Optical properties of dielectric, metal and engineered materials 81
2.25 2.3 2.35
0 0.05 0.1
2.25 2.55 2.85 3.15
0 2 4 6 8
400 500 600 700 800 400 500 600 700 800
400 500 600 700 800 400 500 600 700 800 400 500 600 700 800 400 500 600 700 800 f = 0.005
f = 0.005 f = 0.04
f = 0.04
f = 0.2
f = 0.2
0 2 4 6
0 0.5 1 ( mIεeff)( eRεeff)
λ (nm) λ (nm) λ (nm)
(a) (b) (c)
(e)
(d) (f)
FIGURE 5.4: [(a), (b) and (c)] Real and [(d), (e) and (f)] imaginary parts of ǫef f as function ofλfor a gold-silica composite evaluated using the Bruggeman (solid curve) and the Maxwell-Garnett (dash-dot) formula for three different volume fractions of gold inclusions, namely,f = 0.005 [(a) and (d)],f = 0.04 [(b) and (e)] andf = 0.2 [(c) and (f)]. The dielectric function of gold is taken from Johnson and Christie [13], whileεh= 2.25.
The proper sign in Eq. (5.58) is to be chosen such that Im (ǫef f)>0 to ensure causality.
We now compare the estimates ofǫef f as predicted by the Maxwell-Garnett (MG) and Bruggeman formulas. As mentioned earlier, MG formula is valid only for lower values off, as can be seen from Fig. 5.4. Fig. 5.4 shows the real and imaginary parts ofǫef f as a function of wavelengthλfor different values of volume fraction of gold inclusions in silica. The solid (dash-dot) curve in Fig. 5.4 corresponds to the MG (Bruggeman) formula. It is evident that for lower values off (for example,f = 0.005), both the formulas match very well, but for higher values off they differ drastically. Thus due attention is to be paid when dealing with higher values off of the metal inclusions sinceǫef f
can be completely different as estimated by the two approaches. Throughout the derivation we have considered the inclusions to be spherical, but this can be generalized to other geometries in a straightforward manner. In a similar vein, anisotropic character of the inclusions can be incorporated.
FIGURE 5.5: Characterization of materials based on the electric (ǫ) and magnetic (µ) response.
electric and magnetic susceptibilities through
ǫ=ǫ0(1 +χe) =ǫ0ǫr, (5.59) µ=µ0(1 +χm) =µ0µr. (5.60) Thus electric and magnetic properties of the medium are determined by the relative permittivity and permeabilityǫrandµr, respectively. In 1968 Victor Veselago, in a seminal paper, posed a very interesting problem [17]: Does the standard electromagnetics hold when both ǫr and µr are simultaneously negative? It was shown by Veselago that there is no conceptual problem when both ǫr and µr are simultaneously negative over a certain frequency range.
Of course, such materials do not occur in nature and, if at all, they are to be engineered and fabricated. Such materials were named as negative index materials (NIM) or left-handed materials, since we have to pick the negative sign for n (n = −√ǫrµr) and k, E and H in such materials from a left- handed triplet. In contrast, for standard materials we pick the positive sign for the square root, and we have a right-handed triplet for k, Eand H (see Eqs. (2.25)–(2.28) in Section 2.2). Ignoring the imaginary part of ǫ and µ, the material parameter domain can be split into four quadrants as shown in Fig. 5.5. It is clear from Fig. 5.5 that quadrants II and IV correspond to metal-like behavior. Parameters corresponding to II, meanwhile, are for standard metals and doped semiconductors at low frequencies. QuadrantIV corresponds to ‘magnetic’ metals like some ferrites. ClearlyI corresponds to most dielectric materials andIII refers to negative index materials. The left- handed triplet character for NIMs leads to yet another counterintuitive result regarding the antiparallel nature of the phase velocity and the Poynting vector.
Indeed,S= µ10E×H is antiparallel tok.
It is now clear that NIMs can lead to a host of unexpected and counterintu- itive phenomena. For example, there can be negative refraction leading to the bending of the refracted ray on the same side of the normal. A few other effects are the negative Doppler effect, the anomalous Cherenkov effect, lensing, etc.
Optical properties of dielectric, metal and engineered materials 83 In the context of lensing, a real breakthrough was the theoretical paper by Sir Pendry. He showed that a thin slab of NIM can act as a perfect lens since it can focus not only the propagating waves but also the evanescent waves [18].
