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

Chungpin Liao, Hsien-Ming Chang, Chien-Jung Liao, Jun-Lang Chen and Po-Yu Tsai *National Formosa University (NFU) Advanced Research and Business Laboratory (ARBL) Chakra Energetics, Ltd. Taiwan* 

#### **1. Introduction**

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 ( *<sup>B</sup>* ) (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 ( *<sup>r</sup>* ) 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).

Traditionally, whichever the case, the Brewster angle is a solid property of the material in question with respect to a given light frequency of interest. Namely, there is a one-to-one correspondence between the Brewster angle and the incident light frequency. However, it is one of the purposes of this chapter to show that the Brewster angle of the material in hand can in principle be modified into a new controllable variable, even dynamically, if a postprocess microscopic method called "dipole engineering" is applicable on that 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 (Liao et al., 2006).

Lightwave Refraction and Its Consequences: A Viewpoint of

*p*

*s*

*E E*

*s*

*s*

*E E*

Note that if the medium is characterized by

 

Brewster angle; while in the *r r*

Brewster angle only for the p-wave.

devices, even new applications.

*s*

*E E*

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 217

 

*t r*

 

 

 

 

 

 

*t i t i*

Brewster angle to arise either for the p-wave or the s-wave. That is (Doyle, 1985),

familiar case of light going from vacuum into a piece of glass whose 1 *r r*

 

 

2 cos sin sin

*t i t i*

 

 

sin sin

sin sin

 

 

1 1cos 1 1cos

*r r r r r ti*

 

*i r r r r ti*

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

<sup>2</sup> tan

<sup>2</sup> tan

 

*r r* 

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,

**3. ''Scattering" form Fresnel equations from the dipole source viewpoint** 

Microscopically, all matters including optical materials are made of atoms or molecules, each of which further consists of a positive-charged nucleus (or nuclei) and some orbiting

 

*<sup>p</sup> rr r <sup>B</sup> r r*

 

*<sup>s</sup> rr r <sup>B</sup> r r*

 

 

 1 1cos

*i ti t*

 

*i r r r r ti*

   

 

 

 

 

1

1

(7)

(8)

situation, only the p-wave can. Indeed, in the most

, only the s-wave may experience the

, there is the

 

 

 

 

 

  (4)

(6)

(5)

1 1cos 1 1cos

*r r r r r ti <sup>p</sup> r r r r ti <sup>i</sup>*

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.

#### **2. 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):

$$r^s = \frac{E\_r^s}{E\_i^s} = \frac{n\_i \mu\_{rl} \cos \theta\_i - n\_t \mu\_{ri} \cos \theta\_t}{n\_i \mu\_{rl} \cos \theta\_i + n\_t \mu\_{ri} \cos \theta\_t} \tag{1}$$

$$r^p = \frac{E\_r^p}{E\_i^p} = \frac{\mu\_{ri} n\_t \cos \theta\_i - \mu\_{rl} n\_i \cos \theta\_t}{\mu\_{ri} n\_t \cos \theta\_i + \mu\_{rl} n\_i \cos \theta\_t} \tag{2}$$

where *E* is the electric field, *<sup>r</sup>* is the relative magnetic permeability, 1/2 *<sup>n</sup> r r* is the index of refraction ( *<sup>r</sup>* 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 ( *<sup>B</sup>* ), one of the above reflection coefficients would vanish, then such value of the incident angle is known as the Brewster angle ( *<sup>B</sup>* ). Note that for the most familiar case in which the light wave is incident from vacuum onto a linear nonmagnetic medium ( 1 *<sup>r</sup>* ), 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 *ni* ) 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):

$$\begin{aligned} \frac{E\_l^p}{E\_i^p} &= \left[ \frac{-\sqrt{\mu\_r}}{ (\mu\_r - 1)\sqrt{\varepsilon\_r} + (\varepsilon\_r - 1)\sqrt{\mu\_r}} \cos(\theta\_l - \theta\_i) \right] \\ &\times \left[ \frac{2\cos\theta\_l \sin(\theta\_l - \theta\_i)}{\sin\theta\_l} \right] \end{aligned} \tag{3}$$

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

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

*<sup>s</sup> <sup>s</sup> r i rt i t ri t*

*p r ri t i rt i t <sup>p</sup> ri t i rt i t <sup>i</sup>*

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 (

most familiar case in which the light wave is incident from vacuum onto a linear

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 *ni* ) for simplicity. In addition, to further facilitate our purpose, the Fresnel equations in the equivalent

> 1 1cos

 

*<sup>p</sup> r r r r ti <sup>i</sup>*

*i ti t*

 

 

*En n*

*E n n*

*En n*

*E n n* 

*i i rt i t ri t*

cos cos cos cos

cos cos cos cos

(1)

(2)

 

*<sup>B</sup>* ). Note that for the

(3)

is the

*i* )

 

 

*<sup>B</sup>* ), one of the above reflection coefficients would vanish,

*<sup>r</sup>* is the relative magnetic permeability, 1/2 *<sup>n</sup> r r*

being the relative dielectric coefficient), superscripts "p" and "s"

*<sup>r</sup>* ), only the p-wave possesses a Brewster angle, not the s-wave.

