Abstract

We present a theoretical analysis of the errors of a Zeeman laser interferometer due to the nonorthogonally elliptic polarizations of laser modes. The polarizations make the measured Doppler frequency-shift signal become a modulated and nonharmonic signal and make the phase shift unstable and nonrepeatable when the signal is measured by a phasemeter. If the orthogonality error between two major axes of the laser polarizations is 6°, the relative errors of the phasemeter output will exceed 200% when the measured phase shift is changed from −16.2° to +3.5°. We also introduce a way to eliminate the errors.

© 1992 Optical Society of America

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References

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  1. M. A. Zumberge, “Frequency stability of a Zeeman stabilized laser,” Appl. Opt. 24, 1902–1904 (1985).
    [CrossRef] [PubMed]
  2. G. W. Hopkins, ed. Interferometry, Proc. Soc. Photo-Opt. Instrum. Eng.192, 17 (1979).
  3. H. Qiou, W. Hou, “Direct measurement of air refraction index using interferometric phase measuring techniques,” presented at the International Measurement Confederation Metrology Meeting, Houston, Tex., 1988.
  4. Y. Wu, F. Tiam, T. Yu, X. Yang, “Coherent optical fiber system with Zeeman lasers,” Electron. Lett. 27, 49–50 (1987).
  5. M. Sargent, W. E. Lamb, “Theory of a Zeeman laser II,” Phys. Rev. 164, 450–465 (1967).
    [CrossRef]
  6. M. A. Bouchiat, L. Pottier, “A high-purity circular polarization modulator: application to birefringence and circular dichroism measurements on multidielectric mirrors,” Opt. Commun. 37, 229–233 (1981).
    [CrossRef]
  7. Y. Xie, Y. Wu, “Elliptical polarization and nonorthogonality of stabilized Zeeman laser output,” Appl. Opt. 28, 2043–2046 (1989).
    [CrossRef] [PubMed]
  8. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 1.5.

1989 (1)

1987 (1)

Y. Wu, F. Tiam, T. Yu, X. Yang, “Coherent optical fiber system with Zeeman lasers,” Electron. Lett. 27, 49–50 (1987).

1985 (1)

1981 (1)

M. A. Bouchiat, L. Pottier, “A high-purity circular polarization modulator: application to birefringence and circular dichroism measurements on multidielectric mirrors,” Opt. Commun. 37, 229–233 (1981).
[CrossRef]

1967 (1)

M. Sargent, W. E. Lamb, “Theory of a Zeeman laser II,” Phys. Rev. 164, 450–465 (1967).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 1.5.

Bouchiat, M. A.

M. A. Bouchiat, L. Pottier, “A high-purity circular polarization modulator: application to birefringence and circular dichroism measurements on multidielectric mirrors,” Opt. Commun. 37, 229–233 (1981).
[CrossRef]

Hou, W.

H. Qiou, W. Hou, “Direct measurement of air refraction index using interferometric phase measuring techniques,” presented at the International Measurement Confederation Metrology Meeting, Houston, Tex., 1988.

Lamb, W. E.

M. Sargent, W. E. Lamb, “Theory of a Zeeman laser II,” Phys. Rev. 164, 450–465 (1967).
[CrossRef]

Pottier, L.

M. A. Bouchiat, L. Pottier, “A high-purity circular polarization modulator: application to birefringence and circular dichroism measurements on multidielectric mirrors,” Opt. Commun. 37, 229–233 (1981).
[CrossRef]

Qiou, H.

H. Qiou, W. Hou, “Direct measurement of air refraction index using interferometric phase measuring techniques,” presented at the International Measurement Confederation Metrology Meeting, Houston, Tex., 1988.

Sargent, M.

M. Sargent, W. E. Lamb, “Theory of a Zeeman laser II,” Phys. Rev. 164, 450–465 (1967).
[CrossRef]

Tiam, F.

Y. Wu, F. Tiam, T. Yu, X. Yang, “Coherent optical fiber system with Zeeman lasers,” Electron. Lett. 27, 49–50 (1987).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 1.5.

Wu, Y.

Y. Xie, Y. Wu, “Elliptical polarization and nonorthogonality of stabilized Zeeman laser output,” Appl. Opt. 28, 2043–2046 (1989).
[CrossRef] [PubMed]

Y. Wu, F. Tiam, T. Yu, X. Yang, “Coherent optical fiber system with Zeeman lasers,” Electron. Lett. 27, 49–50 (1987).

Xie, Y.

Yang, X.

Y. Wu, F. Tiam, T. Yu, X. Yang, “Coherent optical fiber system with Zeeman lasers,” Electron. Lett. 27, 49–50 (1987).

Yu, T.

Y. Wu, F. Tiam, T. Yu, X. Yang, “Coherent optical fiber system with Zeeman lasers,” Electron. Lett. 27, 49–50 (1987).

Zumberge, M. A.

