Abstract

Formulas are presented for the use of optical compensators in ellipsometers. Nothing need be known about the compensator characteristics other than the directions of “fast” and “slow” axes. The formulas can be used, without modification, when the compensator is not an exact quarter-wave plate, when multiple internal reflections are present, and when the compensator is absorbing. Also described is a method for calibrating the compensator at the same time it is being used in ellipsometry measurements.

© 1967 Optical Society of America

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  1. R. J. Archer, J. Electrochemical Soc. 104, 619 (1957); Phys. Rev. 110, 354 (1958); J. Opt. Soc. Am. 52, 970 (1962). A. B. Winterbottom, Kgl. Norske Videnskab. Selskab. Skrifter 1, 53 (1955). R. C. Menard, J. Opt. Soc. Am. 52, 427 (1962). F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Stds. 67A, 363 (1963). A. Rothen, Rev. Sci. Instr. 16, 26 (1945); Rev. Sci. Instr. 19, 839 (1948); Rev. Sci. Instr. 28, 283 (1957); Ann. N. Y. Acad. Sci. 53, 1054 (1951). J. B. Bateman and M. W. Harris, Annals of the New York Academy of Sciences 53, 1064 (1951). R. W. Ditchburn and G. A. J. Orchard, Proc. Phys. Soc. (London) 57B, 608 (1954). J. A. Faucher, G. M. McManus, and H. J. Trurnit, J. Opt. Soc. Am. 48, 51 (1958). F. Partovi, J. Opt. Soc. Am. 52, 918 (1962). F. P. Mertens, P. Theroux, and R. C. Plumb, J. Opt. Soc. Am. 53, 788 (1963). K. H. Zaininger and A. G. Revesz, RCA Rev. XXV, 85 (1964). D. K. Burge and H. E. Bennett, J. Opt. Soc. Am. 54, 1428 (1964). D. W. Peterson and N. M. Bashara, J. Opt. Soc. Am. 55, 845 (1965). L. E. Smith and R. R. Stromberg, J. Opt. Soc. Am. 56, 1539 (1966).
    [Crossref] [PubMed]
  2. Various techniques for using wave-plates in conjunction with rotary analyzers are described by: M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, New York, Paris, Los Angeles, 1959), pp. 688–691. G. N. Ramachandran and S. Ramaseshan, Encyclopedia of Physics, S. Flügge, Ed. (Springer-Verlag, Berlin, Göttingen, Heidelberg, 1961), Vol. XXV/1, pp. 34–53. E. N. Cameron, Economic Geology 52, 252 (1957). M. Richartz, Z. Instrumentenk. 73, 205 (1965). R. C. Plumb, J. Opt. Soc. Am. 50, 892 (1960). D. Bergman, J. Opt. Soc. Am. 52, 1080 (1962). A. C. Hall, J. Opt. Soc. Am. 53, 801 (1963). H. G. Jerrard, J. Opt. Soc. Am. 38, 35 (1948). C. A. Skinner, J. Opt. Soc. Am. and Rev. Sci. Instr. 10, 491 (1925). M. Richartz, J. Opt. Soc. Am. 56, 198 (1966).
    [Crossref]
  3. H. G. Jerrard, J. Opt. Soc. Am. 42, 159 (1952). This article is extremely well documented.
    [Crossref]
  4. F. Gabler and P. Sokob, Z. Physik 116, 47 (1940).
    [Crossref]
  5. H. Weinberger and J. Harris, J. Opt. Soc. Am. 54, 552 (1964).
    [Crossref]
  6. D. A. Holmes, J. Opt. Soc. Am. 54, 1115 (1964).
    [Crossref]
  7. R. Bünnagel, Z. Instrumentenk. 69, 79 (1961).
  8. D. A. Holmes, J. Opt. Soc. Am. 54, 1340 (1964); J. Opt. Soc. Am. 55, 209 (1965).
    [Crossref]
