Abstract

The ellipsometric function ρ of a film–substrate system is studied as the film thickness d is kept constant and the angle of incidence ϕ is changed. The generated constant-thickness contours (CTCs) are characterized by an introduced mathematical behavior indicator that represents a group of CTCs. The behavior of each group is developed and studied in the four planes ϕd,X,Z,   and  ρ,where X is the film-thickness exponential function and Z is a previously introduced intermediate plane. In the ϕd plane the film-thickness domain is identified and divided into a sequence of disconnected thickness subdomains (DTSs), depending on only N0  and  N1, and their number depending on the range in which N0/N1 lies. The behavior of the CTCs in the successive planes X, Z, and ρ is then studied in each DTS, and the CTC's space is divided into disconnected subfamilies according to the behavior indicator. Equivalence classes that reduce the infinite number of subfamilies into a finite number are then introduced. The transformation from each plane to the next is studied with the origin of the Z plane mapped onto the point at of the ρ plane, forming a singularity. A multiple-film-thickness inequality is derived to determine the unique solution of the film thickness. The type of reflection being internal or external at both ambient–film and film–substrate interfaces affects the analysis and is also considered. To conclude we introduce the design of polarization-preserving devices and a novel oscillating single-element ellipsometer to fully characterize zero film–substrate systems as examples of applying the knowledge developed here.

© 2006 Optical Society of America

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  1. See, for example, D. Goldstein,Polarized Light (Marcel Dekker, 2003).
  2. H. Zhu, L. Liu, Y. Wen, Z. Lu, and B. Zhang, "High-precision system for automatic null ellipsometric measurement," Appl. Opt. 41, 4536-4540 (2002).
    [CrossRef] [PubMed]
  3. R. M. A. Azzam, A.-R. M. Zaghloul, and N. M. Bashara, "Ellipsometric function of a film-substrate system: applications to the design of reflection-type optical devices and to ellipsometry," J. Opt. Soc. Am. 65, 252-260 (1975).
    [CrossRef]
  4. A. R. M. Zaghloul, "Ellipsometric function of a film-substrate system: applications to the design of reflection-type optical devices and to ellipsometry," Ph.D dissertation (University of Nebraska-Lincoln, 1975).
  5. R. M. A. Azzam, "Ellipsometry of transparent films on transparent substrate," Surf. Sci. 96, 67-80 (1980).
    [CrossRef]
  6. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977), Sec. 4.3.
  7. W. H. Weedon, S. W. McKnight, and A. J. Devaney, "Selection of optimal angles for inversion of multiple-angle ellipsometry and reflectometry equations," J. Opt. Soc. Am. A 8, 1881-1891 (1991).
    [CrossRef]
  8. S. F. Nee and H. E. Bennett, "Accurate null polarimetry for measuring the refractive index of transparent materials," J. Opt. Soc. Am. A 10, 2076-2083 (1993).
    [CrossRef]
  9. S. Li, "Jones-matrix analysis with Pauli matrices: application to ellipsometry," J. Opt. Soc. Am. A 17, 920-926 (2000).
    [CrossRef]
  10. F. Sagnard, F. Bentabet, and C. Vignat, "Theoretical study of method based on ellipsometry for measurement of complex permittivity of materials," Electron. Lett. 36, 1843-1845 (2000).
    [CrossRef]
  11. M. S. A. Yousef and A.-R. M. Zaghloul, "Ellipsometric function of a film-substrate system: characterization and detailed study," J. Opt. Soc. Am. A 6, 355-366 (1989).
    [CrossRef]
  12. M. Ghezzo, "Thickness calculations for a transparent film from ellipsometric measurements," J. Opt. Soc. Am. 58, 362-372 (1968).
    [CrossRef]
  13. R. M. A. Azzam and A.-R. M. Zaghloul, "Determination of the refractive index and thickness of a transparent film on a transparent substrate system from the angles of incidence of zero reflection-induced ellipticity," Opt. Commun. 24, 351-353 (1978).
    [CrossRef]
  14. A.-R. M. Zaghloul, R. M. A. Azzam, and N. M. Bashara, "Inversion of the nonlinear equations of reflection ellipsometry on film-substrate systems," Surf. Sci. 5, 87-96 (1976).
    [CrossRef]
  15. Y. Yoriume, "Method of numerical inversion of the ellipsometry equation for transparent film," J. Opt. Soc. Am. 73, 888-891 (1983).
    [CrossRef]
  16. K. Vedam, R. Rai, F. Lukes, and R. Srinivason, "Simultaneous and independent determination of the refractive index and the thickness of thin films by ellipsometry," J. Opt. Soc. Am. 58, 526-532 (1968).
    [CrossRef]
  17. D. A. Tonova, "Inverse profiling by ellipsometry: a Newton-Kantorovitch algorithm," Opt. Commun. 105, 104-112 (1994).
    [CrossRef]
  18. T. Easwarakhanthan, S. Ravelet, and P. Renard, "An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates," Appl. Surf. Sci. 90, 251-259 (1995).
    [CrossRef]
  19. S. Bosch, J. Perez, and A. Canillas, "Numerical algorithm for spectroscopic ellipsometry of thick transparent films," Appl. Opt. 37, 1177-1179 (1998).
    [CrossRef]
  20. A.-R. M. Zaghloul and M. S. A. Yousef, "Ellipsometric function of a film-substrate system: detailed analysis and closed-form inversion," J. Opt. Soc. Am. A 16, 2029-2044 (1999).
    [CrossRef]
  21. M. Elshazly-Zaghloul and A.-R. M. Zaghloul, "Closed-form inversion of the ellipsometric function of a film-substrate system: absorbing-substrate optical constant," J. Opt. Soc. Am. A 22, 1630-1636 (2005).
    [CrossRef]
  22. S. C. Warnick and M. A. Dahleh, "Ellipsometry as a sensor technology for the control of deposition processes," in Proceedings of the 37th IEEE Control System Society (Institute of Electrical and Electronics Engineers, 1998), Vol. 3, pp. 3162-3167.
  23. I. G. Rosen, T. Parent, B. Fidan, C. Wang, and A. Madhukar, "Design, development, and testing of real-time feedback controllers for semiconductor etching processes using in situ spectroscopic ellipsometry sensing," IEEE Trans. Control Syst. Technol. 10, 64-75 (2002).
    [CrossRef]
  24. W. G. Chinn and N. E. Steenrod, First Concepts of Topology: the Geometry of Mappings of Segments, Curves, Circles, and Disks (Random House, 1966).
  25. M. Holtz, K. Steffens, and E. Weitz, Introduction to Cardinal Arithmetic (Birkhaeuser, 1999).
  26. [x] is defined as the greatest integer that is ≤x (the step function).
  27. A domain [a, b] is divided into a sequence D(i) of disconnected subdomains if ∪D(i) = [a,b] and D(i) ∩ D(j) = { } ∀i ≠ j. Rearrangement may be ascending or descending.
  28. phivT is the angle of total reflection at the ambient-film interface. phivT = 43.23° when N0 = 1.46 and N1 = 1.
  29. The arc length of the exponential function of Eq. (3) as the angle of incidence changes from 0° to 90° is obtained by the definite integral ℒ=(2pid/lambda)∫phiv=0phiv=90((-2N02 sin2 phiv cos phiv)/(N12−N02 sin2 phiv)1/2) dphiv.
  30. For the case of external reflection at the ambient-film interface (N0 < N1) the ratio of any term to the one before in the sequence of Eq. (39) is less than unity. Hence it is a convergent sequence.
  31. In general, a domain is called m simply connected if the boundary of the same consists of m distinct boundaries.
  32. A singular point of a function is isolated if the function is analytic at each point in some deleted neighborhood of that point.
  33. A limit of a function f(x) at a point x = xi exists if and only if lim⁡x-->xi−0f(x)=lim⁡x-->xi+0f(x)=f(xi).
  34. The cases of internal and total reflection at any or both of ambient-film and film-substrate interfaces are, however, beyond the scope of this paper and are considered elsewhere.
  35. A binary relation ℜ on a set is called an equivalence relation on tau provided the following three properties hold: (1) For all a ϵtau, (a, a) ϵ ℜ. (2) For all a and b in tau, if (a, b) ϵ ℜ, then (b, a) ϵ ℜ. (3) For all a, b, and c in tau, if (a, b) ϵ ℜ and (b, c) ϵ ℜ , then (a, c) ϵ ℜ. A relation that satisfies (1) is called reflexive. A relation that satisfies (2) is called symmetric. A relation that satisfies (3) is called transitive.
  36. The determination of drphiv is described in detail in Refs. 3, 4, and 14. See also, D. A. Holmes, "On the calculation of thin-film refractive index and thickness by ellipsometry," Appl. Opt. 6, 168-169 (1967).
  37. One of the possible experimental techniques for scanning the Delta behavior of the rho-CTC is to use the polarizer-surface-analyzer null ellipsometry described in detail in R. M. A. Azzam, A.-R. M. Zaghloul, and N. M. Bashara, "Polarizer-surface-analyzer null ellipsometry for film-substrate systems," J. Opt. Soc. Am. 65, 1464-1471 (1975).
  38. A. R. M. Zaghloul, D. A. Keeling, W. A. Berzett, and J. S. Mason, "Design of reflection retarders by use of nonnegative film-substrate systems," J. Opt. Soc. Am. A 22, 1637-1645, (2005).
    [CrossRef]
  39. A. R. M. Zaghloul, M. Elshazly-Zaghloul, W. A. Berzett, and D. A. Keeling, "Thin-film coatings: an ellipsometric function approach I. Nonnegative transmission systems, polarization devices, coatings, and closed-form design formulas," submitted to J. Opt. Soc. Am. A .
  40. R. M. A. Azzam, "Simultaneous reflection and refraction of light without change of polarization by a single-layer-coated dielectric surface," Opt. Lett. 10, 107-109 (1985).
    [CrossRef] [PubMed]

