Abstract

White light interferometry has been adopted to measure distributed polarization coupling in high-birefringence waveguides. Since the coupling mode is weak compared to the exciting mode, the contrast ratio of the interferogram is very low. This will increase the difficulty of direct detection of the polarization coupling intensity. By rotating the angle between the polarization eigenmodes and the principal axis of the linear polarizer from 45° to 85°, the contrast ratio of the interferogram can be improved more than 10 times. As a result, the measurement sensitivity can be improved more than 100 times.

©2002 Optical Society of America

1. Introduction

White-light interferometry (WLI) has been widely adopted in optical sensors for measurements of strain, stress, twist, temperature, etc. [1–3]. It can provide large dynamic range and high accuracy. Meanwhile, it is not sensitive to optical power fluctuation [4,5]. WLI based optic sensors are immune to electromagnetic interference, suitable for harsh environments, and compatible with optical fiber data communication systems. But the measurement results of WLI based optic sensors may be adversely affected by the distributed polarization coupling (DPC) in high-birefringence fibers (HBFs). Due to the drawbacks in manufacturing process and environmental perturbations, distributed polarization coupling dose exist in HBFs [6,7].

Since polarization mode coupling occurs randomly along the fiber, a scanning Michelson interferometer is adopted to compensate for the optical path difference (OPD) between the exciting mode and coupling mode at each coupling point. The DPC intensity can be as weak as 10-8, the contrast ratio of the output interferogram is too small to be directly detected. Phase modulation or differential signal detection can be adopted for detection of weak mode coupling [8,9], but the complexity of the testing system will increase. This paper describes the rotation angle optimization of the polarization eigenmodes in the white light interferometer to increase the contrast ratio of the output interferogram. As a result, the contrast ratio can be improved more than 10 times and the measurement sensitivity can be improved more than 100 times. This method can also be adopted for the detection of weak mode coupling in other birefringent waveguides.

2. Detection of distributed polarization coupling

The main structure of the distributed polarization coupling detection system is a scanning Michelson interferometer, as shown in Fig. 1. It consists of the following parts:

  1. A superluminescent diode (SLD) as the broadband source.
  2. The high-birefringence fiber or other birefringent waveguides under detection.
  3. A scanning Michelson interferometer.
  4. Optical and electrical devices for manipulation and detection of the light beams.
 

Fig. 1 Structure of the white light interferometer

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The working principle of the distributed polarization coupling detection system can be described as follows:

  1. The linear polarization light from the SLD is coupled into the fiber coil with a polarization-maintaining fiber coupler. The excited mode propagates along the high-birefringence fiber, and the coupling mode is generated at the points where distributed polarization coupling occurs due to the imperfectness of the HBF or environmental and mechanical perturbations.
  2. The angle between the exciting mode and the principal axis of the linear polarizer is rotated to α degree by the half-wave plate after the beam expander, where the diameter of the light beam is expanded to 8~10 mm. The transmitted p-polarization light, with its vibration direction parallel to the injection plane, is split into two beams at the polarization-maintaining beam splitter (PMBS) where the perturbations to the polarization state of the optical beams can be minimized compared to a common beam splitter (BS). One beam is reflected by mirror 1 of the fixed arm of the Michelson interferometer, the other beam is reflected by mirror 2, which is mounted on a linear motion rail of the scanning arm. The beam splitter splits the beam reflected by mirror 1 again; part of the optical power is transmitted to the focusing lens. The beam splitter also splits the reflected beam of mirror 2; part of the optical power is reflected to the focusing lens.
  3. The two light beams interfere with each other at the region between the beam splitter and the focusing lens. The resulting interferogram is focused and detected by the photo detector. The interference optical signal injected on the photo detector is converted to current signal and then converted to digital signal by a high-resolution analog to digital converter (ADC). The output of the ADC is sampled, and the profile of the interferogram is extracted after digital signal processing. The position and intensity of the distributed polarization coupling are calculated.

