Abstract

Through numerical simulations, we point out that introduction of an ellipsometric measurement technique to an absorption-based surface-plasmon resonance (SPR) sensor enhances precision and sensitivity in measuring the imaginary part k of the complex refractive index of the sample. By measuring a pair of ellipsometric ΔΨ parameters, instead of the conventional energy reflectance Rp of p-polarized light in the Kretschmann optical arrangement, we can detect a small change of k that is proportional to that of the concentration of the sample, especially when k1. While one has difficulty in determining the value of k uniquely by the standard technique, when the thickness of Au under the prism is thin (2030  nm), the ellipsometric technique (ET) overcomes the problem. Furthermore, the value of k and the thickness d s of the absorptive sample that is adsorbed on Au can be determined precisely. The ET based on the common-path polarization interferometer is robust against external disturbance such as mechanical vibration and intensity fluctuation of a light source. Although only the p-polarized light is responsible for the SPR phenomenon, we show that the introduction of the ET is significant for quantitative analysis.

© 2007 Optical Society of America

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References

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  1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
  2. B. Liedberg, C. Nylander, and I. Lundstrom, "Surface plasmon resonance for gas detection and biosensing," Sens. Actuators 4, 299-304 (1983).
    [CrossRef]
  3. S. Löfás and B. Johnson, "A novel hydrogel matrix on gold surfaces in surface plasmon resonance sensors for fast and efficient covalent immobilization of ligands," J. Chem. Soc. Chem. Commun. 21, 1526-1528 (1990).
    [CrossRef]
  4. X. Sun, S. Shiokawa, and Y. Matsui, "Experimental studies on biosensing by SPR," Jpn. J. Appl. Phys. 28, 1725-1727 (1989).
    [CrossRef]
  5. H. Kano and S. Kawata, "Surface-plasmon sensor for absorption-sensitivity enhancement," Appl. Opt. 33, 5166-5170 (1994).
    [CrossRef] [PubMed]
  6. A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, "Surface-plasmon resonance spectrometry and characterization of absorbing materals," Appl. Opt. 36, 3314-3320 (1997).
  7. A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, "Sensitivity and detection limit of concentration and adsorption measurements by laser-induced surface-plasmon resonance," Appl. Opt. 36, 6539-6547 (2000).
    [CrossRef]
  8. A. Ikehata, X. Li, T. Itoh, and Y. Ozaki, "High sensitive detection of near-infrared absorption by surface plasmon resonance," Appl. Phys. Lett. 83, 2232-2234 (2003).
    [CrossRef]
  9. K. Kurihara and K. Suzuki, "Theoretical understanding of an absorption-based surface plasmon resonance sensor based on Kretschmann's theory," Anal. Chem. 74, 696-701 (2002).
    [CrossRef] [PubMed]
  10. K. Kurihara, K. Nakamura, E. Hirayama, and K. Suzuki, "Theoretical understanding of an absorption-based surface plasmon resonance sensor applied to sodium ion sensing based on an ion-selective optode membrane," Anal. Chem. 74, 6323-6333 (2002).
    [CrossRef]
  11. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1989).
  12. P. Westphal and A. Bornmann, "Biomolecular detection by surface plasmon enhanced ellipsometry," Sens. Actuators B 84, 278-285 (2002).
    [CrossRef]
  13. I. R. Hooper and J. R. Sambles, "Sensing using differential surface plasmon ellipsometry," J. Appl. Phys. 96, 3004-3011 (2004).
    [CrossRef]
  14. C.-M. Wu, Z.-C. Jian, S.-F. Joe, and L.-B. Chang, "High-sensitivity sensor based on surface plasmon resonance and heterodyne interferometry," Sens. Actuators B 92, 133-136 (2003).
    [CrossRef]
  15. H. P. Ho, W. W. Lam, and S. Y. Wu, "Surface plasmon resonance sensor based on the measurement of differential phase," Rev. Sci. Instrum. 73, 3534-3539 (2002).
    [CrossRef]
  16. S. Y. Wu, H. P. Ho, W. C. Law, and C. Lin, "Highly sensitive differential phase-sensitive surface plasmon biosensor based on the Mach-Zehnder configuration," Opt. Lett. 29, 2378-2380 (2004).
    [CrossRef] [PubMed]
  17. K. Ohta and H. Ishida, "Comparison among several numerical integration methods for Kramers-Kronig transformation," Appl. Spectrosc. 42, 952-957 (1988).
    [CrossRef]
  18. K. Kurihara, K. Nakamura, and K. Suzuki, "Asymmetric SPR sensor response curve-fitting equation for the accurate determination of SPR resonance angle," Sens. Actuators B 86, 49-57 (2002).
    [CrossRef]
  19. G. Margheri, A. Mannori, and F. Quercioli, "High-resolution angular and displacement sensing based on the excitation of surface plasma waves," Appl. Opt. 36, 4521-4525 (1997).
    [CrossRef] [PubMed]
  20. For example, see http://www.biacore.com.
  21. J. Lee, P. I. Rovira, I. An, and R. W. Collins, "Rotating-compensator multichannel ellipsometry: application for real time Stokes vector spectroscopy of thin film growth," Rev. Sci. Instrum. 69, 1800-1810 (1998).
    [CrossRef]
  22. R. Kleim, L. Kuntzler, and A. E. Ghemmaz, "Systematic errors in rotating-compensator ellipsometry," J. Opt. Soc. Am. A 11, 2550-2559 (1994).
    [CrossRef]