 

materials, devices and applications may emerge.

the p and s components (Hecht, 2002):

where *E* is the electric field,

is equal to a particular value (

nonmagnetic medium ( 1

*p*

*E E*

index of refraction ( *<sup>r</sup>*

**2. Brewster angle and "scattering" form of Fresnel equations** 

*s*

*p*

*r*

*r*

"scattering" form (due to Doyle) are retyped here (Doyle, 1985):

2 cos sin sin

 

*t r*

then such value of the incident angle is known as the Brewster angle (

$$\frac{E\_r^p}{E\_i^p} = \left[ \frac{(\mu\_r - 1)\sqrt{\varepsilon\_r} - (\varepsilon\_r - 1)\sqrt{\mu\_r}\cos(\theta\_l + \theta\_i)}{(\mu\_r - 1)\sqrt{\varepsilon\_r} + (\varepsilon\_r - 1)\sqrt{\mu\_r}\cos(\theta\_l - \theta\_i)} \right] \tag{4}$$
 
$$\left[ \sin(\theta\_l - \theta\_i) \right]$$

$$\times \left\lfloor \frac{\sin \left( \theta\_l - \theta\_i \right)}{\sin \left( \theta\_l + \theta\_i \right)} \right\rfloor$$

$$\sim \left\lceil \begin{array}{c} \multicolumn{3}{c}{\prod} \\ \multicolumn{3}{c}{\prod} \end{array} \right\rfloor$$

$$\begin{aligned} \frac{E\_t^s}{E\_i^s} &= \left| \frac{-\sqrt{\mu\_r}}{(\varepsilon\_r - 1)\sqrt{\mu\_r} + (\mu\_r - 1)\sqrt{\varepsilon\_r}\cos(\theta\_t - \theta\_i)} \right| \\ &\times \left[ \frac{2\cos\theta\_l \sin(\theta\_t - \theta\_i)}{\sin\theta\_l} \right] \end{aligned} \tag{5}$$

$$\begin{split} \frac{E\_r^s}{E\_i^s} &= \left[ \frac{(\varepsilon\_r - 1)\sqrt{\mu\_r}}{(\varepsilon\_r - 1)\sqrt{\mu\_r}} + (\mu\_r - 1)\sqrt{\varepsilon\_r}\cos(\theta\_l + \theta\_i) \right] \\ &\times \left[ \frac{\sin(\theta\_l - \theta\_i)}{\sin(\theta\_l + \theta\_i)} \right] \end{split} \tag{6}$$

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),

$$\tan^2 \theta\_{\mathbb{B}}{}^p = \frac{\varepsilon\_r \left(\varepsilon\_r - \mu\_r\right)}{\varepsilon\_r \mu\_r - 1} \tag{7}$$

$$\tan^2 \theta\_B^{-s} = \frac{\mu\_r \left(\mu\_r - \varepsilon\_r\right)}{\varepsilon\_r \mu\_r - 1} \tag{8}$$

Note that if the medium is characterized by *r r* , only the s-wave may experience the Brewster angle; while in the *r r* situation, only the p-wave can. Indeed, in the most familiar case of light going from vacuum into a piece of glass whose 1 *r r* , there is the Brewster angle only for the p-wave.

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.

#### **3. ''Scattering" form Fresnel equations from the dipole source viewpoint**

Microscopically, all matters including optical materials are made of atoms or molecules, each of which further consists of a positive-charged nucleus (or nuclei) and some orbiting

Lightwave Refraction and Its Consequences: A Viewpoint of

*<sup>P</sup> <sup>E</sup>*

cos

 

*rp t i*

fact as a "consequence" of the microscopic induced magnetic dipoles:

*M*

*M*

*im*

*E*

*E*

electric dipoles sources:

through the magnetization *M*

magnetization becomes (Purcell, 1985):

the induced magnetic dipoles too:

due to induced dipole sources *P* and *M* :

In considering the magnetic contribution, 1 *<sup>r</sup>*

*E BB k* 

 

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 219

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

*p t*

<sup>0</sup>

Similarly, there are field contributions from the induced magnetic dipoles, which acted

<sup>0</sup>

Then, using this together with Eq. (13), the Fresnel equation related to *E E t i* / (Eq. (3)), can be converted into an expression for the magnetic component of the incident electric field, in

*M E*

0 0

0 0

*<sup>p</sup> <sup>t</sup> rm*

*p t*

 

sin

. Using the magnetic field strength *HB M*

*i ti*

(13)

*i ip im EEE* (16)

*i ti*

 

*i ti*

 

(12)

<sup>0</sup> and the

*<sup>r</sup>* are adopted in Eq. (10).

(14)

(15)

 

2 cos sin

relation associated with general monochromatic plane waves, the

0 1 *<sup>r</sup> <sup>p</sup> r t r*

and thus 1/2 *n*

sin 2 cos sin

sin 2 cos sin

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:

*p p p*

Namely, by adding Eq. (11) and (14), the incident electric field can be formally expressed as

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

 

  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.

Fig. 1. Orientation of induced dipoles and their fields (Courtesy of W. T. Doyle (Doyle, 1985) ).

#### **3.1 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: *D EP* <sup>0</sup> for linear isotropic materials. Comparing it with *D E* <sup>0</sup> *<sup>r</sup>* , the induced polarization hence emerges as (all in space-time configuration) (Purcell, 1985):

$$
\bar{P} = \varepsilon\_0 \left(\varepsilon\_r - 1\right) \bar{E}\_t^p \tag{9}
$$

When calculating the field contribution from the electric dipoles alone, it is assumed that 1 *<sup>r</sup>* and thus 1/2 *<sup>n</sup> <sup>r</sup>* . Putting this into an equivalent form of the Snell's law (Doyle, 1985):

$$m^2 - 1 = \frac{\sin\left(\theta\_i + \theta\_t\right)\sin\left(\theta\_i - \theta\_t\right)}{\sin^2\theta\_t} \tag{10}$$

and subsequently, together with Eq. (9), into the Fresnel equation relating *E E t i* / , we obtain an expression in which the incident electric field is expressed as a "consequence" of the microscopic sources -- the induced dipoles:

$$E\_{ip}^p = \left[ -\frac{P}{\varepsilon\_0} \cos \left( \theta\_l - \theta\_i \right) \right] \times \left[ \frac{\sin \theta\_l}{2 \cos \theta\_i \sin \left( \theta\_l - \theta\_i \right)} \right] \tag{11}$$

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.

