Abstract

Analytic equations for the degree of polarization of the output lightwave from jointed fibers are derived taking into account the coherence time of light and polarization mode dispersion in the fibers. On the basis of the results, the depolarizing conditions and the principles of optics of the Lyot depolarizer are discussed.

© 1984 Optical Society of America

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References

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  1. B. H. Billings, “A Monochromatic Depolarizer,” J. Opt. Soc. Am. 41, 966 (1951).
    [CrossRef]
  2. A. P. Loeber, “Depolarization of White Light by a Birefringent Crystal. II. The Lyot Depolarizer,” J. Opt. Soc. Am. 72, 650 (1982).
    [CrossRef]
  3. K. Böhm, K. Petermann, E. Weidel, “Performance of Lyot Depolarizers with Birefringent Single-Mode Fibers,” IEEE/OSA J. Lightwave Technol. LT-1, 71 (1983).
    [CrossRef]
  4. W. K. Burns, “Degree of Polarization in the Lyot Depolarizer,” IEEE/OSA J. Lightwave Technol. LT-1, 475 (1983).
    [CrossRef]
  5. H. G. Danielmeyer, H. P. Weber, “Direct Measurement of the Group Velocity of Light,” Phys. Rev. A 3, 1708 (1971).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).
  7. J. Sakai, S. Machida, T. Kimura, “Degree of Polarization in Anisotropic Single-Mode Optical Fibers: Theory,” IEEE J. Quantum Electron. QE-18, 488 (1982).
    [CrossRef]
  8. R. P. Moeller, W. K. Burns, “Depolarised Broadband Source,” Electron. Lett. 19, 187 (1983).
    [CrossRef]
  9. R. Cubeddu, O. Svelto, “Effect of Dispersion on Laser Self-Locking,” Phys. Lett. A 29, 78 (1969).
    [CrossRef]
  10. K. Okamoto, T. Hosaka, Y. Sasaki, “Linearly Single Polarization Fibers with Zero Polarization Mode Dispersion,” IEEE J. Quantum Electron. QE-18, 496 (1982).
    [CrossRef]
  11. K. Okamoto, T. Edahiro, N. Shibata, “Polarization Properties of Single-Polarization Fibers,” Opt. Lett. 7, 569 (1982).
    [CrossRef] [PubMed]

1983 (3)

K. Böhm, K. Petermann, E. Weidel, “Performance of Lyot Depolarizers with Birefringent Single-Mode Fibers,” IEEE/OSA J. Lightwave Technol. LT-1, 71 (1983).
[CrossRef]

W. K. Burns, “Degree of Polarization in the Lyot Depolarizer,” IEEE/OSA J. Lightwave Technol. LT-1, 475 (1983).
[CrossRef]

R. P. Moeller, W. K. Burns, “Depolarised Broadband Source,” Electron. Lett. 19, 187 (1983).
[CrossRef]

1982 (4)

K. Okamoto, T. Hosaka, Y. Sasaki, “Linearly Single Polarization Fibers with Zero Polarization Mode Dispersion,” IEEE J. Quantum Electron. QE-18, 496 (1982).
[CrossRef]

K. Okamoto, T. Edahiro, N. Shibata, “Polarization Properties of Single-Polarization Fibers,” Opt. Lett. 7, 569 (1982).
[CrossRef] [PubMed]

A. P. Loeber, “Depolarization of White Light by a Birefringent Crystal. II. The Lyot Depolarizer,” J. Opt. Soc. Am. 72, 650 (1982).
[CrossRef]

J. Sakai, S. Machida, T. Kimura, “Degree of Polarization in Anisotropic Single-Mode Optical Fibers: Theory,” IEEE J. Quantum Electron. QE-18, 488 (1982).
[CrossRef]

1971 (1)

H. G. Danielmeyer, H. P. Weber, “Direct Measurement of the Group Velocity of Light,” Phys. Rev. A 3, 1708 (1971).
[CrossRef]

1969 (1)

R. Cubeddu, O. Svelto, “Effect of Dispersion on Laser Self-Locking,” Phys. Lett. A 29, 78 (1969).
[CrossRef]

1951 (1)

Billings, B. H.

