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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 4 — Feb. 13, 2012
  • pp: 4776–4783
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Experimental demonstration of light bending at optical frequencies using a non-homogenizable graded photonic crystal

Khanh-Van Do, Xavier Le Roux, Delphine Marris-Morini, Laurent Vivien, and Eric Cassan  »View Author Affiliations


Optics Express, Vol. 20, Issue 4, pp. 4776-4783 (2012)
http://dx.doi.org/10.1364/OE.20.004776


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Abstract

Experimental results on light bending in a non-homogenizable graded photonic crystal operating at optical wavelengths are presented in this paper. A square lattice silicon on insulator photonic crystal made of a two-dimensional chirp of the air-hole filling factor is exploited to produce the bending effect in a near bandgap frequency range. The sensitivity of light paths to wavelength tuning is also exploited to show demultiplexing capability with low insertion loss (<2dB) and low crosstalk (~-20dB). This experimental demonstration opens opportunities for light manipulation using a generalized two-dimensional chirp of photonic crystal lattice parameters. It also constitutes an alternative solution to the use of photonic metamaterials combining dielectric and metallic materials with sub-wavelength unit cells.

© 2012 OSA

1. Introduction

Controlling light paths and beam profiles of guided waves in planar optical structures has received a strong interest for some years. The main proposed approach for this relies on the use of metamaterials [1

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]

4

4. Z. Saïd, A. Sihvola, and A. P. Vinogradov, Metamaterials and Plasmonics: Fundamentals, Modelling, Applications (Springer-Verlag, 2008), pp. 3–10, Chap. 3, p. 106.

]. Coupled with the formalism of transformation optics, this approach has led to several works showing the possibility to mold the flow of electromagnetic waves in almost arbitrary shape waveguiding structures or in cloaking configurations [5

5. A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33(1), 43–45 (2008). [CrossRef] [PubMed]

10

10. W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. 96(4), 041102 (2010). [CrossRef]

]. However, this approach has not solved all the problems. At optical frequencies, strongly anisotropic metamaterials with complicated permittivities and permeabilities are needed, while the potential benefit brought by the use of photonic metamaterials is strongly mitigated by the strong optical losses induced by the use of metals. For this reason, experimental results have been only obtained by reducing the target to the use of broadband all-dielectric structures [11

11. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]

,12

12. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009). [CrossRef]

]. While removing the use of metals, molding of light beams has been in fact reduced in these works to the use of sub-wavelength dielectric structures to control the local average refractive index of planar optical waveguides.

The paper is composed of four parts. Design and fabrication of studied configuration are presented in Section 2. Section 3 is dedicated to the characterization results on light bending effect and demultiplexing operation with low insertion loss and low crosstalk, before conclusion in section 4.

2. Device design and fabrication

The chosen SOI GPhC structure started from the 260nm thick silicon film of the SOI wafer with 2µm of buried oxide and is made of a two-dimensional gradient of air-hole filling factor in a square lattice photonic crystal (PhC) with lattice constant a = 390nm. The variation of hole radius in x-y coordinate is governed by r/a(ρ) = 0.35.exp(-ρ2/2R2), where ρ = √(x2 + y2) is distance from the bottom left corner of GPhC area, as shown in Fig. 1(a)
Fig. 1 Schematic picture of (a) Graded Photonic Crystal (GPhC) structure located in a planar slab optical waveguide with two-dimensional chirp of the air hole filling factor, (b) Distribution of electric field in TE light polarization at normalized frequency a/λ = 0.25 inside the GPhC structure with input/output tapers.
), with R = 62µm. To minimize the light incident angle around zero, the PhC lattice was also rotated by 45°.

Practically, the maximum value of the holes radius is 136nm at the bottom left corner, and the holes radius size is also limited to 85nm for lattice points located above a 0.96R distance from the (0,0) point.

