**5. Feasibilities for experimental analysis of characteristics of the Poynting vector components**

In this section, potentiality of experimental analysis for the time-averaged Poyntig vector is considered. In part, we will show that combined application of conventional interferometry and Stokes-polarimetry should allow of unambiguous determining characteristics of the time-averaged Poynting vector components at each point of electromagnetic field.

One of theoretical aspects in rapidly developing area of the modern optical technology, elaboration of new kinds of optical tweezers [25 - 30, 85] is connected with the fact that the vortex beams and polarized waves (both homogeneous and inhomogeneous) possess an angular momentum [85 - 87]. Existence of controlled angular momentum provides a possibility for controlled rotation of micro-objects locked by corresponding optical traps. Angular momentum of a field can be specified at each spatial point. One may also consider angular momentum averaged over some spatial area. As it is known [86, 88], angular momentum may be divided into a spin momentum associated with circular polarization, and an orbital one produced by specific beam structure. However, density of the angular momentum, *zj* , (at least, of its orbital part) depends upon the location of the axis 0 **r** (i.e., "purchase") with regard to which the parameter *zj* is calculated. As a result, some ambiguity appears. At the same time, another physical value closely associated with the angular momentum, *viz.* the space distribution of characteristics of the time-averaged Poynting vector (more accurately, its transverse component), represents a univocal function of coordinates of each field point.

Distribution of parameters of the time-averaged Poynting vector for Laguerre-Gaussian beams has been considered in [89, 90]. However, behavior of the time-averaged Poynting vector was analyzed only for homogeneously polarized fields and "symmetrical" beams [86, 89].

At the same time, analyzed fields may be more complicated, in part, when their polarization is inhomogeneous. Distribution of the transverse component of the Poynting vector for such fields may be characterized by a set of certain points, i.e., by the net of Poynting vector singularities [87]. The importance of these points comes from the fact that the characteristics of this singular net, such as the Poynting field skeleton, determine qualitative behavior of the Poynting vector at each of the field points [37]. For instance, vortex Poynting singularities [91] are the points, around which circulation of the transverse component of the Poynting vector takes place. In other words, one deals here with the points which are the intersections of the observation plane and the axis of the angular momentum. Thus, spatial distributions of characteristics of the time-averaged Poynting vector components would contain important information on the field, which is concerned with the energy flows [24, 30, 35, 36].

Nevertheless, one can state that no the technique for experimental analysis of the timeaveraged Poynting vector components and their singularities has been developed up to now. Here we would like to demonstrate that the components of the time-averaged Poynting vector can be experimentally analyzed by using conventional optical methods.

It has been shown [37, 87] that the instantaneous components of the Poynting vector may be written as

$$\begin{cases} P\_x \approx \frac{c}{4\pi k} \{E\_x T\_2 - E\_y T\_1\}; \\ P\_y \approx \frac{c}{4\pi k} \{E\_y T\_2 + E\_x T\_1\}; \\ \quad P\_z \approx \frac{c}{4\pi} \{E\_x^2 + E\_y^2\} \end{cases} \tag{14}$$

where

294 Modern Metrology Concerns

One of theoretical aspects in rapidly developing area of the modern optical technology, elaboration of new kinds of optical tweezers [25 - 30, 85] is connected with the fact that the vortex beams and polarized waves (both homogeneous and inhomogeneous) possess an angular momentum [85 - 87]. Existence of controlled angular momentum provides a possibility for controlled rotation of micro-objects locked by corresponding optical traps. Angular momentum of a field can be specified at each spatial point. One may also consider angular momentum averaged over some spatial area. As it is known [86, 88], angular momentum may be divided into a spin momentum associated with circular polarization, and an orbital one produced by specific beam structure. However, density of the angular momentum, *zj* , (at least, of its orbital part) depends upon the location of the axis 0 **r** (i.e., "purchase") with regard to which the parameter *zj* is calculated. As a result, some ambiguity appears. At the same time, another physical value closely associated with the angular momentum, *viz.* the space distribution of characteristics of the time-averaged Poynting vector (more accurately, its transverse component), represents a univocal function

Distribution of parameters of the time-averaged Poynting vector for Laguerre-Gaussian beams has been considered in [89, 90]. However, behavior of the time-averaged Poynting vector was

At the same time, analyzed fields may be more complicated, in part, when their polarization is inhomogeneous. Distribution of the transverse component of the Poynting vector for such fields may be characterized by a set of certain points, i.e., by the net of Poynting vector singularities [87]. The importance of these points comes from the fact that the characteristics of this singular net, such as the Poynting field skeleton, determine qualitative behavior of the Poynting vector at each of the field points [37]. For instance, vortex Poynting singularities [91] are the points, around which circulation of the transverse component of the Poynting vector takes place. In other words, one deals here with the points which are the intersections of the observation plane and the axis of the angular momentum. Thus, spatial distributions of characteristics of the time-averaged Poynting vector components would contain important information on the field, which is concerned with the energy flows [24,

Nevertheless, one can state that no the technique for experimental analysis of the timeaveraged Poynting vector components and their singularities has been developed up to now. Here we would like to demonstrate that the components of the time-averaged Poynting vector can be experimentally analyzed by using conventional optical methods.

