Abstract

Zeroth-order effective-medium theory is applied to sinusoidally modulated volume holograms for which the period of the grating is much smaller than the illumination wavelength. These holograms can be modeled as negative uniaxial films, with the effective ordinary and extraordinary permittivities being determined by the small-period quasi-static approximation. The limits of validity of the small-period approximation are examined as a function of the wavelength-to-grating-period ratio (λ:Λ) by rigorous coupled-wave theory. The limits are shown to be more restrictive for slanted holograms and for conically incident waves than for unslanted holograms and nonconically incident waves.

© 1995 Optical Society of America

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References

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  1. C. W. Haggans, L. Li, R. K. Kostuk, “Effective-medium theory of zeroth-order lamellar gratings in conical mountings,” J. Opt. Soc. Am. A 10, 2217–2225 (1993).
    [CrossRef]
  2. Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, “Antireflec-tion effect in ultrahigh spatial-frequency holographic relief gratings,” Appl. Opt. 26, 1142–1146 (1987).
    [CrossRef] [PubMed]
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1987), p. 706.
  4. E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
    [CrossRef]
  5. L. H. Cescato, E. Gluch, N. Streibl, “Holographic quarter-wave plates,” Appl. Opt. 29, 3286–3290 (1990).
    [CrossRef] [PubMed]
  6. D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
    [CrossRef]
  7. D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, “Homogeneous-layer models for high-spatial-frequency dielectric surface-relief gratings: conical diffraction and antireflection designs,” Appl. Opt. 33, 2695–2706 (1994).
    [CrossRef]
  8. P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
    [CrossRef]
  9. H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).
  10. D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
    [CrossRef]
  11. R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
    [CrossRef]
  12. D. A. G. Bruggeman, “Calculation of various physical constants of heterogeneous substances. Part I. Dielectric constant and conductivity of mixtures of isotropic substances,” Ann. Phys. (Leipzig) 24, 636–679 (1935).
  13. G. B. Smith, “Effective medium theory and angular dispersion of optical constants in films with oblique columnar structure,” Opt. Commun. 71, 279–284 (1989).
    [CrossRef]
  14. I. Hodgkinson, Q. H. Wu, “Effective principal refractive indices and column angles for periodic stacks of thin birefringent films,” J. Opt. Soc. Am. A 10, 2065–2071 (1993).
    [CrossRef]
  15. D. H. Raguin, “Subwavelength structured surfaces: theory and applications,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1993).
  16. J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Act. 29, 1475–1489 (1982).
    [CrossRef]
  17. D. H. Raguin, G. M. Morris, “Analysis of antireflection structured surfaces with continuous one-dimensional surface profiles,” App. Opt. 32, 2582–2598 (1993).
    [CrossRef]
  18. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  19. M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
    [CrossRef]
  20. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged 4th ed. (Academic, New York, 1980), p. 383.
  21. Ref. 21, p. 366.
  22. H. Fu, S. Sugaya, J. K. Erwin, M. Mansuripur, “Measurement of birefringence for optical recording of disk substrates,” Appl. Opt. 10, 1938–1949 (1994).
    [CrossRef]
  23. G. Campbell, T. J. Kim, R. K. Kostuk, “Comparison of methods for determining the bias index of a dichromated gelatin hologram,” Appl. Opt. (to be published).
  24. B. J. Chang, D. C. Leonard, “Dichromated gelatin for the fabrication of holographic optical elements,” Appl. Opt. 18, 2407–2417 (1979).
    [CrossRef] [PubMed]
  25. MULTLYR by M. Mansuripur, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721.

1994 (3)

D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, “Homogeneous-layer models for high-spatial-frequency dielectric surface-relief gratings: conical diffraction and antireflection designs,” Appl. Opt. 33, 2695–2706 (1994).
[CrossRef]

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

H. Fu, S. Sugaya, J. K. Erwin, M. Mansuripur, “Measurement of birefringence for optical recording of disk substrates,” Appl. Opt. 10, 1938–1949 (1994).
[CrossRef]

1993 (4)

1992 (2)

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).

1990 (2)

L. H. Cescato, E. Gluch, N. Streibl, “Holographic quarter-wave plates,” Appl. Opt. 29, 3286–3290 (1990).
[CrossRef] [PubMed]

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

1989 (1)

G. B. Smith, “Effective medium theory and angular dispersion of optical constants in films with oblique columnar structure,” Opt. Commun. 71, 279–284 (1989).
[CrossRef]

1987 (1)

1983 (2)

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
[CrossRef]

1982 (2)

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Act. 29, 1475–1489 (1982).
[CrossRef]

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

1979 (1)

1935 (1)

D. A. G. Bruggeman, “Calculation of various physical constants of heterogeneous substances. Part I. Dielectric constant and conductivity of mixtures of isotropic substances,” Ann. Phys. (Leipzig) 24, 636–679 (1935).