3. Rotation angle optimization of the polarization eigenmodes

If there is a polarization coupling point in the HBF, the exciting mode and the coupling mode at this point can be expressed as follows

Ex0=Ae(t)exp(iφ0)
Ey0=Ac(t)exp(iφ0),

where Ae (t) and Ac (t) are the field amplitudes of the exciting mode and the coupling mode, and φ 0 is the phase of the two optical waves at the coupling point. Due to birefringence, at the end of the fiber, the two ejected waves become

Ex1=Ae(t)exp{i(φ0+kxl)}=Ae(t)exp{i(φ0+k0nxl)}
Ey1=Ac(t)exp{i(φ0+kyl)}=Ac(t)exp{i(φ0+k0nxl+kΔnbl)},

where kx and ky are the wave numbers of the of two eigenmodes, k 0 is the wave number in vacuum, nx is the refractive index of the fast axis of the HBF, Δnb is the refractive index difference of the two axes for the HBF, and l is the distance between the coupling point and the end of the fiber. After the polarizer, the optical signal is converted to linear polarization light, which is described by

Exy0=Ae(t)exp[i(φ0+Δφ1+kxl)]cosα+Ac(t)exp[i(φ0+Δφ1+kyl)]sinα,

where α is the angle between the exciting mode and the principal axis of the linear polarizer, Δφ 1 is optical wave phase difference induced by the polarizer and propagation between the fiber end and the polarizer. After the polarization beam splitter, the light beam is split into two beams

Exy1=Exy2
=22{Ae(t)exp[i(φ0+Δφ1+Δφ2+kxl)]cosα+Ac(t)exp[i(φ0+Δφ1+Δφ2+kyl)]sinα}

where E xy1 and E xy2 are the fields of the linear polarization light beams along the scanning arm and the fixed arm of the Michelson interferometer, respectively; Δφ 2 is the optical wave phase difference induced by propagation between the polarizer and the PMBS. E xy1 and E xy2 propagate along the two arms of the Michelson interferometer until they are reflected by mirror 1 and mirror 2, respectively. The reflected beams propagate back to the PMBS. They are split another time by the PMBS, the two light beams that generate interference can be described respectively as follows

Exyr1=12{Ae(t)exp[i(φ0+Δφ1+Δφ2+kxl+k0δ0+k0Δs)]cosα
+Ac(t)exp[i(φ0+Δφ1+Δφ2+kyl+k0δ0+k0Δs)]sinα},
Exyr2=12{Ae(t)exp[i(φ0+Δφ1+Δφ2+kxl+k0δ0)]cosα
+Ac(t)exp[i(φ0+Δφ1+Δφ2+kyl+k0δ0)]sinα}

where δ 0 is the fixed optical path difference (OPD) between E xy1 and E xyr1, Δs is the OPD between the two arms. The field amplitude of the interference signal is

Eint=Exyr1+Exyr2.

The output current of the photo detector is

Itatal=R0Eint·Eint*=I1+I2+I3+I4+I5,

where R 0 is the responsibility of the photo-detector, the function 〈 〉 denotes time domain averaging, and I 1 ~ I 5 represent for the five interference terms defined by

I1=14R0[Ae2(t)+Ac2(t)]·2cos2α
I2=14R0Ae(t)Ac(t)cos(k0Δnblk0Δs)sin(2α)
I3=14R0[Ae2(t)+Ac2(t)]cos(k0Δs)·2cos2α.
I4=14R02Ae(t)Ac(t)cos(k0Δnbl)sin(2α)
I5=14R0Ae(t)Ac(t)cos(k0Δnbl+k0Δs)sin(2α)

Since in this design a low coherence optical source is adopted, when Δnb l and Δs are both larger than the optical source coherence length Lc , we can get I 3, I 4, I 5 ≈ 0 . Under this condition, the output current is

