Abstract

The bending losses of infrared metallic waveguides are studied by a mode-coupling analysis. Previous results obtained for slab waveguides are confirmed. The high losses observed in practice with circular guides are explained by the near degeneracy between the TE01 and TM11 modes. The method is also used to study losses due to nonsteady-state propagation, such as caused by input mismatch or varying bend radii. Circular metallic waveguides with a thin dielectric lining on the inside could perform better than unlined guides, from the points of view of uniform bending losses and single-mode propagation under arbitrary bends.

© 1981 Optical Society of America

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References

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  1. C. C. Eaglesfield, Proc. Inst. Electr. Eng. Part B 109, 26 (1962).
  2. A. E. Karbowiak, in Electromagnetic Wave Theory, Part 1,J. Brown, Ed. (Pergamon, New York, 1965), p. 419.
  3. E. A. J. Marcatili, R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).
  4. E. Garmire, T. R. McMahon, M. Bass, Appl. Opt. 15, 145 (1976).
    [Crossref] [PubMed]
  5. E. Garmire, Appl. Opt. 15, 3037 (1976).
    [Crossref] [PubMed]
  6. H. Krammer, Appl. Opt. 16, 2163 (1977).
    [Crossref] [PubMed]
  7. Y. Mizushima, T. Sugeta, T. Urisu, H. Nishihara, J. Koyama, Appl. Opt. 19, 3259 (1980).
    [Crossref] [PubMed]
  8. M. Miyagi, Appl. Opt. 20, 1221 (1981).
    [Crossref] [PubMed]
  9. E. Garmire, T. R. McMahon, M. Bass, IEEE J. Quantum Electron., QE-16, 23 (1980).
    [Crossref]
  10. M. E. Marhic, E. Garmire, Appl. Phys. Lett. 38, 743 (1981).
    [Crossref]
  11. We restrict R to values so large that the waveguide does not operate in the whispering-gallery regime described in Refs. 6 and 9.
  12. M. Jouguet, Cables Transm. 2, 133 (1947).
  13. W. J. Albersheim, Bell Syst. Tech. J. 28, 1 (1949).
  14. S. E. Miller, Proc. IRE 40, 1104 (1952).
    [Crossref]
  15. S. E. Miller, Bell Syst. Tech. J. 33, 661 (1954).
  16. H. G. Unger, Bell Syst. Tech. J. 36, 1253 (1957).
  17. H. Nishihara, T. Inoue, J. Koyama, Appl. Phys. Lett. 25, 391 (1974).
    [Crossref]
  18. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), p. 191.
  19. E. A. J. Marcatili, S. E. Miller, Bell Syst. Tech. J. 48, 2161 (1969).
  20. J. R. Beattie, Philos. Mag. 461, 235 (1955).
  21. A. E. Karbowiak, Proc. Inst. Electr. Eng. 102, 698 (1955).
  22. S. P. Morgan, Bell Syst. Tech. J. 36, 1209 (1957).
  23. H. G. Unger, Bell Syst. Tech. J. 36, 1292 (1957).
  24. G. E. Conklin, Bell Syst. Tech. J. 45, 723 (1966).
  25. P. A. Vuorinen, H. E. M. Barlow, Proc. Inst. Electr. Eng. 111, 433 (1964).
    [Crossref]
  26. H. E. M. Barlow, Proc. Inst. Electr. Eng. 104, 403 (1957).
  27. H. E. M. Barlow, in Ref. 2, p. 389; H. E. M. Barlow, H. G. Effemey, S. Taheri, in Ref. 2, p. 399.
  28. H. G. Unger, Bell Syst. Tech. J. 39, 161 (1960).
  29. H. G. Unger, Bell Syst. Tech. J. 41, 745 (1962).
  30. R. O. Miles, R. W. Grow, IEEE J. Quantum Electron. QE-15, 1396 (1979).
    [Crossref]

1981 (2)

M. E. Marhic, E. Garmire, Appl. Phys. Lett. 38, 743 (1981).
[Crossref]

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

1980 (2)

Y. Mizushima, T. Sugeta, T. Urisu, H. Nishihara, J. Koyama, Appl. Opt. 19, 3259 (1980).
[Crossref] [PubMed]

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

1979 (1)

