Exploitation of geometric and propagation phases for spin-dependent rational-multiple complete phase modulation using dielectric metasurfaces

Ata Ur Rahman Khalid; Fu Feng; Naeem Ullah; Xiaocong Yuan; Michael Geoffrey Somekh

doi:10.1364/PRJ.439094

Journals >Photonics Research >Volume 10 >Issue 4 >Page 877 > Article

Photonics Research
Vol. 10, Issue 4, 877 (2022)

Exploitation of geometric and propagation phases for spin-dependent rational-multiple complete phase modulation using dielectric metasurfaces

Ata Ur Rahman Khalid^1、†, Fu Feng^1、†, Naeem Ullah², Xiaocong Yuan^1、4、*, and Michael Geoffrey Somekh^{1、3、5、*}

Author Affiliations

¹Nanophotonics Research Center, Shenzhen Key Laboratory of Micro-Scale Optical Information Technology & Institute of Microscale Optoelectronics, Shenzhen University, Shenzhen 518060, China

²Beijing Engineering Research Center for Mixed Reality and Advanced Display, School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China

³Faculty of Engineering, University of Nottingham, Nottingham NG7 2RD, UK

⁴e-mail: xcyuan@szu.edu.cn

⁵e-mail: mike.somekh@szu.edu.cn

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DOI: 10.1364/PRJ.439094 Cite this Article Set citation alerts

Ata Ur Rahman Khalid, Fu Feng, Naeem Ullah, Xiaocong Yuan, Michael Geoffrey Somekh. Exploitation of geometric and propagation phases for spin-dependent rational-multiple complete phase modulation using dielectric metasurfaces[J]. Photonics Research, 2022, 10(4): 877 Copy Citation Text

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Schematic illustration of a unit cell for rational-multiple phase modulation through merged propagation and geometric phases. For RCP (±σ) light, the silicon meta-atom of H=420 nm rotated at θ(x,y) on glass substrate transfers a ∓2σθ(x,y) geometric phase shift on the cross-polarization channel, i.e., LCP (∓σ) light (left side). The sign of the geometric phase gets reversed, i.e., ±2θ(x,y), while switching the input polarization, i.e., LCP (∓σ) (right side). The propagation phase ±nθ(x,y) can be merged with the PB phase, which yields to rational-multiple complete phase modulation on either RCP or LCP light. In the 2D representation (in the middle), Px=Py=400 nm is the pitch of the unit cell, x and y are the global coordinates, u and v are the local coordinates of the meta-atom, θ(x,y) is the angle between u and x, and L1 and L2 are the spatially varying dimensions of the meta-atom. Herein, the additional parameters +σ and −σ are selected as right and left circular polarization, respectively.

Fig. 1. Schematic illustration of a unit cell for rational-multiple phase modulation through merged propagation and geometric phases. For RCP (±σ) light, the silicon meta-atom of H=420 nm rotated at θ(x,y) on glass substrate transfers a ∓2σθ(x,y) geometric phase shift on the cross-polarization channel, i.e., LCP (∓σ) light (left side). The sign of the geometric phase gets reversed, i.e., ±2θ(x,y), while switching the input polarization, i.e., LCP (∓σ) (right side). The propagation phase ±nθ(x,y) can be merged with the PB phase, which yields to rational-multiple complete phase modulation on either RCP or LCP light. In the 2D representation (in the middle), Px=Py=400 nm is the pitch of the unit cell, x and y are the global coordinates, u and v are the local coordinates of the meta-atom, θ(x,y) is the angle between u and x, and L1 and L2 are the spatially varying dimensions of the meta-atom. Herein, the additional parameters +σ and −σ are selected as right and left circular polarization, respectively.

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Amplitude and phase distribution of the fundamental unit cell for the linearly orthogonal input polarizations. From electric fields Exx and Eyy, the transmission amplitude (top) and phase (bottom) are obtained from parametric sweep of L1 and L2 ranging from 50 to 250 nm. These results indicate the high amplitude as well as complete 2π phase coverage at operating frequency.

Fig. 2. Amplitude and phase distribution of the fundamental unit cell for the linearly orthogonal input polarizations. From electric fields Exx and Eyy, the transmission amplitude (top) and phase (bottom) are obtained from parametric sweep of L1 and L2 ranging from 50 to 250 nm. These results indicate the high amplitude as well as complete 2π phase coverage at operating frequency.

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Fig. 3. Design and operating principle of spin-selective beam deflector and splitter. For RCP (+σ) incident light, the structural parameters are obtained when propagation phase ϕ(x,y)=+6θ(x,y) is combined with geometric phase and meta-atoms are arranged in a super-cell with the phase difference of π/2 (left side). This design works as beam deflector that deflects cross-polarized light at a certain angle θ. For the LCP (−σ), twice the phase multiplication with each of meta-atoms (excluding the fundamental one) turns the unit cells into binary unit cells with phase delay of π, which splits the light (right side).

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Fig. 4. Numerically computed results for beam deflection and splitting. (a) Under the incidence of RCP and LCP light on the meta design, the transmitted electric fields ELCP (left side) and ELCP (right side) after passing through metasurface array in the x–z plane clearly demonstrate the polarization-switchable beam deflection and splitting phenomena. (b) Far-field results of spin-selective beam deflector and splitter. The circularly cross-polarized light deflects under RCP incidence and splits under LCP incidence. The left arrow in the middle indicates incident polarization, and the right arrow indicates the polarization handedness at output.

