Abstract

It is shown that losses and phase constants in the general class of 2-D bent hollow waveguides are expressed by only one normalized parameter including the core width and the bending radius. Numerical analysis is made to obtain losses and phase constants, and simple approximate formulas are presented for several lower-order modes with reasonable accuracy.

© 1981 Optical Society of America

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References

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  1. E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).
  2. H. Nishihara, T. Inoue, J. Koyama, Appl. Phys. Lett. 25, 391 (1974).
    [CrossRef]
  3. E. Garmire, T. McMahon, M. Bass, Appl. Opt. 15, 145 (1976).
    [CrossRef] [PubMed]
  4. E. Garmire, T. McMahon, M. Bass, IEEE J. Quantum Electron. QE-16, 23 (1980).
    [CrossRef]
  5. M. Miyagi, S. Nishida, IEEE Trans. Microwave Theory Tech. MTT-28, 536 (1980).
    [CrossRef]
  6. M. Miyagi, A. Hongo, S. Kawakami, in Technical Digest, Institute of Electronics and Communication Engineers (IECE, Tokyo, 1981), paper OQE80-128, in Japanese; Rec. Electr. Commun. Eng. Conv., Tohoku Univ.50, 45 (1981), in Japanese.
  7. M. Miyagi, Appl. Opt. 20, 1221 (1981).
    [CrossRef] [PubMed]
  8. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970), pp. 366 and 367.
  9. H. Krammer, Appl. Opt. 16, 2163 (1977).
    [CrossRef] [PubMed]
  10. A. Hongo, M. Miyagi, S. Kawakami, in Technical Digest, Institute of Electronics and Communications Engineers (IECE, Tokyo, 1981), paper OQE81-00, to be published.
  11. Ref. 8, p. 446.
  12. S. Kawakami, Hikari Doharo (Optical Waveguides) (Asakura Publishing, Tokyo, 1980), pp. 126–132, in Japanese.

1981 (1)

1980 (2)

E. Garmire, T. McMahon, M. Bass, IEEE J. Quantum Electron. QE-16, 23 (1980).
[CrossRef]

M. Miyagi, S. Nishida, IEEE Trans. Microwave Theory Tech. MTT-28, 536 (1980).
[CrossRef]

1977 (1)

1976 (1)

1974 (1)

H. Nishihara, T. Inoue, J. Koyama, Appl. Phys. Lett. 25, 391 (1974).
[CrossRef]

1964 (1)

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

Bass, M.

E. Garmire, T. McMahon, M. Bass, IEEE J. Quantum Electron. QE-16, 23 (1980).
[CrossRef]

E. Garmire, T. McMahon, M. Bass, Appl. Opt. 15, 145 (1976).
[CrossRef] [PubMed]

Garmire, E.

E. Garmire, T. McMahon, M. Bass, IEEE J. Quantum Electron. QE-16, 23 (1980).
[CrossRef]

E. Garmire, T. McMahon, M. Bass, Appl. Opt. 15, 145 (1976).
[CrossRef] [PubMed]

Hongo, A.

M. Miyagi, A. Hongo, S. Kawakami, in Technical Digest, Institute of Electronics and Communication Engineers (IECE, Tokyo, 1981), paper OQE80-128, in Japanese; Rec. Electr. Commun. Eng. Conv., Tohoku Univ.50, 45 (1981), in Japanese.

A. Hongo, M. Miyagi, S. Kawakami, in Technical Digest, Institute of Electronics and Communications Engineers (IECE, Tokyo, 1981), paper OQE81-00, to be published.

Inoue, T.

H. Nishihara, T. Inoue, J. Koyama, Appl. Phys. Lett. 25, 391 (1974).
[CrossRef]

Kawakami, S.

M. Miyagi, A. Hongo, S. Kawakami, in Technical Digest, Institute of Electronics and Communication Engineers (IECE, Tokyo, 1981), paper OQE80-128, in Japanese; Rec. Electr. Commun. Eng. Conv., Tohoku Univ.50, 45 (1981), in Japanese.

A. Hongo, M. Miyagi, S. Kawakami, in Technical Digest, Institute of Electronics and Communications Engineers (IECE, Tokyo, 1981), paper OQE81-00, to be published.

S. Kawakami, Hikari Doharo (Optical Waveguides) (Asakura Publishing, Tokyo, 1980), pp. 126–132, in Japanese.

Koyama, J.

