Lightwave Refraction and Its Consequences: A Viewpoint of Microscopic Quantum Scatterings by Electric and Magnetic Dipoles

In optics, it is well-known that when a visible light beam, e.g., traveling from air (or more strictly, vacuum) into a piece of smooth flat glass at an angle relative to the normal of the air-glass interface, some proportion of the light will be bounced off at the reflection angle equal to the incident angle. However, when the light beam is with its oscillating electric field parallel to the plane-of-incidence (POI, i.e., the plane constituted by both propagation vectors of the incident and reflected light waves, as well as the interface normal vector) (called the p-wave), there is a particular incident angle at which no bounce-off would occur. This particular angle is known as the Brewster angle ( B  ) (Hecht, 2002). In contrast, when the light beam is with its electric field vector perpendicular to the plane-of-incidence (called the s-wave), no such angle exists (Hecht, 2002). In fact, this is only true for uniform, isotropic, and nonmagnetic (or equivalently, with its relative magnetic permeability ( r  ) equal to unity at the optic frequency of interest) materials such as the above glass piece. Indeed, it is known that for magnetic materials, there may instead exist Brewster angles for the s-waves, while none for the p-waves (as will be demonstrated in Section 2).

In order to reveal the intricacies of the mechanism of the proposed permanent dipole engineering subsequently, existing important result of Doyle (Doyle, 1985) is first thoroughly detailed. That is, the Fresnel equations and Brewster angle formula are to be arrived at intuitively and rigorously obtained, by viewing all light-wave-induced dipole moments (including both electric and magnetic dipoles) as the microscopic sources causing the observed macroscopic optical phenomena at an interface, as compared to the traditional academic "Maxwell" approach ignoring the dipole picture. Then, equipped with suchdeveloped intuitive and quantitative physical picture, the readers are then ready to appreciate the way those optically-responsive, permanent dipoles are externally implemented into a selected host matter and their rendered effects. Namely, the Brewster angle of a selected host material becomes alterable, likely at will, and ultimately new optical materials, devices and applications may emerge.

Brewster angle and "scattering" form of Fresnel equations
Arising from Maxwell's equations (through assuming linear media and adopting monochromatic plane-waves expansion), Fresnel equations provide almost complete quantitative descriptions about the incident, reflected and transmitted waves at an interface, including information concerning energy distribution and phase variations among them (Hecht, 2002). Two of the Fresnel equations are relevant to reflections associated with both the p and s components (Hecht, 2002 where E is the electric field, r  is the relative magnetic permeability,   1/2 rr n   is the index of refraction ( r  being the relative dielectric coefficient), superscripts "p" and "s" stand for the p-wave and s-wave components, while subscripts "i", "r" and "t" denote incident, reflected and transmitted components, respectively. When the incident angle ( i  ) is equal to a particular value ( B  ), one of the above reflection coefficients would vanish, then such value of the incident angle is known as the Brewster angle ( B  ). Note that for the most familiar case in which the light wave is incident from vacuum onto a linear nonmagnetic medium ( 1 r   ), only the p-wave possesses a Brewster angle, not the s-wave.
In the following, to get ready for our proposed idea while without loss of generality, the medium on the incident side is designated to be vacuum (i.e., 1 i n  ) for simplicity. In addition, to further facilitate our purpose, the Fresnel equations in the equivalent "scattering" form (due to Doyle) are retyped here (Doyle, 1985): Namely, the right hand sides of Eq. (3)- (6) are in the form of D  S, with D being the first bracketed term, representing single dipole (electric and magnetic) oscillation; while S being the second term, depicting the collective scattering pattern generated by the whole array of dipoles. While S is nonzero, where D vanishes (Eq. (4) or (6)) is the condition for the Brewster angle to arise either for the p-wave or the s-wave. That is (Doyle, 1985), In the following, an alternative derivation of the "scattering" form Fresnel equations (Eq. (4)-(6)) will be reproduced from (Doyle, 1985) in somewhat details, which stems from the viewpoint of treating induced microscopic (electric and magnetic) dipoles as the effective sources of macroscopic EM waves at the interface. Then, effective ways to implement the proposed permanent dipoles on a host material will be proposed, which will in principle allow us to achieve variant Brewster angles and thus to create novel materials, devices, even new applications.

''Scattering" form Fresnel equations from the dipole source viewpoint
negative-charged electron clouds. When subjected to the EM field of an impinging light wave, the positive and negative charges separate to form induced electric dipoles along the light electric field, while some electrons further move in ways to form induced magnetic dipole moments along the light magnetic field. Note that in this chapter, only far-fields generated by these dipoles are considered and Fig. 1 depicts the relevant orientations.  (Doyle, 1985) ).

