Abstract

An omnidirectional light collector consisting of an axisymmetric spatially gradient refractive index medium can almost perfectly absorb light rays, regardless of where they come from. Based on the conformal mapping with complex gradient-index medium, the omnidirectional light collector, which is here called a dark hole, is able to be designed with an exponential function. The dark hole, however, has a reflective boundary where the Fresnel reflection occurs, which might lessen the absorption efficiency. To design a dark hole with consideration of the Fresnel reflection loss, a method to estimate its absorptance is necessary. Therefore, a formula to calculate the absorptance of the dark hole is derived based on the Lagrangian optics with the etendue conservation. Absorptances calculated using the formula agree well with those calculated using the Mie scattering theory in refractive index small-difference limit, which validates the formula. Absorptance of a dark hole with a silicon core and another dark hole with a complex gradient-index intermediate medium are calculated using the formula to be more than 98.8%. A micro-size dark hole is also shown to efficiently collect light rays with an absorptance of more than 95% using FDTD (finite-difference time-domain) simulation.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

2018 (1)

2017 (1)

2015 (2)

2013 (1)

D. Canavarro, J. Chaves, and M. C. Pereira, “New second-stage concentrators (XX SMS) for parabolic primaries; Comparison with conventional parabolic trough concentrators,” Sol. Energy 92, 98–105 (2013).
[Crossref]

2012 (1)

Z. Chang and G. Hu, “Elastic wave omnidirectional absorbers designed by transformation method,” Appl. Phys. Lett. 101(5), 054102 (2012).
[Crossref]

2011 (2)

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13(2), 024002 (2011).
[Crossref]

X. Zhou, X. Cai, Z. Chang, and G. Hu, “Experimental study on a broadband omnidirectional electromagnetic absorber,” J. Opt. 13(8), 085103 (2011).
[Crossref]

2010 (3)

Q. Cheng, T. J. Cui, W. X. Jiang, and B. G. Cai, “An omnidirectional electromagnetic absorber made of metamaterials,” New J. Phys. 12(6), 063006 (2010).
[Crossref]

W. Lu, J. Jin, Z. Lin, and H. Chen, “A simple design of an artificial electromagnetic black hole,” J. Appl. Phys. 108(6), 064517 (2010).
[Crossref]

A. V. Kildishev, L. J. Prokopeva, and E. E. Narimanov, “Cylinder light concentrator and absorber: theoretical description,” Opt. Express 18(16), 16646–16662 (2010).
[Crossref] [PubMed]

2009 (1)

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[Crossref]

2006 (3)

U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8(7), 118 (2006).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

1994 (1)

1992 (1)

H. A. Yousif and E. Boutros, “A FORTRAN code for the scattering of EM-plane waves by an infinitely long cylinder at oblique incidence,” Comput. Phys. Commun. 69(2-3), 406–414 (1992).
[Crossref]

1983 (1)

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B Condens. Matter 27(2), 985–1009 (1983).
[Crossref]

1980 (1)

1976 (1)

A. Rabl, “Optical and thermal properties of compound parabolic concentrators,” Sol. Energy 18(6), 497–511 (1976).
[Crossref]

1970 (1)

Aguayo, H.

Aspnes, D. E.

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B Condens. Matter 27(2), 985–1009 (1983).
[Crossref]

Boutros, E.

H. A. Yousif and E. Boutros, “A FORTRAN code for the scattering of EM-plane waves by an infinitely long cylinder at oblique incidence,” Comput. Phys. Commun. 69(2-3), 406–414 (1992).
[Crossref]

Cai, B. G.

Q. Cheng, T. J. Cui, W. X. Jiang, and B. G. Cai, “An omnidirectional electromagnetic absorber made of metamaterials,” New J. Phys. 12(6), 063006 (2010).
[Crossref]

Cai, X.