H. Nishihara, T. Inoue, J. Koyama, Appl. Phys. Lett. 25, 391 (1974).
[CrossRef]

Krammer, H.

Marcatili, E. A. J.

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

McMahon, T.

E. Garmire, T. McMahon, M. Bass, IEEE J. Quantum Electron. QE-16, 23 (1980).
[CrossRef]

E. Garmire, T. McMahon, M. Bass, Appl. Opt. 15, 145 (1976).
[CrossRef] [PubMed]

Miyagi, M.

M. Miyagi, Appl. Opt. 20, 1221 (1981).
[CrossRef] [PubMed]

M. Miyagi, S. Nishida, IEEE Trans. Microwave Theory Tech. MTT-28, 536 (1980).
[CrossRef]

M. Miyagi, A. Hongo, S. Kawakami, in Technical Digest, Institute of Electronics and Communication Engineers (IECE, Tokyo, 1981), paper OQE80-128, in Japanese; Rec. Electr. Commun. Eng. Conv., Tohoku Univ.50, 45 (1981), in Japanese.

A. Hongo, M. Miyagi, S. Kawakami, in Technical Digest, Institute of Electronics and Communications Engineers (IECE, Tokyo, 1981), paper OQE81-00, to be published.

Nishida, S.

M. Miyagi, S. Nishida, IEEE Trans. Microwave Theory Tech. MTT-28, 536 (1980).
[CrossRef]

Nishihara, H.

H. Nishihara, T. Inoue, J. Koyama, Appl. Phys. Lett. 25, 391 (1974).
[CrossRef]

Schmeltzer, R. A.

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

H. Nishihara, T. Inoue, J. Koyama, Appl. Phys. Lett. 25, 391 (1974).
[CrossRef]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).

IEEE J. Quantum Electron. (1)

E. Garmire, T. McMahon, M. Bass, IEEE J. Quantum Electron. QE-16, 23 (1980).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. Miyagi, S. Nishida, IEEE Trans. Microwave Theory Tech. MTT-28, 536 (1980).
[CrossRef]

Other (5)

M. Miyagi, A. Hongo, S. Kawakami, in Technical Digest, Institute of Electronics and Communication Engineers (IECE, Tokyo, 1981), paper OQE80-128, in Japanese; Rec. Electr. Commun. Eng. Conv., Tohoku Univ.50, 45 (1981), in Japanese.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970), pp. 366 and 367.

A. Hongo, M. Miyagi, S. Kawakami, in Technical Digest, Institute of Electronics and Communications Engineers (IECE, Tokyo, 1981), paper OQE81-00, to be published.

Ref. 8, p. 446.

S. Kawakami, Hikari Doharo (Optical Waveguides) (Asakura Publishing, Tokyo, 1980), pp. 126–132, in Japanese.

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Figures (7)

Fig. 1
Fig. 1

Bent 2-D hollow waveguide whose refractive index is n(r) and core width 2T.

Fig. 2
Fig. 2

Numerical results for Yp as a function of X (solid lines). Dashed lines represent Yp evaluated by Eqs. (34a) and (34b); the dotted line corresponds to Yp by Eq. (39). Numbers on each curve are mode numbers.

Fig. 3
Fig. 3

Numerical results for Yp as a function of X (solid lines). Dashed lines represent Yp evaluated by Eqs. (34a) and (34b). Numbers on each curve are mode numbers.

Fig. 4
Fig. 4

Numerical results for Yp of the dominant mode as a function of X (solid line). Dashed lines represent Yp predicted by Eqs. (34a) and (34b) whose intersection is located at X = 1.845.

Fig. 5
Fig. 5

Normalized bending loss Yb as a function of X. Solid lines represent numerical results and dashed lines represent Yb predicted by Eq. (42). The dot–dash line is Yb = X. Numbers on each line correspond to mode numbers.

Fig. 6
Fig. 6

Normalized bending loss Yb as a function of X. Solid lines represent numerical results and dashed lines represent Yb predicted by Eq. (42). The dot–dash line is Yb = X. Numbers on each line correspond to mode numbers.

Fig. 7
Fig. 7

Refractive-index profile, symmetrical about the x axis, of a metallic hollow waveguide with inner multilayered dielectrics, where the width of the material with a refractive index of a1n0 or a2n0 is assumed to be δ1T or δ2T except for the nearest layer with a1n0 of δT.

