Abstract

We present a measuring method for any wave plate retardation with fairly high precision that utilizes a laser frequency splitting technique. To avoid strong mode competition in measuring half and full wave plates, we use two separate methods: comparing adjacent longitudinal mode spacing, and phase offset with an additional quarter wave plate. Therefore any wave plate can be characterized by a single instrument, and no complicated experimental arrangement or data analysis is required. The performance of the system is demonstrated by determining the phase retardation of several samples to a precision and repeatability better than λ/104; moreover, an error analysis is proposed.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. S. Klinger, J. W. Lewis, and C. E. Randall, “Polarized Light in Optics and Spectroscopy (Academic, 1990).
  2. H. G. Jerrard, “Optical compensators for measurement of elliptical polarization,” J. Opt. Soc. Am. 38, 35-57 (1948).
    [CrossRef]
  3. R. C. Plumb, “Analysis of elliptically polarized light,” J. Opt. Soc. Am. 50, 892-894 (1960).
    [CrossRef]
  4. B. R. Grunstra and H. B. Perkins, “A method for the measurement of optical retardation angles near 90 degrees,” Appl. Opt. 5, 585-587 (1966).
    [CrossRef] [PubMed]
  5. D. B. Chenault and R. A. Chipman, “Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32, 3513-3519 (1993).
    [CrossRef] [PubMed]
  6. G. C. Nechev, “Analytical phase-measuring technique for retardation measurements,” Appl. Opt. 33, 6621-6625 (1994).
    [CrossRef] [PubMed]
  7. P. A. Williams, A. H. Rose, and C. M. Wang, “Rotating-polarizer polarimeter for accurate retardation measurement,” Appl. Opt. 36, 6466-6472 (1997).
    [CrossRef]
  8. J. E. Hayden and S. D. Jacobs, “Automated spatially scanning ellipsometer for retardation measurements of transparent materials,” Appl. Opt. 32, 6256-6263 (1993).
    [CrossRef] [PubMed]
  9. L.-H. Shyu, C.-L. Chen, and D.-C. Su, “Method for measuring the retardation of a wave plate,” Appl. Opt. 32, 4228-4230(1993).
    [CrossRef] [PubMed]
  10. X. J. Chen, L. S. Yan, and X. S. Yao, “Waveplate analyzer using binary magneto-optic rotators,” Opt. Express 15, 12989-12994 (2007).
    [CrossRef] [PubMed]
  11. Y. Lin, Z. Zhou, and R. Wang, “Optical heterodyne measurement of the phase retardation of a quarter wave plate,” Opt. Lett. 13, 553-555 (1988).
    [CrossRef]
  12. S. Nakadate, “High-precision retardation measurement using phase detection of Young's fringes,” Appl. Opt. 29, 242-246(1990).
    [CrossRef] [PubMed]
  13. M. H. Chiu, C. D. Chen, and D.-C. Su, “Method for determining the fast axis and phase retardation of a wave plate,” J. Opt. Soc. Am. A 13, 1924-1929 (1996).
    [CrossRef]
  14. K. B. Rochford and C. M. Wang, “Accurate interferometric retardance measurements,” Appl. Opt. 36, 6473-6479(1997).
    [CrossRef]
  15. B. Wang and T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847-3854 (1999).
    [CrossRef]
  16. B. Wang and W. Hellman, “Accuracy assessment of linear birefringence measurement system using a Soleil--Babinet compensator,” Rev. Sci. Instrum. 72, 4066-4070 (2001).
    [CrossRef]
  17. S. Cattaneo, O. Zehnder, P. Gunter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurement,” Phys. Rev. Lett. 88, 243901 (2002).
    [CrossRef] [PubMed]
  18. S. Cattaneo and M. Kauranen, “Application of second-harmonic generation to retardation measurements,” J. Opt. Soc. Am. B 20, 520-528 (2003).
    [CrossRef]
  19. S. M. Wilson, V. Vats, and P. H. Vaccaro, “Time-domain method for characterizing retardation plates with high sensitivity and resolution,” J. Opt. Soc. Am. B 24, 2500-2508 (2007).
    [CrossRef]
  20. W. Holzapfel and W. Settgast, “Force to frequency conversion by intracavity photoelastic modulation,” Appl. Opt. 28, 4585-4594 (1989).
    [CrossRef] [PubMed]
  21. W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, “High-resolution, very broadband force measurement by solid-state laser transducers,” Measurement 28, 277-291 (2000).
    [CrossRef]
  22. R. J. Oram, I. D. Latimer, S. P. Spoor, and S. Bocking, “Longitudinal mode separation tuning in 633 nm helium-neon lasers using induced cavity birefringence,” J. Phys. D 26, 1169-1172 (1993).
    [CrossRef]
  23. S. Yang and S. Zhang, “The frequency split phenomenon in a HeNe laser with a rotational quartz plate in its cavity,” Opt. Commun. 68, 55-57 (1988).
    [CrossRef]
  24. Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075(2001).
    [CrossRef]
  25. X. Zong, W. Liu, and S. Zhang, “Measurement of retardations of arbitrary wave plates by laser frequency splitting,” Opt. Eng. 45, 033602 (2006).
    [CrossRef]
  26. S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28(1995).
    [CrossRef]
  27. Z. Shulian and H. Yanmei, “Tuning curves of 70 MHz frequency differences for HeNe standing-wave lasers,” Chin. Phys. Lett. 10, 728-730 (1993).
    [CrossRef]
  28. P. D. Hale and G. W. Day, “Stability of birefringent linear retarders (waveplates),” Appl. Opt. 27, 5146-5153 (1988).
    [CrossRef] [PubMed]

