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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 4 — Feb. 14, 2011
  • pp: 3562–3575
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Squeezing and expanding light without reflections via transformation optics

C. García-Meca, M. M. Tung, J. V. Galán, R. Ortuño, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez  »View Author Affiliations


Optics Express, Vol. 19, Issue 4, pp. 3562-3575 (2011)
http://dx.doi.org/10.1364/OE.19.003562


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Abstract

We study the reflection properties of squeezing devices based on transformation optics. An analytical expression for the angle-dependent reflection coefficient of a generic three-dimensional squeezer is derived. In contrast with previous studies, we find that there exist several conditions that guarantee no reflections so it is possible to build transformation-optics-based reflectionless squeezers. Moreover, it is shown that the design of antireflective coatings for the non-reflectionless case can be reduced to matching the impedance between two dielectrics. We illustrate the potential of these devices by proposing two applications in which a reflectionless squeezer is the key element: an ultra-short perfect coupler for high-index nanophotonic waveguides and a completely flat reflectionless hyperlens. We also apply our theory to the coupling of two metallic waveguides with different cross-section. Finally, we show how the studied devices can be implemented with non-magnetic isotropic materials by using a quasi-conformal mapping technique.

© 2011 OSA

1. Introduction

The ability to squeeze and expand light has many applications in optics, ranging from beam collimation to nanolithography, optical data storage, imaging quality enhancing and efficient coupling to nanoscale structures [1

1. R. Yang, M. A. Abushagur, and Z. Lu, “Efficiently squeezing near infrared light into a 21 nm-by-24 nm nanospot,” Opt. Express 16(24), 20142–20148 (2008). [CrossRef] [PubMed]

,2

2. L. Vivien, S. Laval, E. Cassan, X. Le Roux, and D. Pascal, “2-D taper for low-loss coupling between polarization-insensitive microwaveguides and single-mode optical fibers,” J. Lightwave Technol. 21(10), 2429–2433 (2003). [CrossRef]

]. Transformation optics offers a new way to achieve these effects, since it provides the necessary medium to force electromagnetic fields to undergo the spatial distortion introduced by a certain coordinate transformation [3

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

,4

4. U. Leonhardt and T. G. Philbin, “General Relativity in Electrical Engineering,” N. J. Phys. 8(10), 247 (2006). [CrossRef]

]. Finite embedded coordinate transformations enables us to transfer light alterations, such as bends or shifts, from the transformed media to another one [5

5. 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]

], so it appears that this technique is very adequate for building squeezing devices. A simple two-dimensional (2D) version of a compressing device embedded in free space was studied in [6

6. M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008). [CrossRef] [PubMed]

], showing that unavoidable reflections appear in that case. Reflections imply power loss, which invalidates the utility of squeezers in many situations. The need for antireflective coatings was also observed in [6

6. M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008). [CrossRef] [PubMed]

], although it was not clear at all how to design an antireflective coating for such a complex material and no hint was given in that study. Moreover, the conclusions drawn from the 2D case cannot be generalized to the three-dimensional (3D) one, as there can be fundamental differences between them. In addition, the heuristic condition for no reflections given in [6

6. M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008). [CrossRef] [PubMed]

] only allows us to know whether the device is reflectionless for all angles or not. However, the reflected power depends on the polarization and on the angle, and could be negligible for certain spatial directions. Finally, in some situations it would be desirable that the output medium was different from the input one (in fact, we will take advantage of the squeezer properties in this situation to make a reflectionless device). Therefore, the possibility of achieving reflectionless squeezers is still open and a general study with the aim of obtaining this feature, indispensable for most applications, is lacking.

2. Theory

In an isotropic homogeneous background, only rotations and displacements of the outer boundaries achieve all-angle reflectionless transformation media [7

7. W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background,” arXiv:0806.3231v1 (2008).

]. Thus, to build an all-angle reflectionless squeezer, which has a compressed boundary, it is necessary to consider a more general scenario. Specifically, we will allow the medium that will be transformed into the squeezer to be different from the output medium in original space. This step is the key for some of the results obtained below. It is also important to consider a full problem with the possibility of a 3D transformation as mentioned above. Finally, our aim is to obtain an analytical expression for the total reflection coefficient of the squeezer, so that we can evaluate the reflected power and see if there is any preferred spatial direction for which no reflections occur. The problem under consideration is sketched in Fig. 1
Fig. 1 Sketch of the problem. Cartesian coordinate mesh in the original media is “seen” distorted by the fields in the transformed media.
.

