Abstract

By a combination of quarter-wave plates made of different birefringent materials it is possible to produce achromatic quarter-wave plates whose degree of achromatism is dependant on the dispersions of birefringence and on the thicknesses of the individual quarter-wave plates. These waveplates are widely used in optical instrumentation and the residual errors associated with these devices can be very important in high resolution spectro-polarimetry measurements. The misalignment of optic axis in a double crystal waveplate is one of the main source of error and leads to elliptical eigenpolarization modes in the retarder and the oscillation of its orientation according to the wavelength. This paper will discuss, first, how the characteristics of a quartz-MgF2 quarter-wave plate is affected by such a misalignment. A correlation with the experiment is then achieved in order to highlight the interest of taking a possible tilt error into consideration when doing polarimetric measurements.

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References

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  1. A. M. Title, "Improvement of birefringent filters. 2: Achromatic waveplates," Appl. Opt. 14, 229-237 (1975).
    [PubMed]
  2. P. Hariharan, "Broad-band superachromatic retarders," Meas. Sci. Technol. 9, 1978-1980 (1998).
    [CrossRef]
  3. M. G. Destriau and J. Prouteau, "R?alisation d'un quart d'onde quasi achromatique par juxtaposition de deux lames cristallines de m?me nature," J. Phys. Radium 10, 53-55 (1949).
    [CrossRef]
  4. S. Pancharatnam, "Achromatic combinations of birefringent plates," Proc. Indian. Acad. Sci. 41, 137-144 (1955).
  5. K. Serkowski, Methods of experimental physics vol 12, (N. Carleton, New York, 1974).
  6. V. Chandrasekharan and H. Damany, "Birefringence of sapphire, magnesium fluoride, and quartz in the vacuum ultraviolet, and retardation plates," Appl. Opt. 7, 939-941 (1968).
    [CrossRef] [PubMed]
  7. J. M. Beckers, "Achromatic linear retarders," Appl. Opt. 10, 973-975 (1971).
    [CrossRef] [PubMed]
  8. E. A. West and M. H. Smith, "Polarization errors associated with birefringent waveplates," Proc. SPIE 2265, 260-271 (1994) (San Diego, USA).
    [CrossRef]
  9. J. M. Bennet, Handbook of optics, (Mc Graw-Hill, New York, 1995).
  10. 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]
  11. I. Scierski and F. Ratajczyk, "The Jones matrix of the real dichroic elliptic object," Optik 68, 121-125 (1984).
  12. B. H. Kolner, "Intensity noise enhancement in the half-wave plate/polarizer attenuator," Opt. Lett. 16, 784-786 (1991).
    [CrossRef] [PubMed]
  13. J.F.Nye,Physical Properties of Cristals, (Clarendon, Oxford, 1957).
  14. A. Yariv and P. Yeh, Optical waves in crystals, (J. Wiley and sons, New York, 1984).

Other (14)

A. M. Title, "Improvement of birefringent filters. 2: Achromatic waveplates," Appl. Opt. 14, 229-237 (1975).
[PubMed]

P. Hariharan, "Broad-band superachromatic retarders," Meas. Sci. Technol. 9, 1978-1980 (1998).
[CrossRef]

M. G. Destriau and J. Prouteau, "R?alisation d'un quart d'onde quasi achromatique par juxtaposition de deux lames cristallines de m?me nature," J. Phys. Radium 10, 53-55 (1949).
[CrossRef]

S. Pancharatnam, "Achromatic combinations of birefringent plates," Proc. Indian. Acad. Sci. 41, 137-144 (1955).

K. Serkowski, Methods of experimental physics vol 12, (N. Carleton, New York, 1974).

V. Chandrasekharan and H. Damany, "Birefringence of sapphire, magnesium fluoride, and quartz in the vacuum ultraviolet, and retardation plates," Appl. Opt. 7, 939-941 (1968).
[CrossRef] [PubMed]

J. M. Beckers, "Achromatic linear retarders," Appl. Opt. 10, 973-975 (1971).
[CrossRef] [PubMed]

E. A. West and M. H. Smith, "Polarization errors associated with birefringent waveplates," Proc. SPIE 2265, 260-271 (1994) (San Diego, USA).
[CrossRef]

J. M. Bennet, Handbook of optics, (Mc Graw-Hill, New York, 1995).

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]

I. Scierski and F. Ratajczyk, "The Jones matrix of the real dichroic elliptic object," Optik 68, 121-125 (1984).

B. H. Kolner, "Intensity noise enhancement in the half-wave plate/polarizer attenuator," Opt. Lett. 16, 784-786 (1991).
[CrossRef] [PubMed]

J.F.Nye,Physical Properties of Cristals, (Clarendon, Oxford, 1957).

