## Abstract

We present the design of an invisible metamaterial fibre operating at optical frequencies, which could be fabricated by adapting existing fibre drawing techniques. The invisibility is realised by matching the refractive index of the metamaterial fibre with the surroundings. We present a general recipe for the fabrication of such fibres, and numerically characterise a specific example using hexagonally arranged silver nanowires in a silica background. We find that invisibility is highly sensitive to details of the metamaterial boundary, a problem that is likely to affect most invisibility and cloaking schemes.

©2010 Optical Society of America

## 1. Introduction

Electromagnetic invisibility, i.e. the passing of light through a structure without scattering, distortion, or absorption, is currently of great interest, due to its fascinating implications. A number of structures that are invisible to electromagnetic radiation have been theoretically proposed and experimentally demonstrated in the past few years; from simple devices such as antireflection coatings [1] and absorbing screens [2], to more exotic structures such electromagnetic cloaks, which can hide an object with a plasmonic medium [3–5] or a metamaterial possessing especially engineered effective optical properties [6–10]. In this paper we discuss metamaterials that are themselves optically transparent, as opposed to possessing cloaking properties. Recent proposals of transparent metamaterials are based on alternating layers of natural materials and double-negative metamaterials [11,12] and transformation optics [13].

Here we propose a much simpler approach: optical transparency is achieved by designing a metamaterial with refractive index that matches the surroundings, consisting of an array of sub-wavelength metallic nanocylinders surrounded by a dielectric. Such structures could potentially be fabricated via metamaterial fibre drawing, either via direct co-drawing, in which a macroscopically sized metal-dielectric preform is heated and reduced in size by several orders of magnitude [14,15], or via pumping liquid metal into existing micro- and nano- structured holey photonic crystal fibres (PCFs) [16]. Holey fibres containing feature sizes down to 30nm have recently been reported [17], showing that drawn patterned nanostructures, and thus drawn optical metamaterials, can in principle be produced.

Fibre drawing has already proven extremely successful for the large scale and accurate fabrication of PCFs [18]. In the context of metamaterials, drawing has significant advantages over conventional fabrication techniques, especially in terms of cost (the macroscopic preform is generally an inexpensive dielectric a few cm^{3} in volume) and volume production (one preform can in principle be used to make kilometers of fibre). In contrast, the lithographic techniques used for producing metamaterials at operating frequencies from the terahertz to the visible are generally quite expensive, and produce no more than a few cm^{2} of metamaterial at a time [19]. However, while drawing offers the possibility of creating nanoscale structures in volume, this comes with a constraint – they have to be uniform along one dimension. Recently it has been shown that important functionality can potentially be realised with such fibre-based metamaterials, such as low-loss mid-IR waveguiding [20], sub-wavelength waveguiding [21] and hyperlensing [22].

Here we propose another interesting structure that can be fabricated via drawing of metal cylinders in a dielectric – the optically invisible fibre. Figure 1
illustrates a schematic of the potential fabrication process, whereby a macroscopic metal-dielectric preform is drawn into a fibre that is invisible around a chosen optical wavelength (e.g. *λ* = 633nm for a He:Ne laser) with normal incidence. Our design method is based on matching the effective optical parameters for a metamaterial of 2-dimensional patterned nanocylinders of sub-wavelength pitch to its surroundings. We characterise our invisible fibre via the total scattering cross section *σ _{T}*, showing that

*σ*has a minimum when the real part of the metamaterial effective refractive index matches the surroundings. Taking the example of a fibre to be invisible at 633nm using silver nanocylinders in a silica cylinder of radius

_{T}*R*= 1μm, we study the behaviour of

*σ*with wavelength, incident angle, and polarisation. In this case more than a 95% reduction in

_{T}*σ*can be achieved over ~13nm bandwidth when compared to dielectric and conducting cylinders of the same size, as well as a 95% reduction in

_{T}*σ*for angles that are ± 10° to normal incidence at 633nm. We find that invisibility is highly sensitive to changes in the metamaterial boundary, with best results when the

_{T}*total*filling fraction – including the metamaterial boundaries – matches the bulk filling fraction. Thus, particular attention should be paid to the fibre preform design process. At larger fibre diameters our invisible metamaterial becomes a near-perfect absorber, due its low reflectance and inherently lossy nature.

