Researching | Space–time cloaks through birefringent Goos–Hänchen shifts

Journals >Chinese Optics Letters >Volume 17 >Issue 3 >Page 032701 > Article

Chinese Optics Letters
Vol. 17, Issue 3, 032701 (2019)

Space–time cloaks through birefringent Goos–Hänchen shifts

Humayun Khan¹, Muhammad Haneef^1、*, and Bakhtawar^1、2

Author Affiliations

¹Laboratory of Theoretical Physics, Department of Physics, Hazara University Mansehra, 21300, Pakistan

²Department of Physics, Women University Mardan, 23200, Pakistan

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DOI: 10.3788/COL201917.032701 Cite this Article Set citation alerts

Humayun Khan, Muhammad Haneef, Bakhtawar. Space–time cloaks through birefringent Goos–Hänchen shifts[J]. Chinese Optics Letters, 2019, 17(3): 032701 Copy Citation Text

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Abstract

We report a theoretical demonstration for the creation of space–time holes based on birefringence of reflection, transmission, and the Goos–H chen (GH) shifts from a chiral medium. We observed space–time holes in the reflection, transmission, and their corresponding GH-shifted beams. Two space–time holes are clearly detected in the regions of

0 < t \leq 5 τ_{0}

and

5 w \leq y \leq 5 w

, as well as in the regions of

5 τ_{0} \leq t \leq 0

and

5 w \leq y \leq 5 w

. These space–time holes hide objects and information contents from observers and hackers. The objects and information contents are completely undetectable, and thus events can be cloaked. The results of this paper have potential applications in the invisibility of drone technology and secure communication of information in telecom industries.

H_{0} = \sum_{i = 1}^{4} ℏ ω_{i} | i 〉 〈 i | .

(1)

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H_{I} = \sum_{i = 1}^{4} ℏ | i 〉 〈 i | - \frac{ℏ}{2} [Ω_{c} e^{- i ω_{c} t} | 4 〉 〈 3 | + Ω_{p} e^{- i ω_{p} t} | 4 〉 〈 2 | + Ω_{m} e^{- i ω_{m} t} | 3 〉 〈 1 | + h . c .],

(2)

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\dot{ρ} = - \frac{i}{ℏ} [H_{i}, ρ] - \sum \frac{1}{2} γ_{m n} (a^{†} a ρ + ρ a^{†} a - 2 a ρ a^{†}),

(3)

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{\dot{\tilde{ρ}}}_{31} = (Δ_{p} - \frac{1}{2} γ_{b}) {\tilde{ρ}}_{31} - \frac{i}{2} Ω_{c}^{*} {\tilde{ρ}}_{41} - \frac{i}{2} Ω_{m} ({\tilde{ρ}}_{11} - {\tilde{ρ}}_{33}),

(4)

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{\dot{\tilde{ρ}}}_{41} = [i Δ_{p} - (\frac{1}{2} γ_{1} + γ_{e})] {\tilde{ρ}}_{41} - \frac{i}{2} Ω_{c} {\tilde{ρ}}_{31} - \frac{i}{2} Ω_{p} {\tilde{ρ}}_{21} + \frac{i}{2} Ω_{m} {\tilde{ρ}}_{43},

(5)

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{\dot{\tilde{ρ}}}_{42} = [Δ_{p} - \frac{i}{2} (γ_{e} + γ_{1})] {\tilde{ρ}}_{42} - \frac{i}{2} Ω_{c} {\tilde{ρ}}_{32} - \frac{i}{2} Ω_{p} ({\tilde{ρ}}_{22} - {\tilde{ρ}}_{44}),

(6)

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{\dot{\tilde{ρ}}}_{32} = (i Δ_{p} - \frac{1}{2} γ_{b}) {\tilde{ρ}}_{32} - \frac{i}{2} Ω_{c}^{*} {\tilde{ρ}}_{42} - \frac{i}{2} Ω_{m} {\tilde{ρ}}_{12} .

(7)

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{\tilde{ρ}}_{31}^{(1)} = \frac{Ω_{m}}{2 W_{1}} {\tilde{ρ}}_{11}^{(0)} + \frac{Ω_{p}}{4 W_{1}} \frac{Ω_{c}^{*}}{Δ_{p} - i (γ_{1} + γ_{e}) / 2} {\tilde{ρ}}_{21}^{(0)},

(8)

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{\tilde{ρ}}_{42}^{(1)} = \frac{Ω_{p}}{2 W_{2}} {\tilde{ρ}}_{22}^{(0)} + \frac{Ω_{m}}{4 W_{2}} \frac{Ω_{c}}{Δ_{p} - i γ_{b} / 2} {\tilde{ρ}}_{12}^{(0)},

(9)

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W_{1} = Δ_{p} - \frac{i γ_{b}}{2} - \frac{{| Ω_{c} |}^{2}}{4 (Δ_{p} - i \frac{γ_{1} + γ_{e}}{2})},

(10)

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W_{2} = Δ_{p} - \frac{i (γ_{e} + γ_{1})}{2} - \frac{{| Ω_{c} |}^{2}}{4 (Δ_{p} - i \frac{γ_{b}}{2})} .

