Abstract

The reflective properties of randomly rough surfaces at large incidence angles have been reported due to their potential applications in some of the radiative heat transfer research areas. The main purpose of this work is to investigate the formation mechanism of the specular reflection peak of rough surfaces at large incidence angles. The bidirectional reflectance distribution function (BRDF) of rough aluminum surfaces with different roughnesses at different incident angles is measured by a three-axis automated scatterometer. This study used a validated and accurate computational model, the rigorous coupled-wave analysis (RCWA) method, to compare and analyze the measurement BRDF results. It is found that the RCWA results show the same trend of specular peak as the measurement. This paper mainly focuses on the relative roughness at the range of 0.16<σ/λ<5.35. As the relative roughness decreases, the specular peak enhancement dramatically increases and the scattering region significantly reduces, especially under large incidence angles. The RCWA and the Rayleigh criterion results have been compared, showing that the relative error of the total integrated scatter increases as the roughness of the surface increases at large incidence angles. In addition, the zero-order diffractive power calculated by RCWA and the reflectance calculated by Fresnel equations are compared. The comparison shows that the relative error declines sharply when the incident angle is large and the roughness is small.

© 2014 Optical Society of America

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2012 (2)

S. Schröder, M. Trost, T. Herffurth, A. von Finck, and A. Duparré, “Sophisticated light scattering techniques from the VUV to the IR regions,” Proc. SPIE 8495, 84950V (2012).
[CrossRef]

I. Semenikhin, M. Zanuccoli, M. Benzi, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient RCWA method for simulation of thin film solar cells,” Opt. Quantum Electron. 44, 149–154 (2012).
[CrossRef]

2011 (2)

H. C. Zhou, Z. F. Huang, Q. Cheng, W. Lü, K. Qiu, and C. Chen, “Road surface mirage: a bunch of hot air?” Chin. Sci. Bull. 56, 962–968 (2011).
[CrossRef]

M. Zerrad, M. Lequime, and C. Amra, “Multimodal scattering facilities and modelization tools for a comprehensive investigation of optical coatings,” Proc. SPIE 8169, 81690K (2011).
[CrossRef]

2009 (2)

M. Lequime, M. Zerrad, C. Deumie, and C. Amra, “A goniometric light scattering instrument with high-resolution imaging,” Opt. Commun. 282, 1265–1273 (2009).
[CrossRef]

B. J. Lee and Z. M. Zhang, “Indirect measurements of coherent thermal emission from a truncated photonic crystal structure,” J. Thermophys. Heat Transfer 23, 9–17 (2009).
[CrossRef]

2008 (2)

S. Fahr, C. Rockstuhl, and F. Lederer, “Engineering the randomness for enhanced absorption in solar cells,” Appl. Phys. Lett. 92, 171114 (2008).
[CrossRef]

K. Fu and P.-f. Hsu, “New regime map of the geometric optics approximation for scattering from random rough surfaces,” J. Quant. Spectrosc. Radiat. Transfer 109, 180–188 (2008).
[CrossRef]

2006 (2)

H. J. Lee, Y.-B. Chen, and Z. M. Zhang, “Directional radiative properties of anisotropic rough silicon and gold surfaces,” Int. J. Heat Mass Transfer 49, 4482–4495 (2006).
[CrossRef]

H. J. Lee and Z. M. Zhang, “Measurement and modeling of the bidirectional reflectance of SiO2 coated Si surfaces,” Int. J. Thermophys. 27, 820–839 (2006).
[CrossRef]

2005 (2)

Q. Z. Zhu and Z. M. Zhang, “Correlation of angle-resolved light scattering with the microfacet orientation of rough silicon surfaces,” Opt. Eng. 44, 073601 (2005).
[CrossRef]

H. J. Lee, B. J. Lee, and Z. M. Zhang, “Modeling the radiative properties of semitransparent wafers with rough surfaces and thin-film coatings,” J. Quant. Spectrosc. Radiat. Transfer 93, 185–194 (2005).
[CrossRef]

2004 (1)

Y.-B. Chen, Q. Z. Zhu, T. L. Wright, W. P. King, and Z. M. Zhang, “Bidirectional reflection measurements of periodically microstructured silicon surfaces,” Int. J. Thermophys. 25, 1235–1252 (2004).
[CrossRef]

