Abstract

The effects of the presence of a transparent thin film on a test surface in white-light interferometric surface profiling are investigated. An expression is obtained for the output intensity variations in a Michelson interferometer which includes the effect of multiple reflections within the thin film. The number of reflections that need to be considered to obtain good convergence to the correct solution is discussed.

© 2005 Optical Society of America

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References

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  1. M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit, Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
  2. M. Roy, C.J.R. Sheppard, and P. Hariharan, “Low-coherence interference microscopy using a ferro-electric liquid crystal phase-shifter,” Optics Express,  12, 2512–2516 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2512.
    [Crossref] [PubMed]
  3. P. Hariharan and M. Roy, “Interferometric surface profiling with white-light: effects of surface films,” J. Mod. Opt.,  43, 1797–1800 (1996).
    [Crossref]
  4. S-W. Kim and G.H. Kim, “Thickness profile measurement of transparent thin-film layers by white-light scanning interferometry,” Appl. Opt. 38, 5968–5973 (1999).
    [Crossref]
  5. P. de Groot and X. C. de Lega, “Signal modeling for low-coherence height-scanning interferometric microscopy,” Appl. Opt.,  43, 4821–4830 (2004).
    [Crossref] [PubMed]
  6. P. Hariharan, Optical Interferometry (Academic, San Diego, 2003).
  7. G. W. C. Kaye and T. H. Laby, Tables of Constants in Physics and Chemistry (Longmans, Green & Co Ltd, London, 1959).
  8. L. J. Fried and H. A. Froot, “Thickness measurements of silicon dioxide films over small geometries,” J. Appl. Phys. 39, 5732–5735 (1968).
    [Crossref]

2004 (2)

M. Roy, C.J.R. Sheppard, and P. Hariharan, “Low-coherence interference microscopy using a ferro-electric liquid crystal phase-shifter,” Optics Express,  12, 2512–2516 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2512.
[Crossref] [PubMed]

P. de Groot and X. C. de Lega, “Signal modeling for low-coherence height-scanning interferometric microscopy,” Appl. Opt.,  43, 4821–4830 (2004).
[Crossref] [PubMed]

1999 (1)

1996 (1)

P. Hariharan and M. Roy, “Interferometric surface profiling with white-light: effects of surface films,” J. Mod. Opt.,  43, 1797–1800 (1996).
[Crossref]

1968 (1)

L. J. Fried and H. A. Froot, “Thickness measurements of silicon dioxide films over small geometries,” J. Appl. Phys. 39, 5732–5735 (1968).
[Crossref]

Cohen, F.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit, Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).

Davidson, M.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit, Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).

de Groot, P.

de Lega, X. C.

Fried, L. J.

L. J. Fried and H. A. Froot, “Thickness measurements of silicon dioxide films over small geometries,” J. Appl. Phys. 39, 5732–5735 (1968).
[Crossref]

Froot, H. A.

L. J. Fried and H. A. Froot, “Thickness measurements of silicon dioxide films over small geometries,” J. Appl. Phys. 39, 5732–5735 (1968).
[Crossref]

Hariharan, P.

M. Roy, C.J.R. Sheppard, and P. Hariharan, “Low-coherence interference microscopy using a ferro-electric liquid crystal phase-shifter,” Optics Express,  12, 2512–2516 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2512.
[Crossref] [PubMed]

P. Hariharan and M. Roy, “Interferometric surface profiling with white-light: effects of surface films,” J. Mod. Opt.,  43, 1797–1800 (1996).
[Crossref]

P. Hariharan, Optical Interferometry (Academic, San Diego, 2003).

Kaufman, K.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit, Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).

Kaye, G. W. C.

G. W. C. Kaye and T. H. Laby, Tables of Constants in Physics and Chemistry (Longmans, Green & Co Ltd, London, 1959).

Kim, G.H.

Kim, S-W.

Laby, T. H.

G. W. C. Kaye and T. H. Laby, Tables of Constants in Physics and Chemistry (Longmans, Green & Co Ltd, London, 1959).

Mazor, I.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit, Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).

Roy, M.

M. Roy, C.J.R. Sheppard, and P. Hariharan, “Low-coherence interference microscopy using a ferro-electric liquid crystal phase-shifter,” Optics Express,  12, 2512–2516 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2512.
[Crossref] [PubMed]

P. Hariharan and M. Roy, “Interferometric surface profiling with white-light: effects of surface films,” J. Mod. Opt.,  43, 1797–1800 (1996).
[Crossref]

Sheppard, C.J.R.

M. Roy, C.J.R. Sheppard, and P. Hariharan, “Low-coherence interference microscopy using a ferro-electric liquid crystal phase-shifter,” Optics Express,  12, 2512–2516 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2512.
[Crossref] [PubMed]

Appl. Opt. (2)

J. Appl. Phys. (1)

L. J. Fried and H. A. Froot, “Thickness measurements of silicon dioxide films over small geometries,” J. Appl. Phys. 39, 5732–5735 (1968).
[Crossref]

J. Mod. Opt. (1)

P. Hariharan and M. Roy, “Interferometric surface profiling with white-light: effects of surface films,” J. Mod. Opt.,  43, 1797–1800 (1996).
[Crossref]

Optics Express (1)

M. Roy, C.J.R. Sheppard, and P. Hariharan, “Low-coherence interference microscopy using a ferro-electric liquid crystal phase-shifter,” Optics Express,  12, 2512–2516 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2512.
[Crossref] [PubMed]

Other (3)

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit, Metrology, Inspection and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).

