## Super-thin Mikaelian’s lens of small index as a beam compressor with an extremely high compression ratio |

Optics Express, Vol. 21, Issue 6, pp. 7328-7336 (2013)

http://dx.doi.org/10.1364/OE.21.007328

Acrobat PDF (3198 KB)

### Abstract

Based on a focusing Mikaelian’s lens with small refraction index (0<*n*<<1), an optical device is designed as a super-thin optical beam compressor (e.g., thickness = 3*λ _{0}*) with an extremely high beam compression ratio (more than 19:1). This device can also be used as a beam collimator or a cylindrical-to-plane wave convertor with a much higher transmissivity than a zero-index metamaterial slab. The output beam shows good directionality in both near field and far field. A metamaterial structure is also designed to realize this device and verify its performance with finite element method (FEM).

© 2013 OSA

## 1. Introduction

1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science **305**(5685), 788–792 (2004). [CrossRef] [PubMed]

2. D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **71**(33 Pt 2B), 036609 (2005). [CrossRef] [PubMed]

3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**(5781), 1780–1782 (2006). [CrossRef] [PubMed]

3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**(5781), 1780–1782 (2006). [CrossRef] [PubMed]

4. U. Leonhardt, “Optical conformal mapping,” Science **312**(5781), 1777–1780 (2006). [CrossRef] [PubMed]

5. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. **8**(10), 247 (2006). [CrossRef]

7. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef] [PubMed]

8. D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. **5**(9), 687–692 (2009). [CrossRef]

9. H. Ma, S. Qu, Z. Xu, and J. Wang, “Using photon funnels based on metamaterial cloaks to compress electromagnetic wave beams,” Appl. Opt. **47**(23), 4193–4195 (2008). [CrossRef] [PubMed]

10. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **70**(4), 046608 (2004). [CrossRef] [PubMed]

13. A. Alù, M. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B **75**(15), 155410 (2007). [CrossRef]

17. R. Ilinsky, “Gradient-index meniscus lens free of spherical aberration,” J. Opt. A, Pure Appl. Opt. **2**(5), 449–451 (2000). [CrossRef]

18. M. I. Kotlyar, Y. R. Triandaphilov, A. A. Kovalev, V. A. Soifer, M. V. Kotlyar, and L. O’Faolain, “Photonic crystal lens for coupling two waveguides,” Appl. Opt. **48**(19), 3722–3730 (2009). [CrossRef] [PubMed]

19. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Subwavelength focusing with a Mikaelian planar lens,” Opt. Mem. Neural. Networks **19**(4), 273–278 (2010). [CrossRef]

10. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **70**(4), 046608 (2004). [CrossRef] [PubMed]

*λ*) and good directionality of the output beam. Note that one can also use TO to design light beam compressors and optical collimators [9

_{0}9. H. Ma, S. Qu, Z. Xu, and J. Wang, “Using photon funnels based on metamaterial cloaks to compress electromagnetic wave beams,” Appl. Opt. **47**(23), 4193–4195 (2008). [CrossRef] [PubMed]

22. H. Ma, S. Qu, Z. Xu, and J. Wang, “General method for designing wave shape transformers,” Opt. Express **16**(26), 22072–22082 (2008). [CrossRef] [PubMed]

23. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. **100**(6), 063903 (2008). [CrossRef] [PubMed]

*n*<<1 and use the finite element method (FEM) [24

24. The finite element simulation is conducted by using commercial software COMSOL Multiphysics. http://www.comsol.com/. The number of pixels per wavelength is larger than 20 in our simulation.

## 2. Design method and simulation results

*n*is the refraction index on the optic axis,

_{0}*r*is the radial distance,

*g*is a gradient constant,

*d = π/2g*is the thickness (also focal length) of the ML and

*sech(x)*is the hyperbolic secant function [17

17. R. Ilinsky, “Gradient-index meniscus lens free of spherical aberration,” J. Opt. A, Pure Appl. Opt. **2**(5), 449–451 (2000). [CrossRef]

*n*is chosen to be larger than 1 to ensure

_{0}*n*>1 in the whole device. Utilizing the advantages of metamaterials, we propose to realize an ML with 0

