Abstract

Within the Rayleigh–Rice theory the influence of layer boundary roughness on coherently reflected light is expressed using very complex formulae. Therefore we deal with the simplification of these formulae by employing an approximation based on neglecting cross-correlation effects between both the rough boundaries. It is shown that if the mean distance of the boundaries (mean thickness) is sufficiently large in comparison with the lateral dimensions of the roughness it is possible to describe the individual boundaries of the layers by matrices corresponding to isolated rough surfaces. This fact enables us to simplify the formulae for the optical quantities in a substantial way, which also simplifies the numerical calculation needed for the inverse problem. This statement is illustrated by means of a numerical analysis simulating ellipsometric and reflectometric data of rough silicon dioxide layers placed onto silicon single crystal substrates.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. O. Rice, "Reflection of electromagnetic waves from slightly rough surfaces," Commun. Pure Appl. Math. 4, 351-378 (1951).
    [CrossRef]
  2. G. R. Valenzuela, "Depolarization of EM Waves by Slightly Rough Surfaces," IEEE Trans. Antennas Propag. AP-15, 552-557 (1967).
    [CrossRef]
  3. K. Krishen, "Scattering of Electromagnetic Waves from a layer with rough front and plane back (Small Perturbation Method by Rice)," IEEE Trans. Antennas Propag. AP-18, 573-576 (1970).
    [CrossRef]
  4. R. Schiffer, "Reflectivity of a slightly rough surface," Appl. Opt. 26, 704-712 (1987).
    [CrossRef] [PubMed]
  5. J. I. Larruquert, J. A. Mendez, and J. A. Aznarez, "Far-ultraviolet reflectance measurements and optical-constants of unoxidized aluminium films," Appl. Opt. 34, 4892-4899 (1995).
    [CrossRef] [PubMed]
  6. D. Franta and I. Ohl??ýdal, "Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries," J. Mod. Opt. 45, 903-934 (1998).
    [CrossRef]
  7. D. Franta and I. Ohl??ýdal, "Comparison of Effective Medium Approximation and Rayleigh-Rice Theory Concerning Ellipsometric Characterization of Rough Surfaces," Opt. Commun. 248, 459-467 (2005).
    [CrossRef]
  8. D. Franta and I. Ohlídal, "Influence of Lateral Dimensions of the Irregularities on the Optical Quantities of Rough Surfaces," J. Opt. A-Pure Appl. Opt. 8, 763-774 (2006).
    [CrossRef]
  9. I. Ohlídal and D. Franta, "Ellipsometry of Thin Film Systems," in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 2000), Vol. 41, pp. 181-282.
  10. A. Vasí?ek, Optics of Thin Films (North-Holland, Amsterdam, 1960).
  11. Z. Knittl, Optics of Thin Films (Wiley, London, 1976).
  12. M. Born and E. Wolf, Principles of Optics, 3 ed. (Pergamon Press, Oxford, 1965).
  13. D. E. Aspnes, J. B. Theeten, and F. Hottier, "Investigation of effective-medium models of Microscopic Surface Roughness by Spectroscopic Ellipsometry," Phys. Rev. B 20, 3292-3302 (1979).
    [CrossRef]
  14. I. Ohlídal and F. Lukes, "Ellipsometric parameters of rough surfaces and of a System Substrate-Thin Film with rough boundaries," Opt. Acta 19, 817-843 (1972).
    [CrossRef]
  15. I. Ohlídal, F. Vidzd??a, and M. Ohlídal, "Optical Analysis by means of Spectroscopic Reflectometry of single and double layers with correlated randomly rough boundaries," Opt. Eng. 34, 1761-1768 (1995).
    [CrossRef]
  16. D. Franta, I. Ohlídal, and D. Ne?as, "Optical quantities of rough films calculated by Rayleigh-Rice theory," Phys. Status Solidi C 5, 1395-1398 (2008).
    [CrossRef]

2008 (1)

D. Franta, I. Ohlídal, and D. Ne?as, "Optical quantities of rough films calculated by Rayleigh-Rice theory," Phys. Status Solidi C 5, 1395-1398 (2008).
[CrossRef]

2006 (1)

D. Franta and I. Ohlídal, "Influence of Lateral Dimensions of the Irregularities on the Optical Quantities of Rough Surfaces," J. Opt. A-Pure Appl. Opt. 8, 763-774 (2006).
[CrossRef]

2005 (1)

D. Franta and I. Ohl??ýdal, "Comparison of Effective Medium Approximation and Rayleigh-Rice Theory Concerning Ellipsometric Characterization of Rough Surfaces," Opt. Commun. 248, 459-467 (2005).
[CrossRef]

1998 (1)

D. Franta and I. Ohl??ýdal, "Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries," J. Mod. Opt. 45, 903-934 (1998).
[CrossRef]