  9. A concise description of the various ellipsometer measuring techniques is given by A. Vašíček, in E. Passaglia, R. R. Stromberg, and J. Kruger, Eds., Ellipsometry in the Measurement of Surfaces and Thin Films, Symposium Proceedings (U. S. Department of Commerce, National Bureau of Standards Miscellaneous Publication 256, Washington, D. C., 15Sept1964), pp. 33–36.
  10. H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
    [Crossref]
  11. When converted to a real number, ρoxy can be either positive or negative.
  12. See the papers by Bergman and Hall in Ref. 2.
  13. D. A. Holmes and D. L. Feucht, J. Opt. Soc. Am. 55, 577 (1965).
    [Crossref]
  14. Recall that 0≤ψi≤π/2, 0≤Δi<2π, 0≤ψr<π/2, 0≤Δr<2π, 0≤c<π, −π/2<a≤+π/2, Tc>0, 0≤Δc<2π, 0≤ψ≤π/2, 0≤Δ<2π.
  15. By “calibrated” we simply mean that Tc exp (jΔc) is determined at the particular wavelength of interest.
  16. For example, a Glan–Thompson prism.
  17. Actually Eq. (20) gives four solutions for p; however, all physically distinct positions of the PP axis in Fig. 2(a) can be obtained by limiting p to the range −π/2<p≤+π/2. Equation (20) was derived by setting the imaginary part of ρr equal to zero and solving for tanp.
  18. An interesting consequence of Eq. (20) is that when light is passed through a tandem arrangement of polarizer, wave-plate, wave-plate, it is always possible to rotate the polarizer to two physically distinct settings for which linearly polarized light will emerge from the last wave-plate. This is true regardless of the respective retardations of the two plates and regardless of the relative orientation (the angle c) of their principal axes.
  19. R. R. Alfano and W. H. Woodruff, Appl. Optics 5, 352 (1966), have discussed calibration of a wave plate by ellipsometry; however, they ignored the possibility that Tc can differ from unity, as established experimentally by H. Weinberger and J. Harris, J. Opt. Soc. Am. 54, 552 (1964). When using monochromatic, well-collimated light, a complete calibration of the optical compensator requires determination of both Δc and Tc.
    [Crossref]
  20. We are thus able to compute two “polarization-state transfer functions,” Tc exp(jΔc) and tanψ exp(jΔ), from the measured angles c, a1, p1, a2, and p2. A fringe benefit from our work is that we could set the ellipsometer arms in the straight-through position, replace the reflecting specimen with a transmitting birefringent plate whose principal axes are the sp axes, perform the measurements c, a1, p1, a2, p2 and then we could calibrate both plates.
  21. For a film-covered surface, Δ can lie anywhere in the interval 0≤Δ≤2π. See Ref. 13.
  22. A. B. Winterbottom on p. 102 of the Symposium Proceedings described in Ref. 9.
  23. F. P. Mertens and R. C. Plumb, J. Opt. Soc. Am. 54, 1063 (1964).
    [Crossref]
  24. Theoretical calculations were presented in Ref. 8 which showed that an isotropic lossless plate could be used as a rotary compensator in the infrared. The present work indicates that an isotropic absorbing plate could also be used as a rotary compensator, provided, of course, that sufficient light is transmitted to permit meaningful extinction settings.
  25. A. L. Bloom, Appl. Opt. 5, 1500 (1966).
    [Crossref] [PubMed]
  26. T. J. Bridges and J. W. Klüver, Appl. Opt. 4, 1121 (1965).
    [Crossref]