Figure 1(b) shows the predicted light path within the considered GPhC using two-dimensional Finite-Different Time-Domain (FDTD) simulation. In this simulation, a Gaussian beam waist of 7a (2.6µm) served to excite the electric field in transverse electric (TE) polarization at (x = 0,y = R/2) incident point, which was optimized first using Hamiltonian optics light propagation.

As can be seen, after penetrating into GPhCs region, light path is strongly curved to the right, i.e. towards the region of large radius air holes. It is underlined here that such a property could not be observed with a sub-wavelength corrugation of the photonic crystal area. The photonic structure was in fact designed to operate in band 1 in TE polarization with normalized frequencies a/λ around 0.25, i.e. close to the bandgap between band 1 and band 2 but below the light line to minimize out-off plane losses. In reciprocal space, the exploited iso-frequency curves are centered at the M points of the employed square lattice and not at the Γ point [18

18. E. Cassan, K. V. Do, C. Caer, D. Marris-Morini, and L. Vivien, “Short-wavelength light propagation in graded photonic crystals,” J. Lightwave Technol. 29(13), 1937–1943 (2011). [CrossRef]

].

To estimate the overall transmission efficiency of the studied configuration after the light 90°-turn, two sensors with the same size as the GPhC area were placed at the input and output of the GPhC simulation area, as can be seen in Fig. 1(b). Two slightly different cases were considered: one with the GPhC only, i.e. the configuration in Fig. 1(a) and another one with two additional input and output tapering regions, as shown in Fig. 1(b). These two tapers were designed to minimize the impedance mismatch for electromagnetic waves at the slab waveguide/GPhC two interfaces and thus minimize power reflection. These input tapers contain a linear increase in 20 steps of the air hole radius from r = 85nm (r/a = 0.219) to the hole radius value at the current point along the input interface of the GPhC area (varying itself from 136 to 85nm along the (x = 0,y>0) and (x>0,y = 0) interfaces according to the r/a(ρ) = 0.35.exp(-ρ2/2R2) law). By itself, the reduction of r is responsible for the decrease of the band-edge frequency of the PhC band 1, leading to strong reflection in the considered normalized frequency range around 0.25. A linear increase of the lattice period was thus simultaneously considered to maintain the normalized frequency a/λ below the band-edge limit. In practice, a linear increase of a from 305nm to 390nm at the input of the GPhC area was considered. As a whole, the designed input and output tapers are thus made of a two-dimensional chirp of the air hole radius and a one-dimensional chirp of the lattice period.

Within the proposed configuration, the photonic bandstructure (PBS) varies with the local point in space due the two-dimensional graduality of the structure. Figure 2(a)
Fig. 2 (a) Photonic bandstructure (including band 1, band 2 and light line) of a square lattice photonic crystal made of air holes with r/a = 0.31 normalized radius (air hole size considered at the light incident point), (b) Overall transmission spectrum of the studied configuration calculated using FDTD simulation, and (c) Same data as in (b) but plotted as a function of wavelength, complemented by the transmission spectrum when I/O tapers are introduced to minimize optical insertion losses.
, 2(b), and 2(c) show the calculated bandstructure at the incident point of light (where the air hole radius r/a is around 0.31) in TE light polarization, the calculated overall power transmission as a function of normalized frequency a/λ, and the overall power transmission spectra of the two structures with and without the input/output tapers as function of light wavelength, respectively. As it can be seen in Fig. 2(a), there exists a local photonic bandgap of Δω≈0.02 between band 1 (ω1≈0.26) and band 2 (ω2≈0.28) at the point M (edge corner of the first Brillouin zone). Since light is injected into the studied structure along the Γ-M direction, it is totally reflected for frequencies inside the photonic bandgap. We can see in Fig. 2(a) and 2(b) that a good agreement is obtained between the dispersion diagram and the calculated transmission that drops in the frequency range 0.26<a/λ<0.28. Considering Fig. 2(c), we see that the wavelength range for which light is bended by 90° is roughly extended from 1510nm to 1690nm (i.e ω≈0.23-0.257 in Fig. 2(b)). Wavelengths shorter than λ = 1490nm (ω>0.26) lie in the local photonic bandgap between the first band and the second band of the photonic crystal, which explains the drop in transmission. Transmission after the 90°-turn also drops for wavelengths above λ = 1700nm because we then enter the ‘long-wavelength’ homogenization (or sub-wavelength) approximation of the periodic medium, and light then propagates across the corrugated graded medium mostly in the straight direction. The maximum of 90°-turn transmission power is about −2.5dB for the GPhC configuration without taper (red curve), while it is around −0.5dB when I/O tapers are introduced (blue curve).