P. Hariharan, Optical Interferometry (Academic, San Diego, 2003).

G. W. C. Kaye and T. H. Laby, Tables of Constants in Physics and Chemistry (Longmans, Green & Co Ltd, London, 1959).

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Figures (3)

Fig. 1.
Fig. 1.

Spectrum of the reflected field from the surface of a mirror coated with various thicknesses of a dielectric film from 2nd=0 to 2nd=1.5 λo for multiple reflections (black- 1 reflection, blue- 3 reflections, green- 5 reflections and red- 7 reflections).

Fig. 2.
Fig. 2.

Visibility curves, including the effect of multiple reflections, for various thicknesses of a silica film on silicon (n 1=1.46, n 2=4.05 and k=0.03).

Fig. 3.
Fig. 3.

Convergence of the error in the location of the visibility maximum, with the number of multiple reflections, for different film/substrate reflectances: (a)r 1=0.2,r 2=0.8 (b) r 1=0.4,r 2=0.8 (c) r 1=0.8,r 2=0.8 and (d) r 1=0.4,r 2=0.6.

Equations (27)

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U 1 ( ω ) = U in ( ω ) A ( ω )
U 2 ( ω ) = U in ( ω ) be i ω τ ,
A ( ω ) = r 1 + r 2 ( 1 r 1 2 ) k = 0 ( r 1 r 2 ) k e i ( k + 1 ) ω Δ τ
U out ( ω ) = U 1 ( ω ) + U 2 ( ω )
= U in ( ω ) [ be iωτ + r 1 + r 2 ( 1 r 1 2 ) k = 0 ( r 1 r 2 ) k e i ( k + 1 ) ω Δ τ ] .
S out ( ω ) = U out * ( ω ) U out ( ω ) ,
S out ( ω ) = S in ( ω ) { [ be iωτ + r 1 + r 2 ( 1 r 1 2 ) k = 0 ( r 1 r 2 ) k e i ( k + 1 ) ω Δ τ ]
× [ be iωτ + r 1 + r 2 ( 1 r 1 2 ) k = 0 ( r 1 r 2 ) k e i ( k + 1 ) ω Δ τ ] } .
S out ( ω ) = S in ( ω )
× [ b 2 + r 1 2 + 2 r 1 b cos ω τ + 2 r 2 b ( 1 r 1 2 ) k = 0 ( r 1 r 2 ) k cos [ ( k + 1 ) ω Δ τ ω τ ] ]
+ 2 r 1 r 2 ( 1 r 1 2 ) k = 0 ( r 1 r 2 ) k cos [ ( k + 1 ) ω Δ τ ]
+ r 2 2 ( 1 r 1 2 ) 2 + k = 0 j = 0 ( r 1 r 2 ) k + j e i ω Δ τ ( k j ) ) .
2 b [ r 1 cos ω τ + r 2 ( 1 r 1 2 ) k = 0 ( r 1 r 2 ) k cos [ ( k + 1 ) ω Δ τ ω τ ] ] .
I ( τ ) = S out ( ω ) d ω
I ( τ ) = S in ( ω ) [ 2 b { [ r 1 cos ω τ + r 2 ( 1 r 1 2 ) k = 0 ( r 1 r 2 ) k cos [ ( k + 1 ) ω Δ τ ω τ ] } ] d ω
= Re [ 2 b { r 1 F ( τ ) + r 2 ( 1 r 1 2 ) k = 0 F [ ( k + 1 ) Δ τ τ ] } ] ,
F ( τ ) = S in ( ω ) e i ω τ d ω .
S in ( ω ) = 1 N exp [ ( ω ω 0 ω 1 ) 2 ] ,
F ( τ ) = exp ( i ω 0 τ ) exp [ ( 1 4 ) ω 1 2 τ 2 ] ,
I ( τ ) = Re [ 2 b k = 0 ( r 1 r 2 ) k [ r 1 e i ω 0 ( k Δ τ τ ) e ( 1 4 ) ω 1 2 ( k Δ τ τ ) 2 +
r 2 e i ω 0 ( ( k + 1 ) Δ τ τ ) e ( 1 4 ) ω 1 2 [ ( k + 1 ) Δ τ τ ] 2 ] ] .
S out ( ω ) = S in ( ω ) { [ be i ω τ + r 1 + r 2 ( 1 r 1 2 ) k = 0 ( r 1 r 2 ) k e i ( k + 1 ) ω Δ τ ]
× [ be i ω τ + r 1 + r 2 * ( 1 r 1 2 ) k = 0 ( r 1 r 2 * ) k e i ( k + 1 ) ω Δ τ ] } .
r 2 = ( n 1 n 2 ) 1 i k 2 ( n 1 n 2 ) + 1 + i k 2 .
r 1 = n 0 n 1 n 0 + n 1 ,
I ( τ ) = Re [ b { r 1 F ( τ ) + r 2 ( 1 r 1 2 ) k = 0 ( r 1 r 2 ) k F [ ( k + 1 ) Δ τ τ ]
+ r 1 F ( τ ) + r 2 * ( 1 r 1 2 ) k = 0 ( r 1 r 2 * ) k F [ ( k + 1 ) Δ τ τ ] } ] .

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