Abstract

The most achromatic quarterwave plates appear to be rhomb-type devices. General relations are given for calculating the phase retardation of these devices and are used to determine the variation of the phase retardation with wavelength and with acceptance angle for different rhomb designs. Length-to-aperture ratio, beam deviation, surface coatings, effect of strain birefringence, and other parameters are also considered. The classical version of the Fresnel rhomb can be optimized to improve its performance, and several of the more recent rhomb designs have excellent characteristics. The choice of which type of rhomb to use depends on the requirements of the particular optical system.

© 1970 Optical Society of America

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References

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  1. C. D. West, A. S. Makas, J. Opt. Soc. Amer. 39, 791 (1949).
    [CrossRef]
  2. M. Françon, S. Mallick, J. Vulmière, J. Opt. Soc. Amer. 55, 1553 (1965).
    [CrossRef]
  3. G. Destriau, J. Prouteau, J. Phys. Radium 10, (8) 53 (1949).
    [CrossRef]
  4. S. Pancharatnam, Proc. Indian Acad. Sci. A41, 130, 137 (1955).
  5. C. M. McIntyre, S. E. Harris, J. Opt. Soc. Amer. 58, 1575 (1968).
    [CrossRef]
  6. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, N.Y., 1957), pp. 517–518.
  7. R. J. King, J. Sci. Instrum. 43, 617 (1966).
    [CrossRef]
  8. P. B. Clapham, M. J. Downs, R. J. King, Appl. Opt. 8, 1965 (1969).
    [CrossRef] [PubMed]
  9. F. Mooney, J. Opt. Soc. Amer. 42, 181 (1952).
    [CrossRef]
  10. A. E. Oxley, Phil. Mag. 21, (6) 517 (1911).
  11. V. A. Kizel, Yu. I. Krasilov, V. N. Shamraev, Opt. Spectrosc. 17, 248 (1964).
  12. Yu. I. Krasilov, Opt. Spectrosc. 22, 267 (1967).
  13. See Ref. 6, p. 515.
  14. G. A. Harle, Nature 166, 149 (1950).
    [CrossRef] [PubMed]
  15. F. Abelès, Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1963), Vol. 2, pp. 252–253.
    [CrossRef]
  16. W. Brouwer, American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, N.Y., 1963), p. 6–84.
  17. E. D. Palik, Appl. Opt. 2, 527 (1963).
    [CrossRef]
  18. W. L. Wolfe, S. S. Ballard, K. A. McCarthy, American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, N.Y., 1963), pp. 6–22 and 6–36.
  19. R. J. King, M. J. Downs, Proceedings of the Symposium on Recent Developments in Ellipsometry, Surface Sci. 16, 288 (1969).
    [CrossRef]
  20. H. E. Bennett, J. M. Bennett, Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, N.Y., 1967), Vol. 4, pp. 79–81.

1969 (2)

R. J. King, M. J. Downs, Proceedings of the Symposium on Recent Developments in Ellipsometry, Surface Sci. 16, 288 (1969).
[CrossRef]

P. B. Clapham, M. J. Downs, R. J. King, Appl. Opt. 8, 1965 (1969).
[CrossRef] [PubMed]

1968 (1)

C. M. McIntyre, S. E. Harris, J. Opt. Soc. Amer. 58, 1575 (1968).
[CrossRef]

1967 (1)

Yu. I. Krasilov, Opt. Spectrosc. 22, 267 (1967).

1966 (1)

R. J. King, J. Sci. Instrum. 43, 617 (1966).
[CrossRef]

1965 (1)

M. Françon, S. Mallick, J. Vulmière, J. Opt. Soc. Amer. 55, 1553 (1965).
[CrossRef]

1964 (1)

V. A. Kizel, Yu. I. Krasilov, V. N. Shamraev, Opt. Spectrosc. 17, 248 (1964).

1963 (1)

1955 (1)

S. Pancharatnam, Proc. Indian Acad. Sci. A41, 130, 137 (1955).

1952 (1)

F. Mooney, J. Opt. Soc. Amer. 42, 181 (1952).
[CrossRef]

1950 (1)

G. A. Harle, Nature 166, 149 (1950).
[CrossRef] [PubMed]

1949 (2)

C. D. West, A. S. Makas, J. Opt. Soc. Amer. 39, 791 (1949).
[CrossRef]

G. Destriau, J. Prouteau, J. Phys. Radium 10, (8) 53 (1949).
[CrossRef]

1911 (1)

A. E. Oxley, Phil. Mag. 21, (6) 517 (1911).

Abelès, F.

F. Abelès, Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1963), Vol. 2, pp. 252–253.
[CrossRef]

Ballard, S. S.

W. L. Wolfe, S. S. Ballard, K. A. McCarthy, American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, N.Y., 1963), pp. 6–22 and 6–36.

