Abstract

Polarisation-entangled photon pairs from a two crystal, type-I spontaneous parametric down conversion source are used to make accurate measurements of the ellipsometric angles of a silicon dioxide film on silicon and of internal and external reflection from BK7 glass. Since our source produces an entangled state with some mixture, a novel technique based on quantum tomography was developed to estimate the components of the density matrix for the state before and after reflection from the samples. The ellipsometric angles are readily calculated from these components and experimental measurements made on the samples were found to be in good agreement with their expected values.

© 2006 Optical Society of America

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References

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  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, The Netherlands,1987).
  2. D. E. Aspnes, "Expanding horizons: new developments in ellipsometry and polarimetry," Thin Solid Films 455-456, 3 - 13 (2004).
    [CrossRef]
  3. K. Vedam, "Spectroscopic ellipsometry: a historical overview," Thin Solid Films 313-314, 1 - 9 (1998).
    [CrossRef]
  4. T. E. Jenkins, "Multiple-angle-of-incidence ellipsometry," J. Phys. D Appl. Phys. 32, R45 - R56 (1999).
    [CrossRef]
  5. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773 - R776 (1999).
    [CrossRef]
  6. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, "Measurement of qubits," Phys. Rev. A 64, 052312(2001).
    [CrossRef]
  7. A. Aspect, "Bell’s inequality test: more ideal than ever," Nature (London) 398, 189 - 190 (1999).
    [CrossRef]
  8. J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum Controlled-NOT Gate," Nature (London) 426, 264 - 267 (2003).
    [CrossRef]
  9. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145 - 195 (2002).
    [CrossRef]
  10. A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, andM. C. Teich, "Ellipsometric measurements by use of photon pairs generated by spontaneous parametric downconversion," Opt. Lett. 26, 1717 - 1719 (2001).
    [CrossRef]
  11. A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Entangled-Photon Ellipsometry," J. Opt. Soc. Am. B 19, 656 - 662 (2002).
    [CrossRef]
  12. A. V. Sergienko and G. S. Jaeger, "Quantum information processing and precise optical measurement with entangled-photon pairs," Contemp. Phys. 44, 341 - 356 (2003).
    [CrossRef]
  13. K. C. Toussaint, G. D. Giuseppe, K. J. Bycenski, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum ellipsometry using correlated-photon beams," Phys. Rev. A 70, 023801 (2004).
    [CrossRef]
  14. A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum entanglement and the two-photon Stokes parameters," Opt. Commun. 201, 93 - 98 (2002).
    [CrossRef]
  15. A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, "Nonmaximally Entangled States: Production, Characterization and Utilization," Phys. Rev. Lett. 83, 3103-3107 (1999).
    [CrossRef]
  16. T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, "Two-photon geometric optics," Phys. Rev. A 53, R2804 - R2815 (1996).
    [CrossRef]
  17. D. Dehlinger and M. W. Mitchell, "Entangled photon apparatus for the undergraduate laboratory," Am. J. Phys. 70, 898 - 902 (2002).
    [CrossRef]
  18. D. Dehlinger and M. W. Mitchell, "Entangled photons, nonlocality, and Bell inequalities in the under graduate laboratory," Am. J. Phys. 70, 903 - 910 (2002).
    [CrossRef]
  19. W. J. Munro, K. Nemoto, and A. G. White, "The Bell inequality: a measure of entanglement?" J. Mod. Opt. 48, 1239 - 1246 (2001).
  20. T. S. Larchuk, M. C. Teich, and B. E. A. Saleh, "Statistics of Entangled-Photon Coincidences in Parametric Downconversion," Ann. NY Acad. Sci. 755, 680 - 686 (1995).
    [CrossRef]
  21. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, San Diego, California, 1998).
  22. J. Altepeter, E. Jeffrey, and P. Kwiat, "Phase-compensated ultra-bright source of entangled photons," Opt. Express 13, 8951 - 8959 (2005).
    [CrossRef] [PubMed]

2005 (1)

2004 (2)

D. E. Aspnes, "Expanding horizons: new developments in ellipsometry and polarimetry," Thin Solid Films 455-456, 3 - 13 (2004).
[CrossRef]

