Nanostructured and Subwavelength Waveguides: Fundamentals and Applications

By Maksim Skorobogatiy

Optical waveguides take a prominent role in photonics because they are able to trap and to transport light efficiently between a point of excitation and a point of detection. Moreover, waveguides allow the management of many of the fundamental properties of light and allow highly controlled interaction with other optical systems. For this reason waveguides are ubiquitous in telecommunications, sensing, spectroscopy, light sources, and high power light delivery. Nanostructured and subwavelength waveguides have additional advantages; they are able to confine light at a length scale below the diffraction limit and enhance or suppress light-matter interaction, as well as manage fundamental properties of light such as speed and direction of energy and phase propagation.

This book presents semi-analytical theory and practical applications of a large number of subwavelength and nanostructured optical waveguides and fibers operating in various regions of the electromagnetic spectrum including visible, near and mid-IR and THz. A large number of approximate, while highly precise analytical expressions are derived that describe various modal properties of the planar and circular isotropic, anisotropic, and metamaterial waveguides and fibers, as well as surface waves propagating on planar, and circular interfaces. A variety of naturally occurring and artificial materials are also considered such as dielectrics, metals, polar materials, anisotropic all-dielectric and metal-dielectric metamaterials.

Contents are organized around four major themes:

• Guidance properties of subwavelength waveguides and fibers made of homogeneous, generally anisotropic materials
• Guidance properties of nanostructured waveguides and fibers using both exact geometry modelling and effective medium approximation
• Development of the effective medium approximations for various 1D and 2D nanostructured materials and extension of these approximations to shorter wavelengths
• Practical applications of subwavelength and nanostructured waveguides and fibers 

Nanostructured Subwavelengths and Waveguides is unique in that it collects in a single place an extensive range of analytical solutions which are derived in various limits for many practically important and popular waveguide and fiber geometries and materials.

• Language: English
• Publisher: Wiley
• Release date: May 23, 2012
• ISBN: 9781118343241

Book preview

Series Preface

WILEY SERIES IN MATERIALS FOR ELECTRONIC AND OPTOELECTRONIC APPLICATIONS

This book series is devoted to the rapidly developing class of materials used for electronic and optoelectronic applications. It is designed to provide much-needed information on the fundamental scientific principles of these materials, together with how these are employed in technological applications. The books are aimed at (postgraduate) students, researchers and technologists, engaged in research, development and the study of materials in electronics and photonics, and industrial scientists developing new materials, devices and circuits for the electronic, optoelectronic and communications industries.

The development of new electronic and optoelectronic materials depends not only on materials engineering at a practical level, but also on a clear understanding of the properties of materials, and the fundamental science behind these properties. It is the properties of a material that eventually determine its usefulness in an application. The series therefore also includes such titles as electrical conduction in solids, optical properties, thermal properties, and so on, all with applications and examples of materials in electronics and optoelectronics. The characterization of materials is also covered within the series in as much as it is impossible to develop new materials without the proper characterization of their structure and properties. Structure-property relationships have always been fundamentally and intrinsically important to materials science and engineering.

Materials science is well known for being one of the most interdisciplinary sciences. It is the interdisciplinary aspect of materials science that has led to many exciting discoveries, new materials and new applications. It is not unusual to find scientists with a chemical engineering background working on materials projects with applications in electronics. In selecting titles for the series, we have tried to maintain the interdisciplinary aspect of the field, and hence its excitement to researchers in this field.

Arthur Willoughby

Peter Capper

Safa Kasap

Preface

This book presents semianalytical theory and practical applications of a large number of subwavelengh and nanostructured optical waveguides. Such waveguides currently present a very popular research area as they find themselves at the interfaces of four vibrant domains of modern optics that include: wave propagation in artificial optical materials (metamaterials), propagation in highly anisotropic waveguides, surface waves propagating at the interface with metals (surface plasmon-polaritons) and polar materials (surface phonon-polaritons), as well as guidance with micro- and nanowires.

In the following we call a waveguide subwavelength either if its characteristic size is much smaller than the wavelength of light, or if a typical size of a waveguide guided mode is much smaller than the wavelength of light. Furthermore, we call a waveguide nanostructured if its crossection features a number of subwavelength-size inclusions that significantly alter optical properties of the host material. In this book we focus on waveguides made of nonmagnetic materials including dielectrics, metals, polar materials, and combinations of thereof.

