Abstract

Linear (spectro) polarimetry is usually performed using separate photon flux measurements after spatial or temporal polarization modulation. Such classical polarimeters are limited in sensitivity and accuracy by systematic effects and noise. We describe a spectral modulation principle that is based on encoding the full linear polarization properties of light in its spectrum. Such spectral modulation is obtained with an optical train of an achromatic quarter-wave retarder, an athermal multiple-order retarder, and a polarizer. The emergent spectral modulation is sinusoidal with its amplitude scaling with the degree of linear polarization and its phase scaling with the angle of linear polarization. The large advantage of this passive setup is that all polarization information is, in principle, contained in a single spectral measurement, thereby eliminating all differential effects that potentially create spurious polarization signals. Since the polarization properties are obtained through curve fitting, the susceptibility to noise is relatively low. We provide general design options for a spectral modulator and describe the design of a prototype modulator. Currently, the setup in combination with a dedicated retrieval algorithm can be used to measure linear polarization signals with a relative accuracy of 5%.

© 2009 Optical Society of America

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References

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  1. M. Semel, J.-F. Donati, and D. E. Rees, “Zeeman-Doppler imaging of active stars. 3: Instrumental and technical considerations,” Astron. Astrophys. 278, 231-237 (1993).
  2. K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24, 1475-1477 (1999).
    [CrossRef]
  3. F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric Spectral Intensity Modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474-481 (1999).
    [CrossRef]
  4. S. H. Jones, F. J. Iannarilli, and P. L. Kebabian, “Realization of quantitative-grade fieldable snapshot imaging spectropolarimeter,” Opt. Express 12, 6559-6573 (2004).
    [CrossRef] [PubMed]
  5. A. Taniguchi, K. Oka, H. Okabe, and M. Hayakawa, “Stabilization of a channeled spectropolarimeter by self-calibration,” Opt. Lett. 31, 3279-3281 (2006).
    [CrossRef] [PubMed]
  6. F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
    [CrossRef]
  7. J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, , “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed. Opt. 7, 341-349 (2002).
    [CrossRef] [PubMed]
  8. F. Cremer, W. de Jong, K. Schutte, J. T. Johnson, and B. A. Baertlein, “Surface mine signature modeling for passive polarimetric IR,” Proc. SPIE. 4742, 51-62 (2002).
    [CrossRef]
  9. R. Mayer, R. Priest, C. Steliman, G. Hazel, and A. Schaum, “Detection of camouflaged targets in cluttered backgrounds using fusion of near simultaneous spectral and polarimetric imaging,” Naval Research Lab Tech. Note ADA392956 (Naval Research Laboratory, 2000).
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    [CrossRef] [PubMed]
  13. S. Pancharatnam, “Achromatic combinations of birefringent plates. Part II. An achromatic quarter wave plate,” Proc. Indian Acad. Sci. Sect. A 42, 24-31 (1955).
  14. C. U. Keller, “Instrumentation for astrophysical spectropolarimetry,” in Astrophysical Spectropolarimetry, J. Trujillo-Bueno, F. Moreno-Insertis, and F. Sanchez, eds. (Cambridge University Press, 2002), pp. 303-354
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    [CrossRef]
  16. P. D. Hale and G. W. Day, “Stability of birefringent linear retarders (waveplates),” Appl. Opt. 27, 5146-5152 (1988).
    [CrossRef] [PubMed]
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    [CrossRef]
  20. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948-1950 (1996).
    [CrossRef] [PubMed]
  21. ArcOptix, “Radial polarization converter,” http://www.arcoptix.com/radial_polarization_converter.htm.
  22. F. Snik, “Astronomical applications for “radial polarimetry”,” in Astronomical Polarimetry 2008: Science from Small to Large Telescopes, P. Bastien and N. Manset, eds., ASP Conference Series (Astronomical Society of the Pacific, 2009).

