Conformal transformation optics

Journal name:
Nature Photonics
Year published:
Published online


The field of transformation optics shows that media containing gradients in optical properties are equivalent to curved geometries of spacetime for the propagation of light. Conformal transformation optics — a particular variant of this feature — can be used to design devices with novel functionalities from inhomogeneous, isotropic dielectric media.

At a glance


  1. Conformal invisible cloaks and carpet cloaks.
    Figure 1: Conformal invisible cloaks and carpet cloaks.

    a, Virtual space of the Zhukowski mapping for invisible cloaks. b, Physical space of the Zhukowski mapping for invisible cloaks. c, The electric field, E (the component normal to the x–y plane), pattern for simulation of the transverse electric polarization case (E is the only electric field component) of an incident Gaussian beam at an angle of π/4 to achieve cloaking. The beam's wavelength satisfies the condition , where the radius of each lens is r = 2a and the eigenvalue is chosen to be l = 40 in the numerical simulation. d, Virtual space of conformal carpet cloaks. e, Physical space of conformal carpet cloaks. f, The electric field pattern of an incident Gaussian beam at an angle of π/4 to achieve carpet cloaking. The beam's wavelength can be arbitrarily chosen; here we set it to be the same as in c.

  2. Conformal power mapping and logarithmic mapping.
    Figure 2: Conformal power mapping and logarithmic mapping.

    a, The virtual space of conformal power mapping comprises two Riemann sheets connected by a yellow branch cut. A point source (in red) is placed at w = 1 in the upper half-sheet (blue). b, Physical space of conformal power mapping. The corresponding point source is placed at z = √2. The darker blue or green area indicates a lower refractive index; the brighter blue or green area indicates higher refractive index. c, The electric field pattern of a point source placed at z = 0 looks like two point sources placed at z = i and z = −i in vacuum. In this simulation, only the region inside the black circle is filled with the calculated refractive index, as the refractive index is almost 1 in the region outside the black circle. d, Virtual space of conformal logarithmic mapping. A rectangular region of the whole virtual space is mapped to a quarter segment of an annulus in physical space. The light ray propagates in a straight line because of the flat space (blue). e, Physical space of the conformal logarithmic mapping from d. The darker blue area denotes lower refractive index; the brighter blue area denotes higher refractive index. The light is bent in the curved physical space. f, Electric field pattern inside a waveguide bent by π/2. The wavefront parallel to the horizontal line is gently bent to a vertical orientation in the waveguide. Figure reproduced with permission from: d–f, ref. 61, © 2011 OSA.

  3. Mobius mapping and Schwartz-Christoffel mapping.
    Figure 3: Möbius mapping and Schwartz–Christoffel mapping.

    a, Virtual space of the Möbius mapping. A semicircular region is mapped to another semicircular region. A point source that radiates light rays is placed at the origin. b, Physical space mapped from a by Möbius mapping. This mapping bends radial light rays into parallel rays. The darker blue area denotes lower refractive index; the brighter blue area denotes higher refractive index. c, The electric field pattern radiating from a point source placed at the top of the semicircle. d, Virtual space of Schwartz–Christoffel mapping. A circular region can be mapped to a square region. e, Physical space mapped from d by Schwartz–Christoffel mapping. The square region, with a complex refractive index distribution, bends radial light rays at the origin to become perpendicular to the square boundaries. f, The electric field pattern radiating from a point source at the origin. Figure reproduced with permission from: a–c, ref. 61, © 2011 OSA; d–f, ref. 68, © 2010 APS.

  4. Typical refractive index profiles.
    Figure 4: Typical refractive index profiles.

    a, Maxwell fisheye lens profile, n(r) = 2 / (1 + r2). b, Luneburg lens profile, . c, Eaton lens profile, . d, Virtual space of a mirrored Maxwell fisheye lens is not flat but non-Euclidian: it is a spherical surface with radius 1 (in arbitrary units) and a mirror placed at the equator. The refractive index is uniform at every point on the sphere. e, Physical space of a mirrored Maxwell fisheye lens. f, Ray trajectories in the Miñano lens. The white region is homogeneous. g, Ray trajectories in a modified Miñano lens. In all the above figures, the black circle indicates where the refractive index is 1. Other colourful closed curves from a point source A form an image B. Figure reproduced with permission from: a–c,f–g, ref. 76, © 2011 IOP; d,e, ref. 78, © 2009 IOP.


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  1. College of Physics, Optoelectronics and Energy, Soochow University, Suzhou, Jiangsu 215006, China

    • Lin Xu &
    • Huanyang Chen


Both authors contributed equally to this work.

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