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Journals >Photonics Research >Volume 9 >Issue 12 >Page 2486 > Article

Photonics Research
Vol. 9, Issue 12, 2486 (2021)

Theory of light propagation in arbitrary two-dimensional curved space

Chenni Xu and Li-Gang Wang^*

Author Affiliations

Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou 310027, China

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DOI: 10.1364/PRJ.435993 Cite this Article Set citation alerts

Chenni Xu, Li-Gang Wang. Theory of light propagation in arbitrary two-dimensional curved space[J]. Photonics Research, 2021, 9(12): 2486 Copy Citation Text

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Abstract

As an analog model of general relativity, optics on some two-dimensional (2D) curved surfaces has received increasing attention in the past decade. Here, in light of the Huygens–Fresnel principle, we propose a theoretical frame to study light propagation along arbitrary geodesics on any 2D curved surfaces. This theory not only enables us to solve the enigma of “infinite intensity” that existed previously at artificial singularities on surfaces of revolution but also makes it possible to study light propagation on arbitrary 2D curved surfaces. Based on this theory, we investigate the effects of light propagation on a typical surface of revolution, Flamm’s paraboloid, as an example, from which one can understand the behavior of light in the curved geometry of Schwarzschild black holes. Our theory provides a convenient and powerful tool for investigations of radiation in curved space.

d s^{2} = g_{i j} d x^{i} d x^{j} = [1 + {(\frac{\partial H}{\partial x})}^{2}] d x^{2} + [1 + {(\frac{\partial H}{\partial y})}^{2}] d y^{2} + 2 \frac{\partial H}{\partial x} \frac{\partial H}{\partial y} d x d y,

(1)

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\frac{d^{2} x}{d s^{2}} + Γ_{x x}^{x} {(\frac{d x}{d s})}^{2} + Γ_{y y}^{x} {(\frac{d y}{d s})}^{2} + 2 Γ_{x y}^{x} \frac{d x}{d x} \frac{d y}{d s} = 0,

(2)

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\frac{d^{2} y}{d s^{2}} + Γ_{x x}^{y} {(\frac{d x}{d s})}^{2} + Γ_{y y}^{y} {(\frac{d y}{d s})}^{2} + 2 Γ_{x y}^{y} \frac{d x}{d s} \frac{d y}{d s} = 0 .

(3)

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Φ_{o} (P_{o}) = \sqrt{\frac{1}{i λ}} \int_{Σ_{i}} Φ_{i} (S_{i}) \frac{e^{i k L (S_{i}, P_{o})}}{L (S_{i}, P_{o})} K (S_{i}, P_{o}) d l,

(4)

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d φ = \pm \frac{κ}{r \sqrt{r^{2} - κ^{2}}} \sqrt{1 + {(\frac{d H}{d r})}^{2}} d r,

(5)

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d s^{2} = - (1 - \frac{r_{s}}{r}) c^{2} d t^{2} + {(1 - \frac{r_{s}}{r})}^{- 1} d r^{2} + r^{2} d ψ^{2} + r^{2} \sin^{2} ψ d φ^{2} .

(6)

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I (u, v) = \frac{1}{\cos (\frac{u}{R}) {[1 + R^{2} \tan^{2} (\frac{u}{R}) / z_{r}^{2}]}^{1 / 2}} \times \exp [- \frac{k z_{r} R^{2} v^{2}}{R^{2} \tan^{2} (\frac{u}{R}) + z_{r}^{2}}],

(7)

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σ (u) = σ_{0} \sqrt{\cos^{2} (\frac{u}{R}) + \frac{R^{2}}{z_{r}^{2}} \sin^{2} (\frac{u}{R})},

(8)

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d s^{2} = [1 + {(\frac{d H}{d r})}^{2}] d r^{2} + r^{2} d φ^{2} .

(A1)

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\frac{d^{2} r}{d s^{2}} + \frac{\frac{d H}{d r}}{1 + {(\frac{d H}{d r})}^{2}} \frac{d^{2} H}{d r^{2}} {(\frac{d r}{d s})}^{2} - \frac{r}{1 + {(\frac{d H}{d r})}^{2}} {(\frac{d φ}{d s})}^{2} = 0,

(A2)

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\frac{d^{2} φ}{d s^{2}} + \frac{2}{r} \frac{d r}{d s} \frac{d φ}{d s} = 0 .

