Abstract

Single-polarization characteristics in highly birefringent fibers have been investigated by considering the photoelastic effect in the entire cross section of the fiber. It has been shown by stress analysis that the stress-induced birefringence between the two orthogonal polarization modes in the cladding is much smaller than that in the core and the vicinity of the core–cladding interface, which causes differential bending loss for the two polarization modes. A bending loss formula, which takes into account the photoelastic effect, has been derived for each polarization mode. Differential attenuation characteristics and extinction ratios have been investigated for several bending radii using the bending loss formula.

© 1984 Optical Society of America

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References

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  1. M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-Polarization Operation of Highly Birefringent Bow-Tie Optical Fibers,” Electron. Lett. 19, 246 (1983).
    [CrossRef]
  2. M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend Behavior of Polarizing Optical Fibers,” Electron. Lett. 19, 679 (1983).
    [CrossRef]
  3. J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Mac-Chesney, R. E. Howard, “A Single-Polarization Fiber,” IEEE/OSA J. Lightwave Technol. LT-1, 370 (1983).
    [CrossRef]
  4. A. W. Snyder, F. Ruhl, “New Single-Mode Single-Polarization Optical Fiber,” Electron. Lett. 19, 185 (1983).
    [CrossRef]
  5. A. W. Snyder, F. Ruhl, “Practical Single-Polarization Anisotropic Fibers,” Electron. Lett. 19, 687 (1983).
    [CrossRef]
  6. A. W. Snyder, F. Ruhl, “Single-Mode, Single Polarization Fibers made of Birefringent Materials,” J. Opt. Soc. Am. 73, 1165 (1983).
    [CrossRef]
  7. K. Okamoto, M. P. Varnham, D. N. Payne, “Polarization-Maintaining Optical Fibers with Low Dispersion over a Wide Spectral Range,” Appl. Opt. 22, 2370 (1983).
    [CrossRef] [PubMed]
  8. W. Primak, D. Post, “Photoelastic Constants of Vitreous Silica and its Elastic Coefficient of Refractive Index,” J. Appl. Phys. 30, 779 (1959).
    [CrossRef]
  9. G. W. Scherer, “Stress-Induced Index Profile Distortion in Optical Waveguides,” Appl. Opt. 19, 2000 (1980).
    [CrossRef] [PubMed]
  10. K. Okamoto, T. Hosaka, T. Edahiro, “Stress Analysis of Optical Fibers by a Finite Element Method,” IEEE J. Quantum Electron. QE-17, 2123 (1981).
    [CrossRef]
  11. D. Marcuse, “Curvature Loss Formula for Optical Fibers,” J. Opt. Soc. Am. 66, 216 (1976).
    [CrossRef]

1983

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-Polarization Operation of Highly Birefringent Bow-Tie Optical Fibers,” Electron. Lett. 19, 246 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend Behavior of Polarizing Optical Fibers,” Electron. Lett. 19, 679 (1983).
[CrossRef]

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Mac-Chesney, R. E. Howard, “A Single-Polarization Fiber,” IEEE/OSA J. Lightwave Technol. LT-1, 370 (1983).
[CrossRef]

A. W. Snyder, F. Ruhl, “New Single-Mode Single-Polarization Optical Fiber,” Electron. Lett. 19, 185 (1983).
[CrossRef]

A. W. Snyder, F. Ruhl, “Practical Single-Polarization Anisotropic Fibers,” Electron. Lett. 19, 687 (1983).
[CrossRef]

K. Okamoto, M. P. Varnham, D. N. Payne, “Polarization-Maintaining Optical Fibers with Low Dispersion over a Wide Spectral Range,” Appl. Opt. 22, 2370 (1983).
[CrossRef] [PubMed]

A. W. Snyder, F. Ruhl, “Single-Mode, Single Polarization Fibers made of Birefringent Materials,” J. Opt. Soc. Am. 73, 1165 (1983).
[CrossRef]

1981

K. Okamoto, T. Hosaka, T. Edahiro, “Stress Analysis of Optical Fibers by a Finite Element Method,” IEEE J. Quantum Electron. QE-17, 2123 (1981).
[CrossRef]

1980

1976

1959

W. Primak, D. Post, “Photoelastic Constants of Vitreous Silica and its Elastic Coefficient of Refractive Index,” J. Appl. Phys. 30, 779 (1959).
[CrossRef]

Birch, R. D.

