Abstract

Antireflective surfaces are often realized by minimizing the refractive index contrast between the air and substrate using subwavelength microstructures. In this paper, we introduce another kind of geometry-induced antireflective surface using multireflection to suppress reflection to a very low extent. This surface is composed of micro-∧-shape array with the wedge size much larger than the wavelength of the incident light. Simulation and experimental results show that the micro-∧-shape array can effectively suppress the reflection within wide incident angles and large wavelength ranges. The enhanced light adsorption caused by the greatly increased light path length within the micro-∧-shape array is responsible for the antireflection phenomena. Such antireflective surface may find various applications, in solar cells, for example.

© 2013 Optical Society of America

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2012

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 1–7 (2012).
[CrossRef]

E. Battal, T. A. Yogurt, L. E. Aygun, and A. K. Okyay, “Triangular metallic gratings for large absorption enhancement in thin film Si solar cells,” Opt. Express 20, 9458–9464 (2012).
[CrossRef]

2011

J. G. A. Wehner, E. P. G. Smith, G. M. Venzor, K. D. Smith, A. M. Ramires, B. P. Kolasa, K. R. Olsson, and M. F. Vilela, “HgCdTe photo trapping structure for broadband mid-wavelength infrared absorption,” J. Electron. Mater. 40, 1840–1846 (2011).
[CrossRef]

L. Sainiemi, V. Jokinen, A. Shah, M. shpak, S. Aura, P. Suvanto, and S. Franssila, “Non-reflecting silicon and polymer surfaces by plasma etching and replication,” Adv. Mater. 23, 122–126 (2011).
[CrossRef]

2010

M. Zhu, H. Gao, H. Li, J. Xu, and Y. Chen, “A facile processing way of silica needle arrays with tunable orientation by tube arrays fabrication and etching method,” J. Solid State Chem. 183, 595–599 (2010).
[CrossRef]

E. Garnett and P. Yang, “Light trapping in silicon nanowire solar cell,” Nano Lett. 10, 1082–1087 (2010).
[CrossRef]

Y. M. Song, J. S. Yu, and Y. T. Lee, “Antireflective submicrometer gratings on thin-film silicon solar cells for light-absorption enhancement,” Opt. Lett. 35, 276–278 (2010).
[CrossRef]

2008

2007

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

W. Zhou, M. Tao, L. Chen, and H. Yang, “Microstructured surface design for omnidirectional antireflection coatings on solar cells,” J. Appl. Phys. 102, 103105 (2007).
[CrossRef]

2006

1998

1993

1985

1961

1906

J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. London A 205, 239–248 (1906).

Amotchkina, T. V.

Arnold, W. H.

W. H. Arnold, M. Farnaam, and J. Sliwa, “Titanium nitride as an antireflection coating on highly reflective layers for photolithography,” U.S. patent4,820,611 (11April1989).

Aura, S.

L. Sainiemi, V. Jokinen, A. Shah, M. shpak, S. Aura, P. Suvanto, and S. Franssila, “Non-reflecting silicon and polymer surfaces by plasma etching and replication,” Adv. Mater. 23, 122–126 (2011).
[CrossRef]

Aygun, L. E.

Azzam, R. M. A.

Battal, E.

Berginc, G.

Bouffaron, R.

Chang, Y. H.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Chattopadhyay, S.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Chen, K. H.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Chen, L.

W. Zhou, M. Tao, L. Chen, and H. Yang, “Microstructured surface design for omnidirectional antireflection coatings on solar cells,” J. Appl. Phys. 102, 103105 (2007).
[CrossRef]

Chen, L. C.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Chen, Y.

M. Zhu, H. Gao, H. Li, J. Xu, and Y. Chen, “A facile processing way of silica needle arrays with tunable orientation by tube arrays fabrication and etching method,” J. Solid State Chem. 183, 595–599 (2010).
[CrossRef]

Cox, J. T.

Cui, Y.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 1–7 (2012).
[CrossRef]

Dobrowolski, J. A.

Escoubas, L.

Fan, S.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 1–7 (2012).
[CrossRef]

Farnaam, M.

