Abstract

The plane of polarization of two or more wavelengths can be rotated by a predetermined amount through the use of two or more identical half-wave plates in series, the axes of which are oriented at predetermined angles with respect to the incident plane of polarization. For two wavelengths a rotation of 90° may be accomplished by the use of two plates with their slow axes at angles of 22.5°+δ and 67.5°−δ, respectively. For three wavelengths a 90° rotation is obtained by using three plates at angles of 11.25°+δ, 45°, and 78.75°−δ respectively. The quantity δ is a small angle usually less than 1° which determines the spectral range of achromatization. Identical half-wave plates are easily obtained by cutting a single splitting of mica or a plane parallel sheet of other birefringent material. The wavelength for which the plates have half-wave retardation is not critical. Rather than having a smaller angular aperture than the single half-wave plate, the three-element achromatic rotator has a larger angular aperture. The use of the stereographic projection of the Poincaré sphere for graphical solution of polarized light problems is discussed briefly.

© 1959 Optical Society of America

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References

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  1. See, for example, A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw-Hill Book Company, Inc., New York, 1932), p. 610.
  2. S. Inoué and W. L. Hyde, J. Biophys. Biochem. Cytol. 3, 831 (1957).
  3. A. A. Lebedeff, Rev. opt. 9, 385 (1930), and F. H. Smith, Research (London) 8, 385 (1955).
  4. M. Francon and B. Sergent, Compt. rend. 241, 27 (1955); M. Francon, Optica Acta (Paris) 2, 182 (1955).
    [Crossref]
  5. C. D. West and A. S. Makas [J. Opt. Soc. Am. 39, 791 (1949)], describe some achromatic plates and give references to earlier work.
    [Crossref] [PubMed]
  6. M. P. Lostis, J. phys. radium 18, 51S (1957).
    [Crossref]
  7. M. G. Destriau and J. Prouteau, J. phys. radium 10, 53 (1949).
    [Crossref]
  8. S. Pancharatnam, Proc. Indian Acad. Sci. A41, 130 (1955).
  9. S. Pancharatnam, Proc. Indian Acad. Sci. A41, 137 (1955).
  10. R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941).
    [Crossref]
  11. H. Poincaré, Théorie mathématique de la lumière II (Paris, 1892), Chap. 12.
  12. A good discussion of the Poincaré sphere is given by H. G. Jerrard, J. Opt. Soc. Am. 44, 634 (1954).
    [Crossref]
  13. G. N. Ramachandran and V. Chandrasekharan, Proc. Indian Acad. Sci. A33, 199 (1951). See also G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
    [Crossref]
  14. More information on the stereographic projection is contained in the monograph: W. W. Flexner and G. L. Walker, Military and Naval Maps and Grids (Dryden Press, Inc., New York, 1942). A classical reference is S. L. Penfield, Am. J. Sci. 11, 1, 115 (1901).
    [Crossref]
  15. The Wulff net is described in N. H. Hartshorne and A. Stuart, Crystals and the Polarizing Microscope (Edward Arnold and Company, London, 1952), second edition, p. 35.Wulff net coordinate paper is currently available from the University of Toronto Press, and transparent plastic protractors can be obtained from Ward’s Natural Science Establishment, Inc., Rochester 9, New York, and from N. P. Nies, 969 Skyline Drive, Laguna Beach, California.

1957 (2)

S. Inoué and W. L. Hyde, J. Biophys. Biochem. Cytol. 3, 831 (1957).

M. P. Lostis, J. phys. radium 18, 51S (1957).
[Crossref]

1955 (3)

S. Pancharatnam, Proc. Indian Acad. Sci. A41, 130 (1955).

S. Pancharatnam, Proc. Indian Acad. Sci. A41, 137 (1955).

M. Francon and B. Sergent, Compt. rend. 241, 27 (1955); M. Francon, Optica Acta (Paris) 2, 182 (1955).
[Crossref]

1954 (1)

1951 (1)

G. N. Ramachandran and V. Chandrasekharan, Proc. Indian Acad. Sci. A33, 199 (1951). See also G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
[Crossref]

1949 (2)

1941 (1)

1930 (1)

A. A. Lebedeff, Rev. opt. 9, 385 (1930), and F. H. Smith, Research (London) 8, 385 (1955).

Chandrasekharan, V.

