**7.1 PVDF new Brewster angle**

234 Advanced Photonic Sciences

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

Fig. 15. Observed distinct asymmetric reflection among conjugate light paths under two

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. (28) and (29) are more enhanced under the situation of smaller incident light electric field (or power). Lastly, even though the appearance of FTIR-detected spiky saturation peaks in Figs. 14 (a) and 14 (b) actually imply loss in the signal-to-noise ratios when under heavy attenuation of the incident power, several unsaturated features, e.g., in Figs. 15(a) and 15(b) still remain (wherein 2 sheets of attenuators are employed) (Chen et al., 2008). Such evidenced spectral

attenuator films and at varying incident angles.

reflection caused by distributed dipoles on the poled PVDF films.

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°( *<sup>i</sup>* ) with an accuracy of 0.015°, and then its conjugate range from -50.5°to -59.5°( *<sup>i</sup>* ), 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.

Fig. 16. Configuration of PVDF Brewster angle measurement.

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.


Table 1. PVDF Brewster angles measurement.

Lightwave Refraction and Its Consequences: A Viewpoint of

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 237

Fig. 18. Measured Brewster angles on poled・β-PVDF for two conjugate incident paths, under: (a) 100% laser intensity, (b) 10% laser intensity (only one-tenth of data points are

shown)

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.

Fig. 17. Measured Brewster angles on・β-PVDF for two conjugate incident paths, under: (a) 100% laser intensity, (b) 10% laser intensity (only one-tenth of data points are shown)

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

Fig. 17. Measured Brewster angles on・β-PVDF for two conjugate incident paths, under: (a) 100% laser intensity, (b) 10% laser intensity (only one-tenth of data points are shown)

around 0.129°, and hence may be ignored.

Fig. 18. Measured Brewster angles on poled・β-PVDF for two conjugate incident paths, under: (a) 100% laser intensity, (b) 10% laser intensity (only one-tenth of data points are shown)

Lightwave Refraction and Its Consequences: A Viewpoint of

and major axis *ns* in a relationship expressed as:

index surfaces of the situation on -PVDF (i.e.,

Fig. 19. Construction of the refractive index surface.

the incident angle range of 0°~ 90° (i.e.,

laser power (Tsai et al., 2011).

19), with the angle between them being

cutting perpendicular to *k*

(i.e., 

incident is formed by the light's propagating direction vector *k*

2 1

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 239

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-of-

*n*

However, the whole picture will change considerably in the presence of ordered permanent dipoles. Namely, unlike the traditional elliptic contour in red color in the polar diagram Fig. 20, unconventional contours in blue and green represent 2D (two dimension) refractive

power, respectively; and those in purple and orange colors are on poled- PVDF (i.e.,

 <sup>0</sup> 61.62 °) under 100 % and 10 % laser power, respectively (Tsai et al., 2011). Note that, in Fig. 20, as the ordinate dimension is along the direction of interface (in green) and the abscissa along the norm in real setup, both the I and III quadrants describe the refraction in

2 2  *<sup>i</sup>* ). Hence, the dipole-engineered ones would demonstrate open splittings near the traditional incident angles. Among them, the deviation should be more outstanding for the case with the test film being poled- than -PVDF, and especially when at lower incident

*n cos ns <sup>o</sup> <sup>e</sup>*

will give rise to an elliptic contour which is of the minor axis *no*

*sin*

2 2 , then a slice on the 3D refractive index ellipsoid

(30)

<sup>0</sup> 41.64 °) under 100 % and 10 % laser

*<sup>i</sup>* ), and quadrants II and IV depicts that of 0°~ -90°

and the uniaxis *z*ˆ (see, Fig.

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 ( *<sup>r</sup>* ) for both the - and poled- PVDF films are extracted and tabulated in Table 2 (Tsai et al., 2011).


Table 2. Relative dielectric coefficients through fitting experimental data.

Then, the averaged effective permanent polarization *P*<sup>0</sup> 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: *p p E E t i t <sup>p</sup>* and i ti 2 cos cos cos *<sup>p</sup> tn n n air air <sup>t</sup>* . In the above, 0 *p <sup>i</sup> E S C* , <sup>1</sup> <sup>i</sup> sin sin *<sup>t</sup> n n air t* , with *S* , <sup>0</sup> , and *C* being the irradiating light intensity per unit area, the vacuum permittivity (i.e., <sup>12</sup> 8.85 10 F/m), and the light speed in vacuum, respectively, and <sup>2</sup> *r o n* (Tsai et al., 2011).


