Where, the dyadics £(®) and ju(a>) are the relative permittivity and permeability; and a(w) and ^(ffl) are the dyadics describing the magnetoelectric properties of the material. The permittivity and permeability of free space are s0 = 8.854x10~12 F ■ mTl and ju0 = 4^x10"7 H ■ m~\ respectively. The problem, then, is to determine a single set of four 3x3 complex valued dyadics (£ , a , P , and jS) describing the equivalent homogeneous material to replace the two sets of dyadics representing nanowire and void. The four dyadics are often combined for convenience to form one compact 6x6 matrix as,

In this formalism, the fields are then combined to form a column 6-vector,

so that the right sides of Eq. 15.2 and Eq. 15.3, for instance, are combined to be compactly written as C • F .

Determining average constitutive parameters for composite materials, a process known as homogenisation, is a very active area of research. For all but the simplest cases, great effort must be expended to determine the constitutive parameters for an equivalent homogeneous material, referred to as a homogenised composite material (HCM). Various schemes have been devised to determine the HCM constitutive properties, each with its own limitations (Lakhtakia 1996; Michel 2000; Mackay 2003).

One of the earlier and simplest formalisms, Maxwell Garnett (1904), is limited to situations where one of the materials in the composite takes the form of a small particulate dispersed in a second material, the host, which must make up most of the volume of the composite. As can be seen from the electron micrographs, this is often not the situation for STFs, where nanowire and void can occupy comparable volumes.

Two different modifications of the basic Maxwell Garnett formalism have been developed to solve the problem of homogenizing composites with dense inclusions. One approach is to build up a dense concentration of particles in a number, N, of dilute particle additions or increments and is thus known as the Incremental Maxwell Garnett (IMG) formalism (Lakhtakia 1998). After each addition of particles, the composite is re-homogenised so that the next addition of particles is made into a homogeneous medium. Hence, each homogenisation satisfies the Maxwell Garnett requirement of a system of dilute particles in a host material.

The other modification to the Maxwell Garnett formalism is obtained from the IMG in the limit N^<x>. In this case, a differential equation for the constitutive parameters as a function of concentration of particulate is obtained, replacing the iterative procedure of the IMG formalism. This formalism has been named the Differential Maxwell Garnet (DMG) formalism (Michel et al. 2001). Another approach currently being used to locally homogenise STFs (Sherwin and Lakhtakia 2001; Sherwin et al. 2002), which is not derived from the Maxwell Garnett method, is the Bruggeman formalism (Bruggeman 1935). Unlike the Maxwell Garnett formalism, this method places no restriction on the concentration of inclusions. In fact, this method is carried out by considering both materials of the composite as inclusions in the resultant HCM being sought. The constitutive parameters of the HCM are then adjusted so as to produce zero polarisation in the HCM-inclusion mixture. The HCM thus obtained represents the desired HCM characterizing the composite.

Finally, a more complete formalism that has received some attention is the Strong Property Fluctuation Theory (SPFT) (Tsang and Kong 1981; Zhuck 1994). The method is known for being quite difficult to carry out numerically, but is exact in the unapproximated form. It takes account of spatial correlations between components in the composite. Second order SPFT has been developed for linear bianiostropic HCMs (Mackay et al. 2000, 2001, 2001, 2002). It has also been extended to the weakly non-linear regime (Lakhtakia 2001, Lakhtakia and Mackay 2002). Whichever formalism is chosen, the polarisablity of a single inclusion particle must be calculated as a first step. This can be done relatively simply for ellipsoidally shaped inclusions. Lakhtakia et al. (1999, 2001) have developed a nominal model to describe a STF layer in which both nanowire and void are modeled as ellipsoids. With long, narrow ellipsoids, the aciculate nature of the nanowire can be captured.

It should be pointed out that the development of homogenisation procedures is not at the stage where absolute constitutive properties of a STF can be calculated from first principles. Nonetheless, there are several ways in which calculation of HCMs is useful. Calculation of trends, as various STF parameters are changed, may be useful in the design of STFs for specific purposes. Also, although local homogenisation may not give absolute numbers for the constitutive relations, it does give the form that the constitutive dyadics should take for various types of materials. Some proposed uses of STFs involve infiltrating the porous film with different types of fluids (Lakhtakia 1998; Ertekin and Lakhtakia 1999; Lakhtakia et al. 2001). Here, again, homogenisation is very useful in pointing to the correct form of the constitutive dyadics.

