Artificial Media Optical Properties Subwavelength Scale Philippe Lalanne Centre National de la Recherche Scientifique, Orsay, France Mike Hutley Floating Images Ltd., Hampton, United Kingdom INTRODUCTION In most optical materials, the atomic or molecular structure is so fine that the propagation of light within them may be characterized by their refractive indices. When an object has structure which is larger than the wavelength of light, its influence on the propagation of light may be described by the laws of diffraction, refraction, and reflection. Between these two extremes is a region in which there is structure that is too fine to give rise to diffraction in the usual sense but is too coarse for the medium to be considered as homogenous. For this, a full description can only be achieved through a rigorous solution of Maxwell s electromagnetic equations, and resonance phenomena are often observed. Recent developments in microlithography have extended the possibility of generating subwavelength structures, and it is now possible to produce materials with remarkable new optical properties. OVERVIEW The basic principles of optical design and the physics of reflection, refraction, and diffraction on which it is based have been well understood for a very long time. This knowledge has enabled the successful development of optical science and technology over the last couple of centuries; in recent years, it has developed new levels of sophistication with the availability of computer programs for the optimization of all types of systems. However, until relatively recently, the entire optical technology has been limited by the very reasonable constraint that optical systems should be designed based on the availability of materials. Consider the very simple task of designing an antireflection coating to work at one wavelength at normal incidence. The theory tells us that a single layer will generate two reflected waves, one from the air/layer interface and one from the layer/substrate interface. If the optical thickness of the layer is such that the two are exactly out of phase by p and that the refractive index of the layer is equal to the square root of that of the substrate, the two reflections will have the same amplitude and will cancel exactly. Thus, a single layer will behave as a perfect antireflection coating. Unfortunately, most common optical glasses have a refractive index in the region of 1.55; thus, the layer needs to have a refractive index of 1.245. Sadly, such a material does not, as far as we know, exist; thus, more complex solutions have to be found. Consider, however, what happens if we introduce into a standard material a very fine structure, e.g., a series of holes. If the scale of the structure is substantially smaller than the wavelength of light, it will not be resolved by the light and the light sees a composite material of which the optical properties are between those of air and those of the base material. By varying the fraction of material that is removed, it is possible to control the effective refractive index and add to the range of materials that are available to the optical designer. This principle can be expanded to include composites consisting of several components. Strictly speaking, all structures are three-dimensional (3- D); however, in practice, many examples consist of relatively shallow modulations of an optical surface. It is therefore customary to refer to one-dimensional (1-D), two-dimensional (2-D), and 3-D structures when describing, e.g., a simple diffraction grating, a crossed diffraction grating and a photonic crystal, respectively. In the limit where the wavelength of light is very much greater than the dimensions of the structure, it is possible to regard the material as being homogeneous and possessing an appropriate effective value of refractive index. The approach is known as homogenization. When the dimensions of the structure are close to or larger than the wavelength of light, the optical properties are dominated by the effects of diffraction. However, there is a region between these two extremes where the dimensions are sufficiently small that no diffracted orders propagate, but where it is not possible to apply the simple approximations of a homogeneous medium. This is often referred to as the subwavelength domain for which homogenization techniques do not strictly apply; however, it gives a good physical understanding of the medium properties. Developments in microlithography and 62 Encyclopedia of Optical Engineering DOI: 10.1081/E-EOE 120009537 Copyright D 2003 by Marcel Dekker, Inc. All rights reserved.
