Abstract

The ellipsometric function ρ of a film–substrate system is analyzed through successive transformations from the plane of the two independent variables angle of incidence and film thickness (ϕd plane) to the complex ρ plane. This analysis is achieved by introducing two intermediate planes: the unimodular plane (Zi plane) and the translated ellipsometric plane (ρ* plane). The analysis through the Zi plane leads to classification of the film–substrate systems into two classes: clockwise and counterclockwise. The class of the film–substrate system governs the inversion from the ρ* plane to the Zi-plane. It identifies the number of branch points of ρ*-1 from the ρ* plane to the Zi plane. The branch points of ρ*-1 and its preimage in the ϕd plane are identified and studied. The domain of the double-valued function ρ*-1 is divided into two or four subdomains according to the class of the film–substrate system. In each of these subdomains, the single-valued branch of ρ*-1 is fixed, and we introduce a closed-form solution for the determination of the film thickness of the system. Mathematically, ρ*-1 exists in any domain that does not include the branch points. Hence the exceptive points are divided into two types: removable and essential. The closed-form inversion is obtained for the removable exceptive points. The conformality of both ρ and ρ*, as well as their inverses, leads to identification of the two essential exceptive inversion points, which exist at ϕ=0° and 90°. Accordingly, the closed-form solution is available throughout the ρ plane except at the two points ±1 (corresponding to ϕ=0° and 90°). A study of the extrema of the magnitude and the phase of both ρ and ρ* provides full information on the number of zeros and essential singularities for each of the three categories of film–substrate systems: negative, zero, and positive. Numerical examples are given to illustrate the introduced closed forms. Also, the table of transformation of regions between the ϕd plane and the ρ plane induced by ρ and ρ-1 is given.

© 1999 Optical Society of America

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References

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  1. A.-R. M. Zaghloul, “A study of the ellipsometric function of a film–substrate system and applications to the design of reflection-type optical devices and to ellipsometry,” Ph.D. dissertation (University of Nebraska–Lincoln, Lincoln, Nebraska, 1975).
  2. M. M. Ibrahim, N. M. Bashara, “Parameter-correlation and computational considerations in multiple-angle ellipsometry,” J. Opt. Soc. Am. 61, 1622–1629 (1971).
    [CrossRef]
  3. J. Lekner, “Analytic inversion of ellipsometric data for an unsupported nonabsorbing uniform layer,” J. Opt. Soc. Am. A 7, 1875–1877 (1990).
    [CrossRef]
  4. D. U. Fluckiger, “Analytic methods in the determination of optical properties by spectral ellipsometry,” J. Opt. Soc. Am. A 15, 2228–2232 (1998).
    [CrossRef]
  5. V. G. Polovinkin, S. N. Svitasheva, “Analysis of general ambiguity of inverse ellipsometric problem,” Thin Solid Films 313, 128–131 (1998).
    [CrossRef]
  6. M. Ghezzo, “Thickness calculation for transparent film from ellipsometric measurements,” J. Opt. Soc. Am. 58, 368–372 (1968).
    [CrossRef]
  7. R. M. A. Azzam, “Simple and direct determination of complex refractive index and thickness of unsupported or embedded thin films by combined reflection and transmission ellipsometry at 45° angle of incidence,” J. Opt. Soc. Am. 73, 1080–1082 (1983).
    [CrossRef]
  8. C. G. Galarza, P. P. Khargonekar, N. Layadi, T. L. Vincent, E. A. Rietman, J. T. C. Lee, “A new algorithm for real-time thin-film thickness estimation given in-situ multiwavelength ellipsometry using an extended Kalman filter,” Thin Solid Films 313, 156–160 (1998).
    [CrossRef]
  9. M. S. A. Yousef, 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]
  10. R. M. A. Azzam, A.-R. M. Zaghloul, 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]
  11. The two bilinear transformations of Eqs. (6) and (7) are obtained by translating the centers of the two circles AZ and B/Z of Ref. 9 into the origin of the complex plane.
  12. R. M. A. Azzam, M. E. R. Khan, “Complex reflection coefficients for parallel and perpendicular polarizations of a film–substrate system,” Appl. Opt. 22, 253–264 (1983).
    [CrossRef] [PubMed]
  13. Multiple-valued functions may have singular points that are not removable singularities, poles, or essential singularities. These singular points are called branch points; more precisely, they are points at which the double-valued function becomes a single-valued one.
  14. The two values of ϕ=0° and 90° are deleted from the domain of the angle of incidence because the translated-ellipsometric function ρ* at both values of ϕ is not a conformal mapping (see Subsection 3.C).
  15. The pair of equations that give the same value of x are not the same. The parameters of one of these equations depend on only one of the two fundamental polarizations, either parallel or perpendicular. Those of the second equation depend only on the other fundamental polarization.
  16. From the definition of ln z, where z is a complex variable, we have ln z=∫γ drr+j∫γ dθ=ln|z|+jΔγ arg z. That is, the value of ln z depends not only on the point z but also on the curve γ along which the integral is taken. To overcome the troublesome determination of the curve γ to obtain the closed forms of dr, we use the Cartesian representation of the point x(ρ*).
  17. A.-R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “Inversion of the nonlinear equations of reflection ellipsometry on film–substrate systems,” Surf. Sci. 5, 87–96 (1976).
    [CrossRef]
  18. The polarizer–surface-analyzer null ellipsometry is to be used experimentally to determine the number of the real-axis crossing by ρ for the three categories of the film–substrate system; see Ref. 19. The very simple single-element rotating-polarizer ellipsometry may be used for only the zero and the negative systems; see A.-R. M. Zaghloul, R. M. A. Azzam, “Single-element rotating-polarizer ellipsometer for film–substrate systems,” J. Opt. Soc. Am. 67, 1286–1287 (1977).
    [CrossRef]
  19. R. M. A. Azzam, A.-R. M. Zaghloul, N. M. Bashara, “Polarizer–surface-analyzer null ellipsometry for film–substrate systems,” J. Opt. Soc. Am. 65, 1464–1471 (1975).
    [CrossRef]

