Abstract

Methods for determining the optical constants of a thin plane–parallel slab are developed without using information from other measurements. For the ideal case (negligible absorption and thickness inhomogeneity), a purely geometrical method even avoids the use of relative intensities. The more realistic situations (weak absorption or gain and finite thickness inhomogeneity) are also discussed. The methods are used to determine the refractive index of CdS platelets below the exciton region at room temperature.

© 1980 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).
  2. M. V. Klein, Optics (Wiley, New York, 1970).
  3. H. Mayer, Physick dünner Schichten (Wiss. Verlagsgellschaft, Stuttgart, 1950).
  4. A. Vašiček, Optics of Thin Films (North-Holland, Amsterdam, 1960).
  5. W. A. Pliskin, in Physical Measurement and Analysis of Thin Films, E. M. Murt, W. G. Guldner, Eds. (Plenum, New York, 1969), p. 1.
  6. With increasing η, however, the extreme condition approximates cos2ηz1 = −1; it is correct for φ = 0°.
  7. Equation (6) gives the linear relation with Mi,i+1=(λ/d)(n2-sin2φ0)1/2+(λ/2d)2(2i+1),φ0 = max(φi) and i = 0,1,2… corresponding to the measured extreme angles with φ0 > φ1 > φ2…. From numerical calculations we found that a least squares fit gives n and d to within 1% of the correct value if one assumes φi to be correct to within 0.1° for N ≥ 5 and 1 ≤ n ≤ 3.
  8. For strong absorption reflectivity measurements are inevitable; ellipsometric methods9–12 are clearly favored.
  9. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  10. D. E. Aspnes, in Optical Properties of Solids—New Developments, B. O. Seraphin, Ed. (North-Holland, Amsterdam, 1976), p. 800.
  11. M. M. Gorshkov, Ellipsometry (Soviet Radio, Moscow), 1974).
  12. G. Jungk, “Ellipsometrie—eine Methode zur Untersuchung der elektronischen Struktur fester Körper,” Dissertation, Humboldt U., Berlin (1978).
  13. P. Beckmann, A. Spizzichino, Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  14. H. G. Hecht, J. Res. Nat. Bur. Stand. Sect. A: 80, 567 (1976).
    [CrossRef]
  15. I. Ohlidal, K. Navratil, Thin Solid Films 44, 313 (1977).
    [CrossRef]
  16. J. Bauer, Exp. Tech. Phys. 25, 105 (1977).
  17. In contradiction to the experimental hints one could argue that the observed deviation from Tmax = 1 is due to absorption. However, substitution of the experimental data into Eq. (31) yields n0 = 1.9 for λ = 5700 Å and φ dependent, thus demonstrating the actual influence of thickness inhomogeneities. Naturally, the assumption of the ideal behavior of Sec. II [Tmax = 1 and calculation of n from Tmin of Eq. (3)] leads to a similar incorrect value of n0 = 1.8.
  18. T. M. Bieniewski, S. J. Szyzak, J. Opt. Soc. Am. 53, 496 (1963).
    [CrossRef]

1977 (2)

I. Ohlidal, K. Navratil, Thin Solid Films 44, 313 (1977).
[CrossRef]

J. Bauer, Exp. Tech. Phys. 25, 105 (1977).

1976 (1)

H. G. Hecht, J. Res. Nat. Bur. Stand. Sect. A: 80, 567 (1976).
[CrossRef]

1963 (1)

Aspnes, D. E.

D. E. Aspnes, in Optical Properties of Solids—New Developments, B. O. Seraphin, Ed. (North-Holland, Amsterdam, 1976), p. 800.

Azzam, R. M. A.

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

Bashara, N. M.

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

Bauer, J.

J. Bauer, Exp. Tech. Phys. 25, 105 (1977).

Beckmann, P.

P. Beckmann, A. Spizzichino, Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Bieniewski, T. M.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Gorshkov, M. M.

M. M. Gorshkov, Ellipsometry (Soviet Radio, Moscow), 1974).

Hecht, H. G.