Rather than relying on homogenisation, most calculations of optical properties of STFs have used one of two approaches to obtaining constitutive parameters. One approach is to guess reasonable parameters. This is useful when gross behavior is desired. Furthermore, STFs made of the same materials, in the same way, at different labs will have different detail properties. Thus, STFs for a specific purpose will have to be fine tuned, and absolute numbers are of limited use. The other approach is to use principal indices of refraction extracted from experimental data on CTFs (Hodgkinson and Wu 1998; Hodgkinson et al. 2001; Chiadini and Lakhtakia 2004; Polo and Lakhtakia 2004). This has the added advantage of accounting for higher order influences on constitutive parameters, such as the change in nanowire cross-sectional morphology and material density as a function of %.

So far, only the constitutive parameters of a single electrically thin layer of a STF have been discussed. With the shape and orientation of the nanowire changing over a larger length scale (comparable to optical wavelengths) due to sculpting, the variation of the constitutive properties of the HCM as a function of depth into the film must be described. STFs made of various types of materials have been modeled. For purposes of discussion, a dielectric material will be assumed here since: only one dyadic need be considered, expressions are reasonably compact, and other materials are handled similarly.

The constitutive parameters for the film can be written as,

D(r)=s0gz(z) • sy(Z) • gf • s;1 • z) • eg(z) • E(D, (15.6)

As the material is non-magnetic, Eq. 15.7 takes a simple form with a single scalar, p.o, defining the isotropic magnetic constitutive relation. Eq. 15.6, on the other hand, is much more complex with several factors required to describe the permittivity dyadic.

In this expression, e" , the reference permittivity dyadic, represents the

— reef permittivity dyadic for a layer of STF with the nanowires aligned along a reference direction, the X - axis. In principle, this is the quantity which can be determined by homogenisation. As discussed above, other methods are often used. In any case, for the present situation, it takes the form,

Where, Ux, Uy and Uz are the Cartesian unit vectors. This is the same form of permittivity dyadic used to describe a bulk homogeneous bianisotropic material. This reflects the fact that the principal axes of the nanowire are, in general, all different due to the flattened cross-section of the nanowire. The permittivity along the length of the nanowire is represented by eb. The other two components (ea, ec) represent the permittivities along the principal axes through the cross-section. In canonical models of STFs, the permittivities along the principal axes are assumed to be independent of whereas, in models relying on empirical determinations of the constitutive relations the permittivities are expressed as a function of %. The rotation matrix, given below,

S y(x) = (uxux + Uzuz) cos j + (uzux - uxuz) sin x + uyuy (15.9)

rotates £ by an angle % to account for the tilt of the nanowire in the STF. For

CTFs and chiral STFs, % is constant, while in SNTFs %=%(z). The second rotation matrix rotates the already inclined dyadic about the z axis. For a chiral STF, which has a constant rate of rotation as a function of z, it is written as,

Sz( z) = (wx«x + wy«y) cos(^z/Q) + (uyux - uxuy) sin(^z/Q) + uzuz (5.10)

in order to create the helical structure. For SNTFs, there is no rotation about the z axis and S (z) may be considered to be I, the identity matrix.

With the constitutive relations obtained by one of the methods described above, the propagation of the wave in the STF can be characterised by solving Maxwell's curl equations. The STF will be considered to be an infinite slab occupying the space from z=0 to z=L, with the axis of inhomogeneity along the z direction. As the spectral response of the STF is most often desired, we will assume the fields to have a harmonic time dependence of e"iwt, where w is the angular frequency. In time harmonic form, Maxwell's curl equations are,

With a plane wave incident on the film from z<0, the fields may be expressed in terms of their Fourier amplitudes as

E(r) - [ex(z,K,y)ux + (z,K,¥)uy + ez(z,^)uz]e{lK(xcos^+ysin^)](15.12) H(r) = [A,(z,ff, ^ + Ay(z,k, W)uy + A (z,ff, ^]xcos^+ysi^)] (5.13)

Where, k=k0 is the transverse wavenumber, with k0 the free space wavenumber and 9 the angle of incidence. The plane of incidence is described by the angle y, which is measured from the x direction. Substituting Eq. 15.12 and Eq. 15.13 into the Maxwell curl relations results in two sets of equations. The first is a set of two algebraic equations, which describe the z components of the fields in terms of the x and y components. The second is a set of four coupled ordinary differential equations involving the x and y amplitudes of the two fields. The four differential equations may be written as one compact matrix ordinary differential equation (Berreman 1972; Lakhtakia 1997) in the form,

Where, the field amplitudes have been assembled into the column 4-vector and ^p(z)J, referred to as the kernel, is a complex 4x4 matrix which depends on the form of the constitutive relations describing a particular STF (Lakhtakia 1997; Lakhtakia and Messier 2004).