Bell, J. M.

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Act. 29, 1475–1489 (1982).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1987), p. 706.

Botten, L. C.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Calculation of various physical constants of heterogeneous substances. Part I. Dielectric constant and conductivity of mixtures of isotropic substances,” Ann. Phys. (Leipzig) 24, 636–679 (1935).

Brundrett, D. L.

Campbell, G.

G. Campbell, T. J. Kim, R. K. Kostuk, “Comparison of methods for determining the bias index of a dichromated gelatin hologram,” Appl. Opt. (to be published).

Cescato, L. H.

Chang, B. J.

Collischon, M.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

Craig, M. S.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Derrick, G. H.

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Act. 29, 1475–1489 (1982).
[CrossRef]

Erwin, J. K.

H. Fu, S. Sugaya, J. K. Erwin, M. Mansuripur, “Measurement of birefringence for optical recording of disk substrates,” Appl. Opt. 10, 1938–1949 (1994).
[CrossRef]

Flanders, D. C.

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[CrossRef]

Fu, H.

H. Fu, S. Sugaya, J. K. Erwin, M. Mansuripur, “Measurement of birefringence for optical recording of disk substrates,” Appl. Opt. 10, 1938–1949 (1994).
[CrossRef]

Gaylord, T. K.

Gluch, E.

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

L. H. Cescato, E. Gluch, N. Streibl, “Holographic quarter-wave plates,” Appl. Opt. 29, 3286–3290 (1990).
[CrossRef] [PubMed]

Glytsis, E. N.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged 4th ed. (Academic, New York, 1980), p. 383.

Haggans, C. W.

Haidner, H.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

Hodgkinson, I.

Kim, T. J.

G. Campbell, T. J. Kim, R. K. Kostuk, “Comparison of methods for determining the bias index of a dichromated gelatin hologram,” Appl. Opt. (to be published).

Kimura, Y.

Kipfer, P.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

Kostuk, R. K.

C. W. Haggans, L. Li, R. K. Kostuk, “Effective-medium theory of zeroth-order lamellar gratings in conical mountings,” J. Opt. Soc. Am. A 10, 2217–2225 (1993).
[CrossRef]

G. Campbell, T. J. Kim, R. K. Kostuk, “Comparison of methods for determining the bias index of a dichromated gelatin hologram,” Appl. Opt. (to be published).

Leonard, D. C.

Li, L.

Lindolf, J.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

Mansuripur, M.

H. Fu, S. Sugaya, J. K. Erwin, M. Mansuripur, “Measurement of birefringence for optical recording of disk substrates,” Appl. Opt. 10, 1938–1949 (1994).
[CrossRef]

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

MULTLYR by M. Mansuripur, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721.

Maystre, D.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

McPhedran, R. C.

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Act. 29, 1475–1489 (1982).
[CrossRef]

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Moharam, M. G.

Morris, G. M.

D. H. Raguin, G. M. Morris, “Analysis of antireflection structured surfaces with continuous one-dimensional surface profiles,” App. Opt. 32, 2582–2598 (1993).
[CrossRef]

D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
[CrossRef]

Nevière, M.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Nishida, N.

Ohta, Y.

Ono, Y.

Raguin, D. H.

D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
[CrossRef]

D. H. Raguin, G. M. Morris, “Analysis of antireflection structured surfaces with continuous one-dimensional surface profiles,” App. Opt. 32, 2582–2598 (1993).
[CrossRef]

D. H. Raguin, “Subwavelength structured surfaces: theory and applications,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1993).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged 4th ed. (Academic, New York, 1980), p. 383.

Schwider, J.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

Sheridan, J. T.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

Smith, G. B.

G. B. Smith, “Effective medium theory and angular dispersion of optical constants in films with oblique columnar structure,” Opt. Commun. 71, 279–284 (1989).
[CrossRef]

Stork, W.

H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).

Streibl, N.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

L. H. Cescato, E. Gluch, N. Streibl, “Holographic quarter-wave plates,” Appl. Opt. 29, 3286–3290 (1990).
[CrossRef] [PubMed]

Sugaya, S.