It has been shown [37, 87] that the instantaneous components of the Poynting vector may be

*x xy*

 

 

*<sup>c</sup> P ET ET k <sup>c</sup> P ET ET k <sup>c</sup> P EE*

*y yx*

*z xy*

2 1

{ }; <sup>4</sup>

{ }; <sup>4</sup>

{ }, <sup>4</sup>

2 1

(14)

2 2

analyzed only for homogeneously polarized fields and "symmetrical" beams [86, 89].

of coordinates of each field point.

30, 35, 36].

written as

$$\begin{cases} T\_1 = E\_x \Phi\_x^y \; E\_y \Phi\_y^x + \frac{A\_y^y}{A\_x} E\_{x,\frac{\pi}{2}} - \frac{A\_y^x}{A\_y} E\_{y,\frac{\pi}{2}}; \\\\ T\_2 = E\_x \Phi\_x^x + E\_y \Phi\_y^y + \frac{A\_x^x}{A\_x} E\_{x,\frac{\pi}{2}} + \frac{A\_y^y}{A\_y} E\_{y,\frac{\pi}{2}}; \\\\ E\_i = A\_i \cos(\alpha t + \Phi\_i - kz); \\\\ E\_{i,\frac{\pi}{2}} = A\_i \sin(\alpha t + \Phi\_i - kz), \end{cases} \tag{16}$$

*i xy* , , , *Ai i* – denote, respectively, amplitudes and phases of the corresponding field components, , *l l Ai i* are their partial derivatives, and *k c* / represents the wave number. Note, here the axis *z* coincides with the preferential direction of the wave propagation.

2

One can show after some algebraic transformations that the averaged components of the Poynting vector are as follows:

$$\overline{P}\_x \approx \frac{c}{8\pi k} \{ [A\_x^2 \Phi\_x^x + A\_y^2 \Phi\_y^x] \quad A\_x A\_y (\Phi\_x^y \quad \Phi\_y^y) \cos \Delta - (A\_x A\_y^y + A\_y A\_x^y) \sin \Delta \};$$

$$\overline{P}\_y \approx \frac{c}{8\pi k} \{ [A\_x^2 \Phi\_x^y + A\_y^2 \Phi\_y^y] + A\_x A\_y (\Phi\_x^x \quad \Phi\_y^x) \cos \Delta + (A\_x A\_y^x + A\_y A\_x^x) \sin \Delta \}; \tag{17}$$

$$\overline{P}\_z \approx \frac{c}{8\pi} \{ A\_x^2 + A\_y^2 \}.$$

The second and the third terms in the brackets appearing of the first two rows of Eq. (17) may be rewritten as follows:

$$A\_x A\_y (\Phi\_x^i - \Phi\_y^i) \cos \Delta + (A\_x A\_y^i + A\_y A\_x^i) \sin \Delta = \frac{\partial}{\partial t} (A\_x A\_y \sin \Delta)' \tag{18}$$

where *i xy* , *.* Then, the system of Eqs. (17) is transformed to

$$\begin{cases} \overline{P}\_{\mathbf{x}} \approx -\frac{c}{8\pi k} \{ [A\_{\mathbf{x}}^2 \Phi\_{\mathbf{x}}^{\mathbf{x}} + A\_{\mathbf{y}}^2 \Phi\_{\mathbf{y}}^{\mathbf{x}}] - \frac{\partial}{\partial \mathbf{y}} \{ A\_{\mathbf{x}} A\_{\mathbf{y}} \sin \Delta \} \} \\\\ \overline{P}\_{\mathbf{y}} \approx -\frac{c}{8\pi k} \{ [A\_{\mathbf{x}}^2 \Phi\_{\mathbf{x}}^{\mathbf{y}} + A\_{\mathbf{y}}^2 \Phi\_{\mathbf{y}}^{\mathbf{y}}] + \frac{\partial}{\partial \mathbf{x}} \{ A\_{\mathbf{x}} A\_{\mathbf{y}} \sin \Delta \} \} \\\\ \overline{P}\_{\mathbf{z}} \approx \frac{c}{8\pi} \{ A\_{\mathbf{x}}^2 + A\_{\mathbf{y}}^2 \} \end{cases} \tag{19}$$

As it follows from Eqs. 19, one of the possible ways of the Poynting vector components measurement is complete analysis of orthogonal components:

Measurement of the components intensities (for the defining of components amplitudes and their corresponding derivatives).

Phasometry (interferometry) of the components (for the defining of components phases and their corresponding derivatives).

