Abstract

We discuss a method for measuring electro-optic coefficients by measuring diffraction from a tunable grating. The method involves measuring the changes in the diffraction pattern of a reflection grating, where applied electric fields of alternating direction induce changes in the index of refraction through the electro-optic effect. For certain geometries, these applied fields cause period-doubling effects that produce new peaks in the diffraction pattern. Numerically calculated diffraction patterns are presented for the assumptions of both homogeneous and inhomogeneous fields. Peak splitting, as a function of both the number of slits illuminated and the induced change in the index of refraction, is observed and discussed. Finally, the usefulness of our method for the measurement of electro-optic coefficients is discussed.

© 2001 Optical Society of America

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References

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  1. F. Wang, E. Furman, G. H. Haertling, “Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry,” J. Appl. Phys. 78(1), 9–14 (1995).
  2. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  3. P. Rohl, B. Andress, J. Nordmann, “Electro-optic determination of second and third-order susceptibilities in poled polymer films,” Appl. Phys. Lett. 59, 2793–2795 (1991).
    [Crossref]
  4. F. Wang, K. K. Li, V. Fuflyigin, H. Jiang, J. Zhao, P. Norris, “Thin ferroelectric interferometer for spatial light modulations,” Appl. Opt. 37, 7490–7495 (1998).
    [Crossref]
  5. F. Wang, V. Fuflyigin, A. Osinsky, “Electro-optic properties of oxide ferroelectrics grown on GaN/sapphire,” J. Appl. Phys. 88, 1701–1703 (2000).
    [Crossref]
  6. F. Wang, G. H. Haertling, “Birefringent bistability in (Pb,La)(Zr,Ti)O3 thin films with a ferroelectric-semiconductor interface,” Appl. Phys. Lett. 63, 1730–1733 (1993).
    [Crossref]
  7. R. M. A. Azzam, “Ellipsometry,” in Handbook in Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 27.
  8. D. Trivedi, P. Tayebati, M. Tabat, “Measurement of large electro-optic coefficients in thin films of strontium barium niobate (Sr0.6Ba0.4Nb2O6),” Appl. Phys. Lett. 68, 3227–3229 (1996).
    [Crossref]
  9. P. Tayebati, D. Trivedi, M. Tabat, “Pulsed laser deposition of SBN:75 thin films with electro-optic coefficient of 844pm/V,” Appl. Phys. Lett. 69, 1023–1025 (1996).
    [Crossref]

2000 (1)

F. Wang, V. Fuflyigin, A. Osinsky, “Electro-optic properties of oxide ferroelectrics grown on GaN/sapphire,” J. Appl. Phys. 88, 1701–1703 (2000).
[Crossref]

1998 (1)

1996 (2)

D. Trivedi, P. Tayebati, M. Tabat, “Measurement of large electro-optic coefficients in thin films of strontium barium niobate (Sr0.6Ba0.4Nb2O6),” Appl. Phys. Lett. 68, 3227–3229 (1996).
[Crossref]

P. Tayebati, D. Trivedi, M. Tabat, “Pulsed laser deposition of SBN:75 thin films with electro-optic coefficient of 844pm/V,” Appl. Phys. Lett. 69, 1023–1025 (1996).
[Crossref]

1995 (1)

F. Wang, E. Furman, G. H. Haertling, “Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry,” J. Appl. Phys. 78(1), 9–14 (1995).

1993 (1)

F. Wang, G. H. Haertling, “Birefringent bistability in (Pb,La)(Zr,Ti)O3 thin films with a ferroelectric-semiconductor interface,” Appl. Phys. Lett. 63, 1730–1733 (1993).
[Crossref]

1991 (1)

P. Rohl, B. Andress, J. Nordmann, “Electro-optic determination of second and third-order susceptibilities in poled polymer films,” Appl. Phys. Lett. 59, 2793–2795 (1991).
[Crossref]

Andress, B.

P. Rohl, B. Andress, J. Nordmann, “Electro-optic determination of second and third-order susceptibilities in poled polymer films,” Appl. Phys. Lett. 59, 2793–2795 (1991).
[Crossref]

Azzam, R. M. A.

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

R. M. A. Azzam, “Ellipsometry,” in Handbook in Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 27.

Bashara, N. M.

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

Fuflyigin, V.