X. Zhou, X. Cai, Z. Chang, and G. Hu, “Experimental study on a broadband omnidirectional electromagnetic absorber,” J. Opt. 13(8), 085103 (2011).
[Crossref]

Canavarro, D.

D. Canavarro, J. Chaves, and M. C. Pereira, “New second-stage concentrators (XX SMS) for parabolic primaries; Comparison with conventional parabolic trough concentrators,” Sol. Energy 92, 98–105 (2013).
[Crossref]

Chang, Z.

Z. Chang and G. Hu, “Elastic wave omnidirectional absorbers designed by transformation method,” Appl. Phys. Lett. 101(5), 054102 (2012).
[Crossref]

X. Zhou, X. Cai, Z. Chang, and G. Hu, “Experimental study on a broadband omnidirectional electromagnetic absorber,” J. Opt. 13(8), 085103 (2011).
[Crossref]

Chaves, J.

D. Canavarro, J. Chaves, and M. C. Pereira, “New second-stage concentrators (XX SMS) for parabolic primaries; Comparison with conventional parabolic trough concentrators,” Sol. Energy 92, 98–105 (2013).
[Crossref]

Chen, H.

W. Lu, J. Jin, Z. Lin, and H. Chen, “A simple design of an artificial electromagnetic black hole,” J. Appl. Phys. 108(6), 064517 (2010).
[Crossref]

Cheng, Q.

Q. Cheng, T. J. Cui, W. X. Jiang, and B. G. Cai, “An omnidirectional electromagnetic absorber made of metamaterials,” New J. Phys. 12(6), 063006 (2010).
[Crossref]

Cui, T. J.

Q. Cheng, T. J. Cui, W. X. Jiang, and B. G. Cai, “An omnidirectional electromagnetic absorber made of metamaterials,” New J. Phys. 12(6), 063006 (2010).
[Crossref]

García, H.

Hu, G.

Z. Chang and G. Hu, “Elastic wave omnidirectional absorbers designed by transformation method,” Appl. Phys. Lett. 101(5), 054102 (2012).
[Crossref]

X. Zhou, X. Cai, Z. Chang, and G. Hu, “Experimental study on a broadband omnidirectional electromagnetic absorber,” J. Opt. 13(8), 085103 (2011).
[Crossref]

Jiang, W. X.

Q. Cheng, T. J. Cui, W. X. Jiang, and B. G. Cai, “An omnidirectional electromagnetic absorber made of metamaterials,” New J. Phys. 12(6), 063006 (2010).
[Crossref]

Jin, J.

W. Lu, J. Jin, Z. Lin, and H. Chen, “A simple design of an artificial electromagnetic black hole,” J. Appl. Phys. 108(6), 064517 (2010).
[Crossref]

Kildishev, A. V.

A. V. Kildishev, L. J. Prokopeva, and E. E. Narimanov, “Cylinder light concentrator and absorber: theoretical description,” Opt. Express 18(16), 16646–16662 (2010).
[Crossref] [PubMed]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[Crossref]

Kozminski, K.

Kundtz, N. B.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13(2), 024002 (2011).
[Crossref]

León, N.

Leonhardt, U.

U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8(7), 118 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

Lin, Z.

W. Lu, J. Jin, Z. Lin, and H. Chen, “A simple design of an artificial electromagnetic black hole,” J. Appl. Phys. 108(6), 064517 (2010).
[Crossref]

Lu, C.

Lu, W.

W. Lu, J. Jin, Z. Lin, and H. Chen, “A simple design of an artificial electromagnetic black hole,” J. Appl. Phys. 108(6), 064517 (2010).
[Crossref]

Mattis, R. E.

Mei, Z. L.

Narimanov, E. E.

A. V. Kildishev, L. J. Prokopeva, and E. E. Narimanov, “Cylinder light concentrator and absorber: theoretical description,” Opt. Express 18(16), 16646–16662 (2010).
[Crossref] [PubMed]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[Crossref]

Ohno, H.