Fig. 1. Orientation of induced dipoles and their fields (Courtesy of W. T. Doyle (Doyle, 1985) ).

It is well-known from electromagnetism (Purcell, 1985) that if denoting the induced electric

<sup>0</sup> <sup>1</sup> *<sup>p</sup> P E <sup>r</sup> <sup>t</sup>* 

When calculating the field contribution from the electric dipoles alone, it is assumed that

 <sup>2</sup> 2

and subsequently, together with Eq. (9), into the Fresnel equation relating *E E t i* / , we obtain an expression in which the incident electric field is expressed as a "consequence" of the

sin

<sup>0</sup>

*it it t*

 

sin sin

 

*p t*

, the induced polarization hence emerges as (all in space-time

. Putting this into an equivalent form of the Snell's law (Doyle,

sin

*i ti*

 

2 cos sin

(9)

(10)

for linear isotropic materials.

(11)

, the electric displacement is: *D EP* <sup>0</sup>

1

cos

 

*ip t i*

*<sup>P</sup> <sup>E</sup>*

*n*

**3.1 The p-wave (E parallel to the plane-of-incidence) case** 

polarization by *P*

1 

1985):

 

microscopic sources -- the induced dipoles:

Comparing it with *D E* <sup>0</sup> *<sup>r</sup>*

configuration) (Purcell, 1985):

*<sup>r</sup>* and thus 1/2 *<sup>n</sup> <sup>r</sup>*

$$E\_{rp}^p = \left[\frac{P}{\varepsilon\_0}\cos(\theta\_l + \theta\_i)\right] \times \left[\frac{\sin\theta\_l}{2\cos\theta\_i\sin(\theta\_l + \theta\_i)}\right] \tag{12}$$

Similarly, there are field contributions from the induced magnetic dipoles, which acted through the magnetization *M* . Using the magnetic field strength *HB M* <sup>0</sup> and the *E BB k* relation associated with general monochromatic plane waves, the magnetization becomes (Purcell, 1985):

$$M = \left(\mu\_r - 1\right) \sqrt{\frac{\varepsilon\_0 \varepsilon\_r}{\mu\_0 \mu\_r}} E\_t^p \tag{13}$$

In considering the magnetic contribution, 1 *<sup>r</sup>* and thus 1/2 *n <sup>r</sup>* are adopted in Eq. (10). Then, using this together with Eq. (13), the Fresnel equation related to *E E t i* / (Eq. (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:

$$E\_{im}^p = \left[ -\frac{M\sqrt{\mu\_0}}{\sqrt{\varepsilon\_0}} \right] \times \left[ \frac{\sin \theta\_l}{2 \cos \theta\_i \sin \left(\theta\_l - \theta\_i\right)} \right] \tag{14}$$

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:

$$E\_{rm}^p = \left[ -\frac{M\sqrt{\mu\_0}}{\sqrt{\varepsilon\_0}} \right] \times \left[ \frac{\sin \theta\_t}{2 \cos \theta\_i \sin \left( \theta\_t + \theta\_i \right)} \right] \tag{15}$$

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:

$$E\_i^p = E\_{ip}^p + E\_{im}^p \tag{16}$$

Namely, by adding Eq. (11) and (14), the incident electric field can be formally expressed as due to induced dipole sources *P* and *M* :

Lightwave Refraction and Its Consequences: A Viewpoint of

effects resulting from both the *induced* and *permanent* dipoles.

*t i* 

dipoles responding to the incident lightwave of radian frequency

 

a "DC" polarization will never enter the above equation, and 0

contribution to *<sup>p</sup> Ei* (i.e., Eq. (17)) is now *P P induced i t* cos

relevant optical frequency of interest and the lightwave's incident angle

For instance, in Eq. (17), the cos

an external polarization vector *P*<sup>0</sup>

 

 

each specific

0 0 <sup>0</sup> 1 cos cos *<sup>p</sup> r t it E P <sup>i</sup>*

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 221

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

the fact that the induced polarization *P* (along *<sup>p</sup> Et* ) has only a fractional contribution to *<sup>p</sup> Ei* determined by the vector projection as shown in Fig. 2, for the p-wave situation. Now, if

is introduced within a host material, then, e.g., for the p-wave case, all electric dipoles'

 (see 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 *P*0 really stands for that amount of polarization at the

*<sup>i</sup>* if the added dipoles, and hence the resultant polarization, are linear.