Böhm, K.

K. Böhm, K. Petermann, E. Weidel, “Performance of Lyot Depolarizers with Birefringent Single-Mode Fibers,” IEEE/OSA J. Lightwave Technol. LT-1, 71 (1983).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Burns, W. K.

W. K. Burns, “Degree of Polarization in the Lyot Depolarizer,” IEEE/OSA J. Lightwave Technol. LT-1, 475 (1983).
[CrossRef]

R. P. Moeller, W. K. Burns, “Depolarised Broadband Source,” Electron. Lett. 19, 187 (1983).
[CrossRef]

Cubeddu, R.

R. Cubeddu, O. Svelto, “Effect of Dispersion on Laser Self-Locking,” Phys. Lett. A 29, 78 (1969).
[CrossRef]

Danielmeyer, H. G.

H. G. Danielmeyer, H. P. Weber, “Direct Measurement of the Group Velocity of Light,” Phys. Rev. A 3, 1708 (1971).
[CrossRef]

Edahiro, T.

Hosaka, T.

K. Okamoto, T. Hosaka, Y. Sasaki, “Linearly Single Polarization Fibers with Zero Polarization Mode Dispersion,” IEEE J. Quantum Electron. QE-18, 496 (1982).
[CrossRef]

Kimura, T.

J. Sakai, S. Machida, T. Kimura, “Degree of Polarization in Anisotropic Single-Mode Optical Fibers: Theory,” IEEE J. Quantum Electron. QE-18, 488 (1982).
[CrossRef]

Loeber, A. P.

Machida, S.

J. Sakai, S. Machida, T. Kimura, “Degree of Polarization in Anisotropic Single-Mode Optical Fibers: Theory,” IEEE J. Quantum Electron. QE-18, 488 (1982).
[CrossRef]

Moeller, R. P.

R. P. Moeller, W. K. Burns, “Depolarised Broadband Source,” Electron. Lett. 19, 187 (1983).
[CrossRef]

Okamoto, K.

K. Okamoto, T. Hosaka, Y. Sasaki, “Linearly Single Polarization Fibers with Zero Polarization Mode Dispersion,” IEEE J. Quantum Electron. QE-18, 496 (1982).
[CrossRef]

K. Okamoto, T. Edahiro, N. Shibata, “Polarization Properties of Single-Polarization Fibers,” Opt. Lett. 7, 569 (1982).
[CrossRef] [PubMed]

Petermann, K.

K. Böhm, K. Petermann, E. Weidel, “Performance of Lyot Depolarizers with Birefringent Single-Mode Fibers,” IEEE/OSA J. Lightwave Technol. LT-1, 71 (1983).
[CrossRef]

Sakai, J.

J. Sakai, S. Machida, T. Kimura, “Degree of Polarization in Anisotropic Single-Mode Optical Fibers: Theory,” IEEE J. Quantum Electron. QE-18, 488 (1982).
[CrossRef]

Sasaki, Y.

K. Okamoto, T. Hosaka, Y. Sasaki, “Linearly Single Polarization Fibers with Zero Polarization Mode Dispersion,” IEEE J. Quantum Electron. QE-18, 496 (1982).
[CrossRef]

Shibata, N.

Svelto, O.

R. Cubeddu, O. Svelto, “Effect of Dispersion on Laser Self-Locking,” Phys. Lett. A 29, 78 (1969).
[CrossRef]

Weber, H. P.

H. G. Danielmeyer, H. P. Weber, “Direct Measurement of the Group Velocity of Light,” Phys. Rev. A 3, 1708 (1971).
[CrossRef]

Weidel, E.

