Abstract

Metasurfaces enable us to control the fundamental properties of light with unprecedented flexibility. However, most metasurfaces realized to date aim at modifying plane waves. While the manipulation of nonplanar wavefronts is encountered in a diverse number of applications, their control using metasurfaces is still in its infancy. Here we design a metareflector able to reflect a diverging Gaussian beam back onto itself with efficiency over 90% and focusing at an arbitrary distance. We outline a clear route towards the design of complex metareflectors that can find applications as diverse as optical tweezing, lasing, and quantum optics.

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

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References

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    [Crossref] [PubMed]
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    [Crossref]

2018 (4)

A. E. Minovich and A. V. Zayats, “Geometric-phase metasurfaces based on anisotropic reflection: Generalized design rules,” ACS Photon. 5, 1755–1761 (2018).
[Crossref]

S. Wang, P. Chieh Wu, V.-C. Su, Y.-C. Lai, M.-K. Chen, H. Yu Kuo, B. Han Chen, Y. Han Chen, T.-T. Huang, J.-H. Wang, R.-M. Lin, C.-H. Kuan, T. Li, Z. Wang, S. Zhu, and D. P. Tsai, “A broadband achromatic metalens in the visible,” Nat. Nanotech. 13, 227–232 (2018).
[Crossref]

R. Verre, N. O. Länk, D. Andrén, H. Sípová, and M. Käll, “Large-Scale Fabrication of Shaped High Index Dielectric Nanoparticles on a Substrate and in Solution,” Adv. Opt. Mater. 6, 1701253 (2018).
[Crossref]

O. Tsilipakos, T. Koschny, and C. M. Soukoulis, “Antimatched Electromagnetic Metasurfaces for Broadband Arbitrary Phase Manipulation in Reflection,” ACS Photon. 5, 1101–1107 (2018).
[Crossref]

2017 (4)

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139 (2017).
[Crossref]

M. Khorasaninejad and F. Capasso, “Metalenses: Versatile multifunctional photonic components,” Science 358, eaam8100 (2017).
[Crossref] [PubMed]

Q. Fan, P. Huo, D. Wang, Y. Liang, F. Yan, and T. Xu, “Visible light focusing flat lenses based on hybrid dielectric-metal metasurface reflector-arrays,” Sci. Rep. 7, 1–9 (2017).

H.-H. Hsiao, C. H. Chu, and D. P. Tsai, “Fundamentals and applications of metasurfaces,” Small Methods 1, 1600064 (2017).
[Crossref]

2016 (2)

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref] [PubMed]

V. Ginis, P. Tassin, T. Koschny, and C. M Soukoulis, “Broadband metasurfaces enabling arbitrarily large delay-bandwidth products,” Appl. Phys. Lett. 108, 031601 (2016).
[Crossref]

2015 (2)

X. Ni, Z. J. Wong, M. Mrejen, Y. Wang, and X. Zhang, “An ultrathin invisibility skin cloak for visible light,” Science 349, 1310–1314 (2015).
[Crossref] [PubMed]

P. Moitra, B. A. Slovick, W. Li, I. I. Kravchencko, D. P. Briggs, S. Krishnamurthy, and J. Valentine, “Large-Scale All-Dielectric Metamaterial Perfect Reflectors,” ACS Photon. 2, 692–698 (2015).
[Crossref]

2014 (2)

N. M. Estakhri and A. Alù, “Ultra-thin unidirectional carpet cloak and wavefront reconstruction with graded metasurfaces,” IEEE Antennas Wireless Propag. Lett. 13, 1775–1778 (2014).
[Crossref]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
[Crossref] [PubMed]

2013 (5)

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1232009 (2013).
[Crossref] [PubMed]

J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alù, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013).
[Crossref]

C. Pfeiffer and A. Grbic, “Cascaded metasurfaces for complete phase and polarization control,” Appl. Phys. Lett. 102, 231116 (2013).
[Crossref]

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light Sci. Appl. 2, 1–7 (2013).
[Crossref]

T. Roy, E. T. F. Rogers, and N. I. Zheludev, “Sub-wavelength focusing meta-lens,” Opt. Express 21, 7577–7582 (2013).
[Crossref] [PubMed]

2012 (7)

M. Kang, T. Feng, H.-T. Wang, and J. Li, “Wave front engineering from an array of thin aperture antennas,” Opt. Express 20, 15882 (2012).
[Crossref] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref] [PubMed]

G. T. Oumbé Tékam, V. Ginis, J. Danckaert, and P. Tassin, “Designing an efficient rectifying cut-wire metasurface for electromagnetic energy harvesting,” Appl. Phys. Lett. 110, 083901 (2012).
[Crossref]

