Abstract

A polarization lidar operating at 532  nm was converted into an automatic, polarimetric lidar capable of measuring the entire Stokes vector of backscattered light and its derived quantities. Among these quantities, circular and linear depolarizations were studied as tools for investigating the presence of anisotropic scattering media. Isotropic scatterers show a simple relationship between linear and circular depolarization, a relation that we confirm theoretically and experimentally. Deviations from this relation, which are possible in the presence of anisotropic scatterers such as horizontally oriented ice plates when they are observed with a slant lidar, were studied both numerically and experimentally.

© 2006 Optical Society of America

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References

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  1. K. Sassen, "Polarization in lidar," in Lidar, C. Weitkamp, ed. (Springer-Verlag, 2005), pp. 19-42.
    [CrossRef]
  2. C. M. R. Platt, "Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals," J. Appl. Meteorol. 17, 1220-1224 (1978).
    [CrossRef]
  3. L. R. Bissonette and G. Roy, "Range-height scans of lidar depolarization for characterizing properties and phase of clouds and precipitation," J. Atmos. Ocean. Technol. 18, 1429-1446 (2001).
    [CrossRef]
  4. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  5. J. D. Houston and A. I. Carswell, "Four-component polarization measurement of lidar atmospheric scattering," Appl. Opt. 17, 614-620 (1978).
    [CrossRef] [PubMed]
  6. B. V. Kaul, O. A. Krasnov, A. L. Kuznetzov, and I. V. Samokhvalov, "Polarization sounding of high-altitude aerosol formations," Atmos. Opt. 4, 303-308 (1991).
  7. B. V. Kaul, "Lidar determination of oriented particles in crystal clouds," Atmos. Ocean. Opt. 8, 439-441 (1995).
  8. C. V. M. Van der Mee, J. W. Hovenier, and H. C. van de Hulst, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).
  9. M. I. Mishchenko and J. W. Hovenier, "Depolarization of light by randomly oriented nonspherical particles," Opt. Lett. 20, 1356-1358 (1995).
    [CrossRef] [PubMed]
  10. M. Del Guasta, "Simulation of lidar returns from ideal and deformed hexagonal ice prisms in cold cirrus by means of face tracing," J. Geophys. Res. 106, D12, 12,589-12,602 (2001).
  11. A. H. Auer and D. L. Veal, "The dimensions of ice crystals in natural clouds," J. Atmos. Sci. 26, 919-927 (1970).
    [CrossRef]
  12. P. Yang, B. C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, S. C. Tsay, D. M. Winker, and S. L. Nasiri, "Radiative properties of cirrus clouds in the infrared (8-13 μm) spectral region," J. Quant. Spectrosc. Radiat. Transfer 70, 473-504 (2001).
    [CrossRef]
  13. A. Borovoi, I. Grishin, E. Naats, and U. Oppel, "Backscattering peak of hexagonal ice columns and plates," Opt. Lett. 25, 1388-1390 (2000).
    [CrossRef]
  14. K. Sassen and S. Benson, "A midlatitude cirrus cloud climatology from the facility for atmospheric remote sensing. II. Microphysical properties derived from lidar depolarization," J. Atmos. Sci. 58, 2103-2112 (2001).
    [CrossRef]
  15. F. M. Bréon and B. Dubrulle, "Horizontally oriented plates in clouds," J. Atmos. Sci. 61, 2888-2898 (2004).
    [CrossRef]
  16. H. Chepfer, G. Brogniez, P. Gouloub, F. M. Bréon, and P. H. Flamant, "Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1," J. Quant. Spectrosc. Radiat. Transfer 63, 521-543 (1999).
    [CrossRef]
  17. Y. Takano and K. Jayaweera, "Scattering phase matrix for hexagonal ice crystals computed from ray optics," Appl. Opt. 24, 3254-3263 (1985).
    [CrossRef] [PubMed]
  18. V. E. Derr, N. L. Abshire, R. E. Cupp, and G. T. McNice, "Depolarization of lidar returns from virga and source cloud," J. Appl. Meteorol. 15, 1200-1208 (1976).
    [CrossRef]