## 2. Material design – bulk properties

#### 2.1 Principle

An object in vacuum is invisible if it neither reflects nor absorbs light, and if it induces no changes in phase with respect to its surroundings. Our objective is to realise an optical metamaterial that closely meets such requirements.

The optical properties of a homogeneous material are commonly characterised via its complex impedance *z* and refractive index *n*, related to the relative electric permittivity *ε = n*/*z* and relative magnetic permeability *μ = nz*. The impedance mismatch at an interface causes reflections, while the refractive index characterises the phase velocity in the medium. For example, for a homogeneous slab of length *L*, the transmission and reflection coefficients *t* and *r* for a plane wave of angular frequency *ω* at normal incidence are given by [23]

*c*is the speed of light in vacuum.

It goes without saying that a hypothetical slab with *n* = 1 and *z* = 1 is invisible in vacuum, since *r* = 0 and *t = exp(iωL/c)*. Natural materials possess an impedance and refractive index that are not unity at optical wavelengths, and are thus visible; typically bulk metals possess a permittivity *ε _{m}* < 0 and dielectrics possess

*ε*> 1, with

_{d}*μ*= 1 for both.

However, it is well known that a composite medium composed of a 2-dimensional array of thin metal wires surrounded by a dielectric in the electrostatic limit (feature sizes much smaller than the wavelength) irradiated by TE polarised light (electric field directed along the wires) can be treated as a homogeneous medium with effective permittivity *ε _{eff}* given by

*f*is the area filling fraction of the metal (see, for example, Ref [24].). Thus, by knowing

*ε*and

_{m}*ε*at a given wavelength, it is possible to compute the filling fraction at which the composite medium is invisible by solving for

_{d}*ε*= 1 in Eq. (3), yielding

_{eff}Assuming for now that the composite medium is effectively non-magnetic (*μ _{eff}* = 1), for

*f*satisfying Eq. (4) the effective index and impedance take the values

*n*= 1 and

_{eff}*z*= 1 respectively, as required.

_{eff}By applying the above procedure at a wavelength *λ* = 633nm using silver (*ε _{Ag}* = −17.7 + 0.49

*i*from interpolated values of Ref [25].) and silica (

*ε*= 2.123 from the Sellmeier equation [26]), one obtains that

_{silica}*ε*= 1.00 + 0.03

_{eff}*i*when

*f*= 0.0567. For a hexagonal lattice of cylinders with radius

*r*and centre-to-centre pitch

*d*, i.e.$f=\pi {r}^{2}/(\sqrt{3}{d}^{2}/2)$,one finds that for a cylinder diameter of 2

*r*= 30nm (the smallest drawn feature reported [17]), the pitch must be

*d*= 120nm, in which case

*λ*/

*d*~4 and the electrostatic approximation is not appropriate.

Thus, while Eq. (3) is a good starting point, higher order corrections [27] are necessary to achieve higher precision in calculating the filling fraction required for invisibility.

#### 2.2 Higher order corrections

Here we obtain corrections to the first-order theory by directly calculating *r* and *t* and inverting Eqs. (1) and (2),

*L = d, 2d, 3d*. Following the procedure outlined in [23], using Eq. (5) and (6) we verify that

*z*is independent of

*L*, and unambiguously retrieve

*n*(and thus

*ε*and

*μ*) at a given wavelength.

We use a multipole expansion in combination with the transfer matrix method to calculate *r* and *t* for uniform slabs of hexagonally arranged silver cylinders in a silica background under TE polarisation. Fields around each cylinder are expanded in a multipole basis, where analytic expressions are derived for the cylindrical boundary conditions [28]. Lattice sums are used to relate the field incoming onto each cylinder to the field outgoing from other cylinders within a singer layer of the lattice [29]. The scattering and reflection coefficients of a multi-layer stack of gratings are computed from the coefficients of a single grating layer via a recurrence relation. The permittivity of silica is included in the calculations via a wavelength-dependent Sellmeier expansion [26], whereas for silver we use a ninth-order polynomial fit to measured real and imaginary parts of complex permittivity [25] [Fig. 2(a)
and 2(b)].