O. Tsilipakos, A. C. Tasolamprou, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Pairing Toroidal and Magnetic Dipole Resonances in Elliptic Dielectric Rod Metasurfaces for Reconfigurable Wavefront Manipulation in Reflection,” Adv. Opt. Mater. 6, 1800633 (2012).
[Crossref]

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3, 870 (2012).
[Crossref]

P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators,” Nat. Commun. 3, 692 (2012).
[Crossref] [PubMed]

2011 (4)

P. Y. Chen and A. Alù, “Subwavelength imaging using phase-conjugating nonlinear nanoantenna arrays,” Nano Lett. 11, 5514–5518 (2011).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[Crossref] [PubMed]

M. A. Al-Joumayly and N. Behdad, “Wideband planar microwave lenses using sub-wavelength spatial phase shifters,” IEEE Trans. Antennas Propag. 59, 4542–4552 (2011).
[Crossref]

2010 (2)

P. Padilla, A. Muñoz-Acevedo, M. Sierra-Castañer, and M. Sierra-Pérez, “Electronically reconfigurable transmitarray at Ku band for microwave applications,” IEEE Trans. Antennas Propag. 58, 2571–2579(2010).
[Crossref]

C. G. Ryan, M. R. Chaharmir, J. Shaker, J. R. Bray, Y. M. Antar, and A. Ittipiboon, “A wideband transmitarray using dual-resonant double square rings,” IEEE Trans. Antennas Propag. 58, 1486–1493 (2010).
[Crossref]

2009 (1)

A. Alù, “Mantle cloak: Invisibility induced by a surface,” Phys. Rev. B 80, 245115 (2009).
[Crossref]

2008 (1)

M. A Green, “Self-consistent optical parameters of intrinsic silicon at 300 K including temperature coefficients,” Sol. Energy Mater Sol. Cells 92, 1305–1310 (2008).
[Crossref]

2006 (1)

2004 (1)

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martín, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

2001 (1)

J. A. Encinar, “Design of two-layer printed reflectarrays using patches of variable size,” IEEE Trans. Antennas Propag. 49, 1403–1410 (2001).
[Crossref]

1993 (1)

D. M. Pozar and T. A. Metzler, “Analysis of a reflectarray antenna using microstrip patches of variable size,” Electron. Lett. 29, 657–658 (1993).
[Crossref]

1987 (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[Crossref]

1983 (1)

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

1963 (1)

D. Berry, R. Malech, and W. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag. 11, 645–651 (1963).
[Crossref]

1956 (1)

S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Natl. Acad. Sci. India A 44, 247–262 (1956).
[Crossref]

1951 (1)

Aieta, F.

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139 (2017).
[Crossref]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

Al-Joumayly, M. A.

M. A. Al-Joumayly and N. Behdad, “Wideband planar microwave lenses using sub-wavelength spatial phase shifters,” IEEE Trans. Antennas Propag. 59, 4542–4552 (2011).
[Crossref]

Alù, A.

N. M. Estakhri and A. Alù, “Ultra-thin unidirectional carpet cloak and wavefront reconstruction with graded metasurfaces,” IEEE Antennas Wireless Propag. Lett. 13, 1775–1778 (2014).
[Crossref]

J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alù, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013).
[Crossref]

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3, 870 (2012).
[Crossref]

P. Y. Chen and A. Alù, “Subwavelength imaging using phase-conjugating nonlinear nanoantenna arrays,” Nano Lett. 11, 5514–5518 (2011).
[Crossref] [PubMed]

A. Alù, “Mantle cloak: Invisibility induced by a surface,” Phys. Rev. B 80, 245115 (2009).
[Crossref]

Andrén, D.

R. Verre, N. O. Länk, D. Andrén, H. Sípová, and M. Käll, “Large-Scale Fabrication of Shaped High Index Dielectric Nanoparticles on a Substrate and in Solution,” Adv. Opt. Mater. 6, 1701253 (2018).
[Crossref]

Antar, Y. M.

C. G. Ryan, M. R. Chaharmir, J. Shaker, J. R. Bray, Y. M. Antar, and A. Ittipiboon, “A wideband transmitarray using dual-resonant double square rings,” IEEE Trans. Antennas Propag. 58, 1486–1493 (2010).
[Crossref]

Atwater, H. A.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[Crossref] [PubMed]

Aydin, K.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[Crossref] [PubMed]

Baena, J. D.

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martín, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

Bai, B.

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light Sci. Appl. 2, 1–7 (2013).
[Crossref]

Behdad, N.

M. A. Al-Joumayly and N. Behdad, “Wideband planar microwave lenses using sub-wavelength spatial phase shifters,” IEEE Trans. Antennas Propag. 59, 4542–4552 (2011).
[Crossref]

Belkin, M. A.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3, 870 (2012).
[Crossref]

Bengtsson, J.

