Self-Focusing and the Talbot Effect in Conformal Transformation Optics

Transformation optics has been used to propose various novel optical devices. With the help of metamaterials, several intriguing designs, such as invisibility cloaks, have been implemented. However, as the basic units should be much smaller than the working wavelengths to achieve the effective material parameters, and the sizes of devices should be much larger than the wavelengths of illumination to work within the light-ray approximation, it is a big challenge to implement an experimental system that works simultaneously for both geometric optics and wave optics. In this Letter, by using a gradient-index microstructured optical waveguide, we realize a device of conformal transformation optics (CTO) and demonstrate its self-focusing property for geometry optics and the Talbot effect for wave optics. In addition, the Talbot effect in such a system has a potential application to transfer digital information without diffraction. Our findings demonstrate the photon controlling ability of CTO in a feasible experiment system.

Figure 1

Exponential conformal mapping. (a) Imaging by Maxwell-fish eye in $u\text{−}v$ space. (b) Self-focusing of light rays by Mikaelian lens in $x\text{−}y$ space. (c) A “spiral” light ray in the Riemann surfaces. (d) A sinelike ray in $x\text{−}y$ space.

Figure 2

Schematics and optical measurements of light rays in a conformal waveguide. (a),(c) Schematic view of the microstructured optical waveguide. (b) The effective refractive index calculated from the waveguide thickness profile (see Supplemental Material Sec. III for more details [50]). (d),(f),(e),(g) The experiment results and numerical simulations of the self-focusing effect (d),(f) and sinelike rays (e),(g) in the conformal lens.

Figure 3

Comparison between the ordinary Talbot effect and the conformal Talbot effect. (a),(b) Ordinary Talbot effect in homogeneous medium with infinite periodic source (a) and finite periodic source (b). (c),(d),(e) Self-focusing of the conformal Talbot effect in a Mikaelian lens: (c) experimental results; (d) numerical results; (e) analytical results. The figures in the left column show the schematic of the grating source, the grating period is $D$, the slit width is $d$. An incident laser is used to excite the grating source (denoted as yellow arrows).

Figure 4

Digital coding from different source sequences. (a),(b),(c) Field pattern with thirty-six bits coding sequences: (a) 010001000100010001000100010001000100, (b) 011001100110011001100110011001100110, (c) 011101110111011101110111011101110111. The left pictures schematically show the input coding source sequences denoted by Input1, Input2, Input3. (d),(e),(f) The finite periodic coding source sequences (on top) and the normalized intensity profiles for several propagation distances. For $y=0$, we get the input signal (red solid curve). The encoded result (black solid curve) is the field at $y=L/4$, or the focusing plane. For $y=L/2$, we get the final output result (blue solid curve), which deviates little from the input signal.