Artificial Media Optical Properties Subwavelength Scale 63 associated technologies now make it possible to put these principles into practice and, in particular, to produce artificial media which operate in the resonance domain. As a result, the subject of subwavelength structures now attracts a great deal of research interest with a view to extending the possibilities of waveguides, optical fibers, and electrooptical materials. In the present article, we shall review recent developments in the understanding and technology of subwavelength structures and artificial optical media. We shall start by considering the underlying physics and then describe a selection of real examples by way of illustration. LONG-WAVELENGTH LIMIT OF ARTIFICIAL MEDIA The propagation of electromagnetic waves in composite media with subwavelength inhomogeneities is an old but still very active subject. In principle, the inhomogeneity could be randomly or regularly distributed. However, if it is random, there will in fact be a full spectrum of spatial frequencies present. Furthermore, if the distribution of spatial frequencies is such that there is a significant proportion at wavelengths which are similar to that of the light, then the medium will scatter. Therefore, it is preferable to produce structures which are regular and periodic. In this way, it is possible to control the spatial frequencies that are present and avoid random scattering. Moreover, the theory of composite materials is made easier for periodic structures because a reciprocal lattice analogous to that found for Bloch electron wave in crystals can be introduced to drastically simplify the analysis. Initially, various effective medium approaches like the Maxwell Garnett or Clausius Mossotti approximations [1] were used to determine the dielectric constant of Fig. 1 Examples of 1-D, 2-D and 3-D periodic structures. Black and white regions might correspond to high and low refractive index materials, for instance. For periods sufficiently small compared to the wavelength of the illumination beam, these structured periodic structures may behave as artificial homogeneous materials whose optical properties (refractive index, birefringence, dispersion) are related to the fine geometry of the periodic arrangement. We may talk of refractive index engineering by structuring materials at a subwavelength scale. periodic composite materials like those of Fig. 1. It was later realized that those approaches which rely on a spatial average but which ignore the fine geometry of the inhomogeneity were inadequate even in the long-wavelength limit; that is, when the period is infinitely smaller than the wavelength. The long-wavelength limit is a very important case for which some analytical general results are available from theories known as effective medium theory, homogenization or mean-field theory in the literature. The first important result is the genuine equivalence between periodic artificial media and homogeneous material. For instance, 1-D periodic structures are equivalent to uniaxial crystals with form birefringence; [2] 2-D or 3-D periodic structures are, in general, equivalent to biaxial crystals. This equivalence is far from being trivial from a mathematical point of view, see Ref. [3], for instance. For 1-D periodic structures, the ordinary, n o, and extraordinary, n e, indices of refraction take the simple forms: n o ¼ hei 1=2 and n e ¼ h1=ei 1=2 ð1þ where e denotes the relative permittivity of the periodic structure, and the brackets refer to spatial averaging. For the lamellar two-medium structure of the leftmost part of Fig. 1, Eq. 1 becomes n o ¼ ½f e H þð1 f Þe L Š 1=2 and n e ¼ ½f =e H þð1 f Þ=e L Š 1=2 ð2þ In Eq. 2, the fill factor, f, represents the fraction of high-index material with relative permittivity e H imbedded in the low-index material with relative permittivity e L. Eq. 2 can be obtained by elementary considerations. [2] With the exception of y-polarized waves propagating in the xz plane of 2-D periodic structures for which the effective index is equal to hei 1/2, no closedform expressions for the effective indices are available for 2-D or 3-D periodic structures. Note that, for 2-D periodic structures, simple expressions [4] are known for the upper and lower bounds of the two principal effective indices experienced by waves propagating in the y direction. When the index contrast is small, these bounds are generally quite narrow and their average is a good approximation for the effective index. REAL ARTIFICIAL MEDIA The long-wavelength limit is an academic case. Even with the more advanced nanofabrication facilities, it is not possible to manufacture optical periodic structures that operate in the long-wavelength limit. With the present A