1998

V. G. Polovinkin, S. N. Svitasheva, “Analysis of general ambiguity of inverse ellipsometric problem,” Thin Solid Films 313, 128–131 (1998).
[CrossRef]

C. G. Galarza, P. P. Khargonekar, N. Layadi, T. L. Vincent, E. A. Rietman, J. T. C. Lee, “A new algorithm for real-time thin-film thickness estimation given in-situ multiwavelength ellipsometry using an extended Kalman filter,” Thin Solid Films 313, 156–160 (1998).
[CrossRef]

D. U. Fluckiger, “Analytic methods in the determination of optical properties by spectral ellipsometry,” J. Opt. Soc. Am. A 15, 2228–2232 (1998).
[CrossRef]

1990

1989

1983

1977

1976

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

1975

1971

1968

Azzam, R. M. A.

R. M. A. Azzam, M. E. R. Khan, “Complex reflection coefficients for parallel and perpendicular polarizations of a film–substrate system,” Appl. Opt. 22, 253–264 (1983).
[CrossRef] [PubMed]

R. M. A. Azzam, “Simple and direct determination of complex refractive index and thickness of unsupported or embedded thin films by combined reflection and transmission ellipsometry at 45° angle of incidence,” J. Opt. Soc. Am. 73, 1080–1082 (1983).
[CrossRef]

The polarizer–surface-analyzer null ellipsometry is to be used experimentally to determine the number of the real-axis crossing by ρ for the three categories of the film–substrate system; see Ref. 19. The very simple single-element rotating-polarizer ellipsometry may be used for only the zero and the negative systems; see A.-R. M. Zaghloul, R. M. A. Azzam, “Single-element rotating-polarizer ellipsometer for film–substrate systems,” J. Opt. Soc. Am. 67, 1286–1287 (1977).
[CrossRef]

A.-R. M. Zaghloul, R. M. A. Azzam, 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, 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, A.-R. M. Zaghloul, N. M. Bashara, “Polarizer–surface-analyzer null ellipsometry for film–substrate systems,” J. Opt. Soc. Am. 65, 1464–1471 (1975).
[CrossRef]

Bashara, N. M.

