Abstract

Optical components required for infrared (IR) ellipsometry have distinctly worse characteristics compared to those available for the visible spectrum. The calibration of the optical components used is therefore essential for obtaining reliable results. Here a powerful method is outlined to calibrate simultaneously the polarization characteristics of a source and detector through the synchronous rotation of two polarizers. The performance of this method is to a large degree independent of the quality of (commercially available) polarizers. This renders this method robust and highly suitable for the IR range. Moreover, it is also inherently insensitive toward a nonlinear response of the detector. This enables us to use this method as the first step in the quantification of component imperfections.

© 2009 Optical Society of America

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References

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  1. E. Wold and J. Bremer, “Mueller matrix analysis of infrared ellipsometry,” Appl. Opt. 33, 5982-5993 (1994).
    [CrossRef] [PubMed]
  2. J. H. W. G. den Boer, G. W. M. Kroesen, M. Haverlag, and F. J. de Hoog, “Spectroscopic IR ellipsometry with imperfect components,” Thin Solid Films 234, 323-326 (1993).
    [CrossRef]
  3. M. Luttmann, J.-L. Stehle, C. Defranoux, and J.-P. Piel, “High accuracy IR ellipsometry working with a Ge Brewster angle reflection polarizer and grid analyzer,” Thin Solid Films 313-314, 631-641 (1998).
    [CrossRef]
  4. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1979).
  5. J. H. W. G. den Boer, “Spectroscopic infrared ellipsometry: components, calibration, and application,” Ph.D. dissertation (University of Eindhoven, 1995).
  6. B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer systems,” Phys. Status Solidi C 5, 1031-1035(2008).
    [CrossRef]

2008 (1)

B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer systems,” Phys. Status Solidi C 5, 1031-1035(2008).
[CrossRef]

1998 (1)

M. Luttmann, J.-L. Stehle, C. Defranoux, and J.-P. Piel, “High accuracy IR ellipsometry working with a Ge Brewster angle reflection polarizer and grid analyzer,” Thin Solid Films 313-314, 631-641 (1998).
[CrossRef]

1994 (1)

1993 (1)

J. H. W. G. den Boer, G. W. M. Kroesen, M. Haverlag, and F. J. de Hoog, “Spectroscopic IR ellipsometry with imperfect components,” Thin Solid Films 234, 323-326 (1993).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1979).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1979).

Bremer, J.

de Hoog, F. J.

J. H. W. G. den Boer, G. W. M. Kroesen, M. Haverlag, and F. J. de Hoog, “Spectroscopic IR ellipsometry with imperfect components,” Thin Solid Films 234, 323-326 (1993).
[CrossRef]

Defranoux, C.

M. Luttmann, J.-L. Stehle, C. Defranoux, and J.-P. Piel, “High accuracy IR ellipsometry working with a Ge Brewster angle reflection polarizer and grid analyzer,” Thin Solid Films 313-314, 631-641 (1998).
[CrossRef]

den Boer, J. H. W. G.

J. H. W. G. den Boer, G. W. M. Kroesen, M. Haverlag, and F. J. de Hoog, “Spectroscopic IR ellipsometry with imperfect components,” Thin Solid Films 234, 323-326 (1993).
[CrossRef]

J. H. W. G. den Boer, “Spectroscopic infrared ellipsometry: components, calibration, and application,” Ph.D. dissertation (University of Eindhoven, 1995).

Haverlag, M.

J. H. W. G. den Boer, G. W. M. Kroesen, M. Haverlag, and F. J. de Hoog, “Spectroscopic IR ellipsometry with imperfect components,” Thin Solid Films 234, 323-326 (1993).
[CrossRef]

Herzinger, C. M.

B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer systems,” Phys. Status Solidi C 5, 1031-1035(2008).
[CrossRef]

Johs, B.

B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer systems,” Phys. Status Solidi C 5, 1031-1035(2008).
[CrossRef]

Kroesen, G. W. M.

