Duplex Mikaelian and Duplex Maxwell's Fish-Eye Lenses

Duplex Mikaelian and Duplex Maxwell’s Fish-Eye Lenses

Huiyan Peng, Senlin Liu, Yuze Wu, Yi Yan, Zichun Zhou, Xiaochao Li, Qiaoliang Bao, Lin Xu, and Huanyang Chen

Phys. Rev. Applied 13, 034050 – Published 19 March 2020

Abstract

Here, we report two kinds of absolute optical instruments that can make stigmatic images for geometric optics in two-dimensional space. One is called the duplex Mikaelian lens, which is made by splicing two half-Mikaelian lenses with different periods. The other is an exponential conformal transformer of a duplex Mikaelian lens with a rational number of the ratio of different periods of its two half-Mikaelian lenses, which we call a duplex Maxwell’s fish-eye lens. Duplex Mikaelian lenses have continuous translation symmetry with an arbitrary real number, while duplex Maxwell’s fish-eye lenses have continuous rotation symmetry from zero to 2π. Hence, each duplex Maxwell’s fish-eye lens corresponds to a duplex Mikaelian lens. We further demonstrate the caustic effect of geometric optics in duplex Mikaelian lenses and duplex Maxwell’s fish-eye lenses. In addition, we investigate the Talbot effect of wave optics in the duplex Mikaelian lens based on numeric calculations. Our findings, based on splicing and exponential conformal mapping, enlarge the family of absolute optical instruments.

Received 16 October 2019
Revised 28 December 2019
Accepted 3 March 2020

DOI:https://doi.org/10.1103/PhysRevApplied.13.034050

Physics Subject Headings (PhySH)

Geometrical & wave optics Imaging & optical processing Refraction Transformation optics

Devices

Optical conformal mapping

Atomic, Molecular & Optical

Authors & Affiliations

Huiyan Peng ^1,2,3, Senlin Liu ^1,2, Yuze Wu ^1,2, Yi Yan ^1,2, Zichun Zhou ^1,2, Xiaochao Li^1,2, Qiaoliang Bao⁴, Lin Xu ^3,†, and Huanyang Chen^1,2,*

¹Institute of Electromagnetics and Acoustics and Key Laboratory of Electromagnetic Wave Science and Detection Technology, Xiamen University, Xiamen 361005, China
²Department of Electrical and Electronics Engineering, Xiamen University Malaysia, 43900 Sepang, Selangor, Malaysia
³Institutes of Physical Science and Information Technology and Key Laboratory of Opto-Electronic Information Acquisition and Manipulation of Ministry of Education, Anhui University, Hefei 230601, China
⁴Department of Materials Science and Engineering, and ARC Centre of Excellence in Future Low-Energy Electronics Technologies (FLEET), Monash University, Clayton, Victoria 3800, Australia