ItotalI1+I2,

where I 1 is the direct current converted from the constant part of the optical power, and I 2 is the alternating current converted from the varying part of the optical power. The local polarization-coupling parameter at location l can be defined by [10]

h=IcIe=Ey0·Ey0*Ex0·Ex0*=Ac2(t)Ae2(t),

where Ie and Ic are the optical power intensity of the exciting mode and coupling mode at location l, respectively. Since Ac2(t) ≪ Ae2(t), from Eq. (8) we can get

I2I1=Ae(t)Ac(t)cos(k0Δnblk0Δs)sin(2α)Ae2(t)+Ac2(t)·2cos2α,
h12f(k0Δnblk0Δs)tanα

where f (k 0Δnbl - k 0Δs) is the normalized interferogram of the alternating part. It is dependent on the optical spectrum distribution of the SLD, the location of the polarization point, and the OPD between the two arms of the interferometer. From Eq. (11) we can get

h(l)maxl(I2I1)2cot2α,ΔnblΔsLc

where the function maxl () gets the local maximum of the variant. The contrast ratio of the output interferogram is

R(l)=2I2,maxI1+I2,max,
2h1/2(l)tanα

where I 2,max denotes the maximum of I 2 under the condition |Δnbl - Δs| ≤ Lc . Since the DPC intensity can be as weak as 10-8, the contrast ratio of the interferogram is too low to be directly detected. If the minimum contrast ratio of the interferogram that can be detected by the system is R min, , we can get the minimum detectable value of h -parameter as

hminRmin24cos2α.

The normalized value h min, normal(α) = h min(α)/h min(45°), is dependent on the rotation angle α, see Fig. 2.

 

Fig. 2 Relationship between the normalized minimum detectable h-parameter and the rotation angle

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In previous researches, the rotation angle is set at α = 45° [4,10]. If the rotation angle is set at α = 85°, the contrast ratio of the output interferogram can be improved more than 10 times. Meanwhile, the sensitivity of the DPC detection system can be improved more than 100 times. This method can be adopted for the detection of weak mode coupling in other birefringent waveguides. It can also be adopted in distributed optical fiber sensors to improve the contrast ratio of the output interferogram and relax the requirements on digital signal processing [11]. Although the measurement sensitivity can be further increased with the rotation angle α, this will induce the increase of measurement error. The measurement error is dependent on the angle uncertainty Δα. Their relationship can be described by

Err=cot2(α)cot2(α+Δα)cot2(α+Δα).

The relationship between the measurement error and the angle uncertainty is shown in Fig. 3. The minimum measurement error occurs at α =45° for a given value of rotation angle uncertainty. This is the reason for the previous works to choose α = 45° as the polarization eigenmodes rotation angle. For the detection of weak polarization mode coupling, the rotation angle α should be increased to improve measurement sensitivity. If the rotation angle is selected as α = 85°, the measurement sensitivity can be improved more than 100 times. Meanwhile, the relative measurement error can still be kept under 5% for an angle uncertainty of 0.1°. Further increase of the rotation angle will cause a sharp-increase of the measurement error. So a compromise should be taken between the measurement sensitivity and measurement error.

 

Fig. 3 Relationship between the relative measurement error and the rotation angle

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4. Conclusion

White light interferometry has been adopted to measure the distributed polarization coupling in birefringent waveguides, especially in high-birefringence fibers. The method for rotation angle optimization between the polarization eigenmodes and the principal axis of the linear polarizer has been proposed. By setting the rotation angle at 85°, the contrast ratio of the interferogram can be improved more than 10 times. As a result, the measurement sensitivity of polarization coupling intensity can be improved more than 100 times. Further increase of the rotation angle will cause a sharp-increase of the measurement error. A compromise should be taken between the measurement sensitivity and the measurement error.

Acknowledgments

This work was supported by the Natural Science Fund of Tianjin, and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, China. The authors thank the reviewers for their valuable suggestions.