2004 (2)

2003 (2)

C.-M. Wu, Z.-C. Jian, S.-F. Joe, and L.-B. Chang, "High-sensitivity sensor based on surface plasmon resonance and heterodyne interferometry," Sens. Actuators B 92, 133-136 (2003).
[CrossRef]

A. Ikehata, X. Li, T. Itoh, and Y. Ozaki, "High sensitive detection of near-infrared absorption by surface plasmon resonance," Appl. Phys. Lett. 83, 2232-2234 (2003).
[CrossRef]

2002 (5)

K. Kurihara and K. Suzuki, "Theoretical understanding of an absorption-based surface plasmon resonance sensor based on Kretschmann's theory," Anal. Chem. 74, 696-701 (2002).
[CrossRef] [PubMed]

K. Kurihara, K. Nakamura, E. Hirayama, and K. Suzuki, "Theoretical understanding of an absorption-based surface plasmon resonance sensor applied to sodium ion sensing based on an ion-selective optode membrane," Anal. Chem. 74, 6323-6333 (2002).
[CrossRef]

P. Westphal and A. Bornmann, "Biomolecular detection by surface plasmon enhanced ellipsometry," Sens. Actuators B 84, 278-285 (2002).
[CrossRef]

H. P. Ho, W. W. Lam, and S. Y. Wu, "Surface plasmon resonance sensor based on the measurement of differential phase," Rev. Sci. Instrum. 73, 3534-3539 (2002).
[CrossRef]

K. Kurihara, K. Nakamura, and K. Suzuki, "Asymmetric SPR sensor response curve-fitting equation for the accurate determination of SPR resonance angle," Sens. Actuators B 86, 49-57 (2002).
[CrossRef]

2000 (1)

1998 (1)

J. Lee, P. I. Rovira, I. An, and R. W. Collins, "Rotating-compensator multichannel ellipsometry: application for real time Stokes vector spectroscopy of thin film growth," Rev. Sci. Instrum. 69, 1800-1810 (1998).
[CrossRef]

1997 (2)

A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, "Surface-plasmon resonance spectrometry and characterization of absorbing materals," Appl. Opt. 36, 3314-3320 (1997).