K. Böhm, K. Petermann, E. Weidel, “Performance of Lyot Depolarizers with Birefringent Single-Mode Fibers,” IEEE/OSA J. Lightwave Technol. LT-1, 71 (1983).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Electron. Lett. (1)

R. P. Moeller, W. K. Burns, “Depolarised Broadband Source,” Electron. Lett. 19, 187 (1983).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. Okamoto, T. Hosaka, Y. Sasaki, “Linearly Single Polarization Fibers with Zero Polarization Mode Dispersion,” IEEE J. Quantum Electron. QE-18, 496 (1982).
[CrossRef]

J. Sakai, S. Machida, T. Kimura, “Degree of Polarization in Anisotropic Single-Mode Optical Fibers: Theory,” IEEE J. Quantum Electron. QE-18, 488 (1982).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (2)

K. Böhm, K. Petermann, E. Weidel, “Performance of Lyot Depolarizers with Birefringent Single-Mode Fibers,” IEEE/OSA J. Lightwave Technol. LT-1, 71 (1983).
[CrossRef]

W. K. Burns, “Degree of Polarization in the Lyot Depolarizer,” IEEE/OSA J. Lightwave Technol. LT-1, 475 (1983).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Lett. (1)

Phys. Lett. A (1)

R. Cubeddu, O. Svelto, “Effect of Dispersion on Laser Self-Locking,” Phys. Lett. A 29, 78 (1969).
[CrossRef]

Phys. Rev. A (1)

H. G. Danielmeyer, H. P. Weber, “Direct Measurement of the Group Velocity of Light,” Phys. Rev. A 3, 1708 (1971).
[CrossRef]

Other (1)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

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Figures (6)

Fig. 1
Fig. 1

Schematic configuration of jointed birefringent fibers.

Fig. 2
Fig. 2

Schematic spectrum of a laser diode operating in the multilongitudinal mode.

Fig. 3
Fig. 3

Illustration of the equation f(ττi). If τiτc, f(ττi) becomes zero at around τ = 0.

Fig. 4
Fig. 4

Schematic configuration to obtain the depolarization of light exiting birefringent fibers. (a) and (c) are examples of principles (I) and (II), respectively; (b) shows the Lyot depolarizer with birefringent fibers.

Fig. 5
Fig. 5

Variation of the degree of polarization for linearly polarized incident light as a function of β for several values of α.

Fig. 6
Fig. 6

Explanatory illustration of the Lyot depolarizer consisting of two principles of optics.

Equations (34)