We use this method to calculate *r*, *t,* and extract *n*, *z*, as a function of *f* for two fixed unit cell lengths, *d* = 10nm and *d* = 100nm. The results are shown in Fig. 3(a)
and 3(b): whereas for smaller lattices (*d* = 10nm, i.e. *λ*/*d* ~60), calculations are in excellent agreement with the quasistatic theory of Eq. (3), for larger lattices (*d* = 100nm, i.e. *λ*/*d* ~6) a significant correctionto the first order theory is required. Analytical work on the homogenisation of arrays of metallic cylinders [27] has shown that lowest order dynamic corrections to the static homogenisation theory lead to an additional dipolar magnetic response as well as a quadrupolar electric response, both of same order of magnitude. In particular, the combination of these two effects explains that we have *z* ≠ 1/*n*.

From Fig. 3(b) we conclude that a filling fraction *f =* 0.0659 yields an effective index of 1.00 + 0.01*i* at *λ* = 633nm for a lattice constant of *d* = 100nm, corresponding to a cylinder diameter of 2*r* = 27nm, which can potentially be achieved using fibre drawing techniques.

Figures 3(c) and 3(d) shows the real parts of the optical parameters extracted as a function of wavelength for *d* = 10nm, *f* = 0.0567 and for *d* = 100nm, *f* = 0.0659. In the former case [Fig. 3(c)], all optical parameters are unity at 633nm by design. In the latter case [Fig. 3(d)], the presence of a magnetic response makes it impossible to simultaneously match the refractive index and the impedance to the surroundings. As we will see next, this does not compromise device performance at the desired wavelength. We now characterise its visibility in detail in a fibre geometry.

## 3. Numerical simulations and analysis

We use the multipole method [30,31] to evaluate scattering of waves [32] at optical wavelengths for two-dimensional cylinders of radius *R* made out of the metamaterial characterised in Section 2.2. This method allows to characterise visibility via semi-analytical calculations of the scattering cross section (SCS) for any linear polarisation state, azimuthal angle, and incident angle with respect to the cylinder axis [represented by *δ*, *φ* and *θ* respectively, as shown in the schematic of Fig. 4(a)
]. The scattering cross section *σ* is defined as the total scattered power per unit length at large distances, normalised to the incident power density. In polar coordinates *(ρ,α)* this is written as

**E**and

^{0}**E**are the incident and scattered electric field, respectively. One then obtains the total SCS

^{S}*σ*by integrating over all angles,

_{T}Another parameter for characterising the scattering properties of objects is the equivalent blockage width *W _{eq}* a complex-valued parameter defined as the width of an ideal shadow which produces the same forward-scattered field as the cylinder being observed [33]. It can be shown that

*σ*= 2

_{T}*Re(W*. It follows, for example, that at a given wavelength a perfectly conducting cylinder of radius R possesses

_{eq})*σ*2 in the limit of

_{T}/2R =*R*→∞, whereas a perfectly absorbing cylinder possesses

*σ*/

_{T}*2R =*1 in the same limit. We refer to the vast literature on the scattering cross section for further details [34].

We calculate *σ _{T}* at optical wavelengths for a fibre composed of the metamaterial presented in Section 2.2, formed by 361 hexagonally arranged silver cylinders with radius

*r*= 13.5nm and pitch

*d*= 100nm within a silica cylinder of radius

*R*= 1μm in vacuum. We consider normal incidence, with the electric field parallel to the cylinders (TE polarisation,

*θ*= 90°,

*δ*= 90°). The results are shown in Fig. 4(b): the metamaterial fibre has

*σ*/

_{T}*2R*= −12.6dB at

*λ*= 633nm, corresponding to more than a 97% reduction in shadow when compared to equivalent silver or silica cylinders of the same size. Furthermore, the metamaterial fibre has

*σ*/

_{T}*2R*< 0.1 over a 13nm bandwidth (95% reduction in shadow). Additionally, an effective reduction of

*σ*occurs over ~100nm with respect to either of its constituent materials. The inset of Fig. 4(b) presents the normalised total SCS at normal incidence for

_{T}*λ*= 633nm, as a function of the azimuthal angle

*φ*, demonstrating −12 ± 0.6 dB reduction of total SCS at all angles. The corresponding contour plot of the electric field at

*λ*= 633nm for these three cylinders is shown in Fig. 5(a) –5(c); the designed fibre indeed appears invisible, exhibiting low scattering and loss.