H. G. Hecht, J. Res. Nat. Bur. Stand. Sect. A: 80, 567 (1976).
[CrossRef]

Jungk, G.

G. Jungk, “Ellipsometrie—eine Methode zur Untersuchung der elektronischen Struktur fester Körper,” Dissertation, Humboldt U., Berlin (1978).

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970).

Mayer, H.

H. Mayer, Physick dünner Schichten (Wiss. Verlagsgellschaft, Stuttgart, 1950).

Navratil, K.

I. Ohlidal, K. Navratil, Thin Solid Films 44, 313 (1977).
[CrossRef]

Ohlidal, I.

I. Ohlidal, K. Navratil, Thin Solid Films 44, 313 (1977).
[CrossRef]

Pliskin, W. A.

W. A. Pliskin, in Physical Measurement and Analysis of Thin Films, E. M. Murt, W. G. Guldner, Eds. (Plenum, New York, 1969), p. 1.

Spizzichino, A.

P. Beckmann, A. Spizzichino, Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Szyzak, S. J.

Vašicek, A.

A. Vašiček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Exp. Tech. Phys. (1)

J. Bauer, Exp. Tech. Phys. 25, 105 (1977).

J. Opt. Soc. Am. (1)

J. Res. Nat. Bur. Stand. Sect. A: (1)

H. G. Hecht, J. Res. Nat. Bur. Stand. Sect. A: 80, 567 (1976).
[CrossRef]

Thin Solid Films (1)

I. Ohlidal, K. Navratil, Thin Solid Films 44, 313 (1977).
[CrossRef]

Other (14)

In contradiction to the experimental hints one could argue that the observed deviation from Tmax = 1 is due to absorption. However, substitution of the experimental data into Eq. (31) yields n0 = 1.9 for λ = 5700 Å and φ dependent, thus demonstrating the actual influence of thickness inhomogeneities. Naturally, the assumption of the ideal behavior of Sec. II [Tmax = 1 and calculation of n from Tmin of Eq. (3)] leads to a similar incorrect value of n0 = 1.8.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

M. V. Klein, Optics (Wiley, New York, 1970).

H. Mayer, Physick dünner Schichten (Wiss. Verlagsgellschaft, Stuttgart, 1950).

A. Vašiček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

W. A. Pliskin, in Physical Measurement and Analysis of Thin Films, E. M. Murt, W. G. Guldner, Eds. (Plenum, New York, 1969), p. 1.

With increasing η, however, the extreme condition approximates cos2ηz1 = −1; it is correct for φ = 0°.

Equation (6) gives the linear relation with Mi,i+1=(λ/d)(n2-sin2φ0)1/2+(λ/2d)2(2i+1),φ0 = max(φi) and i = 0,1,2… corresponding to the measured extreme angles with φ0 > φ1 > φ2…. From numerical calculations we found that a least squares fit gives n and d to within 1% of the correct value if one assumes φi to be correct to within 0.1° for N ≥ 5 and 1 ≤ n ≤ 3.

For strong absorption reflectivity measurements are inevitable; ellipsometric methods9–12 are clearly favored.

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

D. E. Aspnes, in Optical Properties of Solids—New Developments, B. O. Seraphin, Ed. (North-Holland, Amsterdam, 1976), p. 800.

M. M. Gorshkov, Ellipsometry (Soviet Radio, Moscow), 1974).

G. Jungk, “Ellipsometrie—eine Methode zur Untersuchung der elektronischen Struktur fester Körper,” Dissertation, Humboldt U., Berlin (1978).

P. Beckmann, A. Spizzichino, Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

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Figures (16)

Fig. 1
Fig. 1

Schematic arrangement for optical studies on a plane–parallel slab.

Fig. 2
Fig. 2

Transmission vs angle of incidence for a slab with d/λ = 10 and n = 2.5 for perpendicular polarization.

Fig. 3
Fig. 3

Transmission vs angle of incidence for a slab with d/λ = 10 and n = 2.5 for parallel polarization.