1966 (2)

R. R. Alfano and W. H. Woodruff, Appl. Optics 5, 352 (1966), have discussed calibration of a wave plate by ellipsometry; however, they ignored the possibility that Tc can differ from unity, as established experimentally by H. Weinberger and J. Harris, J. Opt. Soc. Am. 54, 552 (1964). When using monochromatic, well-collimated light, a complete calibration of the optical compensator requires determination of both Δc and Tc.
[Crossref]

A. L. Bloom, Appl. Opt. 5, 1500 (1966).
[Crossref] [PubMed]

1965 (2)

1964 (4)

1963 (1)

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
[Crossref]

1961 (1)

R. Bünnagel, Z. Instrumentenk. 69, 79 (1961).

1957 (1)

R. J. Archer, J. Electrochemical Soc. 104, 619 (1957); Phys. Rev. 110, 354 (1958); J. Opt. Soc. Am. 52, 970 (1962). A. B. Winterbottom, Kgl. Norske Videnskab. Selskab. Skrifter 1, 53 (1955). R. C. Menard, J. Opt. Soc. Am. 52, 427 (1962). F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Stds. 67A, 363 (1963). A. Rothen, Rev. Sci. Instr. 16, 26 (1945); Rev. Sci. Instr. 19, 839 (1948); Rev. Sci. Instr. 28, 283 (1957); Ann. N. Y. Acad. Sci. 53, 1054 (1951). J. B. Bateman and M. W. Harris, Annals of the New York Academy of Sciences 53, 1064 (1951). R. W. Ditchburn and G. A. J. Orchard, Proc. Phys. Soc. (London) 57B, 608 (1954). J. A. Faucher, G. M. McManus, and H. J. Trurnit, J. Opt. Soc. Am. 48, 51 (1958). F. Partovi, J. Opt. Soc. Am. 52, 918 (1962). F. P. Mertens, P. Theroux, and R. C. Plumb, J. Opt. Soc. Am. 53, 788 (1963). K. H. Zaininger and A. G. Revesz, RCA Rev. XXV, 85 (1964). D. K. Burge and H. E. Bennett, J. Opt. Soc. Am. 54, 1428 (1964). D. W. Peterson and N. M. Bashara, J. Opt. Soc. Am. 55, 845 (1965). L. E. Smith and R. R. Stromberg, J. Opt. Soc. Am. 56, 1539 (1966).
[Crossref] [PubMed]

1952 (1)

1940 (1)

F. Gabler and P. Sokob, Z. Physik 116, 47 (1940).
[Crossref]

Alfano, R. R.

R. R. Alfano and W. H. Woodruff, Appl. Optics 5, 352 (1966), have discussed calibration of a wave plate by ellipsometry; however, they ignored the possibility that Tc can differ from unity, as established experimentally by H. Weinberger and J. Harris, J. Opt. Soc. Am. 54, 552 (1964). When using monochromatic, well-collimated light, a complete calibration of the optical compensator requires determination of both Δc and Tc.
[Crossref]

Archer, R. J.

R. J. Archer, J. Electrochemical Soc. 104, 619 (1957); Phys. Rev. 110, 354 (1958); J. Opt. Soc. Am. 52, 970 (1962). A. B. Winterbottom, Kgl. Norske Videnskab. Selskab. Skrifter 1, 53 (1955). R. C. Menard, J. Opt. Soc. Am. 52, 427 (1962). F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Stds. 67A, 363 (1963). A. Rothen, Rev. Sci. Instr. 16, 26 (1945); Rev. Sci. Instr. 19, 839 (1948); Rev. Sci. Instr. 28, 283 (1957); Ann. N. Y. Acad. Sci. 53, 1054 (1951). J. B. Bateman and M. W. Harris, Annals of the New York Academy of Sciences 53, 1064 (1951). R. W. Ditchburn and G. A. J. Orchard, Proc. Phys. Soc. (London) 57B, 608 (1954). J. A. Faucher, G. M. McManus, and H. J. Trurnit, J. Opt. Soc. Am. 48, 51 (1958). F. Partovi, J. Opt. Soc. Am. 52, 918 (1962). F. P. Mertens, P. Theroux, and R. C. Plumb, J. Opt. Soc. Am. 53, 788 (1963). K. H. Zaininger and A. G. Revesz, RCA Rev. XXV, 85 (1964). D. K. Burge and H. E. Bennett, J. Opt. Soc. Am. 54, 1428 (1964). D. W. Peterson and N. M. Bashara, J. Opt. Soc. Am. 55, 845 (1965). L. E. Smith and R. R. Stromberg, J. Opt. Soc. Am. 56, 1539 (1966).
[Crossref] [PubMed]

Bloom, A. L.

Born, M.

Various techniques for using wave-plates in conjunction with rotary analyzers are described by: M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, New York, Paris, Los Angeles, 1959), pp. 688–691. G. N. Ramachandran and S. Ramaseshan, Encyclopedia of Physics, S. Flügge, Ed. (Springer-Verlag, Berlin, Göttingen, Heidelberg, 1961), Vol. XXV/1, pp. 34–53. E. N. Cameron, Economic Geology 52, 252 (1957). M. Richartz, Z. Instrumentenk. 73, 205 (1965). R. C. Plumb, J. Opt. Soc. Am. 50, 892 (1960). D. Bergman, J. Opt. Soc. Am. 52, 1080 (1962). A. C. Hall, J. Opt. Soc. Am. 53, 801 (1963). H. G. Jerrard, J. Opt. Soc. Am. 38, 35 (1948). C. A. Skinner, J. Opt. Soc. Am. and Rev. Sci. Instr. 10, 491 (1925). M. Richartz, J. Opt. Soc. Am. 56, 198 (1966).
[Crossref]

Brand, F. A.