Fluckiger, D. U.

Galarza, C. G.

C. G. Galarza, P. P. Khargonekar, N. Layadi, T. L. Vincent, E. A. Rietman, J. T. C. Lee, “A new algorithm for real-time thin-film thickness estimation given in-situ multiwavelength ellipsometry using an extended Kalman filter,” Thin Solid Films 313, 156–160 (1998).
[CrossRef]

Ghezzo, M.

Ibrahim, M. M.

Khan, M. E. R.

Khargonekar, P. P.

C. G. Galarza, P. P. Khargonekar, N. Layadi, T. L. Vincent, E. A. Rietman, J. T. C. Lee, “A new algorithm for real-time thin-film thickness estimation given in-situ multiwavelength ellipsometry using an extended Kalman filter,” Thin Solid Films 313, 156–160 (1998).
[CrossRef]

Layadi, N.

C. G. Galarza, P. P. Khargonekar, N. Layadi, T. L. Vincent, E. A. Rietman, J. T. C. Lee, “A new algorithm for real-time thin-film thickness estimation given in-situ multiwavelength ellipsometry using an extended Kalman filter,” Thin Solid Films 313, 156–160 (1998).
[CrossRef]

Lee, J. T. C.

C. G. Galarza, P. P. Khargonekar, N. Layadi, T. L. Vincent, E. A. Rietman, J. T. C. Lee, “A new algorithm for real-time thin-film thickness estimation given in-situ multiwavelength ellipsometry using an extended Kalman filter,” Thin Solid Films 313, 156–160 (1998).
[CrossRef]

Lekner, J.

Polovinkin, V. G.

V. G. Polovinkin, S. N. Svitasheva, “Analysis of general ambiguity of inverse ellipsometric problem,” Thin Solid Films 313, 128–131 (1998).
[CrossRef]

Rietman, E. A.

C. G. Galarza, P. P. Khargonekar, N. Layadi, T. L. Vincent, E. A. Rietman, J. T. C. Lee, “A new algorithm for real-time thin-film thickness estimation given in-situ multiwavelength ellipsometry using an extended Kalman filter,” Thin Solid Films 313, 156–160 (1998).
[CrossRef]

Svitasheva, S. N.

V. G. Polovinkin, S. N. Svitasheva, “Analysis of general ambiguity of inverse ellipsometric problem,” Thin Solid Films 313, 128–131 (1998).
[CrossRef]

Vincent, T. L.

C. G. Galarza, P. P. Khargonekar, N. Layadi, T. L. Vincent, E. A. Rietman, J. T. C. Lee, “A new algorithm for real-time thin-film thickness estimation given in-situ multiwavelength ellipsometry using an extended Kalman filter,” Thin Solid Films 313, 156–160 (1998).
[CrossRef]

Yousef, M. S. A.

Zaghloul, A.-R. M.

M. S. A. Yousef, 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]

The polarizer–surface-analyzer null ellipsometry is to be used experimentally to determine the number of the real-axis crossing by ρ for the three categories of the film–substrate system; see Ref. 19. The very simple single-element rotating-polarizer ellipsometry may be used for only the zero and the negative systems; see A.-R. M. Zaghloul, R. M. A. Azzam, “Single-element rotating-polarizer ellipsometer for film–substrate systems,” J. Opt. Soc. Am. 67, 1286–1287 (1977).
[CrossRef]

A.-R. M. Zaghloul, R. M. A. Azzam, 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, N. M. Bashara, “Polarizer–surface-analyzer null ellipsometry for film–substrate systems,” J. Opt. Soc. Am. 65, 1464–1471 (1975).
[CrossRef]

R. M. A. Azzam, A.-R. M. Zaghloul, 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]

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

Appl. Opt.