H. Fu, S. Sugaya, J. K. Erwin, M. Mansuripur, “Measurement of birefringence for optical recording of disk substrates,” Appl. Opt. 10, 1938–1949 (1994).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1987), p. 706.

Wu, Q. H.

Ann. Phys. (Leipzig) (1)

D. A. G. Bruggeman, “Calculation of various physical constants of heterogeneous substances. Part I. Dielectric constant and conductivity of mixtures of isotropic substances,” Ann. Phys. (Leipzig) 24, 636–679 (1935).

App. Opt. (1)

D. H. Raguin, G. M. Morris, “Analysis of antireflection structured surfaces with continuous one-dimensional surface profiles,” App. Opt. 32, 2582–2598 (1993).
[CrossRef]

Appl. Opt. (6)

Appl. Phys. Lett. (1)

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[CrossRef]

J. Appl. Phys. (1)

M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Act. (1)

J. M. Bell, G. H. Derrick, R. C. McPhedran, “Diffraction gratings in the quasi-static limit,” Opt. Act. 29, 1475–1489 (1982).
[CrossRef]

Opt. Acta (1)

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Opt. Commun. (2)

G. B. Smith, “Effective medium theory and angular dispersion of optical constants in films with oblique columnar structure,” Opt. Commun. 71, 279–284 (1989).
[CrossRef]

E. Gluch, H. Haidner, P. Kipfer, J. T. Sheridan, N. Streibl, “Form birefringence of surface relief gratings and its angular dependence,” Opt. Commun. 89, 173–177 (1992).
[CrossRef]

Opt. Eng. (1)

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

Optik (1)

H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).

Other (6)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1987), p. 706.

D. H. Raguin, “Subwavelength structured surfaces: theory and applications,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1993).

MULTLYR by M. Mansuripur, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged 4th ed. (Academic, New York, 1980), p. 383.

Ref. 21, p. 366.

G. Campbell, T. J. Kim, R. K. Kostuk, “Comparison of methods for determining the bias index of a dichromated gelatin hologram,” Appl. Opt. (to be published).

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Figures (7)

Fig. 1
Fig. 1

Slanted volume hologram with sinusoidal permittivity modulation showing orientation of the normal ⊥ and the tangential ∥ field directions.

Fig. 2
Fig. 2

Volume hologram with slanted fringes (ϕ ≠ 90°) and a conically incident wave (δ ≠ 0°). Also shown are the ordinary and the extraordinary planes.

Fig. 3
Fig. 3

TM–TE phase shift of a normally incident wave transmitted through an unslanted zeroth-order grating as a function of the grating period. The cutoff period is Λcutoff = λ0/n0. Because there is no plane of incidence, the TE direction is considered to be parallel to the grating fringe planes.

Fig. 4
Fig. 4

TM–TE phase shifts predicted by EMT and RCWT for a wave transmitted through an unslanted zeroth-order volume hologram as a function of the angle of incidence within the extraordinary plane for several different grating periods.

Fig. 5
Fig. 5

TM–TE phase shift predicted by EMT and by RCWT for a wave transmitted through an unslanted zeroth-order hologram as a function of the angle of incidence within the ordinary plane.

Fig. 6
Fig. 6

TM–TE phase shift predicted by EMT and by RCWT for a wave transmitted through an unslanted zeroth-order hologram as a function of the angle of incidence at conical incidence (δ = 45°).

Fig. 7
Fig. 7

TM–TE phase shift predicted by EMT and by RCWT for a wave transmitted through a slanted (ϕ = 45°) zeroth-order hologram as a function of the angle of incidence at conical incidence (δ = 45°).

Equations (13)

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

ɛ ( x ) = ɛ 0 + ɛ 1 cos ( K . x ) ,
k = ( 2 π / Λ ) x ̂ ,
D ¯ = 1 Λ 0 Λ D ( x ) d x ,
D ( x ) = ɛ ( x ) E
D ¯ = E ¯ 1 Λ 0 Λ [ ɛ 0 + ɛ 1 cos ( K x ) ] d x = ɛ 0 E ¯ ,
ɛ = ɛ 0 .
E ¯ = 1 Λ 0 Λ E ( x ) d x ,
E ( x ) = D ɛ ( x )
E ¯ ( x ) = D ɛ 0 2 ɛ 1 2 .
ɛ = ɛ 0 2 ɛ 1 2 .
ɛ = ɛ 0 = 1.8496 ,
ɛ = ( 1.8496 2 0.25 2 ) 1 / 2 = 1.8326 .
Δ φ = ( 2 π / λ 0 ) β t ,

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