G. Margheri, A. Mannori, and F. Quercioli, "High-resolution angular and displacement sensing based on the excitation of surface plasma waves," Appl. Opt. 36, 4521-4525 (1997).
[CrossRef] [PubMed]

1994 (2)

1990 (1)

S. Löfás and B. Johnson, "A novel hydrogel matrix on gold surfaces in surface plasmon resonance sensors for fast and efficient covalent immobilization of ligands," J. Chem. Soc. Chem. Commun. 21, 1526-1528 (1990).
[CrossRef]

1989 (1)

X. Sun, S. Shiokawa, and Y. Matsui, "Experimental studies on biosensing by SPR," Jpn. J. Appl. Phys. 28, 1725-1727 (1989).
[CrossRef]

1988 (1)

1983 (1)

B. Liedberg, C. Nylander, and I. Lundstrom, "Surface plasmon resonance for gas detection and biosensing," Sens. Actuators 4, 299-304 (1983).
[CrossRef]

An, I.

J. Lee, P. I. Rovira, I. An, and R. W. Collins, "Rotating-compensator multichannel ellipsometry: application for real time Stokes vector spectroscopy of thin film growth," Rev. Sci. Instrum. 69, 1800-1810 (1998).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1989).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1989).

Bornmann, A.

P. Westphal and A. Bornmann, "Biomolecular detection by surface plasmon enhanced ellipsometry," Sens. Actuators B 84, 278-285 (2002).
[CrossRef]

Chang, L.-B.

C.-M. Wu, Z.-C. Jian, S.-F. Joe, and L.-B. Chang, "High-sensitivity sensor based on surface plasmon resonance and heterodyne interferometry," Sens. Actuators B 92, 133-136 (2003).
[CrossRef]

Collins, R. W.

J. Lee, P. I. Rovira, I. An, and R. W. Collins, "Rotating-compensator multichannel ellipsometry: application for real time Stokes vector spectroscopy of thin film growth," Rev. Sci. Instrum. 69, 1800-1810 (1998).
[CrossRef]

Gershon, P. D.

A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, "Sensitivity and detection limit of concentration and adsorption measurements by laser-induced surface-plasmon resonance," Appl. Opt. 36, 6539-6547 (2000).
[CrossRef]

A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, "Surface-plasmon resonance spectrometry and characterization of absorbing materals," Appl. Opt. 36, 3314-3320 (1997).

Ghemmaz, A. E.

Hirayama, E.

K. Kurihara, K. Nakamura, E. Hirayama, and K. Suzuki, "Theoretical understanding of an absorption-based surface plasmon resonance sensor applied to sodium ion sensing based on an ion-selective optode membrane," Anal. Chem. 74, 6323-6333 (2002).
[CrossRef]

Ho, H. P.

S. Y. Wu, H. P. Ho, W. C. Law, and C. Lin, "Highly sensitive differential phase-sensitive surface plasmon biosensor based on the Mach-Zehnder configuration," Opt. Lett. 29, 2378-2380 (2004).
[CrossRef] [PubMed]

H. P. Ho, W. W. Lam, and S. Y. Wu, "Surface plasmon resonance sensor based on the measurement of differential phase," Rev. Sci. Instrum. 73, 3534-3539 (2002).
[CrossRef]

Hooper, I. R.

I. R. Hooper and J. R. Sambles, "Sensing using differential surface plasmon ellipsometry," J. Appl. Phys. 96, 3004-3011 (2004).
[CrossRef]

Ikehata, A.

A. Ikehata, X. Li, T. Itoh, and Y. Ozaki, "High sensitive detection of near-infrared absorption by surface plasmon resonance," Appl. Phys. Lett. 83, 2232-2234 (2003).
[CrossRef]

Ishida, H.

Itoh, T.

A. Ikehata, X. Li, T. Itoh, and Y. Ozaki, "High sensitive detection of near-infrared absorption by surface plasmon resonance," Appl. Phys. Lett. 83, 2232-2234 (2003).
[CrossRef]

Jian, Z.-C.