Fig. 2. P-wave configuration at the interface and the orientation of embedded permanent

electric dipoles (Courtesy of W. T. Doyle (Doyle, 1985) ).

factor multiplying on *P* (but not on *M* ) is due to

(as the collective result of many imposed electric

at the incident angle

 0 0 cos *<sup>i</sup>* , or

*<sup>i</sup>* . In other words,

*<sup>p</sup> P Et* would be constant for

*i* )

$$E\_i^p = \left| -\frac{P}{\varepsilon\_0} \cos(\theta\_l - \theta\_i) - \frac{M\sqrt{\mu\_0}}{\sqrt{\varepsilon\_0}} \right| \times \left( \frac{\sin \theta\_l}{2 \cos \theta\_i \sin \left(\theta\_l - \theta\_i\right)} \right) \tag{17}$$

As a verification, if putting forms of these sources (i.e., Eq. (9) and (13) in which the induced *<sup>P</sup>* and *<sup>M</sup>* are expressed in term of *<sup>p</sup> Et* ) 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:

$$E\_r^p = E\_{rp}^p + E\_{rm}^p \tag{18}$$

Adding Eq. (12) and (15), the reflected electric field appears as due to the induced dipole sources *P* and *M* :

$$E\_r^p = \left[\frac{P}{\varepsilon\_0}\cos\left(\theta\_l + \theta\_i\right) - \frac{M\sqrt{\mu\_0}}{\sqrt{\varepsilon\_0}}\right] \times \left(\frac{\sin\theta\_l}{2\cos\theta\_l\sin\left(\theta\_l + \theta\_i\right)}\right) \tag{19}$$

Similarly, putting forms of the dipoles sources (i.e., Eq. (9) and (13) in which *P* and *M* are expressed in terms of *<sup>p</sup> Et* ) back into Eq. (19), and using the newly obtained *<sup>p</sup> Ei* vs. *<sup>p</sup> Et* relation (i.e., Eq. (17)), it is found that the obtained reflection coefficient of the p-wave is exactly that of Eq. (4), in "scattering" form.

#### **3.2 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:

$$E\_l^s = \left( -\frac{P}{\varepsilon\_0} - \frac{M\sqrt{\mu\_0}}{\sqrt{\varepsilon\_0}} \cos(\theta\_l - \theta\_i) \right) \times \left( \frac{\sin \theta\_l}{2 \cos \theta\_i \sin \left(\theta\_l - \theta\_i\right)} \right) \tag{20}$$

$$E\_r^s = \left( -\frac{P}{\varepsilon\_0} + \frac{M\sqrt{\mu\_0}}{\sqrt{\varepsilon\_0}} \cos\left(\theta\_l + \theta\_i\right) \right) \times \left(\frac{\sin\theta\_l}{2\cos\theta\_i \sin\left(\theta\_l + \theta\_i\right)}\right) \tag{21}$$

and that the transmission and reflection coefficients are indeed found to be those of Eq. (5) and (6), respectively, in "scattering" form.

#### **4. The proposed permanent dipoles engineering**

#### **4.1 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

As a verification, if putting forms of these sources (i.e., Eq. (9) and (13) in which the induced *<sup>P</sup>* and *<sup>M</sup>* are expressed in term of *<sup>p</sup> Et* ) back into Eq. (17), it is found that the obtained

Adding Eq. (12) and (15), the reflected electric field appears as due to the induced dipole

 

Similarly, putting forms of the dipoles sources (i.e., Eq. (9) and (13) in which *P* and *M* are expressed in terms of *<sup>p</sup> Et* ) back into Eq. (19), and using the newly obtained *<sup>p</sup> Ei* vs. *<sup>p</sup> Et* relation (i.e., Eq. (17)), it is found that the obtained reflection coefficient of the p-wave is

Likewise, following Fig. 1 again, we can also express the incident and reflected electric fields

*<sup>p</sup> <sup>t</sup> <sup>r</sup> t i*

 0

*p t*

0 0

0 0

*P M*

 

**3.2 The s-wave (E perpendicular to the plane-of-incidence) case**

of the s-wave as due to those induced electric and magnetic dipoles:

*i t i*

0 0

0 0

*P M*

*P M*

**4. The proposed permanent dipoles engineering** 

0

0

cos

cos

*s t*

*<sup>s</sup> <sup>t</sup> <sup>r</sup> t i*

 

 

and that the transmission and reflection coefficients are indeed found to be those of Eq. (5)

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

 

*P M*

 

cos

transmission coefficient of the p-wave is exactly that of Eq. (3).

cos

*i t i*

Similarly, conceiving the reflected electric field as:

*E*

exactly that of Eq. (4), in "scattering" form.

*E*

*E*

and (6), respectively, in "scattering" form.

**4.1 Observations and inspirations** 

*E*

sources *P* and *M* :

 0

sin

*i ti*

*<sup>p</sup> p p EE E r rp rm* (18)

sin

*i ti*

 

2 cos sin

2 cos sin

(21)

(20)

2 cos sin

sin

sin

*i ti*

*i ti*

 

 

(19)

(17)

 

2 cos sin

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*t i* factor multiplying on *P* (but not on *M* ) is due to the fact that the induced polarization *P* (along *<sup>p</sup> Et* ) has only a fractional contribution to *<sup>p</sup> Ei* determined by the vector projection as shown in Fig. 2, for the p-wave situation. Now, if an external polarization vector *P*<sup>0</sup> (as the collective result of many imposed electric dipoles responding to the incident lightwave of radian frequency at the incident angle *i* ) is introduced within a host material, then, e.g., for the p-wave case, all electric dipoles' contribution to *<sup>p</sup> Ei* (i.e., Eq. (17)) is now *P P induced i t* cos 0 0 cos *<sup>i</sup>* , or 0 0 <sup>0</sup> 1 cos cos *<sup>p</sup> r t it E P <sup>i</sup>* (see 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 *P*0 really stands for that amount of polarization at the relevant optical frequency of interest and the lightwave's incident angle *<sup>i</sup>* . In other words, a "DC" polarization will never enter the above equation, and 0 *<sup>p</sup> P Et* would be constant for each specific *<sup>i</sup>* if the added dipoles, and hence the resultant polarization, are linear.