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| ψ 〉 = \sqrt{x} | 2 〉 + \sqrt{1 - x} e^{i φ} | 1 〉 .

(12)

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α_{E E} = \frac{N ϱ_{42}^{2} {\tilde{ρ}}_{22}^{(0)}}{ℏ W_{2}},

(13)

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α_{B B} = \frac{N μ_{31}^{2} {\tilde{ρ}}_{11}^{(0)}}{ℏ W_{1}},

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α_{B E} = \frac{N μ_{31} ϱ_{42} | Ω_{c} | e^{i (φ_{2} - φ_{1})} ρ_{21}^{(0)} e^{- i (φ + φ_{c})}}{2 ℏ W_{1} [Δ_{p} - \frac{i (γ_{1} + γ_{e})}{2}]},

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α_{E B} = \frac{N μ_{31} ϱ_{42} | Ω_{c} | e^{i (φ_{1} - φ_{2})} ρ_{21}^{(0)} e^{i (φ + φ_{c})}}{2 ℏ W_{1} [Δ_{p} - \frac{i (γ_{1} + γ_{e})}{2}]},

(16)

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χ_{e} = \frac{α_{E E} + μ_{0} (α_{E B} α_{B E} - α_{B B} α_{E E})}{ϵ_{0} (1 - μ_{0} α_{B B})},

(17)

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χ_{m} = \frac{μ_{0} β_{B B}}{1 - μ_{0} β_{B B}},

(18)

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ξ_{E H} = \frac{c μ_{0} β_{E B}}{1 - μ_{0} β_{B B}},

(19)

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ξ_{H E} = \frac{c μ_{0} β_{B E}}{1 - μ_{0} β_{B B}} .

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n_{r}^{(\pm)} = \sqrt{(1 + χ_{e}) (1 + χ_{m}) - \frac{{(ξ_{E H} + ξ_{H E})}^{2}}{4}} \pm i \frac{ξ_{E H} - ξ_{H E}}{2} .

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R^{\pm} = \frac{\cos α^{\pm} (Q_{0}^{2} - Q_{1}^{2}) Q_{1} Q^{\pm} \sin 2 α_{1} + H_{3} \sin α^{\pm}}{Q_{1} Q_{2} H_{1} \cos α^{\pm} + H_{2} \sin α^{\pm}},

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S_{(t, r)}^{\pm} = - \frac{λ}{2 π | T^{\pm}, R^{\pm} |^{2}} [R e (T^{\pm}, R^{\pm}) \frac{\partial}{\partial k_{y}} I m (T^{\pm}, R^{\pm}) - I m (T^{\pm}, R^{\pm}) \frac{\partial}{\partial k_{y}} R e (T^{\pm}, R^{\pm})] .

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E_{i} {(y, t) |}_{z = 0} = \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} A (k_{y}, Δ_{p}) e^{i (k_{z} z + k_{y} y)} e^{- i Δ_{p} t} d k_{y} d Δ_{p},

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A (k_{y}, Δ_{p}) = \frac{W_{y}}{\sqrt{2 τ_{0}}} e^{- \frac{W_{y}^{2} {(k_{y} - k_{y 0})}^{2}}{4}} e^{- \frac{τ_{0}^{2} Δ_{p}^{2}}{4}} .

(25)

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E_{t} (y, t) = \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} T (k_{y}) A (k_{y}, Δ_{p}) e^{i [k_{z} (z - L) + k_{y} y]} e^{- i Δ_{p} (t - t^{(\pm)})} d k_{y} d Δ_{p},

(26)

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E_{r} (y, t) = \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} R (k_{y}) A (k_{y}, Δ_{p}) e^{i (- k_{z} z + k_{y} y)} e^{- i Δ_{p} (t - t^{(\pm)})} d k_{y} d Δ_{p},

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Humayun Khan, Muhammad Haneef, Bakhtawar. Space–time cloaks through birefringent Goos–Hänchen shifts[J]. Chinese Optics Letters, 2019, 17(3): 032701

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