Abstract

The complex reflection coefficients Rν(ϕ,ζ) of a film–substrate system for the parallel (ν = p) and perpendicular (ν = s) polarizations are examined in detail as functions of the angle of incidence ϕ(0 ≤ ϕ ≤ 90°) and the reduced normalized film thickness ζ(0 ≤ ζ < 1). For definiteness, the reflection of light of wavelength λ = 0.6328 μm by the air–SiO2–Si system is assumed. Families of circles that represent the constant-angle-of-incidence contours, their envelopes, and the associated constant-thickness contours of Rp and Rs are all presented in the complex plane. Furthermore, the amplitude-reflectance and phase-shift functions, |Rν|(ϕ,ζ) and argRν(ϕ,ζ), are plotted vs ζ with ϕ constant and vs ϕ with ζ constant. It is shown that Rp or Rs can assume the same complex value at two different angles of incidence (i.e., the film–substrate system can have identical reflection characteristics for a given polarization at two angles) for certain ranges of film thickness. The distinct case of internal reflection is represented by a separate example.

© 1983 Optical Society of America

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  1. R. M. A. Azzam, A.-R. M. Zaghloul, N. M. Bashara, J. Opt. Soc. Am. 65, 252 (1975).
    [CrossRef]
  2. A.-R. M. Zaghloul, R. M. A. Azzam, Appl. Opt. 21, 739 (1982).
    [CrossRef] [PubMed]
  3. G. Gergely, Ed., Ellipsometric Tables of the Si–SiO2System for Mercury and He–Ne Laser Spectral Lines (Akademiai Kaido, Budapest, 1971).
  4. A. B. Winterbottom, in The Royal Norwegian Scientific Society Report 1 (F. Burns, Trondheim, 1955).
  5. K. D. Naegele, J. Phys. Paris Colloq. C5 38, C5-225 (1977). This paper discusses some of the characteristics of the complex ratio Rν/r02ν of the reflection coefficient of the system Rν to that of the ambient–substrate interface r02ν.
    [CrossRef]
  6. M. D. Williams, Appl. Opt. 21, 747 (1982).
    [CrossRef] [PubMed]
  7. See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.3.
  8. See, for example, A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972), Sec. 8.4.
  9. F. Abelès, J. Phys. Radium 11, 310 (1950).
    [CrossRef]
  10. The envelope is determined as follows: A given value of Re(Rp) > − 1 is assumed, and ϕ is swept in search of a circle that intersects the line Re(Rp) = constant at a point that is most distant from the real axis [i.e., of maximum possible Im(Rp) for the given value of Re(Rp)]. We have found that this procedure works better for this problem than an established method described by G. Zwikker, The Advanced Geometry of Plane Curves and their Applications (Dover, New York, 1963), Chap. 13.
  11. W. L. Wolfe, G. J. Zissis, Eds., The Infrared Handbook (Office of Naval Research, Department of the Navy, Arlington, Va., 1978), p. 7–18.
  12. M. E. Pedinoff, M. Braunstein, O. M. Stafsudd, in Optical Polarimetry, R. M. A. Azzam, D. L. Coffeen, Eds., Proc. Soc. Photo-Opt. Instrum. Eng.112, 74 (1977).
    [CrossRef]
  13. R. M. A. Azzam, Surf. Sci. 96, 67 (1980).
    [CrossRef]

1982 (2)

1980 (1)

R. M. A. Azzam, Surf. Sci. 96, 67 (1980).
[CrossRef]

1977 (1)

K. D. Naegele, J. Phys. Paris Colloq. C5 38, C5-225 (1977). This paper discusses some of the characteristics of the complex ratio Rν/r02ν of the reflection coefficient of the system Rν to that of the ambient–substrate interface r02ν.
[CrossRef]

1975 (1)

1950 (1)

F. Abelès, J. Phys. Radium 11, 310 (1950).
[CrossRef]

Abelès, F.

F. Abelès, J. Phys. Radium 11, 310 (1950).
[CrossRef]

Azzam, R. M. A.

A.-R. M. Zaghloul, R. M. A. Azzam, Appl. Opt. 21, 739 (1982).
[CrossRef] [PubMed]

R. M. A. Azzam, Surf. Sci. 96, 67 (1980).
[CrossRef]

R. M. A. Azzam, A.-R. M. Zaghloul, N. M. Bashara, J. Opt. Soc. Am. 65, 252 (1975).
[CrossRef]

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

Bashara, N. M.

R. M. A. Azzam, A.-R. M. Zaghloul, N. M. Bashara, J. Opt. Soc. Am. 65, 252 (1975).
[CrossRef]

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

Braunstein, M.