Fig. 4
Fig. 4

Transmission vs angle of incidence for very thin films with n = 2.5 for perpendicular polarization. The numbers in parentheses denote the order x0 corresponding to d/λ = 0.6, 1, and 1.4, respectively (see text).

Fig. 5
Fig. 5

Transmission vs angle of incidence for very thin films with n = 2.5 for parallel polarization. The numbers in parentheses denote the order x0 corresponding to d/λ = 0.6, 1, and 1.4, respectively (see text).

Fig. 6
Fig. 6

Transmission vs angle of incidence for a slab with d/λ = 10 and n ^ = 2.5 + i × 10−3 for perpendicular polarization. The envelopes are calculated from Eqs. (18a) and (19), respectively.

Fig. 7
Fig. 7

Transmission vs angle of incidence for a slab with d/λ = 10 and n ^ = 2.5 + i × 10−3 for parallel polarization. The envelopes are calculated from Eqs. (18a) and (19), respectively.

Fig. 8
Fig. 8

Reflectivity vs angle of incidence for a slab with d/λ = 10 and n ^ = 2.5 + i × 10−3 for perpendicular polarization. The envelope is calculated from Eq. (20).

Fig. 9
Fig. 9

Reflectivity vs angle of incidence for a slab with d/λ = 10 and n ^ = 2.5 + i × 10−3 for parallel polarization. The envelope is calculated from Eq. (20).

Fig. 10
Fig. 10

Transmission vs angle of incidence for a slab with d/λ = 10 and n ^ = 2.5 − i × 10−3 for perpendicular polarization. The envelopes are calculated from Eqs. (18a) and (19), respectively.

Fig. 11
Fig. 11

Transmission vs angle of incidence for a slab with d/λ = 10 and n ^ = 2.5 − i × 10−3 for parallel polarization. The envelopes are calculated from Eqs. (18a) and (19), respectively.

Fig. 12
Fig. 12

Reflectivity vs angle of incidence for a slab with d/λ = 10 and n ^ = 2.5 − i × 10−3 for perpendicular polarization. The envelope is calculated from Eq. (20).

Fig. 13
Fig. 13

Reflectivity vs angle of incidence for a slab with d/λ = 10 and n ^ = 2.5 − i × 10−3 for parallel polarization. The envelope is calculated from Eq. (20).

Fig. 14
Fig. 14

Typical λmax(φ) for a CdS platelet.

Fig. 15
Fig. 15

Construction requied to determine Tmax and Tmin for one wavelength from the experiment.

Fig. 16
Fig. 16

Dispersion of the refractive index of CdS: the dots show the results from Bieniewski and Szyzak.18

Equations (49)