Fig. 2. P-wave configuration at the interface and the orientation of embedded permanent electric dipoles (Courtesy of W. T. Doyle (Doyle, 1985) ).

Lightwave Refraction and Its Consequences: A Viewpoint of

Unconventional Brewster angle can be found by Eq. (23) (p-wave case).

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 223

Fig. 3. P-wave configuration and the orientation of embedded permanent electric dipoles

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

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 *QQQ a b* 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.


(Conjugated Incident Light Path, courtesy of W. T. Doyle (Doyle, 1985) ).

**4.4 The surface embedded with a thin distributed double layer** 

powers (Liao et al., 2006), (Haus & Melcher, 1989).

Thus, if the dipole-engineered total contribution is recast in the traditional form, viz., <sup>0</sup> 1 cos *<sup>p</sup> r t it E* , then it is clear that the modified relative dielectric coefficient is equivalently (Liao et al., 2006):

$$\tilde{\varepsilon}\_r = \frac{P\_0}{\varepsilon\_0 E\_t^p} \frac{\cos \left(\theta\_l + \theta\_0\right)}{\cos \left(\theta\_l - \theta\_l\right)} + \varepsilon\_r \tag{22}$$

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 (*<sup>B</sup>* ) would then emerge:

$$\tan^2 \theta\_{\mathcal{B}}{}^p = \frac{\tilde{\varepsilon}\_r \left( \tilde{\varepsilon}\_r - \mu\_r \right)}{\tilde{\varepsilon}\_r \mu\_r - 1} \tag{23}$$

#### **4.2 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 *P*0 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 *P*<sup>0</sup> ) in the frequency range of interest.

#### **4.3 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):

$$\tilde{\varepsilon}\_r = \frac{P\_0}{\varepsilon\_0 E\_t^p} \frac{\cos \left(\theta\_l - \theta\_0\right)}{\cos \left(\theta\_l - \theta\_i\right)} + \varepsilon\_r \tag{24}$$

Thus, if the dipole-engineered total contribution is recast in the traditional form,

cos cos *i r r <sup>p</sup> i t <sup>t</sup>*

0

**4.2 Justification of the effectiveness and meaningfulness of implementing** 

dipole moment (in terms of *P*<sup>0</sup> ) in the frequency range of interest.

**4.3 Different refracted wave powers on two conjugated incident light paths** 

0

*P E*

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)

> cos cos *<sup>i</sup> r r <sup>p</sup> t i <sup>t</sup>*

 0 0

(24)

 

 

*<sup>B</sup>* ) would then emerge:

*P E*

*r t it E* , then it is clear that the modified relative dielectric coefficient

 0 0

0 is the angle between the imposed extra polarization vector and the interface plane

1

(22)

(23)

 

 

(see Fig. 2). Thus, by putting Eq. (22) into the p-wave Brewster angle formula (Eq. (7)), a new

<sup>2</sup> tan

 

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 *P*0 in the above really would have stood for the extra induced dipole effect

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*

*<sup>p</sup> rr r <sup>B</sup> r r*

 

 

viz., <sup>0</sup> 1 cos *<sup>p</sup>*

is equivalently (Liao et al., 2006):

**optically-responsive dipoles** 

becomes (Liao et al., 2006):

resulting from this second substance.

 

 

where 

Brewster angle (

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).

#### **4.4 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:

$$
\left\langle E\_{\parallel} \right\rangle = 0 \quad \text{and} \quad \left\langle H\_{\parallel} \right\rangle = 0 \tag{25}
$$

where *QQQ a b* 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.

Lightwave Refraction and Its Consequences: A Viewpoint of

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 225

Fig. 4. (a) Refraction with p-wave incidence and (b) refraction with s-wave incidence.

Fig. 5. Surface integration of Faraday's law enclosing a section of the double layer and

regions above (a) and below (b) it.

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):

$$\left|E\_{\parallel}\right|^{a} - E\_{\parallel}\left|^{b}\right| \equiv \left\langle E\_{\parallel}\right| = -\frac{1}{\varepsilon\_{0}} \frac{\partial \,\pi\_{s}}{\partial \,\mathbf{x}}\tag{26}$$

where 0 is the dielectric permittivity of free space. An interfacial double layer is composed of a top and bottom layers of equal but opposite surface charges ( *<sup>s</sup>* ) respectively and separated by a tiny distance ( *d* ). It is mathematically described by *<sup>s</sup>* and 0 *d* such that *s s d* stands for the electric dipole moment per unit area on the interface. Eq. (26) is obtained through integrating Faraday's law *E Ht* ( 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 ( *<sup>s</sup>* ) on the interface, which in general is a space-time variable, i.e., *Srt* , (Chen et al., 2008). In other words,

$$\begin{aligned} \left\{ E\_{\parallel} \right\} & \equiv E\_{\parallel} \, \prescript{a}{}{\,} - E\_{\parallel} \, \prescript{b}{}{\,} = \left( E\_{\bar{\imath} \parallel} + E\_{r \parallel} \right) - E\_{t \parallel} \, \Big| \\\ -\frac{1}{\varepsilon\_0} \frac{\partial \, \pi\_s}{\partial \, \chi} (t) & \equiv \mathcal{S} \left( \bar{r} , t \right) \, \prescript{}{}{\,} - \mathcal{S}\_M \left( t \right) \cos \left( \bar{k}\_s \, \cdot \, \bar{r} - a\_s t \right) \end{aligned} \tag{27}$$