The book collects in a single place a large number of analytical solutions that are derived in the long- and short-wavelength limits for a plethora of practically important and popular waveguides and fibres. The waveguides considered in this book include planar and circular isotropic and anisotropic waveguides, as well as surface waves on planar, and circular surfaces. Waveguide materials include dielectrics, metals, polar materials, anisotropic all-dielectric metamaterials, and anisotropic metal/dielectric metamaterials. We then argue that the high-accuracy analytical approximations are valid in a large frequency range and allow analytical expressions for the waveguide fundamental properties such as maximal dispersion, minimal mode area, maximal loss, highest coupling efficiency, bending loss, etc.

After analysis of the basic waveguide structures we consider waveguides made of the nanostructured materials. We argue that in the long-wavelength limit, composite materials can be considered as homogeneous anisotropic dielectrics, and modal properties of waveguides made of such materials can be again detailed analytically.

Finally, many practical application of the nanostructured waveguides are rigorously detailed in this book including, among others, low-loss low-dispersion guidance using porous THz waves, very large modal area single-mode propagation in multifilament fibres, sensing of changes in the analyte refractive index using highly confined surface plasmon-polaritons on the metal/analyte or on the metamaterial/analyte interfaces, field enhancement in subwavelength-size holes as well as field suppression in subwavelength high refractive index rods, long-range propagation of plasmons in thin metallic films, optically transparent conductors and perfect polarisation splitters based of metal/dielectric metamaterials, guidance without cutoff in deeply subwavelength hollow-core metal slot waveguides, leakage spectroscopy of radiative plasmons guided by asymmetric metallic slab waveguides.

Maksim Skorobogatiy

Montréal

January 2012

1

Introduction

Optical waveguides take a prominent role in photonics for the simple reason of enabling light to travel efficiently between a point of excitation and a point of detection while performing some useful function along its way. Optical waveguides are now ubiquitous in telecommunications, sensing, spectroscopy, light sources, high-power light delivery and other high-tech research areas and products. According to the Web of Science database, during the period 2000–2010, on average, more than 5000 research articles per year were published in the area of guided optics, reflecting the fact that design and fabrication of waveguides with various optimal performance parameters is currently a principal goal for many research groups and businesses. Performance limitations of optical waveguides mostly come from two detrimental factors, which are: fabrication imperfections such as inhomogeneity of waveguide cross section, interface roughness and so on, as well as material limitations such as absorption loss, microdefects, and so on. Therefore, a large effort is dedicated to perfection of the fabrication methods and synthesis of the novel optical materials. Another point of concern with optical waveguides is their relatively large size, which for many existing commercial products is larger than the wavelength of operation (typically 400–1600 nm). Thus, compared to the sub-100 nm size of most commercial electronic components, a typical size of the current commercial optical components is at least an order of magnitude larger. Therefore, a strong need exists to miniaturise optical waveguides to make them deeply subwavelength.

This book mainly focuses on two popular trends in the modern guided optics. One trend is a reduction of the waveguide size below that of the wavelength of operation. Depending on design, this strategy can result either in waveguides with modes that are highly localised in a subwavelength-size waveguide core (high refractive-index contrast systems), or in waveguides with modes that are quite delocalised in the waveguide cladding (low refractive-index contrast systems). The former can be used in designing highly compact optical circuits for the optical chip signal processing or even all-optical signal processing, while the latter can be used to build high-sensitivity optical sensors responsive to changes in the analyte surrounding the waveguide, to enable low-loss low-dispersion guidance of mid-IR and far-IR (THz) pulses, and so on.

The second trend is designing artificial optical materials to overcome the limitations of the naturally occurring materials. This is accomplished by blending nanoparticles, nanorods, nanowires and other types of inclusions into the uniform matrix of a host material. Depending on the material, geometry, and relative alignment of the inclusions, the resulted composite materials can exhibit properties that are significantly different from those of a host material. For example, depending on the volume fraction of spherical metallic nanoparticles in a dielectric host material, the resultant composite can behave as a dielectric (positive permittivity) or a metal (negative permittivity). Additionally, if instead of spherical nanoparticles an array of aligned metallic nanowires is used, then the resultant material can be strongly polarisation dependent. In particular, light with its electric field directed along the metallic nanowires will perceive the composite material as metal, while light with its electric field perpendicular to the wire will perceive such material as dielectric. The concept of artificial materials is very appealing as, in principle, it can significantly simplify design and fabrication of the novel optical components.