64 Artificial Media Optical Properties Subwavelength Scale state-of-the-art in nanofabrication, it is only possible to manufacture structures with periods slightly smaller than optical wavelengths. For these real artificial media, one does not have a theorem of equivalence between periodic structures and homogeneous media. On the contrary, the physical properties of real composite materials may sometimes strongly differ from those of homogeneous media. Two different approaches may be distinguished. In the first approach, one looks for closed-form expressions for the effective index by expanding the effective index, n eff, in a power series of the period-to-wavelength ratio, L/l n eff ¼ n ð0þ þ n ð2þ ðl=lþ 2 þ n ð4þ ðl=lþ 4 þ ð3þ where n (0) represents the effective index in the longwavelength limit, and n (2) and n (4) are dimensionless coefficients depending on the microgeometry. Since the pioneer work by Rytov, [5] closed-form expressions up to the order four are available for 1-D two-layered media like that of the left-hand side of Fig. 1. The homogenization of two-component layered media is simplified by the fact that the permittivity is piecewise constant and, thus, that the modes supported by the structure are analytically known. Closed-form expressions for the effective index up to the second orders (and to the fourth order for some specific directions and polarizations) are now available [6,7] for arbitrary 1-D periodic structures. From these expressions, it is concluded that the normal surface of ordinary waves is still a sphere up to the second order (deviation being observed in the fourth-order term only) and that the normal surface of extraordinary waves is no longer an ellipsoid of revolution up to the second order, see Ref. [7] for the general case and Refs. [5,8] for the two-component layered media of Fig. 1. For 2-D periodic structures, almost no results are available: closed-form expressions (requiring, however, a matrix inversion) up to the second order were derived in Ref. [9] for the effective indices experienced by waves propagating in the y direction. In the second approach, one resorts to computation. Because Maxwell s equations for linear dielectric materials are exact, computation plays a crucial role in the analysis and design of periodic artificial media. In principle, any classical numerical method in electromagnetism may be used to compute the effective index; for instance, boundary-matching methods [10] appear quite attractive for special shapes like spheres which do not overlap. In recent years, specific attention has been devoted [11,12] to Fourier expansion techniques which can be used to study any periodic microgeometry and are therefore much wider in scope than the previous approach. Fourier expansion techniques are not exempted from numerical difficulties [13,14] due to the problems of accurately representing permittivity discontinuities with Fourier series. The Fourier plane-wave method of Ref. [15] that incorporates recent Fourier factorization theorems [16] on the product of discontinuous functions appears particularly efficient. SUBSTITUTION OF A SUBWAVELENGTH GRATING BY AN ARTIFICIAL MEDIUM Let us consider the diffraction problem in Fig. 2a, where a subwavelength grating of depth, h, is illuminated at oblique incidence by a linearly polarized plane wave with a free-space wavelength, l. The refractive index of the incident medium is n 1 and that of the substrate is n 2. For the sake of simplicity, we restrict the following discussion to 1-D gratings illuminated under transverse electric (TE) polarization (the electric field vector of the incident plane wave is perpendicular to the plane of incidence). However, the general conclusions we will derive basically holds for any structure including volume or crossed gratings, for arbitrary polarization, and for arbitrary incidence (which may be out of the plane of dispersion) as long as the grating permittivity is independent of y. In this section, we seek to answer the following question: under what conditions may the complex diffraction problem of Fig. 2(a) be approximated to by a simple refraction reflection problem on a homogeneous thin film with an effective refractive index, n eff? Clearly, the expected value for n eff is that seen by a Fig. 2 Replacement of a grating whose permittivity is independent of the y direction by a homogeneous artificial thin film with an effective index, n eff. Is it legitimate?
Artificial Media Optical Properties Subwavelength Scale 65 wave propagating in the xy plane of the 1-D periodic structure of Fig. 1 with a polarization along the z direction and with an oblique angle, y, with respect to the y direction verifying Snell s law n 1 sinðyþ ¼ n eff sinðy 0 Þ Note that for the general case, because the homogeneous thin film is possibly anisotropic, two effective indices experienced by the extraordinary and ordinary waves may be defined for the equivalent thin film. It is important to bear in mind that the replacement of Fig. 2 can only ever be an approximation because in no real situations are the two problems of Fig. 2 strictly equivalent. See Refs. [17,18] where a strict mathematical equivalence is demonstrated in the long-wavelength limit (L/l! 0). We will be concerned in the following by an approximate physical equivalence especially valid for the far-field patterns. The answer to the question is far from being trivial, and many authors have contributed to partially answer it. For 1-D lamellar gratings with high modulation contrasts, see Refs. [9,19,20]; for slanted volume gratings with a sinusoidal modulation, a situation for which the grating permittivity depends on y, see Refs. [7,21]); for 1-D gratings under conical mounts, see Ref. [22]; for 2-D gratings, see Refs. [9,23,24]. In general, three conditions are necessary so that the two problems of Fig. 2 are approximately equivalent. First Condition The first condition is that only the zeroth orders propagate in the substrate and in the incident medium, all the other orders have to be evanescent. Whether a diffraction order propagates or not is given by the grating equation. If only the zeroth transmitted and reflected orders are to propagate, it is immediately deduced from the grating equation that the