Abstract

A geometrical optics treatment shows that whispering-gallery waves following twisted trajectories can be in eigenstates of polarization if σ = (radius of curvature × torsion) is a constant. The Jones vectors of these eigenpolarizations are calculated, along with their propagation constants and power attenuation constants. The curve representing the evolution of an arbitrary input polarization makes a constant angle with the circles passing through the points corresponding to the eigenpolarizations, on the Poincaré sphere or in any one of the equivalent complex-plane representations. Experiments carried out with a cylindrical glass tube give results in good agreement with theoretical conclusions. The theory predicts that the smallest attenuation constant for three-dimensional trajectories in infrared and far-infrared guides should be about (1 + σ2) times larger than that for TE waves along two-dimensional trajectories of the same curvature, providing a practical criterion for the design of guides exhibiting twisted trajectories.

© 1979 Optical Society of America

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References

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  1. E. Garmire, T. McMahon, and M. Bass, “Low-loss optical transmission through bent hollow metal waveguides,” Appl. Phys. Lett. 31, 92–94 (1977).
    [Crossref]
  2. H. Krammer, “Light waves guided by a single curved metallic surface,” Appl. Opt. 17, 316–319 (1978).
    [Crossref] [PubMed]
  3. M. E. Marhic, L. I. Kwan, and M. Epstein, “Optical surface waves along a toroidal metallic guide,” Appl. Phys. Lett. 33, 609–611 (1978);“Invariant properties of helical-circular metallic waveguides,” Appl. Phys. Lett,  33, 874–876 (1978).
    [Crossref]
  4. E. Kreyszig, Introduction to Differential Geometry and Riemannian Geometry (University of Toronto, Toronto, 1968),p. 32.
  5. If n1 is complex, an appropriate attenuation term should be introduced in what follows. We omit this term, together with a phase term, since they have no bearing on the eigenpolarizations, and if needed can easily be taken into account in the expressions of the propagation and attenuation constants.
  6. Three-dimensional curves with σ = const make a constant angle γ = tan−1 σ with a fixed direction, and are called general helixes; Ref. 4, p. 46.
  7. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 1.
  8. H. S. M. Coxeter, Introduction to Geometry (Wiley, New York, 1969), p. 88.
  9. Here,“to reach” should be understood in a physical rather than a mathematical sense, meaning to come so close to the eigenpolarization as to be practically indistinguishable from it.
  10. Another possibility is w± = 0, ∞, which can be viewed as a limiting form of the general statement, since the point at infinity in the w plane belongs, in particular, to the imaginary axis.

1978 (2)

M. E. Marhic, L. I. Kwan, and M. Epstein, “Optical surface waves along a toroidal metallic guide,” Appl. Phys. Lett. 33, 609–611 (1978);“Invariant properties of helical-circular metallic waveguides,” Appl. Phys. Lett,  33, 874–876 (1978).
[Crossref]

H. Krammer, “Light waves guided by a single curved metallic surface,” Appl. Opt. 17, 316–319 (1978).
[Crossref] [PubMed]

1977 (1)

E. Garmire, T. McMahon, and M. Bass, “Low-loss optical transmission through bent hollow metal waveguides,” Appl. Phys. Lett. 31, 92–94 (1977).
[Crossref]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 1.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 1.

Bass, M.

E. Garmire, T. McMahon, and M. Bass, “Low-loss optical transmission through bent hollow metal waveguides,” Appl. Phys. Lett. 31, 92–94 (1977).
[Crossref]

Coxeter, H. S. M.

H. S. M. Coxeter, Introduction to Geometry (Wiley, New York, 1969), p. 88.

Epstein, M.

M. E. Marhic, L. I. Kwan, and M. Epstein, “Optical surface waves along a toroidal metallic guide,” Appl. Phys. Lett. 33, 609–611 (1978);“Invariant properties of helical-circular metallic waveguides,” Appl. Phys. Lett,  33, 874–876 (1978).
[Crossref]

Garmire, E.

E. Garmire, T. McMahon, and M. Bass, “Low-loss optical transmission through bent hollow metal waveguides,” Appl. Phys. Lett. 31, 92–94 (1977).
[Crossref]

Krammer, H.

Kreyszig, E.

E. Kreyszig, Introduction to Differential Geometry and Riemannian Geometry (University of Toronto, Toronto, 1968),p. 32.

Kwan, L. I.

M. E. Marhic, L. I. Kwan, and M. Epstein, “Optical surface waves along a toroidal metallic guide,” Appl. Phys. Lett. 33, 609–611 (1978);“Invariant properties of helical-circular metallic waveguides,” Appl. Phys. Lett,  33, 874–876 (1978).
[Crossref]

Marhic, M. E.