J. Opt. Soc. Am.

M. Ghezzo, “Thickness calculation for transparent film from ellipsometric measurements,” J. Opt. Soc. Am. 58, 368–372 (1968).
[CrossRef]

M. M. Ibrahim, N. M. Bashara, “Parameter-correlation and computational considerations in multiple-angle ellipsometry,” J. Opt. Soc. Am. 61, 1622–1629 (1971).
[CrossRef]

R. M. A. Azzam, A.-R. M. Zaghloul, 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, A.-R. M. Zaghloul, N. M. Bashara, “Polarizer–surface-analyzer null ellipsometry for film–substrate systems,” J. Opt. Soc. Am. 65, 1464–1471 (1975).
[CrossRef]

R. M. A. Azzam, “Simple and direct determination of complex refractive index and thickness of unsupported or embedded thin films by combined reflection and transmission ellipsometry at 45° angle of incidence,” J. Opt. Soc. Am. 73, 1080–1082 (1983).
[CrossRef]

The polarizer–surface-analyzer null ellipsometry is to be used experimentally to determine the number of the real-axis crossing by ρ for the three categories of the film–substrate system; see Ref. 19. The very simple single-element rotating-polarizer ellipsometry may be used for only the zero and the negative systems; see A.-R. M. Zaghloul, R. M. A. Azzam, “Single-element rotating-polarizer ellipsometer for film–substrate systems,” J. Opt. Soc. Am. 67, 1286–1287 (1977).
[CrossRef]

J. Opt. Soc. Am. A

Surf. Sci.

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

Thin Solid Films

V. G. Polovinkin, S. N. Svitasheva, “Analysis of general ambiguity of inverse ellipsometric problem,” Thin Solid Films 313, 128–131 (1998).
[CrossRef]

C. G. Galarza, P. P. Khargonekar, N. Layadi, T. L. Vincent, E. A. Rietman, J. T. C. Lee, “A new algorithm for real-time thin-film thickness estimation given in-situ multiwavelength ellipsometry using an extended Kalman filter,” Thin Solid Films 313, 156–160 (1998).
[CrossRef]

Other

Multiple-valued functions may have singular points that are not removable singularities, poles, or essential singularities. These singular points are called branch points; more precisely, they are points at which the double-valued function becomes a single-valued one.

The two values of ϕ=0° and 90° are deleted from the domain of the angle of incidence because the translated-ellipsometric function ρ* at both values of ϕ is not a conformal mapping (see Subsection 3.C).

The pair of equations that give the same value of x are not the same. The parameters of one of these equations depend on only one of the two fundamental polarizations, either parallel or perpendicular. Those of the second equation depend only on the other fundamental polarization.

From the definition of ln z, where z is a complex variable, we have ln z=∫γ drr+j∫γ dθ=ln|z|+jΔγ arg z. That is, the value of ln z depends not only on the point z but also on the curve γ along which the integral is taken. To overcome the troublesome determination of the curve γ to obtain the closed forms of dr, we use the Cartesian representation of the point x(ρ*).

The two bilinear transformations of Eqs. (6) and (7) are obtained by translating the centers of the two circles AZ and B/Z of Ref. 9 into the origin of the complex plane.

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

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Figures (18)

Fig. 1
Fig. 1

Film–substrate system, where N0, N1, and N2 are the refractive indices of the ambient, film, and substrate, respectively. ϕ and d are the angle of incidence of the light beam and the film thickness, respectively.

Fig. 2
Fig. 2

Unit circle in the X plane: X-CAIC, with clockwise rotation.