Fabrication of sample of this GPhC structure contained two stages. First, strip waveguides were defined using a RAITH150 electron beam lithography process using negative resist. The PhC structure was then separately insolated by means of a lithography process with positive resist. The photoresist patterns were transferred to the 150nm thick top silica cladding layer using a reactive ion etching system. This layer served as a mask to etch the silicon film through a SF6/O2 anisotropic etching process. Our optimized process carefully took into account the proximity effects in the e-beam lithography step.

To give a clear picture of the fabricated GPhC structure, four scanning electron microscopy (SEM) images are shown in Fig. 3
Fig. 3 Scanning electron microscope images of (a), (b) overall view of studied GPhC configuration with input/output tapers; (c) modulation of electron dose used in e-beam lithographic step and some detailed views of local PhC at different locations of the GPhC ; and (d) titled view of input taper of the GPhC area at larger magnification
. Figure 3(a) shows an overview of the fabricated device. Light injection is achieved by a 15µm wide input SOI strip waveguide into the GPhC area, while a large transition is used at the right bottom output to collect optical power coming from the GPhC area. In addition to this main output channel, two additional channels were placed to collect light propagating straightforwardly and after a left side 90°-turn, respectively. This approach allowed us to experimentally verify that negligible output powers were collected in the two straight and left directions. Figure 3(b) shows the full graded PhC area. Even with the chosen large viewing scale, the gradualness introduced in the GPhC region and in the input/output tapers can be guessed. Figure 3(c) shows the modulation of the electron dose used during the e-beam lithographic step to correct the proximity effects, as well as some detailed views of the local PhC at different locations in the GPhC. Last, Fig. 3(d) shows a tilted SEM view of the input taper of the GPhC area.

It was theoretically shown that this configuration is sensitive to wavelength tuning [18

18. E. Cassan, K. V. Do, C. Caer, D. Marris-Morini, and L. Vivien, “Short-wavelength light propagation in graded photonic crystals,” J. Lightwave Technol. 29(13), 1937–1943 (2011). [CrossRef]

]. Different sets of two-wavelength channels were designed for demultiplexing purpose around λ = 1550nm. Such configurations were also considered here. Figure 4
Fig. 4 Scanning electron microscope images of (a) a two-channel wavelength demultiplexer and (b) its zoom at larger magnification.
shows a fabricated structure with two output strip waveguides adjusted at proper positions using FDTD simulation to collect transmission power of two expected wavelength channels.

3. Characterization

To characterize the fabricated device, we used a tunable external cavity laser which provides a wide spectra band from 1390nm to 1620nm. Light from the laser was guided to a polarization controller to provide TE polarization, then through an optical fiber at the entrance facet of structure, and coupled into input a strip waveguide of 3µm width. The first output waveguide has a width of 90µm to cover all the GPhC area width, i.e. to collect the whole light power after the 90°-bend inside the GPhC area. This output waveguide is then slowly reduced to 3µm width with a transition length of 1mm. A microscope objective was served to collect the output transmission power, and the signal was then sent to an all-band optical component tester MT9820A in order to measure the optical transfer function.

To estimate the optical losses in fabricated GPhCs device, we designed and fabricated normalization samples with exactly the same dimensions and input/output beam conditions, but without any GPhC area inside.