M. E. Marhic, L. I. Kwan, and M. Epstein, “Optical surface waves along a toroidal metallic guide,” Appl. Phys. Lett. 33, 609–611 (1978);“Invariant properties of helical-circular metallic waveguides,” Appl. Phys. Lett,  33, 874–876 (1978).
[Crossref]

McMahon, T.

E. Garmire, T. McMahon, and M. Bass, “Low-loss optical transmission through bent hollow metal waveguides,” Appl. Phys. Lett. 31, 92–94 (1977).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

M. E. Marhic, L. I. Kwan, and M. Epstein, “Optical surface waves along a toroidal metallic guide,” Appl. Phys. Lett. 33, 609–611 (1978);“Invariant properties of helical-circular metallic waveguides,” Appl. Phys. Lett,  33, 874–876 (1978).
[Crossref]

E. Garmire, T. McMahon, and M. Bass, “Low-loss optical transmission through bent hollow metal waveguides,” Appl. Phys. Lett. 31, 92–94 (1977).
[Crossref]

Other (7)

E. Kreyszig, Introduction to Differential Geometry and Riemannian Geometry (University of Toronto, Toronto, 1968),p. 32.

If n1 is complex, an appropriate attenuation term should be introduced in what follows. We omit this term, together with a phase term, since they have no bearing on the eigenpolarizations, and if needed can easily be taken into account in the expressions of the propagation and attenuation constants.

Three-dimensional curves with σ = const make a constant angle γ = tan−1 σ with a fixed direction, and are called general helixes; Ref. 4, p. 46.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 1.

H. S. M. Coxeter, Introduction to Geometry (Wiley, New York, 1969), p. 88.

Here,“to reach” should be understood in a physical rather than a mathematical sense, meaning to come so close to the eigenpolarization as to be practically indistinguishable from it.

Another possibility is w± = 0, ∞, which can be viewed as a limiting form of the general statement, since the point at infinity in the w plane belongs, in particular, to the imaginary axis.

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Figures (8)

FIG. 1
FIG. 1

Geometry for the study of whispering-gallery waves; ( p ̂ , b ̂ , t ̂ ) is the moving trihedron at P.

FIG. 2
FIG. 2

(a) Projection of Fig. 1 onto the ( p ̂ , t ̂ ) plane. (b) Projection of Fig. 1 onto the ( p ̂ , b ) plane. The angle is taken to be positive if p ̂ rotates clockwise when P goes from Pm to Pm+1.

FIG. 3
FIG. 3

w-plane representation of the possible loci of w(s) as a function of δ.

FIG. 4
FIG. 4

w′-plane representation of the locus of w′(s) for σ = 0.

FIG. 5
FIG. 5

w-plane representation of the locus of w(s) for σ = 0 and 0 < n < 1. Numerical values are n = (2)−1/2, σ = 1.

FIG. 6
FIG. 6

w′-plane representation of the loci of w ± and w′(s) for 0 ≤ σ ≤ ∞.

FIG. 7
FIG. 7

Right-handed circular helix.

FIG. 8
FIG. 8

Trajectory w(s) and circles wc(sc).

Equations (81)