2005 (2)

A. R. M. Zaghloul, D. A. Keeling, W. A. Berzett, and J. S. Mason, "Design of reflection retarders by use of nonnegative film-substrate systems," J. Opt. Soc. Am. A 22, 1637-1645, (2005).
[CrossRef]

M. Elshazly-Zaghloul and A.-R. M. Zaghloul, "Closed-form inversion of the ellipsometric function of a film-substrate system: absorbing-substrate optical constant," J. Opt. Soc. Am. A 22, 1630-1636 (2005).
[CrossRef]

2003 (1)

See, for example, D. Goldstein,Polarized Light (Marcel Dekker, 2003).

2002 (2)

I. G. Rosen, T. Parent, B. Fidan, C. Wang, and A. Madhukar, "Design, development, and testing of real-time feedback controllers for semiconductor etching processes using in situ spectroscopic ellipsometry sensing," IEEE Trans. Control Syst. Technol. 10, 64-75 (2002).
[CrossRef]

H. Zhu, L. Liu, Y. Wen, Z. Lu, and B. Zhang, "High-precision system for automatic null ellipsometric measurement," Appl. Opt. 41, 4536-4540 (2002).
[CrossRef] [PubMed]

2000 (2)

S. Li, "Jones-matrix analysis with Pauli matrices: application to ellipsometry," J. Opt. Soc. Am. A 17, 920-926 (2000).
[CrossRef]

F. Sagnard, F. Bentabet, and C. Vignat, "Theoretical study of method based on ellipsometry for measurement of complex permittivity of materials," Electron. Lett. 36, 1843-1845 (2000).
[CrossRef]

1999 (2)

1998 (2)

S. Bosch, J. Perez, and A. Canillas, "Numerical algorithm for spectroscopic ellipsometry of thick transparent films," Appl. Opt. 37, 1177-1179 (1998).
[CrossRef]

S. C. Warnick and M. A. Dahleh, "Ellipsometry as a sensor technology for the control of deposition processes," in Proceedings of the 37th IEEE Control System Society (Institute of Electrical and Electronics Engineers, 1998), Vol. 3, pp. 3162-3167.

1995 (1)

T. Easwarakhanthan, S. Ravelet, and P. Renard, "An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates," Appl. Surf. Sci. 90, 251-259 (1995).
[CrossRef]

1994 (1)

D. A. Tonova, "Inverse profiling by ellipsometry: a Newton-Kantorovitch algorithm," Opt. Commun. 105, 104-112 (1994).
[CrossRef]

1993 (1)

1991 (1)

1989 (1)

1985 (1)

1983 (1)

1980 (1)

R. M. A. Azzam, "Ellipsometry of transparent films on transparent substrate," Surf. Sci. 96, 67-80 (1980).
[CrossRef]

1978 (1)

R. M. A. Azzam and A.-R. M. Zaghloul, "Determination of the refractive index and thickness of a transparent film on a transparent substrate system from the angles of incidence of zero reflection-induced ellipticity," Opt. Commun. 24, 351-353 (1978).
[CrossRef]

1976 (1)

A.-R. M. Zaghloul, R. M. A. Azzam, and N. M. Bashara, "Inversion of the nonlinear equations of reflection ellipsometry on film-substrate systems," Surf. Sci. 5, 87-96 (1976).
[CrossRef]

1975 (2)

A. R. M. Zaghloul, "Ellipsometric function of a film-substrate system: applications to the design of reflection-type optical devices and to ellipsometry," Ph.D dissertation (University of Nebraska-Lincoln, 1975).

R. M. A. Azzam, A.-R. M. Zaghloul, and N. M. Bashara, "Ellipsometric function of a film-substrate system: applications to the design of reflection-type optical devices and to ellipsometry," J. Opt. Soc. Am. 65, 252-260 (1975).
[CrossRef]

1968 (2)

1966 (1)

W. G. Chinn and N. E. Steenrod, First Concepts of Topology: the Geometry of Mappings of Segments, Curves, Circles, and Disks (Random House, 1966).

Azzam, R. M. A.

R. M. A. Azzam, "Simultaneous reflection and refraction of light without change of polarization by a single-layer-coated dielectric surface," Opt. Lett. 10, 107-109 (1985).
[CrossRef] [PubMed]

R. M. A. Azzam, "Ellipsometry of transparent films on transparent substrate," Surf. Sci. 96, 67-80 (1980).
[CrossRef]

R. M. A. Azzam and A.-R. M. Zaghloul, "Determination of the refractive index and thickness of a transparent film on a transparent substrate system from the angles of incidence of zero reflection-induced ellipticity," Opt. Commun. 24, 351-353 (1978).
[CrossRef]

A.-R. M. Zaghloul, R. M. A. Azzam, and N. M. Bashara, "Inversion of the nonlinear equations of reflection ellipsometry on film-substrate systems," Surf. Sci. 5, 87-96 (1976).
[CrossRef]

R. M. A. Azzam, A.-R. M. Zaghloul, and N. M. Bashara, "Ellipsometric function of a film-substrate system: applications to the design of reflection-type optical devices and to ellipsometry," J. Opt. Soc. Am. 65, 252-260 (1975).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977), Sec. 4.3.