where the subscripts "i", "r", and "t" represent the incident, reflected, and transmitted components, respectively. In other words, the condition || *E* 0 arises where there is non-uniform distribution of the dipole moment along the projection of the incident light's electric field on the interface. That is, the nonzero *E*|| jump is effectively proportional to the displacement of a longitudinal ( *<sup>s</sup>* varying) or transverse ( *d* varying) mechanical wave *S t* , which is excited by the incident wave and propagating along the double layer with wave vector *<sup>s</sup> k* and frequency *<sup>s</sup>* . Further, such a mechanical wave can additionally be modulated transversely in its dipole length (i.e., *d* ) by a second wave of frequency *<sup>M</sup>* such that <sup>0</sup> *<sup>M</sup>* cos *StS t x M* for the p-wave case, and <sup>0</sup> *StS t M y* cos *<sup>M</sup>* for the swave case. In the above, the modulating wave amplitudes are <sup>0</sup> 0 0 <sup>1</sup> *<sup>s</sup> Sx <sup>x</sup>* 

and <sup>0</sup> 0 0 <sup>1</sup> *<sup>s</sup> Sy <sup>y</sup>* , respectively, with the subscript "0" standing for the root-meansquare amplitude.

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 &

*a b* <sup>1</sup> *<sup>s</sup> EE E*

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

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

, cos

*<sup>s</sup> <sup>M</sup> s s*


where the subscripts "i", "r", and "t" represent the incident, reflected, and transmitted components, respectively. In other words, the condition || *E* 0 arises where there is non-uniform distribution of the dipole moment along the projection of the incident light's electric field on the interface. That is, the nonzero *E*|| jump is effectively proportional to

wave *S t* , which is excited by the incident wave and propagating along the double layer

be modulated transversely in its dipole length (i.e., *d* ) by a second wave of frequency

for the p-wave case, and <sup>0</sup> *StS t M y* cos

, respectively, with the subscript "0" standing for the root-mean-

*E EE EE E*

wave case. In the above, the modulating wave amplitudes are <sup>0</sup>

*a b*

*t Srt S t k r t*

*ir t*

0

is the dielectric permittivity of free space. An interfacial double layer is composed

*d* stands for the electric dipole moment per unit area on the interface. Eq. (26)

*x* 

(

*<sup>s</sup>* ) on the interface, which in general is a space-time variable,

(27)

*<sup>s</sup>* varying) or transverse ( *d* varying) mechanical

*<sup>s</sup>* . Further, such a mechanical wave can additionally

*<sup>M</sup>* for the s-

<sup>1</sup> *<sup>s</sup> Sx <sup>x</sup>*

0 0

(26)

) in the contour integration no

*<sup>s</sup>* ) respectively and

*<sup>s</sup>* and 0 *d* such

being the magnetic


of a top and bottom layers of equal but opposite surface charges (

separated by a tiny distance ( *d* ). It is mathematically described by

is obtained through integrating Faraday's law *E Ht*

side reveals that the two normal sections (i.e., along *E*

*x* 

0 1

and frequency

Melcher, 1989):

where 0 

that *s s* 

dipole moment per unit area (

i.e., *Srt* , (Chen et al., 2008). In other words,

the displacement of a longitudinal (

*<sup>M</sup>* such that <sup>0</sup> *<sup>M</sup>* cos *StS t x M*

0 0

with wave vector *<sup>s</sup> k*

<sup>1</sup> *<sup>s</sup> Sy <sup>y</sup>*

square amplitude.

and <sup>0</sup>

Fig. 4. (a) Refraction with p-wave incidence and (b) refraction with s-wave incidence.

Fig. 5. Surface integration of Faraday's law enclosing a section of the double layer and regions above (a) and below (b) it.

Lightwave Refraction and Its Consequences: A Viewpoint of

than the atomic number of mass of the constituent atoms.

imaginary ( *rI*

orientation (

tan 2 

incident angle (

degrees.

wavelength

to 0 

index ( *nt rR*

 *rI rR* 

*<sup>i</sup>* ). Namely, *<sup>r</sup>*

for low *rI*

do Snell's law, we actually had to vary

cos cos cos cos *<sup>p</sup> rR i t rR i t <sup>r</sup>*

 

of wafers in microelectronic fabrications.

and z, respectively, are given in Fig. 7. Hence, *<sup>r</sup>*

*<sup>t</sup>* (and hence 0

CASTEP-simulated curves of *<sup>r</sup>*

polarization at the incident angle

 

coefficient (i.e., Eq. (2), with *<sup>r</sup>*

the spectral dependence of the relative dielectric coefficient ( *<sup>r</sup>*

 

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 227

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

The adopted simulation procedure was as follows (Liao et al., 2006). CASTEP first simulated

this host lattice through artificially replacing some of its atoms with other elements, or with vacancy defects, hence resulting in the implementation of permanent dipoles of known

corresponding spectral dependence of the new relative dielectric coefficient ( *<sup>r</sup>*

could thus be secured. (However, since CASTEP only simulates intrinsically, viz., it does not

using Snell's law.) Then, using Eq. (22), the value of such-introduced permanent

resultant new Brewster angle was thus obtained through inspecting the modified reflection

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

known 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

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

caused permanent polarization (maximum *P*<sup>0</sup> ) is most likely along the x-direction. The

cases in which the incident light is red and infrared, the involvement of the introduced

vs. light energy (in eV) in the incident directions of x, y,

is, as expected,

*<sup>i</sup>* , in terms of 0 0 *<sup>p</sup> P Et*

) curve against

replaced by *rR*

 