In the remainder of the book we use interchangeably the terms nanostructured materials, materials with subwavelength features, metamaterials and artificial materials. We note that in this book the term metamaterial is understood in a broad sense of an artificially designed material and not in the limited sense of negative refractive index materials.

1.1 CONTENTS AND ORGANISATION OF THE BOOK

The contents of this book are organised around four major themes, which are: guidance properties of subwavelength waveguides and fibres made of homogeneous and, generally, anisotropic materials; description of guidance by various nanostructured waveguides and fibres using the effective medium approximation; development of the effective medium approximations for various 1D and 2D nanostructured materials, understanding of the region of validity of these approximations, and systematic improvement of the effective medium theory at shorter wavelengths; practical applications of subwavelength and nanostructured waveguides and fibres. We now detail briefly the main ideas behind each of these four themes.

First, subwavelength waveguides (both dielectric and metallic) can have strongly confined optical modes characterised by large intensities of the electromagnetic fields inside a waveguide core or in the immediate vicinity of a waveguide core. Examples of such modes can be surface plasmon-polaritons propagating along a planar metallic film or a wire, or the guided modes of a high refractive index dielectric nanowire in a low refractive index cladding. In its simplest implementation subwavelength waveguide can be in the form of a dielectric wire or a slab suspended in air with waveguide size much smaller than the wavelength of light (say λ/6). One might think that the guided mode of such a subwavelength-size waveguide will be strongly delocalised in air with the effective refractive index of a guided mode close to 1. In actuality, even for the modest values of the waveguide core refractive index (say n = 1.45) the air-cladded waveguides can exhibit strong modal confinement, which is beneficial for many practical applications in sensing and nonlinear optics. Even if the mode of a subwavelength waveguide has a strong presence in the cladding region (which is, for example, the case for a low refractive-index contrast core/cladding material combination, or deeply subwavelength waveguides with high refractive-index contrast), this can be of great value for low-loss low-dispersion propagation of short pulses, which is important, for example, in THz time-domain spectroscopy.

Secondly, nanostructured waveguides that are made of composite materials with elemental features (particles, wires, etc.) that are much smaller than the wavelength of light typically operate in the effective medium regime. Examples of nanostructured waveguides include a highly porous fibre that features densely placed air channels in its core, or a planar waveguide with a core containing a periodic sequence of metal/dielectric layers. Commonly, inclusions in composite materials are made of isotropic dielectrics or metals, while sometimes they also include anisotropic polar materials such as piezo- or ferroelectrics. In the long-wavelength limit (with respect to the inclusion size) physically inhomogeneous composite materials can be viewed again as homogeneous, however, anisotropic medium. Typically, such an effective medium is not only strongly anisotropic, but its permittivity (dielectric) tensor can also accept unusual values. For example, highly porous waveguides can have effective permittivity close to that of vacuum, while metal-containing composites can have permittivity smaller than that of vacuum or even negative. Due to the flexibility in the design of material permittivity, waveguides made of such artificial materials can have many interesting properties. Among others, these include: low-loss guidance even if highly absorbing materials are used in the waveguide fabrication, single-mode guidance with very large mode area even when high refractive-index contrast material combination is used to build such waveguides, low dispersion guidance at virtually all frequencies, and so on.

Thirdly, a very interesting regime of operation of nanostructured waveguides is at the limit of validity of effective medium approximation. As we demonstrate in this book, even at shorter wavelengths, composite materials can still be described as homogeneous, however, in a limited sense. In fact, at shorter wavelengths, the effective medium approximation might still hold but only for certain directions of wave propagation, while for the other directions of wave propagation material response can be significantly influenced by the structure of a metamaterial. For example, in a periodic all-dielectric multilayer, wave propagation predominantly along the multilayer plane can be well described by effective medium theory even when size of the individual layers become comparable to the wavelength of light. At the same time, propagation of light perpendicular to the multilayer structure will be influenced by photonic bandgap formation leading to suppression of light propagation in certain frequency ranges due to destructive interference in the periodic stack. Moreover, wave propagation in structured materials can be also modified by the appearance at shorter wavelengths of resonant states strongly confined to the inclusions (or their surfaces) in a metamaterial. For example, while periodic metal/dielectric multilayers can be considered as homogeneous anisotropic dielectrics even at shorter wavelengths, wave propagation in such a medium could be significantly modified due to the appearance of the guided bulk plasmon states localised at the internal metal layers.