M. E. Pedinoff, M. Braunstein, O. M. Stafsudd, in Optical Polarimetry, R. M. A. Azzam, D. L. Coffeen, Eds., Proc. Soc. Photo-Opt. Instrum. Eng.112, 74 (1977).
[CrossRef]

Kyrala, A.

See, for example, A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972), Sec. 8.4.

Naegele, K. D.

K. D. Naegele, J. Phys. Paris Colloq. C5 38, C5-225 (1977). This paper discusses some of the characteristics of the complex ratio Rν/r02ν of the reflection coefficient of the system Rν to that of the ambient–substrate interface r02ν.
[CrossRef]

Pedinoff, M. E.

M. E. Pedinoff, M. Braunstein, O. M. Stafsudd, in Optical Polarimetry, R. M. A. Azzam, D. L. Coffeen, Eds., Proc. Soc. Photo-Opt. Instrum. Eng.112, 74 (1977).
[CrossRef]

Stafsudd, O. M.

M. E. Pedinoff, M. Braunstein, O. M. Stafsudd, in Optical Polarimetry, R. M. A. Azzam, D. L. Coffeen, Eds., Proc. Soc. Photo-Opt. Instrum. Eng.112, 74 (1977).
[CrossRef]

Williams, M. D.

Winterbottom, A. B.

A. B. Winterbottom, in The Royal Norwegian Scientific Society Report 1 (F. Burns, Trondheim, 1955).

Zaghloul, A.-R. M.

Zwikker, G.

The envelope is determined as follows: A given value of Re(Rp) > − 1 is assumed, and ϕ is swept in search of a circle that intersects the line Re(Rp) = constant at a point that is most distant from the real axis [i.e., of maximum possible Im(Rp) for the given value of Re(Rp)]. We have found that this procedure works better for this problem than an established method described by G. Zwikker, The Advanced Geometry of Plane Curves and their Applications (Dover, New York, 1963), Chap. 13.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Phys. Paris Colloq. C5 (1)

K. D. Naegele, J. Phys. Paris Colloq. C5 38, C5-225 (1977). This paper discusses some of the characteristics of the complex ratio Rν/r02ν of the reflection coefficient of the system Rν to that of the ambient–substrate interface r02ν.
[CrossRef]

J. Phys. Radium (1)

F. Abelès, J. Phys. Radium 11, 310 (1950).
[CrossRef]

Surf. Sci. (1)

R. M. A. Azzam, Surf. Sci. 96, 67 (1980).
[CrossRef]

Other (7)

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

See, for example, A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972), Sec. 8.4.

The envelope is determined as follows: A given value of Re(Rp) > − 1 is assumed, and ϕ is swept in search of a circle that intersects the line Re(Rp) = constant at a point that is most distant from the real axis [i.e., of maximum possible Im(Rp) for the given value of Re(Rp)]. We have found that this procedure works better for this problem than an established method described by G. Zwikker, The Advanced Geometry of Plane Curves and their Applications (Dover, New York, 1963), Chap. 13.

W. L. Wolfe, G. J. Zissis, Eds., The Infrared Handbook (Office of Naval Research, Department of the Navy, Arlington, Va., 1978), p. 7–18.

M. E. Pedinoff, M. Braunstein, O. M. Stafsudd, in Optical Polarimetry, R. M. A. Azzam, D. L. Coffeen, Eds., Proc. Soc. Photo-Opt. Instrum. Eng.112, 74 (1977).
[CrossRef]

G. Gergely, Ed., Ellipsometric Tables of the Si–SiO2System for Mercury and He–Ne Laser Spectral Lines (Akademiai Kaido, Budapest, 1971).

A. B. Winterbottom, in The Royal Norwegian Scientific Society Report 1 (F. Burns, Trondheim, 1955).

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Figures (21)

Fig. 1
Fig. 1

Reflection of p- or s-polarized light at an angle ϕ by a film (1)–substrate (2) system. Medium 0 is the ambient, and d is the film thickness.

Fig. 2
Fig. 2

Unit circle of X.

Fig. 3
Fig. 3

CAICs of Rp as a family of circles with the angle of incidence ϕ marked on each contour. The air–SiO2–Si system is assumed at λ = 0.6328 μm.

Fig. 4
Fig. 4

Envelope curves PQ and PQ′ of the CAICs of Fig. 3 and the circle arc Q Q′ of the ϕ = 0 circle.

Fig. 5
Fig. 5

Film thickness period Dϕ vs angle of incidence ϕ for an air ambient and SiO2 film at λ = 0.6328 μm.

Fig. 6
Fig. 6

CTCs of Rp for film thicknesses d = (m/20)D0, m = 0,1,2,…,20. The air–SiO2–Si system is assumed at λ = 0.6328 μm. Along each CTC, ϕ varies from 0 to 90°.