The p-wave (E parallel to the plane-of-incidence) case
It is well-known from electromagnetism (Purcell, 1985) that if denoting the induced electric polarization by P  , the electric displacement is: , the induced polarization hence emerges as (all in space-time configuration) (Purcell, 1985): When calculating the field contribution from the electric dipoles alone, it is assumed that 1 r   and thus Putting this into an equivalent form of the Snell's law (Doyle, 1985): and subsequently, together with Eq. (9), into the Fresnel equation relating / ti EE , we obtain an expression in which the incident electric field is expressed as a "consequence" of the microscopic sources --the induced dipoles: www.intechopen.com where the subscript "p" stands for contribution from the induced electric dipoles. Incorporating Eq. (11) into Eq. (2), the reflected electric field manifests itself as due to the electric dipoles sources: Similarly, there are field contributions from the induced magnetic dipoles, which acted through the magnetization M  . Using the magnetic field strength relation associated with general monochromatic plane waves, the magnetization becomes (Purcell, 1985):  (3)), can be converted into an expression for the magnetic component of the incident electric field, in fact as a "consequence" of the microscopic induced magnetic dipoles: where the subscript "m" stands for contribution from the induced magnetic dipoles. Putting Eq. (14) into Eq. (2), the magnetic component of the reflected electric field appears as due to the induced magnetic dipoles too: When a light wave impinges on an interface, it causes the excitation of electric and magnetic dipoles throughout the second medium, which in turn collectively give rise to reflected, transmitted, and formally, incident waves at the interface. As depicted in Fig. 1, as long as far fields are concerned, the induced electric and magnetic dipoles can be viewed as aligned along the incident electric and magnetic field vectors, respectively. All electric and magnetic fields of interface-relevant waves can be conceived as generated by the co-work of electric and magnetic dipoles. (Of course, for the incident wave, this is only formally true, that is, the incident fields are the "cause" not the "effect" of dipole oscillations.) We thus write: Namely, by adding Eq. (11) and (14), the incident electric field can be formally expressed as due to induced dipole sources P and M : As a verification, if putting forms of these sources (i.e., Eq. (9) and (13) in which the induced P and M are expressed in term of p t E ) back into Eq. (17), it is found that the obtained transmission coefficient of the p-wave is exactly that of Eq. (3).
Similarly, conceiving the reflected electric field as: Adding Eq. (12) and (15), the reflected electric field appears as due to the induced dipole sources P and M : Similarly, putting forms of the dipoles sources (i.e., Eq. (9) and (13) (17)), it is found that the obtained reflection coefficient of the p-wave is exactly that of Eq. (4), in "scattering" form.

The s-wave (E perpendicular to the plane-of-incidence) case
Likewise, following Fig. 1 again, we can also express the incident and reflected electric fields of the s-wave as due to those induced electric and magnetic dipoles: and that the transmission and reflection coefficients are indeed found to be those of Eq. (5) and (6), respectively, in "scattering" form.

Observations and inspirations
From the above elaboration, it becomes obvious that the electric and magnetic dipoles can be much more than mere pedagogical tools for picturing dielectrics and magnetics, as indeed proven by Doyle (Doyle, 1985). In fact, treating them as microscopic EM wave sources from the outset, the "scattering" form of Fresnel equations (i.e., Eq. (3)- (6)), and consequently, the Brewster angle formulas (Eq. (7) and (8)) can all be reproduced. Then, emerging from such www.intechopen.com details come our inspired purposes. Namely, by acquainting ourselves with the role played by these induced dipoles (or, the microscopic scattering sources), it is then intuitively straightforward to learn how new macroscopic optical phenomena, such as the new Brewster angle may be generated if extra anisotropic optically-responsive permanent dipoles were implemented onto the originally isotropic host material in discussion (Liao et al., 2006). In other words, now the notions of P and M are further extended to include the total effects resulting from both the induced and permanent dipoles.
For instance, in Eq. (17), the   cos ti    factor multiplying on P (but not on M ) is due to the fact that the induced polarization P (along p t E ) has only a fractional contribution to is introduced within a host material, then, e.g., for the p-wave case, all electric dipoles' contribution to p i E (i.e., Eq. (17)) is now Fig. 2). Namely, there is now an additional second anisotropic term resulting from externally imposed dipoles whose contribution may not necessarily be less than the induced dipoles of the original isotropic host. Note that the incident light-driven response of these externally imposed permanent dipoles is frequency dependent, and therefore, the above 0 P really stands for that amount of polarization at the relevant optical frequency of interest and the lightwave's incident angle i  . In other words, a "DC" polarization will never enter the above equation, and 0   (Doyle, 1985) ).

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Thus, if the dipole-engineered total contribution is recast in the traditional form, , then it is clear that the modified relative dielectric coefficient is equivalently (Liao et al., 2006): where 0  is the angle between the imposed extra polarization vector and the interface plane (see Fig. 2). Thus, by putting Eq. (22) into the p-wave Brewster angle formula (Eq. (7)), a new Brewster angle ( B  ) would then emerge:

Justification of the effectiveness and meaningfulness of implementing optically-responsive dipoles
A justification of the effectiveness of the proposed permanent dipole engineering is straightforward by noting the following fact. Namely, had the original host material been transformed into a new material by adding in a considerable amount of certain second substance, then 0 P in the above really would have stood for the extra induced dipole effect resulting from this second substance.
However, to this end, an inquiry may naturally arise as to whether the outcome of the proposed dipole-engineering approach being nothing more than having a material of multicomponents from the outset. The answer is clearly no, and there are much more meaningful and practical intentions behind the proposed method. First of all, this is a controllable way to make new materials from known materials without having to largely mess around with typically complicated details of manufacturing processes pertaining to each involved material (if the introduced permanent dipoles are noble enough). Indeed, we have been routinely attempting to create various materials by combining multiple substances, and yet have also been very much limited by problems related to chemical compatibility, phase transition, in addition to many processing and economic considerations. Secondly, permanent dipole engineering would further allow delicate, precise means of manipulating the material properties, such as varying the dipole orientation to render desired optical performance on host materials of choice. Thirdly, all existing techniques known to influence dipoles can be readily applied on the now embedded dipoles to harvest new optical advantages, such as by electrically biasing the dipoles to adjust the magnitude of permanent dipole moment (in terms of 0 P ) in the frequency range of interest.