Bashara, N. M.

A.-R. M. Zaghloul, R. M. A. Azzam, and N. M. Bashara, "Inversion of the nonlinear equations of reflection ellipsometry on film-substrate systems," Surf. Sci. 5, 87-96 (1976).
[CrossRef]

R. M. A. Azzam, A.-R. M. Zaghloul, and N. M. Bashara, "Ellipsometric function of a film-substrate system: applications to the design of reflection-type optical devices and to ellipsometry," J. Opt. Soc. Am. 65, 252-260 (1975).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977), Sec. 4.3.

Bennett, H. E.

Bentabet, F.

F. Sagnard, F. Bentabet, and C. Vignat, "Theoretical study of method based on ellipsometry for measurement of complex permittivity of materials," Electron. Lett. 36, 1843-1845 (2000).
[CrossRef]

Berzett, W. A.

A. R. M. Zaghloul, D. A. Keeling, W. A. Berzett, and J. S. Mason, "Design of reflection retarders by use of nonnegative film-substrate systems," J. Opt. Soc. Am. A 22, 1637-1645, (2005).
[CrossRef]

A. R. M. Zaghloul, M. Elshazly-Zaghloul, W. A. Berzett, and D. A. Keeling, "Thin-film coatings: an ellipsometric function approach I. Nonnegative transmission systems, polarization devices, coatings, and closed-form design formulas," submitted to J. Opt. Soc. Am. A .

Bosch, S.

Canillas, A.

Chinn, W. G.

W. G. Chinn and N. E. Steenrod, First Concepts of Topology: the Geometry of Mappings of Segments, Curves, Circles, and Disks (Random House, 1966).

Dahleh, M. A.

S. C. Warnick and M. A. Dahleh, "Ellipsometry as a sensor technology for the control of deposition processes," in Proceedings of the 37th IEEE Control System Society (Institute of Electrical and Electronics Engineers, 1998), Vol. 3, pp. 3162-3167.

Devaney, A. J.

Easwarakhanthan, T.

T. Easwarakhanthan, S. Ravelet, and P. Renard, "An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates," Appl. Surf. Sci. 90, 251-259 (1995).
[CrossRef]

Elshazly-Zaghloul, M.

M. Elshazly-Zaghloul and A.-R. M. Zaghloul, "Closed-form inversion of the ellipsometric function of a film-substrate system: absorbing-substrate optical constant," J. Opt. Soc. Am. A 22, 1630-1636 (2005).
[CrossRef]

A. R. M. Zaghloul, M. Elshazly-Zaghloul, W. A. Berzett, and D. A. Keeling, "Thin-film coatings: an ellipsometric function approach I. Nonnegative transmission systems, polarization devices, coatings, and closed-form design formulas," submitted to J. Opt. Soc. Am. A .

Fidan, B.

I. G. Rosen, T. Parent, B. Fidan, C. Wang, and A. Madhukar, "Design, development, and testing of real-time feedback controllers for semiconductor etching processes using in situ spectroscopic ellipsometry sensing," IEEE Trans. Control Syst. Technol. 10, 64-75 (2002).
[CrossRef]

Ghezzo, M.

M. Ghezzo, "Thickness calculations for a transparent film from ellipsometric measurements," J. Opt. Soc. Am. 58, 362-372 (1968).
[CrossRef]

Holtz, M.

M. Holtz, K. Steffens, and E. Weitz, Introduction to Cardinal Arithmetic (Birkhaeuser, 1999).

Keeling, D. A.

A. R. M. Zaghloul, D. A. Keeling, W. A. Berzett, and J. S. Mason, "Design of reflection retarders by use of nonnegative film-substrate systems," J. Opt. Soc. Am. A 22, 1637-1645, (2005).
[CrossRef]

A. R. M. Zaghloul, M. Elshazly-Zaghloul, W. A. Berzett, and D. A. Keeling, "Thin-film coatings: an ellipsometric function approach I. Nonnegative transmission systems, polarization devices, coatings, and closed-form design formulas," submitted to J. Opt. Soc. Am. A .

Li, S.

Liu, L.

Lu, Z.

Lukes, F.

Madhukar, A.

I. G. Rosen, T. Parent, B. Fidan, C. Wang, and A. Madhukar, "Design, development, and testing of real-time feedback controllers for semiconductor etching processes using in situ spectroscopic ellipsometry sensing," IEEE Trans. Control Syst. Technol. 10, 64-75 (2002).
[CrossRef]

Mason, J. S.

A. R. M. Zaghloul, D. A. Keeling, W. A. Berzett, and J. S. Mason, "Design of reflection retarders by use of nonnegative film-substrate systems," J. Opt. Soc. Am. A 22, 1637-1645, (2005).
[CrossRef]

McKnight, S. W.

Nee, S. F.

Parent, T.

I. G. Rosen, T. Parent, B. Fidan, C. Wang, and A. Madhukar, "Design, development, and testing of real-time feedback controllers for semiconductor etching processes using in situ spectroscopic ellipsometry sensing," IEEE Trans. Control Syst. Technol. 10, 64-75 (2002).
[CrossRef]

Perez, J.

Rai, R.

Ravelet, S.

T. Easwarakhanthan, S. Ravelet, and P. Renard, "An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates," Appl. Surf. Sci. 90, 251-259 (1995).
[CrossRef]

Renard, P.

T. Easwarakhanthan, S. Ravelet, and P. Renard, "An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates," Appl. Surf. Sci. 90, 251-259 (1995).
[CrossRef]

Rosen, I. G.

I. G. Rosen, T. Parent, B. Fidan, C. Wang, and A. Madhukar, "Design, development, and testing of real-time feedback controllers for semiconductor etching processes using in situ spectroscopic ellipsometry sensing," IEEE Trans. Control Syst. Technol. 10, 64-75 (2002).
[CrossRef]

Sagnard, F.

F. Sagnard, F. Bentabet, and C. Vignat, "Theoretical study of method based on ellipsometry for measurement of complex permittivity of materials," Electron. Lett. 36, 1843-1845 (2000).
[CrossRef]

Srinivason, R.

Steenrod, N. E.

W. G. Chinn and N. E. Steenrod, First Concepts of Topology: the Geometry of Mappings of Segments, Curves, Circles, and Disks (Random House, 1966).

Steffens, K.

M. Holtz, K. Steffens, and E. Weitz, Introduction to Cardinal Arithmetic (Birkhaeuser, 1999).

Tonova, D. A.

D. A. Tonova, "Inverse profiling by ellipsometry: a Newton-Kantorovitch algorithm," Opt. Commun. 105, 104-112 (1994).
[CrossRef]

Vedam, K.

Vignat, C.

F. Sagnard, F. Bentabet, and C. Vignat, "Theoretical study of method based on ellipsometry for measurement of complex permittivity of materials," Electron. Lett. 36, 1843-1845 (2000).
[CrossRef]

Wang, C.

