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We propose a transformational design of an axi-symmetric gradient lens for electromagnetic waves. We show that a metamaterial consisting of toroidal air channels of diameters ranging from 23 nm to 190 nm in a matrix of Polymethylmethacrylate (PMMA) allows for a focussing effect of light over a large bandwidth i.e. [600 – 1000] nm. We finally propose a simplified design of lens allowing for a two-photon lithography implementation.
©2011 Optical Society of America
1. Introduction
There is currently a renewed interest in gradient index (GRIN) lenses in optics and acoustics [1, 2] owing to their links with transformational optics and acoustics. The fascinating topic of transformational photonics is fueled by the interest of scientists in invisibility cloaks [3, 4], but it has in fact a long standing history, as the first paradigm of a GRIN lens can be traced back to the Maxwell fisheye [5] (imagined over one hundred fifty years ago). The first transformation based design which unveiled the underlying physics of the Maxwell fisheye is due to the mathematician Luneburg who in his classical treatise on mathematical optics [6] derived the refractive index of this GRIN lens from the stereographic projection of a sphere on a plane, and this transformation optics approach has been revisited by various authors recently regarding the controversial claim of perfect focussing without negative refraction[7, 8, 9, 10].
2. Gradient index optics design
In this section, we would like to consider the hyperbolic secant profile considered in [1, 2]
where R 0 is the radius of the cylindrical lens, and . Moreover, n 0 and nR are respectively the refractive indices (invariant along the z-axis) at r = 0 (along the optical axis) and at r = R 0 (at the outer edge). Such a design can be deduced from quasi-conformal grids, as detailed for instance in the textbook [11].We show in Fig. 1(a) the variation of the refractive index as a function of the radius, when n 0 = 1.486 (the refractive index of polymethylmethacrylate for wavelength of 700 nm [12]). Figure 1(b) illustrates the focussing power of a three-dimensional cylindrical body with a circular cross-section and the refractive index of Fig. 1(a): a Gaussian plane wave (beam width = 1.5 um) with its electric field in y-direction incident from the bottom at wavelength λ = 700 nm on the heterogeneous isotropic lens is focused onto an image on the other side of the GRIN lens, as confirmed by a slice of the norm of the electric field along the vertical direction in Fig. 1(d). The two-dimensional plot of the real part of the y-component of the electric field exemplifies the scattering by the lens (the z-axis is along the vertical direction, see also Fig. 2(a) for the orientation of axes).
Fig. 1 Numerical validation at wavelength λ = 700 nm for a GRIN lens of thickness and radius R 0 of 1000 nm. (a) Profile of the refractive index versus radius; (b) Three-dimensional plot of the norm of electric field; (c) Side view of the real part of y-component of electric field; (d) Side view of norm of electric field.
Fig. 2 (a) Three-dimensional diagrammatic view of the structured GRIN lens; (b) Side view where R denotes the radius of the lens and r the radius of toroidal air channels; (c) Top view; (d) Table of filling fraction and effective index of refraction.
3. Effective medium design of a structured GRIN lens
We would like to propose a structured design of GRIN lens, which mimics the transformed design of the previous section. We consider a matrix of Polymethylmethacrylate (PMMA) with a refractive index n 0 = 1.486 and we drill some toroidal air channels of varying cross-section. We note that the radius of the channels r increases with the radius of lens R, see Fig. 2(b), and thus the refractive index decreases with increasing radius, in accordance with the refractive index profile in Fig. 1(a). The top view of the device, see Fig. 2(c) emphasizes the simplicity of the device which consists only of five toroidal channels. The three-dimensional view, see Fig. 2(a), shows the layered nature of the device. The table in Fig. 2(d) provides the reader with the exact data leading to the GRIN lens.
We note that the effective permittivity ɛe of the composite medium is given by the classical Maxwell-Garnett formula [13, 14]:
where ɛ 0 and ɛ are the permittivity of material (PMMA) and background (Air), respectively. We consider a ”toroidal unit cell.” Hence, the filling fraction f equal to the ratio of cross-section area of material (PMMA) to the unit cell. We further assume that the hexagonal lattice parameters is d = 200 nm and with r the radius of air channels.4. Illustrative numerical examples
In this section, we solve the full 3D vector Maxwell system using the COMSOL Multiphysics finite element package in order to emphasize the broadband nature of our structured GRIN lens as well as the easiness of its implementation for, say, optical or near infrared applications.