J. H. W. G. den Boer, G. W. M. Kroesen, M. Haverlag, and F. J. de Hoog, “Spectroscopic IR ellipsometry with imperfect components,” Thin Solid Films 234, 323-326 (1993).
[CrossRef]

Luttmann, M.

M. Luttmann, J.-L. Stehle, C. Defranoux, and J.-P. Piel, “High accuracy IR ellipsometry working with a Ge Brewster angle reflection polarizer and grid analyzer,” Thin Solid Films 313-314, 631-641 (1998).
[CrossRef]

Piel, J.-P.

M. Luttmann, J.-L. Stehle, C. Defranoux, and J.-P. Piel, “High accuracy IR ellipsometry working with a Ge Brewster angle reflection polarizer and grid analyzer,” Thin Solid Films 313-314, 631-641 (1998).
[CrossRef]

Stehle, J.-L.

M. Luttmann, J.-L. Stehle, C. Defranoux, and J.-P. Piel, “High accuracy IR ellipsometry working with a Ge Brewster angle reflection polarizer and grid analyzer,” Thin Solid Films 313-314, 631-641 (1998).
[CrossRef]

Wold, E.

Appl. Opt. (1)

Phys. Status Solidi C (1)

B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer systems,” Phys. Status Solidi C 5, 1031-1035(2008).
[CrossRef]

Thin Solid Films (2)

J. H. W. G. den Boer, G. W. M. Kroesen, M. Haverlag, and F. J. de Hoog, “Spectroscopic IR ellipsometry with imperfect components,” Thin Solid Films 234, 323-326 (1993).
[CrossRef]

M. Luttmann, J.-L. Stehle, C. Defranoux, and J.-P. Piel, “High accuracy IR ellipsometry working with a Ge Brewster angle reflection polarizer and grid analyzer,” Thin Solid Films 313-314, 631-641 (1998).
[CrossRef]

Other (2)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1979).

J. H. W. G. den Boer, “Spectroscopic infrared ellipsometry: components, calibration, and application,” Ph.D. dissertation (University of Eindhoven, 1995).

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Figures (6)

Fig. 1
Fig. 1

Setup for characterizing the detector and source with imperfect polarizers. The two virtual polarizers in the setup represent the detector and source polarization dependence and are characterized simultaneously by measuring the intensity as a function of analyzer angle, where A = P and A = P ± 45 ° .

Fig. 2
Fig. 2

Simulated accuracy of the calibration method in which a calibration analysis is made on a simulated signal with a known detector attenuation α D input . Displayed is the relative difference between the result from the calibration method on the test signal α D sim as a function of the value α D input used to construct the test signal.

Fig. 3
Fig. 3

Relative difference between the calibration results α D sim and the input values α D input as a function of polarizer attenuation γ for both calibration methods: separate (lower curve) and simultaneous (upper curve) calibration. The input values for detector/source attenuation were chosen to be 0.8 < α D , S input < 1.0 , and their angles D and S were chosen randomly.

Fig. 4
Fig. 4

Determined attenuation factors and angles of the detector and source, illustrating the components’ polarization dependence.

Fig. 5
Fig. 5

Measured polarizer attenuation γ values for grid polarizers in tandem. The values were determined after calibrating and correcting for the detector and source polarization dependences and nonlinear behavior of the detector.

Fig. 6
Fig. 6

N and C values for air, after correcting for detector and source polarization dependence, nonlinearity in the detector and inefficiencies in the polarizers.