2004 (1)

F. M. Bréon and B. Dubrulle, "Horizontally oriented plates in clouds," J. Atmos. Sci. 61, 2888-2898 (2004).
[CrossRef]

2001 (4)

P. Yang, B. C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, S. C. Tsay, D. M. Winker, and S. L. Nasiri, "Radiative properties of cirrus clouds in the infrared (8-13 μm) spectral region," J. Quant. Spectrosc. Radiat. Transfer 70, 473-504 (2001).
[CrossRef]

K. Sassen and S. Benson, "A midlatitude cirrus cloud climatology from the facility for atmospheric remote sensing. II. Microphysical properties derived from lidar depolarization," J. Atmos. Sci. 58, 2103-2112 (2001).
[CrossRef]

M. Del Guasta, "Simulation of lidar returns from ideal and deformed hexagonal ice prisms in cold cirrus by means of face tracing," J. Geophys. Res. 106, D12, 12,589-12,602 (2001).

L. R. Bissonette and G. Roy, "Range-height scans of lidar depolarization for characterizing properties and phase of clouds and precipitation," J. Atmos. Ocean. Technol. 18, 1429-1446 (2001).
[CrossRef]

2000 (1)

1999 (1)

H. Chepfer, G. Brogniez, P. Gouloub, F. M. Bréon, and P. H. Flamant, "Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1," J. Quant. Spectrosc. Radiat. Transfer 63, 521-543 (1999).
[CrossRef]

1995 (2)

B. V. Kaul, "Lidar determination of oriented particles in crystal clouds," Atmos. Ocean. Opt. 8, 439-441 (1995).

M. I. Mishchenko and J. W. Hovenier, "Depolarization of light by randomly oriented nonspherical particles," Opt. Lett. 20, 1356-1358 (1995).
[CrossRef] [PubMed]

1991 (1)

B. V. Kaul, O. A. Krasnov, A. L. Kuznetzov, and I. V. Samokhvalov, "Polarization sounding of high-altitude aerosol formations," Atmos. Opt. 4, 303-308 (1991).

1986 (1)

C. V. M. Van der Mee, J. W. Hovenier, and H. C. van de Hulst, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).

1985 (1)

1978 (2)

J. D. Houston and A. I. Carswell, "Four-component polarization measurement of lidar atmospheric scattering," Appl. Opt. 17, 614-620 (1978).
[CrossRef] [PubMed]

C. M. R. Platt, "Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals," J. Appl. Meteorol. 17, 1220-1224 (1978).
[CrossRef]

1976 (1)

V. E. Derr, N. L. Abshire, R. E. Cupp, and G. T. McNice, "Depolarization of lidar returns from virga and source cloud," J. Appl. Meteorol. 15, 1200-1208 (1976).
[CrossRef]

1970 (1)

A. H. Auer and D. L. Veal, "The dimensions of ice crystals in natural clouds," J. Atmos. Sci. 26, 919-927 (1970).
[CrossRef]

Abshire, N. L.

V. E. Derr, N. L. Abshire, R. E. Cupp, and G. T. McNice, "Depolarization of lidar returns from virga and source cloud," J. Appl. Meteorol. 15, 1200-1208 (1976).
[CrossRef]

Auer, A. H.

A. H. Auer and D. L. Veal, "The dimensions of ice crystals in natural clouds," J. Atmos. Sci. 26, 919-927 (1970).
[CrossRef]

Baum, B. A.

P. Yang, B. C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, S. C. Tsay, D. M. Winker, and S. L. Nasiri, "Radiative properties of cirrus clouds in the infrared (8-13 μm) spectral region," J. Quant. Spectrosc. Radiat. Transfer 70, 473-504 (2001).
[CrossRef]

Benson, S.

K. Sassen and S. Benson, "A midlatitude cirrus cloud climatology from the facility for atmospheric remote sensing. II. Microphysical properties derived from lidar depolarization," J. Atmos. Sci. 58, 2103-2112 (2001).
[CrossRef]

Bissonette, L. R.