W. H. Arnold, M. Farnaam, and J. Sliwa, “Titanium nitride as an antireflection coating on highly reflective layers for photolithography,” U.S. patent4,820,611 (11April1989).

Flory, F.

Fowler, J.

Franssila, S.

L. Sainiemi, V. Jokinen, A. Shah, M. shpak, S. Aura, P. Suvanto, and S. Franssila, “Non-reflecting silicon and polymer surfaces by plasma etching and replication,” Adv. Mater. 23, 122–126 (2011).
[CrossRef]

Gao, H.

M. Zhu, H. Gao, H. Li, J. Xu, and Y. Chen, “A facile processing way of silica needle arrays with tunable orientation by tube arrays fabrication and etching method,” J. Solid State Chem. 183, 595–599 (2010).
[CrossRef]

Garnett, E.

E. Garnett and P. Yang, “Light trapping in silicon nanowire solar cell,” Nano Lett. 10, 1082–1087 (2010).
[CrossRef]

Garnett, J. C. M.

J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. London A 205, 239–248 (1906).

Halpern, M.

Hass, G.

Henry, R.

Hodgkinson, I.

Hsu, C. H.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Hsu, Y. K.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Huang, Y. F.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Jacobus, G. F.

Javily, K.

Jen, Y. J.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Jokinen, V.

L. Sainiemi, V. Jokinen, A. Shah, M. shpak, S. Aura, P. Suvanto, and S. Franssila, “Non-reflecting silicon and polymer surfaces by plasma etching and replication,” Adv. Mater. 23, 122–126 (2011).
[CrossRef]

Kolasa, B. P.

J. G. A. Wehner, E. P. G. Smith, G. M. Venzor, K. D. Smith, A. M. Ramires, B. P. Kolasa, K. R. Olsson, and M. F. Vilela, “HgCdTe photo trapping structure for broadband mid-wavelength infrared absorption,” J. Electron. Mater. 40, 1840–1846 (2011).
[CrossRef]

Lau, J.

Lee, C. S.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Lee, Y. T.

Leong, J.

Li, H.

M. Zhu, H. Gao, H. Li, J. Xu, and Y. Chen, “A facile processing way of silica needle arrays with tunable orientation by tube arrays fabrication and etching method,” J. Solid State Chem. 183, 595–599 (2010).
[CrossRef]

Liu, T. A.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Lo, H. C.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Marriage, T.

Marsden, D.

Marsden, G.

Masclet, Ph.

Morris, G. M.

Narasimhan, V. K.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 1–7 (2012).
[CrossRef]

Okyay, A. K.

Olsson, K. R.

J. G. A. Wehner, E. P. G. Smith, G. M. Venzor, K. D. Smith, A. M. Ramires, B. P. Kolasa, K. R. Olsson, and M. F. Vilela, “HgCdTe photo trapping structure for broadband mid-wavelength infrared absorption,” J. Electron. Mater. 40, 1840–1846 (2011).
[CrossRef]

Page, L.

Pan, C. L.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Peng, C. Y.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Raguin, D. H.

Ramires, A. M.

J. G. A. Wehner, E. P. G. Smith, G. M. Venzor, K. D. Smith, A. M. Ramires, B. P. Kolasa, K. R. Olsson, and M. F. Vilela, “HgCdTe photo trapping structure for broadband mid-wavelength infrared absorption,” J. Electron. Mater. 40, 1840–1846 (2011).
[CrossRef]

Ruan, Z.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 1–7 (2012).
[CrossRef]

Sainiemi, L.

L. Sainiemi, V. Jokinen, A. Shah, M. shpak, S. Aura, P. Suvanto, and S. Franssila, “Non-reflecting silicon and polymer surfaces by plasma etching and replication,” Adv. Mater. 23, 122–126 (2011).
[CrossRef]

Shah, A.

L. Sainiemi, V. Jokinen, A. Shah, M. shpak, S. Aura, P. Suvanto, and S. Franssila, “Non-reflecting silicon and polymer surfaces by plasma etching and replication,” Adv. Mater. 23, 122–126 (2011).
[CrossRef]

shpak, M.