1995 (2)

I. Ohlídal, F. Vidzd??a, and M. Ohlídal, "Optical Analysis by means of Spectroscopic Reflectometry of single and double layers with correlated randomly rough boundaries," Opt. Eng. 34, 1761-1768 (1995).
[CrossRef]

J. I. Larruquert, J. A. Mendez, and J. A. Aznarez, "Far-ultraviolet reflectance measurements and optical-constants of unoxidized aluminium films," Appl. Opt. 34, 4892-4899 (1995).
[CrossRef] [PubMed]

1987 (1)

1979 (1)

D. E. Aspnes, J. B. Theeten, and F. Hottier, "Investigation of effective-medium models of Microscopic Surface Roughness by Spectroscopic Ellipsometry," Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

1972 (1)

I. Ohlídal and F. Lukes, "Ellipsometric parameters of rough surfaces and of a System Substrate-Thin Film with rough boundaries," Opt. Acta 19, 817-843 (1972).
[CrossRef]

1970 (1)

K. Krishen, "Scattering of Electromagnetic Waves from a layer with rough front and plane back (Small Perturbation Method by Rice)," IEEE Trans. Antennas Propag. AP-18, 573-576 (1970).
[CrossRef]

1967 (1)

G. R. Valenzuela, "Depolarization of EM Waves by Slightly Rough Surfaces," IEEE Trans. Antennas Propag. AP-15, 552-557 (1967).
[CrossRef]

1951 (1)

S. O. Rice, "Reflection of electromagnetic waves from slightly rough surfaces," Commun. Pure Appl. Math. 4, 351-378 (1951).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, J. B. Theeten, and F. Hottier, "Investigation of effective-medium models of Microscopic Surface Roughness by Spectroscopic Ellipsometry," Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

Aznarez, J. A.

Franta, D.

D. Franta, I. Ohlídal, and D. Ne?as, "Optical quantities of rough films calculated by Rayleigh-Rice theory," Phys. Status Solidi C 5, 1395-1398 (2008).
[CrossRef]

D. Franta and I. Ohlídal, "Influence of Lateral Dimensions of the Irregularities on the Optical Quantities of Rough Surfaces," J. Opt. A-Pure Appl. Opt. 8, 763-774 (2006).
[CrossRef]

D. Franta and I. Ohl??ýdal, "Comparison of Effective Medium Approximation and Rayleigh-Rice Theory Concerning Ellipsometric Characterization of Rough Surfaces," Opt. Commun. 248, 459-467 (2005).
[CrossRef]

D. Franta and I. Ohl??ýdal, "Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries," J. Mod. Opt. 45, 903-934 (1998).
[CrossRef]

Hottier, F.

D. E. Aspnes, J. B. Theeten, and F. Hottier, "Investigation of effective-medium models of Microscopic Surface Roughness by Spectroscopic Ellipsometry," Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

Krishen, K.

K. Krishen, "Scattering of Electromagnetic Waves from a layer with rough front and plane back (Small Perturbation Method by Rice)," IEEE Trans. Antennas Propag. AP-18, 573-576 (1970).
[CrossRef]

Larruquert, J. I.

Lukes, F.

I. Ohlídal and F. Lukes, "Ellipsometric parameters of rough surfaces and of a System Substrate-Thin Film with rough boundaries," Opt. Acta 19, 817-843 (1972).
[CrossRef]

Mendez, J. A.

Ne?as, D.

D. Franta, I. Ohlídal, and D. Ne?as, "Optical quantities of rough films calculated by Rayleigh-Rice theory," Phys. Status Solidi C 5, 1395-1398 (2008).
[CrossRef]

Ohl??ýdal, I.

D. Franta and I. Ohl??ýdal, "Comparison of Effective Medium Approximation and Rayleigh-Rice Theory Concerning Ellipsometric Characterization of Rough Surfaces," Opt. Commun. 248, 459-467 (2005).
[CrossRef]

D. Franta and I. Ohl??ýdal, "Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries," J. Mod. Opt. 45, 903-934 (1998).
[CrossRef]

Ohlídal, I.

D. Franta, I. Ohlídal, and D. Ne?as, "Optical quantities of rough films calculated by Rayleigh-Rice theory," Phys. Status Solidi C 5, 1395-1398 (2008).
[CrossRef]

D. Franta and I. Ohlídal, "Influence of Lateral Dimensions of the Irregularities on the Optical Quantities of Rough Surfaces," J. Opt. A-Pure Appl. Opt. 8, 763-774 (2006).
[CrossRef]

I. Ohlídal, F. Vidzd??a, and M. Ohlídal, "Optical Analysis by means of Spectroscopic Reflectometry of single and double layers with correlated randomly rough boundaries," Opt. Eng. 34, 1761-1768 (1995).
[CrossRef]

I. Ohlídal and F. Lukes, "Ellipsometric parameters of rough surfaces and of a System Substrate-Thin Film with rough boundaries," Opt. Acta 19, 817-843 (1972).
[CrossRef]

Rice, S. O.

