Abstract

We measured the second harmonic generation response of a thin film consisting of chiral molecules with four wave plates having different retardation coefficients. By means of the fitting procedure described in a previously reported formalism, we demonstrated that a single set of tensor components of second order surface nonlinearities fits all the data. Our results provide clear experimental evidence for the validity of this method, which can find applications in the studies of chiral structures and achiral anisotropic materials.

© 2009 Optical Society of America

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References

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  1. P. Fischer and F. Hache, “Nonlinear optical spectroscopy of chiral molecules,” Chirality : the pharmacological, biological, and chemical consequences of molecular asymmetry 17, 421-437 (2005).
    [CrossRef]
  2. T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383-1388 (1993).
    [CrossRef]
  3. M. Kauranen, T. Verbiest, J. W. Meijer, E. E. Havinga, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Chiral effects in the second-order optical nonlinearity of a poly(isocyanide) monolayer,” Adv. Mater. 7, 641-644(1995).
    [CrossRef]
  4. L. Hecht and L. D. Barron, “Rayleigh and Raman optical activity from chiral surfaces,” Chem. Phys. Lett. 225, 525-530 (1994).
    [CrossRef]
  5. T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296-8298 (1995).
    [CrossRef]
  6. S. Sioncke, T. Verbiest, and A. Persoons, “Second-order nonlinear optical properties of chiral materials,” Mater. Sci. Eng. R. 42, 115-155 (2003).
    [CrossRef]
  7. J. J. Maki, M. Kauranen, T. Verbiest, and A. Persoons, “Uniqueness of wave-plate measurements in determining the tensor components of second-order surface nonlinearities,” Phys. Rev. B 55, 5021-5026 (1997).
    [CrossRef]
  8. C. Nuckolls, T. J. Katz, T. Verbiest, S. Van Elshocht, H.-G. Kuball, S. Kiesewalter, A. J. Lovinger, and A. Persoons, “Circular dichroism and UV-visible absorption spectra of the Langmuir-Blodgett films of an aggregating helicene,” J. Am. Chem. Soc. 120, 8656-8660 (1998).
    [CrossRef]
  9. C. Nuckolls, T. J. Katz, G. Katz, P. J. Collings, and L. Castellanos, “Synthesis and aggregation of a conjugated helical molecule,” J. Am. Chem. Soc. 121, 79-88 (1999).
    [CrossRef]
  10. X. Li, J. Willits, S. T. Cundiff, I. M. P. Aarts, A. A. E. Stevens, and D. S. Dessau, “Circular dichroism in second harmonic generation from oxidized Si (001),” Appl. Phys. Lett. 89, 022102 (2006).
    [CrossRef]

2006 (1)

X. Li, J. Willits, S. T. Cundiff, I. M. P. Aarts, A. A. E. Stevens, and D. S. Dessau, “Circular dichroism in second harmonic generation from oxidized Si (001),” Appl. Phys. Lett. 89, 022102 (2006).
[CrossRef]

2005 (1)

P. Fischer and F. Hache, “Nonlinear optical spectroscopy of chiral molecules,” Chirality : the pharmacological, biological, and chemical consequences of molecular asymmetry 17, 421-437 (2005).
[CrossRef]

2003 (1)

S. Sioncke, T. Verbiest, and A. Persoons, “Second-order nonlinear optical properties of chiral materials,” Mater. Sci. Eng. R. 42, 115-155 (2003).
[CrossRef]

1999 (1)

C. Nuckolls, T. J. Katz, G. Katz, P. J. Collings, and L. Castellanos, “Synthesis and aggregation of a conjugated helical molecule,” J. Am. Chem. Soc. 121, 79-88 (1999).
[CrossRef]

1998 (1)

C. Nuckolls, T. J. Katz, T. Verbiest, S. Van Elshocht, H.-G. Kuball, S. Kiesewalter, A. J. Lovinger, and A. Persoons, “Circular dichroism and UV-visible absorption spectra of the Langmuir-Blodgett films of an aggregating helicene,” J. Am. Chem. Soc. 120, 8656-8660 (1998).
[CrossRef]

1997 (1)

J. J. Maki, M. Kauranen, T. Verbiest, and A. Persoons, “Uniqueness of wave-plate measurements in determining the tensor components of second-order surface nonlinearities,” Phys. Rev. B 55, 5021-5026 (1997).
[CrossRef]

1995 (2)

T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296-8298 (1995).
[CrossRef]

M. Kauranen, T. Verbiest, J. W. Meijer, E. E. Havinga, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Chiral effects in the second-order optical nonlinearity of a poly(isocyanide) monolayer,” Adv. Mater. 7, 641-644(1995).
[CrossRef]

1994 (1)

L. Hecht and L. D. Barron, “Rayleigh and Raman optical activity from chiral surfaces,” Chem. Phys. Lett. 225, 525-530 (1994).
[CrossRef]

1993 (1)

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383-1388 (1993).
[CrossRef]

Aarts, I. M. P.