Different refracted wave powers on two conjugated incident light paths
If, instead of picking the incidence from the left hand side as depicted in Fig. 2, a conjugate path, i.e., from the right hand side, is taken (see, Fig. 3), then the formula for Eq. (22) becomes (Liao et al., 2006): www.intechopen.com Unconventional Brewster angle can be found by Eq. (23) (p-wave case). Fig. 3. P-wave configuration and the orientation of embedded permanent electric dipoles (Conjugated Incident Light Path, courtesy of W. T. Doyle (Doyle, 1985) ).
In other words, the traditionally fixed Brewster angle of a specific material now not only becomes dependent on the density and orientation of incorporated permanent dipoles, but also on the incident light intensity (more precisely, the incident wave electric field strength). Further, two conjugated incident light paths would give rise to different refracted wave powers (Liao et al., 2006), (Haus & Melcher, 1989).

The surface embedded with a thin distributed double layer
The traditional Fresnel equations in the electromagnetic theory have been used in determining the light power distribution at an interface joining two different media in general. They are known to base upon the so-called "no-jump conditions" ( (Haus & Melcher, 1989), (Hecht, 2002)) wherein the interface-parallel components of the electric and magnetic fields of a plane wave continue seamlessly across the interface, respectively: where ab QQQ  stands for the discontinuity of the physical quantity Q by crossing from the a side to the b side of the interface, and || (or ) is with respect to the interface plane. The general configuration of light incidence can be decomposed into the p-wave and s-wave situations. For the p-wave case, the lightwave's electric field is on the plane of incidence (POI) (see Fig. 4), and for the s-wave situation, it is pointing perpendicularly out of the POI.
When a light beam of frequency  is incident from air (a side) onto a flat smooth dielectric (b side) embedded with a layer of distributed, incident-light-responsive electric dipoles near the surface (see Fig. 5), the above jump condition for the electric field becomes (Haus & Melcher, 1989): ( being the magnetic permeability) over a vanishingly thin strip area enclosing a section of the interface on the plane of incidence (see Fig. 5). Thus, with the integration on the right hand side being null, further applying Stokes' theorem ( (Haus & Melcher, 1989), (Hecht, 2002)) on the left hand side reveals that the two normal sections (i.e., along E   ) in the contour integration no longer cancel each other. This is because the vertical electric field distribution is now nonuniform in the presence of a distributed double layer (see Fig. 5). Thus, instead of continuing across smoothly, the jump in || E is now proportional to the spatial derivative of the electric dipole moment per unit area ( s  ) on the interface, which in general is a space-time variable, Chen et al., 2008). In other words, where the subscripts "i", "r", and "t" represent the incident, reflected, and transmitted components, respectively. In other words, the condition || 0 E  arises where there is non-uniform distribution of the dipole moment along the projection of the incident light's electric field on the interface.
The new power reflection coefficient is then 2 Rr  and the new power transmission coefficient is: 1 TR  , for both the p-and s-wave cases (Hecht, 2002). Therefore, in the presence of a light-responsive, distributed double layer at the interface (e.g., with Fig. 4), the reflected and transmitted lights are expected to be further modulated at the frequency M  . Additionally, this modification to the light reflectivity is incident-power-dependent. Namely, variation in reflectivity is more significant for dimmer incident lights, as implied by the presence of 0 i E in the denominator of the 2 nd term in both Eqs. (28) and (29). Note that these seemingly peculiar behaviors are by no means related to the well-known photoelectric effects, such as those manifested by lightguiding molecules used in liquid crystal displays.
In the absence of the second modulating light, the cos M t  factor is reduced to unity in both Eqs. (28) and (29). As a consequence, the modified light reflection is no longer time-varying, but is either enhanced or decreased depending on the signs of 0 x S or 0 y S in the p-and spolarized cases, respectively. More importantly, asymmetric reflection (or, refraction) would result should the path leading from the incidence to the reflection be reversed (see, e.g., Fig. 4), as the result of a sign change of the corresponding coordinate system. In the following, experimental investigations are conducted on the above-predicted power reflection asymmetry between conjugate light paths, as well as on the inverse dependence of reflectivity upon the incident power.