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend Behavior of Polarizing Optical Fibers,” Electron. Lett. 19, 679 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-Polarization Operation of Highly Birefringent Bow-Tie Optical Fibers,” Electron. Lett. 19, 246 (1983).
[CrossRef]

Edahiro, T.

K. Okamoto, T. Hosaka, T. Edahiro, “Stress Analysis of Optical Fibers by a Finite Element Method,” IEEE J. Quantum Electron. QE-17, 2123 (1981).
[CrossRef]

Hosaka, T.

K. Okamoto, T. Hosaka, T. Edahiro, “Stress Analysis of Optical Fibers by a Finite Element Method,” IEEE J. Quantum Electron. QE-17, 2123 (1981).
[CrossRef]

Howard, R. E.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Mac-Chesney, R. E. Howard, “A Single-Polarization Fiber,” IEEE/OSA J. Lightwave Technol. LT-1, 370 (1983).
[CrossRef]

Mac-Chesney, J. B.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Mac-Chesney, R. E. Howard, “A Single-Polarization Fiber,” IEEE/OSA J. Lightwave Technol. LT-1, 370 (1983).
[CrossRef]

Marcuse, D.

Okamoto, K.

K. Okamoto, M. P. Varnham, D. N. Payne, “Polarization-Maintaining Optical Fibers with Low Dispersion over a Wide Spectral Range,” Appl. Opt. 22, 2370 (1983).
[CrossRef] [PubMed]

K. Okamoto, T. Hosaka, T. Edahiro, “Stress Analysis of Optical Fibers by a Finite Element Method,” IEEE J. Quantum Electron. QE-17, 2123 (1981).
[CrossRef]

Payne, D. N.

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend Behavior of Polarizing Optical Fibers,” Electron. Lett. 19, 679 (1983).
[CrossRef]

K. Okamoto, M. P. Varnham, D. N. Payne, “Polarization-Maintaining Optical Fibers with Low Dispersion over a Wide Spectral Range,” Appl. Opt. 22, 2370 (1983).
[CrossRef] [PubMed]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-Polarization Operation of Highly Birefringent Bow-Tie Optical Fibers,” Electron. Lett. 19, 246 (1983).
[CrossRef]

Pleibel, W.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Mac-Chesney, R. E. Howard, “A Single-Polarization Fiber,” IEEE/OSA J. Lightwave Technol. LT-1, 370 (1983).
[CrossRef]

Post, D.

W. Primak, D. Post, “Photoelastic Constants of Vitreous Silica and its Elastic Coefficient of Refractive Index,” J. Appl. Phys. 30, 779 (1959).
[CrossRef]

Primak, W.

W. Primak, D. Post, “Photoelastic Constants of Vitreous Silica and its Elastic Coefficient of Refractive Index,” J. Appl. Phys. 30, 779 (1959).
[CrossRef]

Ruhl, F.

A. W. Snyder, F. Ruhl, “Single-Mode, Single Polarization Fibers made of Birefringent Materials,” J. Opt. Soc. Am. 73, 1165 (1983).
[CrossRef]

A. W. Snyder, F. Ruhl, “Practical Single-Polarization Anisotropic Fibers,” Electron. Lett. 19, 687 (1983).
[CrossRef]

A. W. Snyder, F. Ruhl, “New Single-Mode Single-Polarization Optical Fiber,” Electron. Lett. 19, 185 (1983).
[CrossRef]

Scherer, G. W.

Sears, F. M.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Mac-Chesney, R. E. Howard, “A Single-Polarization Fiber,” IEEE/OSA J. Lightwave Technol. LT-1, 370 (1983).
[CrossRef]

Simpson, J. R.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Mac-Chesney, R. E. Howard, “A Single-Polarization Fiber,” IEEE/OSA J. Lightwave Technol. LT-1, 370 (1983).
[CrossRef]

Snyder, A. W.

A. W. Snyder, F. Ruhl, “Practical Single-Polarization Anisotropic Fibers,” Electron. Lett. 19, 687 (1983).
[CrossRef]

A. W. Snyder, F. Ruhl, “New Single-Mode Single-Polarization Optical Fiber,” Electron. Lett. 19, 185 (1983).
[CrossRef]

A. W. Snyder, F. Ruhl, “Single-Mode, Single Polarization Fibers made of Birefringent Materials,” J. Opt. Soc. Am. 73, 1165 (1983).
[CrossRef]

Stolen, R. H.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Mac-Chesney, R. E. Howard, “A Single-Polarization Fiber,” IEEE/OSA J. Lightwave Technol. LT-1, 370 (1983).
[CrossRef]

Tarbox, E. J.

