Abstract

An effective ellipsometric technique to determine parameters that characterize second-harmonic optical and magneto-optical effects in centrosymmetric media within the electric-dipole approximation is proposed and outlined in detail. The parameters, which are ratios of components of the nonlinear-surface-susceptibility tensors, are obtained from experimental data related to the state of polarization of the second-harmonic-generated radiation as a function of the angle between the plane of incidence and the polarization plane of the incident, linearly polarized, fundamental radiation. Experimental details of the technique are described. A corresponding theoretical model is given as an example for a single isotropic surface assuming polycrystalline samples. The surfaces of air–Au and air–Ni (in magnetized and demagnetized states) have been investigated ex situ in ambient air, and the results are presented. A nonlinear, least-squares-minimization fitting procedure between experimental data and theoretical formulas has been shown to yield realistic, unambiguous results for the ratios corresponding to each of the above materials. Independent methods for verifying the validity of the fitting parameters are also presented. The influence of temporal variations at the surfaces on the state of polarization (due to adsorption, contamination, or oxidation) is also illustrated for the demagnetized air–Ni surface.

© 2005 Optical Society of America

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References

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  1. W. Hübner, "Electronic theory for nonlinear magneto-optics," in Nonlinear Optics in Metals , K. H. Bennemann, ed. (Clarendon, Oxford, UK, 1998), pp. 268-436.
  2. Th. Rasing, "Nonlinear magneto-optics," J. Magn. Magn. Mater. 175, 35-50 (1997).
    [CrossRef]
  3. A. Kirilyuk, "Nonlinear optics in application to magnetic surfaces and thin films," J. Phys. D 35, R189-R207 (2002).
    [CrossRef]
  4. A. K. Zvezdin and V. A. Kotov, Modern Magneto-Optics and Magneto-Optical Materials (Institute of Physics and Physical Society, London, 1997).
  5. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  6. N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606-622 (1962).
    [CrossRef]
  7. R. Atkinson and N. F. Kubrakov, "Boundary conditions in the simplest model of linear and second harmonic magneto-optical effects," Phys. Rev. B 65, 014432-1-014432-11 (2001).
    [CrossRef]
  8. U. Pustogowa, W. Hübner, and K. H. Bennemann, "Theory for the nonlinear magneto-optical Kerr effect at ferromagnetic transition-metal surfaces," Phys. Rev. B 48, 8607-8618 (1993).
    [CrossRef]
  9. G. A. Reider and T. F. Heinz, "Second-order nonlinear optical effects at surfaces and interfaces: recent advances," in Photonic Probes of Surfaces , P. Halevi, ed. (Elsevier, New York, 1995), pp. 414-478.
  10. S. K. Andersson, M. C. Schanne-Klein, and F. Hache, "Symmetry and phase determination of second-harmonic reflection from calcite surfaces," Phys. Rev. B 59, 3210-3217 (1999).
    [CrossRef]
  11. F. Hache, S. K. Andersson, M. C. Schanne-Klein, and C. Flytzanis, "Investigation of the phase of the second-order susceptibility measured in off-resonant surface second-harmonic generation," Appl. Phys. B 68, 321-323 (1999).
    [CrossRef]
  12. 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]
  13. R. Atkinson and N. F. Kubrakov, "Features of the simplest model of second-harmonic magneto-optical Kerr effects," Appl. Phys. B 74, 697-703 (2002).
    [CrossRef]
  14. A. K. Zvezdin and N. F. Kubrakov, "Nonlinear magneto-optical Kerr effects," JETP 89, 77-85 (1999);
    [CrossRef]
  15. J. E. Sipe, D. J. Moss, and H. M. van Driel, "Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals," Phys. Rev. B 35, 1129-1141 (1987).
    [CrossRef]
  16. J. J. Moré, "The Levenberg-Marquardt algorithm: Implementation and theory," in Lecture Notes in Mathematics 630 , G. A. Watson, ed. (Springer-Verlag, New York, 1978), pp. 104-116.
  17. MATHCAD 11 Software. 1986-2002 Mathsoft Engineering & Education, Inc., http://www.Mathcad.com.
  18. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

2002 (2)

A. Kirilyuk, "Nonlinear optics in application to magnetic surfaces and thin films," J. Phys. D 35, R189-R207 (2002).
[CrossRef]

R. Atkinson and N. F. Kubrakov, "Features of the simplest model of second-harmonic magneto-optical Kerr effects," Appl. Phys. B 74, 697-703 (2002).
[CrossRef]

2001 (1)

R. Atkinson and N. F. Kubrakov, "Boundary conditions in the simplest model of linear and second harmonic magneto-optical effects," Phys. Rev. B 65, 014432-1-014432-11 (2001).
[CrossRef]