L. Sainiemi, V. Jokinen, A. Shah, M. shpak, S. Aura, P. Suvanto, and S. Franssila, “Non-reflecting silicon and polymer surfaces by plasma etching and replication,” Adv. Mater. 23, 122–126 (2011).
[CrossRef]

Simon, J. J.

Sliwa, J.

W. H. Arnold, M. Farnaam, and J. Sliwa, “Titanium nitride as an antireflection coating on highly reflective layers for photolithography,” U.S. patent4,820,611 (11April1989).

Smith, E. P. G.

J. G. A. Wehner, E. P. G. Smith, G. M. Venzor, K. D. Smith, A. M. Ramires, B. P. Kolasa, K. R. Olsson, and M. F. Vilela, “HgCdTe photo trapping structure for broadband mid-wavelength infrared absorption,” J. Electron. Mater. 40, 1840–1846 (2011).
[CrossRef]

Smith, K. D.

J. G. A. Wehner, E. P. G. Smith, G. M. Venzor, K. D. Smith, A. M. Ramires, B. P. Kolasa, K. R. Olsson, and M. F. Vilela, “HgCdTe photo trapping structure for broadband mid-wavelength infrared absorption,” J. Electron. Mater. 40, 1840–1846 (2011).
[CrossRef]

Song, Y. M.

Suvanto, P.

L. Sainiemi, V. Jokinen, A. Shah, M. shpak, S. Aura, P. Suvanto, and S. Franssila, “Non-reflecting silicon and polymer surfaces by plasma etching and replication,” Adv. Mater. 23, 122–126 (2011).
[CrossRef]

Tao, M.

W. Zhou, M. Tao, L. Chen, and H. Yang, “Microstructured surface design for omnidirectional antireflection coatings on solar cells,” J. Appl. Phys. 102, 103105 (2007).
[CrossRef]

Tikhonravov, A. V.

Torchio, Ph.

Trubetskov, M. K.

Venzor, G. M.

J. G. A. Wehner, E. P. G. Smith, G. M. Venzor, K. D. Smith, A. M. Ramires, B. P. Kolasa, K. R. Olsson, and M. F. Vilela, “HgCdTe photo trapping structure for broadband mid-wavelength infrared absorption,” J. Electron. Mater. 40, 1840–1846 (2011).
[CrossRef]

Vilela, M. F.

J. G. A. Wehner, E. P. G. Smith, G. M. Venzor, K. D. Smith, A. M. Ramires, B. P. Kolasa, K. R. Olsson, and M. F. Vilela, “HgCdTe photo trapping structure for broadband mid-wavelength infrared absorption,” J. Electron. Mater. 40, 1840–1846 (2011).
[CrossRef]

Wehner, J. G. A.

J. G. A. Wehner, E. P. G. Smith, G. M. Venzor, K. D. Smith, A. M. Ramires, B. P. Kolasa, K. R. Olsson, and M. F. Vilela, “HgCdTe photo trapping structure for broadband mid-wavelength infrared absorption,” J. Electron. Mater. 40, 1840–1846 (2011).
[CrossRef]

Wishnow, E.

Wollack, E.

Wu, Q. H.

Xie, C.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 1–7 (2012).
[CrossRef]

Xu, J.

M. Zhu, H. Gao, H. Li, J. Xu, and Y. Chen, “A facile processing way of silica needle arrays with tunable orientation by tube arrays fabrication and etching method,” J. Solid State Chem. 183, 595–599 (2010).
[CrossRef]

Yang, H.

W. Zhou, M. Tao, L. Chen, and H. Yang, “Microstructured surface design for omnidirectional antireflection coatings on solar cells,” J. Appl. Phys. 102, 103105 (2007).
[CrossRef]

Yang, P.

E. Garnett and P. Yang, “Light trapping in silicon nanowire solar cell,” Nano Lett. 10, 1082–1087 (2010).
[CrossRef]

Yao, J.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 1–7 (2012).
[CrossRef]

Yao, Y.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 1–7 (2012).
[CrossRef]

Yogurt, T. A.

Yu, J. S.

Zhou, W.