Tables (2)

Tables Icon

Table I Values of the Parameter Ap Describing Best-Fitted Curves of Eq. (39) to the Numerical Results

Tables Icon

Table II Numerical Values X0, Ab, and Xt Which Make Eq. (42) the Best-Fitted Curves to the Numerical Results

Equations (56)

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d 2 E y d r 2 + 1 r d E y d r + [ k 0 2 n 2 ( r ) - ν 2 r 2 ] E y = 0 ,
ν = β R .
E y = J ν ( n 0 k 0 r ) + s Y ν ( n 0 k 0 r )
H θ = - j n 0 k 0 ω μ 0 [ J ν ( n 0 k 0 r ) + s Y ν ( n 0 k 0 r ) ] ,
Z s = E y / H θ ;             r = R ± T ,
J ν ( ρ ± ) + s Y ν ( ρ ± ) J ν ( ρ ± ) + s Y ν ( ρ ± ) = ± j n 0 k 0 ω μ 0 Z s ,
ρ ± = n 0 k 0 ( R ± T ) .
J ν ( ρ + ) - j n 0 k 0 ω μ 0 Z s Y ν ( ρ + ) Y ν ( ρ + ) - j n 0 k 0 ω μ 0 Z s Y ν ( ρ + ) = J ν ( ρ - ) + j n 0 k 0 ω μ 0 Z s J ν ( ρ - ) Y ν ( ρ - ) + j n 0 k 0 ω μ 0 Z s Y ν ( ρ - ) .
J ν ( x ) ~ ( 2 ν ) 1 / 3 A i [ - ( 2 ν ) 1 / 3 ( x - ν ) ] ,
Y ν ( x ) ~ - ( 2 ν ) 1 / 3 B i [ - ( 2 ν ) 1 / 3 ( x - ν ) ] ,
c = ( 2 n 0 k 0 R ) 1 / 3 n 0 k 0 T ,
u 2 = ( n 0 2 k 0 2 - β 2 ) T 2 ,
ξ ± = c - ( u c ) 2 ,
A i ( ξ + ) + j Z s ω μ 0 T c A i ( ξ + ) B i ( ξ + ) + j Z s ω μ 0 T c B i ( ξ + ) = A i ( ξ - ) - j Z s ω μ 0 T c A i ( ξ - ) B i ( ξ - ) - j Z s ω μ 0 T c B i ( ξ - ) ,
( u c ) 2 = ( u 0 c ) 2 + j Z s ω μ 0 T c F ,
F = [ A i ( ξ + 0 ) A i ( ξ + 0 ) - B i ( ξ + 0 ) B i ( ξ + 0 ) ] + [ A i ( ξ - 0 ) A i ( ξ - 0 ) - B i ( ξ - 0 ) B i ( ξ - 0 ) ] [ A i ( ξ + 0 ) A i ( ξ + 0 ) - B i ( ξ + 0 ) B i ( ξ + 0 ) ] - [ A i ( ξ - 0 ) A i ( ξ - 0 ) - B i ( ξ - 0 ) B i ( ξ - 0 ) ] ,
A i ( ξ + 0 ) B i ( ξ + 0 ) = A i ( ξ - 0 ) B i ( ξ - 0 ) ,
ξ ± 0 = c - ( u 0 c ) 2 .
W ( A i , B i ) = A i ( z ) B i ( z ) - A i ( z ) B i ( z ) = 1 π
u 2 = u 0 2 + j Z s ω μ 0 T · A i 2 ( ξ - 0 ) + A i 2 ( ξ + 0 ) A i 2 ( ξ - 0 ) - A i 2 ( ξ + 0 ) c 3 .
β = β 0 - j α ,
α = 1 R · A i 2 ( ξ - 0 ) + A i 2 ( ξ + 0 ) A i 2 ( ξ - 0 ) - A i 2 ( ξ + 0 ) Re ( n 0 k 0 ω μ 0 Z s ) ,
α = c 3 2 u 2 A i 2 ( ξ - 0 ) + A i 2 ( ξ + 0 ) A i 2 ( ξ - 0 ) - A i 2 ( ξ + 0 ) α ,
α = n 0 k 0 u 2 ( n 0 k 0 T ) 3 Re ( n 0 k 0 ω μ 0 Z s ) ,
u = n + 1 2 π