Two cases are considered. In the first one, the output medium is εrd=εr3=F12 so that no antireflective coating is needed. In the second case, εrd=1 and we use a λ/4 dielectric coating with constant εrc. It is known that if εrc2=εr3εrd, reflections are suppressed. In both cases, the calculated relative transmitted power Pt=Pout/Pin is 100% (Pin and Pout are the squeezer input and output power). Without the coating, Pt=63%. We can verify this result with Eq. (3), as Pt=1|R|2. Since incidence is almost normal, we can put kx=0. Substituting the problem data in Eq. (3) we obtain Pt=64%, in very good agreement with numerical results. Note that the different E field intensities in the input and output media are consistent with the conservation of total power flow. It is also worth mentioning that the squeezer provides a compressed version of the fields inside it. This compression is transferred to the outside world near the squeezer. However, once the electromagnetic wave has exited the squeezer, it is subject to the diffraction laws of the output medium. Thus, the Gaussian beam exiting the squeezer will diverge as it propagates. This is mainly observed in Fig. 2(b), as this divergence is faster in air than in the medium with n = 4.

3. Applications

Applications of the proposed reflectionless device are straightforward. Here, two additional potential applications are proposed. The first one is a perfect squeezer-based spot size converter for an efficient coupling between an optical fibre and a high-index nanophotonic waveguide or nanowire, which is one of the most challenging tasks in the field of silicon photonics, due to the large mismatch in mode size of nanowires (sub-wavelength transversal dimensions) and standard single mode fibres (SMF, 10 µm mode diameter). Many solutions have been proposed, following one of these approaches: lateral (in plane) or vertical (out of plane) coupling. The latter requires out of plane diffraction, usually via grating couplers, whose achievable efficiency with conventional designs is lower than 40% [10

10. D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002). [CrossRef]

]. To reach higher efficiencies, highly sophisticated designs are needed, extremely increasing fabrication difficulty [11

11. G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008). [CrossRef]

]. Lateral coupling implies 2D SSC via waveguide inverse tapering down to tens of nanometers wide [2

2. L. Vivien, S. Laval, E. Cassan, X. Le Roux, and D. Pascal, “2-D taper for low-loss coupling between polarization-insensitive microwaveguides and single-mode optical fibers,” J. Lightwave Technol. 21(10), 2429–2433 (2003). [CrossRef]

], so matching the high SMF mode size is very challenging with a single inverted tapering structure. Most actual realizable single stage structures are limited in mode size to approximately 3-4 μm. Efficient coupling to such mode diameters can be achieved by means of lensed or high-numerical-aperture fibres with 3-4 µm mode diameters. Usually, in these structures the required inverted taper is longer than 200 µm and maximum coupling efficiency is lower than 80% [12

12. T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005). [CrossRef]

]. We aim to use our squeezer as an efficient SSC with a dramatically reduced length while achieving 100% coupling efficiency. Thus, the squeezer must compress incoming light fitting its size to that of the waveguide and then deliver the compressed beam to the waveguide without reflections, i.e., match air to the waveguide high-index core. Then, light is kept confined in the waveguide by total internal reflection (TIR) (see Fig. 3
Fig. 3 (a) Gaussian beam propagating in free space. (b) The squeezer couples the beam to a nanophotonic dielectric waveguide.
).

We will assume the telecom wavelength λ= 1.5μm, and a waveguide with sub-wavelength width w = 1 μm and refractive index n = 4, as in the example of Fig. 2 (the problem would be very similar if we used silicon, since nSi = 3.45 in this band).

We limit our study to a 2D case due to the computational complexity of the 3D problem. Given the size of the input beam, a compression F1=4 is enough, which satisfies condition 1 for no reflections. A seamless coupling can be observed in Fig. 4(b)
Fig. 4 (a) Simulation of a 2D Gaussian beam propagating in free space. (b) The beam in (a) is squeezed and perfectly coupled to the nanophotonic waveguide.
.