period-to-wavelength ratio must verify ð4þ structure with a speed c/n eff (all the other modes have a complex propagation constant), this mode travels backward and forward between the two grating boundaries in the same way as multiple-beam interference occurs in a thin film with a refractive index, n eff. Consequently, the zeroth-order reflected and transmitted amplitudes are approximately those of the thin film. If more than one mode propagates in the grating, this picture is no longer valid, see the discussion in Ref. [25]. The fact that for a given frequency only one mode propagates in the grating depends mainly on the grating geometry and weakly on the diffraction geometry. It is actually independent of the refractive indices of the incident medium and of the substrate and weakly depends on n 1 sin(y), a quantity that determines the direction of propagation in the grating (Eq. 4). Fig. 3 shows the domain for which only one mode propagates into the lamellar grating of Fig. 2(a) for normal incidence and for n L =1 and n H = 2.3. The horizontal dashed line represents the largest grating period A L l < 1 maxðn 1 ; n 2 Þþn 1 sinðyþ ð5þ where max holds for the maximum of the arguments. The condition of Eq. 5 that provides a cutoff value solely dependent on the diffraction geometry (n 1 sin(y), the refractive indices of the incident medium and of the substrate) does not depend on the grating geometry. It is necessary but not sufficient. Second Condition The second condition is more subtle and is related to the number of propagating modes that are able to propagate in the xy plane of the 1-D periodic structure of Fig. 1 with a polarization along the z direction and with an oblique angle, y. When only one mode propagates in the periodic Fig. 3 Structural cutoff. Domain of grating parameters (fill factor and period) for which only one mode propagates in the lamellar grating of Fig. 2(a). The computation is performed for TE polarization, for normal incidence and for n L = 1 and n H = 2.3. The fill factor is defined as the fraction of high refractive index material. The horizontal dashed line at abscissa 1/n H represents the largest grating period below which one can achieve a full range of effective indices from n L to n H simply by varying the fill factor from 0 to 1 while preserving a good analogy with artificial thin films.
66 Artificial Media Optical Properties Subwavelength Scale below which one can achieve a full range of effective indices between n L and n H simply by varying the fill factor while preserving the analogy with an artificial thin film. It is called the structural cutoff in Refs. [25,26] to emphasize that it depends mainly on the grating structure and not on the diffraction geometry. It is an important design parameter, in practice, because the larger it is, the more easily can the grating be manufactured. Third Condition The last condition is related to the grating depth. As pointed out in Refs. [20,27], the abovementioned onemode picture is no longer valid for small grating depths. If the grating depth is small enough, the evanescent modes that are created at the top and bottom grating interfaces may tunnel through the grating region and participate with the fundamental to the multiple-beam interference. For dielectric gratings, the impact of evanescent modes on the grating effective properties is significant for grating depths smaller than a quarter wave. [27] The above set of three conditions is rather complex because, basically, all the diffraction problem parameters are involved in the validity of the replacement: the refractive indices of the substrate and of the incident medium from Eq. 3, the grating geometry from the second condition, and the grating depth from the third condition. However, in general, if the above three conditions are fulfilled, in practice, the grating diffraction problem of Fig. 2(a) can be seen, with a good approximation for the far field, as a simple refraction reflection problem on a homogeneous thin film. This drastic reduction in terms of complexity has suggested in the past, and recently, interesting applications [28] for artificial media synthesized by etching surfaces at a subwavelength scale. For example, we note applications exploiting the form birefringence of binary gratings for fabricating wave plates [29,30] or wire-grid polarizers and for characterizing subwavelength lamellar gratings, [31] applications relying on continuous grating profiles which mimic gradient index perpendicular to the substrate mainly for broadband antireflection coating and applications relying on binary profiles which mimic gradient index parallel to the substrate mainly for fabricating blazed diffractive elements. We shall now describe some of these in a little more detail and the interested reader will find a collection of articles on these subjects in Ref. [32]. WIRE-GRID POLARIZER The wire-grid polarizer was probably the earliest device to exploit the form birefringence of subwavelength metallic gratings as it was used by Hertz to test the properties of the newly discovered radio wave in the late 19th century. It consists of a fine grid of parallel metal wires with a spacing that is less than that of the wavelength of light. The oscillating electric field of the incident light tends to induce electric dipoles at the surface and the response of the material to this field determines its optical properties. In a metal, there are electrons which are free to vibrate under the influence of the field, electric