u 0 2 = ( n 0 2 k 0 2 - β 0 2 ) T 2 ,
Z s ω μ 0 Y s ω 0 n 0 2 ,
α = n 0 k 0 u 2 ( n 0 k 0 T ) 3 Re ( ω μ 0 n 0 k 0 Y s ) .
X 2 π 2 c 3 = ( 2 π ) 2 ( n 0 k 0 T ) 3 n 0 k 0 R ,
Y p - 2 π 2 u 0 2 = 2 π 2 ( β 0 2 - n 0 2 k 0 2 ) T 2 ,
Y b α / α 0 ,
β 0 = β 0 [ 1 - 1 6 ( 1 - 15 4 u 2 ) ( n 0 k 0 T u ) 2 ( T R ) 2 ] ;
β 0 = ( 1 + T R ) n 0 k 0 { 1 - 1 2 [ 3 n 0 k 0 R ( n + 3 4 ) π ] 2 / 3 } ,
Y p = { - ( n + 1 ) 2 2 { 1 + 1 3 [ 1 - 15 ( n + 1 ) 2 π 2 ] X 2 ( n + 1 ) 4 } ;             X 1 , X - 2 [ 3 4 ( n + 3 4 ) ] 2 / 3 X 2 / 3 ;             X 1.
( X , Y p ) = [ 0 , - ( n + 1 ) 2 2 ] and [ 4 3 ( n + 3 4 ) 2 , - 2 3 ( n + 3 4 ) 2 ]
Y p = Y 0 - H tanh { A p [ X - 2 3 ( n + 3 4 ) 2 ] } ,
Y p = - ( n + 1 ) 2 2             at X = - ,
Y p = - 2 3 ( n + 3 4 ) 2             at X = .
Y p = { - ( 7 12 n 2 + n + 7 16 ) - ( 1 12 n 2 - 1 16 ) tanh { A p [ X - 2 3 ( n + 3 4 ) 2 ] } ;             X < 4 3 ( n + 3 4 ) 2 , X - 2 [ 3 4 ( n + 3 4 ) ] 2 / 3 X 2 / 3 ; X 4 3 ( n + 3 4 ) 2 .
α = { [ 1 - 2 3 ( 1 - 15 4 u 2 ) ( n 0 k 0 T u ) 4 ( T R ) 2 ] α ; large bending radius , ( n 0 k 0 T u ) 2 ( T R ) α ; small bending radius ,
Y b = { ( n + 1 ) 2 { 1 - 2 3 [ 1 - 15 ( n + 1 ) 2 π 2 ] X 2 ( n + 1 ) 4 } ; X 1 , X ; X 1.
Y b = { ( n + 1 ) 2 { 1 - 2 3 [ 1 - 15 ( n + 1 ) 2 π 2 ] X 2 ( n + 1 ) 4 } ; X < X t , X 0 + ( X - X 0 3 + A b 3 ) 1 / 3 ; X X t ,
Y b = ( X 3 + 1 ) 1 / 3
X 0 ~ n 2 ,
A b ~ n + 1 ,
E y = cos ( u 0 T x - n π 2 ) exp ( - j β z ) .
u 0 tan ( u 0 - n π 2 ) = j ω μ 0 T Z s .
u 0 = u ( 1 + j 1 ω μ 0 T Z s ) ,
u = n + 1 2 π .
β 2 + ( u 0 T ) 2 = ( n 0 k 0 ) 2 ,
α = n 0 k 0 u 2 ( n 0 k 0 T ) 3 Re ( n 0 k 0 ω μ 0 Z s ) .
α = n 0 k 0 u 2 ( n 0 k 0 T ) 3 Re ( ω μ 0 n 0 k 0 Y s )
δ i T = π 2 · 1 ( a i 2 - 1 ) 1 / 2 n 0 k 0             ( i = 1 , 2 ) ,
δ T = π 2 · 1 ( a 1 2 - 1 ) 1 / 2 n 0 k 0 × { 2 ;     TE mode , 1 ;     TM mode ,
α / α H = { ( a 2 2 - 1 a 1 2 - 1 ) m ; TE mode , a 1 4 a 1 2 - 1 · 1 n 2 + κ 2 [ a 1 4 ( a 2 2 - 1 ) a 2 4 ( a 1 2 - 1 ) ] m ; TM mode ,
α H = n 0 k 0 u 2 ( n 0 k 0 T ) 3 × { n n 2 + κ 2 ; TE mode , n ; TM mode .

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