2007 (2)

2006 (1)

X. Zong, W. Liu, and S. Zhang, “Measurement of retardations of arbitrary wave plates by laser frequency splitting,” Opt. Eng. 45, 033602 (2006).
[CrossRef]

2003 (1)

2002 (1)

S. Cattaneo, O. Zehnder, P. Gunter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurement,” Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

2001 (2)

B. Wang and W. Hellman, “Accuracy assessment of linear birefringence measurement system using a Soleil--Babinet compensator,” Rev. Sci. Instrum. 72, 4066-4070 (2001).
[CrossRef]

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075(2001).
[CrossRef]

2000 (1)

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, “High-resolution, very broadband force measurement by solid-state laser transducers,” Measurement 28, 277-291 (2000).
[CrossRef]

1999 (1)

B. Wang and T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847-3854 (1999).
[CrossRef]

1997 (2)

1996 (1)

1995 (1)

S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28(1995).
[CrossRef]

1994 (1)

1993 (5)

1990 (1)

1989 (1)

1988 (3)

1966 (1)

1960 (1)

1948 (1)

Bocking, S.

R. J. Oram, I. D. Latimer, S. P. Spoor, and S. Bocking, “Longitudinal mode separation tuning in 633 nm helium-neon lasers using induced cavity birefringence,” J. Phys. D 26, 1169-1172 (1993).
[CrossRef]

Cattaneo, S.

S. Cattaneo and M. Kauranen, “Application of second-harmonic generation to retardation measurements,” J. Opt. Soc. Am. B 20, 520-528 (2003).
[CrossRef]

S. Cattaneo, O. Zehnder, P. Gunter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurement,” Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

Chen, C. D.

Chen, C.-L.

Chen, X. J.

Chenault, D. B.

Chipman, R. A.

Chiu, M. H.

Day, G. W.

Grunstra, B. R.

Gunter, P.

S. Cattaneo, O. Zehnder, P. Gunter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurement,” Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

Guo, H.

S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28(1995).
[CrossRef]

Hale, P. D.

Han, Y.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075(2001).
[CrossRef]

S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28(1995).
[CrossRef]

Hayden, J. E.

Hellman, W.

B. Wang and W. Hellman, “Accuracy assessment of linear birefringence measurement system using a Soleil--Babinet compensator,” Rev. Sci. Instrum. 72, 4066-4070 (2001).
[CrossRef]

Holzapfel, W.

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, “High-resolution, very broadband force measurement by solid-state laser transducers,” Measurement 28, 277-291 (2000).
[CrossRef]

W. Holzapfel and W. Settgast, “Force to frequency conversion by intracavity photoelastic modulation,” Appl. Opt. 28, 4585-4594 (1989).
[CrossRef] [PubMed]

Jacobs, S. D.