K. C. Toussaint, G. D. Giuseppe, K. J. Bycenski, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum ellipsometry using correlated-photon beams," Phys. Rev. A 70, 023801 (2004).
[CrossRef]

2003 (2)

A. V. Sergienko and G. S. Jaeger, "Quantum information processing and precise optical measurement with entangled-photon pairs," Contemp. Phys. 44, 341 - 356 (2003).
[CrossRef]

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum Controlled-NOT Gate," Nature (London) 426, 264 - 267 (2003).
[CrossRef]

2002 (5)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145 - 195 (2002).
[CrossRef]

A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Entangled-Photon Ellipsometry," J. Opt. Soc. Am. B 19, 656 - 662 (2002).
[CrossRef]

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum entanglement and the two-photon Stokes parameters," Opt. Commun. 201, 93 - 98 (2002).
[CrossRef]

D. Dehlinger and M. W. Mitchell, "Entangled photon apparatus for the undergraduate laboratory," Am. J. Phys. 70, 898 - 902 (2002).
[CrossRef]

D. Dehlinger and M. W. Mitchell, "Entangled photons, nonlocality, and Bell inequalities in the under graduate laboratory," Am. J. Phys. 70, 903 - 910 (2002).
[CrossRef]

2001 (3)

W. J. Munro, K. Nemoto, and A. G. White, "The Bell inequality: a measure of entanglement?" J. Mod. Opt. 48, 1239 - 1246 (2001).

A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, andM. C. Teich, "Ellipsometric measurements by use of photon pairs generated by spontaneous parametric downconversion," Opt. Lett. 26, 1717 - 1719 (2001).
[CrossRef]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, "Measurement of qubits," Phys. Rev. A 64, 052312(2001).
[CrossRef]

1999 (4)

A. Aspect, "Bell’s inequality test: more ideal than ever," Nature (London) 398, 189 - 190 (1999).
[CrossRef]

T. E. Jenkins, "Multiple-angle-of-incidence ellipsometry," J. Phys. D Appl. Phys. 32, R45 - R56 (1999).
[CrossRef]

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773 - R776 (1999).
[CrossRef]

A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, "Nonmaximally Entangled States: Production, Characterization and Utilization," Phys. Rev. Lett. 83, 3103-3107 (1999).
[CrossRef]

1998 (1)

K. Vedam, "Spectroscopic ellipsometry: a historical overview," Thin Solid Films 313-314, 1 - 9 (1998).
[CrossRef]

1995 (1)

T. S. Larchuk, M. C. Teich, and B. E. A. Saleh, "Statistics of Entangled-Photon Coincidences in Parametric Downconversion," Ann. NY Acad. Sci. 755, 680 - 686 (1995).
[CrossRef]

Abouraddy, A. F.

Altepeter, J.

Appelbaum, I.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773 - R776 (1999).
[CrossRef]

Aspect, A.

A. Aspect, "Bell’s inequality test: more ideal than ever," Nature (London) 398, 189 - 190 (1999).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, "Expanding horizons: new developments in ellipsometry and polarimetry," Thin Solid Films 455-456, 3 - 13 (2004).
[CrossRef]

Branning, D.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum Controlled-NOT Gate," Nature (London) 426, 264 - 267 (2003).
[CrossRef]

Bycenski, K. J.

K. C. Toussaint, G. D. Giuseppe, K. J. Bycenski, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum ellipsometry using correlated-photon beams," Phys. Rev. A 70, 023801 (2004).
[CrossRef]

Dehlinger, D.

D. Dehlinger and M. W. Mitchell, "Entangled photon apparatus for the undergraduate laboratory," Am. J. Phys. 70, 898 - 902 (2002).
[CrossRef]

D. Dehlinger and M. W. Mitchell, "Entangled photons, nonlocality, and Bell inequalities in the under graduate laboratory," Am. J. Phys. 70, 903 - 910 (2002).
[CrossRef]

Eberhard, P. H.

A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, "Nonmaximally Entangled States: Production, Characterization and Utilization," Phys. Rev. Lett. 83, 3103-3107 (1999).
[CrossRef]

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773 - R776 (1999).
[CrossRef]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145 - 195 (2002).
[CrossRef]

Giuseppe, G. D.