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
[Crossref]

Bridges, T. J.

Bünnagel, R.

R. Bünnagel, Z. Instrumentenk. 69, 79 (1961).

Feucht, D. L.

Gabler, F.

F. Gabler and P. Sokob, Z. Physik 116, 47 (1940).
[Crossref]

Harris, J.

Hatkin, L.

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
[Crossref]

Holmes, D. A.

Jacobs, H.

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
[Crossref]

Jerrard, H. G.

Klüver, J. W.

Mertens, F. P.

Plumb, R. C.

Sokob, P.

F. Gabler and P. Sokob, Z. Physik 116, 47 (1940).
[Crossref]

Vašícek, A.

A concise description of the various ellipsometer measuring techniques is given by A. Vašíček, in E. Passaglia, R. R. Stromberg, and J. Kruger, Eds., Ellipsometry in the Measurement of Surfaces and Thin Films, Symposium Proceedings (U. S. Department of Commerce, National Bureau of Standards Miscellaneous Publication 256, Washington, D. C., 15Sept1964), pp. 33–36.

Weinberger, H.

Winterbottom, A. B.

A. B. Winterbottom on p. 102 of the Symposium Proceedings described in Ref. 9.

Wolf, E.

Various techniques for using wave-plates in conjunction with rotary analyzers are described by: M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, New York, Paris, Los Angeles, 1959), pp. 688–691. G. N. Ramachandran and S. Ramaseshan, Encyclopedia of Physics, S. Flügge, Ed. (Springer-Verlag, Berlin, Göttingen, Heidelberg, 1961), Vol. XXV/1, pp. 34–53. E. N. Cameron, Economic Geology 52, 252 (1957). M. Richartz, Z. Instrumentenk. 73, 205 (1965). R. C. Plumb, J. Opt. Soc. Am. 50, 892 (1960). D. Bergman, J. Opt. Soc. Am. 52, 1080 (1962). A. C. Hall, J. Opt. Soc. Am. 53, 801 (1963). H. G. Jerrard, J. Opt. Soc. Am. 38, 35 (1948). C. A. Skinner, J. Opt. Soc. Am. and Rev. Sci. Instr. 10, 491 (1925). M. Richartz, J. Opt. Soc. Am. 56, 198 (1966).
[Crossref]

Woodruff, W. H.

R. R. Alfano and W. H. Woodruff, Appl. Optics 5, 352 (1966), have discussed calibration of a wave plate by ellipsometry; however, they ignored the possibility that Tc can differ from unity, as established experimentally by H. Weinberger and J. Harris, J. Opt. Soc. Am. 54, 552 (1964). When using monochromatic, well-collimated light, a complete calibration of the optical compensator requires determination of both Δc and Tc.
[Crossref]

Appl. Opt. (2)

Appl. Optics (1)

R. R. Alfano and W. H. Woodruff, Appl. Optics 5, 352 (1966), have discussed calibration of a wave plate by ellipsometry; however, they ignored the possibility that Tc can differ from unity, as established experimentally by H. Weinberger and J. Harris, J. Opt. Soc. Am. 54, 552 (1964). When using monochromatic, well-collimated light, a complete calibration of the optical compensator requires determination of both Δc and Tc.
[Crossref]

J. Appl. Phys. (1)

H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
[Crossref]