I. G. Rosen, T. Parent, B. Fidan, C. Wang, and A. Madhukar, "Design, development, and testing of real-time feedback controllers for semiconductor etching processes using in situ spectroscopic ellipsometry sensing," IEEE Trans. Control Syst. Technol. 10, 64-75 (2002).
[CrossRef]

Warnick, S. C.

S. C. Warnick and M. A. Dahleh, "Ellipsometry as a sensor technology for the control of deposition processes," in Proceedings of the 37th IEEE Control System Society (Institute of Electrical and Electronics Engineers, 1998), Vol. 3, pp. 3162-3167.

Weedon, W. H.

Weitz, E.

M. Holtz, K. Steffens, and E. Weitz, Introduction to Cardinal Arithmetic (Birkhaeuser, 1999).

Wen, Y.

Yoriume, Y.

Yousef, M. S. A.

Zaghloul, A. R. M.

A. R. M. Zaghloul, D. A. Keeling, W. A. Berzett, and J. S. Mason, "Design of reflection retarders by use of nonnegative film-substrate systems," J. Opt. Soc. Am. A 22, 1637-1645, (2005).
[CrossRef]

A. R. M. Zaghloul, "Ellipsometric function of a film-substrate system: applications to the design of reflection-type optical devices and to ellipsometry," Ph.D dissertation (University of Nebraska-Lincoln, 1975).

A. R. M. Zaghloul, M. Elshazly-Zaghloul, W. A. Berzett, and D. A. Keeling, "Thin-film coatings: an ellipsometric function approach I. Nonnegative transmission systems, polarization devices, coatings, and closed-form design formulas," submitted to J. Opt. Soc. Am. A .

Zaghloul, A.-R. M.

Zhang, B.

Zhu, H.

Appl. Opt. (2)

Appl. Surf. Sci. (1)

T. Easwarakhanthan, S. Ravelet, and P. Renard, "An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates," Appl. Surf. Sci. 90, 251-259 (1995).
[CrossRef]

Electron. Lett. (1)

F. Sagnard, F. Bentabet, and C. Vignat, "Theoretical study of method based on ellipsometry for measurement of complex permittivity of materials," Electron. Lett. 36, 1843-1845 (2000).
[CrossRef]

IEEE Trans. Control Syst. Technol. (1)

I. G. Rosen, T. Parent, B. Fidan, C. Wang, and A. Madhukar, "Design, development, and testing of real-time feedback controllers for semiconductor etching processes using in situ spectroscopic ellipsometry sensing," IEEE Trans. Control Syst. Technol. 10, 64-75 (2002).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (7)

Opt. Commun. (2)

R. M. A. Azzam and A.-R. M. Zaghloul, "Determination of the refractive index and thickness of a transparent film on a transparent substrate system from the angles of incidence of zero reflection-induced ellipticity," Opt. Commun. 24, 351-353 (1978).
[CrossRef]

D. A. Tonova, "Inverse profiling by ellipsometry: a Newton-Kantorovitch algorithm," Opt. Commun. 105, 104-112 (1994).
[CrossRef]

Opt. Lett. (1)

Surf. Sci. (2)

A.-R. M. Zaghloul, R. M. A. Azzam, and N. M. Bashara, "Inversion of the nonlinear equations of reflection ellipsometry on film-substrate systems," Surf. Sci. 5, 87-96 (1976).
[CrossRef]

R. M. A. Azzam, "Ellipsometry of transparent films on transparent substrate," Surf. Sci. 96, 67-80 (1980).
[CrossRef]

Other (18)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977), Sec. 4.3.

See, for example, D. Goldstein,Polarized Light (Marcel Dekker, 2003).

S. C. Warnick and M. A. Dahleh, "Ellipsometry as a sensor technology for the control of deposition processes," in Proceedings of the 37th IEEE Control System Society (Institute of Electrical and Electronics Engineers, 1998), Vol. 3, pp. 3162-3167.

W. G. Chinn and N. E. Steenrod, First Concepts of Topology: the Geometry of Mappings of Segments, Curves, Circles, and Disks (Random House, 1966).

M. Holtz, K. Steffens, and E. Weitz, Introduction to Cardinal Arithmetic (Birkhaeuser, 1999).

[x] is defined as the greatest integer that is ≤x (the step function).

A domain [a, b] is divided into a sequence D(i) of disconnected subdomains if ∪D(i) = [a,b] and D(i) ∩ D(j) = { } ∀i ≠ j. Rearrangement may be ascending or descending.

phivT is the angle of total reflection at the ambient-film interface. phivT = 43.23° when N0 = 1.46 and N1 = 1.

The arc length of the exponential function of Eq. (3) as the angle of incidence changes from 0° to 90° is obtained by the definite integral ℒ=(2pid/lambda)∫phiv=0phiv=90((-2N02 sin2 phiv cos phiv)/(N12−N02 sin2 phiv)1/2) dphiv.

For the case of external reflection at the ambient-film interface (N0 < N1) the ratio of any term to the one before in the sequence of Eq. (39) is less than unity. Hence it is a convergent sequence.

In general, a domain is called m simply connected if the boundary of the same consists of m distinct boundaries.

A singular point of a function is isolated if the function is analytic at each point in some deleted neighborhood of that point.

A limit of a function f(x) at a point x = xi exists if and only if lim⁡x-->xi−0f(x)=lim⁡x-->xi+0f(x)=f(xi).

The cases of internal and total reflection at any or both of ambient-film and film-substrate interfaces are, however, beyond the scope of this paper and are considered elsewhere.

A binary relation ℜ on a set is called an equivalence relation on tau provided the following three properties hold: (1) For all a ϵtau, (a, a) ϵ ℜ. (2) For all a and b in tau, if (a, b) ϵ ℜ, then (b, a) ϵ ℜ. (3) For all a, b, and c in tau, if (a, b) ϵ ℜ and (b, c) ϵ ℜ , then (a, c) ϵ ℜ. A relation that satisfies (1) is called reflexive. A relation that satisfies (2) is called symmetric. A relation that satisfies (3) is called transitive.

The determination of drphiv is described in detail in Refs. 3, 4, and 14. See also, D. A. Holmes, "On the calculation of thin-film refractive index and thickness by ellipsometry," Appl. Opt. 6, 168-169 (1967).

One of the possible experimental techniques for scanning the Delta behavior of the rho-CTC is to use the polarizer-surface-analyzer null ellipsometry described in detail in R. M. A. Azzam, A.-R. M. Zaghloul, and N. M. Bashara, "Polarizer-surface-analyzer null ellipsometry for film-substrate systems," J. Opt. Soc. Am. 65, 1464-1471 (1975).

A. R. M. Zaghloul, "Ellipsometric function of a film-substrate system: applications to the design of reflection-type optical devices and to ellipsometry," Ph.D dissertation (University of Nebraska-Lincoln, 1975).

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Figures (61)

Fig. 1
Fig. 1

Plane curve D ϕ in the ϕ d plane for the air– SiO 2 interface at λ = 632.8   nm as a function of the angle of incidence ϕ , where ϕ is in degrees. Contour A ≡ ([0°, 90°], 100) is the CTC in the ϕ d plane at film thickness d = 100   nm .

Fig. 2
Fig. 2

Real extended domain of D ϕ as a function of the angle of incidence, where ϕ is in degrees. It changes in the interval [ 0 ° , 43.23 ° ] . N 0 = 1.46 , N 1 = 1 , and λ = 632.8   nm . The dashed–dotted line is the asymptote at ϕ T .