Possible implementations of altered Brewster angle demonstrated by quantum mechanical simulations
Numerical experiments for the p-wave case were conducted as an example to evidence the variation of Brewster angles rendered by the proposed dipole engineering. This task very www.intechopen.com much relied on the first-principle quantum mechanical software: CASTEP (Clark et al., 2005). CASTEP is an ab initio quantum mechanical program employing density functional theory (DFT) to simulate the properties of solids, interfaces, and surfaces for a wide range of materials classes including ceramics, semiconductors, and metals. Its first-principle calculations allow researchers to investigate the nature and origin of the electronic, optical, and structural properties of a system without the need for any experimental input other than the atomic number of mass of the constituent atoms.
The adopted simulation procedure was as follows (Liao et al., 2006). CASTEP first simulated the spectral dependence of the relative dielectric coefficient ( r  , including the real ( rR  ) and imaginary (  Two example situations are given here, where dipole engineering can noticeably alter the Brewster angles of a single-crystal silicon wafer under the incidence of a red and an infrared light, respectively (Liao et al., 2006). The red light is of energy 1.98 eV, or, vacuum wavelength  = 0.63 m. The infrared light is of energy 0.825 eV, or,  = 1.5 m. It is wellknown that without the proposed dipole engineering treatment, the single-crystal silicon is opaque to the visible (red) light, while fairly transparent to the infrared light. In fact, for the latter reason, infrared light is routinely applied in the front-to-back side pattern alignment of wafers in microelectronic fabrications.
Here the Si single-crystal unit cell is modified by replacing 2 of its 8 atoms with vacancies (see Fig. 6, regions in dim color are the chopped-out sites). Note that the x-axis corresponds to 0 t    (and hence 0 i    ), while y or z-axis to 90 t    . Fig. 6 shows that the defectcaused permanent polarization (maximum 0 P ) is most likely along the x-direction. The CASTEP-simulated curves of r  vs. light energy (in eV) in the incident directions of x, y, and z, respectively, are given in Fig. 7. Hence, r  is, as expected, i  -dependent. For both cases in which the incident light is red and infrared, the involvement of the introduced www.intechopen.com optically-responsive defect polarization is very much dependent on the its relative orientation with respect to the refractive p-wave's electric field ( p t E ) (see Figs. 8 and 9). Indeed, as evident from Fig. 2, it is most significant when p t E is in the direction of the permanent polarization (maximum 0 P ).   Using the aforementioned calculation procedure, a considerable shift of the original Brewster angle of 77.2 to the new one of 72.5, due to the vacancy defects replacement, is evident in Fig. 10 for the red light case. Note that the reflection for red light decreased www.intechopen.com somewhat post the dipole engineering treatment. For the second IR case, Fig. 9 shows the involved action of the permanent defect polarization. A noticeable shift of the Brewster angle from 74.9 to 78.7 results (see Fig. 11). Moreover, this new Si material manifests higher reflection (or, opaqueness) (see Fig. 11), in addition to relatively high dissipation (see Fig. 7) for the IR light, after the proposed treatment. www.intechopen.com In the above, the shifted Brewster angles for both cases are associated with considerable dissipation caused by the introduced defects, as implied by Fig. 7. In particular, the defectmodified silicon wafer would absorb the total power of the p-wave incident at the Brewster angle. Nonetheless, this should not represent as sure characteristics for the general situations. For example, a B  -altering surface material (to be coated on the original host in a post-process manner) may have its permanent polarization implemented via replacing some of its atoms with other elements, instead of vacancies, and thus may not show such dissipating behavior.

Background on the PVDF material ((Pallathadka, 2006), (Zhang et al., 2002)) and its preparations
Polyvinylidene difluoride, or PVDF (molecular formula: -(CH 2 CF 2 ) n -), is a highly nonreactive, pure thermoplastic and low-melting-point (170C) fluoropolymer. As a specialty plastic material in the fluoropolymer family, it is used generally in applications requiring the highest purity, strength, and resistance to solvents, acids, and bases. With a glass transition temperature (T g ) of about -35 o C, it is typically 50-60% crystalline at room temperature. However, when stretched into thin film, it is known to manifest a large piezoelectric coefficient of about 6-7 pCN -1 , about 10 times larger than those of most other polymers. To enable the material with piezoelectric properties, it is mechanically stretched and then poled with externally applied electric field so as to align the majority of molecular chains ((Pallathadka, 2006) , (Zhang et al., 2002)). These polarized PVDFs fall in 3 categories in general, i.e., alpha (TGTG'), beta (TTTT), and gamma (TTTGTTTG') phases, differentiated by how the chain conformations, of trans (T) and gauche (G), are grouped. FTIR (Fourier transform IR) measurements are normally employed for such differentiation purposes ((Pallathadka, 2006) , (Zhang et al., 2002)). With variable electric dipole contents (or, polarization densities) these PVDF films become ferroelectric polymers, exhibiting efficient piezoelectric and pyroelectric properties, making them versatile materials for sensor and battery applications.
In our experiments, PVDF films of Polysciences (of PA, USA) are subjected to non-uniform mechanical and electric polings to generate -PVDF films of distributed dipolar regions. By applying infrared light beams on these poled -PVDF films, the evidences of enhanced asymmetric refraction at varying incident angles as well as its inverse dependence on the incident power are sought for.

FTIR measurement setup
The adopted experimental setup takes full advantage of the original commercial FTIR measurement structure (Varian 2000 FT-IR) (Chen et al., 2008). The intended investigations are facilitated by putting an extra polarizer (Perkin-Elmer) in front of the detector and some predetermined number of optical attenuators (Varian) before the sample (see, Fig. 12). Figs. 13(a) and 13(b) show the detector calibration results under different numbers of attenuating sheets for both the p-and s-wave incidence, respectively. As is obvious, the detected intensities degraded linearly with the number of attenuators installed and the spectra remain morphologically similar.