*<n<*1. When a light wave normally impinges on an ML with 0

*< n*1, two optical effects can be expected: (i) The size of the focused spot should be larger than that in a traditional case, as the spot size is inversely proportional to the local refraction index [25]; (ii) The output beam is no longer a focused beam but a collimated beam. According to Snell’s law, the output beam should be almost perpendicular to the lens’ surface, if the refraction index of the lens approaches zero. Thus, it can work as an adapter to compress/expand an illumination beam. It should be noted that the spot size of the output beam is limited by the diffraction limit [25], since the ML of inhomogeneous positive refraction index can neither amplify nor convert the evanescent waves [7

_{0}<<7. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef] [PubMed]

*d = π/2g =*6

*λ*and the height is

_{0}*h =*16

*λ*(

_{0}*λ*is the free-space wavelength). The incident wave is a Gaussian beam with waist radius

_{0}*w =*5

*λ*. We can see the performance of the ML changes from a focusing lens to a beam-size compressor, as we change

_{0}*n*from 1 to 0.2 (see Fig. 1). To verify our FEM simulation results, we choose one example (Fig. 1(c)) to verify with an alternative technique – FDTD method (see Fig. 2(d) below). One sees that the FDTD simulation result in Fig. 2(d) agrees well with our FEM simulation result in Fig. 1(c).

_{0}*n*<<1, it can also be used as a collimator due to its focusing property [15]. This has also been verified by FEM as shown in Fig. 2(a). A homogenous slab with refraction index

_{0}*n*= 0 can also be used as a collimator [26

26. Y. Jin and S. L. He, “Impedance-matched multilayered structure containing a zero-permittivity material for spatial filtering,” J. Nonlinear Opt. Phys. Mater. **17**(03), 349–355 (2008). [CrossRef]

*n*and a zero-refraction index slab separately, the transmissivity of the ML (about 5.582%) in Fig. 2(a) is higher than that of the zero-index slab (about 0.011%) in Fig. 2(b). Here the transmissivity is defined as the ratio of the integration of power flux on the output surface with the device to the one without the device. We should note that an ML with

_{0}= 0.2*n*>1 does not work as a collimator but a focusing lens (see Fig. 2(c)) and the focusing length of this lens decreases as

_{0}*n*increases.

_{0}27. J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express **18**(1), 244–252 (2010). [CrossRef] [PubMed]

*n*<<1 has a similar functionality. The device is composed of four regions labeled I, II, III, and IV (see Fig. 3(a) ). The material of regions I and III gradually changes according to Eq. (2) and we can replace y in Eq. (2) with x to obtain the refraction index in regions II and IV. The refraction index distribution in the whole device is shown in Fig. 3(b). If we set a line current at the center of this new device, the output beam is no longer a cylindrical wave but four collimated beams (see Fig. 3(c)). If we fill regions I, II, III and IV with a zero-index material, it can also be a cylindrical-to-plane wave convertor (see Fig. 3(d)). Obviously the advantage of our device is that the intensity of the output field is much higher (one order higher) than that of the device (made by a zero-index material) proposed in [27

_{0}27. J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express **18**(1), 244–252 (2010). [CrossRef] [PubMed]

*n*<<1 and the method to implement the ML using metamaterials. We found this compression function very robust against the lens thickness. As shown in Fig. 4(a) , even if the thickness of an ML with

_{0}*n*= 0.2 is reduced to 3

_{0}*λ*, it still gives a good beam compression performance. However, as the thickness decreases, the transmissivity also decreases (see Fig. 5(a) ). To keep the focusing performance of the ML, we should ensure the medium varies gradually so that the following condition can be fulfilled [28]:where

_{0}*λ = λ*. For a 2D ML, by substituting Eq. (2) into Eq. (3), the above condition can be written as:where

_{0}/n*sinh(x)*is a hyperbolic sine function and

*g*= π/2

*d*. If the thickness

*d*of the ML decreases,

*g*will increase and then

*|1/sinh(gy)|*decreases. For a fixed height, when the ML is much thinner than the wavelength (e.g., less than

*λ*

_{0}/10), condition (4) cannot be satisfied any more. This means the ML will no longer be a graded inhomogeneous medium but some effective medium for the incident wave. Consequently, the focusing property of the ML will disappear. When the thickness of the ML is comparable to the wavelength (e.g., the thickness is λ

_{0}), it is the transition state between a graded inhomogeneous medium and an effective medium.