2003 (2)

Y. Shen, Q. Z. Zhu, and Z. M. Zhang, “A scatterometer for measuring the bidirectional reflectance and transmittance of semiconductor wafers with rough surfaces,” Rev. Sci. Instrum. 74, 4885–4892 (2003).
[CrossRef]

H. Fakhruddin, “Specular reflection from a rough surface,” Phys. Teach. 41, 206–207 (2003).
[CrossRef]

2002 (1)

2001 (2)

D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Interference approach applied to dual-grating dielectric resonant grating reflection filters,” Opt. Lett. 26, 1749–1751 (2001).
[CrossRef]

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

2000 (2)

1995 (1)

1994 (3)

1993 (2)

1988 (1)

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

1985 (1)

1984 (1)

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

1980 (1)

1979 (1)

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

1978 (1)

1969 (1)

C. A. Fenstermaker and F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[CrossRef]

1950 (1)

C. O. Riggs, “Mirage, or regular reflection?” Am. J. Phys. 18, 526 (1950).
[CrossRef]

1921 (1)

A. F. Odell, “Sidewalk mirages,” Science 54, 357 (1921).
[CrossRef]

1920 (1)

F. W. McNair, “A sidewalk mirage,” Science 52, 201 (1920).
[CrossRef]

Adam, J. A.

J. A. Adam, A Mathematical Nature Walk (Princeton University, 2009).

Amra, C.

Aspnes, D. E.

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Bennett, J. M.

J. M. Elson, J. P. Rahn, and J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
[CrossRef]

J. M. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering, 2nd ed. (Optical Society of America, 1999).

Benzi, M.

I. Semenikhin, M. Zanuccoli, M. Benzi, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient RCWA method for simulation of thin film solar cells,” Opt. Quantum Electron. 44, 149–154 (2012).
[CrossRef]

Chen, C.

H. C. Zhou, Z. F. Huang, Q. Cheng, W. Lü, K. Qiu, and C. Chen, “Road surface mirage: a bunch of hot air?” Chin. Sci. Bull. 56, 962–968 (2011).
[CrossRef]

Chen, Y.-B.

H. J. Lee, Y.-B. Chen, and Z. M. Zhang, “Directional radiative properties of anisotropic rough silicon and gold surfaces,” Int. J. Heat Mass Transfer 49, 4482–4495 (2006).
[CrossRef]

Y.-B. Chen, Q. Z. Zhu, T. L. Wright, W. P. King, and Z. M. Zhang, “Bidirectional reflection measurements of periodically microstructured silicon surfaces,” Int. J. Thermophys. 25, 1235–1252 (2004).
[CrossRef]

Cheng, Q.

H. C. Zhou, Z. F. Huang, Q. Cheng, W. Lü, K. Qiu, and C. Chen, “Road surface mirage: a bunch of hot air?” Chin. Sci. Bull. 56, 962–968 (2011).
[CrossRef]

Chew, W. C.

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

Deumie, C.

M. Lequime, M. Zerrad, C. Deumie, and C. Amra, “A goniometric light scattering instrument with high-resolution imaging,” Opt. Commun. 282, 1265–1273 (2009).
[CrossRef]

Dunn, S. C.

Duparré, A.

S. Schröder, M. Trost, T. Herffurth, A. von Finck, and A. Duparré, “Sophisticated light scattering techniques from the VUV to the IR regions,” Proc. SPIE 8495, 84950V (2012).
[CrossRef]

Elson, J. M.

J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

J. M. Elson, J. P. Rahn, and J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
[CrossRef]

Fahr, S.

S. Fahr, C. Rockstuhl, and F. Lederer, “Engineering the randomness for enhanced absorption in solar cells,” Appl. Phys. Lett. 92, 171114 (2008).
[CrossRef]

Fakhruddin, H.

H. Fakhruddin, “Specular reflection from a rough surface,” Phys. Teach. 41, 206–207 (2003).
[CrossRef]

Fenstermaker, C. A.

C. A. Fenstermaker and F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[CrossRef]

Fiegna, C.