3.1. Light bending effect

The normalized 90°-turn light transmission powers through the studied configurations are shown in Fig. 5
Fig. 5 Experimental overall transmission spectra of the studied GPhC configurations without input/output tapers (red line) and with input/output tapers (blue line), respectively.
and can be compared with the theoretical results of Fig. 2. The observed ripples are due to Fabry-Perot resonances at the two edges of the whole sample. Figure 5 shows that light is bended and transmitted through the GPhC area with low loss. Although the electromagnetic exploited modes are situated below the light line (as stated before) whatever the beam position within the GPhC area, out-off-plane optical losses may arise due to the in-plane two-dimensional chirp of the photonic crystal filling factor. This result is an experimental confirmation that out-off-plane losses remain at a small level in the considered configuration.

It can also be seen that the experimental bandwidth is smaller than the theoretical one, and that light bending occurs at slightly shorter wavelengths in comparison with the simulation result. This wavelength shift and spectrum small compression can be understood by the fact that 2D simulation was performed with the effective index approximation.

3.2. Two-channel wavelength demultiplexer

Figure 6
Fig. 6 Operation of two-channel demultiplexing based on GPhCs (a) channel 1: λ1 = 1552nm and channel 2: λ2 = 1616nm; (b) channel 1: λ1 = 1510nm and channel 2: λ2 = 1600nm; and (c) channel 1: λ1 = 1510nm and channel 2: λ2 = 1590nm.
shows the modeling and experimental optical power transmissions of three different GPhC structures each designed to present two wavelength channels around λ = 1550nm (see Fig. 4). In Figs. 6(a), 6(b), and 6(c), dashed lines stand for simulation transmission and rippled lines represent experimental results (related transmission spectra have been normalized by the transmission power spectrum of a sample without GPhCs but the same photonic circuit to input and collect the light). Results show a good agreement between experiment and simulation with a slight wavelength blue-shift, just like already pointed out. Two centered wavelengths of each demultiplexer are well separated and then collected at two output channels with low loss (<2dB) and low crosstalk (less than −20dB).

Beyond the possible operation of these devices as basic demultiplexers (which is not the main aim here), the obtained results show that the light paths within the proposed and studied GPhC area are sensitive to wavelength tuning. A continuous set of optical waveguides located along the output interface would show the progressive shift of the light beam by decreasing the normalized frequency.

As a whole, these results show experimentally that non-homogenizable GPhCs can be used to reconfigure the light path by changing wavelength. Optical beams operating at different wavelengths see different local parameters of the PhC lattice like the hole radius, and definitely do not see the same photonic crystal bandstructure all along the propagation as predicted in [19

19. areV. K. Do, X. L. Roux, C. Caer, D. Marris-Morini, N. Izard, L. Vivien, and E. Cassan, “Wavelength demultiplexer based on a two-dimensional graded photonic crystal,” IEEE Photon. Technol. Lett. 23(15), 1094–1096 (2011). [CrossRef]

].

These results open opportunities to realize more advanced optical functionalities by the use of GPhCs, like the demonstration of different optical functions at different wavelengths (e.g. the same device functioning as a self-collimator or beam splitter at a given wavelength and as superlens at another one), or the re-use of space at different wavelengths.

4. Conclusion

One of the simplest proof-of-concept structures proving the possibility to control electromagnetic fields at optical wavelengths in periodically corrugated media is the 90° turn. This simple device was for example theoretically explored through the formalism of transformation optics applied to photonic metamaterials or the use of all-dielectric sub-wavelength dielectric structures in refs [5

5. A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33(1), 43–45 (2008). [CrossRef] [PubMed]

8

8. Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(10), 104913 (2009). [CrossRef]

,25

25. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18(19), 20321–20333 (2010). [CrossRef] [PubMed]

], respectively. To the best of our knowledge, no experimental demonstration of it has been yet provided at optical wavelengths.

