Abstract

Preparation, measurement, and calculation methods are discussed for the determination of the complex index of refraction, the layer thickness, and induced volume changes of thin layers (due to a phase change, for example). The principle of the calculation is fitting a curve in the reflectance–transmittance plane measured on a range of layer thicknesses, instead of fitting the reflectance and transmittance as a function of independently measured layer thicknesses. This general method is applied to thin films of GaSb and InSb, in which a laser-induced amorphous-to-crystalline transition can be used in optical recording. The information essential for optical recording applications is measured quickly by making use of a stepwise prepared layer thickness distribution, while the complex refractive index and the layer thicknesses can also be calculated unambiguously.

© 1989 Optical Society of America

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References

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  1. D. J. Gravesteijn, H. F. J. J. van Tongeren, M. M. Sens, T. C. J. M. Bertens, C. J. van de Poel, “Phase-Change Optical Data Storage in GaSb,” Appl. Opt. 26, 4772–4776 (1988).
    [CrossRef]
  2. J. C. Manifacier, J. Gasiot, J. P. Fillard, “A Simple Method for the Determination of the Optical Constants n, k and the thickness of a Weakly Absorbing Thin Film,” J. Phys. E. 9, 1002–1004 (1976).
    [CrossRef]
  3. W. E. Case, “Algebraic Method for Extracting Thin-Film Optical Parameters from Spectrophotometer Measurements,” Appl. Opt. 22, 1832–1836 (1983).
    [CrossRef] [PubMed]
  4. M. Chang, U. J. Gibson, “Optical Constant Determinations of Thin Films by a Random Search Method,” Appl. Opt. 24,504–507 (1985).
    [CrossRef] [PubMed]
  5. C. J. van de Poel, “Rapid Crystallization of Thin Solid Films,” J. Mater. Res. 3, 126–132 (1988).
    [CrossRef]
  6. L. Vriens, W. Rippens, “Optical Constants of Absorbing Thin Solid Films on a Substrate,” Appl. Opt. 22, 4105–4110 (1983).
    [CrossRef] [PubMed]
  7. H. Hora, “Stresses in Silicon Crystals from Ion-Implanted Amorphous Regions,” Appl. Phys. A 32, 217–221 (1983).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  9. M. Mansuripur, “Distribution of Light at and Near the Focus of High-Numerical-Aperture Objectives,” J. Opt. Soc. Am. A 3, 2086–2093 (1986).
    [CrossRef]
  10. J. Stuke, G. Zimmer, “Optical Properties of Amorphous 3–5 Compounds,” Phys. Status Solidi B 49, 513–523 (1972).
    [CrossRef]
  11. D. E. Aspnes, A. A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
    [CrossRef]
  12. The value reported in Ref. 1 for na of a-GaSb at λ0 = 780 nm equals 4.6 − 1.2i instead of the printed value 4.6 − 0.2i: a printing error.
  13. Private communications to C. J. van de Poel, see Ref. 5.
  14. G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disk Systems (Hilger, Bristol, 1986).

1988

1986

1985

1983

W. E. Case, “Algebraic Method for Extracting Thin-Film Optical Parameters from Spectrophotometer Measurements,” Appl. Opt. 22, 1832–1836 (1983).
[CrossRef] [PubMed]

L. Vriens, W. Rippens, “Optical Constants of Absorbing Thin Solid Films on a Substrate,” Appl. Opt. 22, 4105–4110 (1983).
[CrossRef] [PubMed]

H. Hora, “Stresses in Silicon Crystals from Ion-Implanted Amorphous Regions,” Appl. Phys. A 32, 217–221 (1983).
[CrossRef]

D. E. Aspnes, A. A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

1976

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A Simple Method for the Determination of the Optical Constants n, k and the thickness of a Weakly Absorbing Thin Film,” J. Phys. E. 9, 1002–1004 (1976).
[CrossRef]

1972

J. Stuke, G. Zimmer, “Optical Properties of Amorphous 3–5 Compounds,” Phys. Status Solidi B 49, 513–523 (1972).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, A. A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Bertens, T. C. J. M.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Bouwhuis, G.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disk Systems (Hilger, Bristol, 1986).

Braat, J.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disk Systems (Hilger, Bristol, 1986).

Case, W. E.

Chang, M.