X. Li, J. Willits, S. T. Cundiff, I. M. P. Aarts, A. A. E. Stevens, and D. S. Dessau, “Circular dichroism in second harmonic generation from oxidized Si (001),” Appl. Phys. Lett. 89, 022102 (2006).
[CrossRef]

Barron, L. D.

L. Hecht and L. D. Barron, “Rayleigh and Raman optical activity from chiral surfaces,” Chem. Phys. Lett. 225, 525-530 (1994).
[CrossRef]

Byers, J. D.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383-1388 (1993).
[CrossRef]

Castellanos, L.

C. Nuckolls, T. J. Katz, G. Katz, P. J. Collings, and L. Castellanos, “Synthesis and aggregation of a conjugated helical molecule,” J. Am. Chem. Soc. 121, 79-88 (1999).
[CrossRef]

Collings, P. J.

C. Nuckolls, T. J. Katz, G. Katz, P. J. Collings, and L. Castellanos, “Synthesis and aggregation of a conjugated helical molecule,” J. Am. Chem. Soc. 121, 79-88 (1999).
[CrossRef]

Cundiff, S. T.

X. Li, J. Willits, S. T. Cundiff, I. M. P. Aarts, A. A. E. Stevens, and D. S. Dessau, “Circular dichroism in second harmonic generation from oxidized Si (001),” Appl. Phys. Lett. 89, 022102 (2006).
[CrossRef]

Dessau, D. S.

X. Li, J. Willits, S. T. Cundiff, I. M. P. Aarts, A. A. E. Stevens, and D. S. Dessau, “Circular dichroism in second harmonic generation from oxidized Si (001),” Appl. Phys. Lett. 89, 022102 (2006).
[CrossRef]

Fischer, P.

P. Fischer and F. Hache, “Nonlinear optical spectroscopy of chiral molecules,” Chirality : the pharmacological, biological, and chemical consequences of molecular asymmetry 17, 421-437 (2005).
[CrossRef]

Hache, F.

P. Fischer and F. Hache, “Nonlinear optical spectroscopy of chiral molecules,” Chirality : the pharmacological, biological, and chemical consequences of molecular asymmetry 17, 421-437 (2005).
[CrossRef]

Havinga, E. E.

M. Kauranen, T. Verbiest, J. W. Meijer, E. E. Havinga, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Chiral effects in the second-order optical nonlinearity of a poly(isocyanide) monolayer,” Adv. Mater. 7, 641-644(1995).
[CrossRef]

Hecht, L.

L. Hecht and L. D. Barron, “Rayleigh and Raman optical activity from chiral surfaces,” Chem. Phys. Lett. 225, 525-530 (1994).
[CrossRef]

Hicks, J. M.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383-1388 (1993).
[CrossRef]

Katz, G.

C. Nuckolls, T. J. Katz, G. Katz, P. J. Collings, and L. Castellanos, “Synthesis and aggregation of a conjugated helical molecule,” J. Am. Chem. Soc. 121, 79-88 (1999).
[CrossRef]

Katz, T. J.

C. Nuckolls, T. J. Katz, G. Katz, P. J. Collings, and L. Castellanos, “Synthesis and aggregation of a conjugated helical molecule,” J. Am. Chem. Soc. 121, 79-88 (1999).
[CrossRef]

C. Nuckolls, T. J. Katz, T. Verbiest, S. Van Elshocht, H.-G. Kuball, S. Kiesewalter, A. J. Lovinger, and A. Persoons, “Circular dichroism and UV-visible absorption spectra of the Langmuir-Blodgett films of an aggregating helicene,” J. Am. Chem. Soc. 120, 8656-8660 (1998).
[CrossRef]

Kauranen, M.