Evidence of asymmetric reflections and enhanced reflectivities for attenuated incident lights
In the reflectivity (   22  00 || ri Rr E E  ) experiments, each electrically-poled -PVDF film (thickness 16-18 m) is subjected to IR light irradiation at varying incident angles and beam www.intechopen.com intensities (via using the above attenuator films) (Chen et al., 2008). Even though a PVDF film encompasses many double layers along its thickness, the reason the above theoretical derivations based on a single double layer (see Fig. 5) should still apply is that there are nonuniform vertical polarizations (or, electric fields) at the surface. In other words, the resultant gradients in the incident-light-responsive planar dipole moment would then render nonzero interfacial jump of || E as addressed by Eq. (26). Since it is the reflectivity that is of interest, at each incident angle and under a specified degree of attenuation, the reflection is normalized with respect to the corresponding reference value in the absence of a PVDF film as shown in Fig. 13 (a) and 13 (b) (Chen et al., 2008). Figs. 14 (a) and 14 (b) illustrate the arising of asymmetric reflections between www.intechopen.com conjugate incident paths (e.g., incidence at 30 versus 330) for both the p-and s-polarized infrared lights under various degrees of intended attenuation (Chen et al., 2008). In the above, the incident angle is defined by rotating clockwise the poled-PVDF sample under top-view of the setup of Fig. 12. Hence, reversing the light reflection path indeed causes a different reflected power to arise, as predicted by the aforementioned theoretical exploration. Note that, however, in traditional FTIR measurements, decrease in the detected intensity has been routinely attributed to increased absorption by PVDF films. Nevertheless, for an obliquely incident light the beam path within PVDF is only slightly larger than that in a normal incident situation. Thus, the resultant infinitesimal increase in PVDF absorption should never be sufficient to account for the detected large difference in reflected power.
Notably, the detected decrease in intensity should instead be attributed to enhanced reflection caused by distributed dipoles on the poled PVDF films. Furthermore, for both p-and s-polarized incident waves, the variations in reflectivity are more significant in the situations where more attenuation is imposed, complying with the above theoretical expectation. Namely, the second terms in the above-derived Eqs. features strongly endorse the above theoretical claim that the reflectivity variation of a dimmer light is more outstanding than that of a brighter one.

PVDF new Brewster angle
The experimental set up is as arranged in Fig. 16, where a light beam of 0.686 mm radius from the He-Ne laser (of the wavelength of 632.8 nm) is converted into p-wave mode after getting through the polarizer (Tsai et al., 2011). Two double convex lenses, with focal lengths being 12.5 cm and 7.5 cm, respectively, are for shrinking down the beam radius to 0.19 mm to reduce the width of light reflection off the PVDF surface. The reflected light is then further focused by a lens (of 2.54 cm focal length) before reaching the diode power detector. An incident angle range is scanned from 50.5°to 59.5°( i  ) with an accuracy of 0.015°, and then its conjugate range from -50.5°to -59.5°( i   ), while the incident light intensity is varied between 100% (8.54 mW) and 10% power by moving an attenuator into or withdrawing from the beam path. The fitted curves for the measured Brewster angles for the two conjugate incident paths, under 100% and 10% laser beam intensities, on both the and poled- films, are shown in Fig. 17 and Fig. 18, respectively (Tsai et al., 2011). It can be seen that Brewster angles measured via the two conjugate incident paths differ considerably. Such difference becomes more outstanding on the poled- PVDF film, and in particular, when the laser beam is attenuated to 10%, as predicted by the theory of the authors (Liao et al., 2006). The typical data are given in Table 1. It is noted that although even the intrinsic  phase PVDF possesses birefringence and this can lead to different Brewster angles as in the above too, the difference degree is at most around 0.129°, and hence may be ignored.  This can be verified by putting into Eq. (23) the known ordinary and extraordinary refractive indices of PVDF and getting the Brewster angles of 54.814° and 54.933°, respectively. Further, the fact that the larger deviation is evidenced in the poled- phase, as compared with that from the  phase, indicates that permanent dipoles are indeed the cause of such alteration in Brewster angles.
By putting the above experimental data (i.e., Table 1) into Eq. (23), the relative dielectric coefficients ( r  ) for both the and poled- PVDF films are extracted and tabulated in Then, the averaged effective permanent polarization   0 P  and orientation 0  can be extracted through trial-and-error (see , Table 3) by putting these coefficients into Eqs. (22) and (24), and using the relations:  Table 3. Extracted effective permanent polarizations and orientations of dipole-engineered PVDF films.
It can be seen that the electro-poling has caused the permanent polarization to increase somewhat, and most of all, its orientation with respect to the interface to add around 20°.

Novel 2D refractive index ellipse
Owing to its intrinsic uniaxial birefringence property ( (Matsukawa et al., 2006), (Yassien et al., 2010)), when a light is incident upon a PVDF film (as formed, without poling), the refracted light is decomposed into an ordinary wave and an extraordinary wave, which correspond to refractive indices of no and ne , respectively. Namely, when the plane-ofincident is formed by the light's propagating direction vector k  and the uniaxis ẑ (see, Fig.   19), with the angle between them being  , then a slice on the 3D refractive index ellipsoid cutting perpendicular to k  will give rise to an elliptic contour which is of the minor axis no and major axis ns in a relationship expressed as:   Experimentally, the following tendency was observed when the test samples were altered from normal to and then to poled- PVDF films. When the incident angles was within the I-