Fig. 3
Fig. 3

Extended imaginary domain of D ϕ as a function of the angle of incidence, where ϕ is in degrees. It changes in the interval ( 43.23 ° , 90 ° ] . N 0 = 1.46 , N 1 = 1 , and λ = 632.8   nm . The dashed–dotted line is the asymptote at ϕ T .

Fig. 4
Fig. 4

Disconnected thickness-subdomains for N 0 = 1 , N 1 = 1.46 ,   and   m = 1 ; Example 1 of Subsection 2. B   ( valid   for N 1 / N 0 = 1.46 ) .

Fig. 5
Fig. 5

Same as in Fig. 4 but for N 0 = 1 , N 1 = 2 / 3 , and m = 1; Example 2 of Subsection 2.B (valid for N 1 / N 0 = 2 / 3 ).

Fig. 6
Fig. 6

Same as in Fig. 4 but for N 0 = 1 , N 1 = 1.46 , m = 3 ; Example 3 of Subsection 2.B ( valid   for   N 1 / N 0 = 1.46 ) .

Fig. 7
Fig. 7

Invariable behavior of D within the successive intervals of N 0 / N 1 for two values of the order of extension m. (The upper steps are for m = 3 , and the lower ones are for m = 1. )

Fig. 8
Fig. 8

Correspondence between the ϕ d plane and the X plane under the transformation x of Eq. (4).

Fig. 9
Fig. 9

Constant-thickness contours of the exponential function x (X-CTCs) traced in the X plane at d = 50 , 140 , 160 ,   and   210   nm; s 1 f 1 , s 2 f 2 , s′f ′ , and   s″f ″ , respectively, where N 0 = 1   and   N 1 = 1.46 at λ = 632.8   nm . The direction of rotation of the X-CTC is also shown.

Fig. 10
Fig. 10

Inverse images x 1 ( j ) , x 1 ( 1 ) , x 1 ( + j ) , and   x 1 ( + 1 ) in the reduced ϕ d plane of the points x = j , 1 , + j ,   and   + 1 of the X plane, respectively. N 0 = 1 , N 1 = 1.46 , and λ = 632.8   nm .

Fig. 11
Fig. 11

Inverse images x 1 ( s ) , x 1 ( f ) , x 1 ( s ) , and   x 1 ( f ) in the reduced ϕ d plane of starting points s , s and finishing points f , f of the two arcs s′f ′   and   s″f ″ , respectively.

Fig. 12
Fig. 12

(a) X-CTC sf at d = 240   nm, where N 0 = 1 , N 1 = 1.46 , at λ = 632.8   nm . (b) Inverse image of sf onto the ϕ d plane. The dashed–dotted line is at the angle ϕ b = 38.85 ° at which x 1 ( sf ) is discontinuous.

Fig. 13
Fig. 13

Arc length in radians of the X-CTC as a function of film thickness d in nanometers for N 0 = 1 , N 1 = 1.46   at   λ = 632.8 nm . The two lines, OA and OB, are on the maximum and the minimum rate of change of the arc length with respect to film thickness, respectively. OA is at N 0 = N 1 , and OB is at N 0 = 0.

Fig. 14
Fig. 14

arg(x) in radians of end points s and f of any X-CTC as a function of film thickness d in nanometers when N 0 = 1   and   N 1 = 1.46 for λ = 632.8   nm . The graphic representation of the members of Ś and the arc length of any X-CTC is shown. This figure could be used to graphically obtain the DTSs of any extended thickness domain.

Fig. 15
Fig. 15

Members of the subfamily of the X-CTCs at λ = 632.8   nm for N 0 = 1   and   N 1 = 1.46 for d   ∈   D ( 1 ) at d = 0, 10, 30, 50, 70, 100, and 108.357   nm . Point x = + 1   is at   d = 0.

Fig. 16
Fig. 16

Same as in Fig. 15 but for d   ∈   D ( 2 ) at d = 110 , 120 , 130 , and 140   nm .

Fig. 17
Fig. 17

Same as in Fig. 15 but for d   ∈   D ( 3 ) at d = 148.717, 150, 160, 180, and 216.713   nm .

Fig. 18
Fig. 18

Same as in Fig. 15 but for d   ∈   D ( 4 ) d = 220, 240, 260, 280, and 290   nm .

Fig. 19
Fig. 19

Domain of the function z in the complex Z plane for the negative system (1, 1.46, 3.85) at λ = 632.8   nm . The circle S (F) is the Z-CAIC at angle of incidence ϕ = 0 ° ( 90 ° ) . The dashed area is not included in the domain.

Fig. 20
Fig. 20

Same as in Fig. 19 but for the zero system (1, 1.46, 2.1316).

Fig. 21
Fig. 21

Same as in Fig. 19 but for the positive system (1, 1.46, 2).

Fig. 22
Fig. 22

Members of the first subfamily of the Z-CTCs for the negative system (1, 1.46, 3.85) at λ = 632.8   nm , where d   ∈   D ( 1 ) at d = 0, 10, 30, 50, 70, 100, and 108.357   nm .

Fig. 23
Fig. 23

Same as in Fig. 22 but for the zero system (1, 1.46, 2.1316).

Fig. 24
Fig. 24

Same as in Fig. 22 but for the positive system (1, 1.46, 2).

Fig. 25
Fig. 25

Members of the second subfamily of the Z-CTCs for the negative system (1, 1.46, 3.85) at λ = 632.8   nm, where d   ∈   D ( 2 ) at d = 110 , 120 , 130 ,   and   140   nm . The dashed Z-CTC, which passes through the origin, is at d = 141.932   nm .

Fig. 26
Fig. 26

Same as in Fig. 25 but for the zero system (1, 1.46, 2.1316).

Fig. 27
Fig. 27

Same as in Fig. 25 but for the positive system (1, 1.46, 2).

Fig. 28
Fig. 28

Members of the third subfamily of the Z-CTCs for the negative system (1, 1.46, 3.85) at λ = 632.8   nm, where d   ∈   D ( 3 ) at d = 148.717, 150, 160, 180, and 216.713   nm .

Fig. 29
Fig. 29

Same as in Fig. 28 but for the zero system (1, 1.46, 2.1316).

Fig. 30
Fig. 30

Same as in Fig. 28 but for the positive system (1, 1.46, 2).

Fig. 31
Fig. 31

Members of the fourth subfamily of the Z-CTCs for the negative system (1, 1.46, 3.85) at λ = 632.8   nm, where d   ∈   D ( 4 ) at d = 220, 240, 260, 280, 290, and 297.434   nm . The two branches of each Z-CTC are shown.

Fig. 32
Fig. 32

Same as in Fig. 31 but for the zero system (1, 1.46, 2.1316).

Fig. 33
Fig. 33

Same as in Fig. 31 but for the positive system (1, 1.46, 2).

Fig. 34
Fig. 34

Z-CTCs for the negative system (1, 1.46, 3.85) at λ = 632.8   nm, which fill the whole domain of z once, shown by the solid curves.

Fig. 35
Fig. 35

Same as in Fig. 34 but for the zero system (1, 1.46, 2.1316). The Z-CTCs are selected from those of Figs. 23, 26, 29, and 32.

Fig. 36
Fig. 36

Same as in Fig. 34 but for the positive system (1, 1.46, 2).