J. J. Maki, M. Kauranen, T. Verbiest, and A. Persoons, “Uniqueness of wave-plate measurements in determining the tensor components of second-order surface nonlinearities,” Phys. Rev. B 55, 5021-5026 (1997).
[CrossRef]

T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296-8298 (1995).
[CrossRef]

M. Kauranen, T. Verbiest, J. W. Meijer, E. E. Havinga, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Chiral effects in the second-order optical nonlinearity of a poly(isocyanide) monolayer,” Adv. Mater. 7, 641-644(1995).
[CrossRef]

Kiesewalter, S.

C. Nuckolls, T. J. Katz, T. Verbiest, S. Van Elshocht, H.-G. Kuball, S. Kiesewalter, A. J. Lovinger, and A. Persoons, “Circular dichroism and UV-visible absorption spectra of the Langmuir-Blodgett films of an aggregating helicene,” J. Am. Chem. Soc. 120, 8656-8660 (1998).
[CrossRef]

Kuball, H.-G.

C. Nuckolls, T. J. Katz, T. Verbiest, S. Van Elshocht, H.-G. Kuball, S. Kiesewalter, A. J. Lovinger, and A. Persoons, “Circular dichroism and UV-visible absorption spectra of the Langmuir-Blodgett films of an aggregating helicene,” J. Am. Chem. Soc. 120, 8656-8660 (1998).
[CrossRef]

Li, X.

X. Li, J. Willits, S. T. Cundiff, I. M. P. Aarts, A. A. E. Stevens, and D. S. Dessau, “Circular dichroism in second harmonic generation from oxidized Si (001),” Appl. Phys. Lett. 89, 022102 (2006).
[CrossRef]

Lovinger, A. J.

C. Nuckolls, T. J. Katz, T. Verbiest, S. Van Elshocht, H.-G. Kuball, S. Kiesewalter, A. J. Lovinger, and A. Persoons, “Circular dichroism and UV-visible absorption spectra of the Langmuir-Blodgett films of an aggregating helicene,” J. Am. Chem. Soc. 120, 8656-8660 (1998).
[CrossRef]

Maki, J. J.

J. J. Maki, M. Kauranen, T. Verbiest, and A. Persoons, “Uniqueness of wave-plate measurements in determining the tensor components of second-order surface nonlinearities,” Phys. Rev. B 55, 5021-5026 (1997).
[CrossRef]

T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296-8298 (1995).
[CrossRef]

Meijer, J. W.

M. Kauranen, T. Verbiest, J. W. Meijer, E. E. Havinga, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Chiral effects in the second-order optical nonlinearity of a poly(isocyanide) monolayer,” Adv. Mater. 7, 641-644(1995).
[CrossRef]

Nolte, R. J. M.

M. Kauranen, T. Verbiest, J. W. Meijer, E. E. Havinga, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Chiral effects in the second-order optical nonlinearity of a poly(isocyanide) monolayer,” Adv. Mater. 7, 641-644(1995).
[CrossRef]

T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296-8298 (1995).
[CrossRef]

Nuckolls, C.

C. Nuckolls, T. J. Katz, G. Katz, P. J. Collings, and L. Castellanos, “Synthesis and aggregation of a conjugated helical molecule,” J. Am. Chem. Soc. 121, 79-88 (1999).
[CrossRef]

C. Nuckolls, T. J. Katz, T. Verbiest, S. Van Elshocht, H.-G. Kuball, S. Kiesewalter, A. J. Lovinger, and A. Persoons, “Circular dichroism and UV-visible absorption spectra of the Langmuir-Blodgett films of an aggregating helicene,” J. Am. Chem. Soc. 120, 8656-8660 (1998).
[CrossRef]

Persoons, A.

S. Sioncke, T. Verbiest, and A. Persoons, “Second-order nonlinear optical properties of chiral materials,” Mater. Sci. Eng. R. 42, 115-155 (2003).
[CrossRef]

C. Nuckolls, T. J. Katz, T. Verbiest, S. Van Elshocht, H.-G. Kuball, S. Kiesewalter, A. J. Lovinger, and A. Persoons, “Circular dichroism and UV-visible absorption spectra of the Langmuir-Blodgett films of an aggregating helicene,” J. Am. Chem. Soc. 120, 8656-8660 (1998).
[CrossRef]

J. J. Maki, M. Kauranen, T. Verbiest, and A. Persoons, “Uniqueness of wave-plate measurements in determining the tensor components of second-order surface nonlinearities,” Phys. Rev. B 55, 5021-5026 (1997).
[CrossRef]

T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296-8298 (1995).
[CrossRef]

M. Kauranen, T. Verbiest, J. W. Meijer, E. E. Havinga, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Chiral effects in the second-order optical nonlinearity of a poly(isocyanide) monolayer,” Adv. Mater. 7, 641-644(1995).
[CrossRef]

Petralli-Mallow, T.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383-1388 (1993).
[CrossRef]

Schouten, A. J.