= 0.63 m. The infrared light is of energy 0.825 eV, or,

*<sup>i</sup>* ), while y or z-axis to 90

was then

) parts) of a chosen host material. Then, dipole engineering was exercised on

<sup>0</sup> ) on the host. Due to this introduced anisotropy, we had to simulate the

has to be used) and Snell's law, the corresponding refractive angle (

case; otherwise, 2 2 cos 2 *nt rR rI*

, including the real ( *rR*

*<sup>i</sup>* -dependent. Using the new medium refractive

*<sup>t</sup>* first instead and went backward to secure

 at 

> 

*<sup>t</sup>* . Fig. 6 shows that the defect-

, and 1, *ni* and 1

) and

) for each

where

*t* )

> *i*

*<sup>i</sup>* , was revealed. The

*ri rt* , becomes

= 1.5 m. It is well-

*<sup>i</sup>* -dependent. For both

*<sup>i</sup>* varying from 0 to 90

 

By combining with the other jump condition, i.e., || *H* 0 , the modified Fresnel reflection coefficient for both the p- and s-wave configurations emerge readily after some algebraic manipulations (Chen et al., 2008):

$$r\_p \equiv \left(\frac{E\_{r0}}{E\_{i0}}\right)\_p = \frac{\left(\frac{n\_t}{\mu\_t}\cos\theta\_i - \frac{n\_i}{\mu\_i}\cos\theta\_t\right)}{\left(\frac{n\_t}{\mu\_t}\cos\theta\_i + \frac{n\_i}{\mu\_i}\cos\theta\_t\right)} - \frac{\frac{n\_t}{\mu\_t}S\_{y0}\cos\alpha\_M t}{\left(\frac{n\_t}{\mu\_t}\cos\theta\_i + \frac{n\_i}{\mu\_i}\cos\theta\_t\right)E\_{i0}}\tag{28}$$

$$r\_s = \underbrace{\left(\frac{E\_{r0}}{E\_{i0}}\right)\_s}\_{s} = \frac{\left(\frac{n\_i}{\mu\_i}\cos\theta\_i - \frac{n\_t}{\mu\_t}\cos\theta\_t\right)}{\left(\frac{n\_i}{\mu\_i}\cos\theta\_i + \frac{n\_t}{\mu\_t}\cos\theta\_t\right)} + \underbrace{\frac{\frac{n\_t}{\mu\_t}S\_{x0}\cos\phi\_M t \cdot \cos\theta\_t}{\mu\_t}}\_{\underbrace{\left(\frac{n\_i}{\mu\_i}\cos\theta\_i + \frac{n\_t}{\mu\_t}\cos\theta\_t\right)}\_{\text{due to distributed double layer}}}\tag{29}$$

The new power reflection coefficient is then <sup>2</sup> *R r* and the new power transmission coefficient is:*T R* 1 , 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 0 *<sup>s</sup> x* or 0 *<sup>s</sup> y* in Fig. 4), the reflected and transmitted lights are expected to be further modulated at the frequency *<sup>M</sup>* . 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 *Ei*<sup>0</sup> in the denominator of the 2nd 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 *Mt* 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 *Sx*<sup>0</sup> or *Sy*<sup>0</sup> 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.

#### **5. 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 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 ( *<sup>r</sup>* , including the real ( *rR* ) and imaginary ( *rI* ) parts) of a chosen host material. Then, dipole engineering was exercised on this host lattice through artificially replacing some of its atoms with other elements, or with vacancy defects, hence resulting in the implementation of permanent dipoles of known orientation (<sup>0</sup> ) on the host. Due to this introduced anisotropy, we had to simulate the corresponding spectral dependence of the new relative dielectric coefficient ( *<sup>r</sup>* ) for each incident angle (*<sup>i</sup>* ). Namely, *<sup>r</sup>* was then *<sup>i</sup>* -dependent. Using the new medium refractive index ( *nt rR* for low *rI* case; otherwise, 2 2 cos 2 *nt rR rI* where tan 2 *rI rR* has to be used) and Snell's law, the corresponding refractive angle (*t* ) could thus be secured. (However, since CASTEP only simulates intrinsically, viz., it does not do Snell's law, we actually had to vary *<sup>t</sup>* first instead and went backward to secure *i* using Snell's law.) Then, using Eq. (22), the value of such-introduced permanent polarization at the incident angle *<sup>i</sup>* , in terms of 0 0 *<sup>p</sup> P Et* at *<sup>i</sup>* , was revealed. The resultant new Brewster angle was thus obtained through inspecting the modified reflection coefficient (i.e., Eq. (2), with *<sup>r</sup>* replaced by *rR* , and 1, *ni* and 1 *ri rt* , becomes cos cos cos cos *<sup>p</sup> rR i t rR i t <sup>r</sup>* ) curve against *<sup>i</sup>* varying from 0 to 90

degrees.