Fillard, J. P.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A Simple Method for the Determination of the Optical Constants n, k and the thickness of a Weakly Absorbing Thin Film,” J. Phys. E. 9, 1002–1004 (1976).
[CrossRef]

Gasiot, J.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A Simple Method for the Determination of the Optical Constants n, k and the thickness of a Weakly Absorbing Thin Film,” J. Phys. E. 9, 1002–1004 (1976).
[CrossRef]

Gibson, U. J.

Gravesteijn, D. J.

Hora, H.

H. Hora, “Stresses in Silicon Crystals from Ion-Implanted Amorphous Regions,” Appl. Phys. A 32, 217–221 (1983).
[CrossRef]

Huijser, A.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disk Systems (Hilger, Bristol, 1986).

Manifacier, J. C.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A Simple Method for the Determination of the Optical Constants n, k and the thickness of a Weakly Absorbing Thin Film,” J. Phys. E. 9, 1002–1004 (1976).
[CrossRef]

Mansuripur, M.

Pasman, J.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disk Systems (Hilger, Bristol, 1986).

Rippens, W.

Schouhamer Immink, K.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disk Systems (Hilger, Bristol, 1986).

Sens, M. M.

Studna, A. A.

D. E. Aspnes, A. A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Stuke, J.

J. Stuke, G. Zimmer, “Optical Properties of Amorphous 3–5 Compounds,” Phys. Status Solidi B 49, 513–523 (1972).
[CrossRef]

van de Poel, C. J.

van Rosmalen, G.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disk Systems (Hilger, Bristol, 1986).

van Tongeren, H. F. J. J.

Vriens, L.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Zimmer, G.

J. Stuke, G. Zimmer, “Optical Properties of Amorphous 3–5 Compounds,” Phys. Status Solidi B 49, 513–523 (1972).
[CrossRef]

Appl. Opt.

Appl. Phys. A

H. Hora, “Stresses in Silicon Crystals from Ion-Implanted Amorphous Regions,” Appl. Phys. A 32, 217–221 (1983).
[CrossRef]

J. Mater. Res.

C. J. van de Poel, “Rapid Crystallization of Thin Solid Films,” J. Mater. Res. 3, 126–132 (1988).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. E.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A Simple Method for the Determination of the Optical Constants n, k and the thickness of a Weakly Absorbing Thin Film,” J. Phys. E. 9, 1002–1004 (1976).
[CrossRef]

Phys. Rev. B

D. E. Aspnes, A. A. Studna, “Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Phys. Status Solidi B

J. Stuke, G. Zimmer, “Optical Properties of Amorphous 3–5 Compounds,” Phys. Status Solidi B 49, 513–523 (1972).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

The value reported in Ref. 1 for na of a-GaSb at λ0 = 780 nm equals 4.6 − 1.2i instead of the printed value 4.6 − 0.2i: a printing error.

Private communications to C. J. van de Poel, see Ref. 5.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. Schouhamer Immink, Principles of Optical Disk Systems (Hilger, Bristol, 1986).

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Figures (7)

Fig. 1
Fig. 1

Measured reflectance as a function of the measured transmittance (points) at certain positions of the prepared sample a-GaSb, N.A. = 0.2, λ0 = 820 nm. Three theoretical curves (solid lines) are plotted with, in the middle, the value obtained from the least-squares method, ñ2 = 4.88 − i1.466, and for (a) δn2 = ±0.1n2 and (b) δk2 = ±0.1k2. Note that n2 determines the height of the extremum E and k2 determines the position of the flank F. The arrow denotes an increasing layer thickness along the curve. The measured reflectance and transmittance as a function of the calculated layer thicknesses are shown in Fig. 5(b)

Fig. 2
Fig. 2

Part of the experimental setup used for measuring the reflectance and transmittance of substrate–incident light. Depending on the positions and distances used, the light reflected from the air–glass interface is collected partially or completely at the detector.

Fig. 3
Fig. 3

Electromagnetic fields at (a) a boundary, (b) a thin layer; arrows denote plane waves, and (c) a thin layer or stack of thin layers on a thick substrate; the values at the boundary of the dashed-line box represents the calculated transmittance and reflectance by means of Eq. (6) of a stack of thin layers or a single thin layer. Now the influence of the substrate should be evaluated.