C.-M. Wu, Z.-C. Jian, S.-F. Joe, and L.-B. Chang, "High-sensitivity sensor based on surface plasmon resonance and heterodyne interferometry," Sens. Actuators B 92, 133-136 (2003).
[CrossRef]

Joe, S.-F.

C.-M. Wu, Z.-C. Jian, S.-F. Joe, and L.-B. Chang, "High-sensitivity sensor based on surface plasmon resonance and heterodyne interferometry," Sens. Actuators B 92, 133-136 (2003).
[CrossRef]

Johnson, B.

S. Löfás and B. Johnson, "A novel hydrogel matrix on gold surfaces in surface plasmon resonance sensors for fast and efficient covalent immobilization of ligands," J. Chem. Soc. Chem. Commun. 21, 1526-1528 (1990).
[CrossRef]

Kano, H.

Kawata, S.

Kleim, R.

Kolomenskii, A. A.

A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, "Sensitivity and detection limit of concentration and adsorption measurements by laser-induced surface-plasmon resonance," Appl. Opt. 36, 6539-6547 (2000).
[CrossRef]

A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, "Surface-plasmon resonance spectrometry and characterization of absorbing materals," Appl. Opt. 36, 3314-3320 (1997).

Kuntzler, L.

Kurihara, K.

K. Kurihara, K. Nakamura, and K. Suzuki, "Asymmetric SPR sensor response curve-fitting equation for the accurate determination of SPR resonance angle," Sens. Actuators B 86, 49-57 (2002).
[CrossRef]

K. Kurihara and K. Suzuki, "Theoretical understanding of an absorption-based surface plasmon resonance sensor based on Kretschmann's theory," Anal. Chem. 74, 696-701 (2002).
[CrossRef] [PubMed]

K. Kurihara, K. Nakamura, E. Hirayama, and K. Suzuki, "Theoretical understanding of an absorption-based surface plasmon resonance sensor applied to sodium ion sensing based on an ion-selective optode membrane," Anal. Chem. 74, 6323-6333 (2002).
[CrossRef]

Lam, W. W.

H. P. Ho, W. W. Lam, and S. Y. Wu, "Surface plasmon resonance sensor based on the measurement of differential phase," Rev. Sci. Instrum. 73, 3534-3539 (2002).
[CrossRef]

Law, W. C.

Lee, J.

J. Lee, P. I. Rovira, I. An, and R. W. Collins, "Rotating-compensator multichannel ellipsometry: application for real time Stokes vector spectroscopy of thin film growth," Rev. Sci. Instrum. 69, 1800-1810 (1998).
[CrossRef]

Li, X.

A. Ikehata, X. Li, T. Itoh, and Y. Ozaki, "High sensitive detection of near-infrared absorption by surface plasmon resonance," Appl. Phys. Lett. 83, 2232-2234 (2003).
[CrossRef]

Liedberg, B.

B. Liedberg, C. Nylander, and I. Lundstrom, "Surface plasmon resonance for gas detection and biosensing," Sens. Actuators 4, 299-304 (1983).
[CrossRef]

Lin, C.

Löfás, S.

S. Löfás and B. Johnson, "A novel hydrogel matrix on gold surfaces in surface plasmon resonance sensors for fast and efficient covalent immobilization of ligands," J. Chem. Soc. Chem. Commun. 21, 1526-1528 (1990).
[CrossRef]

Lundstrom, I.

B. Liedberg, C. Nylander, and I. Lundstrom, "Surface plasmon resonance for gas detection and biosensing," Sens. Actuators 4, 299-304 (1983).
[CrossRef]

Mannori, A.

Margheri, G.

Matsui, Y.

X. Sun, S. Shiokawa, and Y. Matsui, "Experimental studies on biosensing by SPR," Jpn. J. Appl. Phys. 28, 1725-1727 (1989).
[CrossRef]

Nakamura, K.

