Abstract

Reliable control of the deposition process of optical films and coatings frequently requires monitoring the refractive-index profile throughout the layer. In the present research a simple in situ approach is proposed that uses a WKBJ matrix representation of the optical transfer function of a single thin film on a substrate. Mathematical expressions are developed that represent the minima and the maxima envelopes of the curves transmittance versus time and reflectance versus time. The refractive index and the extinction coefficient depth profiles of different films are calculated from simulated spectra as well as from experimental data obtained during the PECVD (plasma-enhanced chemical vapor deposition) of silicon-compound films. Variation in the deposition rate with time is also evaluated from the position of the spectra extrema as a function of time. The physical and mathematical limitations of the method are discussed.

© 1998 Optical Society of America

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References

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  1. B. Bovard, “Rugate filter theory: an overview,” J. Opt. Soc. Am. A 32, 5427–5442 (1993).
  2. S. Ogura, “Starting materials,” in Thin Films for Optical Systems, F. R. Flory, ed. (Marcel Dekker, New York, 1995), Chap. 2, pp. 41–55.
  3. W. G. Sainty, W. D. McFall, D. R. McKenzie, Y. Yin, “Time-dependent phenomena in plasma-assisted chemical vapor deposition of rugate optical films,” Appl. Opt. 34, 5659–5664 (1995).
    [CrossRef] [PubMed]
  4. B. Bovard, F. J. Van Milligen, M. J. Messerly, S. G. Saxe, H. A. Macleod, “Optical constants derivation for an inhomogeneous thin film from in situ transmission measurements,” Appl. Opt. 24, 1803–1807 (1985).
    [CrossRef] [PubMed]
  5. 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]
  6. R. Swanepoel, “Determination of the thickness and the optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983).
    [CrossRef]
  7. J. Mouchart, G. Lagier, B. Pointu, “Détermination des constantes optiques n et k de matériaux faiblement absorbants,” Appl. Opt. 24, 1809–1814 (1985).
    [CrossRef]
  8. D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693–2699 (1996).
    [CrossRef]
  9. L. D. Landau, E. M. Lifchitz, Quantum Mechanics, Nonrelativistic Theory, Course in Theoretical Physics, Vol. 3 (Pergamon, London, 1958).
  10. R. Jacobsson, “Light reflection from films of continuously varying refractive index,” Prog. Opt. 5, 247–286 (1966).
    [CrossRef]
  11. D. Minkov, R. Swanepoel, “Computer drawing of the envelopes of spectra with interference,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. SPIE1782, 212–220 (1993).
    [CrossRef]
  12. M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
    [CrossRef]
  13. H. G. Tompkins, presented at the 44th American Vacuum Society National Symposium, San Jose, Calif. (October 1997).

1996 (1)

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693–2699 (1996).
[CrossRef]

1995 (1)

1993 (1)

B. Bovard, “Rugate filter theory: an overview,” J. Opt. Soc. Am. A 32, 5427–5442 (1993).

1991 (1)

M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
[CrossRef]

1985 (2)

1983 (1)

R. Swanepoel, “Determination of the thickness and the optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983).
[CrossRef]

1976 (1)

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]

1966 (1)

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” Prog. Opt. 5, 247–286 (1966).
[CrossRef]

Bovard, B.

Feldman, A.

M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
[CrossRef]

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]

Gujrathi, S. C.

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693–2699 (1996).
[CrossRef]

Jacobsson, R.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” Prog. Opt. 5, 247–286 (1966).
[CrossRef]

Kahaner, D.

M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
[CrossRef]

Klemberg-Sapieha, J. E.

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693–2699 (1996).
[CrossRef]

Lagier, G.

J. Mouchart, G. Lagier, B. Pointu, “Détermination des constantes optiques n et k de matériaux faiblement absorbants,” Appl. Opt. 24, 1809–1814 (1985).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifchitz, Quantum Mechanics, Nonrelativistic Theory, Course in Theoretical Physics, Vol. 3 (Pergamon, London, 1958).

Leroux, P.

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693–2699 (1996).
[CrossRef]

Lifchitz, E. M.

L. D. Landau, E. M. Lifchitz, Quantum Mechanics, Nonrelativistic Theory, Course in Theoretical Physics, Vol. 3 (Pergamon, London, 1958).

Macleod, H. A.

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]

Martinu, L.

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693–2699 (1996).
[CrossRef]

McClain, M.

M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
[CrossRef]

McFall, W. D.

