TThis is the accepted author version of the article published in
Physica B vol. 407, pp. 4056-4061 (2012). The version of record is available from the publisher’s website from the link: https://doi.org/10.1016/j.physb.2012.01.092 . c (cid:13)
Photonic crystals as metamaterials
S. Foteinopoulou ∗ School of Physics, College of Engineering,Mathematics and Physical Sciences (CEMPS),University of Exeter,Exeter, United Kingdom
Abstract
The visionary work of Veselago had inspired intensive research efforts over the last decade, to-wards the realization of man-made structures with unprecedented electromagnetic (EM) properties.These structures, known as metamaterials, are typically periodic metallic-based resonant structuresdemonstrating effective constitutive parameters beyond the possibilities of natural material. Forexample they can exhibit optical magnetism or simultaneously negative effective permeability andpermittivity which implies the existence of a negative refractive index. However, also periodic di-electric and polar material, known as photonic crystals, can exhibit EM capabilities beyond naturalmaterials. This paper reviews the conditions and manifestations of metamaterial capabilities ofphotonic crystal systems. ∗ Corresponding author: Tel:+44-1392-722101, Fax: +44-1392-264111, e-mail:[email protected] a r X i v : . [ phy s i c s . op ti c s ] J un . Introduction Veselago’s visionary proposal [1] in 1967 entailed the fictitious at that time possibilityof materials with a simultaneously negative permittivity and permeability. He showed thatthese materials would demonstrate negative refraction and possess negative refractive index,and also support EM propagation with antiparallel energy and phase velocities (backwardwaves [2]). The two pioneering works of Sir John Pendry and co-workers [3, 4] in the late 90’sopened up the possibilities for tailoring the effective plasma frequency of a composite wiremedium [3] and artificial magnetism in metallic resonator structures and have set the stagefor the realization of the first negative refractive index metamaterial at GHz frequencies byDavid R. Smith and co-workers [5] in 2000. The emerged field of photonic metamaterialsnow encompasses a variety of theoretically inspired and experimentally realized metallicnanostructures that are functional all the way up to visible frequencies [6–8]. Such artifi-cial meta-structures [6, 7] possess exotic effective constitutive parameters, i.e. permittivity, ε , and permeability µ in some or all propagation directions, which are not available innatural materials. These typically involve a negative effective permeability in frequenciesranging from THz to visible spectrum, which when combined also with a negative effec-tive permittivity lead to an effective negative refracted index and left-handed (backward)EM propagation. Later, metamaterials with effective chiral constitutive relations were alsoconceived, demonstrating capacity for strong optical activity [9–12].Photonic crystals (PCs) on the other hand, are also exhibiting extra-ordinary electro-magnetic responses, uncharacteristic of natural materials. Photonic crystals[13] are mediacomposed of dielectric, polar or metallic building blocks arranged periodically in one-, two-or three-dimensional, but do not include resonator elements that are typical to magneticmetamaterials [4, 6, 7]. Their striking EM behavior encompasses unusual ultra-refraction[14, 15] known as the superprism phenomenon [16], negative refraction [14, 17–21, 23], multi-fringent effects [19, 24, 25], collimation [20, 21, 26, 27], as well as channeling of the dark field[28]. Without a doubt, these extra-ordinary EM responses of photonic crystals opened upnew avenues for beam manipulation[26, 29, 30] and sub-wavelength control[21–23, 28, 31].More recently such functionalities have been demonstrated even in one-dimensional[32] andquasi-crystal[33] arrangements.However, despite their curious electromagnetic response, photonic crystals may not nec-2ssarily act as metamaterials, in the sense of possessing effective photonic properties. In thisarticle, the behavior of photonic crystals as metamaterials is reviewed. Two different classesof behaviors are identified. In Sec. 2 the discussion focuses on the first class of photoniccrystal metamaterial behavior, entailing PCs which may not posses effective constitutiveparameters, but their refractive behavior emulates that of a homogeneous medium with arefractive index n. Such photonic crystal systems are functional at free space wavelengths ofthe order of the structural meta-atom size, typically two to three times larger. Not all PCsare capable to emulate the refractive properties of a homogeneous block; there are stringentconditions enabling such behavior which are discussed briefly in Sec. 2.In Sec. 3 the second class of photonic crystal metamaterials is reviewed. These systemsdo possess effective constitutive parameters. The demonstrated engineered capabilities forthese parameters that go beyond natural materials involve: i) Extreme optical anisotropy[34],-including indefinite permittivity tensors [35] for frequencies ranging from THz to visiblefrequencies. An indefinite permittivity tensor leads to the characteristic EM dispersionwith a hyperbolic surface of wavenormals. Hyperbolic dispersion is not possible in naturalmaterials and is attracting increasing interest [36–43] as it mediates transfer of the darkspatial frequencies of the source, enabling near-field superfocusing [39, 40]. ii) Magneticbehavior -including negative permeability-, mainly at THz and mid-IR frequencies [44–52].The aforementioned types of PC metamaterials are functional at free space wavelengthsmuch larger than the structural meta-atom, typically about 10 times larger or more [38].