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°.

#### **7.2 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

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

By putting the above experimental data (i.e., Table 1) into Eq. (23), the relative dielectric

intensity ( )

extracted through trial-and-error (see, Table 3) by putting these coefficients into Eqs. (22)

intensity per unit area, the vacuum permittivity (i.e., <sup>12</sup> 8.85 10 F/m), and the light speed in


Poled- PVDF 61.62° 9 2 1.7843 10 / *C m*

Table 3. Extracted effective permanent polarizations and orientations of dipole-engineered

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

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

*<sup>r</sup>* ) for both the - and poled- PVDF films are extracted and tabulated in

 *r i* 

 2.008 100 % 1.994 2.062 10 % 2.1

 1.983 100 % 1.948 2.16 10 % 2.239

 ( ) *r i* 

*E E t i t <sup>p</sup>* and i ti 2 cos cos cos *<sup>p</sup>*

*tn n n air air*

> *P*<sup>0</sup>

and orientation

<sup>0</sup> , and *C* being the irradiating light

*<sup>t</sup>* . In the

 

0 can be

of such alteration in Brewster angles.

Parameters Beam

Table 2. Relative dielectric coefficients through fitting experimental data.

, with *S* ,

*r o n* (Tsai et al., 2011).

0

Then, the averaged effective permanent polarization *P*<sup>0</sup>

 , <sup>1</sup> <sup>i</sup> sin sin

*<sup>t</sup> n n air t*


Poled- PVDF

and (24), and using the relations: *p p*

**Parameters**

**7.2 Novel 2D refractive index ellipse** 

vacuum, respectively, and <sup>2</sup>

Table 2 (Tsai et al., 2011).

coefficients (

above, 0 *p <sup>i</sup> E S C* 

PVDF films.

around 20°.

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 *z*ˆ (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:

$$\frac{1}{n\_s^2(\theta)} = \frac{\cos^2 \theta}{n\_o^2} + \frac{\sin^2 \theta}{n\_e^2} \tag{30}$$

However, the whole picture will change considerably in the presence of ordered permanent dipoles. Namely, unlike the traditional elliptic contour in red color in the polar diagram Fig. 20, unconventional contours in blue and green represent 2D (two dimension) refractive index surfaces of the situation on -PVDF (i.e., <sup>0</sup> 41.64 °) under 100 % and 10 % laser power, respectively; and those in purple and orange colors are on poled- PVDF (i.e., <sup>0</sup> 61.62 °) under 100 % and 10 % laser power, respectively (Tsai et al., 2011). Note that, in Fig. 20, as the ordinate dimension is along the direction of interface (in green) and the abscissa along the norm in real setup, both the I and III quadrants describe the refraction in the incident angle range of 0°~ 90° (i.e., *<sup>i</sup>* ), and quadrants II and IV depicts that of 0°~ -90° (i.e., *<sup>i</sup>* ). Hence, the dipole-engineered ones would demonstrate open splittings near the traditional incident angles. Among them, the deviation should be more outstanding for the case with the test film being poled- than -PVDF, and especially when at lower incident laser power (Tsai et al., 2011).

Fig. 19. Construction of the refractive index surface.

Lightwave Refraction and Its Consequences: A Viewpoint of

And, the overall reflectivity in power is: *R r <sup>p</sup>*

respectively; and *k* <sup>0</sup> is incident wave vector; and

permanent dipoles, as will be shown in what follows.

of relative dielectric coefficient of 1.7786

relative dielectric coefficients being *no*

dipolar molecules (Alberts et al., 2007).

magnetic materials (Chang & Liao, 2011).

50°~70°(

equations; and *k icos <sup>i</sup> di <sup>i</sup>* <sup>0</sup>

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 241

coefficients at "0"-"1", "1"-"2", and "2"-"3" interfaces according to the traditional Fresnel

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

Consider a KR-configured SPR measurement setup as an example. It includes: a lens (SF 11)

relative dielectric coefficient of 17.6 0.67*i* (Raether, 1988), a PVDF film (as grown, or , or poled-) as the material under test of thickness of about 15 m ( *d*<sup>2</sup> ), with its original

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

*<sup>i</sup>* ) and its conjugate counterpart paths within the angle range -50°~ -70°(

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

**8. 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

<sup>2</sup> and *ne*

coefficient / layer thickness of the metal film and material under test, respectively.