References and links

1. Libo Yuan, Limin Zhou, and Wei Jin, “Quasi-distributed strain sensing with white-light interferometry: a novel approach,” Opt. Lett. 25, 1074–1076 (2000) [CrossRef]  

2. M. Shlyagin, A. Khomenko, and D. Tentori, “Remote measurement of mode-coupling coefficients in birefringent fiber,” Opt. Lett. 19, 913–915 (1994) [CrossRef]   [PubMed]  

3. Wencai Jing, Yimo Zhang, Ge Zhou, Feng Tang, and Haifeng Li, “Measurement accuracy improvement with PZT scanning for detection of DPC in Hi-Bi fibers,” Opt. Express 10, 685–690 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-685 [CrossRef]   [PubMed]  

4. P Martin, G Le Boudec, and H C Lefevre, “Test apparatus of distributed polarization coupling in fiber gyro coils using white light interferometry,” in Fiber Optic Gyros: 15th Anniversary Conference, Shaoul Ezekiel and Eric Udd, eds., Proc. SPIE 1585, pp. 173–179 (1991)

5. J Tpia-Mercado, A V Khomenko, and A Garcia-Weidner, “Precision and sensitivity optimization for white-light interferometric fiber-optic sensors,” J. Lightwave Technol. 19, 70–74 (2001) [CrossRef]  

6. Kazunori Suzuki, Hirokazu Kubota, Satoki Kawanishi, Masatoshi Tanaka, and Moriyuki Fujita, “Optical properties of a low-loss polarization-maintaining photonic crystal fiber,” Opt. Express 9, 676–680 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-676 [CrossRef]   [PubMed]  

7. Tomasz R Wolinski and Wojtek J Bock, “Simultaneous twist and pressure effects in highly birefringent single-mode Bow-Tie fibers,” J. of Lightwave Technol. 11, 389–394 (1993) [CrossRef]  

8. K Takada, J Noda, and K Okamoto, “Measurement of spatial distribution of mode coupling in birefringent polarization-maintaining fiber with new detection scheme,” Opt. Lett. 11, 680–682 (1986) [CrossRef]   [PubMed]  

9. Ju-Yi Lee and Der-Chin Su, “Central fringe identification by phase quadrature interferometric technique and tunable laser-diode,” Opt. Commun. 198, 333–337 (2001) [CrossRef]  

10. S C Rashleigh, W K Burns, R P Moeller, and R Ulrich, “Polarization holding in birefringent single-mode fibers,” Opt. Lett. 7, 40–42 (1982) [CrossRef]   [PubMed]  

11. Kazuo Hotate, Xueliang Song, and Zuyuan He, “Stress-location measurement along an optical fiber by synthesis of triangle-shaped optical coherence function,” IEEE Photonics Technol. Lett. 13, 233–235 (2001) [CrossRef]  