L. R. Bissonette and G. Roy, "Range-height scans of lidar depolarization for characterizing properties and phase of clouds and precipitation," J. Atmos. Ocean. Technol. 18, 1429-1446 (2001).
[CrossRef]

Borovoi, A.

Bréon, F. M.

F. M. Bréon and B. Dubrulle, "Horizontally oriented plates in clouds," J. Atmos. Sci. 61, 2888-2898 (2004).
[CrossRef]

H. Chepfer, G. Brogniez, P. Gouloub, F. M. Bréon, and P. H. Flamant, "Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1," J. Quant. Spectrosc. Radiat. Transfer 63, 521-543 (1999).
[CrossRef]

Brogniez, G.

H. Chepfer, G. Brogniez, P. Gouloub, F. M. Bréon, and P. H. Flamant, "Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1," J. Quant. Spectrosc. Radiat. Transfer 63, 521-543 (1999).
[CrossRef]

Carswell, A. I.

Chepfer, H.

H. Chepfer, G. Brogniez, P. Gouloub, F. M. Bréon, and P. H. Flamant, "Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1," J. Quant. Spectrosc. Radiat. Transfer 63, 521-543 (1999).
[CrossRef]

Cupp, R. E.

V. E. Derr, N. L. Abshire, R. E. Cupp, and G. T. McNice, "Depolarization of lidar returns from virga and source cloud," J. Appl. Meteorol. 15, 1200-1208 (1976).
[CrossRef]

Del Guasta, M.

M. Del Guasta, "Simulation of lidar returns from ideal and deformed hexagonal ice prisms in cold cirrus by means of face tracing," J. Geophys. Res. 106, D12, 12,589-12,602 (2001).

Derr, V. E.

V. E. Derr, N. L. Abshire, R. E. Cupp, and G. T. McNice, "Depolarization of lidar returns from virga and source cloud," J. Appl. Meteorol. 15, 1200-1208 (1976).
[CrossRef]

Dubrulle, B.

F. M. Bréon and B. Dubrulle, "Horizontally oriented plates in clouds," J. Atmos. Sci. 61, 2888-2898 (2004).
[CrossRef]

Flamant, P. H.

H. Chepfer, G. Brogniez, P. Gouloub, F. M. Bréon, and P. H. Flamant, "Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1," J. Quant. Spectrosc. Radiat. Transfer 63, 521-543 (1999).
[CrossRef]

Gao, B. C.

P. Yang, B. C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, S. C. Tsay, D. M. Winker, and S. L. Nasiri, "Radiative properties of cirrus clouds in the infrared (8-13 μm) spectral region," J. Quant. Spectrosc. Radiat. Transfer 70, 473-504 (2001).
[CrossRef]

Gouloub, P.

H. Chepfer, G. Brogniez, P. Gouloub, F. M. Bréon, and P. H. Flamant, "Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1," J. Quant. Spectrosc. Radiat. Transfer 63, 521-543 (1999).
[CrossRef]

Grishin, I.

Houston, J. D.

Hovenier, J. W.

M. I. Mishchenko and J. W. Hovenier, "Depolarization of light by randomly oriented nonspherical particles," Opt. Lett. 20, 1356-1358 (1995).
[CrossRef] [PubMed]

C. V. M. Van der Mee, J. W. Hovenier, and H. C. van de Hulst, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).

Hu, Y. X.

P. Yang, B. C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, S. C. Tsay, D. M. Winker, and S. L. Nasiri, "Radiative properties of cirrus clouds in the infrared (8-13 μm) spectral region," J. Quant. Spectrosc. Radiat. Transfer 70, 473-504 (2001).
[CrossRef]

Jayaweera, K.

Kaul, B. V.

B. V. Kaul, "Lidar determination of oriented particles in crystal clouds," Atmos. Ocean. Opt. 8, 439-441 (1995).

B. V. Kaul, O. A. Krasnov, A. L. Kuznetzov, and I. V. Samokhvalov, "Polarization sounding of high-altitude aerosol formations," Atmos. Opt. 4, 303-308 (1991).