Fig. 4
Fig. 4

Some examples illustrating the geometric relation pi· li = 0 in the (T,R) plane. (a) Expression corresponds to looking for a global minimum distance, (b) Desired solution during the calculation is positioned in the part of the curve between the two smallest local minimum distances.

Fig. 5
Fig. 5

Results for a-GaSb and c-GaSb, N.A.beam = 0.2, λ0 = 820 nm. (a) Least-squares fit (solid lines) in the (T,R) plane yields ña = 4.88(±1) − 1.466(±2)i, with n1 = 1.523(±2) and ñc = 4.23(±1) − 0.510(±5)i, with n1 = 1.523(±2). (b) Measured reflectance and transmittance (points) as a function of the calculated layer thicknesses obtained from the least-squares fit, σSa = 5 × 10−3 and σSc = 9 × 10−3, respectively, (c) Calculated layer thicknesses as a function of the substrate position, dz = 0.15 mm. (d) Correlation of the amorphous and crystalline layer thicknesses obtained from the least-squares fit with a correlation coefficient of c = 0.998. It results in dc = 0.999(±2)da with σd = 5.41 nm. This indicates the same layer thickness after crystallization.

Fig. 6
Fig. 6

Results for a-InSb and c-InSb, N.A.beam = 0.2, λ0 = 820 nm. (a) Least-squares fit (solid lines) in the (T,R) plane yields ña = 4 82(±1) − 1.95(±1)i, with ñ1 = 1.523(±2) and ñc = 4.07(±1) − 0.755(±5)i, with ñ1 = 1.526(±2). (b) Measured reflectance and transmittance (points) as a function of the calculated layer thicknesses obtained from the least-squares fit, σSa = 1.9 × 10−2 and σSc = 1.3 × 10−2, respectively, (c) Calculated layer thicknesses as a function of the substrate position, dz = 0.15 mm. (d) Correlation of the amorphous and crystalline layer thicknesses obtained from the least-squares fit (c = 0.995). It results in dc = 1.015(±5)da with σd = 6.76 nm. This indicates a small relative layer thickness increase after the crystallization.

Fig. 7
Fig. 7

Accuracy of the calculated layer thickness, using for the indices of refraction, ña = 4.88 − i1.466 and ñc = 4.23 − i0.510, while ΔT = ΔR = 5 × 10−3. Note that at layer thicknesses ∼ Nλ0/(4n2) (N is an integer) and at increasing layer thicknesses the accuracy is small because the partial derivatives as given in Eq. (20) become small.

Tables (1)

Tables Icon

Table I Results Obtained for Magnetron Sputtered Amorphous and Furnace Annealed Crystalline GaSb and InSb Films, λ0 = 820 nm, 15 nm ≤ d2 ≤ 170 nm

Equations (33)