Pendry, J. B.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13(2), 024002 (2011).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Pereira, M. C.

D. Canavarro, J. Chaves, and M. C. Pereira, “New second-stage concentrators (XX SMS) for parabolic primaries; Comparison with conventional parabolic trough concentrators,” Sol. Energy 92, 98–105 (2013).
[Crossref]

Prokopeva, L. J.

Rabl, A.

A. Rabl, “Optical and thermal properties of compound parabolic concentrators,” Sol. Energy 18(6), 497–511 (1976).
[Crossref]

Ramírez, C.

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Smith, D. R.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13(2), 024002 (2011).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Studna, A. A.

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B Condens. Matter 27(2), 985–1009 (1983).
[Crossref]

Toya, K.

Urzhumov, Y. A.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13(2), 024002 (2011).
[Crossref]

Welford, W. T.

Winston, R.

Yousif, H. A.

H. A. Yousif, R. E. Mattis, and K. Kozminski, “Light scattering at oblique incidence on two coaxial cylinders,” Appl. Opt. 33(18), 4013–4024 (1994).
[Crossref] [PubMed]

H. A. Yousif and E. Boutros, “A FORTRAN code for the scattering of EM-plane waves by an infinitely long cylinder at oblique incidence,” Comput. Phys. Commun. 69(2-3), 406–414 (1992).
[Crossref]

Zhou, X.

X. Zhou, X. Cai, Z. Chang, and G. Hu, “Experimental study on a broadband omnidirectional electromagnetic absorber,” J. Opt. 13(8), 085103 (2011).
[Crossref]

Appl. Opt. (4)

Appl. Phys. Lett. (2)

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[Crossref]

Z. Chang and G. Hu, “Elastic wave omnidirectional absorbers designed by transformation method,” Appl. Phys. Lett. 101(5), 054102 (2012).
[Crossref]

Comput. Phys. Commun. (1)

H. A. Yousif and E. Boutros, “A FORTRAN code for the scattering of EM-plane waves by an infinitely long cylinder at oblique incidence,” Comput. Phys. Commun. 69(2-3), 406–414 (1992).
[Crossref]

J. Appl. Phys. (1)

W. Lu, J. Jin, Z. Lin, and H. Chen, “A simple design of an artificial electromagnetic black hole,” J. Appl. Phys. 108(6), 064517 (2010).
[Crossref]

J. Opt. (2)

X. Zhou, X. Cai, Z. Chang, and G. Hu, “Experimental study on a broadband omnidirectional electromagnetic absorber,” J. Opt. 13(8), 085103 (2011).
[Crossref]

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13(2), 024002 (2011).
[Crossref]

J. Opt. Soc. Am. (1)

New J. Phys. (2)

U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8(7), 118 (2006).
[Crossref]

Q. Cheng, T. J. Cui, W. X. Jiang, and B. G. Cai, “An omnidirectional electromagnetic absorber made of metamaterials,” New J. Phys. 12(6), 063006 (2010).
[Crossref]

Opt. Express (4)

Phys. Rev. B Condens. Matter (1)

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B Condens. Matter 27(2), 985–1009 (1983).
[Crossref]

Science (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

Sol. Energy (2)

A. Rabl, “Optical and thermal properties of compound parabolic concentrators,” Sol. Energy 18(6), 497–511 (1976).
[Crossref]

D. Canavarro, J. Chaves, and M. C. Pereira, “New second-stage concentrators (XX SMS) for parabolic primaries; Comparison with conventional parabolic trough concentrators,” Sol. Energy 92, 98–105 (2013).
[Crossref]

Other (5)

M. Born and E. Wolf, Principles of Optics (Cambridge, 1999).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

J. Chaves, Introduction to NONIMAGING OPTICS (CRC, 2008).

R. Winston, J. Minano, and P. Benitez, Nonimaging Optics (Academic, 2005).

RSoft, https://www.synopsys.com/ja-jp/japan/products.html .