Bennett, H. E.

H. E. Bennett, J. M. Bennett, Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, N.Y., 1967), Vol. 4, pp. 79–81.

Bennett, J. M.

H. E. Bennett, J. M. Bennett, Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, N.Y., 1967), Vol. 4, pp. 79–81.

Brouwer, W.

W. Brouwer, American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, N.Y., 1963), p. 6–84.

Clapham, P. B.

Destriau, G.

G. Destriau, J. Prouteau, J. Phys. Radium 10, (8) 53 (1949).
[CrossRef]

Downs, M. J.

P. B. Clapham, M. J. Downs, R. J. King, Appl. Opt. 8, 1965 (1969).
[CrossRef] [PubMed]

R. J. King, M. J. Downs, Proceedings of the Symposium on Recent Developments in Ellipsometry, Surface Sci. 16, 288 (1969).
[CrossRef]

Françon, M.

M. Françon, S. Mallick, J. Vulmière, J. Opt. Soc. Amer. 55, 1553 (1965).
[CrossRef]

Harle, G. A.

G. A. Harle, Nature 166, 149 (1950).
[CrossRef] [PubMed]

Harris, S. E.

C. M. McIntyre, S. E. Harris, J. Opt. Soc. Amer. 58, 1575 (1968).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, N.Y., 1957), pp. 517–518.

King, R. J.

P. B. Clapham, M. J. Downs, R. J. King, Appl. Opt. 8, 1965 (1969).
[CrossRef] [PubMed]

R. J. King, M. J. Downs, Proceedings of the Symposium on Recent Developments in Ellipsometry, Surface Sci. 16, 288 (1969).
[CrossRef]

R. J. King, J. Sci. Instrum. 43, 617 (1966).
[CrossRef]

Kizel, V. A.

V. A. Kizel, Yu. I. Krasilov, V. N. Shamraev, Opt. Spectrosc. 17, 248 (1964).

Krasilov, Yu. I.

Yu. I. Krasilov, Opt. Spectrosc. 22, 267 (1967).

V. A. Kizel, Yu. I. Krasilov, V. N. Shamraev, Opt. Spectrosc. 17, 248 (1964).

Makas, A. S.

C. D. West, A. S. Makas, J. Opt. Soc. Amer. 39, 791 (1949).
[CrossRef]

Mallick, S.

M. Françon, S. Mallick, J. Vulmière, J. Opt. Soc. Amer. 55, 1553 (1965).
[CrossRef]

McCarthy, K. A.

W. L. Wolfe, S. S. Ballard, K. A. McCarthy, American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, N.Y., 1963), pp. 6–22 and 6–36.

McIntyre, C. M.

C. M. McIntyre, S. E. Harris, J. Opt. Soc. Amer. 58, 1575 (1968).
[CrossRef]

Mooney, F.

F. Mooney, J. Opt. Soc. Amer. 42, 181 (1952).
[CrossRef]

Oxley, A. E.

A. E. Oxley, Phil. Mag. 21, (6) 517 (1911).

Palik, E. D.

Pancharatnam, S.

S. Pancharatnam, Proc. Indian Acad. Sci. A41, 130, 137 (1955).

Prouteau, J.

G. Destriau, J. Prouteau, J. Phys. Radium 10, (8) 53 (1949).
[CrossRef]

Shamraev, V. N.

V. A. Kizel, Yu. I. Krasilov, V. N. Shamraev, Opt. Spectrosc. 17, 248 (1964).

Vulmière, J.

M. Françon, S. Mallick, J. Vulmière, J. Opt. Soc. Amer. 55, 1553 (1965).
[CrossRef]

West, C. D.

C. D. West, A. S. Makas, J. Opt. Soc. Amer. 39, 791 (1949).
[CrossRef]

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, N.Y., 1957), pp. 517–518.

Wolfe, W. L.

W. L. Wolfe, S. S. Ballard, K. A. McCarthy, American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, N.Y., 1963), pp. 6–22 and 6–36.

Appl. Opt. (2)

J. Opt. Soc. Amer. (4)

F. Mooney, J. Opt. Soc. Amer. 42, 181 (1952).
[CrossRef]

C. M. McIntyre, S. E. Harris, J. Opt. Soc. Amer. 58, 1575 (1968).
[CrossRef]

C. D. West, A. S. Makas, J. Opt. Soc. Amer. 39, 791 (1949).
[CrossRef]

M. Françon, S. Mallick, J. Vulmière, J. Opt. Soc. Amer. 55, 1553 (1965).
[CrossRef]

J. Phys. Radium (1)

G. Destriau, J. Prouteau, J. Phys. Radium 10, (8) 53 (1949).
[CrossRef]

J. Sci. Instrum. (1)

R. J. King, J. Sci. Instrum. 43, 617 (1966).
[CrossRef]

Nature (1)

G. A. Harle, Nature 166, 149 (1950).
[CrossRef] [PubMed]

Opt. Spectrosc. (2)

V. A. Kizel, Yu. I. Krasilov, V. N. Shamraev, Opt. Spectrosc. 17, 248 (1964).

Yu. I. Krasilov, Opt. Spectrosc. 22, 267 (1967).