226 Advanced Photonic Sciences

By combining with the other jump condition, i.e., || *H* 0 , the modified Fresnel reflection coefficient for both the p- and s-wave configurations emerge readily after some algebraic

*t i t*

 

*i p ti ti*

*i t t*

*i s it it*

*traditional*

*traditional*

 

*r t i t <sup>s</sup>*

*r t t i*

 

cos cos cos

*n n <sup>n</sup> S t*

*ti ti*

*<sup>E</sup> nn nn <sup>E</sup>*

cos cos cos cos

cos cos cos cos

*i t xM t*

*y* in Fig. 4), the reflected and transmitted lights are expected to be

*i t i ti*

cos cos cos cos

*it it*

*<sup>E</sup> nn nn <sup>E</sup>*

 

The new power reflection coefficient is then <sup>2</sup> *R r* and the new power transmission coefficient is:*T R* 1 , 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

reflectivity is incident-power-dependent. Namely, variation in reflectivity is more significant for dimmer incident lights, as implied by the presence of *Ei*<sup>0</sup> in the denominator of the 2nd 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 light-

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 *Sx*<sup>0</sup> or *Sy*<sup>0</sup> 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

**5. Possible implementations of altered Brewster angle demonstrated by** 

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

*n n <sup>n</sup> S t*

*i t y M*

*i t i ti*

0

0

*due to distributed double layer*

*<sup>M</sup>* . Additionally, this modification to the light

*due to distributed double layer*

0

0

*Mt* factor is reduced to unity in both

  (28)

(29)

manipulations (Chen et al., 2008):

*p*

*r*

*r*

0 *<sup>s</sup>* 

 *x* or 0 *<sup>s</sup>* 

further modulated at the frequency

reflectivity upon the incident power.

**quantum mechanical simulations** 

guiding molecules used in liquid crystal displays.

In the absence of the second modulating light, the cos

0 0

0 0

*E*

*E*

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 *<sup>t</sup>* (and hence 0 *<sup>i</sup>* ), while y or z-axis to 90 *<sup>t</sup>* . Fig. 6 shows that the defectcaused permanent polarization (maximum *P*<sup>0</sup> ) is most likely along the x-direction. The CASTEP-simulated curves of *<sup>r</sup>* vs. light energy (in eV) in the incident directions of x, y, and z, respectively, are given in Fig. 7. Hence, *<sup>r</sup>* is, as expected, *<sup>i</sup>* -dependent. For both cases in which the incident light is red and infrared, the involvement of the introduced

Lightwave Refraction and Its Consequences: A Viewpoint of

Fig. 8. The permanent defect polarization (at 0

Fig. 9. The permanent defect polarization (at 0

infrared light of 0.83 eV energy (most significant when 90

red light of 1.98 eV (most significant when 90

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 229

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

*<sup>i</sup>* .)

*<sup>i</sup>* .)

0 ) with respect to the incident angle of a

0 ) with respect to the incident angle of an

optically-responsive defect polarization is very much dependent on the its relative orientation with respect to the refractive p-wave's electric field ( *<sup>p</sup> Et* ) (see Figs. 8 and 9). Indeed, as evident from Fig. 2, it is most significant when *<sup>p</sup> Et* is in the direction of the permanent polarization (maximum *P*<sup>0</sup> ).

Fig. 6. Unit cell of the modified Si crystal with two vacancies (Space group: FD-3M (227); Lattice parameters: a: 5.43, b: 5.43, c: 5.43; α: 90º, β: 90º, γ: 90º).

Fig. 7. The modified relative dielectric coefficient spectra ( *r rR rI i* ), with the refractive light's propagation directions shown for x, y, and z axes, for 0 0 .

optically-responsive defect polarization is very much dependent on the its relative orientation with respect to the refractive p-wave's electric field ( *<sup>p</sup> Et* ) (see Figs. 8 and 9). Indeed, as evident from Fig. 2, it is most significant when *<sup>p</sup> Et* is in the direction of the

Fig. 6. Unit cell of the modified Si crystal with two vacancies (Space group: FD-3M (227);

Lattice parameters: a: 5.43, b: 5.43, c: 5.43; α: 90º, β: 90º, γ: 90º).

Fig. 7. The modified relative dielectric coefficient spectra ( *r rR rI*

light's propagation directions shown for x, y, and z axes, for 0

 

> 0 .

 

*i* ), with the refractive

permanent polarization (maximum *P*<sup>0</sup> ).

Fig. 8. The permanent defect polarization (at 0 0 ) with respect to the incident angle of a red light of 1.98 eV (most significant when 90 *<sup>i</sup>* .)

Fig. 9. The permanent defect polarization (at 0 0 ) with respect to the incident angle of an infrared light of 0.83 eV energy (most significant when 90 *<sup>i</sup>* .)

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

Lightwave Refraction and Its Consequences: A Viewpoint of

**6. Poled PVDF films and asymmetric reflection experiments** 

situations. For example, a

dissipating behavior.

**its preparations** 

battery applications.

incident power are sought for.

**6.2 FTIR measurement setup** 

remain morphologically similar.

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 231

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

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

**6.1 Background on the PVDF material ((Pallathadka, 2006), (Zhang et al., 2002)) and** 

Polyvinylidene difluoride, or PVDF (molecular formula: -(CH2CF2)n-), is a highly nonreactive, pure thermoplastic and low-melting-point (170C) 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 (Tg) of about -35oC, 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

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

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

*<sup>B</sup>* -altering surface material (to be coated on the original host in a

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.

Fig. 10. Shift of the original Brewster angle (77.21) to the new one (72.52) for an incident light of energy 1.98 eV by the proposed vacancy defects introduction.

Fig. 11. Shift of the original Brewster angle (74.9) to the new one (78.7) for an incident light of energy 0.83 eV by the proposed vacancy defects introduction.

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. 10. Shift of the original Brewster angle (77.21) to the new one (72.52) for an incident

Fig. 11. Shift of the original Brewster angle (74.9) to the new one (78.7) for an incident light

light of energy 1.98 eV by the proposed vacancy defects introduction.

of energy 0.83 eV by the proposed vacancy defects introduction.

Fig. 7) for the IR light, after the proposed treatment.

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 *<sup>B</sup>* -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.