0123 2 , where *r <sup>p</sup>*

> <sup>1</sup> *d*<sup>1</sup> and

<sup>01</sup> , *r <sup>p</sup>*

<sup>2</sup> , a silver metal film of 52 nm ( *d*<sup>1</sup> ) thickness of a

<sup>2</sup> , and a buffer layer of air of a relative

are phase angles associated with matters "1" and "2",

<sup>12</sup> , *r <sup>p</sup>*

<sup>2</sup> *d*2 are relative dielectric

<sup>23</sup> are reflection

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

Fig. 20. The polar diagram for refractive index distribution.

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 Iand III- quadrants (i.e., *<sup>i</sup>* ), the evidenced Brewster angles shrunk and the refractive index became smaller (i.e., *<sup>r</sup> <sup>r</sup>* ~ ). On the other hand, when the laser was incident in the II- and IV- quadrants (i.e., *<sup>i</sup>* ), both the Brewster angles and the refractive index ( *<sup>r</sup> <sup>r</sup>* ~ ) switched to larger values. This tendency apparently goes with the above theoretical prediction.

#### **7.3 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 <sup>r</sup> 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)):

$$r\_{0123}^p = \frac{r\_{01}^p + \frac{r\_{12}^p + r\_{23}^p e^{i2\phi\_2}}{1 + r\_{12}^p r\_{23}^p e^{i2\phi\_2}} e^{i2\phi\_1}}{1 + r\_{01}^p \frac{r\_{12}^p + r\_{23}^p e^{i2\phi\_2}}{1 + r\_{12}^p r\_{23}^p e^{i2\phi\_2}} e^{i2\phi\_1}}\tag{31}$$

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-

*<sup>i</sup>* ), both the Brewster angles and the refractive index (

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

called "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

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),

*r*

*p*

01

1

 

1

 

*r*

1

*p*

01

to larger values. This tendency apparently goes with the above theoretical prediction.

*<sup>i</sup>* ), the evidenced Brewster angles shrunk and the refractive index

 *<sup>r</sup> <sup>r</sup>* ~ ) switched

*<sup>r</sup>* ~ ). On the other hand, when the laser was incident in the II- and

*<sup>r</sup> n* ). Among them, the one using prism coupling and in the so-

*e*

2

2 2

2

2

2 2

2

*i*

2

1

(31)

*e*

*i*

2

1

*r r e r r e*

12 23

*p p i p p i*

*r r e r r e*

12 23

12 23

12 23

*p p i p p i*

Fig. 20. The polar diagram for refractive index distribution.

**7.3 Notable indication on traditional SPR measurements** 

minimum in light reflection ((Maier, 2007), (Raether, 1988)).

*r*

*p*

0123

 *r* 

 *<sup>r</sup>* 

and III- quadrants (i.e.,

became smaller (i.e.,

IV- quadrants (i.e.,

refractive indices (

(Raether, 1988)):

And, the overall reflectivity in power is: *R r <sup>p</sup>* 0123 2 , where *r <sup>p</sup>* <sup>01</sup> , *r <sup>p</sup>* <sup>12</sup> , *r <sup>p</sup>* <sup>23</sup> are reflection coefficients at "0"-"1", "1"-"2", and "2"-"3" interfaces according to the traditional Fresnel equations; and *k icos <sup>i</sup> di <sup>i</sup>* <sup>0</sup> are phase angles associated with matters "1" and "2", respectively; and *k* <sup>0</sup> is incident wave vector; and <sup>1</sup> *d*<sup>1</sup> and <sup>2</sup> *d*2 are relative dielectric coefficient / layer thickness of the metal film and material under test, respectively.

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 1.7786 <sup>2</sup> , a silver metal film of 52 nm ( *d*<sup>1</sup> ) thickness of a relative dielectric coefficient of 17.6 0.67*i* (Raether, 1988), a PVDF film (as grown, or , or poled-) as the material under test of thickness of about 15 m ( *d*<sup>2</sup> ), with its original relative dielectric coefficients being *no* <sup>2</sup> and *ne* <sup>2</sup> , 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°( *<sup>i</sup>* ) and its conjugate counterpart paths within the angle range -50°~ -70°( *<sup>i</sup>* ).

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).