2008 (1)

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

2006 (1)

2004 (1)

2002 (2)

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, , “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed. Opt. 7, 341-349 (2002).
[CrossRef] [PubMed]

F. Cremer, W. de Jong, K. Schutte, J. T. Johnson, and B. A. Baertlein, “Surface mine signature modeling for passive polarimetric IR,” Proc. SPIE. 4742, 51-62 (2002).
[CrossRef]

1999 (2)

F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric Spectral Intensity Modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474-481 (1999).
[CrossRef]

K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24, 1475-1477 (1999).
[CrossRef]

1996 (1)

1993 (1)

M. Semel, J.-F. Donati, and D. E. Rees, “Zeeman-Doppler imaging of active stars. 3: Instrumental and technical considerations,” Astron. Astrophys. 278, 231-237 (1993).

1988 (1)

1985 (1)

1971 (1)

1966 (1)

R. J. King, “Quarter-wave retardation systems based on the Fresnel rhomb principle,” J. Sci. Instrum. 43, 617-621(1966).
[CrossRef]

1955 (1)

S. Pancharatnam, “Achromatic combinations of birefringent plates. Part II. An achromatic quarter wave plate,” Proc. Indian Acad. Sci. Sect. A 42, 24-31 (1955).

1949 (1)

1944 (1)

B. Lyot, “Le filtre monochromatique polarisant et ses applications en physique solaire,” Ann. Astrophys. 7, 31-75 (1944).

Aas, C.

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

Baba, J. S.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, , “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed. Opt. 7, 341-349 (2002).
[CrossRef] [PubMed]

Baertlein, B. A.

F. Cremer, W. de Jong, K. Schutte, J. T. Johnson, and B. A. Baertlein, “Surface mine signature modeling for passive polarimetric IR,” Proc. SPIE. 4742, 51-62 (2002).
[CrossRef]

Beckers, J.

Biehl, L. L.

Cameron, B. D.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, , “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed. Opt. 7, 341-349 (2002).
[CrossRef] [PubMed]

Chung, J. R.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, , “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed. Opt. 7, 341-349 (2002).
[CrossRef] [PubMed]

Cote, G. L.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, , “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed. Opt. 7, 341-349 (2002).
[CrossRef] [PubMed]

Cremer, F.

F. Cremer, W. de Jong, K. Schutte, J. T. Johnson, and B. A. Baertlein, “Surface mine signature modeling for passive polarimetric IR,” Proc. SPIE. 4742, 51-62 (2002).
[CrossRef]

Day, G. W.

de Jong, W.

F. Cremer, W. de Jong, K. Schutte, J. T. Johnson, and B. A. Baertlein, “Surface mine signature modeling for passive polarimetric IR,” Proc. SPIE. 4742, 51-62 (2002).
[CrossRef]

de Vries, J.

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

DeLaughter, A. H.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, , “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed. Opt. 7, 341-349 (2002).
[CrossRef] [PubMed]

Donati, J.-F.

M. Semel, J.-F. Donati, and D. E. Rees, “Zeeman-Doppler imaging of active stars. 3: Instrumental and technical considerations,” Astron. Astrophys. 278, 231-237 (1993).

Elmore, D.

S. Guimond and D. Elmore, “Designing effective crystal waveplates requires understanding the engineering tradeoffs,” oemagazine (May 2004).

Evans, J. W.

Ghosh, G.

G. Ghosh, Handbook of Thermo-Optic Coefficients of Optical Materials with Applications (Academic, 1998).

Grant, L.

Guimond, S.

S. Guimond and D. Elmore, “Designing effective crystal waveplates requires understanding the engineering tradeoffs,” oemagazine (May 2004).

Hale, P. D.

Hayakawa, M.

Hazel, G.

R. Mayer, R. Priest, C. Steliman, G. Hazel, and A. Schaum, “Detection of camouflaged targets in cluttered backgrounds using fusion of near simultaneous spectral and polarimetric imaging,” Naval Research Lab Tech. Note ADA392956 (Naval Research Laboratory, 2000).

Hoogeveen, R.

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

Iannarilli, F. J.

