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, 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. SPIE 775, 233–247 (1987).
  2. M.  Roy, C.J.R.  Sheppard, 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, M.  Roy, “Interferometric surface profiling with white-light: effects of surface films,” J. Mod. Opt., 43, 1797–1800 (1996).
    [CrossRef]
  4. S-W.  Kim, 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, 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, T. H.  Laby, Tables of Constants in Physics and Chemistry (Longmans, Green & Co Ltd, London, 1959).
  8. L. J.  Fried, 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, 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, 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, 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, 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, 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. SPIE 775, 233–247 (1987).

Davidson, M.

M.  Davidson, K.  Kaufman, I.  Mazor, 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. SPIE 775, 233–247 (1987).

de Groot, P.

de Lega, X. C.

Fried, L. J.

L. J.  Fried, 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, 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, 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, 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, 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. SPIE 775, 233–247 (1987).

Kaye, G. W. C.

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

Mazor, I.

M.  Davidson, K.  Kaufman, I.  Mazor, 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. SPIE 775, 233–247 (1987).

Roy, M.

M.  Roy, C.J.R.  Sheppard, 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, 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, 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, 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, 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, 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, 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. SPIE 775, 233–247 (1987).

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

G. W. C.  Kaye, 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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