J. Electrochemical Soc. (1)

R. J. Archer, J. Electrochemical Soc. 104, 619 (1957); Phys. Rev. 110, 354 (1958); J. Opt. Soc. Am. 52, 970 (1962). A. B. Winterbottom, Kgl. Norske Videnskab. Selskab. Skrifter 1, 53 (1955). R. C. Menard, J. Opt. Soc. Am. 52, 427 (1962). F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Stds. 67A, 363 (1963). A. Rothen, Rev. Sci. Instr. 16, 26 (1945); Rev. Sci. Instr. 19, 839 (1948); Rev. Sci. Instr. 28, 283 (1957); Ann. N. Y. Acad. Sci. 53, 1054 (1951). J. B. Bateman and M. W. Harris, Annals of the New York Academy of Sciences 53, 1064 (1951). R. W. Ditchburn and G. A. J. Orchard, Proc. Phys. Soc. (London) 57B, 608 (1954). J. A. Faucher, G. M. McManus, and H. J. Trurnit, J. Opt. Soc. Am. 48, 51 (1958). F. Partovi, J. Opt. Soc. Am. 52, 918 (1962). F. P. Mertens, P. Theroux, and R. C. Plumb, J. Opt. Soc. Am. 53, 788 (1963). K. H. Zaininger and A. G. Revesz, RCA Rev. XXV, 85 (1964). D. K. Burge and H. E. Bennett, J. Opt. Soc. Am. 54, 1428 (1964). D. W. Peterson and N. M. Bashara, J. Opt. Soc. Am. 55, 845 (1965). L. E. Smith and R. R. Stromberg, J. Opt. Soc. Am. 56, 1539 (1966).
[Crossref] [PubMed]

J. Opt. Soc. Am. (6)

Z. Instrumentenk. (1)

R. Bünnagel, Z. Instrumentenk. 69, 79 (1961).

Z. Physik (1)

F. Gabler and P. Sokob, Z. Physik 116, 47 (1940).
[Crossref]

Other (13)

A concise description of the various ellipsometer measuring techniques is given by A. Vašíček, in E. Passaglia, R. R. Stromberg, and J. Kruger, Eds., Ellipsometry in the Measurement of Surfaces and Thin Films, Symposium Proceedings (U. S. Department of Commerce, National Bureau of Standards Miscellaneous Publication 256, Washington, D. C., 15Sept1964), pp. 33–36.

Various techniques for using wave-plates in conjunction with rotary analyzers are described by: M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, New York, Paris, Los Angeles, 1959), pp. 688–691. G. N. Ramachandran and S. Ramaseshan, Encyclopedia of Physics, S. Flügge, Ed. (Springer-Verlag, Berlin, Göttingen, Heidelberg, 1961), Vol. XXV/1, pp. 34–53. E. N. Cameron, Economic Geology 52, 252 (1957). M. Richartz, Z. Instrumentenk. 73, 205 (1965). R. C. Plumb, J. Opt. Soc. Am. 50, 892 (1960). D. Bergman, J. Opt. Soc. Am. 52, 1080 (1962). A. C. Hall, J. Opt. Soc. Am. 53, 801 (1963). H. G. Jerrard, J. Opt. Soc. Am. 38, 35 (1948). C. A. Skinner, J. Opt. Soc. Am. and Rev. Sci. Instr. 10, 491 (1925). M. Richartz, J. Opt. Soc. Am. 56, 198 (1966).
[Crossref]

We are thus able to compute two “polarization-state transfer functions,” Tc exp(jΔc) and tanψ exp(jΔ), from the measured angles c, a1, p1, a2, and p2. A fringe benefit from our work is that we could set the ellipsometer arms in the straight-through position, replace the reflecting specimen with a transmitting birefringent plate whose principal axes are the sp axes, perform the measurements c, a1, p1, a2, p2 and then we could calibrate both plates.

For a film-covered surface, Δ can lie anywhere in the interval 0≤Δ≤2π. See Ref. 13.

A. B. Winterbottom on p. 102 of the Symposium Proceedings described in Ref. 9.

When converted to a real number, ρoxy can be either positive or negative.

See the papers by Bergman and Hall in Ref. 2.

Recall that 0≤ψi≤π/2, 0≤Δi<2π, 0≤ψr<π/2, 0≤Δr<2π, 0≤c<π, −π/2<a≤+π/2, Tc>0, 0≤Δc<2π, 0≤ψ≤π/2, 0≤Δ<2π.

By “calibrated” we simply mean that Tc exp (jΔc) is determined at the particular wavelength of interest.

For example, a Glan–Thompson prism.