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R th ( n 2 , k 2 ) R exp = 0 , T th ( n 2 , k 2 ) T exp = 0 , ñ 2 = n 2 i k 2
r s 12 = E s 1 E s 1 + = ñ 1 cos ϕ 1 ñ 2 cos ϕ 2 ñ 1 cos ϕ 1 + ñ 2 cos ϕ 2 , r p 12 = E p 1 E p 1 + = ñ 2 cos ϕ 1 ñ 1 cos ϕ 2 ñ 2 cos ϕ 1 + ñ 1 cos ϕ 2 ,
t s 12 = E s 2 + E s 1 + = 2 ñ 1 cos ϕ 1 ñ 1 cos ϕ 1 + ñ 2 cos ϕ 2 , t p 12 = E p 2 + E p 1 + = 2 ñ 1 cos ϕ 1 ñ 2 cos ϕ 1 + ñ 1 cos ϕ 2 ,
n 0 sin ϕ 0 = ñ 1 sin ϕ 1 = ñ 2 sin ϕ 2 = .
r 13 = r 12 + r 23 exp ( 2 i x ) 1 + r 12 r 23 exp ( 2 i x ) , t 13 = t 12 t 23 exp ( i x ) 1 + r 12 r 23 exp ( 2 i x ) , x = 2 π d 2 λ 0 ñ 2 cos ϕ 2 .
R = I first / I first + = | r | 2 , T = I last + / I first + = | t | 2 ( n cos ϕ ) last ( n cos ϕ ) first ,
D 01 = D inc | 2 d 1 n 1 f ( a f 1 ) ± 1 | ,
R = R 01 + R 13 T 01 2 1 R 13 R 01 , T = T 13 T 01 1 R 13 R 01 , D 01 < D det .
R = R 13 T 01 2 , T = T 13 T 01 , D 01 D det ,
S = g T i = 1 I [ T th i ( n 2 , k 2 , d 2 i ) T exp i 2 ] + g R i = 1 I [ R th i ( n 2 , k 2 , d 2 i ) R exp i ] 2 ,
S n 2 = 0 , S k 2 = 0 ,
S d 2 i = 2 g T ( T th i T exp i ) T th i d 2 i + 2 g R ( R th i R exp i ) R th i d 2 i = 0 , 1 i I .
p i = ( T th i T exp i , R th i R exp i ) , l i = ( T th i d 2 i δ d 2 i , R th i d 2 i δ d 2 i ) , 1 i I ,
p i l i = 0 , 1 i I .
( s A + Δ s ) ( s B Δ s ) d 2 i = d A ,
( s A + Δ s ) > ( s B Δ s ) d 2 i = d A s B + d B s A s A + s B .
s j i = ( T th j T exp i ) 2 + ( R th j R exp i ) 2 .
S = g T i = 1 I [ T th i ( n 1 , n 2 , k 2 , d 2 i ) T exp i ] 2 + g R i = 1 I [ R th i ( n 1 , n 2 , k 2 , d 2 i ) R exp i ] 2
S n 1 = 0 ,
( Δ d i ) 2 = ( T th i d 2 i Δ T i ) 2 + ( R th i d 2 i Δ R i ) 2 [ ( T th i d 2 i ) 2 + ( R th i d 2 i ) 2 ] 2 ( σ exp ) 2 .
( Δ n 2 ) 2 = i = 1 I ( T th i n 2 Δ T i ) 2 + i = 1 I ( R th i n 2 Δ R i ) 2 [ i = 1 I ( T th i n 2 ) 2 + i = 1 I ( R th i n 2 ) 2 ] 2 ( ( σ exp ) I ) 2 ,
( Δ k 2 ) 2 = i = 1 I ( T th i k 2 Δ T i ) 2 + i = 1 I ( R th i k 2 Δ R i ) 2 [ i = 1 I ( T th i k 2 ) 2 + i = 1 I ( R th i k 2 ) 2 ] 2 ( ( σ exp ) I ) 2 .
S a a = i = 1 I ( d a i ) 2 , S a c = i = 1 I d a i d c i , S c c = i = 1 I ( d c i ) 2 ,
α = S a c S a a , c = S a c S a a S c c , σ d = S c c α S a c I 1 ,
( Δ α ) 2 = i = 1 I ( α d a i Δ d a i ) 2 + i = 1 I ( α d c i Δ d c i ) 2 = 1 S a a 2 i = 1 I [ ( d c i 2 α d a i ) 2 ( Δ d a i ) 2 + ( d a i ) 2 ( Δ d c i ) 2 ] ( σ exp I ) 2 .
C R = R c R a R c + R a .
r = ½ ( r c + r a ) + ½ ( r c r a ) cos ( 2 π x / p ) = r 0 + r ± [ exp ( 2 π i x / p ) + exp ( 2 π i x / p ) ] .
I det π D inc 2 [ | r 0 | 2 + 2 MTF ( δ ) | r 0 | 2 + 2 MTF ( δ ) ( r 0 r ± * + r 0 * r ± ) cos ( 2 π υ t / p ) ] + π D inc 2 [ 2 MTF ( 2 δ ) | r ± | 2 cos ( 4 π υ t / p ) ] .
MTF ( δ ) = 2 π arccos ( δ / 2 ) δ / π 1 δ 2 / 4 , 0 δ 2 , MTF ( δ ) = 0 , δ > 2 .
m = 2 MTF ( δ ) r 0 r ± * + r 0 * r ± | r 0 | 2 + 2 MTF ( δ ) | r ± | 2 2 MTF ( δ ) r 0 r ± * + r 0 * r ± | r 0 | 2 .
m 2 MTF ( δ ) ( R c R a ) R c + R a + 2 R c R a cos ( ϕ c ϕ a ) .
d R C
d R m

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