We also calculate *σ _{T}* for the same cylinder possessing an effective

*ε*and

*μ*retrieved in Section 2.2 [Fig. 3(d)], finding excellent agreement with the full multipole calculation [Fig. 4(b)]. The minimum SCS occurs at a wavelength when

*Re(n)*= 1 (minimum phase change), as opposed to when

*Re(z)*= 1 (minimum reflection). Thus, the interference due to a phase change through the fibre, due to a small perturbation in

*n*, has a more significant effect on

*σ*than a reflection due to a small perturbation in

_{T}*z.*This is further illustrated in Fig. 5(d) and 5(e), which compares the scattered electric field magnitude

**|E**| for the metamaterial fibre at λ = 593nm (

^{S}*Re(z)*= 1.00,

*Re(n)*= 1.08) and

*λ*= 633nm (

*Re(z)*= 1.07,

*Re(n)*= 1.00).

## 4. Incident angle, polarisation, and size dependence

We now examine the dependence of *σ _{T}* on polarisation and incident angle with respect to the cylinder axis (

*δ*and

*θ*respectively). Thus far we have considered the case

*δ*= 90°,

*θ*= 90°, corresponding to TE polarisation at normal incidence. Figure 6(a) shows a density plot of total scattering cross section as a function of incident angle, while maintaining the polarisation (i.e. we fix

*δ*= 90° and vary

*θ*). Note that

*σ*< −10dB for

_{Τ}*θ*= 90 ± 10° with respect to the normal at

*λ*= 633nm. As we increase the incident angle, the regions of minimum scattering move to longer wavelengths, and for

*θ*= 90 ± 50° there exists a wavelength range where

*σ*/

_{T}*2R*< 1 over a bandwidth of ~10nm.

Figure 6(b) shows a density plot of total scattering cross section as a function of polarisation, at constant incident angle (i.e. we fix *θ* = 90° and vary *δ*). As expected, the regions of smallest scattering correspond to TE polarisation. As the electric field along the fibre decreases, so does the *σ _{T}*. For purely TM polarisation (magnetic field directed along the wires), the cylinder behaves like a dielectric.

In Section 2 we found that the metamaterial used to design our invisible fibre possesses a refractive index of *n* = 1.00 + 0.01*i* at *λ* = 633nm, corresponding to a loss of 0.43dB/μm; achieving a low-loss transparent metamaterial is thus only truly possible at smaller fibre radii. We now examine the influence of metamaterial fibre dimensions on the total scattering cross section.

Figure 7(a)
shows the normalised total SCS as a function of R, calculated analytically at λ = 633nm using retrieved optical parameters for the designed metamaterial fibre. As we increase cylinder radius *R*, *σ*
_{T}/*2R* reaches unity, thus behaving like a near-perfect absorber. In contrast, the normalised SCS for silver and silica cylinders tends to 2 at large radii, as expected.

Figure 7(b) (crosses) shows the full calculation of the normalised total SCS as a function of *R*, using the multipole method. The fibres are silica cylinders of radius R centred in the origin, containing silver nanocylinders with *r* = 13.5nm arranged in a hexagonal array of pitch *d* = 100nm, such that nanocylinder distance from the origin is *ρ* < *R-r*. Since the number of cylinders *N* increases stepwise with *R*, the total filling fraction *f _{T}* =

*Nr*when increasing

^{2}/R^{2}*R*is not constant, and goes through discontinuities. When

*f*differs from the ideal value of

_{T}*f*, regions at the boundary exist for which the local effective index near the boundary of the metamaterial fibre is not unity, causing increased scattering. Minimal