In determining optical properties such as reflectance or transmission, it is convenient to recast the formulation in terms of a 4x4 matrix, called the matrizant, such that

Where, ff (0)] is the field 4-vector at the surface. Substituting the right hand side of Eq. 15.16 into the Eq. 15.14 yields an equivalent differential equation (Lakhtakia and Messier 2004),

in terms of the matrizant. In this way, the boundary conditions may be stated generally once and for all as,

and the differential equation may be solved solely in terms of the STF properties. The heart of the propagation problem, then, is the solution of Eq. 15.17. In general, p(Z)J is a function of z, and no known closed form solutions of Eq.

15.17 exist. However, there are two special cases for which ^p(z) J is a constant. In these cases, the solution can be written formally as,

One case is that of the CTF, which has a constant [P] because the constitutive relations are constant as a function of z . The other situation is for chiral STFs, but only occurs for normal incidence or, in other words, when k=0. In this case the constitutive relations are not constant due to the helical rotation. However, the problem can be worked in terms of transformed fields under the so-called Oseen transformation (Oseen 1933; Lakhtakia and Weiglhofer 1997) given by

Where, S (Z) is the rotation matrix for chiral STFs described earlier. When written in terms of the transformed fields, the differential equation has a constant kernel for normal incidence. Once the transformed fields are determined, the inverse transformation can be used to obtain the physical fields. The problem is often worked in terms of the transformed fields at oblique angles of incidence, as well. Even though a closed form solution does not exist, the algebra is simplified significantly by using the transformation.

In the more common situation, when the kernel is not constant, several schemes have been used to obtain a solution to Eq. 15.17. One is an exact solution for chiral STFs written as an infinite polynomial series (Lakhtakia and Weiglhofer 1996; Polo and Lakhtakia 2004). In practice, the series must be terminated after a finite number of terms. The convergence of the series may be monitored as each term is added and the summation terminated when the desired accuracy has been obtained. The ability to monitor the convergence as the calculation proceeds may be an advantage over other numerical techniques for which the accuracy of the result is discerned by repeating the entire calculation under different approximation parameters (Polo and Lakhtakia 2004). The method is particularly advantageous if many calculations are to be performed for different situations.

For situations in which [P(z)] is periodic, a perturbational approach has been applied (Weiglhofer and Lakhtakia 1996, 1997) when the STF is weakly anisotropic. For stronger anisotropy, the coupled-wave method has been used (McCall and Lakhtakia 2000, 2001; Wang et al. 2002). A numerical method offering great versatility is the piecewise homogeneity approximation (Abdulhalim et al. 1986; Venugopal and Lakhtakia 2000). In this approximation, the STF is divided into a series of thin slices parallel to the surface. Each slice is treated as if it had a constant permittivity dyadic, with the value of the dyadic taken to be that of the STF at the center of the slice. With the approximation of constant permittivity dyadic, the matrizant for each single slice can be calculated using Eq. 15.19. The matrizant for the entire STF is then obtained as the sequential product of the matrizants of the slices. The accuracy of the method depends on the thickness of the slice used for the calculation, with thinner slices yielding higher accuracy but taking more computation time. Once the matrizant has been determined, the optical properties of the STF can be obtained by solving a boundary value problem with incident, reflected, and transmitted waves ((Lakhtakia and Messier 2004).

The optical characteristics of STFs have been both calculated and measured (McPhun et al. 1998; Wu et al. 2000). By far, the most studied effect is the circular Bragg phenomenon (CBP) of chiral STFs. Most work has been on linear dielectrics. Theoretical calculations have been carried out for non-linear dielectrics (Lakhtakia 1997; Venugopal and Lakhtakia 1998) and gyrotropic materials (Pickett and Lakhtakia 2002; Pickett et al. 2004). The effects of absorption and dispersion in dielectric STFs has also be studied (Venugopal and Lakhtakia 2000).