Fig. 37
Fig. 37

Essential singularity of the ellipsometric function ρ at the point ( ϕ ± , d = ½ D ϕ ± ) . The asymptote at ϕ ± (dashed–dotted line) is also shown. This is for the negative system (1, 1.46, 3.85) at λ = 632.8   nm .

Fig. 38
Fig. 38

Domain of the ellipsometric function ρ for the negative, zero, and positive systems, where the domain of ρ for the positive system ⊂ domain of ρ for the zero system ⊂ domain of ρ for the negative system.

Fig. 39
Fig. 39

Members of the first subfamily of the ρ-CTCs for the negative system (1, 1.46, 3.85) at λ = 632.8   nm, where d   ∈   D ( 1 ) at d = 0, 10, 30, 50, 70, 100, and 108.357   nm .

Fig. 40
Fig. 40

Same as in 39 but for the zero system (1, 1.46, 2.1316).

Fig. 41
Fig. 41

Same as in 39 but for the positive system (1, 1.46, 2).

Fig. 42
Fig. 42

Members of the second subfamily of the ρ-CTCs for the negative system (1, 1.46, 3.85) at λ = 632.8   nm, where d   ∈   D ( 2 ) at d = 110, 120, 130, and 140   nm .

Fig. 43
Fig. 43

Same as in Fig. 42 but for the zero system (1, 1.46, 2.1316). The ρ-CTC at 140   nm is plotted not to crowd the contours around point ρ = + 1.

Fig. 44
Fig. 44

Same as in Fig. 42 but for the positive system (1, 1.46, 2).

Fig. 45
Fig. 45

Members of the third subfamily of the ρ-CTCs for the negative system (1, 1.46, 3.85) at λ = 632.8   nm, where d   ∈   D ( 3 ) at d = 148.717, 150, 160, 180, and 216.713   nm .

Fig. 46
Fig. 46

Same as in Fig. 45 but for the zero system (1, 1.46, 2.1316).

Fig. 47
Fig. 47

Same as in Fig. 45 but for the positive system (1, 1.46, 2).

Fig. 48
Fig. 48

Members of the fourth subfamily of the ρ-CTCs for the negative system (1, 1.46, 3.85) at λ = 632.8   nm, where d   ∈   D ( 4 ) at d = 220, 240, 260, 280, 290, and 297.434   nm . The two branches of each ρ-CTC are also shown.

Fig. 49
Fig. 49

Same as in Fig. 48 but for the zero system (1, 1.46, 2.1316).

Fig. 50
Fig. 50

Same as in Fig. 48 but for the positive system (1, 1.46, 2).

Fig. 51
Fig. 51

The ρ-CTCs that fill the whole domain of ρ once for the negative system (1, 1.46, 3.85) at λ = 632.8   nm . The nonintersection property between any of the ρ-CTCs is evident.

Fig. 52
Fig. 52

The ρ-CTCs that fill the whole domain of ρ once for the zero system (1, 1.46, 2.1316) at λ = 632.8   nm . The dashed branches are of second-order tracing. Also the nonintersection property between any of the ρ-CTCs is evident, except for the point of ρ = + 1 , where all contours meet.

Fig. 53
Fig. 53

The ρ-CTCs that fill the whole domain of ρ twice for the positive system (1, 1.46, 2) at λ = 632.8   nm . Notice the intersection between the ρ-CTCs.

Fig. 54
Fig. 54

The two complementary families of contours, ρ-CAICs and ρ-CTCs, that fill the whole domain of ρ once, each, for the negative system (1, 1.46, 3.85) at λ = 632.8   nm . Note the uniqueness of the point of intersection between any ρ-CAIC (solid closed contours) and any ρ-CTC (dashed contours).

Fig. 55
Fig. 55

The two complementary families of contours, ρ-CAICs and ρ-CTCs, that fill the whole domain of ρ once, each, for the zero system (1, 1.46, 2.1316) at λ = 632.8   nm . Point ξ is an example of the intersection between the two families [ ϕ = 45 °   and   d = 110 nm,   d   ∈   D ( 2 ) ] .

Fig. 56
Fig. 56

The two complementary families of contours, ρ-CAICs and ρ-CTCs, that fill the whole domain of ρ twice, each, for the positive system (1, 1.46, 2) at λ = 632.8   nm .

Fig. 57
Fig. 57

Intersection between the ρ-CTC at d and the ρ - CAICs at ϕ 1 , ϕ 2 ,   or     ϕ 3 .

Fig. 58
Fig. 58

CTC in the ρ plane for a film–substrate system with d = 650   nm at λ = 632.8   nm (1, 1.46, 2.1316; the zero system).

Fig. 59
Fig. 59

Ellipsometric angle ψ ( ° ) as changed with the angle of incidence ϕ 0 ( ° ) for the film–substrate system of Fig. 58.

Fig. 60
Fig. 60

Same as in Fig. 59 but for the ellipsometric angle Δ ( ° ) .

Fig. 61
Fig. 61

Oscillating single-element ellipsometer, the X-measuring ellipsometer: L, laser (light) source; P, polarizer; S, sample; D, signal detector; ϕ 0 , angle of incidence.

Tables (4)

Tables Icon

Table 1 Relation between the Behavior Indicator of the X-CTC and That of the Z-CTC within Each Member of the Disconnected Thickness Subdomains for the Case of External Reflection at the Film–Substrate Interface

Tables Icon

Table 2 Same as Table 1 but for Internal Reflection at the Film–Substrate Interface

Tables Icon

Table 3 Relation between the Behavior Indicator of the Z-CTC and That of the ρ-CTC within Each of the Disconnected Thickness Subdomains for the Case of External Reflection at both the Ambient–Film and Film–Substrate Interfaces

Tables Icon

Table 4 Same as in Table 3 but for Internal Reflection at Both Interfaces

Equations (178)