Fig. 3
Fig. 3

(a) Image of the X-CAIC under the bilinear transformation z1(x) for any category of the film–substrate system at any angle of incidence. The relation between the directions of rotation is shown (same direction of rotation). (b) Same as in (a) but for the bilinear transformation z2(x) where the film–substrate system is positive: N0=1, N1=1.46, and N2=2, at λ=632.8 nm. (c) Same as in (a) but for the bilinear transformation z2(x) where the film–substrate system is zero: N0=1, N1=1.46, and N2=2.1316 at λ=632.8 nm. The upper image (point z2=+1) is at normal incidence, ϕ=0. The lower image is at any angle of incidence ϕ different from zero. (d) Same as in (a) but for the bilinear transformation z2(x) where the film–substrate system is negative: N0=1, N1=1.46, and N2=3.85, at λ=632.8 nm. The upper image is at angle of incidence ϕ<ϕ±. The middle image (point z2=+1) is at ϕ=ϕ±. The lower image is at ϕ>ϕ± (ϕ±=70.559°).

Fig. 4
Fig. 4

(a) Members of the clockwise class of ρ-CAIC’s that correspond to the zero film–substrate system, where N0=1, N1=1.46, and N2=2.1316 at λ=632.8 nm. (b) Same as in (a) but corresponding to the positive film–substrate system, where N0=1, N1=1.46, and N2=2 at λ=632.8 nm. (c) Same as in (a) but corresponding to the negative film–substrate system, where ϕ(ϕ±, 90°) and N0=1, N1=1.46, and N2=3.85 at λ=632.8 nm. (d) Members of the counterclockwise class of ρ-CAIC’s that correspond to the negative film–substrate system, where ϕ(0°, ϕ±°) and N0=1, N1=1.46, and N2=3.85 at λ=632.8 nm.

Fig. 5
Fig. 5

The two auxiliary circles ξmin and ξmax superimposed on the ρ*-CAIC for the zero film–substrate system: N0=1, N1=1.46, and N2=2.1316 at ϕ=45° and λ=632.8 nm. The two branch points of z1(ρ*) are A1 and A2 (clockwise class).

Fig. 6
Fig. 6

Same as in Fig. 5 but for the positive system: N0=1, N1=1.46, and N2=2 at ϕ=45° and λ=632.8 nm (clockwise class).

Fig. 7
Fig. 7

Same as in Fig. 5 but for the negative system: N0=1, N1=1.46, and N2=3.85 at ϕ>ϕ±, ϕ=75° and λ=632.8 nm (clockwise class).

Fig. 8
Fig. 8

The two auxiliary circles ξmin and ξmax superimposed on the ρ*-CAIC for the negative system: N0=1, N1=1.46, and N2=3.85 at ϕ<ϕ±, ϕ=40°, and λ=632.8 nm. Notice that each of the two auxiliary circles intersects the ρ*-CAIC at two different points. These points—α1, α2, α3, and α4—are the branch points of zi(ρ*) (counterclockwise class).

Fig. 9
Fig. 9

Behavior of the discriminant fi as a function of the film thickness d for the positive system: N0=1, N1=1.46, and N2=2 at ϕ=45° and λ=632.8 nm, where d(0, Dϕ) (clockwise class). In this and subsequent figures, values given in angstroms are to be read as 10× the values in nanometers.

Fig. 10
Fig. 10

Same as in Fig. 9 but for the zero system: N0=1, N1=1.46, and N2=2.1316.

Fig. 11
Fig. 11

Same as in Fig. 9 but for the negative system: N0=1, N1=1.46, and N2=3.85 at ϕ>ϕ±, where ϕ=75°.

Fig. 12
Fig. 12

The two nonconcurrent lines (— . — . — . —) that divide the ρ*-CAIC (domain of zi(ρ*) ) into the four disconnected subdomains D1, D2, D3, and D4 for the negative system: N0=1, N1=1.46, and N2=3.85 at ϕ<ϕ± at λ=632.8 nm, where ϕ=40°. The four branch points and the subdomains are also shown (counterclockwise class).

Fig. 13
Fig. 13

Same as in Fig. 9 but for the negative system: N0=1, N1=1.46, and N2=3.85 at ϕ=40°<ϕ± and λ=632.8 nm (counterclockwise class).