In the present work, a third theoretical approach was considered, relying on an all-dielectric structure to minimize optical losses and operating in the non-homogenizable regime with two purposes in mind: i) making possible the reconfiguration of light paths by wavelength tuning, ii) leading to easier fabrication due to larger lattice constants if compared with sub-wavelength all-dielectric structures. We choose for this a silicon on insulator graded photonic crystal made of a square lattice with a two-dimensional chirp of the air hole filling factor optimized with an Hamiltonian optics light propagation approach. Experimental results of 90° light bending were first provided. A good agreement was obtained between experimental results and FDTD simulation. Additionally, the sensitivity of light paths to wavelength was experimentally proved. A two-channel wavelength configuration was considered for this, showing low crosstalk (~-20dB). As a whole, the reported results also showed low optical loss in the fabricated graded photonic crystal structures (estimated below 2dB). This experimental demonstration opens future opportunities for light manipulation using a combination of unusual dispersive phenomena in PhCs and the additional degree of freedom brought by a generalized two-dimensional chirp of the PhC lattice cell parameters. This is also an alternative solution to the use of photonic metamaterials combining dielectric and metallic materials with sub-wavelength unit cells.

References and links

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.

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

3.

N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (John Wiley and Sons, 2006), pp. 211–221.

4.

Z. Saïd, A. Sihvola, and A. P. Vinogradov, Metamaterials and Plasmonics: Fundamentals, Modelling, Applications (Springer-Verlag, 2008), pp. 3–10, Chap. 3, p. 106.

5.

A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33(1), 43–45 (2008). [CrossRef] [PubMed]

6.

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]

7.

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]

8.

Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(10), 104913 (2009). [CrossRef]

9.

N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009). [CrossRef] [PubMed]

10.

W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. 96(4), 041102 (2010). [CrossRef]

11.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]

12.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009). [CrossRef]

13.

P. S. J. Russel, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38(8), 1599–1619 (1991). [CrossRef]

14.

P. S. J. Russel and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol. 17(11), 1982–1988 (1999). [CrossRef]

15.

E. Centeno, D. Cassagne, and J.-P. Albert, “Mirage and superbending effect in two-dimensional graded photonic crystals,” Phys. Rev. B 73(23), 235119 (2006). [CrossRef]

16.

K. Ren and X. Ren, “Controlling light transport by using a graded photonic crystal,” Appl. Opt. 50(15), 2152–2157 (2011). [CrossRef] [PubMed]

17.

E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, and J. M. Lourtioz, “Graded photonic crystals curve the flow of light: An experimental demonstration by the mirage effect,” Appl. Phys. Lett. 92(13), 133501 (2008). [CrossRef]

18.

E. Cassan, K. V. Do, C. Caer, D. Marris-Morini, and L. Vivien, “Short-wavelength light propagation in graded photonic crystals,” J. Lightwave Technol. 29(13), 1937–1943 (2011). [CrossRef]

19.

areV. K. Do, X. L. Roux, C. Caer, D. Marris-Morini, N. Izard, L. Vivien, and E. Cassan, “Wavelength demultiplexer based on a two-dimensional graded photonic crystal,” IEEE Photon. Technol. Lett. 23(15), 1094–1096 (2011). [CrossRef]

20.

Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: extended Hamiltonian method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036612 (2004). [CrossRef] [PubMed]

21.

E. Centeno and D. Cassagne, “Graded photonic crystals,” Opt. Lett. 30(17), 2278–2280 (2005). [CrossRef] [PubMed]

22.

F. Grillot, L. Viv, S. Laval, and E. Cassan, “Propagation loss in single-mode ultrasmall square silicon-on-insulator optical waveguides,” J. Lightwave Technol. 24(2), 891–896 (2006). [CrossRef]

23.

E. Cassan, S. Laval, S. Lardenois, and A. Koster, “On-chip optical interconnects with compact and low-loss light distribution in silicon-on-insulator rib waveguides,” IEEE Sel. Top. Quantum Electron. 9(2), 460–464 (2003).

24.