McKenzie, D. R.

Messerly, M. J.

Minkov, D.

D. Minkov, R. Swanepoel, “Computer drawing of the envelopes of spectra with interference,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. SPIE1782, 212–220 (1993).
[CrossRef]

Mouchart, J.

J. Mouchart, G. Lagier, B. Pointu, “Détermination des constantes optiques n et k de matériaux faiblement absorbants,” Appl. Opt. 24, 1809–1814 (1985).
[CrossRef]

Ogura, S.

S. Ogura, “Starting materials,” in Thin Films for Optical Systems, F. R. Flory, ed. (Marcel Dekker, New York, 1995), Chap. 2, pp. 41–55.

Pointu, B.

J. Mouchart, G. Lagier, B. Pointu, “Détermination des constantes optiques n et k de matériaux faiblement absorbants,” Appl. Opt. 24, 1809–1814 (1985).
[CrossRef]

Poitras, D.

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693–2699 (1996).
[CrossRef]

Sainty, W. G.

Saxe, S. G.

Swanepoel, R.

R. Swanepoel, “Determination of the thickness and the optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983).
[CrossRef]

D. Minkov, R. Swanepoel, “Computer drawing of the envelopes of spectra with interference,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. SPIE1782, 212–220 (1993).
[CrossRef]

Tompkins, H. G.

H. G. Tompkins, presented at the 44th American Vacuum Society National Symposium, San Jose, Calif. (October 1997).

Van Milligen, F. J.

Yin, Y.

Ying, X.

M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
[CrossRef]

Appl. Opt. (3)

Comput. Phys. (1)

M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
[CrossRef]

J. Opt. Soc. Am. A (1)

B. Bovard, “Rugate filter theory: an overview,” J. Opt. Soc. Am. A 32, 5427–5442 (1993).

J. Phys. E (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]

R. Swanepoel, “Determination of the thickness and the optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983).
[CrossRef]

Opt. Eng. (1)

D. Poitras, P. Leroux, J. E. Klemberg-Sapieha, S. C. Gujrathi, L. Martinu, “Characterization of homogeneous and inhomogeneous Si-based optical coatings deposited in dual-frequency plasma,” Opt. Eng. 35, 2693–2699 (1996).
[CrossRef]

Prog. Opt. (1)

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” Prog. Opt. 5, 247–286 (1966).
[CrossRef]

Other (4)

D. Minkov, R. Swanepoel, “Computer drawing of the envelopes of spectra with interference,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. SPIE1782, 212–220 (1993).
[CrossRef]

H. G. Tompkins, presented at the 44th American Vacuum Society National Symposium, San Jose, Calif. (October 1997).

L. D. Landau, E. M. Lifchitz, Quantum Mechanics, Nonrelativistic Theory, Course in Theoretical Physics, Vol. 3 (Pergamon, London, 1958).

S. Ogura, “Starting materials,” in Thin Films for Optical Systems, F. R. Flory, ed. (Marcel Dekker, New York, 1995), Chap. 2, pp. 41–55.

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Figures (11)

Fig. 1
Fig. 1

Multiple-wavelength optical monitor mounted on a plasma deposition chamber.

Fig. 2
Fig. 2

(a) Typical spectra evolution measured with the optical monitor. (b) Closer view of the reflectance modulation with time at λ = 789 nm.

Fig. 3
Fig. 3

Linear refractive-index profile: (a) reflectance versus time. (b) Model (dashed curve) and calculated (solid curve) n(t) profiles (λ = 800 nm).

Fig. 4
Fig. 4

Linear refractive-index profile: (a) A (t) profile; (b) k(t) profile (dashed line, model; solid curve, calculated).

Fig. 5
Fig. 5

Linear refractive-index profile: (a) Deposition rate values between successive extrema. (b) Physical position in the layer as a function of deposition time (dashed line, model; solid curve, calculated).

Fig. 6
Fig. 6

Experimental data for an inhomogeneous SiN1.3 film on glass: (a) reflectance evolution, (b) calculated n(z) profile, (c) calculated deposition rate variation (λ = 789 nm).

Fig. 7
Fig. 7

Experimental data for an inhomogeneous SiN1.3 film on silicon: (a) reflectance evolution, (b) calculated n(z) profile, (c) calculated deposition rate variation (λ = 826 nm).

Fig. 8
Fig. 8

Experimental data for an inhomogeneous SiN1.3 film on glass: Evolution of the refractive index dispersion within the layer (same film as in Fig. 6).