M. Kauranen, T. Verbiest, J. W. Meijer, E. E. Havinga, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Chiral effects in the second-order optical nonlinearity of a poly(isocyanide) monolayer,” Adv. Mater. 7, 641-644(1995).
[CrossRef]

T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296-8298 (1995).
[CrossRef]

Sioncke, S.

S. Sioncke, T. Verbiest, and A. Persoons, “Second-order nonlinear optical properties of chiral materials,” Mater. Sci. Eng. R. 42, 115-155 (2003).
[CrossRef]

Stevens, A. A. E.

X. Li, J. Willits, S. T. Cundiff, I. M. P. Aarts, A. A. E. Stevens, and D. S. Dessau, “Circular dichroism in second harmonic generation from oxidized Si (001),” Appl. Phys. Lett. 89, 022102 (2006).
[CrossRef]

Teerenstra, M. N.

T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296-8298 (1995).
[CrossRef]

M. Kauranen, T. Verbiest, J. W. Meijer, E. E. Havinga, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Chiral effects in the second-order optical nonlinearity of a poly(isocyanide) monolayer,” Adv. Mater. 7, 641-644(1995).
[CrossRef]

Van Elshocht, S.

C. Nuckolls, T. J. Katz, T. Verbiest, S. Van Elshocht, H.-G. Kuball, S. Kiesewalter, A. J. Lovinger, and A. Persoons, “Circular dichroism and UV-visible absorption spectra of the Langmuir-Blodgett films of an aggregating helicene,” J. Am. Chem. Soc. 120, 8656-8660 (1998).
[CrossRef]

Verbiest, T.

S. Sioncke, T. Verbiest, and A. Persoons, “Second-order nonlinear optical properties of chiral materials,” Mater. Sci. Eng. R. 42, 115-155 (2003).
[CrossRef]

C. Nuckolls, T. J. Katz, T. Verbiest, S. Van Elshocht, H.-G. Kuball, S. Kiesewalter, A. J. Lovinger, and A. Persoons, “Circular dichroism and UV-visible absorption spectra of the Langmuir-Blodgett films of an aggregating helicene,” J. Am. Chem. Soc. 120, 8656-8660 (1998).
[CrossRef]

J. J. Maki, M. Kauranen, T. Verbiest, and A. Persoons, “Uniqueness of wave-plate measurements in determining the tensor components of second-order surface nonlinearities,” Phys. Rev. B 55, 5021-5026 (1997).
[CrossRef]

T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296-8298 (1995).
[CrossRef]

M. Kauranen, T. Verbiest, J. W. Meijer, E. E. Havinga, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Chiral effects in the second-order optical nonlinearity of a poly(isocyanide) monolayer,” Adv. Mater. 7, 641-644(1995).
[CrossRef]

Willits, J.

X. Li, J. Willits, S. T. Cundiff, I. M. P. Aarts, A. A. E. Stevens, and D. S. Dessau, “Circular dichroism in second harmonic generation from oxidized Si (001),” Appl. Phys. Lett. 89, 022102 (2006).
[CrossRef]

Wong, T. M.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383-1388 (1993).
[CrossRef]

Yee, H. I.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383-1388 (1993).
[CrossRef]

Adv. Mater. (1)

M. Kauranen, T. Verbiest, J. W. Meijer, E. E. Havinga, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Chiral effects in the second-order optical nonlinearity of a poly(isocyanide) monolayer,” Adv. Mater. 7, 641-644(1995).
[CrossRef]

Appl. Phys. Lett. (1)

X. Li, J. Willits, S. T. Cundiff, I. M. P. Aarts, A. A. E. Stevens, and D. S. Dessau, “Circular dichroism in second harmonic generation from oxidized Si (001),” Appl. Phys. Lett. 89, 022102 (2006).
[CrossRef]

Chem. Phys. Lett. (1)