Jerrard, H. G.

Kauranen, M.

S. Cattaneo and M. Kauranen, “Application of second-harmonic generation to retardation measurements,” J. Opt. Soc. Am. B 20, 520-528 (2003).
[CrossRef]

S. Cattaneo, O. Zehnder, P. Gunter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurement,” Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

Klinger, D. S.

D. S. Klinger, J. W. Lewis, and C. E. Randall, “Polarized Light in Optics and Spectroscopy (Academic, 1990).

Kobusch, M.

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, “High-resolution, very broadband force measurement by solid-state laser transducers,” Measurement 28, 277-291 (2000).
[CrossRef]

Latimer, I. D.

R. J. Oram, I. D. Latimer, S. P. Spoor, and S. Bocking, “Longitudinal mode separation tuning in 633 nm helium-neon lasers using induced cavity birefringence,” J. Phys. D 26, 1169-1172 (1993).
[CrossRef]

Lewis, J. W.

D. S. Klinger, J. W. Lewis, and C. E. Randall, “Polarized Light in Optics and Spectroscopy (Academic, 1990).

Li, K.

S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28(1995).
[CrossRef]

Li, Y.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075(2001).
[CrossRef]

Lin, Y.

Liu, W.

X. Zong, W. Liu, and S. Zhang, “Measurement of retardations of arbitrary wave plates by laser frequency splitting,” Opt. Eng. 45, 033602 (2006).
[CrossRef]

Nakadate, S.

Nechev, G. C.

Neuschaefer-Rube, S.

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, “High-resolution, very broadband force measurement by solid-state laser transducers,” Measurement 28, 277-291 (2000).
[CrossRef]

Oakberg, T. C.

B. Wang and T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847-3854 (1999).
[CrossRef]

Oram, R. J.

R. J. Oram, I. D. Latimer, S. P. Spoor, and S. Bocking, “Longitudinal mode separation tuning in 633 nm helium-neon lasers using induced cavity birefringence,” J. Phys. D 26, 1169-1172 (1993).
[CrossRef]

Perkins, H. B.

Plumb, R. C.

Randall, C. E.

D. S. Klinger, J. W. Lewis, and C. E. Randall, “Polarized Light in Optics and Spectroscopy (Academic, 1990).

Rochford, K. B.

Rose, A. H.

Settgast, W.

Shulian, Z.

Z. Shulian and H. Yanmei, “Tuning curves of 70 MHz frequency differences for HeNe standing-wave lasers,” Chin. Phys. Lett. 10, 728-730 (1993).
[CrossRef]

Shyu, L.-H.

Spoor, S. P.

R. J. Oram, I. D. Latimer, S. P. Spoor, and S. Bocking, “Longitudinal mode separation tuning in 633 nm helium-neon lasers using induced cavity birefringence,” J. Phys. D 26, 1169-1172 (1993).
[CrossRef]

Su, D.-C.

Vaccaro, P. H.

Vats, V.

Wang, B.

B. Wang and W. Hellman, “Accuracy assessment of linear birefringence measurement system using a Soleil--Babinet compensator,” Rev. Sci. Instrum. 72, 4066-4070 (2001).
[CrossRef]

B. Wang and T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847-3854 (1999).
[CrossRef]

Wang, C. M.

Wang, R.

Williams, P. A.

Wilson, S. M.

Xu, X.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075(2001).
[CrossRef]

Yan, L. S.

Yang, S.

S. Yang and S. Zhang, “The frequency split phenomenon in a HeNe laser with a rotational quartz plate in its cavity,” Opt. Commun. 68, 55-57 (1988).
[CrossRef]

Yanmei, H.

Z. Shulian and H. Yanmei, “Tuning curves of 70 MHz frequency differences for HeNe standing-wave lasers,” Chin. Phys. Lett. 10, 728-730 (1993).
[CrossRef]

Yao, X. S.

Zehnder, O.

S. Cattaneo, O. Zehnder, P. Gunter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurement,” Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

Zhang, S.