I. Semenikhin, M. Zanuccoli, M. Benzi, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient RCWA method for simulation of thin film solar cells,” Opt. Quantum Electron. 44, 149–154 (2012).
[CrossRef]

Fu, K.

K. Fu and P.-f. Hsu, “New regime map of the geometric optics approximation for scattering from random rough surfaces,” J. Quant. Spectrosc. Radiat. Transfer 109, 180–188 (2008).
[CrossRef]

Gove, P. B.

P. B. Gove, Webster’s Third New International Dictionary of the English Language Unabridged (G. & C. Merriam Company, 1964).

Gralak, B.

Grann, E. B.

Herffurth, T.

S. Schröder, M. Trost, T. Herffurth, A. von Finck, and A. Duparré, “Sophisticated light scattering techniques from the VUV to the IR regions,” Proc. SPIE 8495, 84950V (2012).
[CrossRef]

Hottier, F.

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Hsu, P.-f.

K. Fu and P.-f. Hsu, “New regime map of the geometric optics approximation for scattering from random rough surfaces,” J. Quant. Spectrosc. Radiat. Transfer 109, 180–188 (2008).
[CrossRef]

Huang, Z. F.

H. C. Zhou, Z. F. Huang, Q. Cheng, W. Lü, K. Qiu, and C. Chen, “Road surface mirage: a bunch of hot air?” Chin. Sci. Bull. 56, 962–968 (2011).
[CrossRef]

Jacob, D. K.

Jurich, M.

King, W. P.

Y.-B. Chen, Q. Z. Zhu, T. L. Wright, W. P. King, and Z. M. Zhang, “Bidirectional reflection measurements of periodically microstructured silicon surfaces,” Int. J. Thermophys. 25, 1235–1252 (2004).
[CrossRef]

Knotts, M. E.

Kosa, T.

T. Kosa and P. Palffy-Muhoray, “Mirage mirror on the wall,” Am. J. Phys. 68, 1120–1122 (2000).
[CrossRef]

Lederer, F.

S. Fahr, C. Rockstuhl, and F. Lederer, “Engineering the randomness for enhanced absorption in solar cells,” Appl. Phys. Lett. 92, 171114 (2008).
[CrossRef]

Lee, B. J.

B. J. Lee and Z. M. Zhang, “Indirect measurements of coherent thermal emission from a truncated photonic crystal structure,” J. Thermophys. Heat Transfer 23, 9–17 (2009).
[CrossRef]

H. J. Lee, B. J. Lee, and Z. M. Zhang, “Modeling the radiative properties of semitransparent wafers with rough surfaces and thin-film coatings,” J. Quant. Spectrosc. Radiat. Transfer 93, 185–194 (2005).
[CrossRef]

Lee, H. J.

H. J. Lee, Y.-B. Chen, and Z. M. Zhang, “Directional radiative properties of anisotropic rough silicon and gold surfaces,” Int. J. Heat Mass Transfer 49, 4482–4495 (2006).
[CrossRef]

H. J. Lee and Z. M. Zhang, “Measurement and modeling of the bidirectional reflectance of SiO2 coated Si surfaces,” Int. J. Thermophys. 27, 820–839 (2006).
[CrossRef]

H. J. Lee, B. J. Lee, and Z. M. Zhang, “Modeling the radiative properties of semitransparent wafers with rough surfaces and thin-film coatings,” J. Quant. Spectrosc. Radiat. Transfer 93, 185–194 (2005).
[CrossRef]

Lequime, M.

M. Zerrad, M. Lequime, and C. Amra, “Multimodal scattering facilities and modelization tools for a comprehensive investigation of optical coatings,” Proc. SPIE 8169, 81690K (2011).
[CrossRef]

M. Lequime, M. Zerrad, C. Deumie, and C. Amra, “A goniometric light scattering instrument with high-resolution imaging,” Opt. Commun. 282, 1265–1273 (2009).
[CrossRef]

Liu, L. H.

W. J. Zhang, J. M. Zhao, and L. H. Liu, “Experimental study of the effective BRDT of a copper foam sheet,” in Proceedings of the 7th International Symposium on Radiative Transfer, Kuşadasi, Turkey (2013), paper RAD-13-SH4.

Lü, W.