G. N. Ramachandran and V. Chandrasekharan, Proc. Indian Acad. Sci. A33, 199 (1951). See also G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
[Crossref]

Destriau, M. G.

M. G. Destriau and J. Prouteau, J. phys. radium 10, 53 (1949).
[Crossref]

Flexner, W. W.

More information on the stereographic projection is contained in the monograph: W. W. Flexner and G. L. Walker, Military and Naval Maps and Grids (Dryden Press, Inc., New York, 1942). A classical reference is S. L. Penfield, Am. J. Sci. 11, 1, 115 (1901).
[Crossref]

Francon, M.

M. Francon and B. Sergent, Compt. rend. 241, 27 (1955); M. Francon, Optica Acta (Paris) 2, 182 (1955).
[Crossref]

Hardy, A. C.

See, for example, A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw-Hill Book Company, Inc., New York, 1932), p. 610.

Hartshorne, N. H.

The Wulff net is described in N. H. Hartshorne and A. Stuart, Crystals and the Polarizing Microscope (Edward Arnold and Company, London, 1952), second edition, p. 35.Wulff net coordinate paper is currently available from the University of Toronto Press, and transparent plastic protractors can be obtained from Ward’s Natural Science Establishment, Inc., Rochester 9, New York, and from N. P. Nies, 969 Skyline Drive, Laguna Beach, California.

Hyde, W. L.

S. Inoué and W. L. Hyde, J. Biophys. Biochem. Cytol. 3, 831 (1957).

Inoué, S.

S. Inoué and W. L. Hyde, J. Biophys. Biochem. Cytol. 3, 831 (1957).

Jerrard, H. G.

Jones, R. C.

Lebedeff, A. A.

A. A. Lebedeff, Rev. opt. 9, 385 (1930), and F. H. Smith, Research (London) 8, 385 (1955).

Lostis, M. P.

M. P. Lostis, J. phys. radium 18, 51S (1957).
[Crossref]

Makas, A. S.

Pancharatnam, S.

S. Pancharatnam, Proc. Indian Acad. Sci. A41, 130 (1955).

S. Pancharatnam, Proc. Indian Acad. Sci. A41, 137 (1955).

Perrin, F. H.

See, for example, A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw-Hill Book Company, Inc., New York, 1932), p. 610.

Poincaré, H.

H. Poincaré, Théorie mathématique de la lumière II (Paris, 1892), Chap. 12.

Prouteau, J.

M. G. Destriau and J. Prouteau, J. phys. radium 10, 53 (1949).
[Crossref]

Ramachandran, G. N.

G. N. Ramachandran and V. Chandrasekharan, Proc. Indian Acad. Sci. A33, 199 (1951). See also G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
[Crossref]

Sergent, B.

M. Francon and B. Sergent, Compt. rend. 241, 27 (1955); M. Francon, Optica Acta (Paris) 2, 182 (1955).
[Crossref]

Stuart, A.

The Wulff net is described in N. H. Hartshorne and A. Stuart, Crystals and the Polarizing Microscope (Edward Arnold and Company, London, 1952), second edition, p. 35.Wulff net coordinate paper is currently available from the University of Toronto Press, and transparent plastic protractors can be obtained from Ward’s Natural Science Establishment, Inc., Rochester 9, New York, and from N. P. Nies, 969 Skyline Drive, Laguna Beach, California.

Walker, G. L.

More information on the stereographic projection is contained in the monograph: W. W. Flexner and G. L. Walker, Military and Naval Maps and Grids (Dryden Press, Inc., New York, 1942). A classical reference is S. L. Penfield, Am. J. Sci. 11, 1, 115 (1901).
[Crossref]

West, C. D.