S. H. Jones, F. J. Iannarilli, and P. L. Kebabian, “Realization of quantitative-grade fieldable snapshot imaging spectropolarimeter,” Opt. Express 12, 6559-6573 (2004).
[CrossRef] [PubMed]

F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric Spectral Intensity Modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474-481 (1999).
[CrossRef]

Johnson, J. T.

F. Cremer, W. de Jong, K. Schutte, J. T. Johnson, and B. A. Baertlein, “Surface mine signature modeling for passive polarimetric IR,” Proc. SPIE. 4742, 51-62 (2002).
[CrossRef]

Jones, S. H.

S. H. Jones, F. J. Iannarilli, and P. L. Kebabian, “Realization of quantitative-grade fieldable snapshot imaging spectropolarimeter,” Opt. Express 12, 6559-6573 (2004).
[CrossRef] [PubMed]

F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric Spectral Intensity Modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474-481 (1999).
[CrossRef]

Karalidi, T.

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

Kato, T.

Kebabian, P.

F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric Spectral Intensity Modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474-481 (1999).
[CrossRef]

Kebabian, P. L.

Keller, C. U.

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

C. U. Keller, “Instrumentation for astrophysical spectropolarimetry,” in Astrophysical Spectropolarimetry, J. Trujillo-Bueno, F. Moreno-Insertis, and F. Sanchez, eds. (Cambridge University Press, 2002), pp. 303-354

King, R. J.

R. J. King, “Quarter-wave retardation systems based on the Fresnel rhomb principle,” J. Sci. Instrum. 43, 617-621(1966).
[CrossRef]

Laan, E.

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

Lyot, B.

B. Lyot, “Le filtre monochromatique polarisant et ses applications en physique solaire,” Ann. Astrophys. 7, 31-75 (1944).

Mayer, R.

R. Mayer, R. Priest, C. Steliman, G. Hazel, and A. Schaum, “Detection of camouflaged targets in cluttered backgrounds using fusion of near simultaneous spectral and polarimetric imaging,” Naval Research Lab Tech. Note ADA392956 (Naval Research Laboratory, 2000).

Navarro, R.

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

Oka, K.

Okabe, H.

Oomen, G.

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, “Achromatic combinations of birefringent plates. Part II. An achromatic quarter wave plate,” Proc. Indian Acad. Sci. Sect. A 42, 24-31 (1955).

Priest, R.

R. Mayer, R. Priest, C. Steliman, G. Hazel, and A. Schaum, “Detection of camouflaged targets in cluttered backgrounds using fusion of near simultaneous spectral and polarimetric imaging,” Naval Research Lab Tech. Note ADA392956 (Naval Research Laboratory, 2000).

Rees, D. E.

M. Semel, J.-F. Donati, and D. E. Rees, “Zeeman-Doppler imaging of active stars. 3: Instrumental and technical considerations,” Astron. Astrophys. 278, 231-237 (1993).

Robinson, B. F.

Schadt, M.

Schaum, A.

R. Mayer, R. Priest, C. Steliman, G. Hazel, and A. Schaum, “Detection of camouflaged targets in cluttered backgrounds using fusion of near simultaneous spectral and polarimetric imaging,” Naval Research Lab Tech. Note ADA392956 (Naval Research Laboratory, 2000).

Schutte, K.

F. Cremer, W. de Jong, K. Schutte, J. T. Johnson, and B. A. Baertlein, “Surface mine signature modeling for passive polarimetric IR,” Proc. SPIE. 4742, 51-62 (2002).
[CrossRef]

Scott, H. E.

F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric Spectral Intensity Modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474-481 (1999).
[CrossRef]

Semel, M.

M. Semel, J.-F. Donati, and D. E. Rees, “Zeeman-Doppler imaging of active stars. 3: Instrumental and technical considerations,” Astron. Astrophys. 278, 231-237 (1993).

Snik, F.

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

F. Snik, “Astronomical applications for “radial polarimetry”,” in Astronomical Polarimetry 2008: Science from Small to Large Telescopes, P. Bastien and N. Manset, eds., ASP Conference Series (Astronomical Society of the Pacific, 2009).

Stalder, M.

Stam, D. M.