S. O. Rice, "Reflection of electromagnetic waves from slightly rough surfaces," Commun. Pure Appl. Math. 4, 351-378 (1951).
[CrossRef]

Schiffer, R.

Theeten, J. B.

D. E. Aspnes, J. B. Theeten, and F. Hottier, "Investigation of effective-medium models of Microscopic Surface Roughness by Spectroscopic Ellipsometry," Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

Valenzuela, G. R.

G. R. Valenzuela, "Depolarization of EM Waves by Slightly Rough Surfaces," IEEE Trans. Antennas Propag. AP-15, 552-557 (1967).
[CrossRef]

Appl. Opt. (2)

Commun. Pure Appl. Math. (1)

S. O. Rice, "Reflection of electromagnetic waves from slightly rough surfaces," Commun. Pure Appl. Math. 4, 351-378 (1951).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

G. R. Valenzuela, "Depolarization of EM Waves by Slightly Rough Surfaces," IEEE Trans. Antennas Propag. AP-15, 552-557 (1967).
[CrossRef]

K. Krishen, "Scattering of Electromagnetic Waves from a layer with rough front and plane back (Small Perturbation Method by Rice)," IEEE Trans. Antennas Propag. AP-18, 573-576 (1970).
[CrossRef]

J. Mod. Opt. (1)

D. Franta and I. Ohl??ýdal, "Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries," J. Mod. Opt. 45, 903-934 (1998).
[CrossRef]

J. Opt. A-Pure Appl. Opt. (1)

D. Franta and I. Ohlídal, "Influence of Lateral Dimensions of the Irregularities on the Optical Quantities of Rough Surfaces," J. Opt. A-Pure Appl. Opt. 8, 763-774 (2006).
[CrossRef]

Opt. Acta (1)

I. Ohlídal and F. Lukes, "Ellipsometric parameters of rough surfaces and of a System Substrate-Thin Film with rough boundaries," Opt. Acta 19, 817-843 (1972).
[CrossRef]

Opt. Commun. (1)

D. Franta and I. Ohl??ýdal, "Comparison of Effective Medium Approximation and Rayleigh-Rice Theory Concerning Ellipsometric Characterization of Rough Surfaces," Opt. Commun. 248, 459-467 (2005).
[CrossRef]

Opt. Eng. (1)

I. Ohlídal, F. Vidzd??a, and M. Ohlídal, "Optical Analysis by means of Spectroscopic Reflectometry of single and double layers with correlated randomly rough boundaries," Opt. Eng. 34, 1761-1768 (1995).
[CrossRef]

Phys. Rev. B (1)

D. E. Aspnes, J. B. Theeten, and F. Hottier, "Investigation of effective-medium models of Microscopic Surface Roughness by Spectroscopic Ellipsometry," Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

Phys. Status Solidi C (1)

D. Franta, I. Ohlídal, and D. Ne?as, "Optical quantities of rough films calculated by Rayleigh-Rice theory," Phys. Status Solidi C 5, 1395-1398 (2008).
[CrossRef]

Other (4)

I. Ohlídal and D. Franta, "Ellipsometry of Thin Film Systems," in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 2000), Vol. 41, pp. 181-282.

A. Vasí?ek, Optics of Thin Films (North-Holland, Amsterdam, 1960).

Z. Knittl, Optics of Thin Films (Wiley, London, 1976).

M. Born and E. Wolf, Principles of Optics, 3 ed. (Pergamon Press, Oxford, 1965).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1.

Schematic diagram of the j-th layer of the isotropic thin film system.

Fig. 2.
Fig. 2.

Schematic diagram representing thin film system containing only one layer.

Fig. 3.
Fig. 3.

Schematic diagrams of the different types of the rough layers.

Fig. 4.
Fig. 4.

Spectral dependencies of the components of the Poincaré vector P⃗ for the incidence angle of 65° and reflectance R corresponding to rough layer calculated by the RRT for the IRL model (d=25 nm, σ=5 nm and τ=50 nm). The fits of the RRT data were performed by formulae belonging to the IRA and smooth layer.

Fig. 5.
Fig. 5.

The thickness dependencies of the relative differences (σ′-σ)/σ, (τ′-τ)/τ and (d′-d)/d between the fitted parameters and parameters selected for the RRT simulation and quantity χ expressing the quality of the fits for the different models of rough layer (σ=5 nm and τ=50 nm).

Fig. 6.
Fig. 6.