2. Photonic crystals as metamaterials with effective propagation propertieswithout effective constitutive parameters
Photonic crystals generally demonstrate highly complex refractive behavior that is typ-ically multifringent. The native modes of propagation in these systems are the so calledFloquet-Bloch waves (FB waves) [53], which simultaneously satisfy Maxwell’s equation andBloch’s theorem [19, 54]. The FB wave consists of a sum of infinite number of plane waves.However such sum represents a unique propagating mode characterized by a certain energypropagation velocity v e ; in other words, the FB wave is an entity [19]. It is this planewave sum character of the FB wave that is responsible for the characteristic “wiggly” phasefronts of EM propagation within a PC medium. Multifringent behavior in PCs arises fromthe simultaneous coupling of many FB waves under certain conditions of illumination. How-3ver, the pioneering work of Notomi in 2000 [17] showed that it is possible, in PCs with ahigh refractive index contrast, to have coupling to a single only FB wave. Furthermore, heshowed that at the plane of incidence such PCs possessed a circular contour of wavenormals,just like homogeneous conventional dielectrics do. He termed such contours of wavenormalsas Equi-frequency contours (EFC). The refractive index corresponding to such contour wassimply, | n p | = ck/ω , with k being the radius of the EFC contour in wave vector space. Thisrefractive index would yield correctly the magnitude of the refracted angle of a beam hit-ting the PC interface from Snell’s law. Strikingly, refraction was positive for positive slopebands, where v g · k > v g being the group velocity and k the wave vector and negativefor negative slope bands where v g · k < v g essentially represents the direction ofthe Poynting vector, S (but averaged within the structural unit cell) [19]. It so becomes FIG. 1: (Color online) (a) The PC metamaterial design functional at free space wavelength λ free =2 . n ∼ − ∼ -45 deg. for an incident beam at 45 deg.at the PC metamaterial interface. (d) Transmission versus incident angle for a 12-layer-thickstructure. (e) Far field superfocusing through the latter structure, with the incident source placedat d s = 2 . λ free from the first interface. The image is formed at d i ∼ . λ . So, d s + d i ∼ . n g , yielding the magnitude of the propagation velocity( v e = c/ | n g | ) of an EM wave of frequency ω relates to the phase index, n p as: n g = ω d | n p | dω + | n p | (1)just like in a homogeneous material.These works implied it is possible for dielectric PCs to emulate, as far as refractionand propagation velocities are concerned, a homogeneous medium with a refractive index n p ( ω ), which can take negative values. In these respects, such PCs act as metamaterials. Aplethora of works focused on the quest and analysis of PC designs possessing such capacitydemonstrated by almost isotropic surfaces of wave normals (or contours for 2D structures)[18, 33, 55–62]. An example of such PC metamaterial studied by R. Moussa et al. [62]is depicted in Fig. 1(a) consisting of alumina rods in air and operating for electric fieldparallel to the rods axis. The associated EFC is shown in Fig. 1(b) and compared withthat of vacuum, for the operational frequency that corresponds to a free space wavelengthequal to 2.8985 a,- with a being the lattice constant designated in Fig. 1(a). Fig. 1(c)demonstrates the expected negative refraction at ∼ −
45 deg. for an incident beam of 45deg., since the refractive index n p is close to -1.However, although isotropic EFSs (or EFCs in 2D structures) manifest the possibilityfor a metamaterial behavior (with respect to refraction and propagation), there are sev-eral hovering caveats one must take careful consideration of. The extensive analysis of S.Foteinopoulou et al. [19] in various 2D PCs revealed that such metamaterial behavior issubject to the satisfaction of several stringent conditions. These are determined by requir-ing single beam propagation emanating from direct refraction and not umklapp coupling.Meeting all these conditions entails careful engineering, which in general restricts meta-material behavior only for certain interface cuts which must be along symmetry directions.Band regions were several bands are present or one band that varies non-monotonically withwavevector lead to multifringence and are so inappropriate for PC metamaterial designs.Having the metamaterial capacity of effective refraction and propagation makes thesePCs highly attractive for a multitude of applications that rely on such properties. Theadvantage is that unlike their homogenized resonant metamaterial counterparts, PCs are5 IG. 2: (Color online) 2D photonic crystals with infinitely long rods along y. In (a) E-polarizationincidence is shown. In (b) H-polarization incidence is shown inherently lossless; so there is no loss of EM energy during propagation. The disadvantageis that these are not a true homogenized medium, thus even structures with n p = −