W. Zhou, M. Tao, L. Chen, and H. Yang, “Microstructured surface design for omnidirectional antireflection coatings on solar cells,” J. Appl. Phys. 102, 103105 (2007).
[CrossRef]

Zhu, M.

M. Zhu, H. Gao, H. Li, J. Xu, and Y. Chen, “A facile processing way of silica needle arrays with tunable orientation by tube arrays fabrication and etching method,” J. Solid State Chem. 183, 595–599 (2010).
[CrossRef]

Adv. Mater.

L. Sainiemi, V. Jokinen, A. Shah, M. shpak, S. Aura, P. Suvanto, and S. Franssila, “Non-reflecting silicon and polymer surfaces by plasma etching and replication,” Adv. Mater. 23, 122–126 (2011).
[CrossRef]

Appl. Opt.

J. Appl. Phys.

W. Zhou, M. Tao, L. Chen, and H. Yang, “Microstructured surface design for omnidirectional antireflection coatings on solar cells,” J. Appl. Phys. 102, 103105 (2007).
[CrossRef]

J. Electron. Mater.

J. G. A. Wehner, E. P. G. Smith, G. M. Venzor, K. D. Smith, A. M. Ramires, B. P. Kolasa, K. R. Olsson, and M. F. Vilela, “HgCdTe photo trapping structure for broadband mid-wavelength infrared absorption,” J. Electron. Mater. 40, 1840–1846 (2011).
[CrossRef]

J. Opt. Soc. Am.

J. Solid State Chem.

M. Zhu, H. Gao, H. Li, J. Xu, and Y. Chen, “A facile processing way of silica needle arrays with tunable orientation by tube arrays fabrication and etching method,” J. Solid State Chem. 183, 595–599 (2010).
[CrossRef]

Nano Lett.

E. Garnett and P. Yang, “Light trapping in silicon nanowire solar cell,” Nano Lett. 10, 1082–1087 (2010).
[CrossRef]

Nat. Commun.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 1–7 (2012).
[CrossRef]

Nat. Nanotechnol.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Philos. Trans. R. Soc. London A

J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. London A 205, 239–248 (1906).

Other

W. H. Arnold, M. Farnaam, and J. Sliwa, “Titanium nitride as an antireflection coating on highly reflective layers for photolithography,” U.S. patent4,820,611 (11April1989).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Optical model of the ∧-shape array. (b) Schematic graph of multiple reflection between adjacent wedges. (c) Calculated reflectance of planar silica surface and the ∧-shape array. (d) Calculated transmittance of planar silica surface and the ∧-shape array.

Fig. 2.
Fig. 2.

(a) Reflection times as a function of the ∧-shape’s sharpness. (b) Reflection times as a function of the incident angle.

Fig. 3.
Fig. 3.

(a) Scattering electric field of ∧-shape array with incident wavelengths 400, 900, and 1400 nm. (b) Calculated scattering electric field in the middle cross section of the groove from 3 to 1 μm on the x axis. (c) Scattering electric field of ∧-shape array with incident wavelengths 2000, 2200, and 2400 nm. (d) Calculated scattering electric field in the middle cross section of the groove from 3 μm to 1 μm on the x axis (the red rectangular box covers the whole ∧-shape structure and the substrate).

Fig. 4.
Fig. 4.

(a) Scanning electron microscope image (collected with a 70° tilted angle) of the silica needle array. (b) Enlarged observation of the silica needle array. (c) Measured reflectance of the planar silica and silica needle. (d) 3D image of reflectance of the planar silica as a function of the incident wavelength and angle. (e) 3D image of reflectance of the silica needle array as a function of the incident wavelength and angle.

Equations (7)

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θ reflect 1 = π 2 θ 0 .
θ reflect n = π 2 ( 2 n 1 ) θ 0 .
n k = [ π 4 θ 0 + 1 2 ] ,
2 k = [ π 2 θ 0 + 1 ] .
θ reflect 1 = π 2 ( θ + θ 0 ) .
θ reflect n = π 2 [ θ + ( 2 n 1 ) θ 0 ] .
2 k = [ π 2 θ 2 θ 0 + 1 ] .

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