R. O. Miles, R. W. Grow, IEEE J. Quantum Electron. QE-15, 1396 (1979).
[Crossref]

1977 (1)

1976 (2)

1974 (1)

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

1969 (1)

E. A. J. Marcatili, S. E. Miller, Bell Syst. Tech. J. 48, 2161 (1969).

1966 (1)

G. E. Conklin, Bell Syst. Tech. J. 45, 723 (1966).

1964 (2)

P. A. Vuorinen, H. E. M. Barlow, Proc. Inst. Electr. Eng. 111, 433 (1964).
[Crossref]

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

1962 (2)

H. G. Unger, Bell Syst. Tech. J. 41, 745 (1962).

C. C. Eaglesfield, Proc. Inst. Electr. Eng. Part B 109, 26 (1962).

1960 (1)

H. G. Unger, Bell Syst. Tech. J. 39, 161 (1960).

1957 (4)

H. G. Unger, Bell Syst. Tech. J. 36, 1253 (1957).

H. E. M. Barlow, Proc. Inst. Electr. Eng. 104, 403 (1957).

S. P. Morgan, Bell Syst. Tech. J. 36, 1209 (1957).

H. G. Unger, Bell Syst. Tech. J. 36, 1292 (1957).

1955 (2)

J. R. Beattie, Philos. Mag. 461, 235 (1955).

A. E. Karbowiak, Proc. Inst. Electr. Eng. 102, 698 (1955).

1954 (1)

S. E. Miller, Bell Syst. Tech. J. 33, 661 (1954).

1952 (1)

S. E. Miller, Proc. IRE 40, 1104 (1952).
[Crossref]

1949 (1)

W. J. Albersheim, Bell Syst. Tech. J. 28, 1 (1949).

1947 (1)

M. Jouguet, Cables Transm. 2, 133 (1947).

Albersheim, W. J.

W. J. Albersheim, Bell Syst. Tech. J. 28, 1 (1949).

Barlow, H. E. M.

P. A. Vuorinen, H. E. M. Barlow, Proc. Inst. Electr. Eng. 111, 433 (1964).
[Crossref]

H. E. M. Barlow, Proc. Inst. Electr. Eng. 104, 403 (1957).

H. E. M. Barlow, in Ref. 2, p. 389; H. E. M. Barlow, H. G. Effemey, S. Taheri, in Ref. 2, p. 399.

Bass, M.

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

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

Beattie, J. R.

J. R. Beattie, Philos. Mag. 461, 235 (1955).

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), p. 191.

Conklin, G. E.

G. E. Conklin, Bell Syst. Tech. J. 45, 723 (1966).

Eaglesfield, C. C.

C. C. Eaglesfield, Proc. Inst. Electr. Eng. Part B 109, 26 (1962).

Garmire, E.

M. E. Marhic, E. Garmire, Appl. Phys. Lett. 38, 743 (1981).
[Crossref]

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

E. Garmire, Appl. Opt. 15, 3037 (1976).
[Crossref] [PubMed]

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

Grow, R. W.

R. O. Miles, R. W. Grow, IEEE J. Quantum Electron. QE-15, 1396 (1979).
[Crossref]

Inoue, T.

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

Jouguet, M.

M. Jouguet, Cables Transm. 2, 133 (1947).

Karbowiak, A. E.

A. E. Karbowiak, Proc. Inst. Electr. Eng. 102, 698 (1955).

A. E. Karbowiak, in Electromagnetic Wave Theory, Part 1,J. Brown, Ed. (Pergamon, New York, 1965), p. 419.

Koyama, J.

Krammer, H.

Marcatili, E. A. J.

E. A. J. Marcatili, S. E. Miller, Bell Syst. Tech. J. 48, 2161 (1969).

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

Marhic, M. E.

M. E. Marhic, E. Garmire, Appl. Phys. Lett. 38, 743 (1981).
[Crossref]

McMahon, T. R.

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

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

Miles, R. O.

R. O. Miles, R. W. Grow, IEEE J. Quantum Electron. QE-15, 1396 (1979).
[Crossref]

Miller, S. E.

E. A. J. Marcatili, S. E. Miller, Bell Syst. Tech. J. 48, 2161 (1969).

S. E. Miller, Bell Syst. Tech. J. 33, 661 (1954).

S. E. Miller, Proc. IRE 40, 1104 (1952).
[Crossref]

Miyagi, M.