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R = ρ 2 exp ( 2 η z 2 ) + ρ 2 exp ( - 2 η z 2 ) - 2 ρ 2 cos 2 η z 1 exp ( 2 η z 2 ) + ρ 4 exp ( - 2 η z 2 ) - 2 ρ 2 cos ( 2 ϕ + 2 η z 1 ) ,
T = τ 1 2 r 2 2 exp ( 2 η z 2 ) + ρ 4 exp ( - 2 η z 2 ) - 2 ρ 2 cos ( 2 ϕ + 2 η z 1 ) ,
ρ 2 ( π ) = [ ( n 2 - k 2 ) cos φ - z 1 ] 2 + ( 2 n k cos φ - z 2 ) 2 [ ( n 2 - k 2 ) cos φ + z 1 ] 2 + ( 2 n k cos φ + z 2 ) 2 ρ 2 ( σ ) = ( cos φ - z 1 ) 2 + z 2 2 ( cos φ + z 1 ) 2 + z 2 2 , tan ϕ ( π ) = 2 cos φ ( n 2 - k 2 ) z 2 - 2 n k z 1 z 1 2 + z 2 2 - ( n 2 + k 2 ) 2 cos 2 φ , tan ϕ ( σ ) = 2 z 2 cos φ z 1 2 + z 2 2 - cos 2 φ ,
τ 1 2 ( π ) = 4 ( n 2 + k 2 ) 2 cos 2 φ [ ( n 2 - k 2 ) cos φ + z 1 ] 2 + ( 2 n k cos φ + z 2 ) 2 , τ 2 2 ( π ) = 4 ( z 1 2 + z 2 2 ) [ ( n 2 - k 2 ) cos φ + z 1 ] 2 + ( 2 n k cos φ + z 2 ) 2 , τ 1 2 ( σ ) = 4 cos 2 φ ( cos φ + z 1 ) 2 + z 2 2 , τ 2 2 ( σ ) = 4 ( z 1 2 + z 2 2 ) ( cos φ + z 1 ) 2 + z 2 2 .
T ( σ ) = 8 z 1 2 cos 2 φ 8 z 1 2 cos 2 φ + ( z 1 2 - cos 2 φ ) 2 ( 1 - cos 2 η z 1 ) ,
1 - cos 2 η z 1 = η z 1 cos 2 φ n 2 + cos 2 φ sin 2 η z 1 ,
( 1 - cos 2 η z 1 ) ( z 1 2 - n 4 cos 2 φ ) = η z 1 cos 2 φ sin 2 η z 1 × ( z 1 2 - n 4 cos 2 φ ) 2 ( n 2 + cos 2 φ ) ( z 1 2 - n 4 cos 2 φ ) + 2 ( n 4 - 1 ) z 1 2 cos 2 φ ;
2 ( d / λ ) ( n 2 - sin 2 φ ) 1 / 2 = x             ( x - ^ integer ) .
N = integer of { 2 d λ [ n - ( n 2 - 1 ) 1 / 2 ] + 1 }
λ 2 d 2 = 2 ( M j k - M i j )             and             n 2 = sin 2 φ i + [ 3 M i j - M j k ] 2 8 ( M j k - M i j ) ,
x = 2 d λ 0 ( n 0 2 - sin 2 φ ) 1 / 2 .
( z 1 2 - cos 2 φ 2 z 1 cos φ ) 2 sin 2 η z 1 = M
n 0 δ n 1 2 [ sin 2 ( φ 0 + δ φ ) - sin 2 φ 0 ] + δ λ λ 0 ( n 0 2 - sin 2 φ 0 ) ( 1 + δ λ 2 λ 0 ) .
M 0 1 - T min T min = R max T min ,
n 2 = sin 2 φ + cos 2 φ { 1 + 2 M 0 [ 1 + ( 1 + 1 M 0 ) 1 / 2 ] } .
λ 2 d = ( n 2 - sin 2 φ 2 ) 1 / 2 - ( n 2 - sin 2 φ 1 ) 1 / 2 .
n = tan φ B .
T = 8 z 1 2 cos 2 φ 8 z 1 2 cos 2 φ + ( z 1 2 - cos 2 φ ) 2 ( 1 - cos 2 π x 0 n z 1 )
d λ = x 0 2 n
2 d λ ( ɛ eff - sin 2 φ ) 1 / 2 = x
ɛ eff ( π ) = ɛ eff ( σ ) ,
ɛ e f f ( σ ) = ɛ , ɛ eff ( π ) = ɛ + sin 2 φ ( ɛ - ɛ ) ,
ɛ eff ( σ ) = ɛ sin 2 ϑ + ɛ cos 2 ϑ , ɛ eff ( π ) = ɛ sin 2 φ + ( ɛ cos 2 ϑ + ɛ sin 2 ϑ ) cos 2 φ ,