However, the phase difference between orthogonal components is included in relations (19). Note that:


Consequently, the phase difference cannot be measured within the enough accuracy by the components analysis. Such value may be easily measured by the Stokes polarimetry [83]. Stocks parameters for monochromatic wave have the form [17]:

$$\begin{aligned} s\_0 &= A\_x^2 + A\_y^2\\ s\_1 &= A\_x^2 - A\_y^2\\ s\_2 &= 2A\_x A\_y \cos \Delta\\ s\_3 &= 2A\_x A\_y \sin \Delta. \end{aligned} \tag{20}$$

Value may be determined as:

$$
\Delta = \tan^{-1} \frac{\mathbf{s}\_3}{\mathbf{s}\_2} \,\prime \tag{21}
$$

and the Eqs. 19 may be used for the analysis of Poynting vector components. If one takes into account Eqs. (20), the relations (19) are transformed to the form:

$$\overline{P}\_{\mathbf{x}} \approx -\frac{c}{16\pi k} \{ [(s\_0 + s\_1)\Phi\_{\mathbf{x}}^{\mathbf{x}} + (s\_0 - s\_1)\Phi\_{\mathbf{y}}^{\mathbf{x}}] - \frac{\overline{\alpha}s\_3}{\overline{\alpha}\mathbf{y}} \}$$

$$\overline{P}\_{\mathbf{y}} \approx -\frac{c}{16\pi k} \{ [(s\_0 + s\_1)\Phi\_{\mathbf{x}}^{\mathbf{y}} + (s\_0 - s\_1)\Phi\_{\mathbf{y}}^{\mathbf{y}}] + \frac{\overline{\alpha}s\_3}{\overline{\alpha}\mathbf{x}} \}\tag{22}$$

$$\overline{P}\_{\mathbf{z}} \approx \frac{c}{8\pi}s\_0$$

Thus, components of the Poynting vector may be defined by the Stokes parameters and derivatives of components phases. Note, that phasometry of only one component (let us *y* component for certainty) is necessary, because a phase of the orthogonal component is determined as *<sup>x</sup> <sup>y</sup>* . Obviously, in practical sense, the phasometry of the smooth component (without any singularity in the analyzed area) is preferable.

As it is known (see, for example, [92] which is the closest for the topic of our study), three kinds of measurement are necessary for defining of phase of scalar field at each field point:

Intensity of component field <sup>2</sup> *i i I A* .

Intensity of referent wave <sup>2</sup> *r r I A* .

Intensity of total field *sI* .

However, the phase difference between orthogonal components is included in relations

i. as it is known [17] measurement of the "absolute" phase in optics is "problematic". Only phase difference between object wave and referent one is fixed. In other words,

ii. the experimental arrangement for simultaneous measuring of component phase with

Consequently, the phase difference cannot be measured within the enough accuracy by the components analysis. Such value may be easily measured by the Stokes polarimetry [83].

> *sAA sAA s AA s AA*

 

0

1 2 3

2 2

*x y x y x y x y*

2 2

1 3 2

tan *s s*

and the Eqs. 19 may be used for the analysis of Poynting vector components. If one takes

*x x y*

*<sup>c</sup> <sup>s</sup> P ss ss*

*<sup>c</sup> <sup>s</sup> P ss ss*

 

*y x y*

*z*

01 01

{[( ) ( ) ] } <sup>16</sup>

*k y*

*x x*

*y y*

01 01

8

*<sup>c</sup> P s*

Thus, components of the Poynting vector may be defined by the Stokes parameters and derivatives of components phases. Note, that phasometry of only one component (let us *y* component for certainty) is necessary, because a phase of the orthogonal component is determined as *<sup>x</sup> <sup>y</sup>* . Obviously, in practical sense, the phasometry of the smooth

As it is known (see, for example, [92] which is the closest for the topic of our study), three kinds of measurement are necessary for defining of phase of scalar field at each field

{[( ) ( ) ] } <sup>16</sup>

0

*k x*

(20)

, (21)

3

3

(22)

2 cos 2 sin .

the phase is measured within the constant component.

the same constant components is practically impossible.

Stocks parameters for monochromatic wave have the form [17]:

into account Eqs. (20), the relations (19) are transformed to the form:

component (without any singularity in the analyzed area) is preferable.

*i i I A* .

*r r I A* .

Intensity of component field <sup>2</sup>

Intensity of referent wave <sup>2</sup>

Intensity of total field *sI* .

(19). Note that:

Value may be determined as:

point:

The component phase (within the constant component) may be derived from the relation:

$$\Phi\_i = \arccos \frac{I\_s - \left(I\_i + I\_r\right)}{2\sqrt{I\_i I\_s}} \cdot \tag{23}$$

Naturally, the phase derivatives *<sup>k</sup> <sup>i</sup>* , included in Eqs. (19), (22) are independent on the constant component.

Thus, joint applying of the conventional interferometry and Stokes-polarimetry allow us unambiguously determine the characteristics of the Poynting vector components at each point of an optical field.