Such a perfect lens can offer superresolution beating the diffraction limit. In contrast, a standard lens can focus only the propagating part and thus cannot override the Rayleigh criterion. On the design front, the key suggestions again come from Sir Pendry and his coworkers [19, 20]. Designing materials with negativeǫis not difficult, since we know that metals possess large negativeǫ at low frequencies. The job was to reduce the plasma frequency. With a ‘di- lute’ metal in the form of array of wires, this was achieved in 1996 [19]. After about three years, in 1999, Sir Pendry theoretically showed how to achieve a magnetic material from nonmagnetic constituents [20]. Split-ring resonators (SRR) emerged as the core units in the artificial magnetic materials that could offerµ <0. A realization of the clever arrangement of wires and SRRs led to the first implementation of NIMs in the microwave range [21]. Fishnet struc- tures emerged as the likely candidates to push the frequency range to near-IR and visible [22], since there were enormous difficulties (both conceptual and practical) in scaling down the SRRs to the optical domain.
The major difficulties in metamaterials (the Greek wordmeta means ‘be- yond,’ these are materials beyond what is available in nature) and especially in NIM research have been the losses in such materials. In fact, some of them exploit the material resonances, which are associated with large losses. Most of the theoretical predictions on the prefect lensing, etc. assume low or no losses in the structure. It has been shown theoretically that inclusion of losses can wash out the sub-wavelength features, and it is difficult to achieve super- resolution even with the parameters of the best NIM structures available to date [23]. Several schemes to overcome the devastating effects of losses have been proposed and most efforts have not yet achieved a ‘perfect’ metamaterial at optical frequencies.
Volumes have been written on such engineered materials and excellent reviews and monographs exist [11, 24, 25, 26]. In fact, one of the first meta- materials was conceived by Sir J. C. Bose, when he showed that a twisted medium can affect the polarization of light [27]. Here we introduced the read- ers to this fascinating area of engineered materials or metamaterials. Such metamaterials hold the key to many practical problems like perfect lensing, superresolution, invisibility cloaks, etc. Some of these effects are discussed briefly in Chapter 14.
Chapter 6
More on polarized light
6.1 State of polarization of light waves . . . 86 6.1.1 Jones vector representation of pure polarization states . 86 6.1.2 Partially polarized states . . . 91 6.1.3 Concept of 2×2 coherency matrix . . . 93 6.1.4 Stokes parameters: Intensity-based representation of
polarization states . . . 95 6.1.5 The Poincar´e sphere representation of Stokes
polarization parameters . . . 98 6.1.6 Decomposition of mixed polarization states . . . 98 6.2 Interaction of polarized light with material media . . . 100 6.2.1 Basic medium polarimetry characteristics . . . 101 6.2.2 Relationship between Jones and Mueller matrices . . . 102 6.2.3 Jones matrices for nondepolarizing interactions:
Examples and parametric representation . . . 105 6.2.4 Standard Mueller matrices for basic interactions
(diattenuation, retardance, depolarization): Examples
and parametric representation . . . 107 6.3 Experimental polarimetry and representative applications . . . 113 6.3.1 Stokes vector (light-measuring) polarimeters . . . 114 6.3.2 Mueller matrix (sample-measuring) polarimeter . . . 116 In Chapter 4 it was noted that the experiments carried out by Fresnel and Young led to the discovery of transverse character of light that could satisfacto- rily describe the phenomenon of interference using polarized light. The basic definition of state of polarization of light via the polarization ellipse of the transverse EM field was briefly introduced in Chapter 4. When such polarized light passes through any anisotropic medium, its polarization state is trans- formed depending upon medium characteristics. The variations in the state of polarization of a wave thus enable us to characterize the system under con- sideration. A number of mathematical formalisms have been developed over the years to deal with the propagation of polarized light and its interaction with optical systems. Among these, the Jones calculus and the Stokes-Mueller calculus have been the most widely used. The former is a field-based model that assumes coherent addition of the phase and amplitude of EM waves, and the latter is an intensity-based model that instead utilizes the incoherent addition of wave intensities. In this chapter we define the various states of
86 Wave Optics: Basic Concepts and Contemporary Trends
polarization of light waves and discuss the interaction of such polarized light with material media using the Jones and Stokes-Mueller calculus. We also briefly introduce the concepts of polarimetric measurements and touch upon representative applications of experimental polarimetry.