Fig. 7
Fig. 7

Same as Fig. 6 for thicknesses d = D0 + (m/10)(D90 − D0), m = 0,1,2,…,10. Along each CTC, ϕ varies from ϕ ¯, Eq. (12), to 90°.

Fig. 8
Fig. 8

Amplitude reflectance |Rp| (a) and reflection phase shift δp (b) vs normalized film thickness ζ with the angle of incidence ϕ as a parameter marked by each curve for the air–SiO2–Si system at λ = 0.6328 μm.

Fig. 9
Fig. 9

Amplitude reflectance |Rp| (a) and reflection phase shift δp (c) vs angle of incidence ϕ along the CTCs of Fig. 6. (b) and (d) depict the same quantities for the CTCs of Fig. 7.

Fig. 10
Fig. 10

CAICs of Rs (a) and their envelope (b) for the air–SiO2–Si system at λ = 0.6328 μm.

Fig. 11
Fig. 11

CTCs of Rs for the air–SiO2–Si system at λ = 0.6328 μm. (a) d = (m/20)D0, m = 0,1,2,…,20, and (b) d = D0 + (m/10)(D90D0), m = 0,1,2,…,10.

Fig. 12
Fig. 12

Amplitude reflectance |Rs| (a) and reflection phase shift δs (b) vs normalized film thickness ζ with the angle of incidence ϕ as a parameter marked by each curve for the air–SiO2–Si system at λ = 0.6328 μm.

Fig. 13
Fig. 13

Amplitude reflectance |Rs| (a) and reflection phase shift δs (c) vs angle of incidence ϕ along the CTCs of Fig. 11(a). (b) and (d) depict the same quantities for the CTCs of Fig. 11(b).

Fig. 14
Fig. 14

CAICs of Rp for partial internal reflection below the critical angle for the Ge–ThF4–air system at λ = 10.6 μm. The distinct circles that lie within the ϕ = 0 circle correspond to ϕ = 4, 6, 8, 10, and 12° in order of decreasing radius.

Fig. 15
Fig. 15

Associated CTCs of Rp for d = (m/5)D0, m = 0,1,…, 5.

Fig. 16
Fig. 16

CAICs of Rs for partial internal reflection below the critical angle by the Ge–ThF4–air system at λ = 10.6 μm. The circles that lie between the ϕ = 0 and ϕ = 13.5° correspond to angles ϕ = 2, 4, 6, 8, 10, 11.65, 12, 12.5, and 13° in order of increasing radius.

Fig. 17
Fig. 17

CTCs of Rs associated with Fig. 16 for d = (m/5)D0, m =

Fig. 18
Fig. 18

Family of overlapping (a) and nonoverlapping (b) circle arcs that represent the CAICs of Rp and Rs, respectively, for total internal reflection above the critical angle by the Ge–ThF4–air system at 10.6 μm.

Fig. 19
Fig. 19

CTC of Rp for SiO2 film thickness d = 78.4 nm showing a small loop. The air–SiO2–Si system is assumed at λ = 0.6328 μm.

Fig. 20
Fig. 20

CTCs of Rp for the air–SiO2–Si system for film thicknesses of (a) 0.5948 μm, (b) 1.041 μm, and (c) 1.487 μm. Each CTC intersects itself and hence shows one or more loops.

Fig. 21
Fig. 21

Same as in Fig. 20 for Rs.

Equations (16)

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R ν = r 01 ν + r 12 ν X 1 + r 01 ν r 12 ν X , ν = p , s ,
X = exp ( j 2 π ζ ) .
ζ = d / D ϕ
D ϕ = λ 2 ( N 1 2 N 0 2 sin 2 ϕ ) 1 / 2
N 1 sin ϕ = N 0 sin ϕ .
R ν = R ν ( ϕ , ζ ) , ν = p , s .
0 ϕ 90 ° ,
0 ζ < 1.
R ν ( ϕ , ζ ) = R ν ( ϕ , ζ + m ) ,
| R ν | = | R ν | ( ϕ , ζ ) ,
δ ν = arg R ν ( ϕ , ζ ) , ν = p , s
ϕ = ϕ B 01 = tan 1 1.46 = 55.59 ° ,
ϕ ¯ = arcsin [ ( N 1 / N 0 ) 2 ( λ / 2 d N 0 ) 2 ] 1 / 2 .
ϕ c = arcsin ( N 2 / N 0 ) = arcsin ( 0.25 ) = 14.48 ° .
csc 2 ϕ 0 = ( N 0 / N 2 ) 2 + ( N 0 / N 1 ) 2 ,
R p = r 02 p = 0.413.

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