Nanophotonics 1 (2012): 181–198 © 2012 Science Wise Publishing & De Gruyter • Berlin • Boston. DOI 10.1515/nanoph-2012-0020 Review _ Physics of the zero-n photonic gap: fundamentals and latest developments Lei Zhou 1, *, Zhengyong Song 1 , Xueqin Huang 2 and PCs are artifi cial materials with a periodic modulation on C.T. Chan 2 the dielectric constant, which can create a photonic band gap (PBG) via Bragg scatterings, inside which no propagating 1 State Key Laboratory of Surface Physics , Key Laboratory photonic mode exists [1 – 3] . PCs have attracted intensive of Micro and Nano Photonic Structures (Ministry of studies in the last two decades due to their unique EM prop- Education) and Physics Department, Fudan University, erties and potential applications. The existence of a forbid- Shanghai 200433 , China , e-mail: [email protected] 2 Physics Department , The Hong Kong University of Science den frequency band in a PC alters dramatically the properties and Technology, Clear Water Bay, Kowloon, Hong Kong , of light, enabling control of spontaneous emission in quan- China tum devices and light manipulation for photonic informa- tion technology [10 – 12] . However, such a Bragg gap is very * Corresponding author sensitive to the periodicity of the system, the incident angle Edited by Nader Engheta, University of Pennsylvania, and polarization [transverse electric (TE) wave or transverse Philadelphia, Pennsylvania, USA magnetic (TM) wave] of the input light, dictated by the Bragg mechanism. As a result, the PBG frequency is inversely pro- Abstract portional to the lattice constant, and thus the size of a PC can- not be made very compact (at least several wavelengths in A short overview is presented on the research works related each dimension) and randomness will destroy the band gap – to the zero-n gap, which appears as the volume-averaged [13 – 16] . To make photonic devices that are compact in size refraction index vanishes in photonic structures containing and robust against disorders, new PBG mechanism needs to both positive and negative-index materials. After introducing be found. – the basic concept of the zero-n gap based on both rigorous MTMs belong to another class of artifi cial EM materials mathematics and numerical simulations, the unique proper- [4 – 9] . These materials are composites consisting of local ties of such a band gap are discussed, including its robust- resonant EM microstructures in subwavelength scales, such ness against weak disorder, wide-incidence-angle operation that the whole medium can be viewed as a homogeneous and scaling invariance, which do not belong to a conventional one [17] exhibiting arbitrary values of electric permittivity Bragg gap. We then describe the simulation and experimental ε and magnetic permeability μ . The much expanded param- – verifi cations on the zero-n gap and its extraordinary proper- eter freedom makes MTMs an ideal platform to manipulate ties in different frequency domains. After that, the unusual EM wave propagations, leading to many interesting phe- – photonic and physical effects discovered based on the zero-n nomena such as negative refraction [8, 9, 18, 19] , super gap and their potential applications are reviewed, including imaging [20 – 23] , invisibility cloaking [24 – 28] , and so beam manipulations and nonlinear effects. Before concluding on. In particular, when both ε and μ are negative, such a – this review, several interesting ideas inspired from the zero-n medium is also called a left-handed material (LHM), since gap works will be introduced, including the zero-phase gaps, E, H, and k of a plane wave propagating inside it form zero-permittivity and zero-permeability gaps, complete band a left-handed set instead of a right-handed one in a con- gaps, and zero-refraction-index materials with Dirac-Cone ventional medium. The unusual EM properties of a LHM dispersion. were fi rst theoretically studied by Veselago in 1968 [4] , who found that the direction of energy fl ow is opposite to Keywords: left-handed materials; meta materials; photonic that of the wavevector k inside a LHM, so that an EM wave band gaps; photonic crystals. will be bent negatively when it passes through an inter- face between a normal medium and a LHM. As a result, 1. Introduction such a medium is also said to possess a negative refrac- tion index ( n ) [8, 9, 29] . Other peculiar EM properties of Over the last several decades, considerable interests appear in the LHM include reversed Doppler effect [30] and reversed employing artifi cial electromagnetic (EM) materials to con- Cherenkov radiation [31] . Veselago ’ s proposal of LHM did trol light propagations as desired, which cannot be achieved not attract immediate attention since it is well accepted that with naturally existing materials. Such artifi cial EM materi- a