Fig. 14
Fig. 14

Loci of points in the reduced ϕd plane that generate values of ρ* such that |ρ*| = |ξmax|. The loci are the inverse images of the branch points of zi(ρ*) of the complex ρ* plane, for both the zero film–substrate system—N0=1, N1=1.46, and N2=2.1316—and the positive film–substrate system: N0=1, N1=1.46, and N2=2, at λ=632.8 nm (clockwise class).

Fig. 15
Fig. 15

Loci of points in the reduced ϕd plane that generate values of ρ* such that |ρ*| = |ξmax| (---) and |ρ*| = |ξmin| (—). The loci are the inverse images of the branch points of zi(ρ*) of the complex ρ* plane, for the negative film–substrate system: N0=1, N1=1.46, and N2=3.85 at λ=632.8 nm (counterclockwise class).

Fig. 16
Fig. 16

Behavior of the constant-thickness contour in the complex ρ plane at d=100 nm for the negative film–substrate system: N0=1, N1=1.46, and N2=3.85 at λ=632.8 nm.

Fig. 17
Fig. 17

Same as in Fig. 16 but for d=400 nm.

Fig. 18
Fig. 18

The one-to-one correspondence of both ρ and ρ-1 between regions of the ϕd plane and the ρ plane for (a) the positive system, (b) the zero system, and (c) the negative system. The corresponding image of any region by either ρ or ρ-1 has the same inclined shading.

Equations (163)