Fig. 9
Fig. 9

Half-period-sinus refractive-index profile: (a) model (dashed curve) and calculated (solid curve) n(z) profiles, (b) WKBJ validity condition from Eq. (18) (dashed line, 1/λ, λ = 800 nm).

Fig. 10
Fig. 10

Effect of ζ on n(t) profiles (dashed line, model profile; solid curves, calculated profiles, λ = 800 nm).

Fig. 11
Fig. 11

Ex situ optical analysis of SiN1.3 on glass (same sample as in Fig. 6): (a) n(z) profile calculated by the envelope method [solid curve, same as Fig. 6(b)] and homogeneous one-layer model (dotted curve) used to fit the ex situ measurements in (b). (b) Ex situ reflectance data (gray curve), and reflectance calculated using the profiles in (a) (black solid curve, graded profile; dotted curve, optimized homogeneous profile). (c) In situ reflectance measurement (gray curve) and reflectance calculated using the profiles in (a) (black solid curve, graded profile; dotted curve, optimized homogeneous profile).

Equations (22)

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= n in n out 1 / 2 cos   δ i n in n out 1 / 2 sin   δ i n in n out 1 / 2 sin   δ n out n in 1 / 2 cos   δ ,
T = n s n out n 0 n in t in t out   exp δ 1 + r out r ˘ in   exp 2 δ 2 ,
R = r out + r ˘ in   exp 2 δ 1 + r out r ˘ in   exp 2 δ 2 .
A = exp - 4 π / λ 0 d   k z d z .
n in = N t + η N t 2 - n 0 2 n s 2 1 / 2 1 / 2 , N t = n 0 2 + n s 2 2 - 2 n 0 n s T max - T min T max + T min ,
n out z = 2 n in n s n 0 n s 2 - n in 2 T max - T min T max T min + n 0 1 + 4 n in 2 n s 2 n in 2 - n s 2 2 T max - T min T max T min 2 1 / 2 ,
A z = r in r out T max / T min 1 / 2 - 1 T max / T min 1 / 2 + 1 ,
R min = r out 2 - 2 r out r ˘ in A + | r ˘ in | 2 A 2 1 - 2 r out r ˘ in A + r out 2 | r ˘ in | 2 A 2 ,
R max = r out 2 + 2 r out r ˘ in A + | r ˘ in | 2 A 2 1 + 2 r out r ˘ in A + r out 2 | r ˘ in | 2 A 2 .
n in = n 0 n s 1 / 2 N r + η N r 2 - n s 2 + k s 2 n s 2 1 / 2 ,
N r = + 1 R min - - 1 R max + 2 - 1 R max - + 1 R min + 2 .
r out z = - - 1 2 B A - ζ C - 1 2 - 4 + B A - ζ C 2 1 / 2 1 / 2 ,
A z = 1 / 2   1 - r out 4 R max - R min r out r ˘ in r out 2 2 R min R max - R max - R min + 2 - R min - R max ,
A = | r ˘ in | 2 R max - R min 2 , B = 2 r ˘ in 2 2 - R max - R min × 2 R min R max - R min - R max , C = 2 + B 2 A 2 - D A 1 / 2 , D = - 2 | r ˘ in | 2 R max - R min 2 + 8 r ˘ in 2 R max + R min 2 + 2 1 - R max - R min + 2 R max 2 R min 2 × 1 R min + 1 R max - 1 , ζ = + 1 if 1 + r out 4 r out 2 > B A , - 1 if 1 + r out 4 r out 2 < B A .
k z = - λ 4 π d d z ln   A z .
m λ / 4 = 0 d   n z d z
λ 4 z 2 - z 1 t 2 - t 1 t 1 t 2   n t d t ,
Δ n in n in n in 2 - n 0 2 n s 2 - n in 2 2 n 0 n s + n in 2 n 0 n s - n in 2 T max + T min T max - T min Δ T T ,
Δ n out n out T max + T min T max - T min Δ T T + n 0 C n 1 + C n 2 1 / 2 × n in 2 + n s 2 n in 2 - n s 2 Δ n in n in ,
C n = 2 n in n s n in 2 - n s 2 T max - T min T max T min .
Δ A A 2 T max T min 1 / 2 T max - T min Δ T T + 2 n 0 n out n out 2 - n 0 2 Δ n out n out + 2 n s n in n in 2 - n s 2 Δ n in n in .
1 n 2 z   n z     1 λ .

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