A. Yariv and P. Yeh, Optical waves in crystals, (J. Wiley and sons, New York, 1984).

Supplementary Material (1)

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Figures (13)

Figure. 1.
Figure. 1.

Simulation of the retardance (degrees) of a quartz-MgF2 quarter-wave plate according to the wavelength (nm), the thicknesses of the quartz and the MgF2 are 239.1 µm and 197.1 µm respectively.

Figure 2.
Figure 2.

Schematic of the misalignment : (F1, S1) represent the fast and slow axes of the quartz, (F2, S2) the fast and slow axes of the MgF2 and (F, S) those of the quartz-MgF2 quarter-wave plate.

Figure 3.
Figure 3.

Parameters of an elliptical state of polarization J.

Figure 4.
Figure 4.

Ellipticity of the eigenvectors of a quartz-MgF2 quarter-wave plate versus wavelength and for different values of the tilt between the two plates.

Figure 5.
Figure 5.

Azimuth of the eigenvector J1 calculated for several tilt errors and according to the wavelength.

Figure 6.
Figure 6.

Calculated retardance of a 239.1 µm quartz plate divided by 2π (red curve, bellow) and those of a 197.1 µmMgF2 plate divided by 2π (blue curve, above) as a function of the wavelength.

Figure 7.
Figure 7.

Calculated ellipticity (degrees) of the eigenpolarization modes (blue curve, above) and the orientation of the fast axis (red curve, bellow) of the retarder with a tilt error of 0.72°.

Figure 8.
Figure 8.

(54Ko) Eigenpolarization modes of a quartz-MgF2 quarter-wave plate (9Ko version). Colors are in respect with the wavelength of the incident light.

Figure 9.
Figure 9.

Schematic layout of the experiment. P1 and P2 are Glan-polarizers. L is the quartz-MgF2 quarter-wave plate.

Figure 10.
Figure 10.

Intensity detected versus the azimuth-angle α (degrees) of an elliptic birefringent object between two orthogonal polarizers and the ellipticity ε of its eigenpolarization modes.

Figure 11.
Figure 11.

Experimental set-up.

Figure 12.
Figure 12.

Experimental azimuth (blue curve) and theoretical azimuth calculated for a misalignment of 0.72° and thicknesses of 239.1 µm for the quartz plate and 197.1 µm for the MgF2 plate (red curve).

Figure 13.
Figure 13.

Experimental azimuth (blue curve) and the best fit obtained for a misalignment of 0.78° and thicknesses of 242 µm for the quartz plate and 172 µm for the MgF2 plate (red curve).

Tables (1)

Tables Icon

Tableau 1. Characteristics of the retarder calculated by fitting experimental and calculated values on different spectral range.

Equations (11)

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λ 1 4 = d q Δ n q ( λ 1 ) d m Δ n m ( λ 1 )
λ 2 4 = d q Δ n q ( λ 2 ) d m Δ n m ( λ 2 )
[ M ] = [ R ( θ ) ] [ M m ] [ R ( θ ) ] [ M q ]
[ R ( θ ) ] = [ cos θ sin θ sin θ cos θ ]
[ M q ] = [ 1 0 0 e i δ q ]
[ M m ] = [ 1 0 0 e i δ m ]
δ q = 2 π λ Δ n q d q
δ m = 2 π λ Δ n m d m
J 1 = [ a b + ic ] , J 2 = [ b + ic a ]
sin ( ε ) = sin ( 2 υ ) sin ( ϕ )
tan 2 α = 2 a b a 2 b 2 c 2

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