L. Hecht and L. D. Barron, “Rayleigh and Raman optical activity from chiral surfaces,” Chem. Phys. Lett. 225, 525-530 (1994).
[CrossRef]

Chirality : the pharmacological, biological, and chemical consequences of molecular asymmetry (1)

P. Fischer and F. Hache, “Nonlinear optical spectroscopy of chiral molecules,” Chirality : the pharmacological, biological, and chemical consequences of molecular asymmetry 17, 421-437 (2005).
[CrossRef]

J. Am. Chem. Soc. (2)

C. Nuckolls, T. J. Katz, T. Verbiest, S. Van Elshocht, H.-G. Kuball, S. Kiesewalter, A. J. Lovinger, and A. Persoons, “Circular dichroism and UV-visible absorption spectra of the Langmuir-Blodgett films of an aggregating helicene,” J. Am. Chem. Soc. 120, 8656-8660 (1998).
[CrossRef]

C. Nuckolls, T. J. Katz, G. Katz, P. J. Collings, and L. Castellanos, “Synthesis and aggregation of a conjugated helical molecule,” J. Am. Chem. Soc. 121, 79-88 (1999).
[CrossRef]

J. Chem. Phys. (1)

T. Verbiest, M. Kauranen, J. J. Maki, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Linearly polarized probes of surface chirality,” J. Chem. Phys. 103, 8296-8298 (1995).
[CrossRef]

J. Phys. Chem. (1)

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383-1388 (1993).
[CrossRef]

Mater. Sci. Eng. R. (1)

S. Sioncke, T. Verbiest, and A. Persoons, “Second-order nonlinear optical properties of chiral materials,” Mater. Sci. Eng. R. 42, 115-155 (2003).
[CrossRef]

Phys. Rev. B (1)

J. J. Maki, M. Kauranen, T. Verbiest, and A. Persoons, “Uniqueness of wave-plate measurements in determining the tensor components of second-order surface nonlinearities,” Phys. Rev. B 55, 5021-5026 (1997).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the experimental setup: λ / 2 , half-wave plate; P, polarizer; WP, wave plate; L1 and L2, lenses; VIS-F, visible light filter; S, sample; A, analyzer; BS, beam splitter; C, chopper; PD, photodiode; IR-F, infrared filter; PMT, photomultiplier tube; M-CON, motion controller. The inset shows the chemical structure of the sample, a chiral helicene molecule.

Fig. 2
Fig. 2

Intensity of the fundamental light at 800 nm versus the angle of rotation of four different wave plates. The experimental data are indicated with open symbols, and the lines correspond to theoretical fits.

Fig. 3
Fig. 3

Intensity of the normalized second harmonic signal versus the angle of rotation of four different wave plates. The experimental data are indicated with open symbols, and the lines correspond to theoretical fits.

Tables (2)

Tables Icon

Table 1 Fitting Parameters from the Fits in Fig. 2

Tables Icon

Table 2 Fitting Parameters from the Fits in Fig. 3

Equations (4)

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

I S , P ( 2 ω ) = | f S , P E P 2 ( ω ) + g S , P E S 2 ( ω ) + h S , P E P ( ω ) E S ( ω ) | 2 ,
I ( 2 ω ) = A { 1 4 [ ( f 1 + g 1 ) sin 2 δ 2 2 f 1 cos 2 δ 2 4 f 2 cos δ 2 sin δ 2 cos 2 θ WP + ( f 1 g 2 ) sin 2 δ 2 cos 4 θ WP + h 1 sin 2 δ 2 sin 4 θ WP ] 2 + 1 4 [ ( f 2 + g 2 ) sin 2 δ 2 2 f 2 cos 2 δ 2 + 4 f 1 cos δ 2 sin δ 2 cos 2 θ WP + ( f 2 g 2 ) sin 2 δ 2 cos 4 θ WP + 2 h 1 cos δ 2 sin 2 θ WP ] 2 } .
[ E P ( ω ) E S ( ω ) ] = [ cos 2 β cos β sin β cos β sin β sin 2 β ] [ cos δ 2 + i sin δ 2 cos 2 θ i sin δ 2 sin 2 θ i sin δ 2 sin 2 θ cos δ 2 i sin δ 2 cos 2 θ ] [ cos α sin α ] ,
I P ( ω ) = cos 2 δ 2 + sin 2 δ 2 cos 2 2 ( θ WP + θ 0 ) ,

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