Numerical calculations reveal that Pt=100% again, where Pout has been evaluated at the right waveguide end. Without the squeezer, Pt<40% and decreases with z due to an inefficient mode matching. To extend the application to larger compression factors, antireflective coatings are necessary. In a 3D problem, compression in both transversal directions is demanded. Since in this application we have normal incidence, a squeezer fulfilling condition 3 with the proper antireflective coating can be employed. As for its size, the squeezer can be as short as desired. Nonetheless, the necessary constitutive parameters become extreme as we reduce d1. A surprisingly small length below 10 μm, far below the current state of the art, is enough to achieve a set of parameters with very moderate values for the 2D and 3D cases. Although fabrication of the 3D squeezer would be challenging, it is worth pointing out that the quasi-conformal mapping technique introduced in [13

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

] would provide a practically realizable non-magnetic isotropic implementation for the 2D squeezer, as we will see in section 4. Regardless of its application as SSC, our squeezer-TIR waveguide device presents the important advantage of using an isotropic homogeneous dielectric as the guiding element, as opposed to previously proposed squeezers based on transformation optics [14

14. B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B 79(8), 85103 (2009). [CrossRef]

], where complex materials are needed.

Now we apply our theory to this example. To a first approximation, we assume that the effective indices in waveguides W1 and W2 are those of the filling medium. Therefore, in Fig. 6(b), εr3=n32=1. Obviously, reflections appear because we are not fulfilling condition 1. This can be easily solved by filling waveguide W2 with a material n3=2 [Fig. 6(c)]. In this case, no modulation of the electric field is observed, showing that reflections have been suppressed. Alternatively, if we want waveguide W2 to be empty, we can design and use a dielectric antireflective coating following the procedure described above. This way, a perfect coupling between both waveguides is achieved. In Fig. 6(d), we depict the scattering parameter S 11 as a function of the refractive index of the filling medium of waveguide W2, for the case where the coupler is used [as in Figs. 6(b) and 6(c)]. A pronounced minimum is clearly seen very close to n3=2, validating our choice of effective indices.

4. Practical implementation

We can see in Fig. 7 that there are almost no reflections in this case either. Specifically, we obtained from the simulation a reflected power of 0.0036%. Thus, we have shown that it is possible to couple both waveguides with only a certain refractive index distribution. In this specific implementation, the required refractive index varies between 0.86 and 2.1 [see Fig. 7(b)]. Although it is possible to implement a refractive index below unity, it is preferable to have a refractive index range higher or equal to unity. For that purpose, we can just approximate the index by unity in the small regions where it is lower than one. Another possibility is to divide the whole index distribution by its minimum value (0.86). This does not affect the functionality of the device and introduces very weak reflections (as shown by numerical calculations) because the index mismatch is very low. We can of course completely avoid reflections by using the appropriate anti-reflective coating.

5. Conclusion

In summary, we have derived an analytical expression for the reflection coefficient of an optically transformed embedded squeezer. We have found the conditions for no reflections, showing how antireflective coatings can be used in non-reflectionless cases. This study has allowed us to design an ultra-short perfect SSC, as well as a completely flat reflectionless hyperlens. In addition, we have shown how to eliminate the reflections that appear when transformation optics-based devices are used to couple metallic waveguides with different cross-section. Finally, we have proposed a non-magnetic isotropic implementation of the constructed 2D squeezers/expanders, which only requires a spatially varying refractive index distribution.

Appendix A. Derivation of the reflection and transmission coefficients

In this section we derive Eq. (3). We start from macroscopic Maxwell's equations (assuming that the time dependence of the fields is of the formeiωt):

×E=iωB×H=iωB
(A.1)

Together with the constitutive relations:

D=ε.EB=μ.H
(A.2)

with:

ε=(ε11ε12ε13ε21ε22ε23ε31ε32ε33)μ=(μ11μ12μ13μ21μ22μ23μ31μ32μ33)
(A.3)

Now we define the tensors (analogous expressions are defined for μtt, μtz and μzt):

εtt=(ε11ε120ε21ε220000)εtz=(00ε1300ε23000)εzt=(000000ε31ε320)
(A.4)

and separate Maxwell's equations in their transverse and longitudinal components in Cartesian coordinates:

ikt×Ezz^+z^×Etz=iωμttHt+iωμtzHzz^
(A.5)
ikt×Et=iωμztHt+iωμ33Hzz^
(A.6)
ikt×Hzz^+z^×Htz=iωεttEtiωεtzEzz^
(A.7)
ikt×Ht=iωεztEtiωε33Ezz^
(A.8)

whereEt=Exx^+Eyy^, Ht=Hxx^+Hyy^, kt=kxx^+kyy^ and we have assumed a spatial dependence of the fields of the form eikr. From Eqs. (A.6) and (A.8), the longitudinal components of the fields can be expressed as a function of the transverse ones:

Hzz^=1ωμ33kt×Et1μ33μztHt
(A.9)
Ezz^=1ωε33kt×Ht1ε33εztEt
(A.10)