Finally, subwavelength and nanostructured waveguides enable many interesting practical applications. Consistent with the rest of the book, we use analytical and semianalytical methods to present and explain such applications. Among others we detail: low-loss low-dispersion guidance using porous THz waves, very large modal area single-mode propagation in multifilament fibres, sensing of changes in the analyte refractive index using highly confined surface plasmon-polaritons, field enhancement in subwavelength-size holes as well as field suppression in subwavelength high refractive index rods, long-range propagation of plasmons in thin metallic films, optically transparent conductors and perfect polarisation splitters based on metal/dielectric metamaterials, guidance without cutoff in deeply subwavelength hollow-core metal slot waveguides, leakage spectroscopy of radiative plasmons guided by asymmetric metallic slab waveguides. From simple to more complex, highlighting connections between adjacent sections we develop in this book the following major topics.

1.2 STEP-INDEX SUBWAVELENGTH WAVEGUIDES MADE OF ISOTROPIC MATERIALS

These are the simplest waveguides considered in this book. They are in the form of a planar slab or a cylindrical fibre (Figures 1.1(a) and (b)) having core refractive index ncore. The core region is surrounded by cladding of refractive index nclad. The famous example of such waveguides is a telecommunication fibre that uses a very small refractive-index contrast ( , and ) [1] to guide a single mode at . As we show later in the book, to guarantee single-mode operation of such a fibre the core radius should be smaller than a certain maximal value . Therefore, to ensure the strongest modal confinement in the fibre core (Figure 1.1(b)), which also results in the reduction of macrobending loss, one typically chooses the fibre core radius smaller but close to the cutoff radius acutoff. Because of the very low refractive-index contrast used in the telecommunication fibre, a typical core diameter is therefore large and is on the order of , which is much larger than the wavelength of light.

Figure 1.1 (a) Modes of a subwavelength waveguide are strongly present in the cladding region, with a little power guided in the waveguide core. (b) Modes of a regular waveguide (core size is comparable to or larger than the wavelength of light) are localised strongly in the waveguide core, with a little power guided in the waveguide cladding. (c) When introducing a deeply subwavelength low refractive index channel into a core of a regular waveguide, fields in such a channel could be greatly enhanced due to conservation of the transverse component of the electric displacement field.

ch01fig001.eps

When reducing the fibre core size far below the value of a cutoff radius acutoff (see Figure 1.1(a)), while keeping the wavelength of operation constant, the waveguide remains single mode, however, for too small a core radius the mode becomes strongly delocalised in the cladding. In fact, mode size tends to infinity when the core size goes to zero as for planar waveguides and exponentially fast for circular fibres. Remarkably, if the fibre core shape is perfectly circular with no roughness, and no bending is present in the fibre, there always exists at least one doubly degenerate guided mode even for infinitesimally small core radii.

In the limit of short wavelengths (high frequencies) a waveguide becomes multimode, while the fundamental mode becomes well localised in the fibre core with mode size comparable to the core size. An interesting question then is about the smallest mode size possible for a waveguide that still operates in a single-mode regime. Theoretically, one can design a single-mode waveguide of arbitrarily small size with the fundamental mode well localised in the waveguide core by using very high refractive-index contrast between the core and cladding materials. In practice, the maximal attainable refractive-index contrast is limited by the material availability. For example, in the near-IR frequency range, for the materials with relatively small optical absorption, the highest refractive index available has values between 3 (chalcogenide glass) and 4 (various forms of silicon).

We now consider in more detail the case of a step-index slab waveguide. The single-mode criterion for a slab waveguide dictates that the waveguide core size should be smaller than a certain maximal value that we call a cutoff size . Later in the book we show that for a waveguide operating in the vicinity of its single-mode cutoff frequency, the mode sizes of both the TE and TM modes are smaller than . From this expression it follows that by insuring that the refractive-index contrast between the two materials is larger than n²core−n²clad>0.4720, then the mode size becomes smaller than the wavelength of light. Such refractive-index contrast is easily achievable with almost any choice of a practical core material, when the cladding material is air. As an example, for a single-mode dielectric slab suspended in air with a core made of silica , high refractive index plastic , or chalcogenide glass , the corresponding mode sizes will be , or , respectively.