4.1. Original structured lens
Let us now compute the electromagnetic field for an incident plane wave on the structured GRIN lens described in Fig. 2(a). The total energy density computed along the segment [α, β] shown in Fig. 3(c). We note that the energy is concentrated around the point γ of the segment [α, β], and this corresponds to the focussing point in Fig. 3(c). The three-dimensional plot in Fig. 3(b) clearly shows the rotational invariance of the field in the azimuthal direction. Finally, the slice of the plot of the norm of the electric field in the vertical plane shows the focussing effect.
Fig. 3 Numerical validation at wavelength λ = 700 nm for a GRIN lens of thickness of 600 nm and radius R 0 of 1000 nm. (a) The total energy density varies with optical axis (αβ). Dashed box indicates the location of GRIN lens; (b) Three-dimensional plot of the real part of the the y-component of electric field; (c) Side view of the y-component of electric field; (d) Side view of norm of electric field.
An interesting feature of the multi-layered GRIN lens is that the focal plane β comes closer to the lens when we increase the number of layers, see Fig. 4(a), whereby we considered a GRIN lens with 3 and 5 layers. Moreover, the concentration of energy is more pronounced for 5 layers than for 3 layers, see Fig. 4(b). The concentrating effect of the GRIN lens is emphasized by Fig. 4(d), in accordance with the slice of the plot of the norm of the electric field along the vertical plane, see Fig. 4(c).
Fig. 4 (a) The total energy density varies with the distance from lens along optical axis; (b) The distributions of total energy density on the focal plane β in panel (c); (c) Side view of total energy density; (d) The distributions of total energy density from panel (c).
4.2. Robustness of focussing effect versus wavelength
Importantly, our device works over a large bandwidth as its structured design is based upon effective medium theory, see Fig. 5. Interestingly, the smaller the wavelength the more pronounced the focussing effect (and the farther the focal plane from the lens).
Fig. 5 Total energy density for wavelengths of (a) 600 nm; (b) 850 nm; (c) 1000 nm; (d) Variation of total energy density versus z axis. Dashed box indicates the location of lens.
4.3. Robustness of focussing effect versus geometric perturbation
Finally, we investigated the effect of some air channel removal on the focussing by our GRIN lens, see Fig. 6(a,b,c), where it should be noted that the refractive index profile of the lens, see Fig. 6(d), is lower or equal than that of the material (PMMA) on either sides (the slab is of infinite transverse extent): This demonstrates that the GRIN lens does not work as a waveguide. Importantly, two photon-lithography could be implemented in order to manufacture the lens, according to Fig. 6(e,f), whereby a double split ring design with two air cuts of 50 nm in thickness, is shown to work equally well.
Fig. 6 Calculated magnitude of energy density (J/m 3) for a wavelength of 700 nm (a) Only PMMA; (b) Effective structure with inner air channels removed; (c) Complete structure; (d) The effective refractive index for each toroidal unit cell (dotted green line and solid red lines are the effective index in panels (b) and (c), respectively). The dashed blue curve is the ideal index from equation (1) for comparison. (e) Three-dimensional diagrammatic view of the split ring GRIN lens; (f) Three-dimensional plot of the energy density.
5. Conclusion
In this paper, we have investigated the focussing features of a cylindrical GRIN lens. We proposed a practical design with a matrix of PMMA drilled with toroidal air channels. The fact that our design allows for a tunable position of the focal plane through a variation of number of layers is also advantageous. We point out that our design works over a finite bandwidth, as it is deduced from an effective medium model, and it is also robust versus geometric perturbations.
Acknowledgments
This work was funded by the European Commission through the Erasmus Mundus Joint Doctorate Programme Europhotonics (Grant No. 159224-1-2009-1-FR-ERA MUNDUS-EMJD).
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