The thickness dependencies of the relative differences (σ′-σ)/σ, (τ′-τ)/τ and (d′-d)/d between the fitted parameters and parameters selected for the RRT simulation and quantity χ expressing the quality of the fits for the IRL model of rough layer corresponding to the different values of the autocorrelation length τ (σ=2 nm).

Fig. 7.
Fig. 7.

Spectral dependencies of the differences between the values of the optical quantities P⃗ and R calculated using the RRT and using the approximations: EMA, SDT, EMA&SDT and smooth layer. The IRL model was used for the RRT calculation with the following parameters: d=25 nm, σ=5 nm and τ=50 nm.

Fig. 8.
Fig. 8.

Spectral dependencies of the components of the Poincaré vector P⃗ for the incidence angle of 65° and reflectance R corresponding to rough layer calculated by the RRT for the IRL model (d=500 nm, σ=5 nm and τ=50 nm). The fits of the RRT data were performed by formulae belonging to the IRA, EMA&SDT and smooth layer.

Fig. 9.
Fig. 9.

Spectral dependencies of the differences between the values of the optical quantities P→ and R calculated using the RRT and using the approximations: IRA, IRA&SDT and smooth layer. The IRL model was used for the RRT calculation with the following parameters: d=25 nm, σ=5 nm and τ=50 nm.

Tables (2)

Tables Icon

Table 1. The values of the parameters found using the individual approximations by fitting the RRT simulated data calculated for the IRL model with the following parameters: d=25 nm, σ=5 nm and τ=50 nm.

Tables Icon

Table 2. The values of the parameters found using the individual approximations by fitting the RRT simulated data calculated for the IRL model with the following parameters: d=500 nm, σ=5 nm and τ=50 nm.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

A q , j 1 = B qj A qj ,
A qj = ( A ̂ qj + A ̂ qj ) .
B qj = 1 t ̂ qj ( 1 r ̂ qj r ̂ qj 1 ) .
T j = ( exp ( i X ̂ j ) 0 0 exp ( i X ̂ j ) ) ,
X ̂ j = 2 π λ d j n ̂ j 2 sin 2 θ 0 .
M q = B q 1 T 1 B q 2 T 2 B q 3 . . . B q , N T N B q , N + 1 .
r ̂ q = M q , 21 M q , 11 and t ̂ q = 1 M q , 11 .
M q = B q 1 T 1 B q 2 = B q ( SB ) ( n ̂ a , n ̂ f ) T ( d , n ̂ f ) B q ( SB ) ( n ̂ f , n ̂ s ) ,
B qj = 1 t ̂ qj + ( 1 r ̂ qj r ̂ qj + t ̂ qj + t ̂ qj r ̂ qj + r ̂ qj ) ,
r ̂ qj + = r ̂ qj and t ̂ qj + t ̂ qj r ̂ qj + r ̂ qj _ = 1
r ̂ qj = r ̂ qj ( 0 ) + σ j 2 + + f ̂ qj ( K x , K y ) w j ( K x n 0 k 0 sin θ 0 , K y ) d K x d K y ,
r ̂ q = r ̂ q ( 0 ) + σ 1 2 + + f ̂ 1 q ( K x , K y ) w 1 ( K x n 0 k 0 sin θ 0 , K y ) d K x d K y
+ σ 2 2 + + f ̂ 2 q ( K x , K y ) w 2 ( K x n 0 k 0 sin θ 0 , K y ) d K x d K y
+ σ 1 σ 2 + + f ̂ 12 q ( K x , K y ) w 12 ( K x n 0 k 0 sin θ 0 , K y ) d K x d K y ,
w 12 ( K x , K y ) = c 12 ( K x , K y ) w 1 ( K x , K y ) w 2 ( K x , K y ) ,
c 12 ( K x , K y ) 1 .
M q = 1 t ̂ q + ( 1 r ̂ q r ̂ q + t ̂ q + t ̂ q r ̂ q + r ̂ q ) .
M q = B q ( RL ) ( σ 1 , w 1 , σ 2 , w 2 , c 12 , d , n ̂ a , n ̂ f , n ̂ s ) ,
M q = B q 1 T 1 B q 2 = B q ( RB ) ( σ 1 , w 1 , n ̂ a , n ̂ f ) T ( d , n ̂ f ) B q ( RB ) ( σ 2 , w 2 , n ̂ f , n ̂ s ) ,
w ( K x , K y ) = w ( K ) = τ 2 4 π e K 2 τ 2 4 ,
P = ( I s , I cII , I cIII ) = ( sin 2 Ψ sin Δ , sin 2 Ψ cos Δ , cos 2 Ψ ) .
Q ( σ , τ , d ) = i = 1 N i P ( RRT ) P ( IRA ) ( σ , τ , d ) W i
+ j = 1 N j [ R ( RRT ) R ( IRA ) ( σ , τ , d ) ] W j ,
χ = Q 3 N i + N j ,

Metrics