Appl. Opt. (2)

Electron. Lett. (1)

Y. Wu, F. Tiam, T. Yu, X. Yang, “Coherent optical fiber system with Zeeman lasers,” Electron. Lett. 27, 49–50 (1987).

Opt. Commun. (1)

M. A. Bouchiat, L. Pottier, “A high-purity circular polarization modulator: application to birefringence and circular dichroism measurements on multidielectric mirrors,” Opt. Commun. 37, 229–233 (1981).
[CrossRef]

Phys. Rev. (1)

M. Sargent, W. E. Lamb, “Theory of a Zeeman laser II,” Phys. Rev. 164, 450–465 (1967).
[CrossRef]

Other (3)

G. W. Hopkins, ed. Interferometry, Proc. Soc. Photo-Opt. Instrum. Eng.192, 17 (1979).

H. Qiou, W. Hou, “Direct measurement of air refraction index using interferometric phase measuring techniques,” presented at the International Measurement Confederation Metrology Meeting, Houston, Tex., 1988.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 1.5.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of a Zeeman laser interferometer; ϕ is the phasemeter.

Fig. 2
Fig. 2

Theoretical curves of the reference signal I1 with different B and cos α values, where Δω = 2π × 1.5 MHz.

Fig. 3
Fig. 3

(a) Theoretical curves of the measured signal I2 with B = 0.8, 0.9, 1.0 and cos α = 0.9. These curves illustrate that the I2 is modulated and nonharmonic because of the dichroism of the laser cavity, where Δω = 2π × 1.5 MHz, dΔϕ/dt = 2π × 1 MHz. (b) Theoretical curves of I2 with B = 0.9, 1.0 and cos α = 0.7, 0.8, and 1.0. These curves describe the influence of the birefringence of the laser cavity on the measured signals, where Δω = 2π × 1.5 MHz, dΔϕ/dt = 2π × 1 MHz.

Fig. 4
Fig. 4

Output of the phasemeter as a function of the phase shift Δϕ in degrees.

Fig. 5
Fig. 5

Curves of the relative errors Er of the phasemeter with Δϕ in degrees.

Equations (16)

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L = 0 t ( λ / 2 ) Δ f d t ( λ / 2 ) N ,
L = 0 t ( λ / 2 ) ( d ϕ / d t ) d t = 0 t ( λ / 2 ) d ϕ ,
θ = tan 1 | 2 cos α / ( 1 B 2 ) | ,
[ E x E y ] ± = ( 1 / 2 ) A E i [ 0 ± i B exp ( i α ) ] exp ( i ω ± t ) ,
T 1 = ( 1 / 2 ) [ 1 i 1 + i 1 + i 1 i ] .
T 2 = ( 1 / 2 2 ) [ 1 1 1 1 ] .
T 3 = ( 1 / 2 2 ) [ 1 1 1 1 ] [ 1 0 0 i ] [ exp ( i ϕ 1 ) 0 0 exp ( i ϕ 2 ) ] .
I 1 = 2 I ( 1 + B 2 ) + 2 I ( 1 B 2 ) cos ( Δ ω t ) 4 I B cos α sin ( Δ ω t ) ,
I 2 = 2 I [ ( 1 + B 2 ) + ( 1 B 2 ) cos ( Δ ϕ ) ] + 2 I ( 1 B 2 ) cos ( Δ ϕ ) + I ( 1 + 2 B cos α + B 2 ) × cos ( Δ ω t + Δ ϕ ) + I ( 1 2 B cos α + B 2 ) cos ( Δ ω t Δ ϕ ) .
V e ( t ) = K m [ I 1 ( t ) I 2 ( t ) ] = K m I 2 [ 2 ( 1 B 2 ) cos ( Δ ω t Δ ϕ ) 4 B ( 1 B 2 ) cos α × sin ( Δ ω t Δ ϕ ) + ( 1 B 4 ) cos ( Δ ϕ ) + 8 B 2 cos 2 α sin ( Δ ϕ ) + 2 ( 1 B 2 ) ] ,
V e ( t ) = K m I 2 [ 2 ( 1 B 4 ) cos ( Δ ϕ ) + 8 B 2 cos 2 α sin ( Δ ϕ ) + 2 ( 1 B 2 ) 2 ] .
V e ( t ) = 8 K m I 2 sin ( Δ ϕ ) .
E r = | 4 sin ( Δ ϕ ) ( 1 B 2 ) cos ( Δ ϕ ) 4 B 2 cos 2 α sin ( Δ ϕ ) ( 1 B 2 ) 2 | / | ( 1 B 4 ) cos ( Δ ϕ ) + 4 B 2 cos 2 α sin ( Δ ϕ ) + ( 1 B 2 ) 2 | .
E r = tan 2 α .
T 0 = [ 2 cos β 1 / ( n cos β 1 + cos β 2 ) 0 0 2 cos β 1 / ( cos β 1 + n cos β 2 ) ] × [ 2 n cos β 2 / ( n cos β 1 + cos β 2 ) 0 0 2 n cos β 2 / ( n cos β 2 + cos β 1 ) ] ,
tan 2 β 1 = { [ ( n B ) / ( n B 1 ) ] γ 1 } / ( 1 n 2 ) , B < 1 , γ = 2 B > 1 , γ = 2 .

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