1, canhave large reflections [63]. A high number of works with PCs falling in this category havefocused on emulating Pendry’s perfect lens [64] as far as the propagating components ofthe input source are concerned [33, 57, 58, 60, 62]. An example is shown in Fig. 1(e) forthe design of the work of Ref. [62] seen in Fig. 1(a), which was conceived to satisfy thestringent single beam metamaterial conditions at lower frequencies were reflections can beconsiderably smaller. Indeed, as shown in Fig. 1(d) a high transmission is found for almostall incident angles up to 70 deg. through a structure of 12 rods along the propagationdirection. Note the superfocusing demonstrated in Fig. 1(e) is in the far field with thesource placed at a distance 2.83 times the free space wavelength from the first interface.The potential for exploitation of effective propagation properties is enormous and goesbeyond superlensing applications. Recently, it has been proposed that PCs can have a zerophase index[65, 66] emulating zero refractive index materials [37]. Also, in an arrangementof slow varying building block size they lead to a spatial varying phase index which can beexploited for superbending [67], photonic mirage effects [68, 69] and transformation opticsdevices such as beam shifters or cloak-like scattering [70]. Accordingly, in view of theirinherently lossless properties such photonic crystal metamaterials are highly attractive forthese type of applications. The considerable challenge to be met is optimization of in-coupling efficiency. Surface texturing seems a promising avenue in this direction [62, 71].
3. Photonic crystals as metamaterials possessing extra-ordinary effective con-stitutive parameters ε E in the directions along the optical axis (y), and permittivity ε H perpendicularto the optical axis, where ε E and ε H represent the field-averaging and the Maxwell-Garnettvalues respectively. Indicatively, the optical anisotropy | ε H − ε E | /ε E for a square PC latticeof Si rods in air and filling ratio of f = 0 .
30% would be about 60%. Evidently, 2D PCcomposites in the effective medium regime provide a platform for extreme engineering ofoptical anisotropy.How extreme? Lately, it was demonstrated that when plasmonic rods are used as the PCbuilding blocks which have a negative permittivity at optical frequencies, the anisotropy canbe so extreme that permittivity is negative along the optical axis and positive perpendicularto that [36, 37, 39, 40]. Very recently, S. Foteinopoulou et al. [38], explored the possibilityto transfer such possibility in the THz and mid-IR regions, where dire need for optical set-up components exists, with the incorporation of polar instead of plasmonic materials. Allthe aforementioned structures possess an unusual hyperbolic surface of wave normals forthe extraordinary mode, which is not encountered in natural media. The response of suchPC metamaterials to the ordinary mode is metallic-like with typically a small skin depth,rendering the extra-ordinary mode with hyperbolic dispersion as the dominant mode[38].Not all PC composites that behave as effective metamaterials are subject to the field-averaging/Maxwell Garnett relations, which generally apply for extremely subwavelengthmeta-atoms. More complicated effective medium theories have been developed by variousgroups [73–77] to extend the frequency range of their validity, including theories predictingmagnetic behavior from non-magnetic inclusions [48]. Magnetic behavior in such compositesemanates from the Mie resonances on the individual meta-atom building blocks [48, 52].When the refractive index contrast between the structural building blocks and matrix isvery high then such magnetic behavior can be so strong that it enables a negative effectivepermeability µ [48–50]. In 2D structures effective negative permeability has been reported for7 IG. 3: (Color online) Retrieved refractive index [panel (a)], permittivity function [panel (b)]and permeability function [panel (c)] for E-waves through the LiF/NaCl composite. The real(imaginary) part of the retrieved parameters is shown as solid (dashed) lines. In (d) the real partof the bulk permittivity for NaCl, and LiF are depicted (solid and dotted lines respectively). a certain direction for high index dielectric [51] and polar material composites [44–47, 51, 52].As an example of the possibility of magnetic behavior the effective constitutive param-eters for a 2D square PC array of LiF circular rods, -3 µ m in diameter, in a NaCl matrixand spaced 5 µ m apart are shown in Fig. 3. We observe versus free space wavelength,the effective refractive index n [in (a)], the effective permittivity ε [in (b)] and the effectivepermeability µ in [in (c)] for E-polarized waves and normal incidence [ θ I = 0 in Fig. 2(a)].The solid line represent the real part and the dotted lines the imaginary part of the afore-mentioned quantities. In other words, the effective permittivity represents the ε y elementof the permittivity tensors and the effective permeability represents the µ z element of thepermeability tensor, where the direction x , y and z are defined in Fig. 2. A very strong, yetnon-negative magnetic behavior is seen around 40 µ m, where there is a high permittivitycontrast between rods and background of about 50 to 1. Fig. 3(d) shows the real part ofthe permittivity as