 / 2 α R = { ( 1 - b 2 1 - c 2 ) 1 / 2 ( 8 π ) 1 / 2 v x 3 ( 1 - 1 - c 2 1 - a 2 u x 2 v x 2 ) 1 / 4 u x 2 w x 2 ( 1 + 1 w x ) ( 1 + u x 2 v x 2 ) - 1 / 2 exp ( - F ) ; v x v x e ( 1 - b 2 1 - a 2 ) 1 / 2 exp { - ˆ 3 [ 1 - ( 12 q - 3 2 ˆ π ) 2 / 3 + 4 ˆ ] 3 / 2 } ; v x v x e
F = ˆ 3 [ 1 - 1 - c 2 1 - a 2 u x 2 v x 2 - ( 12 q - 3 2 ˆ π ) 2 / 3 + ( 1 + u x 2 v x 2 ) - 1 / 2 4 ˆ ] 3 / 2
ˆ = 2 n 0 k 0 ( 1 - a 2 ) 3 / 2 R
v x e = 1 2 ( 1 - c 2 1 - a 2 ) 1 / 2 1 / 2 ( u x + π ) .
u c v c = ( 4 q - 3 4 q - 1 ) 1 / 6 ( 1 - a 2 1 - b 2 ) 1 / 2 η / ( 1 + η 2 ) 1 / 2
η = ( 12 q - 3 2 ˆ π ) 1 / 3 ( 1 + u x 2 v x 2 ) - 1 / 2 × [ 1 - 1 3 ( 6 ˆ ) 1 / 3 ( 1 + u x 2 v x 2 ) - 1 / 2 ( 4 q - 1 4 π ) - 2 / 3 ] .