Notable indication on traditional SPR measurements
Facilitated by its very high-Q resonance angle, the surface plasmon resonance (SPR) type of techniques, and their variations, are known to be very sensitive tools for measurements of refractive indices (   r r n  ). Among them, the one using prism coupling and in the socalled "Kretschmann-Raether (KR) configuration" ( (Maier, 2007), (Raether, 1988)), is probably the most widely adopted practice. That is, the KR-configured materials from top down is arranged to be: prism (the "0" matter), metal film (the "1" matter), air gap, target under test (the "2" matter), and buffer layer (the "3" matter). When a p-wave is incident at a so-called "resonance angle" onto the topmost KR-configured plane (i.e., at "0"-"1" interface), a surface resonant plasma wave is excited at the metal-air interface, leading to a minimum in light reflection ( (Maier, 2007), (Raether, 1988)).
The overall reflection coefficient (of an incident p-wave) off this 4-material KR configuration can be derived from using the Fabry-Perot interference principle and is ( (Maier, 2007), (Raether, 1988)): However, it is found in the above experiments that in the presence of permanent dipoles, not only is the Brewster angle dependent on the incident light power as well as the dipole orientation, but also that two conjugate incident light paths result in distinctively different refractions. Therefore, although the form of Eq. (31) remains the same in the presence of permanent dipoles, values of local reflection coefficients involved can vary considerably from those of their classical counterparts. In other words, the traditional confidence in SPR type of measurements may be in jeopardy when the material under test is embedded with permanent dipoles, as will be shown in what follows.
Consider a KR-configured SPR measurement setup as an example. It includes: a lens (SF 11) of relative dielectric coefficient of   7786 . 1 2 , a silver metal film of 52 nm ( d1 ) thickness of a relative dielectric coefficient of i 67 . 0 6 . 17   (Raether, 1988), a PVDF film (as grown, or , or poled-) as the material under test of thickness of about 15 m ( d 2 ), with its original relative dielectric coefficients being no 2 and ne 2 , and a buffer layer of air of a relative dielectric coefficient of about 1. This configuration is then subjected to the irradiation of a light beam , from a 632.8 nm wavelength He-Ne laser, of the incident angles ranging within 50°~70°(  i ) and its conjugate counterpart paths within the angle range -50°~ -70°(   i ).
The numerical calculation result based on the above setup is shown in Fig. 21 and indicates the following (Tsai et al., 2011). The birefringence (i.e., ne besides no ) of asgrown PVDF suffices to give a maximal SPR resonance angle deviation of about 0.5° (away from 58°). This deviation of resonance angles is considerably amplified in the -(2.5°) and poled- (4°) cases, owing to the increase of permanent polarization density and alignment. In such sensitive SPR type of measurements, these large deviations of resonance angles represent large distortions in the light reflectivity, as illustrated in Fig.  21. Notably, it also affirms the theoretical prediction (see, Eq. (24)) that the reflectivity coefficient is inversely proportional to the strength of the incident (or, transmitted) light electric field. All these findings indicate that traditional SPR type of measurements needs to exercise precaution when the material under test is embedded with permanent dipoles. For example, most living cells are with cell walls made of two opposite double layers of dipolar molecules (Alberts et al., 2007).

The quasistatic macroscopic mixing theory for magnetic permeability
Although mixing formulas for the effective-medium type of approximations for the dielectric permittivities in the infinite-wavelength (i.e., quasistatic) limit (Lamb et al., 1980), such as the Maxwell Garnett formula (Garnett, 1904), have been popularly applied in the whole spectral range of electromagnetic fields, their magnetic counterpart has seldom been addressed up to this day. The current effort is thus to derive such an equation to approximately predict the final permeability as the result of mixing together several magnetic materials (Chang & Liao, 2011

A brief review of the derivation leading to the Maxwell Garnett and Bruggeman formulas
Historically, an isotropic host material was hypothesized to embed with a collection of spherical homogeneous inclusions. With the molecular polarization of a single molecule of such inclusions being denoted α, the following relation was established within the linear range (Cheng, 1989): www.intechopen.com where m p  was the induced dipole moment and m E  was the polarizing electric field intensity at the location of the molecule. Since the treatment was aiming for uniform spherical inclusions, the polarizability became a scalar, such that m E  was expressed as (Purcell, 1985): Here E  was the average field within the bulk host, p E  was the electric field at this molecular location caused by all surrounding concentric spherical shells of the bulk, and near E  was due to asymmetry within the inclusion. In those cases of interest where either the structure of the inclusion was regular enough, such as a cubical or spherical particulate, or all incorporated molecules were randomly distributed, near E  became essentially zero. It was further approximated that (Purcell, 1985), to be elaborated later, with P  being the polarization density associated with a uniformly polarized sphere, and ε 0 being the permittivity in free space. Hence, given Eq. (32), with the number density of such included molecules denoted as n, and m Pn p    (Cheng, 1989), the polarization density was further expressed as: However, it was well-known that for isotropic media where ε r stood for the relative permittivity (i.e., the electric field at the center of a uniformly polarized sphere (with P  being its polarization density) was Then, a relation known as the Lorentz-Lorenz formula readily followed ( (Lorenz, 1880), (Lorentz, 1880)): In those special cases where the permittivity of each tiny included particle was ε s and the host material was vacuum (ε r =1), such that n = V -1 (V being the volume of the spherical inclusions), and Eq. (35) would have to satisfy (Garnett, 1904): Combining Eqs. (35) and (36) gave the effective permittivity (ε eff ) of the final mixture (Garnett, 1904): with f = nV being the volume ratio of the embedded tiny particles (0 ≤ f ≤ 1) within the final mixture. If, instead of vacuum, the host material was with a permittivity of ε h , Eq. (37) was then generalized to the famous Maxwell Garnett mixing formula: www.intechopen.com For the view in which the inclusion was no longer treated as a perturbation to the original host material, Bruggeman managed to come up with a more elegant form wherein different ingredients were assumed to be embedded within a host (Bruggeman, 1935). By utilizing Eqs. (35) and (36), he had: where f i and ε i are the volume ratio and permittivity of the i-th ingredient.