X. Zong, W. Liu, and S. Zhang, “Measurement of retardations of arbitrary wave plates by laser frequency splitting,” Opt. Eng. 45, 033602 (2006).
[CrossRef]

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075(2001).
[CrossRef]

S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28(1995).
[CrossRef]

S. Yang and S. Zhang, “The frequency split phenomenon in a HeNe laser with a rotational quartz plate in its cavity,” Opt. Commun. 68, 55-57 (1988).
[CrossRef]

Zhang, Y.

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075(2001).
[CrossRef]

Zhou, Z.

Zong, X.

X. Zong, W. Liu, and S. Zhang, “Measurement of retardations of arbitrary wave plates by laser frequency splitting,” Opt. Eng. 45, 033602 (2006).
[CrossRef]

Appl. Opt. (10)

B. R. Grunstra and H. B. Perkins, “A method for the measurement of optical retardation angles near 90 degrees,” Appl. Opt. 5, 585-587 (1966).
[CrossRef] [PubMed]

D. B. Chenault and R. A. Chipman, “Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32, 3513-3519 (1993).
[CrossRef] [PubMed]

G. C. Nechev, “Analytical phase-measuring technique for retardation measurements,” Appl. Opt. 33, 6621-6625 (1994).
[CrossRef] [PubMed]

P. A. Williams, A. H. Rose, and C. M. Wang, “Rotating-polarizer polarimeter for accurate retardation measurement,” Appl. Opt. 36, 6466-6472 (1997).
[CrossRef]

J. E. Hayden and S. D. Jacobs, “Automated spatially scanning ellipsometer for retardation measurements of transparent materials,” Appl. Opt. 32, 6256-6263 (1993).
[CrossRef] [PubMed]

L.-H. Shyu, C.-L. Chen, and D.-C. Su, “Method for measuring the retardation of a wave plate,” Appl. Opt. 32, 4228-4230(1993).
[CrossRef] [PubMed]

S. Nakadate, “High-precision retardation measurement using phase detection of Young's fringes,” Appl. Opt. 29, 242-246(1990).
[CrossRef] [PubMed]

K. B. Rochford and C. M. Wang, “Accurate interferometric retardance measurements,” Appl. Opt. 36, 6473-6479(1997).
[CrossRef]

W. Holzapfel and W. Settgast, “Force to frequency conversion by intracavity photoelastic modulation,” Appl. Opt. 28, 4585-4594 (1989).
[CrossRef] [PubMed]

P. D. Hale and G. W. Day, “Stability of birefringent linear retarders (waveplates),” Appl. Opt. 27, 5146-5153 (1988).
[CrossRef] [PubMed]

Chin. Phys. Lett. (1)

Z. Shulian and H. Yanmei, “Tuning curves of 70 MHz frequency differences for HeNe standing-wave lasers,” Chin. Phys. Lett. 10, 728-730 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

J. Phys. D (1)

R. J. Oram, I. D. Latimer, S. P. Spoor, and S. Bocking, “Longitudinal mode separation tuning in 633 nm helium-neon lasers using induced cavity birefringence,” J. Phys. D 26, 1169-1172 (1993).
[CrossRef]

Measurement (1)

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, “High-resolution, very broadband force measurement by solid-state laser transducers,” Measurement 28, 277-291 (2000).
[CrossRef]

Opt. Commun. (1)

S. Yang and S. Zhang, “The frequency split phenomenon in a HeNe laser with a rotational quartz plate in its cavity,” Opt. Commun. 68, 55-57 (1988).
[CrossRef]

Opt. Eng. (2)

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071-1075(2001).
[CrossRef]

X. Zong, W. Liu, and S. Zhang, “Measurement of retardations of arbitrary wave plates by laser frequency splitting,” Opt. Eng. 45, 033602 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

S. Zhang, H. Guo, K. Li, and Y. Han, “Laser longitudinal mode splitting phenomenon and its applications in laser physics and active metrology sensors,” Opt. Lasers Eng. 23, 1-28(1995).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

S. Cattaneo, O. Zehnder, P. Gunter, and M. Kauranen, “Nonlinear optical technique for precise retardation measurement,” Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (2)

B. Wang and T. C. Oakberg, “A new instrument for measuring both the magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847-3854 (1999).
[CrossRef]

B. Wang and W. Hellman, “Accuracy assessment of linear birefringence measurement system using a Soleil--Babinet compensator,” Rev. Sci. Instrum. 72, 4066-4070 (2001).
[CrossRef]

Other (1)

D. S. Klinger, J. W. Lewis, and C. E. Randall, “Polarized Light in Optics and Spectroscopy (Academic, 1990).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Block diagram of anisotropic laser with intracavity rotator and retarder.