*n*<<1 can be extremely high. For example in Fig. 4(a), when the full width of the incident plane wave is 90

_{0}*λ*, the full width of the output beam is 4.77

_{0}*λ*(the compression ratio is almost 19:1). If we increase the size of the input beam, the size of the output beam is almost unchanged (see Fig. 5(b)). Thus, we can obtain an extremely high beam compression ratio simply by increasing the height of the ML. A single aperture (e.g., a diaphragm) can also be used as a beam compressor with an extremely high compression ratio [25]. Compared with the present ML with 0<

_{0}*n*<<1, a single aperture has two drawbacks: (i) The energy efficiency is quite low. An aperture simply blocks the edge part of the rays, while the ML can guide almost all the rays (including the edge part) to the optical axis. (ii) The directionality of the output beam is not good due to the diffraction of the light beam when passing through an aperture, while the output beam of an ML with

_{0}*n*= 0.2 shows a very good directionality in both near-field (see Fig. 4(a)) and far-field regions (see Fig. 4(b)). For a fair comparison, we also simulate the case when a plane wave with a full width of 90

_{0}*λ*impinges on a diaphragm with a hole width of 4.77

_{0}*λ*(see Fig. 4(f)). In this case, the output beam does not have good directionality or uniformity as compared with our ML in Fig. 4(a).

_{0}*λ*and thus Eq. (4) is satisfied. In this case, if a lightwave impinges on the ML, the reflections only occur at its front and back surfaces. Since the local reflection coefficients at the ML’s front and back surfaces are different (unlike the FP resonance of a homogenous dielectric slab), the peak transmissivity is not 100% in this case. If the thickness of the ML is too small and the wavelength

_{0}*λ*remains unchanged (then Eq. (4) cannot be satisfied), the refraction index will no longer vary gradually. In this case, ML can be treated as some effective medium with very low refraction index, and thus the transmissivity is very low. As the ML’s thickness d increases, the region with the refraction index not very close to zero also increases. Thus, the transmissivity increases with the increasing thickness. While the output beam is mainly focused on the optical axis, the size of the output beam is mainly determined by the refraction index

_{0}*n*

_{0}on the optic axis. That is the reason why the size of the output beam changes little as the thickness changes (see Fig. 5(c)). With increasing thickness of the ML, the transmissivity increases and the compression ratio keeps unchanged. In practice, we can choose a resonant frequency as the working frequency or use a thick device to obtain a high transmissivity. For a super-thin device, we can improve its transmissivity by using impedance matched materials with

*ε = μ = n*(see Fig. 4(c)).

*λ*

_{0}before the ML (see Fig. 6 ). For a ML with height 90

*λ*

_{0}and width 3

*λ*

_{0}, the edge part of the incident beam contributes little to the output beam (cf. Figure 6(a) and 6(b)). For the impedance matched ML with height 90

*λ*

_{0}and width 3

*λ*

_{0}, the edge part of the incident beam really contributes and makes an obvious difference to the output beam (cf. Figure 6(c) and 6(d)).

## 3. Experimental design

*n*, thickness

_{0}= 0.3*d =*3

*λ*and height

_{0}*h*=

*10λ*for TE polarization working at 10

_{0}*GHz*(

*λ*= 30 mm). The whole ML is divided into 30 square units in height (y direction), 9 square units in thickness (

_{0}*x*direction) and is infinitely long in the

*z*direction. The size of each square unit is 10mm × 10mm (about

*λ*

_{0}/3<

*λ*/10) to make the effective medium theory valid. An infinitely long cylindrical copper wire with diameter

*D*is set at the center of each square unit (see Fig. 7(a) ). We use standard commercial software CST-microwave Studio to extract each unit’s

_{r}*S*parameters and use these

*S*parameters to retrieve the effective refractive index

*n*and impedance

*z*[29

29. R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **76**(2), 026606 (2007). [CrossRef] [PubMed]

*n*of each metamaterial unit. Then we choose different diameters of cylindrical wires in different units (see Fig. 7(b)) according to the ML’s refraction index distribution. The radii of the cylindrical wires are identical along the

*x*direction and change gradually along the

*y*direction. FEM shows a good beam compression performance of the designed metamaterial structure (see Fig. 8 ). To reduce the fabrication work, we make some approximation (with step changes) of this ML. The approximate parameters are also shown in Fig. 7(b). The reduced device also performs well the beam compression (see Fig. 8 (c)). As the refraction index is close to zero in the whole device, the local wavelength

*λ = λ*is much larger than the size of each unit, and thus a slight reduction in unit cell can hardly influence the performance of the device. The designed wavelength is 30 mm, while FEM simulation results show that both the non-reduced and reduced devices can work in the band between

_{0}/n*λ*= 29mm to

_{0}*λ*

_{0}= 31mm.