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Figures (14)

Fig. 1
Fig. 1 Schematic diagram of the original (virtual) complex plane. The horizontal axis indicates a real part x of z, and the vertical axis indicates an imaginary part y of z. The original complex plane is set to a semi-infinite region where y is set to from 0 to y0 with the entire x-axis.
Fig. 2
Fig. 2 Schematic diagram of the projected (physical) plane. Horizontal axis indicates a real part u of w, and vertical axis indicates an imaginary part v of w. In the projected plane, lines of x = constant in the original plane are transformed into lines along radial directions. On the other hand, lines of y = constant in the original plane are transformed into circles.
Fig. 3
Fig. 3 Perspective view of the gradient refractive index (GRIN) distribution on the projected plane. The refractive index is color contoured on the left, and the extinction coefficient K is color contoured on the right. On the right of the figure, there is a gap boundary where an extinction coefficient K has a gap at the central core boundary.
Fig. 4
Fig. 4 Perspective view of the complex gradient refractive index (CGRIN) distribution on the projected plane. The refractive index is color contoured on the left, and the extinction coefficient K is color contoured on the right. On the right of the figure, there is a gap boundary where an extinction coefficient K has a gap at the periphery of the dark hole.
Fig. 5
Fig. 5 Light ray incident on gap boundary where extinction coefficient has a gap. The incident angle on the surface of the gap boundary is Θi. The origin of the coordinate system is O. The complex refractive indexes across the boundary are n + 0i in the outer region and n + iK in the inner region.
Fig. 6
Fig. 6 The Fresnel reflectances calculated using Eqs. (10) and (11). Horizontal axis indicates incident angle, and vertical axis indicates reflectance. The ratios of K/n are set to 0.046, 0.6, 1.0, 1.5, and 2.0 respectively. The reflectances of s- and p-polarization are found to decrease with smaller ratio of the K/n. On the other hand, it can be found that the reflectances increase with larger ratio K/n and larger incident angle Θ, which means that the Fresnel reflectance should be considered at the gap boundaries.
Fig. 7
Fig. 7 The cylindrical coordinates (r, θ) taken for the light ray path (Q) in the u-v plane with initial point (Q)0. The refractive index field is axisymmetric about the origin O where the field is constant n0 with the radius of more than r0. An angle β is defined as an angle between the light ray path and v-axis at the far initial point. A point (H) is an intersection of an extrapolation along the initial light ray direction with the v = 0 line. The angle β can be set to zero without loss of generality, which means that light rays are incident on the dark hole in a direction parallel to the v-axis.
Fig. 8
Fig. 8 Light ray paths in the u-v cross-sectional plane. Horizontal axis indicates u-axis normalized by r0 and vertical axis indicates v-axis normalized by r0.
Fig. 9
Fig. 9 S- and p-polarization absorptances calculated by both the formula (etendue conservation) and the Mie scattering theory (Mie theory) with respect to Kcore/ncore. Horizontal axis indicates the ratio of Kcore/ncore, and vertical axis indicates absorptance Qabs. The solid line indicates absorptance calculated by the formula for s-polarization, and the dashed line indicates that for p-polarization. The circle mark indicates absorptance calculated by the Mie theory for s-polarization, and the cross mark indicates that for p-polarization.
Fig. 10
Fig. 10 Electromagnetic wave propagating in the cross-sectional plane in steady state. The diameter of the dark hole is set to 5.0 μm. The source of the wave is placed at (0.0 μm, −2.5 μm) with a width of 10.0 μm. The refractive index in the original coordinate, ϕ0, is set to 1.0 + 0.0i, and the complex refractive index in the central core is 4.94 + 0.23i. The wavelength is set to 0.42 μm.
Fig. 11
Fig. 11 Absorption density qabs defined as normalized absorbed power per unit area in the cross-sectional plane in steady state. The qabs is color-contoured in log scale. The diameter of the dark hole is set to 5.0 μm. The parameters are the same as those used in Fig. 10.
Fig. 12
Fig. 12 Electromagnetic wave propagating in the cross-sectional plane for several time durations. The diameter of the dark hole, 2r0, is set to 5.0 μm. The source of the electromagnetic wave is placed at (2.0 μm, −2.5 μm) with a width of 3.0 μm. The original refractive index, ϕ0, is set to 1.0 + 0.001i. The central core radius is 0.5 μm. The optical wavelength is set to 0.42 μm.
Fig. 13
Fig. 13 Electromagnetic wave propagating in the cross-sectional plane in steady state. The diameter of the dark hole is set to 5.0 μm. The source of the wave is placed at (0.0 μm, −2.5 μm) with a width of 10.0 μm. The complex refractive index in the original coordinate, ϕ0, is set to 1.0 + 0.0465i, and the complex refractive index of the central core is 4.94 + 0.23i. The wavelength is set to 0.42 μm.
Fig. 14
Fig. 14 Absorption density qabs defined as normalized absorbed power per unit area in the cross-sectional plane in steady state. The qabs is color-contoured in log scale. The diameter of the dark hole is set to 5.0 μm. The parameters are the same as those used in Fig. 13.

