Abstract

A direct spectral phase function calculation method based on spectral phase shifting is described. We show experimentally that the direct phase function calculation method can provide a simple and fast solution in calculating the spectral phase function, while maintaining the same level of accurate measurement capability as that based on the Fourier transform approach.

© 2004 Optical Society of America

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References

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  14. The Levenberg-Marquardt algorithm is available as lsqnonlin function by a commercial S/W MATLAB.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

I. Yamaguchi, �??Surface tomography by wavelength scanning interferometry,�?? Opt. Eng. 39, 40-46 (2000).
[CrossRef]

Opt. Lett.

Other

K. Creath, �??Temporal phase measuring methods,�?? in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993).

The Levenberg-Marquardt algorithm is available as lsqnonlin function by a commercial S/W MATLAB.

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of the 3-D spectral scanning interferometric thickness profilometer.

Fig. 2.
Fig. 2.

Measured samples: (a) Fresnel Zone Plate (FZP) with no thin films and (b) rectangular patterned Si wafer upon which a SiO2 thin film is deposited and planarized by a chemical mechanical planarization process.

Fig. 3.
Fig. 3.

(a) Computation time required to obtain the phase value of a specific point versus the number of data N (*: the Fourier transform method, □: the direct phase calculation method) and (b) Ratio between the computation time of the Fourier transform method and that of the direct method versus the number of data N.

Fig. 4.
Fig. 4.

(a)–(b): Spectral phase function ϕ(k) at a specific coordinate (x,y) (a) for the object in Fig. 2(a) and (b) for the object in Fig. 2(b) (Solid line: Fourier transform method, *: proposed direct phase calculation method). (c)–(d): Phase differences between the calculated phase functions obtained by the two different approaches (c) for the object in Fig. 2(a) and (d) for the object in Fig. 2(b).

Fig. 5.
Fig. 5.

Three-dimensional thickness profile measurement results by use of the proposed direct spectral phase calculation method (a) for the object in Fig. 2(a) and (b) for the object in Fig. 2(b).

Fig. 6.
Fig. 6.

Two-dimensional section profiles of Fig. 5(a) and (b), respectively. The solid line represents the result obtained by use of the direct calculation method and the dotted line is obtained by use of the Fourier transform method.

Equations (9)

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I x y k = i 0 x y k { 1 + γ x y k cos [ ϕ x y k ] }
I k p 1 p 2 p 3 n 1 n 2 n 3
= i 0 ( k ) { 1 + γ ( k ) cos [ 2 k p 1 + ψ k p 2 p 3 n 1 n 2 n 3 ] }
= i 0 ( k ) { 1 + γ ( k ) cos [ 2 k c p 1 + ψ x y k p 2 p 3 n 1 n 2 n 3 + 2 s 0 δk + 2 p 1 δk ] } }
I m k p 1 p 2 p 3 n 1 n 2 n 3
= i 0 ( k ) { 1 + γ ( k ) cos ( 2 k c p 1 + ψ k p 2 p 3 n 1 n 2 n 3 + 2 s 0 ( m 3 ) Δ k ) }
ϕ ( k c ) = 2 k c p 1 + ψ k c p 2 p 3 n 1 n 2 n 3 = tan 1 [ 1 cos ( 4 Δ k s 0 ) sin ( 2 Δ k s 0 ) ( I 2 I 4 2 I 3 I 5 I 1 ) ]
p 1 x y = δϕ x y k 2 δk
η p 1 p 2 p 3 = i = 1 N [ ϕ model k i p 1 p 2 p 3 ϕ measured ( k i ) ] 2

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