We thus conclude that using even the simplest step-index dielectric waveguides that feature high refractive-index contrast between their core and cladding materials it is possible to ensure a single-mode operation, and deeply subwavelength mode size. Moreover, when using air cladding nclad=1 in the practical realisation of such waveguides, then almost any choice of a core material (as long as ncore>1.21) will result in a mode with subwavelength size. Although simple air-clad cylindrical fibres are relatively straightforward to analyse theoretically [2, 3], experimentally (for applications in the visible and near-IR ) they are somewhat difficult to realise and to work with due to the fragile nature of suspended-in-air submicrometre diameter dielectric wires [4, 5]. Suspended-in-air subwavelength cylindrical fibres [6, 7] and slab waveguides [8] have been recently used in THz region ( ), where the resultant fibres and planar waveguides are robust free-standing structures.

Due to strong localisation of the modal fields in high refractive index subwavelength waveguides, they have been recently investigated for many applications in sensing, coupling to nanophotonic devices, nonlinear optics, signal processing, and so on. An overview of the theory, fabrication and applications of dielectric nanowires can be found in Ref. [3].

To calculate numerically the dispersion relations of the planar waveguides and circular fibres we briefly, but fully, present two standard transfer matrix formulations [9, 10] to treat planar and cylindrical multilayer waveguides.

1.3 FIELD ENHANCEMENT IN THE LOW REFRACTIVE INDEX DISCONTINUITY WAVEGUIDES

Another surprisingly simple way of localising light in subwavelength structures is by introducing a subwavelength air-filled hole into the centre of a core of a step-index waveguide [11] (see Figure 1.1(c)). For the modes of certain polarisations one can then observe a strong increase in the field intensity inside the hole, compared to the average field intensity in the fibre core. Alternatively, by placing a subwavelength air-filled slot into the core of a planar waveguide, for a TM-polarised light one can also observe strong enhancement in the air-filled slot. This effect is easy to explain by considering the boundary conditions at the interface between the fibre core and the air inclusion. Denoting to be the component of the electric field perpendicular to the interface (in the core region), then a corresponding electric-field component in the air region can be found from the condition of continuity of the displacement field across the interface . Furthermore, assuming that a waveguide mode has a dominant component of the electric field perpendicular to the core/air inclusion interface, then the density of the electromagnetic energy in the air inclusion will be larger than the density of the electromagnetic energy in the core. As an example, the field and electromagnetic energy enhancement factors in the subwavelength air channel running through a fibre core made of silica , or chalcogenide glass , will be 2.2, and 7.9, respectively. Experimentally, field enhancement in the low refractive index subwavelength inclusions have been demonstrated in planar slot waveguides and optical fibres with nanoholes both in the near-IR spectral region [12, 13] and in the THz region [11].

1.4 POROUS WAVEGUIDES AND FIBRES

Although strong field localisation can be achieved in slot waveguides, the relative electromagnetic energy in the low refractive index inclusion compared to the total modal energy is still very small. This is due to the simple fact that the size of a subwavelength inclusion is much less than the size of a waveguide core. To increase the fraction of mode propagating in air, one could think of simply including more of the low refractive index subwavelength inclusions into the waveguide core (Figures 1.2(a) and (b)). Such fibres were first introduced in Refs. [14, 15] where a periodic array of deeply subwavelength air holes in a circular fibre core was analysed. It was established that such a highly porous fibre retains a large fraction of the electromagnetic energy in the air holes inside of the core region. The necessary condition for this is that the thickness of veins separating air holes should be much smaller than the air hole size. As an example, in a fibre core of size comparable to the wavelength of light ∼λ, one would use air holes of size ∼ , separated from each other with material veins of size ∼ . It was later predicted that when using strongly asymmetric air inclusions (ellipses, rectangles, etc.), the resultant porous fibre could be highly birefringent, or even support a true single mode [16].

Figure 1.2 Highly porous fibres featuring arrays of low refractive index inclusions in the high refractive index material. (a) Planar stacks. (b) Fibres with arrays of cylindrical inclusions.

ch01fig002.eps

There are several practical advantages offered by porous fibres. First, in porous fibres the effective refractive index of the core material is greatly reduced compared to that of a solid material. Therefore, porous fibres can remain single mode even for relatively large core radii, which, in turn, greatly simplifies coupling to external light sources. For example, a typical porous fibre with 70% porosity by volume made of a polyethylene (PE) plastic ( ) would have an effective refractive index of . As a result, the largest core diameter at which the air-clad fibre remains single mode will be

, while the corresponding mode size will be . In the THz frequency range (1 THz corresponds to ) a typical beam size of a collimated source is quite large ∼1 mm, which is ∼3 times larger than the wavelength of light. A highly porous single-mode fibre presented in the example has a mode size that matches well the size of an external beam, therefore, the efficiency of coupling into the fundamental mode will be high. In contrast, a solid air-clad single-mode fibre would have a maximal diameter of , while the corresponding mode size will be ∼λ, which is significantly smaller than the beam size of a typical THz source. As a result of this mismatch, coupling to strongly localised modes in high refractive index subwavelength dielectric wires in THz frequency range is challenging.