dipoles are induced which then reradiate in such a way that the light is reflected. In a dielectric, there are no free electrons and the light is transmitted. In a wire-grid polarizer, the electrons are free to oscillate along the wires but not perpendicular to them because the wire width is smaller than the wavelength. Therefore, for light polarized parallel to the wires (TE polarization case), it behaves as a conductor and reflects, whereas for light of the orthogonal polarization, it behaves as a dielectric and transmits. This result may simply be derived by inserting into Eq. 2 a large negative value for the relative permittivity, e H, of the metal and assuming e L = 1. For wires of high conductivity, the transmission is significantly greater than the proportion of the area occupied by the gaps between the wires. There are various ways of producing wire-grid polarizers. For the far infrared, it is possible to wind a wire around a suitable former and produce a free-standing grid. For shorter wavelengths in the near infrared, it is necessary to form them on a transparent substrate such as zinc selenide or KRS5. In this case, it is possible either to rule a diffraction grating directly into the surface of the substrate or to record in photoresist a fine, straight line interference pattern. Either way, this provides a corrugated surface which may be metallized by vacuum deposition at an oblique angle in such a way that only the tips of the corrugations are coated and the net result is a series of very fine parallel wires. [33] It is also possible to apply the techniques of photolithography combined with laser or electron beam writing to generate a pattern of metallic strips. In general, the greater the wavelength-toperiod ratio, the better the wire-grid polarizer performs. For most applications, a ratio of 10:1 would be desirable. As the resolution of lithographic processes improves, the smaller it is possible to make the period of the grid and the shorter the wavelength at which it will function as a polarizer. They are now operating in the visible [34 36] and even in the ultraviolet regions. [37] However, the performance is degraded by the low conductivity of metals in the visible and the ultraviolet regions of the spectrum. ANTIREFLECTION COATING The feature of an optical surface which gives rise to unwanted Fresnel reflections is the sudden transition, or
Artificial Media Optical Properties Subwavelength Scale 67 impedance mismatch, from one optical medium to another. If the transition can be made more gradual and extended over at least a significant fraction of a wavelength, the reflection can be significantly reduced. This has been achieved in glass by treating the surface with acid. Under the appropriate circumstances, material may be leached out of the glass to leave a more open structure in the region of the surface and a gradually more dense structure as one penetrates into the glass. As this occurs at a molecular level, it is on a scale much finer than the wavelength of light and is equivalent to a gradual change of refractive index and the reflection of the surface is reduced to a level as low as that achieved with very complex multilayer antireflection coatings. An analogue of the leached glass antireflection surface is to be found on the eye of the night-flying moth. The cornea is covered with a fine regular hexagonal array of protuberances which have a period of about 200 nm and a similar depth and a cross section that is approximately sinusoidal. This natural corrugation was discovered by Bernhard in 1967 who postulated [38] that it results in a gradual change of refractive index which reduced the reflection over a wide spectral and angular bandwidth and significantly improved the moth s camouflage. The effect was first copied by Clapham and Hutley [39] by recording finely spaced optical interference fringes in two orthogonal directions in photoresist. This generated an egg box periodic modulation in the surface. The antireflection properties of such a surface match those of multilayer interference coatings over a spectral region which covers the visible and the near infrared. Although photoresist is a convenient material in which to make the moth-eye structure, it is very soft and highly absorbant in the blue; thus, it is unsuitable for most practical applications. However, the structure may be copied by electroforming in nickel. Such a copy appears very dark blue, almost black, because almost all of the light is absorbed. From the nickel copy, it is possible to copy the form of the surface into very highly polymerized plastic through the process of UV embossing and into slightly less hard plastics by hot pressing. [40] Recently, Hutley and Gombert [41] have developed the technology to the stage where large areas (a substantial fraction of 1 m 2 ) are available and their team is finding commercial applications such as on the reverse side of Fresnel lenses in overhead projectors. The problem of surface reflection is particularly acute for semiconductors which have very high values of refractive index in the visible and the near-infrared spectral regions. The reflection losses at normal incidence are 40%; to reduce them would be of interest for solarcell applications and other optoelectronic devices. Binary pillar structures (with antireflection performance comparable to those achieved by a single-layer coating) have Fig. 4 Antireflection corrugated surface. Scanning electron micrograph of an antireflection surface generated