natural material shows no magnetism at high frequencies als include photonic crystals (PCs) [1 – 3] and metamaterials [32] . A breakthrough appeared in 1999 when Pendry et al. (MTMs) [4 – 9] , operating based on the Bragg and local reso- [6] showed that a split ring resonator (SRR) could provide nance mechanisms, respectively. magnetic responses at any desired frequency. People then _ 182 L. Zhou et al.: Physics of the zero-n photonic gap: fundamentals and latest developments successfully fabricated LHMs by combining SRR and elec- where c is the speed of light, nx()= εμ () x () x and tric resonators such as metallic wires [5] , and demonstrated that such a medium can indeed exhibit a negative refractive μ()x Zx()= are respectively the refraction index and index [8] . Since then, many efforts were devoted to explor- ε()x ing other intriguing EM manipulation phenomena based on impedance at frequency ω . Imposing the periodicity con- MTMs. straint E()xa+= eExika (), we fi nd that the dispersion relation While PCs and MTMs are constructed based on different is determined by physical principles and thus the developments of two fi elds appear quite independent, we notice that there are particular Tr [ T ( ω )] = 2cos ka , (2) sub-fi elds that combine the concepts from both sides and then ω × new physics emerge from the interplays between two physi- where T ( ) is a 2 2 transfer matrix. For a double-layer unit- cal principles. For example, employing PCs with anoma- cell, a simple calculation yields lous dispersions to mimic the negative refraction behaviors ⎛⎞⎛naω Z Z ⎞ ωωω=+12 of LHMs attracted much attention in early developments of Tr[ T ( )] 2cos⎜⎟ -⎜⎟ -2 sin( n11 d / c )sin( n 2 d 2 / c ), ⎝⎠cZZ⎝⎠ MTMs [33 – 35] . On the other hand, PCs built with LHMs 21 were found to exhibit many unusual properties. In particu- (3) lar, it was fi rst discovered in [36] that a new type of PBG where n , Z , d are, respectively, the refractive index, imped- called zero-n– gap arises in a PC consisting of both right- i i i ance, and thickness of the i th layer. Clearly, the fi rst term in handed materials (RHMs) and LHMs. Such a zero-n– gap Eq. (3) represents the solution in a homogeneous medium exhibits many unusual physical properties that do not exist with average refractive index nndnda=+()/, while the in a conventional Bragg PBG, such as central-frequency scal- 11 2 2 second term is responsible for gap opening if there is an ing invariance and robustness against weak disorders. Later, impedance mismatch. For the special case of matched imped- many other unconventional physical properties of the zero-n– ance (Z = Z = Z , a constant), the dispersion relation is given gap were discovered, and the existence of such a special PBG 1 2 0 by ka = n–ωa/c. In general case (Z ≠ Z ), when was experimentally verifi ed in different frequency regimes. 1 2 ωπ=∈ More recently, several interesting ideas were inspired from nacm / ( m integers) , (4) the zero-n– gap works, resulting in new sub-branches in pho- tonic research. Eq. (3) becomes In this paper, we will briefl y review the fundamental phys- – ⎛⎞ZZ ics and latest developments of the zero-n gap research. We =+12 +2 ω ≥ Tr() T 2⎜⎟ -2sin( n11 d /) c 2. (5) have no intention to make this paper a comprehensive over- ⎝⎠ZZ 21 view on the development of the whole sub-fi eld, but rather try to make it concise and idea-inspiring. This review is organized Except that as follows. We fi rst introduce the basic concept and some pre- n d ω / c = p π ( p ∈ integers), (6) liminary properties of the zero-n– gap in Section 2, and then 1 1 summarize in Section 3 the available efforts from both full- Eq. (5) implies that Eq. (2) has no real solution for k , indicat- – wave simulations and experiments on verifying the zero-n gap ing the opening of a band gap. Equation (4) is the familiar and its extraordinary physical properties. Section 4 is devoted Bragg condition, which can have multiple solutions for con- to reviewing the unusual physical effects discover ed based ventional PBG materials. However, if we mix both positive – on the zero-n gap materials, and Sections 5 and 6 summarize and negative- n materials to form a periodic structure, there is – some interesting ideas inspired from the zero-n gap research. an extra possibility that We conclude this review in Section 7. n– = 0, (7) which also leads to Eq. (5) and thus a spectral gap. This 2. Basic concept and extraordinary properties new type of band gap differs fundamentally from the usual of the zero-n– gap Bragg gap, as we shall demonstrate in the subsequent We fi rst present a proof that in a one-dimensional (1D) lay- sections. – = ered stack containing both positive- and negative-n materi- The n 0 condition for the spectral gap is not limited to als, n– = 0 implies the existence of a PBG [36] .