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E X ( t , Ψ ) = 1 2 · Σ n E n · exp i [ 2 π ( ν + n δ ν ) t + Ψ n ] ,
E Y ( t , Ψ ) = 1 2 · Σ n E n · exp i [ 2 π ( ν + n δ ν ) ( t - τ ) + Ψ n ] ,
[ E X 1 i ( t , Ψ ) E Y 1 i ( t , Ψ ) ] = ( cos α sin α - sin α cos α ) [ E X ( t , Ψ ) E Y ( t , Ψ ) ] .
Φ n X 1 = - Z 1 [ β X 1 + 2 π δ ν · d β X 1 d ω · n + ½ ( 2 π δ ν ) 2 · d 2 β X 1 d ω 2 · n 2 ] ,
Φ n Y 1 = - Z 1 [ β Y 1 + 2 π δ ν · d β Y 1 d ω · n + ½ ( 2 π δ ν ) 2 · d 2 β Y 1 d ω 2 · n 2 ] ,
E X 1 0 ( t , Ψ ) = E X 1 i ( t , Ψ + Φ X 1 ) ,
E Y 1 0 ( t , Ψ ) = E Y 1 i ( t , Ψ + Φ Y 1 ) .
[ E X 2 i ( t , Ψ ) E Y 2 i ( t , Ψ ) ] = ( cos β sin β - sin β cos β ) [ E X 1 0 ( t , Ψ ) E Y 1 0 ( t , Ψ ) ] .
Φ n X 2 = - Z 2 [ β X 2 + 2 π δ ν · d β X 2 d ω · n + ½ ( 2 π δ ν ) 2 · d 2 β X 2 d ω 2 · n 2 ] ,
Φ n Y 2 = - Z 2 [ β Y 2 + 2 π δ ν · d β Y 2 d ω · n + ½ ( 2 π δ ν ) 2 · d 2 β Y 2 d ω 2 · n 2 ] ,
E X 2 0 ( t , Ψ ) = E X 2 i ( t , Ψ + Φ X 2 ) ,
E Y 2 0 ( t , Ψ ) = E Y 2 i ( t , Ψ + Φ Y 2 ) .
E X 2 0 ( t , Ψ ) = cos β [ cos α · E X ( t , Ψ + Φ X 1 + Φ X 2 ) + sin α · E Y ( t , Ψ + Φ X 1 + Φ X 2 ) ] + sin β [ - sin α · E X ( t , Ψ + Φ Y 1 + Φ X 2 ) + cos α · E Y ( t , Ψ + Φ Y 1 + Φ X 2 ) ] ,
E Y 2 0 ( t , Ψ ) = - sin β [ cos α · E X ( t , Ψ + Φ X 1 + Φ Y 2 ) + sin α · E Y ( t , Ψ + Φ Y 1 + Φ Y 2 ) ] + sin β [ - sin α · E X ( t , Ψ + Φ X 1 + Φ Y 2 ) + cos α · E Y ( t , Ψ + Φ Y 1 + Φ Y 2 ) ] .
J x x = 1 + sin 2 α · cos 2 β · f ( τ ) - sin 2 α · sin 2 β · f ( τ + τ 1 ) + cos 2 α · sin 2 β · f ( τ - τ 1 ) ,
J y y = 1 - sin 2 α · cos 2 β · f ( τ ) + sin 2 α · sin 2 β · f ( τ + τ 1 ) - cos 2 α · sin 2 β · f ( τ - τ 1 )
J x y = - ½ sin 2 α · sin 2 β · Γ ( τ + τ 2 ) · exp [ - j ( 2 π ν τ + Z 2 Δ β 2 ) ] - ½ sin 2 α · sin 2 β · Γ ( τ - τ 2 ) · exp [ j ( 2 π ν τ - Z 2 Δ β 2 ) ] + sin 2 α · sin 2 β · Γ ( τ + τ 1 - τ 2 ) · exp [ j ( 2 π ν τ + Z 1 Δ β 1 - Z 2 Δ β 2 ) ] - cos 2 α . sin 2 β · Γ ( τ - τ 1 + τ 2 ) · exp [ - j ( 2 π ν τ - Z 1 Δ β 1 + Z 2 Δ β 2 ) ] + cos 2 α · cos 2 β · Γ ( τ - τ 1 - τ 2 ) · exp [ j ( 2 π ν τ - Z 1 Δ β 1 - Z 2 Δ β 2 ) ] - sin 2 α · cos 2 β · Γ ( τ + τ 1 + τ 2 ) · exp [ - j ( 2 π ν τ + Z 1 Δ β 1 + Z 2 Δ β 2 ) ] ,