z 1 ( n 2 - sin 2 φ ) 1 / 2 ,             z 2 n k z 1             and             ( exp ± 2 η z 2 ) 1 ± 2 η z 2
T max ( σ ) = z 1 2 cos φ z 1 2 cos φ + η n k ( z 1 2 + cos 2 φ ) ,             R min ( σ ) = 0.
R min ( σ ) = ( z 1 2 - cos 2 φ ) 2 z 1 4 [ sin 2 φ + cos 2 φ + ( 2 cos φ η n k ) 2 ] .
T min ( σ ) = 4 z 1 2 cos 2 φ ( z 1 2 + cos 2 φ ) 2 + 4 η n k cos φ ( z 1 2 + cos 2 φ ) ,
R max ( σ ) = ( z 1 2 - cos 2 φ ) 2 ( z 1 2 + cos 2 φ ) 2 + 4 η n k cos φ ( z 1 2 + cos 2 φ ) .
2 ϕ ( σ ) 2 n k cos φ z 1 ( n 2 - 1 ) 2 η z 1 .
M 1 ( φ ) T max ( π ) T max ( σ )
α d = 2 n z 1 2 cos φ ( M 1 - 1 ) z 1 2 ( n 2 - M 1 ) - n 2 cos 2 φ ( n 2 M 1 - 1 ) ,
α d 2 × 10 - 2 ,
M 2 ( φ 1 φ 2 ) T max ( φ 1 ) T max ( φ 2 ) ,
α d = 2 ( M 2 - 1 ) ( n 2 - sin 2 φ 1 ) cos φ 1 ( n 2 - sin 2 φ 2 ) cos φ 2 n [ ( n 2 - sin 2 φ 1 ) cos φ 1 ( n 2 - sin 2 φ 2 + cos 2 φ 2 ) - M 2 ( n 2 - sin 2 φ 2 ) cos φ 2 ( n 2 - sin 2 φ 1 + cos 2 φ 1 ) ]
k > 0 :             δ δ φ T max ( σ ) < 0 ,             k < 0 :             δ δ φ T max ( σ ) > 0.
cos 2 η i z 1 = { 1 - β i 2 for maxima - 1 + β i 2 for minima ,
T max = 1 N i = 1 N 8 z 1 2 cos 2 φ 8 z 1 2 cos 2 φ + ( z 1 2 - cos 2 φ ) 2 β i 2 1 - ( z 1 2 - cos 2 φ ) 2 8 z 1 2 cos 2 φ σ ,
T min = 1 N i = 1 N 8 z 1 2 cos 2 φ 8 z 1 2 cos 2 φ + ( z 1 2 - cos 2 φ ) 2 ( 2 - β i 2 ) 8 z 1 2 cos 2 φ 8 z 1 2 cos 2 φ + 2 ( z 1 2 - cos 2 φ ) 2 + 8 z 1 2 cos 2 φ ( z 1 2 - cos 2 φ ) 2 [ 8 z 1 2 cos 2 φ + 2 ( z 1 2 - cos 2 φ ) 2 ] 2 σ ,
σ = 1 N i = 1 N β i 2
8 z 1 2 cos 2 φ 8 z 1 2 cos 2 φ + 2 ( z 1 2 - cos 2 φ ) 2 = 1 2 ( 1 - T max ) { [ 1 + 4 T min ( 1 - T max ) ] 1 / 2 - 1 } ,
σ = 8 cos 2 φ z 1 2 ( z 1 2 - cos 2 φ ) 2 ( 1 - T max ) .
T min ( β = 0 , k > 0 ) < T min ( β = 0 , k = 0 ) < T min ( β 0 , k = 0 ) .
n 2 = c = ( c 2 + f ) 1 / 2 ,
η n k = z 1 2 cos φ z 1 2 + cos 2 φ 1 - T max T max ,
c = cos 2 φ [ 1 + tan 2 φ + 2 ( 1 T min - 1 T max ) ] , f = 4 sin 2 φ cos 2 [ 1 T max - 1 T min ] - 1.
x 0 - 1 = ( n 0 2 - sin 2 φ 2 n 0 2 - sin 2 φ 1 ) 1 / 2 - 1.
λ 0 2 d = ( n 0 2 - sin 2 φ 2 ) 1 / 2 - ( n 0 2 - sin 2 φ 1 ) 1 / 2 ,
n 2 = sin 2 φ + ( x i λ max 2 d ) 2 ,
Mi,i+1=(λ/d)(n2-sin2φ0)1/2+(λ/2d)2(2i+1),

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