Krasnov, O. A.

B. V. Kaul, O. A. Krasnov, A. L. Kuznetzov, and I. V. Samokhvalov, "Polarization sounding of high-altitude aerosol formations," Atmos. Opt. 4, 303-308 (1991).

Kuznetzov, A. L.

B. V. Kaul, O. A. Krasnov, A. L. Kuznetzov, and I. V. Samokhvalov, "Polarization sounding of high-altitude aerosol formations," Atmos. Opt. 4, 303-308 (1991).

McNice, G. T.

V. E. Derr, N. L. Abshire, R. E. Cupp, and G. T. McNice, "Depolarization of lidar returns from virga and source cloud," J. Appl. Meteorol. 15, 1200-1208 (1976).
[CrossRef]

Mishchenko, M. I.

Naats, E.

Nasiri, S. L.

P. Yang, B. C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, S. C. Tsay, D. M. Winker, and S. L. Nasiri, "Radiative properties of cirrus clouds in the infrared (8-13 μm) spectral region," J. Quant. Spectrosc. Radiat. Transfer 70, 473-504 (2001).
[CrossRef]

Oppel, U.

Platt, C. M. R.

C. M. R. Platt, "Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals," J. Appl. Meteorol. 17, 1220-1224 (1978).
[CrossRef]

Roy, G.

L. R. Bissonette and G. Roy, "Range-height scans of lidar depolarization for characterizing properties and phase of clouds and precipitation," J. Atmos. Ocean. Technol. 18, 1429-1446 (2001).
[CrossRef]

Samokhvalov, I. V.

B. V. Kaul, O. A. Krasnov, A. L. Kuznetzov, and I. V. Samokhvalov, "Polarization sounding of high-altitude aerosol formations," Atmos. Opt. 4, 303-308 (1991).

Sassen, K.

K. Sassen and S. Benson, "A midlatitude cirrus cloud climatology from the facility for atmospheric remote sensing. II. Microphysical properties derived from lidar depolarization," J. Atmos. Sci. 58, 2103-2112 (2001).
[CrossRef]

K. Sassen, "Polarization in lidar," in Lidar, C. Weitkamp, ed. (Springer-Verlag, 2005), pp. 19-42.
[CrossRef]

Takano, Y.

Tsay, S. C.

P. Yang, B. C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, S. C. Tsay, D. M. Winker, and S. L. Nasiri, "Radiative properties of cirrus clouds in the infrared (8-13 μm) spectral region," J. Quant. Spectrosc. Radiat. Transfer 70, 473-504 (2001).
[CrossRef]

van de Hulst, H. C.

C. V. M. Van der Mee, J. W. Hovenier, and H. C. van de Hulst, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

Van der Mee, C. V. M.

C. V. M. Van der Mee, J. W. Hovenier, and H. C. van de Hulst, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).

Veal, D. L.

A. H. Auer and D. L. Veal, "The dimensions of ice crystals in natural clouds," J. Atmos. Sci. 26, 919-927 (1970).
[CrossRef]

Winker, D. M.

P. Yang, B. C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, S. C. Tsay, D. M. Winker, and S. L. Nasiri, "Radiative properties of cirrus clouds in the infrared (8-13 μm) spectral region," J. Quant. Spectrosc. Radiat. Transfer 70, 473-504 (2001).
[CrossRef]

Wiscombe, W. J.

P. Yang, B. C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, S. C. Tsay, D. M. Winker, and S. L. Nasiri, "Radiative properties of cirrus clouds in the infrared (8-13 μm) spectral region," J. Quant. Spectrosc. Radiat. Transfer 70, 473-504 (2001).
[CrossRef]

Yang, P.