Mizushima, Y.

Morgan, S. P.

S. P. Morgan, Bell Syst. Tech. J. 36, 1209 (1957).

Nishihara, H.

Schmeltzer, R. A.

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

Sugeta, T.

Unger, H. G.

H. G. Unger, Bell Syst. Tech. J. 41, 745 (1962).

H. G. Unger, Bell Syst. Tech. J. 39, 161 (1960).

H. G. Unger, Bell Syst. Tech. J. 36, 1292 (1957).

H. G. Unger, Bell Syst. Tech. J. 36, 1253 (1957).

Urisu, T.

Vuorinen, P. A.

P. A. Vuorinen, H. E. M. Barlow, Proc. Inst. Electr. Eng. 111, 433 (1964).
[Crossref]

Appl. Opt. (5)

Appl. Phys. Lett. (2)

M. E. Marhic, E. Garmire, Appl. Phys. Lett. 38, 743 (1981).
[Crossref]

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

Bell Syst. Tech. J. (10)

S. E. Miller, Bell Syst. Tech. J. 33, 661 (1954).

H. G. Unger, Bell Syst. Tech. J. 36, 1253 (1957).

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

W. J. Albersheim, Bell Syst. Tech. J. 28, 1 (1949).

S. P. Morgan, Bell Syst. Tech. J. 36, 1209 (1957).

H. G. Unger, Bell Syst. Tech. J. 36, 1292 (1957).

G. E. Conklin, Bell Syst. Tech. J. 45, 723 (1966).

E. A. J. Marcatili, S. E. Miller, Bell Syst. Tech. J. 48, 2161 (1969).

H. G. Unger, Bell Syst. Tech. J. 39, 161 (1960).

H. G. Unger, Bell Syst. Tech. J. 41, 745 (1962).

Cables Transm. (1)

M. Jouguet, Cables Transm. 2, 133 (1947).

IEEE J. Quantum Electron. (2)

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

R. O. Miles, R. W. Grow, IEEE J. Quantum Electron. QE-15, 1396 (1979).
[Crossref]

Philos. Mag. (1)

J. R. Beattie, Philos. Mag. 461, 235 (1955).

Proc. Inst. Electr. Eng. (3)

A. E. Karbowiak, Proc. Inst. Electr. Eng. 102, 698 (1955).

P. A. Vuorinen, H. E. M. Barlow, Proc. Inst. Electr. Eng. 111, 433 (1964).
[Crossref]

H. E. M. Barlow, Proc. Inst. Electr. Eng. 104, 403 (1957).

Proc. Inst. Electr. Eng. Part B (1)

C. C. Eaglesfield, Proc. Inst. Electr. Eng. Part B 109, 26 (1962).

Proc. IRE (1)

S. E. Miller, Proc. IRE 40, 1104 (1952).
[Crossref]

Other (4)

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), p. 191.

A. E. Karbowiak, in Electromagnetic Wave Theory, Part 1,J. Brown, Ed. (Pergamon, New York, 1965), p. 419.

We restrict R to values so large that the waveguide does not operate in the whispering-gallery regime described in Refs. 6 and 9.

H. E. M. Barlow, in Ref. 2, p. 389; H. E. M. Barlow, H. G. Effemey, S. Taheri, in Ref. 2, p. 399.

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Figures (2)

Fig. 1
Fig. 1

Slab waveguide.

Fig. 2
Fig. 2

Linear taper transition. The radius of curvature is infinite before z = 0, constant and equal to Rf after z = L, and given by Eq.

Tables (3)

Tables Icon

Table I Calculation of Critical Bending Radii RC for TE01

Tables Icon

Table II Changes in α and β Due to a Thin Dielectric Layer

Tables Icon

Table III Calculation of Critical Bending Radii Rc for TE01 in a Lined Guide

Equations (46)