References

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  1. Libo Yuan, Limin Zhou, and Wei Jin, “Quasi-distributed strain sensing with white-light interferometry: a novel approach,” Opt. Lett. 25, 1074–1076 (2000)
    [Crossref]
  2. M. Shlyagin, A. Khomenko, and D. Tentori, “Remote measurement of mode-coupling coefficients in birefringent fiber,” Opt. Lett. 19, 913–915 (1994)
    [Crossref] [PubMed]
  3. Wencai Jing, Yimo Zhang, Ge Zhou, Feng Tang, and Haifeng Li, “Measurement accuracy improvement with PZT scanning for detection of DPC in Hi-Bi fibers,” Opt. Express 10, 685–690 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-685
    [Crossref] [PubMed]
  4. P Martin, G Le Boudec, and H C Lefevre, “Test apparatus of distributed polarization coupling in fiber gyro coils using white light interferometry,” in Fiber Optic Gyros: 15th Anniversary Conference, Shaoul Ezekiel and Eric Udd, eds., Proc. SPIE 1585, pp. 173–179 (1991)
  5. J Tpia-Mercado, A V Khomenko, and A Garcia-Weidner, “Precision and sensitivity optimization for white-light interferometric fiber-optic sensors,” J. Lightwave Technol. 19, 70–74 (2001)
    [Crossref]
  6. Kazunori Suzuki, Hirokazu Kubota, Satoki Kawanishi, Masatoshi Tanaka, and Moriyuki Fujita, “Optical properties of a low-loss polarization-maintaining photonic crystal fiber,” Opt. Express 9, 676–680 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-676
    [Crossref] [PubMed]
  7. Tomasz R Wolinski and Wojtek J Bock, “Simultaneous twist and pressure effects in highly birefringent single-mode Bow-Tie fibers,” J. of Lightwave Technol. 11, 389–394 (1993)
    [Crossref]
  8. K Takada, J Noda, and K Okamoto, “Measurement of spatial distribution of mode coupling in birefringent polarization-maintaining fiber with new detection scheme,” Opt. Lett. 11, 680–682 (1986)
    [Crossref] [PubMed]
  9. Ju-Yi Lee and Der-Chin Su, “Central fringe identification by phase quadrature interferometric technique and tunable laser-diode,” Opt. Commun. 198, 333–337 (2001)
    [Crossref]
  10. S C Rashleigh, W K Burns, R P Moeller, and R Ulrich, “Polarization holding in birefringent single-mode fibers,” Opt. Lett. 7, 40–42 (1982)
    [Crossref] [PubMed]
  11. Kazuo Hotate, Xueliang Song, and Zuyuan He, “Stress-location measurement along an optical fiber by synthesis of triangle-shaped optical coherence function,” IEEE Photonics Technol. Lett. 13, 233–235 (2001)
    [Crossref]

2002 (1)

2001 (4)

J Tpia-Mercado, A V Khomenko, and A Garcia-Weidner, “Precision and sensitivity optimization for white-light interferometric fiber-optic sensors,” J. Lightwave Technol. 19, 70–74 (2001)
[Crossref]

Kazunori Suzuki, Hirokazu Kubota, Satoki Kawanishi, Masatoshi Tanaka, and Moriyuki Fujita, “Optical properties of a low-loss polarization-maintaining photonic crystal fiber,” Opt. Express 9, 676–680 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-676
[Crossref] [PubMed]

Ju-Yi Lee and Der-Chin Su, “Central fringe identification by phase quadrature interferometric technique and tunable laser-diode,” Opt. Commun. 198, 333–337 (2001)
[Crossref]

Kazuo Hotate, Xueliang Song, and Zuyuan He, “Stress-location measurement along an optical fiber by synthesis of triangle-shaped optical coherence function,” IEEE Photonics Technol. Lett. 13, 233–235 (2001)
[Crossref]

2000 (1)

1994 (1)

1993 (1)

Tomasz R Wolinski and Wojtek J Bock, “Simultaneous twist and pressure effects in highly birefringent single-mode Bow-Tie fibers,” J. of Lightwave Technol. 11, 389–394 (1993)
[Crossref]

1991 (1)

P Martin, G Le Boudec, and H C Lefevre, “Test apparatus of distributed polarization coupling in fiber gyro coils using white light interferometry,” in Fiber Optic Gyros: 15th Anniversary Conference, Shaoul Ezekiel and Eric Udd, eds., Proc. SPIE 1585, pp. 173–179 (1991)

1986 (1)

1982 (1)

Bock, Wojtek J

Tomasz R Wolinski and Wojtek J Bock, “Simultaneous twist and pressure effects in highly birefringent single-mode Bow-Tie fibers,” J. of Lightwave Technol. 11, 389–394 (1993)
[Crossref]

Burns, W K

Fujita, Moriyuki

Garcia-Weidner, A

He, Zuyuan

Kazuo Hotate, Xueliang Song, and Zuyuan He, “Stress-location measurement along an optical fiber by synthesis of triangle-shaped optical coherence function,” IEEE Photonics Technol. Lett. 13, 233–235 (2001)
[Crossref]

Hotate, Kazuo

Kazuo Hotate, Xueliang Song, and Zuyuan He, “Stress-location measurement along an optical fiber by synthesis of triangle-shaped optical coherence function,” IEEE Photonics Technol. Lett. 13, 233–235 (2001)
[Crossref]

Jin, Wei

Jing, Wencai

Kawanishi, Satoki

Khomenko, A V

Khomenko, A.

