## 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 ($ϕ–d$ 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 $ϕ–d$ 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 $ϕ–d$ plane and the ρ plane induced by ρ and $ρ-1$ is given.

© 1999 Optical Society of America

### 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 (3)

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]

#### 1976 (1)

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]

#### Bashara, N. M.

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]

#### 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]

#### 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]

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]

#### 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]

#### Zaghloul, A.-R. M.

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]

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).

#### Surf. Sci. (1)

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 (2)

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 (6)

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).

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.

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*).

### Cited By

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### Figures (18)

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

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

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 (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 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 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

(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 (b) Same as in (a) but corresponding to the positive film–substrate system, where $N0=1,$ $N1=1.46,$ and $N2=2$ at (c) Same as in (a) but corresponding to the negative film–substrate system, where and $N0=1,$ $N1=1.46,$ and $N2=3.85$ at (d) Members of the counterclockwise class of ρ-CAIC’s that correspond to the negative film–substrate system, where and $N0=1,$ $N1=1.46,$ and $N2=3.85$ at

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 The two branch points of $z1(ρ*)$ are $A1$ and $A2$ (clockwise class).

Fig. 6

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

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 (clockwise class).

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 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

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 where (clockwise class). In this and subsequent figures, values given in angstroms are to be read as 10× the values in nanometers.

Fig. 10

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

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

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 where $ϕ=40°.$ The four branch points and the subdomains are also shown (counterclockwise class).

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 (counterclockwise class).

Fig. 14

Loci of points in the reduced $ϕ–d$ plane that generate values of $ρ*$ such that 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 (clockwise class).

Fig. 15

Loci of points in the reduced $ϕ–d$ plane that generate values of $ρ*$ such that (---) and (—). 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 (counterclockwise class).

Fig. 16

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

Fig. 17

Same as in Fig. 16 but for

Fig. 18

The one-to-one correspondence of both ρ and $ρ-1$ between regions of the $ϕ–d$ 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)

$ρ=Rp/Rs,$
$x=exp(-j2β).$
$β=πd/Dϕ,$
$z1(x)=ab¯+x1+abx,$
$z2(x)=g¯+cxc+gx,$
$J1=1-a2|b|2,$
$J2=c2-|g|2.$
$ρ=R1z1(x)+R2z2(x)+R3,$
$R1=(1-a2)b(cg-ab)(1-a2|b|2)(g-abc),$
$R2=(1-c2)g(ag-bc)(|g|2-c2)(g-abc),$
$R3=ac-R1ab¯-R2g¯/c.$
$x1±$
$=-a(bg¯-b¯g)±[a2(bg¯-b¯g)2+4|abc-g|2]1/22(abc-g).$
$ρ=(R1+R2)z+R3, at x=x1±.$
$x2±=-(ab¯g+abg¯+2c)±[(ab¯g+abg¯+2c)2-4|g+abc|2]1/22(g+abc).$
$ρ=(R1-R2)z+R3, at x=x2±.$
$x2±=-(abg+c)±[(c2-g2)(1-a2b2)]1/2(g+abc),$
$ρ*=R1z1+R2z2=|ρ*|exp(δΔ*).$
$ρ*=ρ-R3=R1z1+R2z2,$
$K1=|R2|2-|R1|2-|ρ*|2,$
$K2=|R1|2-|R2|2-|ρ*|2.$
$f1=f2=(|ρ*|+|R1|+|R2|)(|ρ*|-|R1|+|R2|)×(|ρ*|+|R1|-|R2|)(|ρ*|-|R1|-|R2|).$
$ρ*=ρ-R3,$
$z1-(ρ*) is the branch if ρ*∈DU,$
$z1+(ρ*) is the branch if ρ*∈DL.$
$Δ1*<Δ2*<Δ3*<Δ4*,$
$z1+(ρ*) is the branch if ρ*∈D1,$
$z1-(ρ*) is the branch if ρ*∈D2,$
$z1+(ρ*) is the branch if ρ*∈D3,$
$z1-(ρ*) is the branch if ρ*∈D4.$
$(ii) ρ* at |ρ*|→∞,$
$(iii) ρ* at ϕ=0° or 90°.$
$zi(ρ*)=zi+(ρ*)=zi-(ρ*)=-Ki2ρ¯*Ri, i=1,2.$
$ρ→±∞ at c+gx=0.$
$∂ρ∂x=∂ρ*∂x=0 at ϕ=0° or 90°.$
$Ax2+Bx+C=0,$
$A=bc[g2(1-c2)-ab(1-g2)],$
$B=b[g(1-a2)(1+c2)-c(1-g2)(1+a2)],$
$C=c[b(1-a2)-a(1-g2)].$
$x=-B±B2-4AC2A.$
$d=dr+μDϕ,$
$dmax=(1+n)λ4[N1-(N12-N02)1/2].$
$μ≤dmax-drϕDϕ,$
$the MFT.$
$ψ=41.968°, Δ=40.033° at ϕ′=80°,$
$ψ=43.326°, Δ=20.001° at ϕ″=85°.$
$dmax=399.26 nm.$
$A(80°)∩A(85°)={100} nm.$
$ψ=44.249°, Δ=40.191°, at ϕ′=80°,$
$ψ=43.953°, Δ=20.044°, at ϕ″=85°.$
$dr=106.45178 nm.$
$dr=103.55971 nm.$
$dmax=798.51984 nm.$
$A(80°)∩A(85°)={400} nm.$
$ψ=45.571°, Δ=-20.056°, at ϕ′=40°,$
$ψ=44.458°, Δ=-38.759°, at ϕ″=50°.$
$D=dr=100 nm.$
$x=-(a/b).$
$ρ=ρ*+R3,$
$∂ρ∂d=∂ρ*∂d.$
$ϕ=ϕp, dr=Dϕp.$
$ϕ=ϕp, dr=Dϕp.$
$ϕ=ϕp2, dr=Dϕp2.$