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Subwavelength Imaging in Photonic Crystals

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Subwavelength Imaging in Photonic Crystals PHYSICAL REVIEW B 68, 045115 ͑2003͒ Subwavelength imaging in photonic crystals Chiyan Luo,* Steven G. Johnson, and J. D. Joannopoulos Department of Physics and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA J. B. Pendry Condensed Matter Theory Group, The Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom ͑Received 5 December 2002; revised manuscript received 14 February 2003; published 29 July 2003͒ We investigate the transmission of evanescent waves through a slab of photonic crystal and explore the recently suggested possibility of focusing light with subwavelength resolution. The amplification of near-field waves is shown to rely on resonant coupling mechanisms to surface photon bound states, and the negative refractive index is only one way of realizing this effect. It is found that the periodicity of the photonic crystal imposes an upper cutoff to the transverse wave vector of evanescent waves that can be amplified, and thus a photonic-crystal superlens is free of divergences even in the lossless case. A detailed numerical study of the optical image of such a superlens in two dimensions reveals a subtle and very important interplay between propagating waves and evanescent waves on the final image formation. Particular features that arise due to the presence of near-field light are discussed. DOI: 10.1103/PhysRevB.68.045115 PACS number͑s͒: 78.20.Ci, 42.70.Qs, 42.30.Wb I. INTRODUCTION important to note that the transmission considered here dif- fers fundamentally from its conventional implication of en- Negative refraction of electromagnetic waves, initially ergy transport, since evanescent waves need not carry energy proposed in the 1960s,1 recently attracted strong research in their decaying directions. Thus, it is possible to obtain interest2–16 and generated some heated debate.17–29 In par- transmission amplitudes for evanescent waves greatly ex- ticular, much attention was focused on the intriguing possi- ceeding unity without violating energy conservation. Here, bility of superlensing suggested in Ref. 5: a slab of uniform we discuss two mechanisms linking amplification of evanes- ‘‘left-handed material’’ with permittivity ⑀ϭϪ1 and perme- cent waves to the existence of bound slab photon states. ability ␮ϭϪ1 is capable of capturing both the propagating These bound states are decoupled from the continuum of and evanescent waves emitted by a point source placed in propagating waves, thus our findings are distinct from the front of the slab and refocusing them into a perfect point effect of Fano resonances31 in electromagnetism, which were image behind the slab. While the focusing effect of propa- recently studied in the context of patterned periodic gating waves can be appreciated from a familiar picture of structures32–36 and surface-plasmon-assisted energy rays in geometric optics, it is amazing that perfect recovery transmission.37–39 As for the problem of negative refraction, of evanescent waves may also be achieved via amplified we have found that the concept of superlensing does not in transmission through the negative-index slab. Some of the general require a negative refractive index. Moreover, we discussion of this effect relies on an effective-medium argue that the surface Brillouin zones of both photonic crys- model6 that assigns a negative ⑀ and a negative ␮ to a peri- tals and periodic effective media provide a natural upper cut- odic array of positive-index materials: i.e., a photonic off to the transverse wave vector of evanescent waves that crystal.30 Such an effective-medium model holds for large- can be amplified, and thus no divergences exist at large trans- scale phenomena involving propagating waves, provided that verse wave vectors24 in photonic crystals or in effective the lattice constant a is only a small fraction of the free-space media.27 As with effective media, the lattice constant of pho- wavelength in the frequency range of operation. However, in tonic crystals must always be smaller than the details to be the phenomenon of superlensing, in which the subwave- imaged. We derive the ultimate limit of superlens resolution length features themselves are of central interest, an in terms of the photonic-crystal surface periodicity and infer effective-medium model places severe constraints on the lat- that resolution arbitrarily smaller than the wavelength should tice constant a: it must be smaller than the subwavelength be possible in principle, provided that sufficiently high di- details we are seeking to resolve. The question of whether electric contrast can be obtained. We pursue these ideas in and to what extent superlensing would occur in the more realistic situations by calculating the bound photon states in general case of photonic crystals still remains unclear. carefully designed two-dimensional ͑2D͒ and 3D