Actually Eq. (20) gives four solutions for p; however, all physically distinct positions of the PP axis in Fig. 2(a) can be obtained by limiting p to the range −π/2<p≤+π/2. Equation (20) was derived by setting the imaginary part of ρr equal to zero and solving for tanp.

An interesting consequence of Eq. (20) is that when light is passed through a tandem arrangement of polarizer, wave-plate, wave-plate, it is always possible to rotate the polarizer to two physically distinct settings for which linearly polarized light will emerge from the last wave-plate. This is true regardless of the respective retardations of the two plates and regardless of the relative orientation (the angle c) of their principal axes.

Theoretical calculations were presented in Ref. 8 which showed that an isotropic lossless plate could be used as a rotary compensator in the infrared. The present work indicates that an isotropic absorbing plate could also be used as a rotary compensator, provided, of course, that sufficient light is transmitted to permit meaningful extinction settings.

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Figures (3)

Fig. 1
Fig. 1

In (a), the cross-hatched portion represents a reflection specimen with an assumed perfectly flat surface. An incident electromagnetic plane wave has electric-field components whose complex amplitudes are denoted by Eip and Eis. The component Eip is in the plane of incidence and is considered positive when pointing away from the reflecting surface. The component Eis is perpendicular to the plane of incidence and is considered positive when pointing into the paper. The angle of incidence is i. The corresponding electric-field components of the reflected plane wave are Erp and Ers.

Fig. 2
Fig. 2

In (a) is shown the sp reference system for the incident light. The incident light is first passed through a polarizer whose electric-field transmission axis is shown as PP. Then the light is passed through a wave plate whose principal axes are shown as the mutually orthogonal xy axes. The x axis can be either the “fast” axis or the “slow” axis; the choice is immaterial. The positive x axis is located with respect to the positive s axis by the angle c where 0≤c<π. The PP axis is located with respect to the x axis by the angle p where −π/2<p≤+π/2. The elliptically polarized light emerging from the wave plate is then reflected from the specimen surface. If the reflected light is linearly polarized, then the electric-field oscillation axis is shown as EE in (b), which shows the reference frame for the reflected light. The EE axis is located with respect to the positive s axis by the angle e where −π/2<e≤+π/2. To extinguish the reflected linearly polarized light, the electric-field transmission axis of an analyzer must be perpendicular to EE. Hence, the analyzer transmission axis AA is located by the angle a where a=eπ/2 if e>0, a=e + π/2 if e≤0 and −π/2<a≤+ π/2. In this figure, p is negative, e is positive, and a is negative. Note that the definitions of a and e are different from those used in Sec. 2 and Fig. 1. The distinction between the p axis and the p angle is carefully maintained in the text. In taking a measurement with the ellipsometer, the compensator orientation is held fixed at the angle c and the polarizer (PP axis) and the analyzer (AA axis) are rotated until extinction of the reflected light is achieved.

Fig. 3
Fig. 3

Locus of ρrxy as a function of c for selected values of ρr. The real part of ρrxy is plotted on the horizontal axis and the imaginary part of ρrxy is plotted on the vertical axis. Equation (9) was used in the calculation. The circle numbers identify the selected value of ρr as follows: (1) ρr=0.5ej22.5°, (2) ρr=0.5ej45°, (3) ρr=0.5ej67.5°, (4) ρr=1ej22.5°, (5) ρr=1ej67.5°. The dot (·) on each curve identifies the starting point when c=0. At the starting point, ρrxy=ρr. As c increases from zero, ρrxy travels around the circle in the direction indicated by the arrow on the circle. When c reaches 180°, then ρrxy arrives back at the starting point ρr, thus all points on the circle can be obtained by restricting c to the range 0≤c<π.

Equations (30)