An additional advantage of a porous fibre is a greatly reduced absorption loss. In particular, in the THz frequency region, most materials exhibit very high absorption losses well in excess of 100 dB/m. In porous fibres the modal field is mostly localised in the low-loss air region, therefore, effective modal losses can be more than an order of magnitude less than that of a fibre material.

Additionally, when comparing porous and solid-core fibres featuring modes of the same mode size we find that bending loss of a porous fibre is considerably smaller than the bending loss of a solid fibre. This is a very interesting feature of porous fibres, and multifilament fibres discussed in the next section. In fact, at a fixed frequency of operation, for the fundamental modes of the same size, the diameter of a porous fibre is always much larger than the diameter of a solid-core fibre. As a result, a porous fibre has a larger fraction of the mode concentrated in its core compared to a solid-core fibre. Due to stronger modal confinement by porous fibres, their bending losses are superior to those of solid-core fibres.

Finally, porous subwavelength fibres made of highly nonlinear chalcogenide glass were recently demonstrated as a viable platform for building mid-IR light sources [17]. This is due to the possibility of modal dispersion engineering offered by the porosity parameter, as well as due to strong confinement in the subwavelength fibre core that promotes nonlinear effects.

Experimentally, porous subwavelength fibres have been fabricated and extensively studied in the THz region [7, 18–20]. It was confirmed that porous fibres have significantly lower absorption losses than their solid-core counterparts, and that coupling to porous fibres is very efficient. Further experimental investigations are underway to study bending loss in such fibres and the possibility of their use in THz imaging spectroscopy.

1.5 MULTIFILAMENT CORE FIBRES

The principle of operation of a multifilament core fibre (MCF) is similar in spirit to that of a porous fibre. MCF feature a set of higher refractive index inclusions in a lower refractive index background. One can either use an array of deeply subwavelength ( ) high refractive index dielectric wires, or an array of much larger wires (with the size ) with a refractive index that is only very slightly higher than that of a background. Similarly to the porous fibres, MCF fibre core can be considered as a dielectric of certain effective refractive index that is only slightly higher than that of a background material. As a result, one can realise single-mode fibres featuring modes of very large mode area, which is beneficial for various high power applications [21, 22]. Moreover, it was demonstrated that multifilament core fibres have much lower bending losses when compared to a single solid-core fibre featuring a fundamental mode of the same size [23].

1.6 NANOSTRUCTURED WAVEGUIDES AND EFFECTIVE MEDIUM APPROXIMATION

For the sake of optical modeling, nanostructured waveguides can be typically substituted by equivalent waveguides made from uniform, while generally anisotropic, effective materials. For example, a highly porous air-clad fibre featuring a periodic array of subwavelength air channels can be approximated as a step-index air-clad fibre of the same diameter with a core made of the uniform low refractive index anisotropic dielectric. In a similar manner, the properties of a multifilament core fibre can be inferred [23] from the properties of an equivalent low refractive-index contrast step-index fibre. Substitution of a geometrically complex waveguide having deeply subwavelength features with a geometrically simple waveguide made of an effective uniform material is possible because metamaterials featuring deeply subwavelength inclusions behave in all respects as uniform, generally, anisotropic media.

To deduce effective optical properties (dielectric tensor) of a medium comprising periodic array of deeply subwavelength objects placed in a uniform background, one can use the Bloch theorem. Namely, in a periodic structure extended waves are of the Bloch type, with a spatial dependence of their harmonic electromagnetic fields F of frequency ω given by:

(1.1)

Numbered Display Equation

where function Uk(r) is periodic with periodicity of the underlying lattice (R is any lattice vector). Importantly, there is also a nonlinear dispersion relation that relates the values of the modal wavevector k and the operation frequency ω. For nonmagnetic nonabsorbing materials, in the regime of low frequencies when the operational wavelength is much longer than the lattice period, from the time-reversal symmetry [9] it follows that the dispersion relation can be presented in the following form:

(1.2)

Numbered Display Equation

where a is a characteristic size of a unit cell. In the long-wavelength