## 4. Discussion and conclusions

*n*<<1 and a single aperture has been discussed in section 2. Compared with an ML, a telescope system used as a compressor has some drawbacks: (i) The whole device is not compact. The thickness of a telescope system is often much larger than the working wavelength (e.g., the thickness is about a million of the wavelength). (ii) The compression ratio is low (e.g., 5:1). When used as a collimator, our ML has higher alignment accuracy due to its aberration-free feature. Compared with a zero-index slab for a collimator, the transmissivity of our ML with 0<

_{0}*n*<<1 is much larger.

_{0}23. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. **100**(6), 063903 (2008). [CrossRef] [PubMed]

*x, y, z*) is a coordinate point in the reference space and (

*x’, y’, z’*) is a coordinate point in the physical space.

*d*is the thickness of the device and

*α*is the compression ratio. The corresponding material parameters can be determined by TO [3

3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**(5781), 1780–1782 (2006). [CrossRef] [PubMed]

23. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. **100**(6), 063903 (2008). [CrossRef] [PubMed]

32. W. X. Tang, Y. Hao, and F. Medina, “Broadband extraordinary transmission in a single sub-wavelength aperture,” Opt. Express **18**(16), 16946–16954 (2010). [CrossRef] [PubMed]

33. J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. **101**(20), 203901 (2008). [CrossRef] [PubMed]

*α*requires more complicated materials. For our ML used as a compressor, the size of the output beam hardly changes when the size of the incident beam increases (see Fig. 5(b)), and thus it can obtain an extremely high compression ratio by increasing the height of the device. The performance of the TO-based device described in Eq. (6) is also verified by FEM as shown in Fig. 4(d). Under the same condition, the output beam of our ML shows a better directionality in both near field and far field, compared with the TO-based device (cf. Figure 4 (b) and 4(e)).

*n*<<1 has many advantages: super-thin thickness, extremely high beam compression ratio, and very good directionality of the output beam. We have also designed a metamaterial structure composed of copper wires to realize such a 2D ML with

_{0}*n*= 0.3. The present idea can be extended to terahertz and infrared bands. The ML with 0<

_{0}*n*<<1 may have many potential applications in integrated optics, aligning optical instruments (e.g., binoculars), optical couplers for planar lightwave circuits, free space optical communication, etc.

_{0}## Acknowledgments

## References and links

1. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science |

2. | D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

3. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

4. | U. Leonhardt, “Optical conformal mapping,” Science |

5. | U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. |

6. | U. Leonhardt and T. G. Philbin, |

7. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

8. | D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. |

9. | H. Ma, S. Qu, Z. Xu, and J. Wang, “Using photon funnels based on metamaterial cloaks to compress electromagnetic wave beams,” Appl. Opt. |

10. | R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

11. | Y. Jin and S. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Opt. Express |

12. | V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. |

13. | A. Alù, M. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B |

14. | A. L. Mikaelian, “Application of stratified medium for waves focusing,” Dokl. Akad. Nauk SSSR |

15. | A.L. Mikaelian, “Self-focusing medium with variable index of refraction,” in Progress in Optics |

16. | A. L. Mikaelian, “General method of inhomogeneous media calculation by the given ray traces,” Dokl. Akad. Nauk |

17. | R. Ilinsky, “Gradient-index meniscus lens free of spherical aberration,” J. Opt. A, Pure Appl. Opt. |

18. | M. I. Kotlyar, Y. R. Triandaphilov, A. A. Kovalev, V. A. Soifer, M. V. Kotlyar, and L. O’Faolain, “Photonic crystal lens for coupling two waveguides,” Appl. Opt. |

19. | V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Subwavelength focusing with a Mikaelian planar lens,” Opt. Mem. Neural. Networks |

20. | Y. R. Triandaphilov and V. V. Kotlyar, “Photonic crystal Mikaelian lens,” Opt. Mem. Neural. Networks |