K. C. Toussaint, G. D. Giuseppe, K. J. Bycenski, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum ellipsometry using correlated-photon beams," Phys. Rev. A 70, 023801 (2004).
[CrossRef]

Jaeger, G. S.

A. V. Sergienko and G. S. Jaeger, "Quantum information processing and precise optical measurement with entangled-photon pairs," Contemp. Phys. 44, 341 - 356 (2003).
[CrossRef]

James, D. F. V.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, "Measurement of qubits," Phys. Rev. A 64, 052312(2001).
[CrossRef]

A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, "Nonmaximally Entangled States: Production, Characterization and Utilization," Phys. Rev. Lett. 83, 3103-3107 (1999).
[CrossRef]

Jeffrey, E.

Jenkins, T. E.

T. E. Jenkins, "Multiple-angle-of-incidence ellipsometry," J. Phys. D Appl. Phys. 32, R45 - R56 (1999).
[CrossRef]

Kwiat, P.

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, "Measurement of qubits," Phys. Rev. A 64, 052312(2001).
[CrossRef]

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773 - R776 (1999).
[CrossRef]

A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, "Nonmaximally Entangled States: Production, Characterization and Utilization," Phys. Rev. Lett. 83, 3103-3107 (1999).
[CrossRef]

Larchuk, T. S.

T. S. Larchuk, M. C. Teich, and B. E. A. Saleh, "Statistics of Entangled-Photon Coincidences in Parametric Downconversion," Ann. NY Acad. Sci. 755, 680 - 686 (1995).
[CrossRef]

Mitchell, M. W.

D. Dehlinger and M. W. Mitchell, "Entangled photon apparatus for the undergraduate laboratory," Am. J. Phys. 70, 898 - 902 (2002).
[CrossRef]

D. Dehlinger and M. W. Mitchell, "Entangled photons, nonlocality, and Bell inequalities in the under graduate laboratory," Am. J. Phys. 70, 903 - 910 (2002).
[CrossRef]

Munro, W. J.

W. J. Munro, K. Nemoto, and A. G. White, "The Bell inequality: a measure of entanglement?" J. Mod. Opt. 48, 1239 - 1246 (2001).

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, "Measurement of qubits," Phys. Rev. A 64, 052312(2001).
[CrossRef]

Nemoto, K.

W. J. Munro, K. Nemoto, and A. G. White, "The Bell inequality: a measure of entanglement?" J. Mod. Opt. 48, 1239 - 1246 (2001).

O’Brien, J. L.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum Controlled-NOT Gate," Nature (London) 426, 264 - 267 (2003).
[CrossRef]

Pryde, G. J.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum Controlled-NOT Gate," Nature (London) 426, 264 - 267 (2003).
[CrossRef]

Ralph, T. C.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum Controlled-NOT Gate," Nature (London) 426, 264 - 267 (2003).
[CrossRef]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145 - 195 (2002).
[CrossRef]

Saleh, B. E. A.

K. C. Toussaint, G. D. Giuseppe, K. J. Bycenski, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum ellipsometry using correlated-photon beams," Phys. Rev. A 70, 023801 (2004).
[CrossRef]

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum entanglement and the two-photon Stokes parameters," Opt. Commun. 201, 93 - 98 (2002).
[CrossRef]

A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Entangled-Photon Ellipsometry," J. Opt. Soc. Am. B 19, 656 - 662 (2002).
[CrossRef]

A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, andM. C. Teich, "Ellipsometric measurements by use of photon pairs generated by spontaneous parametric downconversion," Opt. Lett. 26, 1717 - 1719 (2001).
[CrossRef]

T. S. Larchuk, M. C. Teich, and B. E. A. Saleh, "Statistics of Entangled-Photon Coincidences in Parametric Downconversion," Ann. NY Acad. Sci. 755, 680 - 686 (1995).
[CrossRef]

Sergienko, A. V.