Fig. 2
Fig. 2

Measurement scheme for frequency difference of (a) splitting modes and (b) adjacent modes.

Fig. 3
Fig. 3

Laser intensity tuning curve (a) with or (b) without polarization flipping occurring. PPL, parallel polarized light; VPL, vertical polarized light.

Fig. 4
Fig. 4

Spectral scheme of two adjacent longitudinal modes oscillating within lasing bandwidth.

Fig. 5
Fig. 5

Experimental setup of wave plate measuring system. M1, M2, resonator mirrors; T, discharge tube; W, window plate; WP, wave plate; CWP, compensated wave plate; PZT, piezoelectric transducer; TS, temperature sensor; BS, beam splitter; D1, D2, photoelectric detectors; P, polarizer; APD, avalanche photodiode; HFA, high frequency amplifier; FC, frequency counter; CB, controlling board; PC, personal computer.

Fig. 6
Fig. 6

Measurement results for QWP sample at temperature 25.28 ° C . (a) Frequency difference (FD) of split modes Δ v and adjacent modes Δ v . (b) Measured phase retardation of QWP.

Fig. 7
Fig. 7

Measurement results for HWP sample 1 at temperature 25.96 ° C . (a) Frequency difference (FD) of two longitudinal mode spacings Δ 1 and Δ 2 . (b) Measured phase retardation of HWP.

Fig. 8
Fig. 8

Measurement results for HWP sample 2 at temperature 26.92 ° C . (a) Phase retardation of sum WP (both the HWP and compensated WP). (b) Phase retardation of compensated WP.

Fig. 9
Fig. 9

Equivalent wave plate retardation of the offset QWP and different HWPs with respect to azimuth angle of the two fast axes.

Fig. 10
Fig. 10

Measurement results of a wave plate 80 ° in a period of (a)  360 s and (b) 9 h. MR, measurement results; CR, compensated results at a nominal temperature 25 ° C ; ET, the environmental temperature.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

R ( ρ ) = [ cos ρ sin ρ sin ρ cos ρ ] ,
V ( δ , φ ) = [ cos 2 φ · exp ( j δ 2 ) + sin 2 φ · exp ( j δ 2 ) j sin 2 φ sin δ 2 j sin 2 φ sin δ 2 cos 2 φ · exp ( j δ 2 ) + sin 2 φ · exp ( j δ 2 ) ] ,
E 1 = exp ( j 2 k l ) R ( ρ ) V ( δ , φ ) V ( δ , φ ) R ( ρ ) E 1 E 2 = exp ( j 2 k l ) V ( δ , φ ) R ( ρ ) R ( ρ ) V ( δ , φ ) E 2 ,
v 1 , 2 = Δ ( m ± δ 2 π )
E 1 y E 1 x = ± 1 cos ( 2 ρ + 2 φ ) sin ( 2 ρ + 2 φ ) , E 2 y E 2 x = ± 1 cos ( 2 φ ) sin ( 2 φ ) ,
Δ v = Δ δ π ,
δ = Δ v Δ π ,
δ = Δ v Δ v + Δ v π .
cos δ = cos δ 1 cos δ 2 sin δ 1 sin δ 2 cos [ 2 ( φ 1 φ 2 ) ] ,
tan ρ = sin δ 1 2 sin δ 2 2 sin [ 2 ( φ 1 φ 2 ) ] cos δ 1 2 cos δ 2 2 sin δ 1 2 sin δ 2 2 cos [ 2 ( φ 1 φ 2 ) ] .
Δ v = 1 2 ( Δ 1 Δ 2 ) , Δ = 1 2 ( Δ 1 + Δ 2 ) .
δ = π * ( 1 + Δ 1 Δ 2 Δ 1 + Δ 2 ) ,
δ = π * ( 2 + Δ 1 Δ 2 Δ 1 + Δ 2 ) .

Metrics