H. C. Zhou, Z. F. Huang, Q. Cheng, W. Lü, K. Qiu, and C. Chen, “Road surface mirage: a bunch of hot air?” Chin. Sci. Bull. 56, 962–968 (2011).
[CrossRef]

Mattsson, L.

J. M. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering, 2nd ed. (Optical Society of America, 1999).

McCrackin, F. L.

C. A. Fenstermaker and F. L. McCrackin, “Errors arising from surface roughness in ellipsometric measurement of the refractive index of a surface,” Surf. Sci. 16, 85–96 (1969).
[CrossRef]

McNair, F. W.

F. W. McNair, “A sidewalk mirage,” Science 52, 201 (1920).
[CrossRef]

Moharam, M. G.

Nevière, M.

O’Donnell, K. A.

Odell, A. F.

A. F. Odell, “Sidewalk mirages,” Science 54, 357 (1921).
[CrossRef]

Palffy-Muhoray, P.

T. Kosa and P. Palffy-Muhoray, “Mirage mirror on the wall,” Am. J. Phys. 68, 1120–1122 (2000).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

Pommet, D. A.

Popov, E.

Qiu, K.

H. C. Zhou, Z. F. Huang, Q. Cheng, W. Lü, K. Qiu, and C. Chen, “Road surface mirage: a bunch of hot air?” Chin. Sci. Bull. 56, 962–968 (2011).
[CrossRef]

Rabolt, J. F.

Rahn, J. P.

Riggs, C. O.

C. O. Riggs, “Mirage, or regular reflection?” Am. J. Phys. 18, 526 (1950).
[CrossRef]

Roche, P.

Rockstuhl, C.

S. Fahr, C. Rockstuhl, and F. Lederer, “Engineering the randomness for enhanced absorption in solar cells,” Appl. Phys. Lett. 92, 171114 (2008).
[CrossRef]

Sangiorgi, E.

I. Semenikhin, M. Zanuccoli, M. Benzi, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient RCWA method for simulation of thin film solar cells,” Opt. Quantum Electron. 44, 149–154 (2012).
[CrossRef]

Schröder, S.

S. Schröder, M. Trost, T. Herffurth, A. von Finck, and A. Duparré, “Sophisticated light scattering techniques from the VUV to the IR regions,” Proc. SPIE 8495, 84950V (2012).
[CrossRef]

Semenikhin, I.

I. Semenikhin, M. Zanuccoli, M. Benzi, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient RCWA method for simulation of thin film solar cells,” Opt. Quantum Electron. 44, 149–154 (2012).
[CrossRef]

Shen, Y.

Y. Shen, Q. Z. Zhu, and Z. M. Zhang, “A scatterometer for measuring the bidirectional reflectance and transmittance of semiconductor wafers with rough surfaces,” Rev. Sci. Instrum. 74, 4885–4892 (2003).
[CrossRef]

Stover, J. C.

J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed. (SPIE, 1995), Chap. 6.

J. C. Stover, Optical Scattering: Measurement and Analysis (SPIE, 2012).

Swalen, J. D.

Tayeb, G.

Theeten, J. B.

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Thorsos, E. I.

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

Torricini, D.

Trost, M.

S. Schröder, M. Trost, T. Herffurth, A. von Finck, and A. Duparré, “Sophisticated light scattering techniques from the VUV to the IR regions,” Proc. SPIE 8495, 84950V (2012).
[CrossRef]

von Finck, A.

S. Schröder, M. Trost, T. Herffurth, A. von Finck, and A. Duparré, “Sophisticated light scattering techniques from the VUV to the IR regions,” Proc. SPIE 8495, 84950V (2012).
[CrossRef]

Vyurkov, V.

I. Semenikhin, M. Zanuccoli, M. Benzi, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient RCWA method for simulation of thin film solar cells,” Opt. Quantum Electron. 44, 149–154 (2012).
[CrossRef]

Warnick, K. F.

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media 11, R1–R30 (2001).
[CrossRef]

Woyk, E.

Wright, T. L.

Y.-B. Chen, Q. Z. Zhu, T. L. Wright, W. P. King, and Z. M. Zhang, “Bidirectional reflection measurements of periodically microstructured silicon surfaces,” Int. J. Thermophys. 25, 1235–1252 (2004).
[CrossRef]

Zanuccoli, M.