Berry, D.

D. Berry, R. Malech, and W. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag. 11, 645–651 (1963).
[Crossref]

Berry, M. V.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[Crossref]

Beruete, M.

F. Falcone, T. Lopetegi, M. A. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martín, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

Blanchard, R.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref] [PubMed]

Boltasseva, A.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1232009 (2013).
[Crossref] [PubMed]

Bonache, J.

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A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1232009 (2013).
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P. Padilla, A. Muñoz-Acevedo, M. Sierra-Castañer, and M. Sierra-Pérez, “Electronically reconfigurable transmitarray at Ku band for microwave applications,” IEEE Trans. Antennas Propag. 58, 2571–2579(2010).
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P. Moitra, B. A. Slovick, W. Li, I. I. Kravchencko, D. P. Briggs, S. Krishnamurthy, and J. Valentine, “Large-Scale All-Dielectric Metamaterial Perfect Reflectors,” ACS Photon. 2, 692–698 (2015).
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J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alù, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013).
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Figures (12)

Fig. 1
Fig. 1 Concept and design of the reflective metasurface: (a) Spherical concave mirror (left) and metasurface with the same behavior as a concave mirror (right) reflecting an incoming beam. (b) Gaussian reflective metasurface reflecting a Gaussian beam back onto itself. (c) Top view of the metasurface designed to reflect a circularly polarized Gaussian beam. All the nanofins have the same dimensions and their angle of rotation changes with the square of their distance from the center. (d) Unit cell consisting of a nanofin on top of a substrate. The efficiency of the metasurface can be optimized by adjusting the height (t), length (L), and width (w) of the nanofins and the unit cell size (P). The 2π phase coverage can be obtained by varying the angle of rotation (θ).
Fig. 2
Fig. 2 Optimized hybrid and purely dielectric metasurfaces for reflection of focused beams: (a) Schematic of the hybrid structure with TiO2 nanofins on an Al mirror. (b) Spectra of the cross-polarized reflectance |S31|2 fordifferent sizes of the unit cell P. (c) Phase coverage of the hybrid structure for P = 380 nm at selected wavelengths. (d) Schematic of the fully dielectric metasurface with Si nanofins on top of a Si substrate separated by a SiO2 spacer. (e) Spectra of the cross-polarized reflectance |S31|2 for different unit cell sizes P. (f) Phase coverage of the dielectric metasurface with Si nanofins for the same parameters as in (e) for P = 400 nm at selected wavelengths.
Fig. 3
Fig. 3 Reflection phase response of the nanofins: (a) Phase coverage of the hybrid dielectric-metal structure with TiO2 nanofins for λ = 1064 nm, P = 380 nm, and values of the other parameters as in Fig. 2(b). (b) Relation between the nanofin angle of rotation and the desired reflection phase used to determine the rotation of every individual fin to obtain the reflection phase distribution in (c). Red points correspond to the data from (a) and the blue line to the interpolation of this data. (c) Phase shift distribution required to build a metareflector for a Gaussian mirror of radius 12.57 μm and NA = 0.24 for a wavelength of λ = 1064 nm and focal length of f = 50λ as a function of the distance to the center of the mirror discretized with the unit cell size P = 380 nm in the x and y directions.
Fig. 4
Fig. 4 Simulation results of the hybrid Gaussian metasurface built with TiO2 nanofins on top of an Al substrate, optimized at the trapping wavelength λ = 1064 nm with unit cell size P = 380 nm and the values of the other parameters as in Fig. 2(b). This mirror has a radius of 12.57 μm, and a focal length f = 50λ = 53 μm. Top: Intensity distribution of the incident Gaussian field (Ii) as a function of the distance z scaled with λ (a),at z = 1350 nm (b), at z = 25λ (c) and at the focal distance z = f = 50λ (d). Bottom: Intensity distribution of the cross-polarized reflected field (Ic) as a function of the distance z scaled with λ (e), at z = 1350 nm (f), at z = 25λ (g), and at the focal distance z = f = 50λ (h).
Fig. 5
Fig. 5 Phase coverage of hybrid structures made of Si (a) and TiO2 (b) nanofins of width w = 80 nm, heights t = 200 nm (a) and t = 800 nm (b), and length L = 350 nm patterned on an Al substrate of thickness ts = 300 nm with unit cell size P = 380 nm.
Fig. 6
Fig. 6 Spectra of the cross-polarized reflectance |S31|2 of the dielectric structure made of Si nanofins of width w = 80 nm, height t = 200 nm, lengths L = 250 nm (a), L = 300 nm (b) and L = 350 nm (c) and angle of rotation θ = 0 on top of Si substrate of thickness ts = 700 nm with a SiO2 spacer of thickness td = 300 nm. |S31|2 increases by increasing L or the unit cell size P.
Fig. 7
Fig. 7 Phase coverage of the dielectric structure for the same parameters as in Fig. 6(a) [Fig. 7(a)], Fig. 6(b) [Fig. 7(b)], and Fig. 6(c) [Fig. 7(c)] but keeping the unit cell size fixed to P = 400 nm and for wavelength values within the plateau of constant reflectance |S31|2. The radial coordinate is |S31|2 and the angular coordinate is the phase angle of S31.
Fig. 8
Fig. 8 Spectra of the [(a)-(c)] cross- and [(d)-(f)] co-polarized reflected and [(g)-(i)] cross- and [(j)-(l)] co- polarized transmitted intensities, and [(m)-(o)] absortion loss for [(a), (d), (g), (j), (m)] the dielectric structure, [(b), (e), (h), (k), (n)] the dielectric structure without the Si substrate and [(c), (f), (i), (l), (o)] the standalone Si nanofin in air, for the parameter values as in Fig. 6. The outliers at the smallest wavelengths are due to a too coarse mesh—a finer mesh is not possible with the available computer resources. For wavelengths above 650 nm, the accuracy is better than 1%.
Fig. 9
Fig. 9 Intensity of the (a) incident and (b) cross-polarized reflected field at the initial plane z = 1350 nm used in the 2-step method for the hybrid structure with TiO2 nanofins on Al with parameter values as in Fig. 4 of the main text.
Fig. 10
Fig. 10 Simulation results of the dielectric mirror built with Si nanofins on top of a poly-Si substrate with a SiO2 spacer, optimized at the typical trapping wavelength λ = 700 nm with the values of the other parameters as in Fig. 7(b). This mirror has a radius of 8.2 μm and a focal length f = 35 μm, giving a numerical aperture NA = 0.23. Top: Intensity distribution of the incident field (Ib) as a function of the distance z scaled with λ (a), at z = 550 nm (b), at z = 25λ (c) and at the focal distance z = f = 50λ (d). Bottom: Intensity distribution of the cross-polarized reflected field (Ic) as a function of the distance z scaled with λ (e), at z = 550 nm (f), at z = 25λ (g) and at the focal distance z = f = 50λ (h).
Fig. 11
Fig. 11 Simulation results of the hybrid Gaussian mirror built with Si nanofins on top of an Al substrate, optimized at the typical trapping wavelength λ = 1064 nm with the values of the other parameters as in Fig. 5(a). This mirror has a radius of 12.57 μm and a focal length f = 50λ, giving a numerical aperture NA = 0.24. Top: Intensity distribution of the incident Gaussian field (Ii) as a function of the distance z scaled with λ(a), at z = 850 nm (b), at z = 25λ (c) and at the focal distance z = f = 50λ (d). Bottom: Intensity distribution of the cross-polarized reflected field (Ic) as a function of the distance z scaled with λ (e), at z = 850 nm (f), at z = 25λ (g) and at the focal distance z = f = 50λ (h).
Fig. 12
Fig. 12 Phase of the ideal Gaussian field (blue line) and cross-polarized reflected field (red dotted line) at the plane z = 1350 nm for the hybrid structure with TiO2 nanofins on Al with parameter values as in Fig. 4 of the main text.