Kubota, Hirokazu

Le Boudec, G

P Martin, G Le Boudec, and H C Lefevre, “Test apparatus of distributed polarization coupling in fiber gyro coils using white light interferometry,” in Fiber Optic Gyros: 15th Anniversary Conference, Shaoul Ezekiel and Eric Udd, eds., Proc. SPIE 1585, pp. 173–179 (1991)

Lee, Ju-Yi

Ju-Yi Lee and Der-Chin Su, “Central fringe identification by phase quadrature interferometric technique and tunable laser-diode,” Opt. Commun. 198, 333–337 (2001)
[Crossref]

Lefevre, H C

P Martin, G Le Boudec, and H C Lefevre, “Test apparatus of distributed polarization coupling in fiber gyro coils using white light interferometry,” in Fiber Optic Gyros: 15th Anniversary Conference, Shaoul Ezekiel and Eric Udd, eds., Proc. SPIE 1585, pp. 173–179 (1991)

Li, Haifeng

Martin, P

P Martin, G Le Boudec, and H C Lefevre, “Test apparatus of distributed polarization coupling in fiber gyro coils using white light interferometry,” in Fiber Optic Gyros: 15th Anniversary Conference, Shaoul Ezekiel and Eric Udd, eds., Proc. SPIE 1585, pp. 173–179 (1991)

Moeller, R P

Noda, J

Okamoto, K

Rashleigh, S C

Shlyagin, M.

Song, Xueliang

Kazuo Hotate, Xueliang Song, and Zuyuan He, “Stress-location measurement along an optical fiber by synthesis of triangle-shaped optical coherence function,” IEEE Photonics Technol. Lett. 13, 233–235 (2001)
[Crossref]

Su, Der-Chin

Ju-Yi Lee and Der-Chin Su, “Central fringe identification by phase quadrature interferometric technique and tunable laser-diode,” Opt. Commun. 198, 333–337 (2001)
[Crossref]

Suzuki, Kazunori

Takada, K

Tanaka, Masatoshi

Tang, Feng

Tentori, D.

Tpia-Mercado, J

Ulrich, R

Wolinski, Tomasz R

Tomasz R Wolinski and Wojtek J Bock, “Simultaneous twist and pressure effects in highly birefringent single-mode Bow-Tie fibers,” J. of Lightwave Technol. 11, 389–394 (1993)
[Crossref]

Yuan, Libo

Zhang, Yimo

Zhou, Ge

Zhou, Limin

IEEE Photonics Technol. Lett. (1)

Kazuo Hotate, Xueliang Song, and Zuyuan He, “Stress-location measurement along an optical fiber by synthesis of triangle-shaped optical coherence function,” IEEE Photonics Technol. Lett. 13, 233–235 (2001)
[Crossref]

J. Lightwave Technol. (1)

J. of Lightwave Technol. (1)

Tomasz R Wolinski and Wojtek J Bock, “Simultaneous twist and pressure effects in highly birefringent single-mode Bow-Tie fibers,” J. of Lightwave Technol. 11, 389–394 (1993)
[Crossref]

Opt. Commun. (1)

Ju-Yi Lee and Der-Chin Su, “Central fringe identification by phase quadrature interferometric technique and tunable laser-diode,” Opt. Commun. 198, 333–337 (2001)
[Crossref]

Opt. Express (2)

Opt. Lett. (4)

Proc. SPIE (1)

P Martin, G Le Boudec, and H C Lefevre, “Test apparatus of distributed polarization coupling in fiber gyro coils using white light interferometry,” in Fiber Optic Gyros: 15th Anniversary Conference, Shaoul Ezekiel and Eric Udd, eds., Proc. SPIE 1585, pp. 173–179 (1991)