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ϕ d   plane   x X   plane   z Z   plane   ρ ρ   plane.
ρ = A z + B / z + C ,
z = c + g x 1 + a b x ,
x = exp ( j 2 π d / D ϕ ) ,
D ϕ = λ 2 ( N 1     2 N 0     2 sin 2 ϕ ) 1 2 ,
A = ( a 2 1 ) b ( a b c g ) ( g a b c ) 2 ,
B = ( 1 c 2 ) g ( a g b c ) ( g a b c ) 2 ,
C = ( g + a b c ) ( a c g + b ) 2 b g ( a 2 + c 2 ) ( g a b c ) 2 .
i : D ϕ d ,
i n : ϕ - D ϕ   plane   ϕ d   plane .
Ś = Ś ( 0 ) Ś ( 9 0 ) ,
Ś ( 0 ) = ( 0 , m D 90 ) { ( n / 2 ) D 0 : n = 1 , 2 , } ,
Ś ( 90 ) = ( 0 , m D 90 ) { ( n / 2 ) D 90 : n = 1 , 2 , } .
Ś = ( Ś ( 0 ) Ś ( 90 ) ) ,
Ś = Ś ( 0 ) + Ś ( 90 ) ( Ś ( 0 ) Ś ( 90 ) ) .
Ś ( 0 ) = [ 2 m 1 - ( N 0 / N 1 ) 2 ] .
Ś ( 90 ) = 2 m 1         N 0 , N 1 .
( Ś ( 0 ) Ś ( 90 ) ) = Ś ( 90 ) ,
if   ( N 0 / N 1 ) 2 = 1 ( 1 / n ) 2 ,
n = 2 , 3 , ,
( Ś ( 0 ) Ś ( 90 ) ) = 0 ,
if   ( N 0 / N 1 ) 2 1 ( 1 / n ) 2 ,
n = 2 , 3 , .
Ś = { d i : i = 1 , 2 , , k } ,
D = { D ( i ) : i = 1 , 2 , , K + 1 } { { d } : d Ś ( 0 ) Ś ( 90 ) } .
Ś ( 0 ) Ś ( 9 0 ) = {     } .
D = ( Ś + 1 ) + ( Ś ( 0 ) Ś ( 9 0 ) ) ,
D = [ 0 , m D 90 ] ,
D = {     } .
ϕ T = sin 1 ( N 1 / N 0 ) .
( j , j λ 2 N 0     2 N 1     2 ] .
D ϕ = λ 2 N 1 2 - N 0 2 sin 2 ϕ , ϕ [ 0 , ϕ T ] ,
D ϕ = j λ 2 N 0     2 sin 2 ϕ N 1       2 , ϕ ( ϕ T , 90 ] .
Ś ( 0 ) = [ 2.74 ] = 2 Eq. ( 11 ) Ś ( 0 ) = { ½ D 0 , D 0 } .
Ś ( 90 ) = 1 Eq . ( 12 ) Ś ( 90 ) = { ½ D 90 } .
1 1 ( N 0 / N 1 ) 2 ( = 1.37 )
( Ś ( 0 ) Ś ( 90 ) ) = 0 Ś ( 0 ) Ś ( 90 ) = {       } .
Ś = 3.
Ś = { ½ D 0 , ½ D 90 , D 0 } .
D = 4 ,
D ( 1 ) = { d : 0 d ½ D 0 } = [ 0 , ½ D 0 ] ,
D ( 2 ) = { d : ½ D 0 < d < ½ D 90 } = ( ½ D 0 , ½ D 90 ) ,
D ( 3 ) = { d : ½ D 90 d D 0 } = [ ½ D 90 , D 0 ] ,
D ( 4 ) = { d : D 0 < d D 90 } = ( D 0 , D 90 ] .
Ś ( 0 ) = [ 4 ] Eq . ( 11 ) Ś ( 0 ) = { ½ D 0 , D 0 , 3 / 2 D 0 , 2 D 0 } .
Ś ( 90 ) = 1 Eq . ( 12 ) Ś ( 90 ) = { ½ D 90 } .
1 1 ( N 0 / N 1 ) 2 ( = 2 )
( Ś ( 0 ) Ś ( 90 ) ) = Ś ( 90 ) = 1 Ś ( 0 ) Ś ( 90 ) = { D 0 } .
Ś = 4.
Ś = { ½ D 0 , D 0 , 3 / 2 D 0 , 2 D 0 } .
D = 6.
D ( 1 ) = { d : 0 d ½ D 0 } = [ 0 , ½ D 0 ] ,
D ( 2 ) = { d : ½ D 0 < d < D 0 } = ( ½ D 0 , D 0 ) ,
D ( 3 ) = { d : d = D 0 } = { D 0 } ,
D ( 4 ) = { d : D 0 < d 3 / 2 D 0 } = ( D 0 , 3 / 2 D 0 ] ,
D ( 5 ) = { d : 3 / 2 D 0 < d < 2 D 0 } = ( 3 / 2 D 0 , 2 D 0 ) ,
D ( 6 ) = { d : d = 2 D 0 = D 90 } = { D 90 } .
Ś ( 0 ) = [ 8.23 ] = 8 Eq . ( 11 ) Ś ( 0 ) = { ½ D 0 , D 0 , 3 / 2 D 0 , 2 D 0 , 5 / 2 D 0 , 3 D 0 , 7 / 2 D 0 , 4 D 0 } .
Ś ( 90 ) = 5 Eq . ( 12 ) Ś ( 90 ) = { ½ D 90 , D 90 , 3 / 2 D 90 , 2 D 90 , 5 / 2 D 90 } .
1 1 - ( N 0 / N 1 ) 2 ( = 1.37 )
( Ś ( 0 ) Ś ( 90 ) ) = 0 Ś ( 0 ) Ś ( 90 ) = {       } .
Ś = 13.
Ś = { ½ D 0 , ½ D 90 , D 0 , D 90 , 3 / 2 D 90 , 2 D 0 , 3 / 2 D 90 , 5 / 2 D 0 , 2 D 90 , 3 D 0 , 5 / 2 D 90 , 7 / 2 D 0 , 4 D 0 } .
D = 14 ,
D ( 1 ) = [ 0 , ½ D 0 ] ,
D ( 2 ) = ( ½ D 0 , ½ D 90 ) ,
D ( 14 ) = ( 4 D 0 , 3 D 90 ] .
3 2 > 1 1 ( N 0 / N 1 ) 2 1 5 3 > N 0 N 1 0 ;
Ś ( 0 ) = 2     N 0 N 1 [ 0 , 5 3 ) .
2 > 1 1 ( N 0 N 1 ) 2 3 2 3 2 > N 0 N 1 5 3 ;
Ś ( 0 ) = 3 N 0 N 1 [ 5 3 , 3 2 ) .
k + 2 2 > 1 1 ( N 0 / N 1 ) 2 k + 1 2 k ( k + 4 ) ( k + 2 ) > N 0 N 1 ( k + 1 ) ( k + 3 ) ( k + 1 ) ;
Ś ( 0 ) = ( k + 1 ) N 0 N 1 [ ( k + 1 ) ( k + 3 ) k + 1 , k ( k + 4 ) k + 2 ) .
[ ( k + 1 ) ( k + 3 ) ( k + 1 ) , k ( k + 4 ) ( k + 2 ) ) , k = 1 , 2 , 3 , 4 .
N 0 / N 1 = 1 ( 1 / n ) 2 , n = 2 , 3 , 4 ,
( Ś ( 0 ) Ś (90 ) ) = Ś ( 90 ) .
D = Ś ( 0 ) + 2.
D = ( k + 3 ) N 0 N 1 [ ( k + 1 ) ( k + 3 ) ( k + 1 ) , k ( k + 4 ) k + 2 ) .
q > 2 m ( 1 1 ( N 0 / N 1 ) 2 1 ) q 1.
D = 4 m + q 1.
D = 4 m + q 1 N 0 N 1 [ ( q 1 ) ( 4 m + q 1 ) ( 2 m + q 1 ) , q ( 4 m + q ) ( 2 m + q ) ) .
= 4 π ( N 1 N 1    2 N 0     2 ) d / λ .
d″ = d + δd ,
s′f s″f {     }
δ d ( N 1 N 1     2 N 0     2 N 1     2 N 0     2 ) d .
s′f s″f = {     } .
ϕ b = sin 1 [ ( N 1 N 0 ) 2 ( λ 2 N 0 d ) 2 ] 1 / 2 .
d - seq= { d n : d n = D 0 [ 1 ( N 0 N 1 ) 2 ] n 1 2 , n = 1 , 2 , } .