*σ*is thus achieved at values of

_{T}*R*for which

*f*= 0.659. In particular, we observe the lowest normalised SCS of−24dB for a metamaterial cylinder of

_{T}*R*= 230nm. Despite possessing significantly lower SCS compared to silver and silica cylinders of the same size,

*σ*is strongly sensitive to total filling fraction, since scattering occurs when the local filling fraction at the metamaterial boundary significantly deviates from that required by design. For example, the inset of Fig. 7 shows a contour plot of the electric field for

_{T}*R*= 470nm (containing 73 nanocylinders, thus

*f*= 0.602 and

_{T}*σ*/

_{T}*2R*= −6.2dB) and

*R*= 480nm (containing 85 nanocylinders, thus

*f*= 0.672 and a much reduced normalised total SCS,

_{T}*σ*/

_{T}*2R*= −19.4dB).

Note that the metamaterial fibre with *R* = 3μm already contains 3241 silver nanocylinders. Due to memory constraints, we could not perform a full calculation of the metamaterial total SCS for *R* > 3μm.

## 6. Conclusions and future outlook

In conclusion, we have designed and characterised a metamaterial fibre which exhibits a strong reduction in scattering cross section compared to metal and dielectric cylinders of equivalent size. The design procedure can be appropriately adapted to create invisible fibres at any optical wavelength by changing the metal/dielectric filling fraction, or create fibres with other arbitrary permittivity between that of the background and of the metal. Our device operates optimally at the designed wavelength under TE polarisation and normal incidence, yet excellent performance is maintained for ~10nm of bandwidth, or within a ± 10° variation in incident angle with respect to normal incidence, where a reduction in shadow of more than 95% is maintained. Additionally, visibility is significantly reduced over 100nm bandwidth with respect to its constituent materials. Optical parameters obtained from extraction procedures give a good estimate of the total scattering cross section, unless imperfect termination of the nanostructure at the edges causes additional scattering. As with other invisible metamaterials, our device is intrinsically limited by loss, being invisible at small radii, yet behaving like a near-perfect absorber for larger fiber radii. The feasibility of co-drawing metal-dielectric microstructured fibres has already been demonstrated [14,15], and so has the feasibility of drawing air-dielectric nanoscopic features [17]; future work will focus on combining these techniques to produce nanoscale metallic features. While being harder to fabricate, the concept we presented can be extended to produce other 3-dimensional optically invisible structures, such as spheres, which would have a wider tolerance on incident angle and polarisation.

## Acknowledgements

This material is based on research sponsored by the Air Force Research Laboratory, under Agreement No. FA2386-09-1-4084. B.T.K. acknowledges support from an Australian Research Council Future Fellowship.

## References and links

**1. **J. Ward, “Towards invisible glass,” Vacuum **22**(9), 369–375 (1972). [CrossRef]

**2. **R. L. Fante, M. T. McCormack, T. D. Syst, and M. A. Wilmington, “Reflection properties of the Salisbury screen,” IEEE Trans. Antenn. Propag. **36**(10), 1443–1454 (1988). [CrossRef]

**3. **A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **72**(1), 016623 (2005). [CrossRef] [PubMed]

**4. **A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express **15**(6), 3318–3332 (2007). [CrossRef] [PubMed]

**5. **B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. **103**(15), 153901 (2009). [CrossRef] [PubMed]

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

**7. **D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science **314**(5801), 977–980 (2006). [CrossRef] [PubMed]

**8. **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]

**9. **T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science **328**(5976), 337–339 (2010). [CrossRef] [PubMed]

**10. **N. A. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express **15**(10), 6314–6323 (2007). [CrossRef] [PubMed]

**11. **W.-H. Sun, Y. Lu, R.-W. Peng, L.-S. Cao, D. Li, X. Wu, and M. Wang, “Omnidirectional transparency induced by matched impedance in disordered metamaterials,” J. Appl. Phys. **106**(1), 013104 (2009). [CrossRef]

**12. **Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A **78**(2), 023813 (2008). [CrossRef]

**13. **C. Yang, J. Yang, M. Huang, J. Shi, and J. Peng, “Electromagnetic cylindrical transparent devices with irregular cross section,” Radioengineering **19**, 136–140 (2010).