Compt. rend. (1)

M. Francon and B. Sergent, Compt. rend. 241, 27 (1955); M. Francon, Optica Acta (Paris) 2, 182 (1955).
[Crossref]

J. Biophys. Biochem. Cytol. (1)

S. Inoué and W. L. Hyde, J. Biophys. Biochem. Cytol. 3, 831 (1957).

J. Opt. Soc. Am. (3)

J. phys. radium (2)

M. P. Lostis, J. phys. radium 18, 51S (1957).
[Crossref]

M. G. Destriau and J. Prouteau, J. phys. radium 10, 53 (1949).
[Crossref]

Proc. Indian Acad. Sci. (3)

S. Pancharatnam, Proc. Indian Acad. Sci. A41, 130 (1955).

S. Pancharatnam, Proc. Indian Acad. Sci. A41, 137 (1955).

G. N. Ramachandran and V. Chandrasekharan, Proc. Indian Acad. Sci. A33, 199 (1951). See also G. N. Ramachandran and S. Ramaseshan, J. Opt. Soc. Am. 42, 49 (1952).
[Crossref]

Rev. opt. (1)

A. A. Lebedeff, Rev. opt. 9, 385 (1930), and F. H. Smith, Research (London) 8, 385 (1955).

Other (4)

H. Poincaré, Théorie mathématique de la lumière II (Paris, 1892), Chap. 12.

More information on the stereographic projection is contained in the monograph: W. W. Flexner and G. L. Walker, Military and Naval Maps and Grids (Dryden Press, Inc., New York, 1942). A classical reference is S. L. Penfield, Am. J. Sci. 11, 1, 115 (1901).
[Crossref]

The Wulff net is described in N. H. Hartshorne and A. Stuart, Crystals and the Polarizing Microscope (Edward Arnold and Company, London, 1952), second edition, p. 35.Wulff net coordinate paper is currently available from the University of Toronto Press, and transparent plastic protractors can be obtained from Ward’s Natural Science Establishment, Inc., Rochester 9, New York, and from N. P. Nies, 969 Skyline Drive, Laguna Beach, California.

See, for example, A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw-Hill Book Company, Inc., New York, 1932), p. 610.

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Figures (4)

Fig. 1
Fig. 1

(a) The two-element achromatic rotator. θ1 and θ2 are the azimuths of the slow axes of the two half-wave plates. (b) The three-element achromatic rotator. θ1′, θ2′, and θ3′ are defined as θ1 and θ2 above. OA is the azimuth of the plane of polarization of the incident light, OB is that of the emerging light.

Fig. 2
Fig. 2

A model of the Poincaré sphere. The necessary motions are obtained by rotation of the sphere about the vertical axis and the rotation of a perforated ring about a fixed horizontal axis. A pencil point inserted in a hole in the perforated ring will trace the action of a birefringent plate. In the right half of the picture a mirror is positioned to permit the side view of the sphere to be photographed simultaneously. Shown by heavy black lines are the actions of the two plates of an achromatic rotator with δ=1°. The point P represents the plane of polarization of the incident light, A and B the slow axes of the two wave-plates, and Q the final plane of polarization of the wavelengths λ1 and λ2.

Fig. 3
Fig. 3

The action of the three-element achromatic 90° rotator portrayed on a stereographic map of the Poincaré sphere. The complete circle is the equatorial circle. The essential features of stereographic projection are given in the text.

Fig. 4
Fig. 4

Photographs taken through (a) a single wave plate, (b) a two-element achromatic 90° rotator, and (c) a three-element achromatic 90° rotator, each between parallel Polaroid sheet polarizers. The darker the area, the more effective is the rotator for the corresponding angular zone. The exposures were four seconds each on a single roll of Tri-X film which was developed for four minutes in D-19. The printing of the three negatives was done simultaneously on a paper of number 1 contrast. The circles were at 5°, 10°, and 15° to the axis of the system, measured from the camera objective. Half-wave plates of mica were used. In (a) the characteristic biaxial pattern can be seen. In (c) the pattern is complex, but the angular aperture is greater than in either (a) or (b).

Tables (2)

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Table I The achromatic 90° rotators.

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Table II Orientations of plates for a desired rotation of α.a

Equations (11)

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θ 1 = 22.5° + δ
θ 2 = 67.5° - δ ,
θ 1 = 11.25° + δ
θ 2 = 45°
θ 3 = 78.75° - δ
θ 1 = 7.5° + δ ,
θ 2 = 30° ,
θ 3 = 60° ,
θ 4 = 82.5° - δ .
2 element :             cos Δ = 1 - [ sin 2 2 θ 1 ] - 1 ,
3 element :             cos Δ = 1 - [ sin 2 θ 1 ( 1 + sin 2 θ 1 ) ] - 1 .