P. Yang, B. C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, S. C. Tsay, D. M. Winker, and S. L. Nasiri, "Radiative properties of cirrus clouds in the infrared (8-13 μm) spectral region," J. Quant. Spectrosc. Radiat. Transfer 70, 473-504 (2001).
[CrossRef]

Appl. Opt. (2)

Astron. Astrophys. (1)

C. V. M. Van der Mee, J. W. Hovenier, and H. C. van de Hulst, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).

Atmos. Ocean. Opt. (1)

B. V. Kaul, "Lidar determination of oriented particles in crystal clouds," Atmos. Ocean. Opt. 8, 439-441 (1995).

Atmos. Opt. (1)

B. V. Kaul, O. A. Krasnov, A. L. Kuznetzov, and I. V. Samokhvalov, "Polarization sounding of high-altitude aerosol formations," Atmos. Opt. 4, 303-308 (1991).

J. Appl. Meteorol. (2)

C. M. R. Platt, "Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals," J. Appl. Meteorol. 17, 1220-1224 (1978).
[CrossRef]

V. E. Derr, N. L. Abshire, R. E. Cupp, and G. T. McNice, "Depolarization of lidar returns from virga and source cloud," J. Appl. Meteorol. 15, 1200-1208 (1976).
[CrossRef]

J. Atmos. Ocean. Technol. (1)

L. R. Bissonette and G. Roy, "Range-height scans of lidar depolarization for characterizing properties and phase of clouds and precipitation," J. Atmos. Ocean. Technol. 18, 1429-1446 (2001).
[CrossRef]

J. Atmos. Sci. (3)

A. H. Auer and D. L. Veal, "The dimensions of ice crystals in natural clouds," J. Atmos. Sci. 26, 919-927 (1970).
[CrossRef]

K. Sassen and S. Benson, "A midlatitude cirrus cloud climatology from the facility for atmospheric remote sensing. II. Microphysical properties derived from lidar depolarization," J. Atmos. Sci. 58, 2103-2112 (2001).
[CrossRef]

F. M. Bréon and B. Dubrulle, "Horizontally oriented plates in clouds," J. Atmos. Sci. 61, 2888-2898 (2004).
[CrossRef]

J. Geophys. Res. (1)

M. Del Guasta, "Simulation of lidar returns from ideal and deformed hexagonal ice prisms in cold cirrus by means of face tracing," J. Geophys. Res. 106, D12, 12,589-12,602 (2001).

J. Quant. Spectrosc. Radiat. Transfer (2)

P. Yang, B. C. Gao, B. A. Baum, Y. X. Hu, W. J. Wiscombe, S. C. Tsay, D. M. Winker, and S. L. Nasiri, "Radiative properties of cirrus clouds in the infrared (8-13 μm) spectral region," J. Quant. Spectrosc. Radiat. Transfer 70, 473-504 (2001).
[CrossRef]

H. Chepfer, G. Brogniez, P. Gouloub, F. M. Bréon, and P. H. Flamant, "Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1," J. Quant. Spectrosc. Radiat. Transfer 63, 521-543 (1999).
[CrossRef]

Opt. Lett. (2)

Other (2)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

K. Sassen, "Polarization in lidar," in Lidar, C. Weitkamp, ed. (Springer-Verlag, 2005), pp. 19-42.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Polarimetric lidar.

Fig. 2
Fig. 2

(a) Scattering geometry used. (b) Rotation of a horizontally oriented plate (I = 60, a = 80, AR = 0.37) about its axis (v) and the off-zenith lidar angle (Φ) leading to parallel (β p ) and perpendicular (β s ) backscatter when the plate is illuminated with vertical linear polarization. Units are square meters per steradian for β and degrees for the angles.

Fig. 3
Fig. 3

Simulations for ideal hexagonal oriented plates: (a) δl as a function of laser linear polarization direction α (AR = 0.22); δl for random (RND) orientation is also shown. (b) Simulated (δl, δ c ) points for α = 0 as obtained for several ARs and sizes. Open symbols show results for the same particles but in random orientation.

Fig. 4
Fig. 4

Deformed, oriented hexagonal plates: (a) linear depolarization δl as a function of the laser linear polarization direction α (AR = 0.22). δl for random (RND) orientation is also shown. (b) Simulated (δl, δ c ) points for α = 0 for several ARs and sizes. Open symbols show results for the same particles but in random orientation.