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

Steliman, C.

R. Mayer, R. Priest, C. Steliman, G. Hazel, and A. Schaum, “Detection of camouflaged targets in cluttered backgrounds using fusion of near simultaneous spectral and polarimetric imaging,” Naval Research Lab Tech. Note ADA392956 (Naval Research Laboratory, 2000).

Taniguchi, A.

ter Horst, R.

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

Vanderbilt, V. C.

Ann. Astrophys. (1)

B. Lyot, “Le filtre monochromatique polarisant et ses applications en physique solaire,” Ann. Astrophys. 7, 31-75 (1944).

Appl. Opt. (3)

Astron. Astrophys. (1)

M. Semel, J.-F. Donati, and D. E. Rees, “Zeeman-Doppler imaging of active stars. 3: Instrumental and technical considerations,” Astron. Astrophys. 278, 231-237 (1993).

J. Biomed. Opt. (1)

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, , “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed. Opt. 7, 341-349 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Sci. Instrum. (1)

R. J. King, “Quarter-wave retardation systems based on the Fresnel rhomb principle,” J. Sci. Instrum. 43, 617-621(1966).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Proc. Indian Acad. Sci. Sect. A (1)

S. Pancharatnam, “Achromatic combinations of birefringent plates. Part II. An achromatic quarter wave plate,” Proc. Indian Acad. Sci. Sect. A 42, 24-31 (1955).

Proc. SPIE (1)

F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric Spectral Intensity Modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474-481 (1999).
[CrossRef]

Proc. SPIE. (2)

F. Snik, T. Karalidi, C. U. Keller, E. Laan, R. ter Horst, R. Navarro, D. M. Stam, C. Aas, J. de Vries, G. Oomen, and R. Hoogeveen, “SPEX, An in-orbit spectropolarimeter for planetary exploration,” Proc. SPIE. 7010, 701015 (2008).
[CrossRef]

F. Cremer, W. de Jong, K. Schutte, J. T. Johnson, and B. A. Baertlein, “Surface mine signature modeling for passive polarimetric IR,” Proc. SPIE. 4742, 51-62 (2002).
[CrossRef]

Other (6)

R. Mayer, R. Priest, C. Steliman, G. Hazel, and A. Schaum, “Detection of camouflaged targets in cluttered backgrounds using fusion of near simultaneous spectral and polarimetric imaging,” Naval Research Lab Tech. Note ADA392956 (Naval Research Laboratory, 2000).

ArcOptix, “Radial polarization converter,” http://www.arcoptix.com/radial_polarization_converter.htm.

F. Snik, “Astronomical applications for “radial polarimetry”,” in Astronomical Polarimetry 2008: Science from Small to Large Telescopes, P. Bastien and N. Manset, eds., ASP Conference Series (Astronomical Society of the Pacific, 2009).

C. U. Keller, “Instrumentation for astrophysical spectropolarimetry,” in Astrophysical Spectropolarimetry, J. Trujillo-Bueno, F. Moreno-Insertis, and F. Sanchez, eds. (Cambridge University Press, 2002), pp. 303-354

S. Guimond and D. Elmore, “Designing effective crystal waveplates requires understanding the engineering tradeoffs,” oemagazine (May 2004).

G. Ghosh, Handbook of Thermo-Optic Coefficients of Optical Materials with Applications (Academic, 1998).

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic setup of the spectral modulator. The solid and dashed lines of the retarders represent the (orthogonal) fast and slow axis, respectively. (b) Illustration of the spectral modulation principle on the Poincaré sphere.

Fig. 2
Fig. 2

Design of a retroreflecting quarter-wave retarder prism based on 3 TIRs in a fused silica rhomb. The path of light with an angle of 2 ° from normal incidence is indicated in light gray.