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= τ d s ,
d s = 2 ρ ϕ ,
E = ( E E ) .
E m = ( r 0 0 r ) E m .
E m + 1 = ( cos sin sin cos ) E m = ( r cos r sin r sin r cos ) E m .
E m + 1 = r E m ,
( r cos r r sin r sin r cos r ) ( E E ) = ( 0 0 ) ,
| r cos r r sin r sin r cos r | = 0 ,
r ± = 2 1 cos { r + r ± [ ( r r ) 2 4 r r tan 2 ] 1 / 2 } .
r = 1 2 ϕ n 2 ( n 2 1 ) 1 / 2
r = 1 2 ϕ ( n 2 1 ) 1 / 2 ,
r + r = 2 2 ϕ ( n 2 + 1 ) ( n 2 1 ) 1 / 2
r r = 2 ϕ ( n 2 1 ) 1 / 2 .
= τ d s = 2 τ ρ ϕ = 2 σ ϕ ,
σ = τ ρ .
sin tan 2 σ ϕ ,
cos 1 .
r ± = 1 ϕ [ ( n 2 + 1 ) ( n 2 1 ) 1 / 2 ± ( n 2 1 4 σ 2 ) 1 / 2 ] .
( E E ) ± = ( r sin r ± r cos )
( E E ) ± = ( r cos r ± r sin ) ,
( E E ) ± = ( 2 σ [ 1 2 ϕ ( n 2 1 ) 1 / 2 ] ( n 2 1 ) 1 / 2 ( n 2 1 4 σ 2 ) 1 / 2 )
( E E ) ± = ( ( n 2 1 ) 1 / 2 ± ( n 2 1 4 σ 2 ) 1 / 2 2 σ [ 1 2 ϕ n 2 ( n 2 1 ) 1 / 2 ] ) .
( E E ) ± = ( 1 A B )
( E E ) ± = ( A ± B 1 ) ,
A = ( n 2 1 ) 1 / 2 / 2 σ
B = ( n 2 1 4 σ 2 ) 1 / 2 / 2 σ = ( A 2 1 ) 1 / 2 .
r ± = 1 + i β ± d s ,
β ± = i ( 2 ρ ) 1 [ ( n 2 + 1 ) ( n 2 1 ) 1 / 2 ± ( n 2 1 4 σ 2 ) 1 / 2 ] .
E ± + d E ± = r ± E ± = ( 1 + i β ± d s ) E ± ,
E ± = E ± 0 exp ( i 0 s β ± d s ) ,
P ± = P ± 0 exp ( 2 0 s β ± i d s ) = P ± 0 exp ( 0 s α ± d s ) ,
α ± = 2 β ± i = ρ 1 Re [ ( n 2 + 1 ) ( n 2 1 ) 1 / 2 ± ( n 2 1 4 σ 2 ) 1 / 2 ] .
E ( 0 ) = E + 0 + E 0 .
E ( s ) = E + 0 exp ( i 0 s β + d s ) + E 0 exp ( i 0 s β d s ) .
E + E * = 0 .
w ( s ) = E / E .
w m + 1 = ( E E ) m + 1 = r cos ( E / E ) m + r sin r sin ( E / E ) m + r cos = r cos w m + r sin r sin w m + r cos ,
r = 1 ρ 1 μ d s
r = 1 ρ 1 μ d s ,
μ = n 2 ( n 2 1 ) 1 / 2
μ = ( n 2 1 ) 1 / 2 .
w + d w = ( 1 ρ 1 μ d s ) w + τ d s τ wds + 1 ρ 1 μ d s ( w ρ 1 μ w ds + τ d s ) ( 1 + τ w ds + ρ 1 μ d s ) w ρ 1 μ w ds + τ d s + τ w 2 d s + ρ 1 μ w ds ,
d w w 2 2 A w + 1 = τ d s .
w 0 w d w w 2 2 A w + 1 = 0 s τ d s = T ( s ) ,
w 2 2 A w + 1 = ( w w + ) ( w w ) ,
w ± = A ± B ,
w w + w w = w 0 w + w 0 w e ( w + w ) T ( s ) = w 0 w + w 0 w e 2 B T ( s ) .
w = A B coth { B T ( s ) coth 1 [ ( w 0 A ) / B ] } .
w + w = 1 ,
| w w + | / | w w | = | w 0 w + | / | w 0 w | .
s t = ρ | Re ( n 2 1 4 σ 2 ) 1 / 2 | 1 .
w + w * + 1 = 0 .
w = 0
w + = ,
α = 2 ρ 1 Re [ ( n 2 1 ) 1 / 2 ] = α TE
α + = 2 ρ 1 Re [ n 2 ( n 2 1 ) 1 / 2 ] = α TM .
s t = ρ | Re [ ( n 2 1 ) 1 / 2 ] | 1 .
w ± = i [ A i ± ( A i 2 + 1 ) 1 / 2 ] ,
σ c = 2 1 ( n 2 1 ) 1 / 2 .
s t = ρ / 2 ( σ c 2 σ 2 ) 1 / 2 ,
w ± = A r ± i ( 1 A r 2 ) 1 / 2
α ± = ρ 1 ( n 2 + 1 ) ( n 2 1 ) 1 / 2 ,
| A | 1 .
| w + | 2 | A | 1 , | w | 1 / 2 | A | 1 ,
α + 2 ρ 1 Re [ n 2 ( n 2 1 ) 1 / 2 ] = α TM ,
α ( 1 + σ 2 ) 2 ρ 1 Re [ ( n 2 1 ) 1 / 2 ] = ( 1 + σ 2 ) α TE .
x = a cos t , y = a sin t , z = b t ,
ρ = ( a 2 + b 2 ) / a
τ = b / ( a 2 + b 2 ) ,
σ = b / a .
s = ( a 2 + b 2 ) 1 / 2 t .
T ( 2 π ) = exp ( 2 π ( 1 + l 2 / 4 π 2 a 2 ) 1 / 2 n 2 + 1 ( n 2 1 ) 1 / 2 ) .
Δ = ( β + r β r ) s ,
d w d s = τ ( w w + ) ( w w )
w w + w w = w 0 w + w 0 w e 2 B T ( s ) .
w c w + w c w = w w + w w e 2 T ( s c ) ,
d w c d s c = w w + w w ( d w c d s c e 2 T ( s c ) + ( w c w ) 2 τ e 2 T ( s c ) ) .
d w c d s c | 0 = 2 τ w + w ( w w + ) ( w w ) ,
d w d s | s / d w c d s c | 0 = w + w 2 = B .
Re [ ( n 2 1 ) 1 / 2 ] 0
Re [ ( n 2 1 4 σ 2 ) 1 / 2 ] 0 .