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-Polarization Operation of Highly Birefringent Bow-Tie Optical Fibers,” Electron. Lett. 19, 246 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend Behavior of Polarizing Optical Fibers,” Electron. Lett. 19, 679 (1983).
[CrossRef]

Varnham, M. P.

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend Behavior of Polarizing Optical Fibers,” Electron. Lett. 19, 679 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-Polarization Operation of Highly Birefringent Bow-Tie Optical Fibers,” Electron. Lett. 19, 246 (1983).
[CrossRef]

K. Okamoto, M. P. Varnham, D. N. Payne, “Polarization-Maintaining Optical Fibers with Low Dispersion over a Wide Spectral Range,” Appl. Opt. 22, 2370 (1983).
[CrossRef] [PubMed]

Appl. Opt.

Electron. Lett.

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Single-Polarization Operation of Highly Birefringent Bow-Tie Optical Fibers,” Electron. Lett. 19, 246 (1983).
[CrossRef]

M. P. Varnham, D. N. Payne, R. D. Birch, E. J. Tarbox, “Bend Behavior of Polarizing Optical Fibers,” Electron. Lett. 19, 679 (1983).
[CrossRef]

A. W. Snyder, F. Ruhl, “New Single-Mode Single-Polarization Optical Fiber,” Electron. Lett. 19, 185 (1983).
[CrossRef]

A. W. Snyder, F. Ruhl, “Practical Single-Polarization Anisotropic Fibers,” Electron. Lett. 19, 687 (1983).
[CrossRef]

IEEE J. Quantum Electron.

K. Okamoto, T. Hosaka, T. Edahiro, “Stress Analysis of Optical Fibers by a Finite Element Method,” IEEE J. Quantum Electron. QE-17, 2123 (1981).
[CrossRef]

IEEE/OSA J. Lightwave Technol.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Mac-Chesney, R. E. Howard, “A Single-Polarization Fiber,” IEEE/OSA J. Lightwave Technol. LT-1, 370 (1983).
[CrossRef]

J. Appl. Phys.

W. Primak, D. Post, “Photoelastic Constants of Vitreous Silica and its Elastic Coefficient of Refractive Index,” J. Appl. Phys. 30, 779 (1959).
[CrossRef]

J. Opt. Soc. Am.

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Figures (6)

Fig. 1
Fig. 1

Dispersion characteristics for typical highly birefringent Panda fiber.

Fig. 2
Fig. 2

Refractive-index profiles for two orthogonal polarizxation modes in birefringent fiber along several azimuthal directions (a) θ = 0°, (b) θ = 22°, (c) θ = 49°, and (d) θ = 90°.

Fig. 3
Fig. 3

Equivalent-index profiles for two polarization modes obtained from Fig. 2. β x /k and β y /k are normalized propagation constants at λ = 0.85 μm.

Fig. 4
Fig. 4

Comparison of dispersion characteristics determined by the equivalent-index method with those by the perturbation method (Fig. 1).

Fig. 5
Fig. 5

Bending loss properties of two polarization modes for several bending radii (a) R = 1.5 cm, (b) R = 2.5 cm, and (c) R = 3.5 cm.

Fig. 6
Fig. 6

Extinction ratios for two polarization modes and bending losses for the guided (x-polarization) mode for several bending radii.

Equations (19)

