Goos–H&#228;nchen shifts in reflective phase-gradient-produced metasurfaces

Junxian Shi; Jingshan Qi; Linyong Qian; Caiqin Han; Changchun Yan

doi:10.3788/COL201816.061602

Journals >Chinese Optics Letters >Volume 16 >Issue 6 >Page 061602 > Article

Chinese Optics Letters
Vol. 16, Issue 6, 061602 (2018)

Goos–Hänchen shifts in reflective phase-gradient-produced metasurfaces

Junxian Shi, Jingshan Qi, Linyong Qian, Caiqin Han, and Changchun Yan^*

Author Affiliations

Jiangsu Key Laboratory of Advanced Laser Materials and Devices, School of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, China

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DOI: 10.3788/COL201816.061602 Cite this Article Set citation alerts

Junxian Shi, Jingshan Qi, Linyong Qian, Caiqin Han, Changchun Yan. Goos–Hänchen shifts in reflective phase-gradient-produced metasurfaces[J]. Chinese Optics Letters, 2018, 16(6): 061602 Copy Citation Text

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Abstract

We studied Goos–H nchen (GH) shifts on a reflective phase-gradient-produced metasurface. Their analytical solutions were achieved for both TE and TM polarizations utilizing the generalized Snell’s law. The calculated results show that the GH shifts are evidently affected by phase gradients and incident angles, which means that a certain range of GH shifts can be realized as long as an incident angle, phase gradient, and frequency are properly chosen. This offers an effective method for the control of GH shifts.

E^{(1)} (x, y, t) = E_{z}^{0} \exp (- i ω t) {\exp {i [k_{x}^{(1)} x + k_{y}^{(1)} y]} + R_{TE} \exp {i [k_{x}^{(2)} x + k_{y}^{(2)} y]}},

(1)

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E^{(2)} (x, y, t) = T_{TE} E_{z}^{0} \exp {i [k_{x}^{(3)} x + k_{y}^{(3)} y - ω t]},

(2)

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k_{y}^{(1) *} = k_{y}^{(1)} + \frac{d ϕ}{d y} = k_{y}^{(2) *} = k_{y}^{(3) *} .

(3)

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1 + R_{TE} = T_{TE},

(4)

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\frac{1}{μ_{1}} [k_{x}^{(1)} + R_{TE} k_{x}^{(2)}] = \frac{1}{μ_{2}} T_{TE} k_{x}^{(3)} .

(5)

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R_{TE} = \frac{μ_{2} k_{x}^{(1)} - μ_{1} k_{x}^{(3)}}{μ_{2} k_{x}^{(1)} + μ_{1} k_{x}^{(3)}} = r_{TE} \exp (i δ_{TE}),

(6)

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θ_{c} = \arcsin (\pm \frac{n_{2}}{n_{1}} - \frac{λ_{0}}{2 π n_{1}} \frac{d ϕ}{d y}) .

(7)

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Y_{TE}^{*} = 2 μ_{1} μ_{2} \frac{k_{y}^{(1)} ω^{2} ε_{1} μ_{1} - k_{y}^{(1)} ω^{2} ε_{2} μ_{2} + c ω^{2} ε_{1} μ_{1} + c^{2} k_{y}^{(1)} + c k_{y}^{(1) 2}}{μ_{2}^{2} [ω^{2} ε_{1} μ_{1} - k_{y}^{(1) 2}] + μ_{1}^{2} {{[k_{y}^{(1)} + c]}^{2} - ω^{2} ε_{2} μ_{2}}} \times \frac{1}{\sqrt{{[k_{y}^{(1)} + c]}^{2} - ω^{2} ε_{2} μ_{2}} \sqrt{ω^{2} ε_{1} μ_{1} - k_{y}^{(1) 2}}},

(8)

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Y_{TM}^{*} = 2 ε_{1} ε_{2} \frac{k_{y}^{(1)} ω^{2} ε_{1} μ_{1} - k_{y}^{(1)} ω^{2} ε_{2} μ_{2} + c ω^{2} ε_{1} μ_{1} + c^{2} k_{y}^{(1)} + c k_{y}^{(1) 2}}{ε_{2}^{2} [ω^{2} ε_{1} μ_{1} - k_{y}^{(1) 2}] + ε_{1}^{2} {{[k_{y}^{(1)} + c]}^{2} - ω^{2} ε_{2} μ_{2}}} \times \frac{1}{\sqrt{{[k_{y}^{(1)} + c]}^{2} - ω^{2} ε_{2} μ_{2}} \sqrt{ω^{2} ε_{1} μ_{1} - k_{y}^{(1) 2}}} .

(9)

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η = \frac{Y^{*} - Y_{0}}{Y_{0}},

(10)

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Junxian Shi, Jingshan Qi, Linyong Qian, Caiqin Han, Changchun Yan. Goos–Hänchen shifts in reflective phase-gradient-produced metasurfaces[J]. Chinese Optics Letters, 2018, 16(6): 061602

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