Equations (27)

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w= r 0 exp( i z r 0 ),
u 2 + v 2 = ( r 0 exp( y r 0 ) ) 2 r 2 ,
ρ r r 0 ,
ϕ=n+iK= ϕ 0 / | dw dz | = ϕ 0 ρ ,
ϕ 0 = n 0 +i K 0 .
ρ core = n 0 n core ,
ϕ= n 0 +i K 0 ρ ,
ρ core = n 0 n core ρ1.
K 0 = K core n core ,
F s = | cos Θ i ( 1+iK/n ) 1 ( 1/ ( 1+iK/n ) ) 2 sin 2 Θ i cos Θ i +( 1+iK/n ) 1 ( 1/ ( 1+iK/n ) ) 2 sin 2 Θ i | 2 ,
F p = | ( 1+iK/n )cos Θ i 1 ( 1/ ( 1+iK/n ) ) 2 sin 2 Θ i ( 1+iK/n )cos Θ i + 1 ( 1/ ( 1+iK/n ) ) 2 sin 2 Θ i | 2 .
ϕ= n 0 r 0 r .
d dv ( n r 2 θ ˙ 1+ r 2 θ ˙ 2 )=0,
θ ˙ dθ dr .
rdθ dr =r θ ˙ =constant.
rdθ dr =tanΘ,
tan Θ 0 = H r 0 2 H 2 .
ρ= r r 0 =exp[ cos Θ 0 sin Θ 0 ( π Θ 0 θ ) ].
dG= n 2 dScosΘdΩ.
d G 0 = n 0 2 d S 0 d Ω 0 = n 0 2 ρ 2 dScosΘdΩ=dG.
dB=I( ρ ) F s,p ( Θ )dScosΘdΩ.
A s,p =1 dB I( ρ )dScosΘdΩ =1 F s,p ( Θ )I( ρ )dScosΘdΩ I( ρ )dScosΘdΩ .
A s,p =1 F s,p ( Θ )( I 0 / n 0 2 ) n 0 2 d S 0 d Ω 0 ( I 0 / n 0 2 ) n 0 2 d S 0 d Ω 0 =1 F s,p ( Θ ) I 0 d S 0 d Ω 0 I 0 d S 0 d Ω 0 .
A s,p =1 0 r 0 F s,p ( Θ( H ) )dH 0 r 0 dH ,
Θ=arctan( H r 0 2 H 2 ).
q abs = ( Re( E× H )/2 ) r r 0 dudv[ ( Re( E× H )/2 ) ] ,
η Q 2 r 0 I in = r r 0 dudv[ ( Re( E× H )/2 ) ] 2 r 0 I in .

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