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E i p / E i s ρ i = tan ψ i exp ( j Δ i ) .
0 ψ i π / 2 , 0 Δ i < 2 π .
E r p / E r s ρ r = tan ψ r exp ( j Δ r ) ,
0 ψ r π / 2 , 0 Δ r < 2 π .
R p E r p / E i p ,             R s E r s / E i s ,
R p / R s ρ = tan ψ exp ( j Δ ) ,
ρ = ρ r / ρ i .
ρ r x y E r y / E r x .
ρ r x y = ( ρ r - tan c ) / ( ρ r tan c + 1 ) .
ρ 0 x y = T c exp ( j Δ c ) ρ r x y .
T c exp ( j Δ c ) = cos ( β x d ) + j K x sin ( β x d ) cos ( β y d ) + j K y sin ( β y d ) ,
β k = 2 π n k / λ ,             k = x , y ,
K k = ( n 2 k + 1 ) / 2 n k ,             k = x , y ,
Δ c + δ r = m π ,
tan c = cot 2 ψ r sin Δ c sin ( Δ r - Δ c ) ± [ cot 2 2 ψ r sin 2 Δ c - sin ( Δ r - Δ c ) sin ( Δ r + Δ c ) ] 1 2 sin ( Δ r - Δ c ) .
tan e E 0 y / E 0 x ( at compensation ) .
- cot a 1 - cot a 2 = T c exp ( j Δ c ) [ ( ρ r - tan c 1 ) / ( ρ r tan c 1 + 1 ) ] T c exp ( j Δ c ) [ ( ρ r - tan c 2 ) / ( ρ r tan c 2 + 1 ) ] .
ρ r = { cos ( c 2 + c 1 ) sin ( a 2 - a 1 ) ± j [ sin 2 ( c 2 - c 1 ) sin 2 ( a 2 + a 1 ) - sin 2 ( a 2 - a 1 ) ] 1 2 } × [ sin ( c 2 - c 1 ) sin ( a 2 + a 1 ) - sin ( c 2 + c 1 ) sin ( a 2 - a 1 ) ] - 1 .
1.2 exp ( j 80 ° ) ,             1.2 exp ( j 100 ° ) ,             1.2 exp ( j 260 ° ) ,             1.2 exp ( j 280 ° ) ,             0.8 exp ( j 80 ° ) ,             0.8 exp ( j 100 ° ) ,                                                                                         0.8 exp ( j 260 ° ) ,             0.8 exp ( j 280 ° ) .
ρ r = m exp ( j 10 ° + j 30 ° n ) ,
T c exp ( j Δ c ) = ( - cot a k ) ( ρ r tan c k + 1 ) ( ρ r - tan c k ) ,
ρ r = ρ · tan c + tan p T c exp ( j Δ c ) 1 - tan c tan p T c exp ( j Δ c ) .
tan p = ( sin ( Δ + Δ c ) - tan 2 c sin ( Δ - Δ c ) ± { [ sin ( Δ + Δ c ) - tan 2 c sin ( Δ - Δ c ) ] 2 + 4 tan 2 c sin 2 Δ } 1 2 ) × ( 2 T c tan c sin Δ ) - 1 .
tan e = ρ r ( at compensation ) .
T c exp ( j Δ c ) = cot p k · 1 + ρ tan c tan a k tan c - ρ tan a k ,
ρ = ( sin ( a 1 - a 2 ) sin ( p 1 + p 2 ) + cos 2 c sin ( a 1 + a 2 ) × sin ( p 1 - p 2 ) ± j { - [ sin ( a 1 - a 2 ) sin ( p 1 + p 2 ) + cos 2 c sin ( a 1 + a 2 ) sin ( p 1 - p 2 ) ] 2 - sin 2 2 c sin 2 a 1 × sin 2 a 2 sin 2 ( p 1 - p 2 ) } 1 2 ) × [ 2 sin 2 c sin a 1 sin a 2 sin ( p 2 - p 1 ) ] - 1 .
c = 40° , 45° , 50° ;             ρ = m exp ( j 32.5° + j 45° n ) ;         m = 1 2 , 2 ;                                                 n = 0 , 1 , 2 , 7 ; T c exp ( j Δ c ) = 1.1 exp ( j + j 10° n ) ;             n = 0 , 1 , 2 , 35.
ρ r x y - j csc Δ r csc 2 ψ r = { csc 2 Δ r csc 2 2 ψ r - 1 } 1 2 exp ( j θ ) ,
ϕ r = arctan ( sin 2 ( ψ r + c ) - sin 2 c sin 2 ψ r cos 2 ( Δ r / 2 ) sin 2 ( ψ r + c ) + sin 2 c sin 2 ψ r cos 2 ( Δ r / 2 ) ) 1 2 ,
δ r = arctan ( sin Δ r cos Δ r cos 2 c - sin 2 c cot 2 ψ r ) .