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β x = β ( 0 ) - k [ ( C 1 + C 2 ) σ x 0 X ( v ) + 2 C 2 σ y 0 Y ( v ) ] ,
β y = β ( 0 ) - k [ 2 C 2 σ x 0 X ( v ) + ( C 1 + C 2 ) σ y 0 Y ( v ) ] ,
n x ( r , θ ) = n ( r , θ ) - C 1 σ x ( r , θ ) - C 2 [ σ y ( r , θ ) + σ z ( r , θ ) ] ,
n y ( r , θ ) = n ( r , θ ) - C 1 σ y ( r , θ ) - C 2 [ σ z ( r , θ ) + σ x ( r , θ ) ] ,
n x ( a r r 1 ) = a r 1 0 2 π [ n - C 1 σ x - C 2 ( σ y + σ z ) ] r d r d θ a r 1 0 2 π r d r d θ ,
n y ( a r r 1 ) = a r 1 0 2 π [ n - C 1 σ y - C 2 ( σ z + σ x ) ] r d r d θ a r 1 0 2 π r d r d θ .
α B , ν = 10 ln ( 10 ) · 2 α ν             ( ν = x or y ) ,
2 α ν = π u ν 2 exp [ - / 3 2 ( σ ν 3 / β ν 2 ) R ] 2 σ ν 3 / 2 v ν 2 R a 2 T ( v ) v ,
T ( v ν ) = u ν 2 v ν 2 D ν 2 { [ J 0 2 ( u ν ) + J 1 2 ( u ν ) ] + D ν 2 ( r 1 a ) 2 [ K 1 2 ( w ν ) + K 0 2 ( w ν ) ] + S ν } .
v ν = k a n 1 ν 2 - n 2 ν 2 ( k = 2 π / λ ; λ is the wavelength in vacuum ) ,
u ν = k 2 n 1 ν 2 - β ν 2 · a ,
w v = β ν 2 - k 2 n 2 ν 2 · a ,
σ ν = w ν / a ,
S ν = { ( r 1 a ) 2 [ B ν K 0 ( q ν ) + C ν I 0 ( q ν ) ] 2 - ( r 1 a ) 2 [ - B ν K 1 ( q ν ) + C ν I 1 ( q ν ) ] 2 - [ B ν K 0 ( p ν ) + C ν I 0 ( p ν ) ] 2 + [ - B ν K 1 ( p ν ) + C ν I 1 ( p ν ) ] 2 , ( β ν > k n ν int ) ( r 1 a ) 2 [ - B ν ln ( r 1 a ) + C ν ] 2 + ( r 1 a ) 2 B ν [ - B ν ln ( r 1 a ) + C ν ] + / 2 1 ( r 1 a ) 2 B ν 2 - C ν 2 - B ν C ν - / 2 1 B ν 2 ,             ( β ν = k n ν int ) ( r 1 a ) 2 [ - B ν π 2 Y 0 ( q ν ) + C ν J 0 ( q ν ) ] 2 + ( r 1 a ) 2 [ B ν π 2 Y 1 ( q ν ) - C ν J 1 ( q ν ) ] 2 - [ - B ν π 2 Y 0 ( p ν ) + C ν J 0 ( p ν ) ] 2 - [ B ν π 2 Y 1 ( p ν ) - C ν J 1 ( p ν ) ] 2 ,             ( β ν < k n ν int ) ,
B ν = { [ P ν I 1 ( p ν ) J 0 ( u ν ) + I 0 ( p ν ) u ν J 1 ( u ν ) ] , ( β ν > k n ν int ) u ν J 1 ( u ν ) , ( β ν = k n ν int ) , [ J 0 ( p ν ) u ν J 1 ( u ν ) - p ν J 1 ( p ν ) J 0 ( u ν ) ] , ( β ν < k n ν int ) ,
C ν = { [ p ν K 1 ( p ν ) J 0 ( u ν ) - K 0 ( p ν ) u ν J 1 ( u ν ) ] , ( β ν > k n ν int ) , J 0 ( u ν ) , ( β ν = k n ν int ) , π 2 [ Y 0 ( p ν ) u ν J 1 ( u ν ) - p ν Y 1 ( p ν ) J 0 ( u ν ) ] , ( β ν < k n ν int ) ,
D ν = { [ p ν I 1 ( p ν ) J 0 ( u ν ) + I 0 ( p ν ) u ν J 1 ( u ν ) ] [ q ν I 1 ( q ν ) K 0 ( w ν ) + I 0 ( q ν ) w ν K 1 ( w ν ) ] , ( β ν > k n ν int ) , u ν J 1 ( u ν ) w ν K 1 ( w ν ) ,             ( β ν = k n ν int ) , [ J 0 ( p ν ) u ν J 1 ( u ν ) - p ν J 1 ( p ν ) J 0 ( u ν ) ] [ J 0 ( q ν ) w ν K 1 ( w ν ) - q ν J 1 ( q ν ) K 0 ( w ν ) ] , ( β ν < k n ν int ) ,
p ν = β ν 2 - ( k n ν int ) 2 · a ,
q ν = β ν 2 - ( k n ν int ) 2 · r 1 .

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