K. Kurihara, K. Nakamura, E. Hirayama, and K. Suzuki, "Theoretical understanding of an absorption-based surface plasmon resonance sensor applied to sodium ion sensing based on an ion-selective optode membrane," Anal. Chem. 74, 6323-6333 (2002).
[CrossRef]

K. Kurihara, K. Nakamura, and K. Suzuki, "Asymmetric SPR sensor response curve-fitting equation for the accurate determination of SPR resonance angle," Sens. Actuators B 86, 49-57 (2002).
[CrossRef]

Nylander, C.

B. Liedberg, C. Nylander, and I. Lundstrom, "Surface plasmon resonance for gas detection and biosensing," Sens. Actuators 4, 299-304 (1983).
[CrossRef]

Ohta, K.

Ozaki, Y.

A. Ikehata, X. Li, T. Itoh, and Y. Ozaki, "High sensitive detection of near-infrared absorption by surface plasmon resonance," Appl. Phys. Lett. 83, 2232-2234 (2003).
[CrossRef]

Quercioli, F.

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

Rovira, P. I.

J. Lee, P. I. Rovira, I. An, and R. W. Collins, "Rotating-compensator multichannel ellipsometry: application for real time Stokes vector spectroscopy of thin film growth," Rev. Sci. Instrum. 69, 1800-1810 (1998).
[CrossRef]

Sambles, J. R.

I. R. Hooper and J. R. Sambles, "Sensing using differential surface plasmon ellipsometry," J. Appl. Phys. 96, 3004-3011 (2004).
[CrossRef]

Schuessler, H. A.

A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, "Sensitivity and detection limit of concentration and adsorption measurements by laser-induced surface-plasmon resonance," Appl. Opt. 36, 6539-6547 (2000).
[CrossRef]

A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, "Surface-plasmon resonance spectrometry and characterization of absorbing materals," Appl. Opt. 36, 3314-3320 (1997).

Shiokawa, S.

X. Sun, S. Shiokawa, and Y. Matsui, "Experimental studies on biosensing by SPR," Jpn. J. Appl. Phys. 28, 1725-1727 (1989).
[CrossRef]

Sun, X.

X. Sun, S. Shiokawa, and Y. Matsui, "Experimental studies on biosensing by SPR," Jpn. J. Appl. Phys. 28, 1725-1727 (1989).
[CrossRef]

Suzuki, K.

K. Kurihara, K. Nakamura, E. Hirayama, and K. Suzuki, "Theoretical understanding of an absorption-based surface plasmon resonance sensor applied to sodium ion sensing based on an ion-selective optode membrane," Anal. Chem. 74, 6323-6333 (2002).
[CrossRef]

K. Kurihara and K. Suzuki, "Theoretical understanding of an absorption-based surface plasmon resonance sensor based on Kretschmann's theory," Anal. Chem. 74, 696-701 (2002).
[CrossRef] [PubMed]

K. Kurihara, K. Nakamura, and K. Suzuki, "Asymmetric SPR sensor response curve-fitting equation for the accurate determination of SPR resonance angle," Sens. Actuators B 86, 49-57 (2002).
[CrossRef]

Westphal, P.

P. Westphal and A. Bornmann, "Biomolecular detection by surface plasmon enhanced ellipsometry," Sens. Actuators B 84, 278-285 (2002).
[CrossRef]

Wu, C.-M.

C.-M. Wu, Z.-C. Jian, S.-F. Joe, and L.-B. Chang, "High-sensitivity sensor based on surface plasmon resonance and heterodyne interferometry," Sens. Actuators B 92, 133-136 (2003).
[CrossRef]

Wu, S. Y.