The magnetic flux density at the center of a uniformly magnetized sphere
surface current can be expected to appear on the surface of a uniformly magnetized sphere (wherein M  is the finalized net anti-responsive magnetization vector, see Fig. 22). In Fig. 22,  S K is the induced anti-reactive surface current density (in A/m) on the sphere's surface. By integrating all surface current density on strips of the sphere's surface the magnetic flux density ( c B  ) at the center of a uniformly magnetized sphere is obtained to be (Lorrain & Corson, 1970):

Magnetic permeabilities formula mixing
Now, this time consider an isotropic host material embedded with a collection of spherical homogeneous magnetic particles. Given the magnetic flux density at the location of a single molecule of the inclusions being m B  , the following relation holds in general: In the special case where the host material is vacuum (μ r = 1) and the permeability of the spherical particles is μ s , n = V -1 (V being the volume of a spherical particle), and Eq. (47 Where f = nV is the volume ratio of the embedded particles within the mixture (0 ≤ f ≤ 1). In the more general situations where the host is no longer vacuum but of the permeability μ h , then the more general mixing formula of permeabilities becomes (Chang & Liao, 2011): As with Bruggeman's approach for dielectrics (Bruggeman, 1935), the derived magnetic permeability formula can be generalized to the multi-component form (Chang & Liao, 2011): where f i and μ i denote the volume ratios and permeabilities of the involved different inclusions, respectively. Or (Chang & Liao, 2011), Although the actual mixing procedures can vary widely such that substantial deviations may result between the theoretical and measured values, Eq. (52) should still serve as a valuable guide when designing magnetic materials or composites.