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Figures (3)

Fig. 1
Fig. 1 Structure of the white light interferometer
Fig. 2
Fig. 2 Relationship between the normalized minimum detectable h-parameter and the rotation angle
Fig. 3
Fig. 3 Relationship between the relative measurement error and the rotation angle

Equations (27)

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E x 0 = A e ( t ) exp ( i φ 0 )
E y 0 = A c ( t ) exp ( i φ 0 ) ,
E x 1 = A e ( t ) exp { i ( φ 0 + k x l ) } = A e ( t ) exp { i ( φ 0 + k 0 n x l ) }
E y 1 = A c ( t ) exp { i ( φ 0 + k y l ) } = A c ( t ) exp { i ( φ 0 + k 0 n x l + k Δ n b l ) } ,
E xy 0 = A e ( t ) exp [ i ( φ 0 + Δ φ 1 + k x l ) ] cos α + A c ( t ) exp [ i ( φ 0 + Δ φ 1 + k y l ) ] sin α ,
E xy 1 = E xy 2
= 2 2 { A e ( t ) exp [ i ( φ 0 + Δ φ 1 + Δ φ 2 + k x l ) ] cos α + A c ( t ) exp [ i ( φ 0 + Δ φ 1 + Δ φ 2 + k y l ) ] sin α }
E xyr 1 = 1 2 { A e ( t ) exp [ i ( φ 0 + Δ φ 1 + Δ φ 2 + k x l + k 0 δ 0 + k 0 Δ s ) ] cos α
+ A c ( t ) exp [ i ( φ 0 + Δ φ 1 + Δ φ 2 + k y l + k 0 δ 0 + k 0 Δ s ) ] sin α } ,
E xyr 2 = 1 2 { A e ( t ) exp [ i ( φ 0 + Δ φ 1 + Δ φ 2 + k x l + k 0 δ 0 ) ] cos α
+ A c ( t ) exp [ i ( φ 0 + Δ φ 1 + Δ φ 2 + k y l + k 0 δ 0 ) ] sin α }
E int = E xyr 1 + E xyr 2 .
I tatal = R 0 E int · E int * = I 1 + I 2 + I 3 + I 4 + I 5 ,
I 1 = 1 4 R 0 [ A e 2 ( t ) + A c 2 ( t ) ] · 2 cos 2 α
I 2 = 1 4 R 0 A e ( t ) A c ( t ) cos ( k 0 Δ n b l k 0 Δ s ) sin ( 2 α )
I 3 = 1 4 R 0 [ A e 2 ( t ) + A c 2 ( t ) ] cos ( k 0 Δ s ) · 2 cos 2 α .
I 4 = 1 4 R 0 2 A e ( t ) A c ( t ) cos ( k 0 Δ n b l ) sin ( 2 α )
I 5 = 1 4 R 0 A e ( t ) A c ( t ) cos ( k 0 Δ n b l + k 0 Δ s ) sin ( 2 α )
I total I 1 + I 2 ,
h = I c I e = E y 0 · E y 0 * E x 0 · E x 0 * = A c 2 ( t ) A e 2 ( t ) ,
I 2 I 1 = A e ( t ) A c ( t ) cos ( k 0 Δ n b l k 0 Δ s ) sin ( 2 α ) A e 2 ( t ) + A c 2 ( t ) · 2 cos 2 α ,
h 1 2 f ( k 0 Δ n b l k 0 Δ s ) tan α
h ( l ) max l ( I 2 I 1 ) 2 cot 2 α , Δ n b l Δ s L c
R ( l ) = 2 I 2 , max I 1 + I 2 , max ,
2 h 1 / 2 ( l ) tan α
h min R min 2 4 cos 2 α .
Err = cot 2 ( α ) cot 2 ( α + Δ α ) cot 2 ( α + Δ α ) .

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