(A3)

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\frac{d φ}{d s} = \frac{κ}{r^{2}},

(A4)

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\frac{d r}{d s} = \pm \frac{1}{\sqrt{1 + {(\frac{d H}{d r})}^{2}}} \sqrt{1 - \frac{κ^{2}}{r^{2}}} .

(A5)

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\frac{d φ}{d r} = \pm \frac{κ}{r^{2} \sqrt{1 - \frac{κ^{2}}{r^{2}}}} \sqrt{1 + {(\frac{d H}{d r})}^{2}} .

(A6)

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\frac{1}{\sqrt{g}} \partial_{i} (\sqrt{g} g^{i j} \partial_{j} Φ) + k^{2} Φ = 0,

(C1)

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\frac{\partial^{2} Φ}{\partial u^{2}} - \frac{\tan (\frac{u}{R})}{R} \frac{\partial Φ}{\partial u} + \frac{1}{R^{2} \cos^{2} (\frac{u}{R})} \frac{\partial^{2} Φ}{\partial v^{2}} + k^{2} Φ = 0,

(C2)

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\frac{\partial^{2} Ψ}{\partial u^{2}} + \frac{1}{R^{2} \cos^{2} (\frac{u}{R})} \frac{\partial^{2} Ψ}{\partial v^{2}} + k_{eff}^{2} Ψ = 0,

(C3)

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\frac{\partial^{2} Ξ}{\partial u^{2}} ≪ 2 k \frac{\partial Ξ}{\partial u},

(C4)

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2 i k \frac{\partial ϕ}{\partial u} + \frac{1}{R^{2} \cos^{2} (\frac{u}{R})} \frac{\partial^{2} ϕ}{\partial v^{2}} = 0 .

(C5)

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ϕ (u, v) = \exp [i α (u) + \frac{i k v^{2}}{2 β (u)}],

(C6)

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- 2 k \frac{\partial α (u)}{\partial u} + \frac{1}{R^{2} \cos^{2} (\frac{u}{R})} \frac{i k}{β (u)} = 0,

(C7)

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\frac{\partial β (u)}{\partial u} - \frac{1}{R^{2} \cos^{2} (\frac{u}{R})} = 0 .

(C8)

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β (u) = \frac{1}{R} \tan (\frac{u}{R}) - \frac{i z_{r}}{R^{2}},

(C9)

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α (u) = i {\frac{1}{4} \ln [\frac{z_{r}^{2} + R^{2} \tan^{2} (\frac{u}{R})}{z_{r}^{2}}] + \frac{i}{2} \arctan [\frac{R \tan (\frac{u}{R})}{z_{r}}]},

(C10)

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Φ (u, v) = {[\cos (\frac{u}{R})]}^{\frac{1}{2}} {[\frac{z_{r}^{2} + R^{2} \tan^{2} (\frac{u}{R})}{z_{r}^{2}}]}^{\frac{1}{4}} \times \exp {- \frac{k z_{r} R^{2} v^{2}}{2 [z_{r}^{2} + R^{2} \tan^{2} (\frac{u}{R})]}} \times \exp {- \frac{i}{2} \arctan [\frac{R \tan (\frac{u}{R})}{z_{r}}]} \times \exp [\frac{i k R^{2} v^{2}}{2} \frac{R \tan (\frac{u}{R})}{R^{2} \tan^{2} (\frac{u}{R}) + z_{r}^{2}}] \times \exp [\frac{i}{2 k} \int Δ (u^{'}) d u^{'}] \exp (i k u),

(C11)

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I (u, v) = Φ^{*} (u, v) Φ (u, v) = {[\cos (\frac{u}{R})]}^{- 1} {[1 + \frac{R^{2} \tan^{2} (\frac{u}{R})}{z_{r}^{2}}]}^{- \frac{1}{2}} \times \exp [- \frac{k z_{r} R^{2} v^{2}}{z_{r}^{2} + R^{2} \tan^{2} (\frac{u}{R})}]

(C12)

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σ (u) = σ_{0} \sqrt{\cos^{2} (\frac{u}{R}) + \frac{R^{2}}{z_{r}^{2}} \sin^{2} (\frac{u}{R})} .

(C13)

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Chenni Xu, Li-Gang Wang. Theory of light propagation in arbitrary two-dimensional curved space[J]. Photonics Research, 2021, 9(12): 2486

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