L. Vivien, S. Lardenois, D. Pascal, S. Laval, E. Cassan, J. L. Cercus, A. Koster, J. M. Fédéli, and M. Heitzmann, “Experimental demonstration of a low-loss optical H-tree distribution using silicon-on-insulator microwaveguides,” Appl. Phys. Lett. 85(5), 701–703 (2004). [CrossRef]

25.

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18(19), 20321–20333 (2010). [CrossRef] [PubMed]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(130.5296) Integrated optics : Photonic crystal waveguides

ToC Category:
Photonic Crystals

History
Original Manuscript: November 23, 2011
Revised Manuscript: January 15, 2012
Manuscript Accepted: February 6, 2012
Published: February 10, 2012

Citation
Khanh-Van Do, Xavier Le Roux, Delphine Marris-Morini, Laurent Vivien, and Eric Cassan, "Experimental demonstration of light bending at optical frequencies using a non-homogenizable graded photonic crystal," Opt. Express 20, 4776-4783 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4776


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References

  1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004). [CrossRef] [PubMed]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  3. N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (John Wiley and Sons, 2006), pp. 211–221.
  4. Z. Saïd, A. Sihvola, and A. P. Vinogradov, Metamaterials and Plasmonics: Fundamentals, Modelling, Applications (Springer-Verlag, 2008), pp. 3–10, Chap. 3, p. 106.
  5. A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett.33(1), 43–45 (2008). [CrossRef] [PubMed]
  6. 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]
  7. M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express16(15), 11555–11567 (2008). [CrossRef] [PubMed]
  8. Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys.105(10), 104913 (2009). [CrossRef]
  9. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express17(17), 14872–14879 (2009). [CrossRef] [PubMed]
  10. W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett.96(4), 041102 (2010). [CrossRef]
  11. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8(7), 568–571 (2009). [CrossRef] [PubMed]
  12. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3(8), 461–463 (2009). [CrossRef]
  13. P. S. J. Russel, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt.38(8), 1599–1619 (1991). [CrossRef]
  14. P. S. J. Russel and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol.17(11), 1982–1988 (1999). [CrossRef]
  15. E. Centeno, D. Cassagne, and J.-P. Albert, “Mirage and superbending effect in two-dimensional graded photonic crystals,” Phys. Rev. B73(23), 235119 (2006). [CrossRef]
  16. K. Ren and X. Ren, “Controlling light transport by using a graded photonic crystal,” Appl. Opt.50(15), 2152–2157 (2011). [CrossRef] [PubMed]
  17. E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, and J. M. Lourtioz, “Graded photonic crystals curve the flow of light: An experimental demonstration by the mirage effect,” Appl. Phys. Lett.92(13), 133501 (2008). [CrossRef]
  18. E. Cassan, K. V. Do, C. Caer, D. Marris-Morini, and L. Vivien, “Short-wavelength light propagation in graded photonic crystals,” J. Lightwave Technol.29(13), 1937–1943 (2011). [CrossRef]
  19. areV. K. Do, X. L. Roux, C. Caer, D. Marris-Morini, N. Izard, L. Vivien, and E. Cassan, “Wavelength demultiplexer based on a two-dimensional graded photonic crystal,” IEEE Photon. Technol. Lett.23(15), 1094–1096 (2011). [CrossRef]
  20. Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: extended Hamiltonian method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(3), 036612 (2004). [CrossRef] [PubMed]
  21. E. Centeno and D. Cassagne, “Graded photonic crystals,” Opt. Lett.30(17), 2278–2280 (2005). [CrossRef] [PubMed]
  22. F. Grillot, L. Viv, S. Laval, and E. Cassan, “Propagation loss in single-mode ultrasmall square silicon-on-insulator optical waveguides,” J. Lightwave Technol.24(2), 891–896 (2006). [CrossRef]
  23. E. Cassan, S. Laval, S. Lardenois, and A. Koster, “On-chip optical interconnects with compact and low-loss light distribution in silicon-on-insulator rib waveguides,” IEEE Sel. Top. Quantum Electron.9(2), 460–464 (2003).
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