- seq = { n : n = ( 4 n / λ ) ( N 1 [ 1 ( N 0 / N 1 ) 2 ] 1 / 2 ) d n , n = 1 , 2 , } .
n = 1 n = 2 π .
n = 1 d n = λ 2 ( N 1 N 1       2 N 0       2 ) .
arg ( x ) | s = 4 π N 1 d / λ ,
arg ( x ) | f = 4 π d ( N 1     2 N 0     2 ) 1 / 2 / λ .
d 1 = ( λ / 4 N 1 ) ( = ½ D 0 ) ,
d 2 = λ 4 N 1 2 N 0 2 ( = ½ D 90 ) .
{ d 1 , d 2 } Ś .
d 3 = λ 2 N 1 ( = D 0 ) ,
d 4 = λ 2 N 1     2 N 0     2 ( = D 90 ) .
{ d 3 , d 4 } Ś .
D ( 1 ) = [ 0 , ½ D 0 ] ,
D ( 2 ) = ( ½ D 0 , ½ D 90 ) ,
D ( 3 ) = [ ½ D 90 , D 0 ] ,
D ( 4 ) = ( D 0 , D 90 ] .
β = β ( N 1 / N 0 , m ) .
β ( 2 3 < N 1 N 0 < , 1 ) = { 0 , 10 , 1 , 01 }
( = { + , + , , + } ) ,
β ( 2 / 3 , 1 ) = { 0 , 10 , 1 , 01 , 101 , 0101 } ( = { + , + , , + , + + , + + } ) .
x ( N 0 , N 1 , ϕ , d ) ,
z ( N 0 , N 1 , N 2 , ϕ , d ) ,
z ¯ = c + g x ¯ 1 + a b x ¯ .
z z ¯ = ( g a b c ) | 1 + a b x | 2 ( x x ¯ ) ,
Im ( z ) = ( g a b c ) | 1 + a b x | 2   Im ( x ) .
( g a b c ) = 2 N 1 cos ϕ 1 ( N 2     2 N 1    2 ) [ N 0 cos ϕ 2 ( cos 2 ϕ 1 + cos 2 ϕ ) + N 2 cos ϕ ( N 0     2 + N 1     2 ) ( N 0 sin ϕ N 1 N 2 ) 2 ] ( N 1 cos ϕ + N 0 cos ϕ 1 ) ( N 2 cos ϕ 1 + N 1 cos ϕ 2 ) ( N 0 cos ϕ + N 1 cos ϕ 1 ) ( N 1 cos ϕ 1 + N 2 cos ϕ 2 ) .
i f   N 2 > ( N 1     2 / N 0 ) ( N 0 , N 1 , N 2 )   is   a   negative   system,
if  N 2 = ( N 1     2 / N 0 ) ( N 0 , N 1 , N 2 )   is   a   zero   system,   and
if  N 2 < ( N 1     2 / N 0 ) ( N 0 , N 1 , N 2 )   is   a   positive   system .
( N 0 , N 1 , N 2 ) = ( 1 , 1.46 , 3.85 ) , negative   system ,
( N 0 , N 1 , N 2 ) = ( 1 , 1.46 , 2.1316 ) , zero   system , and
( N 0 , N 1 , N 2 ) = ( 1 , 1.46 , 2 ) , positive   system ,
S F =  { } .
ϕ ± = sin 1 ( N 0     2 N 2     2 N 1     4 N 0     4 2 N 0     2 N 2    2 ) 1 / 2 .
d = λ 4 [ N 0     2 2 N 1     2 + N 2     2 ( N 1     2 N 0     2 ) ( N 2    2 N 1     2 ) ] 1 / 2 .
ρ = 1 at   ϕ = 0 ° ,
ρ = A z + B / z + C ( 0 ° ,   90 ° ) ,
ρ = + 1 at   ϕ = 90 °
ρ ¯ = A z ¯ + B / z ¯ + C .
ρ ρ ¯ = ( A B | z | 2 ) ( z z ¯ ) ;
I m ( ρ ) = ( A B | z | 2 ) I m ( z ) .
ρ ( z ( x = + 1 ) ) = ( 1 , + 1 ) real   N 0 , N 1 , and   N 2 .
( ϕ 1 , d 1 ) ( ϕ 2 , d 2 ) ρ ( ϕ 1 , d 1 ) ρ ( ϕ 2 , d 2 ) .
ρ ( ϕ , d ) = one-to-one   mapping   ( ϕ , d ) ½ D ϕ ,
ρ ( ϕ , d ) = infinite-to-one   mapping   ( ϕ , d ) ½ D ϕ .
( ϕ 1 , d 1 ) ( ϕ 2 , d 2 ) ρ ( ϕ 1 , d 1 ) = ρ ( ϕ 2 , d 2 ) .
τ = { τ : τ    is a ρ-CTC } .
τ 1 τ 2 .
[ 1 ] , [ 01 ] , [ 0 ] , [ 10 ] , ( [ ] , [ + ] , [ + ] , [ + ] , ) .
β = ( τ / ) D .
D = 14 ,
β = 6.
ρ = tan ψ   exp ( j Δ ) .
d c ϕ = d r ϕ + μ D ϕ .
L d c ϕ G .
L d r ϕ + μ D ϕ G .
( L d r ϕ ) / D ϕ μ ( G d r ϕ ) / D ϕ .
A ( ϕ ) = { d c i ϕ : i = 1 , 2 , , n } .
d = A ( ϕ ) A ( ϕ ) .
d r 20 = 123.689   nm , d r 60 = 31.156   nm .
D ( 9 ) = [ 5 / 2 D 0 , 2 D 90 ] ,
D ( 13 ) = [ 7 / 2 D 0 , 4 D 0 ] ,
D ( 17 ) = [ 7 / 2 D 90 , 5 D 0 ] .
( 5 / 2 D 0 123.689 ) D 20 μ ( 5 D 0 123.689 ) D 20 . .
A ( 20 ) = { 569.52 , 792.43 , 1015.35 } .
5 / 2 D 0 31.156 D 60 μ 5 D 0 31.156 D 60 .
1.9 μ 3.9.
A ( 60 ) = { 569.52 , 838.699 } .
d = A ( 20 ) A ( 60 ) = 569.52   nm .
max = π ( 1 + the   number   of   crossing   times   of   the   real   axis ) ;
d max = ( 1 + n ) λ 4 ( N 1 N 1     2 N 0     2 ) .
m = [ d max D 90 ] + 1.
max = 3 π ,
d max = 1197.78   nm .
D 90 = 297.434   nm ,
m = [ 4.03 ] + 1 = 5.
d c ϕ = d r ϕ + μ D ϕ d max ,
μ  d max d r ϕ D ϕ .
ϕ = 20 ° ineq . ( 83 ) μ = 4.82,
A ( 20 ) = { 346.604 , 569.52 , 792.43 , 1015.35 } ,
ϕ = 60 ° ineq . ( 83 ) μ 4.33 ,
A ( 60 ) = { 300.34 , 569.52 , 838.699 , 1107.88 } .
A ( 20 ) A ( 60 ) = { 569.52 } .
d = λ 4 ( N 1     2 N 0     2 sin 2 ϕ 0 ) 1 / 2 ( ϕ 0 = sin 1 [ ( N 1 N 0 ) 2 ( λ 4 N 0 d ) 2 ] 1 / 2 ) .
N 1 = [ ( λ 4 d ) 2 + N 0     2 sin 2 ϕ 0 ] 1 / 2 .
I = p       2 cos 2 ( P ) + s       2 sin 2 ( P ) ,
p = s ,
I = p       2 + s       2 ,
P 1 45 , P 2 45 ° .
N 2 = N 1     2 / N 0 .

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