a function of free space wavelength for the LiF rods (dotted line) andthe NaCl matrix (solid line). Notice, that clearly the strength of the magnetic behaviorincreases with the permittivity contrast between rods and matrix.The effective constitutive parameters are determined from an alternative retrievalmethod that relies on the information contained in r/t ratio, with r and t being the complex8eflectivity and transmissivity. In particular it can be shown [38] that the effective refractiveindex can be retrieved from: n = cωL cos − (cid:16) rt (cid:17) L (cid:16) rt (cid:17) L + lπ , (2)where L is the thickness of the structure, ω the frequency of the EM wave, and c thevelocity of light and l signifies the branch solution. The correct branch is identified afterperforming the retrieval for several thicknesses, L (and shown in panel (a) of Fig. 3). Afterthe correct branch for n is determined, the impedance, z , can be obtained from: z − z = (cid:0) rt (cid:1) Li sin ( n ωLc ) , (3)by choosing the root that satisfies passivity, i.e. Re ( z ) >
0. Then, the effective permittivityand permeabilities depicted in panels (b) and (c) of Fig. 3 respectively can be easily obtainedfrom ε = nz and µ = nz .Retrieval is off-course meaningful only when the structure does behave as a bulk effectivemedium, and may or may not agree with simple or more developed effective medium models.It is essential to have available characterization tools that test the validity of metamaterialdescription of the 2D composite. Properly constructed functions of r/t [38], can be shownthat should be length independent for a homogeneous medium. Thus testing the variance ofthese for a large ensemble of thicknesses provides a quantitative measure of effective mediumvalidity. This is however possible only for inherently lossless structures as losses make quicklytransmission zero. However, in the same work [38] it was shown that properly constructedfunctions of r/t show distinct angular profile that serves as a signature of effective mediumbehavior. One of them, entailing a sin signature becomes too sensitive to numerical error forhigh permittivities and/or very high losses. The second of these constructed test functions,is however robust even under these conditions and demands a flat angular profile as evidenceof metamaterial behavior. In particular, if θ I is the incident angle, the function E ( θ I ) [38] E ( θ I ) = c ω L ( Im [∆( θ I )] Re [∆( θ I )] /π + lIm [∆( θ I )]) , (4)9ith ∆( θ I ) = cos − (cid:16) rt (cid:17) L (cid:16) rt (cid:17) L θ I , (5)and l the order of the branch should be flat. It can be shown that when magneticbehavior is present then, E ( θ I ) = Im ( εµ ). Such flat profile test for two characteristic freespace wavelengths of 45 µ m and 60 µ m is shown in the insets of Fig. 4, where the flat profileof E ( θ I ) can be clearly seen. The calculated values for the flat E for other frequenciesare shown in Fig. 4 as diamonds. Notice, the excellent agreement of the latter with theexpected Im ( εµ ) value (solid line in the figure), ascertaining the metamaterial behavior ofthe structure. FIG. 4: (Color online) Flat profile test. The diamonds represent the values of the flat E ( θ I )function at various free space wavelengths and are compared with the expected value of Im ( εµ )(solid line). The actual function E ( θ I ) versus incident angle θ I is depicted for two cases in theinsets. The analysis of Ref. [38] suggests that for 2D composites it is sufficient to check effectivemedium behavior under the illumination conditions of Fig. 2(a) and Fig. 2(b). If theeffective medium criteria apply for these illuminations simulateously, then the entire PCblock acts as a uniaxial metamaterial for an arbitrary illumination.To recap, extra-ordinary metamaterial behaviors for this class of PCs entails magneticbehavior in some direction(s) and extreme anisotropy with even hyperbolic dispersion. These10an be engineered also for the THz and mid-IR regime to open new avenues for beam shapingand manipulation capabilities in these frequencies. Applications exploiting thus far suchhyperbolic dispersion are near-field superlenses[39, 40] and angle-dependent polarizationfiltering [38]. The challenge to be met is constructing composites with a high figure ofmerit (FOM) given by Re ( k z ) /Im ( k z ) with k z being the wave vector along the propagationdirection. The results of Ref. [38] are promising in this directions as a FOM exceeding 10has been reported even in a frequency regime around the polariton resonance of one of theconstituents.
4. Conclusions
Metamaterial behavior of photonic crystals was reviewed and two different classes of dif-ferent characteristics were identified and discussed. In the first class, structures with strictengineering at wavelengths comparable to the structural meta-atom can possess effectivepropagation properties without possessing effective refractive index. In the second class,deep subwavelength non-magnetic PC composites can demonstrate metamaterial constitu-tive parameters including magnetic behavior in some or all direction and hyperbolic EMwave dispersion.
Acknowledgement
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