S. Y. Wu, H. P. Ho, W. C. Law, and C. Lin, "Highly sensitive differential phase-sensitive surface plasmon biosensor based on the Mach-Zehnder configuration," Opt. Lett. 29, 2378-2380 (2004).
[CrossRef] [PubMed]

H. P. Ho, W. W. Lam, and S. Y. Wu, "Surface plasmon resonance sensor based on the measurement of differential phase," Rev. Sci. Instrum. 73, 3534-3539 (2002).
[CrossRef]

Anal. Chem. (2)

K. Kurihara and K. Suzuki, "Theoretical understanding of an absorption-based surface plasmon resonance sensor based on Kretschmann's theory," Anal. Chem. 74, 696-701 (2002).
[CrossRef] [PubMed]

K. Kurihara, K. Nakamura, E. Hirayama, and K. Suzuki, "Theoretical understanding of an absorption-based surface plasmon resonance sensor applied to sodium ion sensing based on an ion-selective optode membrane," Anal. Chem. 74, 6323-6333 (2002).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

A. Ikehata, X. Li, T. Itoh, and Y. Ozaki, "High sensitive detection of near-infrared absorption by surface plasmon resonance," Appl. Phys. Lett. 83, 2232-2234 (2003).
[CrossRef]

Appl. Spectrosc. (1)

J. Appl. Phys. (1)

I. R. Hooper and J. R. Sambles, "Sensing using differential surface plasmon ellipsometry," J. Appl. Phys. 96, 3004-3011 (2004).
[CrossRef]

J. Chem. Soc. Chem. Commun. (1)

S. Löfás and B. Johnson, "A novel hydrogel matrix on gold surfaces in surface plasmon resonance sensors for fast and efficient covalent immobilization of ligands," J. Chem. Soc. Chem. Commun. 21, 1526-1528 (1990).
[CrossRef]

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

X. Sun, S. Shiokawa, and Y. Matsui, "Experimental studies on biosensing by SPR," Jpn. J. Appl. Phys. 28, 1725-1727 (1989).
[CrossRef]

Opt. Lett. (1)

Rev. Sci. Instrum. (2)

J. Lee, P. I. Rovira, I. An, and R. W. Collins, "Rotating-compensator multichannel ellipsometry: application for real time Stokes vector spectroscopy of thin film growth," Rev. Sci. Instrum. 69, 1800-1810 (1998).
[CrossRef]

H. P. Ho, W. W. Lam, and S. Y. Wu, "Surface plasmon resonance sensor based on the measurement of differential phase," Rev. Sci. Instrum. 73, 3534-3539 (2002).
[CrossRef]

Sens. Actuators (1)

B. Liedberg, C. Nylander, and I. Lundstrom, "Surface plasmon resonance for gas detection and biosensing," Sens. Actuators 4, 299-304 (1983).
[CrossRef]

Sens. Actuators B (3)

P. Westphal and A. Bornmann, "Biomolecular detection by surface plasmon enhanced ellipsometry," Sens. Actuators B 84, 278-285 (2002).
[CrossRef]

K. Kurihara, K. Nakamura, and K. Suzuki, "Asymmetric SPR sensor response curve-fitting equation for the accurate determination of SPR resonance angle," Sens. Actuators B 86, 49-57 (2002).
[CrossRef]

C.-M. Wu, Z.-C. Jian, S.-F. Joe, and L.-B. Chang, "High-sensitivity sensor based on surface plasmon resonance and heterodyne interferometry," Sens. Actuators B 92, 133-136 (2003).
[CrossRef]

Other (3)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1989).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

For example, see http://www.biacore.com.

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Figures (9)

Fig. 1
Fig. 1

Two optical models used for numerical simulations:(a) Kretschmann optical arrangement for the three-layer system (prism–Au–sample), and (b) that for the four-layer system (prism–Au–sample–external medium).

Fig. 2
Fig. 2

(a) Wavelength dependence of n, n ( λ ) ; (b) that of k, k ( λ ) . k ( λ ) is given by a Lorentzian curve with a position of the absorption maximum λ = 632.8   nm , an FWHM of 100   nm , and its height k as a parameter. n ( λ ) is derived from Kramers–Kronig transformation of k ( λ ) , where n ¯ is an average of the refractive index.