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

E a ( z ) = E a ( 0 ) exp ( - γ a z ) , and E b ( z ) = E b ( 0 ) exp ( - γ b z ) ,
d E a d z + γ a E a - i C E b = 0 ,
d E b d z + γ b E b - i C E a = 0.
E a ( z ) = E a ( 0 ) exp ( - γ z ) , and E b ( z ) = E b ( 0 ) exp ( - γ z ) .
[ ( γ a - γ ) - i C - i C ( γ b - γ ) ] [ E a ( 0 ) E b ( 0 ) ] = [ 0 0 ] .
γ 2 = ( γ a + γ b ) γ + γ a γ b + C 2 = 0 ,
γ ± = 1 2 { γ a + γ b ± [ ( γ a - γ b ) 2 - 4 C 2 ] 1 / 2 } .
γ a = α a + i β a ,             γ b = α b + i β b ,             γ ± = α ± + i β ± ,
Δ α = α a - α b ,             Δ β = β a - β b ,
α ± = 1 2 ( α a + α b ± 2 - 1 / 2 P ) ,
β ± = 1 2 ( β a + β b ± 2 1 / 2 Δ α Δ β P - 1 ) ,
P = { ( Δ α 2 - Δ β 2 - 4 C 2 + [ ( Δ α 2 - Δ β 2 - 4 C 2 ) 2 + 4 Δ α 2 Δ β 2 ] 1 / 2 } 1 / 2 .
α p = p 2 λ 2 2 w 3 Re ( ν - 1 ) ,             p = 1 , 2 , ,
β p = β 0 - π p 2 λ 4 w 2 .
C p , q = π ( 1 3 + 2 π 2 p 2 ) 1 / 2 w λ R .
Δ α = 3 λ 2 2 w 3 Re ( ν - 1 ) ,
Δ β = 3 π λ 4 w 2 ,
C 1 , 2 = 2.30 w λ R .
α - 1 2 [ α a + α b - Δ α Δ β ( Δ β 2 + 4 C 2 ) - 1 / 2 ] .
α - = α a [ 1 + Δ α α a ( C Δ β ) 2 ]
= α a [ 1 + ( R c R ) 2 ] ,
R c = 1.69 ( w 3 / λ 2 ) .
ν m = ( 1 - i ) ( σ / 2 ω 0 ) 1 / 2 ,
α p q TM = ρ - 1 Re ( ν m - 1 ) ,
β p q TM = β 0 [ 1 - 1 2 ( r p q β 0 ρ ) 2 ] ,
α p q TE = ρ - 1 Re ( ν m - 1 ) [ ( r p q β 0 ρ ) 2 + p 2 ( r p q ) 2 - p 2 ] ,
β p q TE = β 0 [ 1 - 1 2 ( r p q β 0 ρ ) 2 ] ,
α - = 1 2 [ α a + α b - ( Δ α 2 - 4 C 2 ) 1 / 2 ] ,
C = C β 0 ρ R c - 1 = ( α a α b - 2 α a 2 ) 1 / 2 ( α a α b ) 1 / 2 .
R c C β 0 ρ [ α a α b Δ β 2 + ( α b 2 ) 2 ] - 1 / 2 .
η = E a · E - * d s 2 ( E a 2 d s ) ( E - 2 d s ) ,
E ± = E a E a + E b E b ,
E b ( z ) E a ( z ) = E b ( 0 ) E a ( 0 ) = γ a - γ i C = i C γ b - γ = ( γ a - γ γ b - γ ) 1 / 2 = ( Δ γ ( Δ γ ) 2 - 4 C 2 - Δ γ ( Δ γ ) 2 - 4 C 2 ) 1 / 2 = ( 1 1 - u - 1 1 - u ) 1 / 2 = 1 1 - u i u = - i u 1 ± 1 - u ,
u = ( 2 C Δ γ ) 2 .
E = ( 1 + 1 - u ) E a + i u E b .
E a · E b * = E b · E a * d s = 0 , E a 2 d s = E b 2 d s = 1 ,
η = [ 1 + u ( 1 + 1 - u ) - 2 ] - 1 ,
η = 1 2 [ 1 + 1 - u ] for u > 0.
η 1 - u 4 = 1 - | C Δ γ | 2 .
R - 1 ( z ) = z L R f - 1 .
C ( z ) = C β 0 ρ R - 1 ( z ) = C β 0 ρ ( R f L ) - 1 z .
L Δ β - 1 ,
ζ = t ρ 1 ,
.
Δ β = β b - β a = Δ β + δ β b - δ β a .
Δ β r p q β 0 ρ 2 or r p q β 0 ρ 2 ,

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