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Fig. 5. Schematic of the working principle of the rational-multiple spin-switchable SOC. For RCP incoming light, the LCP light emerging from the back side of the design can carry OAM with any arbitrary value of topological charge value ℓ. By switching the incident light to LCP, the predesigned device can switch the topological charge value multiple of m depending upon the design feature. In this representation, the design is composed of silicon meta-atoms with azimuthal angle α and radius r on the glass substrate.

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Fig. 6. Numerical simulation results for polarization-switchable rational-multiple SOC designs. Five different designs with different structural parameters for OAM mode with topological charge value ℓ1=2 are implemented; the specific design can switch to 1.5, 2, 2.5, 3, or −3 times of ℓ1 by switching the handedness of incident light. (a), (c), (e), (g), (i) Near-field amplitude and phase of each design and far-field result of each design; (b), (d), (f), (h), (j) their corresponding rational-multiple results.

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Fig. 7. Phase modulations when the meta-atom is rotated at θ=π/8. (a) The PB phase provides equal phase modulation with opposite sign; (b) 3 times phase modulation when n=4, and (c) 3 times phase modulation with opposite sign when n=1.

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Serial No.	Propagation/Structural/Retardation/Dynamic Phase and Geometric Phase	Phase Modulation
1	$ϕ_{R C P \to L C P} (x, y) = 0 - 2 θ (x, y) = - 2 θ (x, y)$
	$ϕ_{L C P \to R C P} (x, y) = 0 + 2 θ (x, y) = 2 θ (x, y)$	Equal with opposite sign
2	$ϕ_{R C P \to L C P} (x, y) = \pm 0.4 θ (x, y) - 2 θ (x, y) = - 1.6 θ (x, y) / − 2.4 θ (x, y)$
	$ϕ_{L C P \to R C P} (x, y) = \pm 0.4 θ (x, y) + 2 θ (x, y) = 2.4 θ (x, y)) / 1.6 θ (x, y)$	$1.5 \times$ with opposite sign
3	$ϕ_{R C P \to L C P} (x, y) = \pm θ (x, y) - 2 θ (x, y) = - θ (x, y) / − 3 θ (x, y)$
	$ϕ_{L C P \to R C P} (x, y) = \pm θ (x, y) + 2 θ (x, y) = 3 θ (x, y)) / θ (x, y)$	$3 \times$ with opposite sign
4	$ϕ_{R C P \to L C P} (x, y) = \pm 2 θ (x, y) - 2 θ (x, y) = 0 (x, y) / − 4 θ (x, y)$
	$ϕ_{L C P \to R C P} (x, y) = \pm 2 θ (x, y) + 2 θ (x, y) = 4 θ (x, y)) / 0 (x, y)$	$0 \times$
5	$ϕ_{R C P \to L C P} (x, y) = \pm 2.5 θ (x, y) - 2 θ (x, y) = 0.5 θ (x, y) / − 4.5 θ (x, y)$
	$ϕ_{L C P \to R C P} (x, y) = \pm 2.5 θ (x, y) + 2 θ (x, y) = 4.5 θ (x, y)) / − 0.5 θ (x, y)$	$9 \times$
6	$ϕ_{R C P \to L C P} (x, y) = \pm 3 θ (x, y) - 2 θ (x, y) = θ (x, y) / − 5 θ (x, y)$
	$ϕ_{L C P \to R C P} (x, y) = \pm 3 θ (x, y) + 2 θ (x, y) = 5 θ (x, y)) / − θ (x, y)$	$5 \times$
7	$ϕ_{R C P \to L C P} (x, y) = \pm 4 θ (x, y) - 2 θ (x, y) = 2 θ (x, y) / − 6 θ (x, y)$
	$ϕ_{L C P \to R C P} (x, y) = \pm 4 θ (x, y) + 2 θ (x, y) = 6 θ (x, y)) / − 2 θ (x, y)$	$3 \times$
8	$ϕ_{R C P \to L C P} (x, y) = \pm 4.67 θ (x, y) - 2 θ (x, y) = 2.67 θ (x, y) / − 6.67 θ (x, y)$
	$ϕ_{L C P \to R C P} (x, y) = \pm 4.67 θ (x, y) + 2 θ (x, y) = 6.67 θ (x, y)) / − 2.67 θ (x, y)$	$\sim 2.5 \times$
9	$ϕ_{R C P \to L C P} (x, y) = \pm 6 θ (x, y) - 2 θ (x, y) = 4 θ (x, y) / − 8 θ (x, y)$
	$ϕ_{L C P \to R C P} (x, y) = \pm 6 θ (x, y) + 2 θ (x, y) = 8 θ (x, y)) / − 4 θ (x, y)$	$2 \times$
10	$ϕ_{R C P \to L C P} (x, y) = \pm 10 θ (x, y) - 2 θ (x, y) = 8 θ (x, y) / − 12 θ (x, y)$
	$ϕ_{L C P \to R C P} (x, y) = \pm 10 θ (x, y) + 2 θ (x, y) = 12 θ (x, y)) / − 8 θ (x, y)$	$1.5 \times$