J y x = - ½ sin 2 α · sin 2 β · Γ ( τ + τ 2 ) · exp [ j ( 2 π ν τ + Z 2 Δ β 2 ) ] - ½ sin 2 α · sin 2 β · Γ ( τ - τ 2 ) · exp [ - j ( 2 π ν τ - Z 2 Δ β 2 ) ] + sin 2 α · sin 2 β · Γ ( τ + τ 1 - τ 2 ) · exp [ - j ( 2 π ν τ + Z 1 Δ β 1 - Z 2 Δ β 2 ) ] - cos 2 α . sin 2 β · Γ ( τ - τ 1 + τ 2 ) · exp [ j ( 2 π ν τ - Z 1 Δ β 1 + Z 2 Δ β 2 ) ] + cos 2 α · cos 2 β · Γ ( τ - τ 1 - τ 2 ) · exp [ - j ( 2 π ν τ - Z 1 Δ β 1 - Z 2 Δ β 2 ) ] - sin 2 α · cos 2 β · Γ ( τ + τ 1 + τ 2 ) · exp [ j ( 2 π ν τ + Z 1 Δ β 1 + Z 2 Δ β 2 ) ] ,
f ( τ - τ i ) = exp [ - ( Δ ν π 2 ) 2 · ( τ - τ i ) 2 ] · cos ( 2 π ν τ - Δ β i Z i ) ( i - 1 , 2 ) = Γ ( τ - τ i ) · cos ( 2 π ν τ - Δ β i Z i ) ,
τ i = Z i ( d β X i d ω - d β Y i d ω ) ,
Δ β i = ( β X i - β Y i ) Z i .
P = 1 - 4 J ( J x x + J y y ) 2 ,
P = [ cos 2 2 η { cos 2 2 β + sin 2 2 β [ Γ ( τ 2 ) ] 2 } + sin 2 2 η { ½ sin 2 2 β · [ Γ ( τ 1 ) ] 2 + cos 4 β [ Γ ( τ 1 + τ 2 ) ] 2 + sin 4 β [ Γ ( τ 1 - τ 2 ) ] 2 } - sin 4 η · sin 2 β · Γ ( τ 2 ) × [ cos 2 β · Γ ( τ 1 + τ 2 ) - sin 2 β · Γ ( τ 1 - τ 2 ) ] cos ( Z 1 Δ β 1 ) + ½ sin 2 2 η · sin 2 2 β · { [ Γ ( τ 1 ) ] 2 - Γ ( τ 1 - τ 2 ) · Γ ( τ 1 + τ 2 ) } cos ( 2 Z 1 Δ β 1 ) ] 1 / 2 ,
τ 1 , τ 2 , τ 2 - τ 1 τ C
J x x = 1 + sin 2 α · cos 2 β · f ( τ ) , J y y = 1 - sin 2 α · cos 2 β · f ( τ ) , J x y = J y x = 0.
P = sin 2 α · cos 2 β · f ( τ ) .
sin 2 α · cos 2 β · f ( τ ) = 0.
I = J x x + ( J y y - J x x ) sin 2 θ + ½ ( J x y + J y x ) sin 2 θ = E 0 2 · π 1 / 2 4 · Δ ν δ ν · [ 1 + sin 2 α · cos 2 β · cos 2 θ · f ( τ ) ] ,
J x x = E X 2 0 ( t , Ψ ) · E X 2 0 ( t , Ψ ) * , J y y = E Y 2 0 ( t , Ψ ) · E Y 2 0 ( t , Ψ ) * , J x y = E X 2 0 ( t , Ψ ) · E Y 2 0 ( t , Ψ ) * , J y x = E Y 2 0 ( t , Ψ ) · E X 2 0 ( t , Ψ ) * .
f ( τ - τ i ) = { γ ( τ i ) } - 1 / 4 Δ ν · exp [ - ( π Δ ν ) 2 4 γ ( τ i ) ( τ - τ i ) 2 ] · cos [ ϕ ( τ - τ i ) ] ,
ϕ ( τ - τ i ) = 2 π ν τ - Δ β i z i + π 4 · d τ i d ω · ( τ - τ i ) 2 8 γ ( τ i ) - ½ arctan [ ½ · ( 2 π Δ ν ) 2 · d τ i d ω ] ,
γ ( τ i ) = π 4 4 · ( d τ i d ω ) 2 + ( 1 Δ ν ) 4 .
γ ( τ i ) = ( 1 Δ ν ) 4 .
f ( τ - τ i ) = exp { - ( Δ ν π 2 ) 2 · ( τ - τ i ) 2 } · cos ( 2 π ν τ - Δ β i Z i ) ( i = 1 , 2 ) = Γ ( τ - τ i ) · cos ( 2 π ν τ - Δ β i Z i ) , ϕ ( τ - τ i ) = 2 π ν τ - Δ β i Z i .

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