slabs of Recently, we showed that all-angle negative refraction AANR photonic crystals and present a comprehensive nu- ͑AANR͒ is possible using a photonic crystal with a positive merical study of a 2D superlensing structure. A subtle and index, and we demonstrated imaging with a resolution below very important interplay between propagating waves and or on the order of wavelength in this approach.12,15 In this evanescent waves on image formation is revealed, which paper, we investigate the possibility of photonic-crystal su- makes the appearance of the image of a superlens substan- perlensing in detail by studying the transmission of evanes- tially different from that of a real image behind a conven- cent waves through a slab of such a photonic crystal. It is tional lens. These numerical results confirm the qualitative 0163-1829/2003/68͑4͒/045115͑15͒/$20.0068 045115-1 ©2003 The American Physical Society LUO, JOHNSON, JOANNOPOULOS, AND PENDRY PHYSICAL REVIEW B 68, 045115 ͑2003͒ overall transmission coefficient through the slab structure can be written as ϭ Ϫ ͒Ϫ1 ͑ ͒ t t p-a͑1 Tk,pr p-aTk,pr p-a Tk,pta-p . 1 ͑ ͒ In Eq. 1 , ta-p and t p-a are the transmission coefficients through the individual interfaces from air to the photonic crystal and from the photonic crystal to air, respectively, r p-a is the reflection coefficient on the photonic-crystal/air bound- ary, and Tk,p is the translation matrix that takes the fields from zϭϪh to zϭ0 inside the photonic crystal. When h is an integral multiple of the crystal’s z period, Tk,p is diagonal with elements eikzh for a crystal eigenmode of Bloch wave FIG. 1. Illustration of light-wave transmission through a photo- vector kϩk zˆ, with Im k у0. In what follows, we discuss ͑ ͒ z z nic crystal. a Transmission through a slab of photonic crystal. The the possibility of amplification in t, beginning with the diag- incident field Fin and the reflected field Frefl are measured at the ϭϪ onal element that describes the zeroth-order transmission— left surface of the slab (z h), and the transmitted field Ftrans is i.e., transmission of waves with k in the first surface Bril- measured at the right surface of the slab (zϭ0). ͑b͒ Transmission louin zone, referred to as t . through a boundary surface between air and a semi-infinite photonic 00 Generally speaking, Eq. ͑1͒ describes a transmitted wave crystal. The incident field F , the reflected field F , and the in refl that is exponentially small for large enough ͉k͉. This can be transmitted field Ftrans are all measured at the air/photonic-crystal interface. In both figures, k is the transverse wave vector of light seen in the special case of a slab of uniform material with permittivity ⑀ and permeability ␮, where all the matrices in and as indicates the surface periodicity. Eq. ͑1͒ can be expressed in the basis of a single plane wave ϭ discussions in this paper and can be readily compared to and thus reduce to a number. In particular, Tk,p exp(ikzh) ϭexp(ihͱ⑀␮␻2/c2Ϫ͉k͉2), which becomes exponentially experimental data. ͱ This paper is organized as follows. Section II is a general small as ͉k͉ goes above ⑀␮␻/c. Equation ͑1͒ then becomes discussion for amplified transmission of evanescent waves a familiar elementary expression through a photonic-crystal slab. Section III considers the t t eikzh implementation of superlensing in photonic crystals and its ϭ p-a a-p ͑ ͒ t00 . 2 ultimate resolution limit and, also, describes qualitatively the Ϫ 2 2ikzh 1 r p-ae expected appearance of the image of a superlens. Section IV presents numerical results for superlensing in a model pho- Equation ͑2͒ has an exponentially decaying numerator, while tonic crystal. Section V discusses further aspects of super- for fixed r p-a the denominator approaches 1. Thus, waves lensing in photonic crystals, and Sec. VI summarizes the with large enough ͉k͉ usually decay during transmission in paper. accordance with their evanescent nature. There exist, however, two separate mechanisms by which the evanescent waves can be greatly amplified through trans- II. ORIGIN OF NEAR-FIELD AMPLIFICATION mission, a rather unconventional phenomenon. The first Let us first consider the transmission of a light wave mechanism, as proposed in Ref. 5, is based on the fact that through a slab of lossless dielectric structure ͓Fig. 1͑a͔͒, pe- under appropriate conditions, the reflection and transmission riodic in the transverse direction, at a definite frequency ␻ coefficients through individual interfaces can become diver- ͑with free-space wavelength ␭ϭ2␲c/␻) and a definite gent and may thus be called a single-interface resonance. For transverse wave vector k. Let the slab occupy the spatial example, under the condition of single-interface resonance, !ϱ ͑ ͒ region ϪhϽzϽ0, and in this paper we also assume a mirror ta-p ,t p-a ,r p-a , and in the denominator of Eq. 2 the 2 ͑ ͒ symmetry with respect to the plane zϭϪh/2 at the center of term r p-aexp(2ikzh) dominates over 1.
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