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ρ=Rp/Rs,
Rν=r01ν+r12νx1+r01νr12νx,ν=p, s,
x=exp(-j2β).
β=πd/Dϕ,
Dϕ=λ2 (N12-N02 sin2 ϕ)-1/2
z1(x)=ab¯+x1+abx,
z2(x)=g¯+cxc+gx,
|z1(x)| = |z2(x)| =1.
J1=1-a2|b|2,
J2=c2-|g|2.
ρ=R1z1(x)+R2z2(x)+R3,
R1=(1-a2)b(cg-ab)(1-a2|b|2)(g-abc),
R2=(1-c2)g(ag-bc)(|g|2-c2)(g-abc),
R3=ac-R1ab¯-R2g¯/c.
x1±
=-a(bg¯-b¯g)±[a2(bg¯-b¯g)2+4|abc-g|2]1/22(abc-g).
ρ=(R1+R2)z+R3,atx=x1±.
x2±=-(ab¯g+abg¯+2c)±[(ab¯g+abg¯+2c)2-4|g+abc|2]1/22(g+abc).
ρ=(R1-R2)z+R3,atx=x2±.
x2±=-(abg+c)±[(c2-g2)(1-a2b2)]1/2(g+abc),
ρ=tan ψ exp(jΔ),
ρ*=R1z1+R2z2=|ρ*|exp(δΔ*).
zi(ρ*)=-Ki±(Ki2-4|ρ*Ri|2)1/22ρ¯*Ri,i=1, 2,
ρ*=ρ-R3=R1z1+R2z2,
K1=|R2|2-|R1|2-|ρ*|2,
K2=|R1|2-|R2|2-|ρ*|2.
zi+(ρ*)=-Ki+(Ki2-4|ρ*Ri|2)1/22ρ¯*Ri,i=1, 2,
zi-(ρ*)=-Ki-(Ki2-4|ρ*Ri|2)1/22ρ¯*Ri,i=1, 2.
|zi+(ρ*)| = |zi-(ρ*)| =1.
fi=Ki2-4|ρ*Ri|2,i=1, 2,
f1=f2=(|ρ*|+|R1|+|R2|)(|ρ*|-|R1|+|R2|)×(|ρ*|+|R1|-|R2|)(|ρ*|-|R1|-|R2|).
||R2z2|-|R1z1||  |ρ*|  |R1z1|+|R2z2|.
||R2|-|R1||  |ρ*|  |R1|+|R2|.
|ξmax| = |R1|+|R2|,
|ξmin| = ||R2|-|R1||.
ρ*=ρ-R3,
ρ=tan ψ exp(jΔ).
DU=ρ* : ρ* lies in the upper part with respect
to the cutting line and
DL=ρ* : ρ* lies in the lower part with respect
to the cutting line.
z1-(ρ*)isthebranchifρ*DU,
z1+(ρ*)isthebranchifρ*DL.
(z1(x), z2(x))=(z1-(ρ*), z2+(ρ*))  ρ*DU,
(z1(x), z2(x))=(z1+(ρ*), z2-(ρ*))  ρ*DL.
xU(ρ*)=ab¯-z1-(ρ*)abz1-(ρ*)-1g¯-cz2+(ρ*)gz2+(ρ*)-c ρ*DU,
xL(ρ*)=ab¯-z1+(ρ*)abz1+(ρ*)-1g¯-cz2-(ρ*)gz2+(ρ*)-c ρ*DL.
dr=Dϕ2π cos-1 (Re xU(ρ*)) ρ*DU,
dr=Dϕ2π [2π-cos-1(Re xL(ρ*))] ρ*DL,
ρ1*=ρ*||ρ*| = |ξmax|=(Re ρ1*, Im ρ1*),
ρ2*=ρ*||ρ*| = |ξmax|=(Re ρ2*, Im ρ2*).
I-Im ρ1*R-Re ρ1*-Im ρ2*-Im ρ1*Re ρ2*-Re ρ1*=0,
DU={ρ* : Im ρ*>0},
DL={ρ* : Im ρ*<0}.
Δ1*<Δ2*<Δ3*<Δ4*,
Δn*=arg αn,
D1={ρ* : Δ1*<arg ρ*<Δ2*},
D2={ρ* : Δ2*<arg ρ*<Δ3*},
D3={ρ* : Δ3*<arg ρ*<Δ4*},
D4={ρ* : Δ4*<arg ρ*<Δ1*+2π}.
z1+(ρ*)isthebranchifρ*D1,
z1-(ρ*)isthebranchifρ*D2,
z1+(ρ*)isthebranchifρ*D3,
z1-(ρ*)isthebranchifρ*D4.
(z1(x), z2(x))(z1+(ρ*), z2-(ρ*)) ρ*D1D3,
(z1(x), z2(x))(z1-(ρ*), z2+(ρ*)) ρ*D2D4.
x13(ρ*)=ab¯-z1+(ρ*)abz1+(ρ*)-1g¯-cz2-(ρ*)gz2-(ρ*)-c ρ*D1D3,
x24(ρ*)=ab¯-z1-(ρ*)abz1-(ρ*)-1g¯-cz2+(ρ*)gz2+(ρ*)-c ρ*D2D4.
dr=Dϕ2π cos-1[Re x13(ρ*)] ρ*D1,
dr=Dϕ2π cos-1[Re x24(ρ*)] ρ*D2,
dr=Dϕ2π [2π-cos-1(Re x13(ρ*))] ρ*D3,