Equations (8)

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M ̲ pol ( α ) = 1 2 [ 1 + α 1 α 0 0 1 α 1 + α 0 0 0 0 2 α 0 0 0 0 2 α ] .
I ̲ D = M ̲ pol ( α D ) · R ̲ ( D A ) · M ̲ pol ( γ ) · R ̲ ( A P ) · M ̲ pol ( γ ) · R ̲ ( P S ) · M ̲ pol ( α S ) · I ̲ S .
I D 0 , A = P = I 0 4 · [ { ( 1 + α D ) ( 1 + α S ) + 1 2 ( 1 α D ) ( 1 α S ) cos ( 2 ( D S ) ) } 1 + { ( 1 α D ) ( 1 + α S ) cos ( 2 D ) + ( 1 + α D ) ( 1 α S ) cos ( 2 S ) } cos ( 2 A ) + { ( 1 α D ) ( 1 + α S ) sin ( 2 D ) + ( 1 + α D ) ( 1 α S ) sin ( 2 S ) } sin ( 2 A ) + { 1 2 ( 1 α D ) ( 1 α S ) cos ( 2 ( D + S ) ) } cos ( 4 A ) + { 1 2 ( 1 α D ) ( 1 α S ) sin ( 2 ( D + S ) ) } sin ( 4 A ) ] .
I D 0 , A = P 45 ° = I 0 4 · [ { ( 1 + α D ) ( 1 + α S ) + 1 2 ( 1 α D ) ( 1 α S ) sin ( 2 ( S D ) ) } 1 + { ( 1 α D ) ( 1 + α S ) cos ( 2 D ) + ( 1 + α D ) ( 1 α S ) sin ( 2 S ) } cos ( 2 A ) + { ( 1 α D ) ( 1 + α S ) sin ( 2 D ) ( 1 + α D ) ( 1 α S ) cos ( 2 S ) } sin ( 2 A ) + { 1 2 ( 1 α D ) ( 1 α S ) sin ( 2 ( D + S ) ) } cos ( 4 A ) { 1 2 ( 1 α D ) ( 1 α S ) cos ( 2 ( D + S ) ) } sin ( 4 A ) ] ,
I D 0 , A = P + 45 ° = I 0 4 · [ { ( 1 + α D ) ( 1 + α S ) + 1 2 ( 1 - α D ) ( 1 α S ) sin ( 2 ( D S ) ) } 1 + { ( 1 α D ) ( 1 + α S ) cos ( 2 D ) ( 1 + α D ) ( 1 α S ) sin ( 2 S ) } cos ( 2 A ) + { ( 1 α D ) ( 1 + α S ) sin ( 2 D ) + ( 1 + α D ) ( 1 α S ) cos ( 2 S ) } sin ( 2 A ) { 1 2 ( 1 α D ) ( 1 α S ) sin ( 2 ( D + S ) ) } cos ( 4 A ) + { 1 2 ( 1 α D ) ( 1 α S ) cos ( 2 ( D + S ) ) } sin ( 4 A ) ] .
I D 0 = I 0 4 ( 1 cos 2 Ψ cos 2 P ) · ( 1 + cos 2 P cos 2 Ψ 1 cos 2 Ψ cos 2 P cos 2 A + sin 2 Ψ cos Δ sin 2 P 1 cos 2 Ψ cos 2 P sin 2 A ) · [ ( 1 2 + α S 2 ) + ( 1 2 - α S 2 ) cos 2 ( P - S ) ] · [ ( 1 2 + α D 2 ) + ( 1 2 - α D 2 ) cos 2 ( D A ) ] I D 0 = I 0 4 ( 1 cos 2 Ψ cos 2 P ) ( 1 + α cos 2 A + β sin 2 A ) · F correction ( α D , D , α S , S , P , A ) .
α = ( 1 γ ) 2 cos 2 P ( 1 + γ ) ( 1 γ ) cos 2 Ψ ( 1 + γ ) 2 ( 1 + γ ) ( 1 + γ ) cos 2 Ψ cos 2 P , β = ( 1 γ ) 2 sin 2 Ψ cos Δ sin 2 P ( 1 + γ ) 2 ( 1 + γ ) ( 1 γ ) cos 2 Ψ cos 2 P .
N = 1 γ 1 + γ cos 2 P 1 + γ 1 γ α 1 α cos 2 P , C = β sin 2 P ( ( 1 + γ 1 γ ) 2 cos 2 2 P 1 α cos 2 P ) .

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