with an electron beam writer. The surface is composed of a square grid of 350-nm-high pyramids etched into silicon. The grating period is 150 nm. Under illumination at normal incidence, the reflectivity of the antireflection surface does not exceed 4% over a broad spectral domain from 200 nm to 1.1 mm. For comparison, the reflectivity of bare silicon substrate is above 40% in this spectral domain. More details can be found in Ref. [45]. (Courtesy of Yoshiaki Kanamori, University of Tohoku.) been investigated [42] for the 10 25 mm region of the spectrum by conventional lithography techniques and reflectivities as low as 0.5% have been experimentally demonstrated [28] for the 10.6-mm line of the CO 2 laser. For visible antireflection coating, moth-eye-type structures with a 260-nm period have been fabricated [43] on silicon substrates with overall performance comparable to two-layer coatings. More recently, using electron-beam lithography, high aspect-ratio pyramid profiles similar to optimal ones predicted by electromagnetic analysis [44] have also been produced [45] (Fig. 4). Quite apart from their optical performance, structures of this type offer the additional advantages that there are no problems with adhesion or with diffusion of one material into another. Because they are monolithic and introduce no foreign material, they also tend to be more stable and durable than multilayer dielectric materials, particularly when used with high-powered lasers. [46] On the other hand, they are generally more sensitive to mechanical damage. BLAZED GRATINGS In order to maximize the efficiency with which a grating or other diffracting component directs light into a chosen order of diffraction, it is necessary that it be blazed. A
68 Artificial Media Optical Properties Subwavelength Scale That is to say that the form of the groove profile is that of a prism which refracts light into the same direction as that of the chosen order of diffraction. This may be explained by considering a grating working in the first diffracted order; thus, there is an optical path difference of one wavelength for light passing through adjacent grooves. As we consider different parts of the wavefront, the geometry of the diffraction process introduces a linear variation of phase of 2p from one side of a groove to the other. In order to achieve maximum amplitude in the diffracted wave, it is necessary to keep the grooves in phase so that all grooves interfere constructively and then to introduce a sudden change of 2p at the groove boundaries. A triangular prism produces such a variation of optical path across the grating period by varying the relative proportion of the path in air and in the material of the prism. Gratings with a triangular profile are often referred to as échellettes and have been the standard way of maximizing the efficiency of spectroscopic gratings for the last 90 years. Unfortunately, for gratings of high dispersion (and for diffractive optical components of high numerical aperture), the angles of diffraction and, hence, the prism angles become quite high. It then happens that each groove casts a shadow on its neighbor, the groove is vignetted, and energy is lost. However, the same prism effect can be achieved with a material of constant thickness but in which there is a progressive variation of refractive index. If such a structure were fabricated, vignetting would still occur (the optical paths within the grooves would be curved, as in a mirage). Such a variation of effective index can also be achieved with an artificial optical medium composed of binary structures consisting of a series of microscopic pillars of varying dimension, provided that the interpillar distances are small enough compared to the wavelength of light. For these structures, if the interpillar distances are not too small (let us say about the structural cutoff for the considered wavelength), practically no vignetting is observed. [26] In fact, each pillar acts as a waveguide [47] which conducts the light from one side of the grating to the other without shadowing, but introduces a phase lag which depends upon the pillar dimensions. At the exit surface of the grating, there is effectively an array of coherent phased emitters. Binary blazed gratings of this form have been made in small areas using electron beam writing and have demonstrated efficiencies that are superior to those of the equivalent conventional blazed gratings. Gratings have been made with periods as small as 0.99 mm blazed in transmission for red light, which in a conventional transmission grating would require facet angles in the region of 40. Not only have efficiencies in excess of 80% been measured for both polarizations, but this has been Fig. 5 Blazed binary lens. Scanning electron micrograph located not far from a corner of a 20 off-axis diffractive lens designed for operation at 860 nm. The lens has a square aperture of 200 200 mm and a focal of 400 mm. The Fresnel zone widths are in between 1.7 and 9 mm. The pillars etched into a TiO 2 thin film deposited on a glass substrate are 990 nm high and are located on a square grid with a 405-nm period. The corresponding aspect ratio for the thinner pillars is approximately 10. Tested with a vertical-cavity surface-emitting laser emitting a nearly Gaussian beam, an 80% first-order efficiency was measured. More details can be found in Ref. [26]. maintained over a much wider range of angles of incidence than would have been possible with a conventional grating. [48] Fig. 5 shows an electron micrograph of a typical structure. PHOTONIC CRYSTALS In the practical examples considered so far, waves were propagating in 1-D or 2-D structures parallel or nearly parallel to one of the translation-invariant directions of the artificial media. Very interesting optical properties are also obtained for waves propagating perpendicularly to that direction. For example, consider propagation along the x direction in the 1-D structure of Fig. 1. The light sees a series of layers with alternating high and low refractive indices. Multiple reflections and refractions occur at the interfaces and along with interference, light may be