dr=Dϕ2π [2π-cos-1(Re x24(ρ*))] ρ*D4.
(i)ρ*at|ρ*| = |ξmin|orat|ρ*| = |ξmax|,
(ii)ρ*at|ρ*|,
(iii)ρ*atϕ=0°or90°.
zi(ρ*)=zi+(ρ*)=zi-(ρ*)=-Ki2ρ¯*Ri,i=1,2.
zi(ρ*)=-ρ*R¯i|Ri|(|R2|-|R1|)at|ρ*| = |ξmin|,
zi(ρ*)=ρ*R¯i|Ri|(|R2|+|R1|)at|ρ*| = |ξmax|,
dr=j Dϕ2π ln(x1+)atarg ρ*=Δ1*,
dr=j Dϕ2π ln(x1-)atarg ρ*=Δ3*,
dr=j Dϕ2π ln(x2+)atarg ρ*=Δ2*,
dr=j Dϕ2π ln(x2-)atarg ρ*=Δ4*.
ρ±atc+gx=0.
dr=j Dϕ2π ln-cg.
ρx=ρ*x=0atϕ=0°or90°.
Ax2+Bx+C=0,
A=bc[g2(1-c2)-ab(1-g2)],
B=b[g(1-a2)(1+c2)-c(1-g2)(1+a2)],
C=c[b(1-a2)-a(1-g2)].
x=-B±B2-4AC2A.
|x| =-B±B2-4AC2A1,
d=dr+μDϕ,
dmax=(1+n)λ4[N1-(N12-N02)1/2].
μdmax-drϕDϕ,
A(ϕ)={d : d=dr+μDϕ}, and μ satisfies
theMFT.
ψ=41.968°,Δ=40.033°atϕ=80°,
ψ=43.326°,Δ=20.001°atϕ=85°.
Im ρ*=0.5785
Im ρ*>0 ρ*DU.
Im ρ*=0.322,
Im ρ*>0 ρ*DU.
dmax=399.26nm.
μ1.02  μ=0or1.
A(80°)={100, 393.548}.
μ1.01  μ=0or1.
A(85°)={100, 396.44}.
A(80°)A(85°)={100}nm.
ψ=44.249°,Δ=40.191°,atϕ=80°,
ψ=43.953°,Δ=20.044°,atϕ=85°.
Im ρ*=0.628°,
>0 ρ*DU.
dr=106.45178nm.
Im ρ*=0.331
>0 ρ*DU.
dr=103.55971nm.
dmax=798.51984nm.
μ<2.36  μ=0, 1,or2.
A(80°)={106.45178, 400, 693.5478}.
μ2.34  μ=0, 1,or2.
A(85)={103.55971, 400, 696. 4397}.
A(80°)A(85°)={400}nm.
ψ=45.571°,Δ=-20.056°,atϕ=40°,
ψ=44.458°,Δ=-38.759°,atϕ=50°.
α1=(0.241, 0)atϕ=40°,
=(0.484, 0)atϕ=50°,
α2=(0.193, 0.329)atϕ=40°,
=(0.437, 0.554)atϕ=50°,
α3=(-0.241, 0)atϕ=40°,
=(-0.484, 0)atϕ=50°,
α4=(-0.193, 0.329)atϕ=40°,
=(-0.437, 0.554)atϕ=50°.
D1={ρ* : 0°<arg ρ*<59.581°},
D2={ρ* : 59.581°<arg ρ*<180°},
D3={ρ* : 180°<arg ρ*<300.419°},
D4={ρ* : 300.419°<arg ρ*<360°}.
D1={ρ* : 0°<arg ρ*<51.737°},
D2={ρ* : 51.737°<arg ρ*<180°},
D3={ρ* : 180°<arg ρ*<308.262°},
D4={ρ* : 308.262°<arg ρ*<360°}.
arg ρ*|ϕ=40°=1.475°  arg ρ*|ϕ=40°D1|ϕ=40°,
arg ρ*|ϕ=50°=1.078°  arg ρ*|ϕ=50°D1|ϕ=50°.
dr=100nmatbothϕ = 40°andϕ=50°.
D=dr=100nm.
d (arg ρ*)0.
d |ρ*| =0atx1±and/orx2±.
d (arg ρ)=0atIm1ρ d (ρ)=0.
x=-(a/b).
sin2 ϕp=N22N02+N22.
d |ρ| =0atRe1ρ d (ρ)=0.
ρ=ρ*+R3,
ρd=ρ*d.
exp[j(Δ-Δ*)]=|ρ|d+j|ρ*| Δ*d/
|ρ|d+j|ρ| Δd.
d |ρ| =d |ρ*| =0.
ϕdplaneXplaneZiplaneρ*plane  ρplane.
ϕ=ϕ±,dr=12 Dϕ±,
ϕ=ϕp,dr=Dϕp.
ϕ=ϕp,dr=Dϕp.
ϕ=ϕp1,dr=12 Dϕp1,
ϕ=ϕp2,dr=Dϕp2.
cos ϕp1=N0N2 cos2 ϕ1cos ϕ2,
ln z=γ drr+jγ dθ=ln|z|+jΔγ arg z.

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