Upon substitution of Eq. (A.5) and (A.7) in Eq. (A.9) and (A.10), we arrive to:

Etz=(iμ33z^×Iμtzkt×Iiε33z^×Ikt×Iεzt)Et++(iωz^×Iμtt+iωμ33z^×Iμtzμztiωε33z^×Ikt×Ikt×I)Ht
(A.11)
Htz=(iωz^×Iεttiωε33z^×Iεtzεzt+iωμ33z^×Ikt×Ikt×I)Et++(iε33z^×Iεtzkt×Iiμ33z^×Ikt×Iμzt)Ht
(A.12)

where I=x^x^+y^y^+z^z^. As stated above, we will limit ourselves to k y = 0. In addition, the problem is simplified due to the fact that both the auxiliary layer and the outer medium are characterized by diagonal constitutive parameters. Given these simplifications and considering that z=ikz, Eq. (A.11) and (A.12) reduce to:

ikz(EtHt)=A(EtHt)
(A.13)
A=(000ikx2ε33ω+iωμ2200iωμ1100ikx2μ33ωiωε2200iωε11000)
(A.14)

This is an eigenvalue problem with four solutions. From the eigenvalues ikz we find the four possible values of kz (two for TE polarization and two for TM polarization), together with their corresponding eigenvectors (polarization states):

TE:kz1,2=±μ11μ33(ω2ε22μ33kx2)Et=Ey^Ht=EkZ1,2ωμ11x^
(A.15)
TM:kz3,4=±ε11ε33(ω2μ22ε33kx2)Et=Ex^Ht=Eωε11kZ3,4y^
(A.16)

Now we particularize Eq. (A.15) and (A.16) for the parameters of the auxiliary layer and the outer medium. The parameters of the former correspond to a transformation medium associated with the transformationxi=x(i)/F(i), which leads to:

ε2ij=ε0ξijμ2ij=μ0ξij
(A.17)
ξij=F1F2F3(1F10001F20001F3)T(1F10001F20001F3)=(F2F3F1000F1F3F2000F1F2F3)
(A.18)

This agrees with Eq. (2). In the case of the isotropic outer medium, we have:

ε3ij=ε3δijμ3ij=μ3δij
(A.19)

Substituting Eqs. (A.17)(A.19) into Eqs (A.15)(A.16), we have:

TE:kz,aux1,2=±ξ11ξ33(ω2ε0μ0ξ22ξ33kx2)Et=Ey^Ht=Ekz,aux1,2ωμ0ξ11x^
(A.20)
TM:kz,aux3,4=±ξ11ξ33(ω2ε0μ0ξ22ξ33kx2)Et=Ex^Ht=Eωε0ξ11kz,aux3,4y^
(A.21)

for the auxiliary layer, and:

TE:kz,out1,2=±ω2ε3μ3kx2Et=Ey^Ht=Ekz,out1,2ωμ3x^
(A.22)
TM:kz,out3,4=±ω2ε3μ3kx2Et=Ex^Ht=Eωε3kz,out3,4y^
(A.23)

for the outer medium. Finally, we obtain the reflection coefficients for both TE and TM excitation. To this end, we suppose and incident field (TE or TM) in the auxiliary layer propagating towards the outer medium and see if there are TE or TM reflected and transmitted waves, demanding equality of the tangential fields at the boundary between both media. For simplicity, we assume that z = 0 at the boundary. For TE excitation we have in matrix notation:

(01kz,aux1ωμ0ξ110)+R11(01kz,aux1ωμ0ξ110)+R21(100ωε0ξ11kz,aux1)=T11(01kz,out1ωμ30)+T21(100ωεkz,out1)
(A.24)

from which we deduce that:

R21=T21=0
(A.25)
R11=μ3kz,aux1μ0ξ11kz,out1μ3kz,aux1+μ0ξ11kz,out1T11=2μ3kz,aux1μ3kz,aux1+μ0ξ11kz,out1
(A.26)

R11 and T11 are the TE reflection and transmission coefficients for TE excitation. R21and T21 are the cross-polarization reflection and transmission coefficients from TE excitation to TM polarized waves. Analogously, for the TM case we have:

(100ωε0ξ11kz,aux1)+R12(01kz,aux1ωμ0ξ110)+R22(100ωε0ξ11kz,aux1)=T12(01kz,out1ωμ30)+T22(100ωεkz,out1)
(A.27)

and:

R12=T12=0
(A.28)
R22=ξ11ε0kz,out1ε3kz,aux1ξ11ε0kz,out1+ε3kz,aux1T22=2ξ11ε0kz,out1ξ11ε0kz,out1+ε3kz,aux1
(A.29)

RTE=R11=μr3ω2ε0μ0F12kx2F2ω2ε3μ3kx2μr3ω2ε0μ0F12kx2+F2ω2ε3μ3kx2
(A.30)
RTM=R22=F2ω2ε3μ3kx2εr3ω2ε0μ0F12kx2F2ω2ε3μ3kx2+εr3ω2ε0μ0F12kx2
(A.31)

Acknowledgements

Financial support by the Spanish MICINN under contract CONSOLIDER EMET (CSD2008-00066) and PROMETEO-2010-087 R&D Excellency Program (NANOMET) is gratefully acknowledged. C. G.-M., R. O. and F.J. R.-F. acknowledge financial support from grants FPU of MICINN, FPI of U.P.V. and FPI of Generalitat Valenciana, respectively.

References and links

1.

R. Yang, M. A. Abushagur, and Z. Lu, “Efficiently squeezing near infrared light into a 21 nm-by-24 nm nanospot,” Opt. Express 16(24), 20142–20148 (2008). [CrossRef] [PubMed]

2.

L. Vivien, S. Laval, E. Cassan, X. Le Roux, and D. Pascal, “2-D taper for low-loss coupling between polarization-insensitive microwaveguides and single-mode optical fibers,” J. Lightwave Technol. 21(10), 2429–2433 (2003). [CrossRef]

3.

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

4.

U. Leonhardt and T. G. Philbin, “General Relativity in Electrical Engineering,” N. J. Phys. 8(10), 247 (2006). [CrossRef]

5.

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]

6.

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008). [CrossRef] [PubMed]

7.

W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background,” arXiv:0806.3231v1 (2008).

8.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B. I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005). [CrossRef]

9.

E. Hecht, Optics, (Addison Wesley, 4th edition, 2001).

10.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002). [CrossRef]

11.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008). [CrossRef]

12.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005). [CrossRef]

13.

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

14.

B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B 79(8), 85103 (2009). [CrossRef]

15.

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008). [CrossRef] [PubMed]

16.

Y. Xiong, Z. Liu, and X. Zhang, “A simple design of flat hyperlens for lithography and imaging with half-pitch resolution down to 20 nm,” Appl. Phys. Lett. 94(20), 203108 (2009). [CrossRef]

17.

A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. 32(23), 3432–3434 (2007). [CrossRef] [PubMed]

18.

D. P. Gaillot, C. Croënne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal–dielectric planar hyperlens,” N. J. Phys. 10(11), 115039 (2008). [CrossRef]

19.

P. H. Tichit, S. N. Burokur, and A. de Lustrac, “Waveguide taper engineering using coordinate transformation technology,” Opt. Express 18(2), 767–772 (2010). [CrossRef] [PubMed]

20.

X. Zang and C. Jiang, “Manipulating the field distribution via optical transformation,” Opt. Express 18(10), 10168–10176 (2010). [CrossRef] [PubMed]

21.

Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express 18(6), 6089–6096 (2010). [CrossRef] [PubMed]

OCIS Codes
(220.3630) Optical design and fabrication : Lenses
(230.0230) Optical devices : Optical devices
(160.3918) Materials : Metamaterials

ToC Category:
Physical Optics

History
Original Manuscript: October 26, 2010
Revised Manuscript: December 30, 2010
Manuscript Accepted: January 3, 2011
Published: February 9, 2011

Citation
C. García-Meca, M. M. Tung, J. V. Galán, R. Ortuño, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, "Squeezing and expanding light without reflections via transformation optics," Opt. Express 19, 3562-3575 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3562


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References

  1. R. Yang, M. A. Abushagur, and Z. Lu, “Efficiently squeezing near infrared light into a 21 nm-by-24 nm nanospot,” Opt. Express 16(24), 20142–20148 (2008). [CrossRef] [PubMed]
  2. L. Vivien, S. Laval, E. Cassan, X. Le Roux, and D. Pascal, “2-D taper for low-loss coupling between polarization-insensitive microwaveguides and single-mode optical fibers,” J. Lightwave Technol. 21(10), 2429–2433 (2003). [CrossRef]
  3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  4. U. Leonhardt and T. G. Philbin, “General Relativity in Electrical Engineering,” N. J. Phys. 8(10), 247 (2006). [CrossRef]
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