1999 (3)

A. K. Zvezdin and N. F. Kubrakov, "Nonlinear magneto-optical Kerr effects," JETP 89, 77-85 (1999);
[CrossRef]

S. K. Andersson, M. C. Schanne-Klein, and F. Hache, "Symmetry and phase determination of second-harmonic reflection from calcite surfaces," Phys. Rev. B 59, 3210-3217 (1999).
[CrossRef]

F. Hache, S. K. Andersson, M. C. Schanne-Klein, and C. Flytzanis, "Investigation of the phase of the second-order susceptibility measured in off-resonant surface second-harmonic generation," Appl. Phys. B 68, 321-323 (1999).
[CrossRef]

1997 (2)

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]

Th. Rasing, "Nonlinear magneto-optics," J. Magn. Magn. Mater. 175, 35-50 (1997).
[CrossRef]

1993 (1)

U. Pustogowa, W. Hübner, and K. H. Bennemann, "Theory for the nonlinear magneto-optical Kerr effect at ferromagnetic transition-metal surfaces," Phys. Rev. B 48, 8607-8618 (1993).
[CrossRef]

1987 (1)

J. E. Sipe, D. J. Moss, and H. M. van Driel, "Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals," Phys. Rev. B 35, 1129-1141 (1987).
[CrossRef]

1962 (1)

N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606-622 (1962).
[CrossRef]

Andersson, S. K.

S. K. Andersson, M. C. Schanne-Klein, and F. Hache, "Symmetry and phase determination of second-harmonic reflection from calcite surfaces," Phys. Rev. B 59, 3210-3217 (1999).
[CrossRef]

F. Hache, S. K. Andersson, M. C. Schanne-Klein, and C. Flytzanis, "Investigation of the phase of the second-order susceptibility measured in off-resonant surface second-harmonic generation," Appl. Phys. B 68, 321-323 (1999).
[CrossRef]

Atkinson , R.

R. Atkinson and N. F. Kubrakov, "Features of the simplest model of second-harmonic magneto-optical Kerr effects," Appl. Phys. B 74, 697-703 (2002).
[CrossRef]

R. Atkinson and N. F. Kubrakov, "Boundary conditions in the simplest model of linear and second harmonic magneto-optical effects," Phys. Rev. B 65, 014432-1-014432-11 (2001).
[CrossRef]

Bennemann, K. H.

U. Pustogowa, W. Hübner, and K. H. Bennemann, "Theory for the nonlinear magneto-optical Kerr effect at ferromagnetic transition-metal surfaces," Phys. Rev. B 48, 8607-8618 (1993).
[CrossRef]

Bloembergen , N.

N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606-622 (1962).
[CrossRef]

Flytzanis, C.

F. Hache, S. K. Andersson, M. C. Schanne-Klein, and C. Flytzanis, "Investigation of the phase of the second-order susceptibility measured in off-resonant surface second-harmonic generation," Appl. Phys. B 68, 321-323 (1999).
[CrossRef]

Hache, F.

S. K. Andersson, M. C. Schanne-Klein, and F. Hache, "Symmetry and phase determination of second-harmonic reflection from calcite surfaces," Phys. Rev. B 59, 3210-3217 (1999).
[CrossRef]

F. Hache, S. K. Andersson, M. C. Schanne-Klein, and C. Flytzanis, "Investigation of the phase of the second-order susceptibility measured in off-resonant surface second-harmonic generation," Appl. Phys. B 68, 321-323 (1999).
[CrossRef]

Hübner, W.

U. Pustogowa, W. Hübner, and K. H. Bennemann, "Theory for the nonlinear magneto-optical Kerr effect at ferromagnetic transition-metal surfaces," Phys. Rev. B 48, 8607-8618 (1993).
[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]

Kirilyuk, A.

A. Kirilyuk, "Nonlinear optics in application to magnetic surfaces and thin films," J. Phys. D 35, R189-R207 (2002).
[CrossRef]

Kubrakov, N. F.

R. Atkinson and N. F. Kubrakov, "Features of the simplest model of second-harmonic magneto-optical Kerr effects," Appl. Phys. B 74, 697-703 (2002).
[CrossRef]

R. Atkinson and N. F. Kubrakov, "Boundary conditions in the simplest model of linear and second harmonic magneto-optical effects," Phys. Rev. B 65, 014432-1-014432-11 (2001).
[CrossRef]

A. K. Zvezdin and N. F. Kubrakov, "Nonlinear magneto-optical Kerr effects," JETP 89, 77-85 (1999);
[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]

Moss, D. J.

J. E. Sipe, D. J. Moss, and H. M. van Driel, "Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals," Phys. Rev. B 35, 1129-1141 (1987).
[CrossRef]

Pershan, P. S.