^†xuin@ahu.edu.cn
^*kenyon@xmu.edu.cn

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Issue

Vol. 13, Iss. 3 — March 2020

Subject Areas

Optics

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Images

Figure 1
Gradient refractive index and light trajectories in the Mikaelian lens $| E z |$ and the corresponding Maxwell’s fish-eye lens. (a) Mikaelian lens with ${p = 2}$ is in z space, which is divided into infinite number ribbon regions (marked with integer $j$ ), with a reference period length $L = 2 π a$ between dashed lines. Its gradient refractive index, written as Eq. (1), is shown with the contour plot in complex plane $z = x + y i$ . Two red and blue light rays launch from the red source at position coordinate $(0, 0)$ . In the reference period length, the red source can make three more images marked with two blue points and a red point. (b) Schematic plot of the Riemann surface of exponential conformal mapping. Branch cuts are denoted by dashed lines, which are mapped from the dashed lines of z space. Chiral rotated surface is used to lift a third space dimension for a better illustration of branch cuts. (c) Corresponding gradient refractive index and light trajectories in the Maxwell’s fish-eye lens, which is mapped from the ribbon region marked with $j = 1$ of the part “a” figure by exponential conformal mapping. Indeed, it is also the projection from multiple w space of the Riemann surface of (b). Radius a denoted with a black circle is mapped from the y axis of z space. (d) Mikaelian lens with ${p = 2/3}$ is in z space. Two additional purple lines denote boundaries of the large period. (e) Light rays in the Riemann surface of Maxwell’s fish-eye lens with ${p = 2/3}$ . (f) Light rays in Maxwell’s fish-eye lens with ${p = 2/3}$ .
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Figure 2
Gradient refractive index and light trajectories in the duplex Mikaelian lens with ${p, q}$ and the corresponding duplex Maxwell’s fish-eye lens. (a) Mikaelian lens with ${2, 1}$ in z space. Its gradient refractive index is without mirror symmetry along the y axis, as shown with the contour plot. During the reference period length $L = 2 π a$ , two red and blue light rays launch from the red source at position coordinate $(0, 0)$ in the region marked with $j = 1$ , which can make two more images. (b) Schematic plot of the Riemann surface and light trajectories of the duplex Maxwell’s fish-eye lens with ${2, 1}$ . (c) Projection to w space of the Riemann surface of (b). (d) Mikaelian lens with ${1, 2}$ in z space. (e) Schematic plot of the Riemann surface and light trajectories of the duplex Maxwell’s fish-eye lens with ${1, 2}$ . (f) Projection to w space of the Riemann surface of (e).
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Figure 3
(a) Colored light rays in the Mikaelian lens with ${p = 1}$ emitted from and focused on red and blue points, which are off the y axis. (b) Colored light rays in the Mikaelian lens with ${p = 2}$ emitted from and focused on red and blue points, which are off the y axis. (c) Colored light rays in the duplex Mikaelian lens with ${p = 2, q = 1}$ emitted from and focused on red points, which are off the y axis. Blue dashed curves denote caustic curves. (d) Colored light rays in the Maxwell’s fish-eye lens with ${p = 1}$ emitted from and focused on red and blue points, which are off radius a. (e) Colored light rays in the Maxwell’s fish-eye lens with ${p = 2}$ emitted from and focused on red and blue points, which are off radius a. (f) Colored light rays in the duplex Maxwell’s fish-eye lens with ${p = 2, q = 1}$ emitted from and focused on red points, which are off radius a. Blue dashed squarelike curves denote caustic curves.
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Figure 4
(a) Colored light rays in the duplex Maxwell’s fish-eye lens with ${p = 3, q = 1}$ emitted from and focused on red points, which are outside radius a. Blue dashed triangularlike curve inside radius a is caustic curve. (b) Colored light rays in the duplex Maxwell’s fish-eye lens with ${p = 3, q = 1}$ emitted from and focused on red and blue points, which are on radius a. (c) Colored light rays in the duplex Maxwell’s fish-eye lens with ${p = 3, q = 1}$ emitted from and focused on red points, which are inside radius a. Blue dashed triangularlike curve outside radius a is the caustic curve. (d) Colored light rays in the duplex Maxwell’s fish-eye lens with ${p = 1, q = 3}$ emitted from and focused on red points, which are outside radius a. Blue dashed clover-shaped curve inside radius a is the caustic curve. (e) Colored light rays in the duplex Maxwell’s fish-eye lens with ${p = 1, q = 3}$ emitted from and focused on red and blue points, which are on radius a. (f) Colored light rays in the duplex Maxwell’s fish-eye lens with ${p = 1, q = 3}$ emitted from and focused on red points, which are inside radius a. Blue dashed clover-shaped curve outside radius a is the caustic curve.
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Figure 5
Talbot effect in the Mikaelian lens and duplex Mikaelian lens. (a) Normal value of electromagnetic field $| E z |$ (with units of V/m) out of the plane plotted in the Mikaelian lens of Fig. 1. (b) Electromagnetic field plotted in the duplex Mikaelian lens of Fig. 2. Metallic grating source with its magnitude satisfying $a b s (1 - y)$ is set at the x axis. (c) Electromagnetic field out of the plane at every half period in the Mikaelian lens. (d) Electromagnetic field out of the plane at every period in the duplex Mikaelian lens.
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