Equations (11)

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E ( ρ , ϕ , z ) = E 0 ( z ) exp  ( ρ 2 w 2 ( z ) ) exp  ( i k ρ 2 2 R ( z ) ) exp  ( i ψ ( z ) ) ,
Φ inc ( ρ , ϕ , z ) = A ( z ) k ρ 2 + Φ inc ( ρ = 0 , ϕ , z ) ,
R ( z ) = z [ 1 + ( z 0 z ) 2 ]
Φ refl ( ρ , ϕ , z ) = k ρ 2 ( A ( z ) 1 / R ) + Φ inc ( ϕ , z ) = k ρ 2 ( A ( z ) ) + Φ inc ( ϕ , z ) .
A ( z ) = A ( z ) 1 R .
1 R ( z ) = 1 R ( z ) 2 R .
Φ mirror = k ρ 2 R ( z ) .
R ( z ) = a z [ 1 + ( z 0 / z ) 2 ] = b R ( z ) ,
a b = z [ 1 + ( z 0 / z ) 2 ] z [ 1 + ( z 0 / z ) 2 ] .
R = 2 b b 1 R ( z ) = 2 a b 1 z [ 1 + ( z 0 / z ) 2 ] .
Φ mirror = ( 1 b ) k ρ 2 2 b R ( z ) = ( 1 b ) k ρ 2 2 a z [ 1 + ( z 0 / z ) 2 ] .

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