21. | D. V. Nesterenko, “Metal-dielectric Mikaelian’s lens,” Computer Optics |

22. | H. Ma, S. Qu, Z. Xu, and J. Wang, “General method for designing wave shape transformers,” Opt. Express |

23. | M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. |

24. | The finite element simulation is conducted by using commercial software COMSOL Multiphysics. http://www.comsol.com/. The number of pixels per wavelength is larger than 20 in our simulation. |

25. | M. Born and E. Wolf, |

26. | Y. Jin and S. L. He, “Impedance-matched multilayered structure containing a zero-permittivity material for spatial filtering,” J. Nonlinear Opt. Phys. Mater. |

27. | J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express |

28. | Y. A. Kravtsov and Y. I. Orlov, |

29. | R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

30. | F. J. Duarte and L. W. Hillman, |

31. | F. J. Duarte, |

32. | W. X. Tang, Y. Hao, and F. Medina, “Broadband extraordinary transmission in a single sub-wavelength aperture,” Opt. Express |

33. | J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. |

**OCIS Codes**

(110.2760) Imaging systems : Gradient-index lenses

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 18, 2013

Revised Manuscript: February 18, 2013

Manuscript Accepted: March 4, 2013

Published: March 15, 2013

**Citation**

Fei Sun, Yun Gui Ma, Xiaochen Ge, and Sailing He, "Super-thin Mikaelian’s lens of small index as a beam compressor with an extremely high compression ratio," Opt. Express **21**, 7328-7336 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-6-7328

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### References

- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004). [CrossRef] [PubMed]
- D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(33 Pt 2B), 036609 (2005). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
- U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys.8(10), 247 (2006). [CrossRef]
- U. Leonhardt and T. G. Philbin, Geometry and Light: Science of Invisibility (Dover, 2010).
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys.5(9), 687–692 (2009). [CrossRef]
- H. Ma, S. Qu, Z. Xu, and J. Wang, “Using photon funnels based on metamaterial cloaks to compress electromagnetic wave beams,” Appl. Opt.47(23), 4193–4195 (2008). [CrossRef] [PubMed]
- R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(4), 046608 (2004). [CrossRef] [PubMed]
- Y. Jin and S. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Opt. Express18(16), 16587–16593 (2010). [CrossRef] [PubMed]
- V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett.102(13), 133902 (2009). [CrossRef] [PubMed]
- A. Alù, M. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B75(15), 155410 (2007). [CrossRef]
- A. L. Mikaelian, “Application of stratified medium for waves focusing,” Dokl. Akad. Nauk SSSR81, 569–571 (1951).
- A.L. Mikaelian, “Self-focusing medium with variable index of refraction,” in Progress in Optics XVII, 283–346 (1980).
- A. L. Mikaelian, “General method of inhomogeneous media calculation by the given ray traces,” Dokl. Akad. Nauk83(2), 219 (1952).
- R. Ilinsky, “Gradient-index meniscus lens free of spherical aberration,” J. Opt. A, Pure Appl. Opt.2(5), 449–451 (2000). [CrossRef]
- M. I. Kotlyar, Y. R. Triandaphilov, A. A. Kovalev, V. A. Soifer, M. V. Kotlyar, and L. O’Faolain, “Photonic crystal lens for coupling two waveguides,” Appl. Opt.48(19), 3722–3730 (2009). [CrossRef] [PubMed]
- V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Subwavelength focusing with a Mikaelian planar lens,” Opt. Mem. Neural. Networks19(4), 273–278 (2010). [CrossRef]
- Y. R. Triandaphilov and V. V. Kotlyar, “Photonic crystal Mikaelian lens,” Opt. Mem. Neural. Networks17(1), 1–7 (2008).
- D. V. Nesterenko, “Metal-dielectric Mikaelian’s lens,” Computer Optics35(1), 47 (2011) (in Russian; not accessible to the authors, but mentioned by a reviewer).
- H. Ma, S. Qu, Z. Xu, and J. Wang, “General method for designing wave shape transformers,” Opt. Express16(26), 22072–22082 (2008). [CrossRef] [PubMed]
- M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett.100(6), 063903 (2008). [CrossRef] [PubMed]
- The finite element simulation is conducted by using commercial software COMSOL Multiphysics. http://www.comsol.com/ . The number of pixels per wavelength is larger than 20 in our simulation.
- M. Born and E. Wolf, Principles of Optics, 5th edition (Pergamon, 1975).
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