Fig. 5
Fig. 5

(a) Ordinary lidar profiles Ip and Is as obtained with linear laser polarization, and (δl, δ c ) for a sample cloud measurement. (b) Scatter plot of (δl, δ c ) for the entire 17 February 2003 cloud event.

Fig. 6
Fig. 6

(a) Ordinary lidar profiles Ip and Is as obtained with linear laser polarization, and δl and δ c depolarizations (Dep) for a sample measurement. (b) Scatter plot of (δl, δ c ) for the whole 4 November 2004 cloud event.

Fig. 7
Fig. 7

29 September 2005 cloud event. Top, left to right, ordinary parallel signal Ip and linear (δl) depolarization; bottom, circular (δ c ) depolarization and δl−δ c ∕(2 + δ c ) as an indicator of displacement from Eq. (12).

Fig. 8
Fig. 8

l, δ c ) scatter plot for the whole cloud event of 29 September 2004. The simulated points for ideal plates [Fig. 3(b)] are shown for comparison.

Tables (1)

Tables Icon

Table 1 Specifications of Our Lidar

Equations (19)

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

δ l = I s I p = I Q I + Q
v 1 = G 1 K R 2 [ 1 0 0 0 ] * M POL 1 * M FE * M DM * M ATM * S 0 ,
v 2 = G 2 K R 2 [ 1 0 0 0 ] * M POL 2 * M FE * M DM * M ATM * S 0 ,
M POL 1 = [ 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ] , PMT   1  channel       M POL 2 = [ 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ] PMT   2  channel .
M FE ON = [ 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ] , M FE OFF = [ 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ] .
M FE = [ 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ] , OFF–OFF–ON M FE = [ 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ] , OFF–OFF–OFF
M FE = [ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ] . ON–OFF–ON
Cells  OFF OFF O F F : G v 1 v 2 = I Q I + Q
Cells  ON OFF O N : G v 1 v 2 = I U I + U ,
Cells  OFF OFF O N : G v 1 v 2 = I V I + V ,
S I = [ 1 Q / I U / I V / I ] = [ 1 ( 1 G v 1 / v 2 ) ( 1 + G v 1 / v 2 ) ( OFF OFF OFF ) ( 1 G v 1 / v 2 ) ( 1 + G v 1 / v 2 ) ( ON OFF ON ) ( 1 G v 1 / v 2 ) ( 1 + G v 1 / v 2 ) ( OFF OFF ON ) ] .
δ c = v 2 G v 1 = I + V I V .
M ATM = [ a 1 0 0 0 0 a 2 0 0 0 0 a 2 0 0 0 0 a 1 2 a 2 ] .
δ c = 2 δ l 1 δ l .
M DM = [ 1 0 0 0 0 1 0 0 0 0 cos ( v ) sin ( v ) 0 0 sin ( v ) cos ( v ) ] ,
M FE i = [ 1 0 0 0 0 C 2 + S 2 cos ( γ i + π 2 ) C S [ 1 cos ( γ i + π 2 ) ] S sin ( γ i + π 2 ) 0 C S [ 1 cos ( γ i + π 2 ) ] S 2 + C 2 cos ( γ i + π 2 ) C sin ( γ i + π 2 ) 0 S sin ( γ i + π 2 ) C sin ( γ i + π 2 ) cos ( γ i + π 2 ) ] ,
[ C = cos [ 2 ( π 4 + d ω ) ] S = sin [ 2 ( π 4 + d ω ) ] ON   i th   cell ,
[ C = cos ( 2 d ω ) S = sin ( 2 d ω ) OFF   i th   cell .
M FE = [ 1 0 0 0 0 0.303 0.02 0.95 0 0.04 1 0.01 0 0.95 0.03 0.3 ] OFF–OFF–ON , [ 1 0 0 0 0 1 0.08 0.03 0 0.02 0.14 0.99 0 0.08 0.99 0.14 ] OFF–OFF–OFF , [ 1 0 0 0 0 0.36 0.88 0.30 0 0.93 0.31 0.19 0 0.07 0.35 0.93 ] ON–OFF–ON .

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