N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606-622 (1962).
[CrossRef]

Persoons, A.

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]

Pustogowa, U.

U. Pustogowa, W. Hübner, and K. H. Bennemann, "Theory for the nonlinear magneto-optical Kerr effect at ferromagnetic transition-metal surfaces," Phys. Rev. B 48, 8607-8618 (1993).
[CrossRef]

Rasing, Th.

Th. Rasing, "Nonlinear magneto-optics," J. Magn. Magn. Mater. 175, 35-50 (1997).
[CrossRef]

Schanne-Klein, M. C.

F. Hache, S. K. Andersson, M. C. Schanne-Klein, and C. Flytzanis, "Investigation of the phase of the second-order susceptibility measured in off-resonant surface second-harmonic generation," Appl. Phys. B 68, 321-323 (1999).
[CrossRef]

S. K. Andersson, M. C. Schanne-Klein, and F. Hache, "Symmetry and phase determination of second-harmonic reflection from calcite surfaces," Phys. Rev. B 59, 3210-3217 (1999).
[CrossRef]

Sipe, J. E.

J. E. Sipe, D. J. Moss, and H. M. van Driel, "Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals," Phys. Rev. B 35, 1129-1141 (1987).
[CrossRef]

van Driel, H. M.

J. E. Sipe, D. J. Moss, and H. M. van Driel, "Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals," Phys. Rev. B 35, 1129-1141 (1987).
[CrossRef]

Verbiest, T.

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]

Zvezdin , A. K.

A. K. Zvezdin and N. F. Kubrakov, "Nonlinear magneto-optical Kerr effects," JETP 89, 77-85 (1999);
[CrossRef]

Appl. Phys. B (2)

F. Hache, S. K. Andersson, M. C. Schanne-Klein, and C. Flytzanis, "Investigation of the phase of the second-order susceptibility measured in off-resonant surface second-harmonic generation," Appl. Phys. B 68, 321-323 (1999).
[CrossRef]

R. Atkinson and N. F. Kubrakov, "Features of the simplest model of second-harmonic magneto-optical Kerr effects," Appl. Phys. B 74, 697-703 (2002).
[CrossRef]

J. Magn. Magn. Mater. (1)

Th. Rasing, "Nonlinear magneto-optics," J. Magn. Magn. Mater. 175, 35-50 (1997).
[CrossRef]

J. Phys. D (1)

A. Kirilyuk, "Nonlinear optics in application to magnetic surfaces and thin films," J. Phys. D 35, R189-R207 (2002).
[CrossRef]

JETP (1)

A. K. Zvezdin and N. F. Kubrakov, "Nonlinear magneto-optical Kerr effects," JETP 89, 77-85 (1999);
[CrossRef]

Phys. Rev. (1)

N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606-622 (1962).
[CrossRef]

Phys. Rev. B (5)

R. Atkinson and N. F. Kubrakov, "Boundary conditions in the simplest model of linear and second harmonic magneto-optical effects," Phys. Rev. B 65, 014432-1-014432-11 (2001).
[CrossRef]

U. Pustogowa, W. Hübner, and K. H. Bennemann, "Theory for the nonlinear magneto-optical Kerr effect at ferromagnetic transition-metal surfaces," Phys. Rev. B 48, 8607-8618 (1993).
[CrossRef]

J. E. Sipe, D. J. Moss, and H. M. van Driel, "Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals," Phys. Rev. B 35, 1129-1141 (1987).
[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]

S. K. Andersson, M. C. Schanne-Klein, and F. Hache, "Symmetry and phase determination of second-harmonic reflection from calcite surfaces," Phys. Rev. B 59, 3210-3217 (1999).
[CrossRef]

Other (7)

W. Hübner, "Electronic theory for nonlinear magneto-optics," in Nonlinear Optics in Metals , K. H. Bennemann, ed. (Clarendon, Oxford, UK, 1998), pp. 268-436.

J. J. Moré, "The Levenberg-Marquardt algorithm: Implementation and theory," in Lecture Notes in Mathematics 630 , G. A. Watson, ed. (Springer-Verlag, New York, 1978), pp. 104-116.

MATHCAD 11 Software. 1986-2002 Mathsoft Engineering & Education, Inc., http://www.Mathcad.com.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

G. A. Reider and T. F. Heinz, "Second-order nonlinear optical effects at surfaces and interfaces: recent advances," in Photonic Probes of Surfaces , P. Halevi, ed. (Elsevier, New York, 1995), pp. 414-478.

A. K. Zvezdin and V. A. Kotov, Modern Magneto-Optics and Magneto-Optical Materials (Institute of Physics and Physical Society, London, 1997).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

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Figures (6)

Fig. 1
Fig. 1

(a) Configuration of the semi-infinite magnetic medium and the quantities used in the theoretical model. (b) Illustration of the definition of the state of polarization.