Fig. 3
Fig. 3

The conventional angular spectra R p ( ϕ ) for the three-layer system for k = 0, 0.001, 0.003, 0.005, and 0.01, where (a) λ = 582.8 , (b) λ = 632.8 , and (c) λ = 682.8   nm . The thickness of Au is d m = 50 (upper figures) and 30   nm (lower figures).

Fig. 4
Fig. 4

Δ Ψ plots obtained from the ellipsometric method for the three-layer system, where each plot corresponds to that in Fig. 3:(a) λ = 582.8 , (b) 632.8, and (c) 682.8   nm and d m = 50 (upper figures) and 30   nm (lower figures). In each plot, the value of k was varied from 0 to 0.3 by 0.001 for three fixed incident angles ϕ, angular positions of which were indicated by vertical dashed lines in Fig. 3.

Fig. 5
Fig. 5

Polarization ellipse of the reflected light of λ = 632.8   nm for each k value in the three-layer system:(a) d m = 30 and ϕ = 70.2 ° and (b) d m = 50   nm and ϕ = 71.8 ° . An arrow depicted on each ellipse indicates whether the direction of the polarization is clockwise or counterclockwise.

Fig. 6
Fig. 6

R p ( ϕ ) (upper figures) and Δ Ψ plots (lower figures) for the four-layer system:(a) λ = 582.8 , (b) 632.8, and (c) 682.8   nm , where d m = 50   nm and d s = 20   nm . For each R p ( ϕ ) plot, the value of k was varied:k = 0, 0.005, 0.01, 0.05, and 0.1. For the Δ Ψ plots, the value of k was varied from 0 to 0.3 by 0.001 for three fixed incident angles ϕ, angular positions of which were indicated by vertical dashed lines in each upper figures.

Fig. 7
Fig. 7

R p ( ϕ ) (upper figures) and Δ Ψ plots (lower figures) for the four-layer system for (a) k = 0.005, (b) 0.05, and (c) 0.1, where d m = 50   nm and λ = 632.8   nm . In each R p ( ϕ ) plot, d s was changed: d s = 20 , 25 , and 30   nm . In each Δ Ψ plot, d s was varied from 20 to 30   nm by a 1 .0   nm step for ϕ = 74.3 ° .

Fig. 8
Fig. 8

(a) Polarization ellipse of the reflected light of λ = 632.8   nm for k = 0, 0.005, 0.01, 0.03, 0.05, and 0.1 in the four-layer system, where d m = 50   nm , d s = 20   nm , and ϕ = 73.9 ° , and (b) that for d s = 20 , 25 , and 30   nm , where d m = 50   nm , k = 0.005, and ϕ = 74.3 ° .

Fig. 9
Fig. 9

Plots of R p , Δ, and Ψ as functions of k (a) for the three-layer ( ϕ = 71.77 ° ) and (b) for the four-layer system ( ϕ = 73.82 ° , d s = 20   nm , and k = 0.005), where λ = 632.8   nm and d m = 50   nm . (c) Plots of R p , Δ, and Ψ as functions of d s for the four-layer system, where λ = 632.8   nm , d m = 50   nm , k = 0.005, and ϕ = 73.82 ° .

Tables (1)

Tables Icon

Table 1 Comparison of K Factors (a) for the Three-Layer and (b) for the Four-Layer System and that of (c) L factors for the Four-Layer System a

Equations (2)

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

S = ( S 0 , S 1 , S 2 , S 3 ) = ( 1 , cos 2 ε cos 2 θ , cos 2 ε sin 2 θ , sin 2 ε ) = ( 1 , cos 2 Ψ , sin 2 Ψ cos Δ , sin 2 Ψ sin Δ ) ,
[ E x E y ] = A exp ( i δ ) [ cos θ cos ε i sin θ sin ε sin θ cos ε + i cos θ sin ε ] ,

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