I. Semenikhin, M. Zanuccoli, M. Benzi, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient RCWA method for simulation of thin film solar cells,” Opt. Quantum Electron. 44, 149–154 (2012).
[CrossRef]

Zerrad, M.

M. Zerrad, M. Lequime, and C. Amra, “Multimodal scattering facilities and modelization tools for a comprehensive investigation of optical coatings,” Proc. SPIE 8169, 81690K (2011).
[CrossRef]

M. Lequime, M. Zerrad, C. Deumie, and C. Amra, “A goniometric light scattering instrument with high-resolution imaging,” Opt. Commun. 282, 1265–1273 (2009).
[CrossRef]

Zhang, W. J.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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W. J. Zhang, J. M. Zhao, and L. H. Liu, “Experimental study of the effective BRDT of a copper foam sheet,” in Proceedings of the 7th International Symposium on Radiative Transfer, Kuşadasi, Turkey (2013), paper RAD-13-SH4.

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[CrossRef]

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Figures (13)

Fig. 1.
Fig. 1.

Schematic of the experimental arrangement.

Fig. 2.
Fig. 2.

Definitions of the BRDF and the stage coordinates systems.

Fig. 3.
Fig. 3.

Geometry of the rough surface diffraction problem analyzed.

Fig. 4.
Fig. 4.

Electric field intensity distribution of random rough surface (silicon, σ=1.0μm, τ=2.0μm) at oblique incident plane wave (θ=30°, λ=1.033μm, TE) calculated by (a) RCWA and (b) FDTD.

Fig. 5.
Fig. 5.

Magnetic field intensity and vector distribution of random rough surface (silicon, σ=1.0μm, τ=2.0μm) at oblique incident plane wave (θ=30°, λ=1.033μm, TE) calculated by (a) RCWA and (b) FDTD.

Fig. 6.
Fig. 6.

BRDF×cosθr of the rough aluminum surfaces (Sample 1) at different incident angles measured by the TAAS: (a) θi=30° and (b) θi=75°.

Fig. 7.
Fig. 7.

BRDF×cosθr of the rough aluminum surfaces with different roughnesses at different incident angles measured by the TAAS.

Fig. 8.
Fig. 8.

Comparison of the reflectance from two different gold one-dimensional rough surfaces solved by the RCWA and measurement in Ref. [10]. Perfect reflection of incident power is denoted by 0.5.

Fig. 9.
Fig. 9.

Reflectance from the rough surface (σ=3.0μm, τ=10.0μm) of different diffracted orders at different incident angles: (a) TE and (b) TM.

Fig. 10.
Fig. 10.

Average spectral reflectance ((RTE+RTM)/2) from different rough surfaces: (a) σ=3.0μm, τ=10.0μm; (b) σ=0.4μm, τ=10.0μm; (c) σ=0.4μm, τ=2.0μm; and (d) σ=0.1μm, τ=1.0μm.

Fig. 11.
Fig. 11.

Scattering region of three different rough surfaces at different incidence angles.

Fig. 12.
Fig. 12.

Comparison of the TIS solved by the RCWA and the Rayleigh criterion: (a) σ=3.0μm, τ=10.0μm; (b) σ=0.4μm, τ=10.0μm; (c) σ=0.4μm, τ=2.0μm; and (d) σ=0.1μm, τ=1.0μm.

Fig. 13.
Fig. 13.

Relative errors between the zero-order diffractive power calculated by RCWA and the reflectance calculated by Fresnel equations: (a) σ=3.0μm, τ=10.0μm; (b) σ=0.4μm, τ=10.0μm; (c) σ=0.4μm, τ=2.0μm; and (d) σ=0.1μm, τ=1.0μm.

Tables (1)

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Table 1. Dimensions and Roughness Parameters of Samples

Equations (6)

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

θ=arccos[cos(αγ)sinβ],
φ=arccos[sinβsin(γα)sinθ].
BRDF(θi,φi:θr,φr)·cosθr=VrVR1a1Ωr,
TISscattered powerspecularly reflected power=PsPo(4πσcosθiλ)2,
σλ0,then1TIS(λi4πσcosθi)2.
cosθi0,then1TIS(λi4πσcosθi)2.

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