Phil. Mag. (1)

A. E. Oxley, Phil. Mag. 21, (6) 517 (1911).

Proc. Indian Acad. Sci. (1)

S. Pancharatnam, Proc. Indian Acad. Sci. A41, 130, 137 (1955).

Surface Sci. (1)

R. J. King, M. J. Downs, Proceedings of the Symposium on Recent Developments in Ellipsometry, Surface Sci. 16, 288 (1969).
[CrossRef]

Other (6)

H. E. Bennett, J. M. Bennett, Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, N.Y., 1967), Vol. 4, pp. 79–81.

W. L. Wolfe, S. S. Ballard, K. A. McCarthy, American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, N.Y., 1963), pp. 6–22 and 6–36.

See Ref. 6, p. 515.

F. Abelès, Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1963), Vol. 2, pp. 252–253.
[CrossRef]

W. Brouwer, American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, N.Y., 1963), p. 6–84.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, N.Y., 1957), pp. 517–518.

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Figures (6)

Fig. 1
Fig. 1

Variation of phase retardation with wavelength for the following λ/4 plates: (a) crystal quartz, (b) mica, (c) stretched plastic film, (d) apophyllite, (e) crystal quartz–calcite, and (f) Fresnel rhomb.

Fig. 2
Fig. 2

Internal phase changes on reflection, δp, δs, and Δ = δpδs, as a function of angle of incidence θ for a glass prism of refractive index 1.511. The polarizing angle θ ¯ and critical angle θc are also indicated.

Fig. 3
Fig. 3

Types of “rhombs”: (a) Fresnel rhomb, (b) optimized Fresnel rhomb similar to (a) except that the rhomb angle is 51.5° and one reflecting surface is coated with 200 Å of MgF2, (c) Mooney rhomb, (d) achromatic device 1 (AD-1), (e) achromatic device 2 (AD-2), (f) form of AD-2 similar to (d) except that the angle is changed to 72.2° and the base surface is coated with 275 Å of MgF2, and (g) achromatic device (AD) (two glass prisms and aluminum mirror). The useful apertures are indicated by arrows.

Fig. 4
Fig. 4

Variation of phase retardation with wavelength for (a) Fresnel rhomb, (b) coated glass Fresnel rhomb, (c) Mooney rhomb, and (g) AD-1.

Fig. 5
Fig. 5

Variation of phase retardation with wavelength for (a) Fresnel rhomb, (e) AD-2, (f) coated AD-2, and (g) AD.

Fig. 6
Fig. 6

Variation of phase retardation with external angle of incidence on the rhomb face for the rhombs shown in Fig. 3. Negative angles correspond to smaller internal angles, positive angles to larger internal angles.

Tables (2)

Tables Icon

Table I Optical Constants Used in the Calculations for the AD

Tables Icon

Table II Properties of Achromatic Rhombs

Equations (9)

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tan ( δ p / 2 ) = [ n ( n 2 sin 2 θ 1 ) 1 2 ] / cos θ ,
tan ( δ s / 2 ) = ( n 2 sin 2 θ 1 ) 1 2 / n cos θ ,
tan ( Δ / 2 ) tan ( δ p / 2 δ s / 2 ) = [ cos θ ( n 2 sin 2 θ 1 ) 1 2 ] / n sin 2 θ ,
sin θ Δ max = [ 2 / ( n 2 + 1 ) ] 1 2 .
Δ max = 2 tan 1 [ ( n 2 1 ) / 2 n ] .
tan δ s = 2 η o s b / η o s 2 ( a 2 + b 2 ) ,
tan δ p = 2 η o p d / ( c 2 + d 2 ) η o p 2 .
η o s = n o cos θ o , η o p = n o / cos θ o , a 2 + b 2 = [ ( n 1 2 k 1 2 n o 2 sin 2 θ o ) 2 + 4 n 1 2 k 1 2 ] 1 2 , b = { [ ( a 2 + b 2 ) / 2 ] [ n 1 2 k 1 2 n o 2 sin 2 θ o ) / 2 ] } 1 2 , d = b { 1 [ n 2 o sin 2 θ o / ( a 2 + b 2 ) ] } ,
c 2 + d 2 = ( n 1 2 + k 1 2 ) 2 / ( a 2 + b 2 ) .

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