**14. **J. Hou, D. Bird, A. George, S. Maier, B. T. Kuhlmey, and J. C. Knight, “Metallic mode confinement in microstructured fibres,” Opt. Express **16**(9), 5983–5990 (2008). [CrossRef] [PubMed]

**15. **A. Tuniz, B. T. Kuhlmey, R. Lwin, A. Wang, J. Anthony, R. Leonhardt, and S. C. Fleming, “Drawn metamaterials with plasmonic response at terahertz frequencies,” Appl. Phys. Lett. **96**(19), 191101 (2010). [CrossRef]

**16. **M. A. Schmidt, L. N. Prill Sempere, H. K. Tyagi, C. G. Poulton, and P. S. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys. Rev. B **77**(3), 033417 (2008). [CrossRef]

**17. **Y. Ruan, H. Ebendorff-Heidepriem, and T. M. Monro, “Subwavelength soft glass fibres with extremely small hole size for field enhancement,” in *Proceedings of the Australasian Conference on Optics, Lasers and Spectroscopy and Australian Conference on Optical Fibre Technology* (Adelaide, Australia, 2009).

**18. **J. C. Knight, “Photonic crystal fibres,” Nature **424**(6950), 847–851 (2003). [CrossRef] [PubMed]

**19. **A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: recent advances and outlook,” Metamaterials (Amst.) **2**(1), 1–17 (2008). [CrossRef]

**20. **M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express **17**(17), 14851–14864 (2009). [CrossRef] [PubMed]

**21. **E. J. Smith, Z. Liu, Y. Mei, and O. G. Schmidt, “Combined surface plasmon and classical waveguiding through metamaterial fiber design,” Nano Lett. **10**(1), 1–5 (2010). [CrossRef]

**22. **A. Tuniz, P. Chen, B. T. Kuhlmey, and S. C. Fleming, “Design of an optical hyperlens with metallic nanocylinders,” in *Proceedings of META '10, 2nd International Conference on Metamaterials, Photonc Crystals and Plasmonics* (Cairo, Egypt, 2010).

**23. **P. Markos, and C. M. Soukoulis, *Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials* (Princeton University Press, 2008), Chap. 14.

**24. **R. C. McPhedran, N. A. Nicorovici, and L. C. Botten, “The TEM mode and homogenization of doubly periodic structures,” J. Electromagn. Waves Appl. **11**(7), 981–1012 (1997). [CrossRef]

**25. **P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

**26. **I. H. Malitson, “Interspecimen Comparison of the Refractive Index of Fused Silica,” J. Opt. Soc. Am. **55**(10), 1205–1208 (1965). [CrossRef]

**27. **R. C. McPhedran, C. G. Poulton, N. A. Nicorovici, and A. B. Movchan, “Low frequency corrections to the static effective dielectric constant of a two-dimensional composite material,” Proc. R. Soc. Lond. A **452**(1953), 2231–2245 (1996). [CrossRef]

**28. **L. C. Botten, N. A. P. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A **17**(12), 2165–2176 (2000). [CrossRef]

**29. **R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and K. A. Grubits, “Lattice sums for gratings and arrays,” J. Math. Phys. **41**(11), 7808–7816 (2000). [CrossRef]

**30. **T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. De Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**(10), 2322–2330 (2002). [CrossRef]

**31. **B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B **19**(10), 2331–2340 (2002). [CrossRef]

**32. **D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A **11**(9), 2526–2538 (1994). [CrossRef]

**33. **P. S. Kildal, A. A. Kishk, and A. Tengs, “Reduction of forward scattering from cylindrical objects using hardsurfaces,” IEEE Trans. Antenn. Propag. **44**(11), 1509–1520 (1996). [CrossRef]

**34. **E. F. Knott, J. F. Shaeffer, and M. T. Tuley, *Radar cross section* (New York: SciTech Publishing, 2004), Chapter 3.