K. C. Toussaint, G. D. Giuseppe, K. J. Bycenski, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum ellipsometry using correlated-photon beams," Phys. Rev. A 70, 023801 (2004).
[CrossRef]

A. V. Sergienko and G. S. Jaeger, "Quantum information processing and precise optical measurement with entangled-photon pairs," Contemp. Phys. 44, 341 - 356 (2003).
[CrossRef]

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum entanglement and the two-photon Stokes parameters," Opt. Commun. 201, 93 - 98 (2002).
[CrossRef]

A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Entangled-Photon Ellipsometry," J. Opt. Soc. Am. B 19, 656 - 662 (2002).
[CrossRef]

A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, andM. C. Teich, "Ellipsometric measurements by use of photon pairs generated by spontaneous parametric downconversion," Opt. Lett. 26, 1717 - 1719 (2001).
[CrossRef]

Teich, M. C.

K. C. Toussaint, G. D. Giuseppe, K. J. Bycenski, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum ellipsometry using correlated-photon beams," Phys. Rev. A 70, 023801 (2004).
[CrossRef]

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum entanglement and the two-photon Stokes parameters," Opt. Commun. 201, 93 - 98 (2002).
[CrossRef]

A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Entangled-Photon Ellipsometry," J. Opt. Soc. Am. B 19, 656 - 662 (2002).
[CrossRef]

T. S. Larchuk, M. C. Teich, and B. E. A. Saleh, "Statistics of Entangled-Photon Coincidences in Parametric Downconversion," Ann. NY Acad. Sci. 755, 680 - 686 (1995).
[CrossRef]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145 - 195 (2002).
[CrossRef]

Toussaint, K. C.

Vedam, K.

K. Vedam, "Spectroscopic ellipsometry: a historical overview," Thin Solid Films 313-314, 1 - 9 (1998).
[CrossRef]

Waks, E.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773 - R776 (1999).
[CrossRef]

White, A. G.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum Controlled-NOT Gate," Nature (London) 426, 264 - 267 (2003).
[CrossRef]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, "Measurement of qubits," Phys. Rev. A 64, 052312(2001).
[CrossRef]

W. J. Munro, K. Nemoto, and A. G. White, "The Bell inequality: a measure of entanglement?" J. Mod. Opt. 48, 1239 - 1246 (2001).

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773 - R776 (1999).
[CrossRef]

A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, "Nonmaximally Entangled States: Production, Characterization and Utilization," Phys. Rev. Lett. 83, 3103-3107 (1999).
[CrossRef]

Zbinden, H.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145 - 195 (2002).
[CrossRef]

Am. J. Phys. (2)

D. Dehlinger and M. W. Mitchell, "Entangled photon apparatus for the undergraduate laboratory," Am. J. Phys. 70, 898 - 902 (2002).
[CrossRef]

D. Dehlinger and M. W. Mitchell, "Entangled photons, nonlocality, and Bell inequalities in the under graduate laboratory," Am. J. Phys. 70, 903 - 910 (2002).
[CrossRef]

Ann. NY Acad. Sci. (1)

T. S. Larchuk, M. C. Teich, and B. E. A. Saleh, "Statistics of Entangled-Photon Coincidences in Parametric Downconversion," Ann. NY Acad. Sci. 755, 680 - 686 (1995).
[CrossRef]

Contemp. Phys. (1)

A. V. Sergienko and G. S. Jaeger, "Quantum information processing and precise optical measurement with entangled-photon pairs," Contemp. Phys. 44, 341 - 356 (2003).
[CrossRef]

J. Mod. Opt. (1)

W. J. Munro, K. Nemoto, and A. G. White, "The Bell inequality: a measure of entanglement?" J. Mod. Opt. 48, 1239 - 1246 (2001).