Fig. 2
Fig. 2

Schematic diagram of the nonlinear system. A, analyser; P, polarizer; B, beam splitter; D, photomultiplier; ES, electromagnet and sample stage; F1,2, 800 nm or 400 nm filter; L, lens; Ti, femtosecond laser; Q, quartz plate; λ/2, half-wave plate.

Fig. 3
Fig. 3

Real and imaginary parts of χ˜ as a function of ψ and φ for the Au sample at (a) φ=45°, (b) ψ=45°. (c) Intensities Isp(φ) and Ipp(φ); the fitting procedure involved the data given in Table 1.

Fig. 4
Fig. 4

Real and imaginary parts of χ˜ as a function of ψ and φ for the demagnetized Ni sample at (a) φ=45°, (b) ψ=45°. (c) Intensities Isp(φ) and Ipp(φ); the fitting procedure involved the data given in Table 1.

Fig. 5
Fig. 5

Real and imaginary parts of χ˜±(ψ) for the magnetized Ni sample (longitudinal configuration). The parameters are given in Table 1.

Fig. 6
Fig. 6

Real and imaginary parts of χ˜(ψ) for two demagnetized Ni samples: (a) 1 h old, filled triangles; (b) 12 months old, open triangles.

Tables (1)

Tables Icon

Table 1 Parameters Used in the Model of the Nonlinear Effects for the Wavelength of 800 nm

Equations (25)

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

χ(ψ)=-Es(i)Ep(i)=tan ψ,
χ˜=tan θ˜+i tan η˜1+i tan θ˜ tan η˜
E1=2ζζ+γEs(i),E2=2n0ζγn2ζ+n02γEp(i),
D3=-20n0n2αζn2ζ+n02γEp(i),
0-1PiNS=χijkFjFk+χijklFjFkml,
F1=E1,F2=E2,
F3=0-1D3.
E˜s=ik0S0-1ζ+γ˜P1NS,
E˜p=-ik0Sn00-1n˜2ζ+n02γ˜(γ˜P2NS+αn˜2P3NS),
P1NSP2NSP3NS=0(Aˆ+m1Bˆ1-m2Bˆ2+m3Bˆ3)F12F22F322F2F32F1F32F1F2.
Aˆ=0000χ1130000χ11300χ311χ311χ333000,
Bˆ1=00000χ1121χ2111χ2221χ2331000000χ323100,
Bˆ2=χ2221χ2111χ233100000000χ11210000χ32310,
Bˆ3=000χ1233000000-χ12330000000.
χ˜±(ψ)=a1 sin 2ψ±b1(1-cos 2ψ)r3±(1+cos 2ψ)r45(a2+a3r1+a5r2)(1+cos 2ψ)+a4(1-cos 2ψ)r1±(b4r3+r46)sin 2ψ,
a1=2αγ-1ξξ˜,a2=-2αγ-1,a3=αn˜2γ˜-1,  a4=αn˜2γ˜-1ξ2,a5=α3n˜2γ˜-1γ-2,  r45=b2r4+b3r5,r46=b5r4+b6r6,  b1=-ξ2ξ˜,b2=-ξ˜,b3=-α2γ-2ξ˜,  b4=ξ,b5=-ξ,b6=-2α2n˜2γ˜-1γ-1ξ,  ξ=n2ζ+n02γ/n0γ(ζ+γ),  ξ˜=(n˜2ζ+n02γ˜)/n0γ˜(ζ+γ˜).
r1=χ311n2χ113,r2=n2χ333χ113,r3=χ2221n2χ113,
r4=χ2111n2χ113,r5=n2χ2331χ113,r6=χ3231χ113.
Sre±=1N-1 k=0N[Re χ˜k±-Re χ˜±(ψk)]2=0,
Sim±=1N-1 k=0N[Im χ˜k±-Im χ˜±(ψk)]2=0,
χ2221/χ311=αn˜2γ˜-1ξ˜-1χ˜±(π/2).
χ˜±(0)=±r45(a2+a3r1+a5r2)-1,
Isp=Cspα2ζ4|(γ+ζ)2(n˜2ζ+n02γ˜)|-2
Ipp=Cppαζ2γ2 2γ-1γ˜-n˜2(r1+α2γ-2r2)(n2ζ+n02γ)2(n˜2ζ+n02γ˜)2,
χ˜±(ψ)=a1 sin 2ψ±c1(1+cos 2ψ)r7(a2+a3r1+a5r2)(1+cos 2ψ)+a4(1-cos 2ψ)r1±c2r7 sin 2ψ,

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