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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 17 — Aug. 15, 2011
  • pp: 16154–16159
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Focussing light through a stack of toroidal channels in PMMA

Tieh-Ming Chang, Sebastien Guenneau, Jérôme Hazart, and Stefan Enoch  »View Author Affiliations


Optics Express, Vol. 19, Issue 17, pp. 16154-16159 (2011)
http://dx.doi.org/10.1364/OE.19.016154


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Abstract

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 OSA

1. Introduction

2. Gradient index optics design

In this section, we would like to consider the hyperbolic secant profile considered in [1

1. S. -C. S. Lin, T. J. Huang, J. H. Sun, and T. T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B 79, 094302 (2009). [CrossRef]

, 2

2. A. Climente, D. Torrent, and J. Sanchez-Dehesa, “Sound focusing by gradient index sonic lenses,” Appl. Phys. Lett. 97, 104103 (2010). [CrossRef]

]
n(r)=n0sech(αr)whereα=1R0cosh1(n0nR),
(1)
where R 0 is the radius of the cylindrical lens, and r=x2+y2. 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

11. C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-index optics: Fundamentals and applications (Springer, New York, 2002).

].

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

12. S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mat. 29, 1481–1490 (2007). [CrossRef]

]). 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 note that the effective permittivity ɛe of the composite medium is given by the classical Maxwell-Garnett formula [13

13. J. C. M. Garnett, “Colours in Metal Glasses and in Metallic Films,” Phil. Trans. Royal Soc. London, Ser. A 203, 385 (1904). [CrossRef]

, 14

14. X. Hu, C. T. Chan, J. Zi, M. Li, and K. -M. Ho, “Diamagnetic Response of Metallic Photonic Crystals at Infrared and Visible Frequencies,” Phys. Rev. Lett. 96, 223901 (2006). [CrossRef] [PubMed]

]:
ɛeɛɛe+ɛ=ɛ0ɛɛ0+ɛf
(2)
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 f=12πr2/3d2 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).

References and links

1.

S. -C. S. Lin, T. J. Huang, J. H. Sun, and T. T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B 79, 094302 (2009). [CrossRef]

2.

A. Climente, D. Torrent, and J. Sanchez-Dehesa, “Sound focusing by gradient index sonic lenses,” Appl. Phys. Lett. 97, 104103 (2010). [CrossRef]

3.

J. B. Pendry, D. Schurig, and D. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

4.

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]

5.

C. A. Swainson (alias J. C. Maxwell), “Problems,” Cambridge Dublin Math. J. 8, 188–189 (1854).

6.

R. K. Luneburg, Mathematical theory of optics (University of California Press, Berkeley, 1964).

7.

U. Leonhardt, “Perfect imaging without negative refraction,” N. J. Phys. 11(9), 093040 (2009). [CrossRef]

8.

S. Guenneau, A. Diatta, and R. C. McPhedran, “Focussing: coming to the point in metamaterials,” J. Modern Opt. 57, 511–527 (2010). [CrossRef]

9.

P. Benitez, J. C. Minano, J. C. Gonzalez, and C. Juan, “Perfect focussing of scalar wave fields in three dimensions,” Opt. Express 18(8), 7650–7663 (2010). [CrossRef] [PubMed]

10.

R. MerlinComment on, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A 82(5), 057801 (2010). [CrossRef]

11.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-index optics: Fundamentals and applications (Springer, New York, 2002).

12.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mat. 29, 1481–1490 (2007). [CrossRef]

13.

J. C. M. Garnett, “Colours in Metal Glasses and in Metallic Films,” Phil. Trans. Royal Soc. London, Ser. A 203, 385 (1904). [CrossRef]

14.

X. Hu, C. T. Chan, J. Zi, M. Li, and K. -M. Ho, “Diamagnetic Response of Metallic Photonic Crystals at Infrared and Visible Frequencies,” Phys. Rev. Lett. 96, 223901 (2006). [CrossRef] [PubMed]

OCIS Codes
(080.2740) Geometric optics : Geometric optical design
(110.2760) Imaging systems : Gradient-index lenses
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(050.2065) Diffraction and gratings : Effective medium theory

ToC Category:
Metamaterials

History
Original Manuscript: May 4, 2011
Revised Manuscript: June 23, 2011
Manuscript Accepted: June 23, 2011
Published: August 9, 2011

Citation
Tieh-Ming Chang, Sebastien Guenneau, Jérôme Hazart, and Stefan Enoch, "Focussing light through a stack of toroidal channels in PMMA," Opt. Express 19, 16154-16159 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16154


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References

  1. S. -C. S. Lin, T. J. Huang, J. H. Sun, and T. T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B 79, 094302 (2009). [CrossRef]
  2. A. Climente, D. Torrent, and J. Sanchez-Dehesa, “Sound focusing by gradient index sonic lenses,” Appl. Phys. Lett. 97, 104103 (2010). [CrossRef]
  3. J. B. Pendry, D. Schurig, and D. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]
  4. U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]
  5. C. A. Swainson (alias J. C. Maxwell), “Problems,” Cambridge Dublin Math. J. 8, 188–189 (1854).
  6. R. K. Luneburg, Mathematical theory of optics (University of California Press, Berkeley, 1964).
  7. U. Leonhardt, “Perfect imaging without negative refraction,” N. J. Phys. 11(9), 093040 (2009). [CrossRef]
  8. S. Guenneau, A. Diatta, and R. C. McPhedran, “Focussing: coming to the point in metamaterials,” J. Modern Opt. 57, 511–527 (2010). [CrossRef]
  9. P. Benitez, J. C. Minano, J. C. Gonzalez, and C. Juan, “Perfect focussing of scalar wave fields in three dimensions,” Opt. Express 18(8), 7650–7663 (2010). [CrossRef] [PubMed]
  10. R. MerlinComment on, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A 82(5), 057801 (2010). [CrossRef]
  11. C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-index optics: Fundamentals and applications (Springer, New York, 2002).
  12. S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mat. 29, 1481–1490 (2007). [CrossRef]
  13. J. C. M. Garnett, “Colours in Metal Glasses and in Metallic Films,” Phil. Trans. Royal Soc. London, Ser. A 203, 385 (1904). [CrossRef]
  14. X. Hu, C. T. Chan, J. Zi, M. Li, and K. -M. Ho, “Diamagnetic Response of Metallic Photonic Crystals at Infrared and Visible Frequencies,” Phys. Rev. Lett. 96, 223901 (2006). [CrossRef] [PubMed]

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