Fig. 3
Fig. 3

(a) Thermal test results for the combination of MgF 2 (with variable thickness) and sapphire as a function of wavelength ( 550 750 nm ). Various thickness ratios ( MgF 2 :   Al 2 O 3 ) from 1.7 to 3.7 have been measured. The vertical axis represents the relative variation of the overall retardance of the crystal combination with temperature. The crosses represent the determined zero crossings of the data for a certain wavelength. The dashed line represents the thickness ratio 1 : 2.2 . (b) The corresponding theoretical curve with literature values for the thermo-optic constants. The dashed line here represents the zero crossing of the curve.

Fig. 4
Fig. 4

The upper panel shows a modulated spectrum obtained with the prototype described in the text (solid curve). The dotted curve represents the unmodulated intensity spectrum obtained after adding the signals from two orthogonal orientations of the polarizer. The normalized modulation signal is shown below. The lower panel shows the results for P L ( λ ) and ϕ L ( λ ) as obtained by the retrieval algorithm. Offsets from the ideal values of P L ( λ ) ( = 1 ) and ϕ L ( λ ) (constant) are clearly observed. The solid curve corresponds to the signal shown in the upper panel using the dual beam method, whereas the dotted curve shows the results from the single beam method ( ϕ L is the same in both cases). The dashed curves represent rotations of the input polarization in steps of 45 ° , which are reproduced by the ϕ L measurements. The dashed curve for P L ( λ ) represents the measurement with the input polarization direction coinciding with the K-prism axis, for which the measurement efficiency is significantly higher than for the situation with input at 45 ° from that (solid curves).

Equations (18)

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s 1 = t ( I + Q ) , s 2 = t ( I Q ) , s 3 = t ( I + U ) , s 4 = t ( I U ) ,
P L = Q 2 + U 2 I ,
ϕ L = 1 2 arctan U Q .
S ( λ ) = 1 2 s 0 ( λ ) × [ 1 ± P L ( λ ) cos ( 2 π · δ ( λ , T ) λ + 2 · ϕ L ( λ ) ) ] ,
s 0 ( λ ) = I 0 ( λ ) t ( λ ) ,
δ TIR = 2   arctan ( cos θ n ( λ ) sin 2 θ 1 n ( λ ) sin 2 θ ) .
θ c < α , β , γ < 90 ° ,
α + β + γ = 180 ° ;
δ TIR ( α , λ 0 ) + δ TIR ( β , λ 0 ) + δ TIR ( γ , λ 0 ) = π 2 ;
d δ TIR ( α , λ 0 ) d θ d δ TIR ( β , λ 0 ) d θ + d δ TIR ( γ , λ 0 ) d θ = 0.
δ ( λ , T ) = δ 1 ( λ , T ) ± δ 2 ( λ , T ) = | n e , 1 ( λ , T ) n o , 1 ( λ , T ) | d 1 ± | n e , 2 ( λ , T ) n o , 2 ( λ , T ) | d 2 ,
γ 1 δ 1 ( λ 0 ) ± γ 2 δ 2 ( λ 0 ) = 0 ,
γ i = 1 δ i ( λ 0 ) d δ i ( λ 0 ) d T = 1 d i d d i d T + 1 n e , i ( λ 0 ) n o , i ( λ 0 ) d ( n e , i ( λ 0 ) n o , i ( λ 0 ) ) d T .
δ ( λ , ζ , η ) = δ 1 ( λ ) [ 1 + ζ 2 2 n o , 1 ( cos 2 η n e , 1 sin 2 η n o , 1 ) ] δ 2 ( λ ) [ 1 + ζ 2 2 n o , 2 ( sin 2 η n e , 2 cos 2 η n o , 2 ) ] | n e , 2 n o , 2 > 0 ; δ 2 ( λ ) [ 1 + ζ 2 2 n o , 2 ( cos 2 η n e , 2 sin 2 η n e , 2 ) ] | n e , 2 n o , 2 < 0.
ϕ L , meas = ϕ L , 0 + π λ d δ d T Δ T .
δ = k λ ( k 1 2 ) ( λ + Δ λ 2 ) ( k + 1 2 ) ( λ Δ λ 2 ) ;
Δ λ = λ 2 δ ( 1 + λ 2 / 4 δ 2 ) .
ϕ L , est = π · λ ϕ Δ λ ( λ ) .

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