Materials that can rotate the plane of polarization of incident plane po- larized light are said to be optically active. Such materials can be crystalline as well as amorphous. The classical example of the setup leading to rotation
Left
Right Right
LCP RCP
FIGURE 4.5: Fresnel prism.
of the plane of polarization is as follows. A uniaxial material like quartz, with the optic axis along the direction of propagation, is placed between a polar- izer and a crossed analyzer. In absence of the quartz crystal, light is blocked by the polarizer-analyzer system. Insertion of the quartz leads to some finite transmission. A rotation of the analyzer by an angle φ again extinguishes transmission. This implies that after exiting the quartz crystal, the polariza- tion plane has undergone a rotation by an angle φ. Experiments show that different samples of quartz exhibit different types of rotation, either clockwise (right-handed) or counterclockwise (left-handed). One type happens to be the mirror image of the other. Experiment also shows thatφ=αdwithα∼1/λ2. For example, ford= 1 mm,φyellow = 20◦ and φviolet = 50◦. Thus there is a significant dependence of optical activity on the wavelength.
Optical activity is also observed in some materials like sugar, nicotine, camphor, etc. In this caseφ=αdc′, whereα∼1/λ2 is the rotation constant, dis the length of the cell andc′is the concentration. Similar experiments form the foundation of how to determine the concentration of the optically active materials. It has many chemical and biological applications. The explanation of the rotation of plane of polarization in optically active media was first given by Fresnel, who first showed the similarity between double refraction and op- tical activity. The explanation is based on the fact that any linearly polarized light can be thought of as a superposition of left and right circular waves as in Fig. 4.6. Fresnel further assumed that in an optically active medium, the right and left waves propagate with different velocities. On this basis all the optically active media can be divided in two classes, namely,right (vR> vL) and left (vR < vL). For experimental demonstration of the validity of this assumption, Fresnel prepared a special prism as in Fig. 4.5. The inequality nR< nL holds for the first and third prisms, while for the secondnR > nL. The angle of refraction of the RCP is smaller than that of LCP when the wave leaves the first prism. While passing through the second prism the RCP com- ponent is refracted to a higher degree than the LCP and the angle between the two directions increases. Finally, the spatial separation between the rays increases after passing the last prism.
We now explain the rotation of plane of polarization. Due to different velocities of the right and left components, the time taken to traverse the same length of the material will be different for these components. Hence the
66 Wave Optics: Basic Concepts and Contemporary Trends )
b ( )
a (
FIGURE 4.6: Explanation of the rotation of plane of polarization. The dashed line gives the direction of oscillation of the electric field. (a) and (b) represent the polarizations at the input and the output faces, respectively.
angle spanned by the right component will be different from that of the left component. The symmetry direction is then given by the arithmetic average of these two angles (see the dashed line in Fig. 4.6). Thus the angle of rotation will be given by
φ=φR−φR+φL
2 = φR−φL
2 . (4.58)
The right and left circular components for a plane polarized light at the input face of the uniaxial material can be written as
ER=E0(ˆx−iy)eˆ i(kRz−ωt), (4.59) EL=E0(ˆx+iy)eˆ i(kLz−ωt). (4.60) Note that in the above equations we have incorporated Fresnel’s postulate that inside the material the right and left circular waves have different wave vectors or different velocities. At the output face of the slab with thicknessh, thexandy components of the electric field can be written as
Ex= 2E0exp
i
kR+kL
2 h−ωt
cos
kR−kL
2 h
, (4.61)
Ey=−2E0exp
i
kR+kL
2 h−ωt
sin
kR−kL
2 h
, (4.62)
so that
Ey
Ex
=−tan
kR−kL
2 h
. (4.63)
It is clear that this ratio is a real number and hence for arbitrary thickness of the slab, light remains linearly polarized. The direction ofE now makes an
angleφwith thex-axis, which is given by φ=−kR−kL
2 h=−k0nR−nL
2 h=−ω c
nR−nL
2 h. (4.64)
For a right (left) material, the rotation is in the clockwise (counterclockwise) direction.
Chapter 5
Optical properties of dielectric, metal and engineered materials
5.1 Linear response theory and dielectric response . . . 70 5.1.1 Time domain picture . . . 70 5.1.2 Frequency domain picture . . . 71 5.2 Kramers-Kronig relations . . . 72 5.3 Dispersion in metals: Drude model . . . 75 5.4 Planar composites and motivation for metal-dielectric
structures . . . 76 5.5 Metal-dielectric composites . . . 78 5.5.1 Maxwell-Garnett theory . . . 79 5.5.2 Bruggeman theory for multicomponent composite
medium . . . 80 5.6 Metamaterials and negative index materials . . . 81 Material properties can be determined from the real-time response of the system. For example, for a dielectric, the dipoles constituting the medium take finite time to respond to the quickly oscillating radiation (∼ 1015 Hz in optical domain) passing through the dielectric. We present a brief sketch of the linear response function theory in order to understand the frequency dependence of the dielectric function ǫ(ω). We discuss this for ǫ(ω), while similar arguments can be developed for magnetic response. While the details were worked out for the Lorentz model of the dielectric in Section 2.3.1, here we develop the response for metals (known as the Drude model). In dealing with the susceptibilities, we pay due attention to causality and its manifestation in the form of Kramers-Kronig relations.
Most of our attention is focused on some of the properties of composite materials, which are formed by two or more constituents. It so happens that the effective dielectric and material properties of the composites can turn out to be better than those of the constituents. There are several such examples from linear and nonlinear optics [9, 10], where the composites are engineered to have the desired optical response or to have larger nonlinear effective sus- ceptibilities. Our goal will be to understand the mechanism of how the prop- erties of the composites can be manipulated. There can be different kinds of composites from a geometrical viewpoint. For example, we can talk about metallic/dielectric sub-wavelength layers stacked together or we can have
heterostructures with tiny metal (dielectric) inclusions embedded in a (metal) dielectric host. Of late such nano-composites are very much in focus because of their interesting properties and wide application potentials. We first look at the two-component layered composites to highlight the remarkable possibili- ties with such structures. Indeed, in Section 5.4 we show that a metal-dielectric composite can exhibit extremely large anisotropy though both the constituents are isotropic in nature. We then use the local field modification to obtain the effective dielectric function of the heterostructures. Note that some of these issues have been discussed in detail in textbooks [3] or in monographs [11].
We follow these sources to arrive at the Clausius-Mossotti relation and the effective dielectric function for two or multi-component heterogeneous media.
Special attention is paid to metal inclusion in a dielectric host since this is used to a large extent for many other problems in other chapters. The interest in such metal nano-composites stems from the fact that they can support lo- calized plasmon resonances. The excitation of the localized plasmons can lead to large local fields that are needed for low-threshold optical processes (see Chapter 10 and Section 10.2.3 for more details and applications).