A practical method to secure magnetic permeability in optical regimes
As widely known, electronic polarization is involved in the absorption of electromagnetic wave within materials and this mechanism is represented by permittivity even in light-wave frequencies. However, refractive index which describes light absorption and reflection is calculated in terms of permittivity and permeability (accounting for magnetization). But permeability spectra in light frequencies are hardly available for most materials. In contrast with it, permeabilities for various materials are fairly well documented at microwave frequencies (see, for example, (Goldman, 1999) and (Jorgensen, 1995)). It is noted that more often than not an electronic permittivity spectrum (ε r (ω)) is secured by measuring its corresponding refractive index (N(ω)) while bluntly assuming its relative permeability (μ r ) to be unity. Obviously, this approach is one-sided and inappropriate. In particular, as we are entering the nanotech era, many new possibilities should emerge and surprise us with their novel optical permeabilities, for example, the originally nonmagnetic manganese crystal can be made ferromagnetic once its lattice constant is varied (Hummel, 1985). In this section, a method is proposed such that reliable optical permeability values can be obtained numerically.
The magnetic permeability of a specific material emerges fundamentally from wavefunctions of its electrons. In the popular density functional theory (DFT) approach, the www.intechopen.com role of these electron wavefunctions is taken by the one-electron spinorbitals, called the Kohn-Sham orbitals ( os  ) (see, e.g., (Shankar, 1994) and (Atkins & Friedman, 1997)). A spinorbital is a product of an orbital wavefunction and a spin wavefunction: where o  is the orbital part, and s  is the spin part of os  .
In the density functional theory, these spinorbitals are solutions to the equation: where  r   is the local charge probability density function. Then, the exact ground-state electronic energy as well as other electronic properties of this n-electron system are known to be unique functions of ( (Shankar, 1994), (Atkins & Friedman, 1997)). Further, the overall wavefunction satisfying the Pauli exclusion principle is often expressed in terms of the Slater determinant ( (Shankar, 1994), (Atkins & Friedman, 1997) Hence, the primitive transition rate of a typical electronic excitation within a material of interest can now be expressed in terms of the Fermi's golden rule ( (Shankar, 1994), (Atkins & Friedman, 1997) ) are the electron states (energies) before and after the perturbation (H 1 ) takes place, respectively; ω is the radian frequency of the propagating light wave; ħ is Planck constant divided by 2 ; and C is a proportional constant. H 1 in Eq. (56) can be expressed as (Shankar, 1994): where 0 A  is the vector potential of the injected light wave; M P  is the momentum operator of electrons in the material of interest. M P  represents electrons' straight-line motion which constitute electronic polarization. Thus substituting Eq. (57) into Eq. (56) will result in R total proportional to the imaginary part of ε r (i.e. ε rI ). At the same time (Shankar, 1994 Applying a spin-orbit decomposition (see Eqs. (53) and (55) with C o and C s being constant coefficients.
As aforementioned, despite being termed "electronic transition rate" to account for the lightwave absorption within a material, the primitive transition rate R total is in essence a series of delta functions situated at varying frequencies (or energies, see Eq. (56)), and thus is too spikey to be real. In fact, this spikey nature results from our attempt to describe the dynamics of multitudes of electrons by a limited number of Kohn-Sham orbitals. Therefore, R total ought to be "smoothed" prior to being converted to a realistic absorption spectrum.
Here, Gaussian functions are adopted to replace all delta functions. The smoothed R total , R o and R s are now denoted as R' total , R' o and R' s , respectively, and thus from Eq. (62) we have: It is a common practice in the first-principle quantum mechanical calculations (such as by the computer codes CASTEP and DMol 3 ((Clark et al., 2005), (Delley, 1990)) that R' o is normally set equal to ε rI simulated with the "spin polarized" option in these kind of codes turned off (as in the nonmagnetic case).
Substituting Eq. (58) into Eq. (64) gives R' s . And then the product of R' o and R' s , R' total , is proportional to μ rI (the scaling factor is defined as C M ). With μ r being linear and causal, there is an exact one-to-one correspondence between its real and imaginary parts as prescribed by the Kramers-Kronig relation (see, e.g., (Landau & Lifshitz, 1960)). Thus, the real part of μ r (i.e., μ rR ) is readily available once μ rI is numerically obtained. To make all this happen, the calculation of R' s is now in order.
Within the formulation of R' s spectrum (see its unsmoothed form in Eq. (64) ((Clark et al., 2005), (Delley, 1990)) simulations by selecting the "spin polarized" option. In evaluating the bracketed term in Eq. (64)  As mentioned, in those cases where the "spin polarized" option in, e.g., CASTEP is turned off, the resultant ε rI spectra actually gives R' o in Eq. (65). With both R' s and R' o being revealed, the optical μ r spectrum of interest can be secured within a proportional constant C M , and then the Kramers-Kronig relation. Finally, this universal constant C M is uncovered by comparing the erected μ rI spectrum with existing data covering from low to lightwave frequencies (Lide & Frederikse, 1994).
The calculated refractive index spectrum of iron crystal from using the proposed approach is shown in Fig. 23. A comparison with the known refractive index spectrum of iron crystal exposed by a linearly polarized light (Lide & Frederikse, 1994) (see Fig. 24) reveals that both figures are relatively close in features. Accordingly, the universal proportional constant C M obtained from comparing the two N I curves is about 13.7. The completed iron μ r spectrum is provided in Fig. 25 after applying R' o = ε rI , Eq. (66) and then the Kramers-Kronig relation. Therefore, the optical μ r and refractive index spectra of all materials, including those to be developed, can now be fully explored for the first time. Incidentally, to further challenge the proposed procedure, we erroneously leave the "spin polarized" action of CASTEP on when simulating the copper crystal, a knowingly www.intechopen.com nonmagnetic material. It turned out that the numerically converged spin density was a minimal 1.35 x 10 -4 μ B /atom, showing that the proposed procedure is robust against erroneous initial conditions.

Conclusion
This chapter contains a clear, yet rigorous, picture as to how the real physical causes behind all these macroscopic optical phenomena -i.e., the microscopic electric and magnetic dipoleswork to come up with such macroscopic results. Both the electric and magnetic dipoles react to the imposed EM wave in such a way that actually results in the Fresnel equations.
In arriving at the more intuitive "scattering form" of the Fresnel equations, microscopic EMinduced physical electric and magnetic dipoles were rigorously employed as the source of electromagnetic waves by Doyle et al. Motivated by such an approach, the authors started to speculate how the incorporation of permanent dipoles might affect many macroscopic optical phenomena, e.g., the Brewster angle of a specific optical material. Among its predictions, the traditionally fixed Brewster angle of a specific material now not only becomes dependent on the density and orientation of incorporated permanent dipoles, but also on the incident light intensity (more precisely, the incident wave electric field strength). Further, two conjugated incident light paths would give rise to different refracted wave powers. This kind of microscopic approaches are called "dipole engineering".
Theoretical elaboration and then IR experiments on poled polyvinylidene fluoride (PVDF) films were conducted to verify the emergence of asymmetric reflections at varying incident angles, as well as the inverse dependence of reflectivity upon the impinging light intensity. In addition, experiments on dipole-engineered PVDF films show that by way of adding/reducing permanent dipole density and varying orientations, the aforementioned theoretical predictions can be evidenced unambiguously in the visible light range. Further, effective polarization density can be quantified from the above experiments subjected to different dipole engineering processes. As a result, the traditionally elliptic contour of a slanted two dimensional section of the refractive index ellipsoid now manifests symmetric www.intechopen.com open splittings at near the traditional incident angle. It implies that severe challenge to the accuracy of traditional surface plasmon resonance measurements may arise in the presence of permanent dipoles of various morphologies, such as in the forms of nano-particles or membrane double layers.
Traditional approach of obtaining electric permittivity through directly measuring a material's refractive index begins to appear flawed at dawn of the nanotech era. On the other hand, achieving novel magnetics and thus new materials of variable refractive indices starts to become meaningful. However, unlike its electric counterpart, data for optical magnetic permeability is hardly available for most materials. An effective method is proposed to secure practically accurate optical permeabilities through manipulating electronic transition rates generated by first-principle quantum mechanical simulations.