F. Wang, V. Fuflyigin, A. Osinsky, “Electro-optic properties of oxide ferroelectrics grown on GaN/sapphire,” J. Appl. Phys. 88, 1701–1703 (2000).
[Crossref]

F. Wang, K. K. Li, V. Fuflyigin, H. Jiang, J. Zhao, P. Norris, “Thin ferroelectric interferometer for spatial light modulations,” Appl. Opt. 37, 7490–7495 (1998).
[Crossref]

Furman, E.

F. Wang, E. Furman, G. H. Haertling, “Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry,” J. Appl. Phys. 78(1), 9–14 (1995).

Haertling, G. H.

F. Wang, E. Furman, G. H. Haertling, “Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry,” J. Appl. Phys. 78(1), 9–14 (1995).

F. Wang, G. H. Haertling, “Birefringent bistability in (Pb,La)(Zr,Ti)O3 thin films with a ferroelectric-semiconductor interface,” Appl. Phys. Lett. 63, 1730–1733 (1993).
[Crossref]

Jiang, H.

Li, K. K.

Nordmann, J.

P. Rohl, B. Andress, J. Nordmann, “Electro-optic determination of second and third-order susceptibilities in poled polymer films,” Appl. Phys. Lett. 59, 2793–2795 (1991).
[Crossref]

Norris, P.

Osinsky, A.

F. Wang, V. Fuflyigin, A. Osinsky, “Electro-optic properties of oxide ferroelectrics grown on GaN/sapphire,” J. Appl. Phys. 88, 1701–1703 (2000).
[Crossref]

Rohl, P.

P. Rohl, B. Andress, J. Nordmann, “Electro-optic determination of second and third-order susceptibilities in poled polymer films,” Appl. Phys. Lett. 59, 2793–2795 (1991).
[Crossref]

Tabat, M.

D. Trivedi, P. Tayebati, M. Tabat, “Measurement of large electro-optic coefficients in thin films of strontium barium niobate (Sr0.6Ba0.4Nb2O6),” Appl. Phys. Lett. 68, 3227–3229 (1996).
[Crossref]

P. Tayebati, D. Trivedi, M. Tabat, “Pulsed laser deposition of SBN:75 thin films with electro-optic coefficient of 844pm/V,” Appl. Phys. Lett. 69, 1023–1025 (1996).
[Crossref]

Tayebati, P.

P. Tayebati, D. Trivedi, M. Tabat, “Pulsed laser deposition of SBN:75 thin films with electro-optic coefficient of 844pm/V,” Appl. Phys. Lett. 69, 1023–1025 (1996).
[Crossref]

D. Trivedi, P. Tayebati, M. Tabat, “Measurement of large electro-optic coefficients in thin films of strontium barium niobate (Sr0.6Ba0.4Nb2O6),” Appl. Phys. Lett. 68, 3227–3229 (1996).
[Crossref]

Trivedi, D.

P. Tayebati, D. Trivedi, M. Tabat, “Pulsed laser deposition of SBN:75 thin films with electro-optic coefficient of 844pm/V,” Appl. Phys. Lett. 69, 1023–1025 (1996).
[Crossref]

D. Trivedi, P. Tayebati, M. Tabat, “Measurement of large electro-optic coefficients in thin films of strontium barium niobate (Sr0.6Ba0.4Nb2O6),” Appl. Phys. Lett. 68, 3227–3229 (1996).
[Crossref]

Wang, F.

F. Wang, V. Fuflyigin, A. Osinsky, “Electro-optic properties of oxide ferroelectrics grown on GaN/sapphire,” J. Appl. Phys. 88, 1701–1703 (2000).
[Crossref]

F. Wang, K. K. Li, V. Fuflyigin, H. Jiang, J. Zhao, P. Norris, “Thin ferroelectric interferometer for spatial light modulations,” Appl. Opt. 37, 7490–7495 (1998).
[Crossref]

F. Wang, E. Furman, G. H. Haertling, “Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry,” J. Appl. Phys. 78(1), 9–14 (1995).

F. Wang, G. H. Haertling, “Birefringent bistability in (Pb,La)(Zr,Ti)O3 thin films with a ferroelectric-semiconductor interface,” Appl. Phys. Lett. 63, 1730–1733 (1993).
[Crossref]

Zhao, J.