As with other periodic systems, chiral STFs exhibit strong reflection in narrow bands of wavelengths. The reflection bands, or Bragg peaks, are centered at a series of wavelengths forming a harmonic series. The range of wavelengths over which the reflection occurs is referred to as the Bragg regime or alternatively the Bragg zone. Normally, all of the peaks are very weak, except the one at the longest wavelength, 1st order Bragg peak, and the one at half that wavelength, 2nd order Bragg peak. These phenomena have been systematically studied for a range of parameters describing both the incident plane wave and the chiral STF by Venugopal and Lakhtakia (1999-2000). Several generalisations can be cited. Reflection of light in the 1st Bragg regime is like reflection from a normal mirror. The state of linear polarisation is maintained upon reflection; while the state of circular polarisation is reversed upon reflection. Unlike a normal mirror, however, the reflection only occurs in a restricted band of frequencies. Additionally, the reflection in the 1st Bragg regime is strongest at large angles of incidence and goes to zero as the angle of incidence approaches zero. As reflection in the 1st Bragg regime is similar to normal reflection, it has attracted little attention. Reflection in the 2nd Bragg regime is quite different. It is characterised by strong reflection of circularly polarised light matching the handedness of the material, a phenomenon known as the circular Bragg phenomenon (CBP). Light of the opposite circular polarisation is largely transmitted. Furthermore, unlike reflection from a normal mirror in which the handedness of the light is reversed upon reflection, the reflected light in the CBP maintains the same circular polarisation as the incident light. Because of these unusual properties, the CBP has received considerable attention. Fig. 15.4 shows an example of the calculated optical remittance spectrums (reflectances and transmittances) , in the vicinity of the 2nd order Bragg peak, for a typical chiral STF. In this example, the wave is at normal incidence and the chiral STF is right handed. The remittances are designated by: RRR, RRL, Rlr, Rll and TRR, TRL, TLR, TLL. Here, Rrl, for instance, represents the fraction of the incident power density that is reflected as right circularly polarised (RCP) light when the incident wave is left circularly polarised (LCP). Both RRL and RLR are referred to as cross-polarised reflectances, since they describe a change in polarisation state upon reflection. On the other hand, Rrr and RLL are referred to as co-polarised reflectances, since the polarisation state is preserved on reflection. The same terminology is used to describe the transmittances. In Fig. 15.4(a) the Bragg regime can be clearly seen in the spectrum of RRR between X0=775 nm and X0=815 nm, where the reflectance is very high. Small but significant cross-polarised reflectances and transmittances can also be seen within the Bragg regime. The co-polarised reflectance RLL however, remains very small within the Bragg regime, demonstrating the preferential reflection of RCP light from a right-handed chiral STF. The CBP is also evident in the transmission spectrum, shown in Fig. 15.4(b), with a dip in TRR over the wavelengths corresponding to the peak in the RRR spectrum. TLL remains high in the Bragg regime indicating that LCP light is mainly transmitted. Outside of the Bragg regime, all of the remittances exhibit oscillations. These correspond to the Fabry-Perot oscillations (Hecht 2002) seen in normal homogeneous material and are a result of interference between the waves reflected from the two surfaces of the material. They can be related to material thickness and average index of refraction. Investigations of CBP characteristics as a function of various parameters yield several trends. At normal incidence, four parameters are useful: Q, %, ec and g d, where sd = £a£b!{£a cos2 % + £b sin2 z) (15.21)

With these quantities, the central wavelength A.^ of the Bragg peak and the full-width-at-half-maximum of the peak A^^ can be estimated for weakly absorbing chiral STFs as tf *n(\£c I1'2 +\ed I1'2 ] (15.22)

It should be noted that when | sc |=| sd |, referred to as the pseudo-isotropic point, the width of the Bragg regime is zero and the CBP disappears. As the pseudo-isotropic point may have practical applications, it has been characterised by calculation for some selected materials using experimentally derived constitutive parameters (Lakhtakia 2002). The height of the Bragg peak is dependant on the number of helical cycles in the film and the degree of local birefringence. Greater peak heights are obtained for larger local birefringence. The peak height saturates as the thickness of the film increases, with the thickness of the film required to obtain a particular peak height dependant on the details of the parameters describing the particular chiral STF. With light incident at oblique angles, the Bragg peak shifts to shorter wavelengths, becomes narrower, and is reduced in height, as the angle of incidence is increased.

Fig. 15.4. Optical remittances of a right handed chiral STF with Q=200 nm, L=60Q, X=63.6°, sa=1.9025, sb=2.0199, (a) reflectances (b) transmittances (Adapted from Polo and Lakhtakia 2004)

Fig. 15.4. Optical remittances of a right handed chiral STF with Q=200 nm, L=60Q, X=63.6°, sa=1.9025, sb=2.0199, (a) reflectances (b) transmittances (Adapted from Polo and Lakhtakia 2004)

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