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

J. Phys. D Appl. Phys. (1)

T. E. Jenkins, "Multiple-angle-of-incidence ellipsometry," J. Phys. D Appl. Phys. 32, R45 - R56 (1999).
[CrossRef]

Nature (London) (2)

A. Aspect, "Bell’s inequality test: more ideal than ever," Nature (London) 398, 189 - 190 (1999).
[CrossRef]

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum Controlled-NOT Gate," Nature (London) 426, 264 - 267 (2003).
[CrossRef]

Opt. Commun. (1)

A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum entanglement and the two-photon Stokes parameters," Opt. Commun. 201, 93 - 98 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (3)

K. C. Toussaint, G. D. Giuseppe, K. J. Bycenski, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum ellipsometry using correlated-photon beams," Phys. Rev. A 70, 023801 (2004).
[CrossRef]

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773 - R776 (1999).
[CrossRef]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, "Measurement of qubits," Phys. Rev. A 64, 052312(2001).
[CrossRef]

Phys. Rev. Lett. (1)

A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, "Nonmaximally Entangled States: Production, Characterization and Utilization," Phys. Rev. Lett. 83, 3103-3107 (1999).
[CrossRef]

Rev. Mod. Phys. (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145 - 195 (2002).
[CrossRef]

Thin Solid Films (2)

D. E. Aspnes, "Expanding horizons: new developments in ellipsometry and polarimetry," Thin Solid Films 455-456, 3 - 13 (2004).
[CrossRef]

K. Vedam, "Spectroscopic ellipsometry: a historical overview," Thin Solid Films 313-314, 1 - 9 (1998).
[CrossRef]

Other (3)

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, "Two-photon geometric optics," Phys. Rev. A 53, R2804 - R2815 (1996).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, The Netherlands,1987).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, San Diego, California, 1998).

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Figures (2)

Fig. 1.
Fig. 1.

Entangled photon ellipsometer, based on Ref. [10].

Fig. 2.
Fig. 2.

Experimental apparatus: M1, M2 beam steering mirrors, BBO the two crystal source, HWP halfwave plates, QWP quarterwave plates, PBS polarising beam splitters, IF interference filters, D detectors. The dashed section of the figure shows the position of the equipment when the sample is removed for calibration.

Tables (3)

Tables Icon

Table 1. Experimental and expected values for the SiO2 – Si system

Tables Icon

Table 2. Experimental and expected values for BK7 glass

Tables Icon

Table 3. Experimental and expected values for total internal reflection in a glass prism

Equations (22)

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tan ψ = r p r s ,
Δ = arg ( r p r s )
Ψ = 1 1 + ε 2 ( H H + ε e V V )
Ψ = 1 1 + ( ε tan ψ ) 2 ( H H + ε tan ψ e i ( ϕ Δ ) V V ) ,
N c = C cos θ 1 cos θ 2 + ε tan ψ e i ( ϕ Δ ) sin θ 1 sin θ 2 2 ,
Ψ = c HH H H + c HV H V + c VH VH + c VV V V
ρ = w i i ( c HHi 2 c HHi c HVi * c HHi c VHi * c HHi c VVi * c HHi * c HVi c VHi 2 c HVi c VHi * c VHi c VVi * c HHi * c VHi c HVi * c VHi c VHi 2 c VHi c VVi * c HHi * c VVi c HVi * c VVi c VHi * c VVi c VVi 2 ) ,
ρ 11 = i w i c HHi 2
ρ 44 = i w i c VVi 2
ρ 41 = i w i c HHi * c VVi .
ψ r = R ( r p c HH H H + r s c HV HV + r p c VH VH + r s c VV VV )
ρ r 11 = R r p 2 ρ 11 ,
ρ r 44 = R r s 2 ρ 44 ,
tan ψ = ρ r 11 ρ 44 ρ 11 ρ r 44 .
ρ r 41 = R r s * r p ρ 41
= R r s r p e i Δ ρ 41 ,
Δ = arg ( ρ 41 ρ r 41 ) = arg ( ρ 41 ) arg ( ρ r 41 ) .
C ρ 11 = C tr ( ρ H H ) = N c ( H H ) ,
C ρ 44 = C tr ( ρ V V ) = N c ( V V ) ,
2 Cℜ ( ρ 41 ) = N c ( D D ) + N c ( A A ) N c ( R R ) N c ( L L )
2 Cℑ ( ρ 41 ) = N c ( D L ) + N c ( A R ) N c ( L A ) N c ( R D )
N c = C 2 ( cos 2 θ 1 + ε 2 sin 2 θ 1 + 2 ε cos ( ϕ ) cos θ 1 sin θ 1 ) .

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