Appl. Opt. (1)

Appl. Phys. Lett. (4)

F. Wang, G. H. Haertling, “Birefringent bistability in (Pb,La)(Zr,Ti)O3 thin films with a ferroelectric-semiconductor interface,” Appl. Phys. Lett. 63, 1730–1733 (1993).
[Crossref]

D. Trivedi, P. Tayebati, M. Tabat, “Measurement of large electro-optic coefficients in thin films of strontium barium niobate (Sr0.6Ba0.4Nb2O6),” Appl. Phys. Lett. 68, 3227–3229 (1996).
[Crossref]

P. Tayebati, D. Trivedi, M. Tabat, “Pulsed laser deposition of SBN:75 thin films with electro-optic coefficient of 844pm/V,” Appl. Phys. Lett. 69, 1023–1025 (1996).
[Crossref]

P. Rohl, B. Andress, J. Nordmann, “Electro-optic determination of second and third-order susceptibilities in poled polymer films,” Appl. Phys. Lett. 59, 2793–2795 (1991).
[Crossref]

J. Appl. Phys. (2)

F. Wang, V. Fuflyigin, A. Osinsky, “Electro-optic properties of oxide ferroelectrics grown on GaN/sapphire,” J. Appl. Phys. 88, 1701–1703 (2000).
[Crossref]

F. Wang, E. Furman, G. H. Haertling, “Electro-optic measurements of thin-film materials by means of reflection differential ellipsometry,” J. Appl. Phys. 78(1), 9–14 (1995).

Other (2)

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

R. M. A. Azzam, “Ellipsometry,” in Handbook in Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 27.

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Figures (7)

Fig. 1
Fig. 1

(a) With the probe beam incident on the sample from the substrate side, reflection and transmission occur simultaneously (for a transparent sample and substrate). An experimental setup is shown schematically along with the qualitative features of the electric field distribution. (b) An expanded view of the electric field lines. The horizontal arrows show the in-plane electric field components; the vertical arrows show the normal field components.

Fig. 2
Fig. 2

Wave front is changed after reflection by the periodic grating depending on assumptions made about the electric field. The dashed lines correspond to the wave front with no applied field. The flat lines (A, B) correspond to the wave front for a homogeneous field. The tilted lines 1, 2, 3, and 4 correspond to the wave front with an inhomogeneous field with linear corrections. For each case, the wave fronts are assumed to be near the sample.

Fig. 3
Fig. 3

Diagram showing the far-field diffraction pattern with no applied field. The longitudinal coordinate was set up in the same direction as that of specular reflection. The number (m) of the grating pairs illuminated is 20. Other parameters are as follows: wavelength, 0.6328 µm; incident angle, 10 deg; electrode width, 10 µm; electrode spacing (center to center), 20 µm.

Fig. 4
Fig. 4

When the homogeneous field approximation is used with Δn = ±0.02, new peaks appear. The strength of the new peaks depends on the value of |Δn|. Here the number of grating pairs illuminated is 20. Other parameters are as given in Fig. 3.

Fig. 5
Fig. 5

When the inhomogeneous field approximation is used with only linear corrections, splitting of the peaks occurs, and the symmetry between positive and negative orders is broken. For this calculation, Δn = ±0.00006 and the number of grating pairs illuminated is 20. Other parameters are as given in Fig. 3.

Fig. 6
Fig. 6

Using the inhomogeneous field approximation, we show the dependence of the diffraction pattern on the number of grating pairs illuminated for a constant Δn = ±0.00006. The central peak structure depends strongly on the number of grating pairs illuminated so that, when the beam is focused, the diffraction pattern is changed dramatically. Other parameters are as given in Fig. 3.

Fig. 7
Fig. 7

Using the inhomogeneous field approximation with the number of grating pairs illuminated constant, we show the dependence of the structure of the central peak on the change in the refractive index. As Δn changes from 3.6 × 10-5 to 6 × 10-5, the split in the central peak is clearly visible. Other parameters are as given in Fig. 3.

Equations (6)

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

Iθ=sinka sinθ/2ka sinθ/22sin2mδ/22m sinδ/22,
Iθ=sinka sinθ/2ka sinθ/22sinmδ/2m sinδ/22×cosΦ0/22,
Φ0=k2a2cosθ/2+αsinθ/2+k2ΔnE2t.
Eiθ=sinka/2sinθ-θi2ka/2sinθ-θi2×sinmδi/2m sinδi/2expj m-12 δi,
Φ11=0, Φ12=k2.5a2 cosθ2+αsinθ2+Φ0, Φ13=k0.5a2 cosθ2+αsinθ2, Φ14=k2a2 cosθ2+αsinθ2+Φ0,
Iθ=i=14EiθexpjΦ1i×i=14EiθexpjΦ1i*142,

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