Artificial Media Optical Properties Subwavelength Scale 69 reflected back for wavelengths approximately equal to twice the period. The width of the reflectance band is defined by the wavelengths between which the reflectance increases as layers are added. In general, low absorption and high reflectivity are obtained. Another way to understand the reflectance band is to consider that the medium, supposedly infinite in the x direction, prevents the propagation of light in a certain range of wavelengths. For these wavelengths, the effective index computed with the abovementioned methods is complex. Its imaginary part does not imply any heat dissipation because the alternate layers are made of transparent materials; it just signifies that waves cannot propagate, and its value is simply inversely proportional to the penetration depth (see Ref. [49] for 1-D structures and Ref. [15] for 2-D and 3-D structures). The range of wavelengths for which propagation is forbidden is called the bandgap [49] by analogy with the energy gap experienced by electrons submitted to a periodic potential in semiconductors. As the propagation direction deviates progressively from the normal to the layers, the gap shifts in frequency and becomes more and more narrow. By increasing the degree of symmetry, a 2-D bandgap can open for all propagation directions in the xz plane for the 2-D structure of Fig. 1. Ultimately, structures with 3-D periodicity may offer a full omnidirectional bangap: In a given frequency range, no wave can propagate in all directions for any polarization. [50,51] The potential of 3-D photonic crystal to fully control the spontaneous emission of atoms and to influence optoelectronic emitting devices was first realized by Yablonovitch [52] and John [53] in 1987. It took some years for the scientific community to react, but the topic is now the subject of intense research activity. Photonic crystal with 3-D periodicity are difficult to manufacture especially in the visible or near infrared. Fortunately, 2-D periodic structures which are easier to manufacture already exhibit some useful properties. For instance, if a single-point defect in the periodicity is created by removing one cylinder in the 2-D structure of Fig. 1, a wave confined in the air gap by the multidirectional Bragg reflector constituted by the surrounding pillars is able to propagate along the y-direction without attenuation. In this case, the light is actually confined in the low refractive index region. This is the contrary of the standard waveguide for which guidance is achieved by total internal reflection in the high-index material. [54] New generation of fibers working on that principle have been actually fabricated by almost standard preform and stretching techniques [55] (Fig. 6). Because the light sees essentially no material (vacuum), transmissions of waves at power levels not possible in conventional waveguides appear feasible without damage, leading to greatly increased threshold powers for nonlinear optics. Such fibers are also able to conduct light around relatively Fig. 6 Scanning electron micrograph of an air-core photonic crystal fiber. More details can be found in Ref. [55]. (Courtesy of P.S.-J. Russell, University of Bath.) sharp bends without significant radiation losses. This is because the photonic crystal structure surrounding the air core supports only evanescent waves and, therefore, prevents the loss of energy in radiative modes. This property can also be exploited for compact optoelectronic integrated circuits based on photonic crystal waveguides supporting sharp turns without loss. [50,56] CONCLUSION In many cases, it is possible, by engineering the structure of a medium on a scale smaller than that of the wavelength of light, to control its optical properties. By producing such an artificial medium, it is possible to extend the range of possibilities that are available to the optical designer. In some cases, it is possible to consider a material with subwavelength structure as homogeneous and to characterize its performance by an effective value of refractive index. In other cases, the situation is more complex and can only satisfactorily be described by a rigorous solution of Maxwell s equations. We have seen that the principle of artificial media is well established today and have considered a selection of practical examples. However, the subject is currently receiving a considerable amount of research and development interest because the technology of microfabrication is advancing very rapidly. It is becoming increasingly practical to generate in a complete range of materials, arrays of pillars, ridges, and holes that are deep, narrow, and smooth. Therefore, it seems realistic to expect that in the near future, we shall see a range of high-performance A
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