**4.1 U and P singularities in partially spatially coherent combined beams**

Let us consider vector singularities in partially coherent optical beams by giving the following simple instructive example. Mutually incoherent and orthogonally polarized Laguerre-Gaussian mode LG01 and a plane wave are coaxially mixed. Such components can be obtained from one laser (using a computer-generated hologram for forming LG01 mode) in interferometric arrangement with optical delay, *l* , considerably exceeding a coherence length of the used laser, *l* , or using two different lasers. Intensity of a plane wave is set deliberately to be less than the peak intensity of the mode, see Fig. 12.

Thus, we consider two-component mixture co-directional orthogonally polarized beams, one of which contains a common phase singularity, *viz.* optical vortex. Interference between such beams with forming common interference fringes is excluded by two reasons: (i) specified mutual incoherence of the components: (ii) polarization orthogonality of them. Note, even only the second condition *per si* determines that, independently on the degree of mutual coherence of two beams over whole interval from zero (for optical path difference exceeding the coherence length) to unity (for zero optical path difference) *visually* observed and *photometrically* measured pattern remains unchangeable. However, more delicate polarization analysis of the combined beam enables to differentiate two limiting cases, *viz.* completely coherent and completely incoherent mixing of orthogonally polarized components.

undetermined [18]. Vector skeletons of coherent inhomogeneously polarized fields were elaborated in details in papers [70 - 72]. By crossing L lines, handedness is step-like changed into opposite one; by crossing C point, azimuth of polarization is changed into orthogonal one. These types of singularities are blurred in quantum-mechanical description being "camouflaged" by so-called quantum vacuum [69], though the distance of influence of such

All mentioned singularities disappear in the case of partially coherent wave fields (though they remain in each completely coherent component, mode in a set of which partially coherent radiation is decomposed. Instead (beside) of them, new singularities appear inherent just in partially coherent fields. Let us emphasize, that singularities of partially coherent fields have formed the novel topic in the field of singular optics just at the

For that, two situations arise again: (i) scalar case when polarization can be ignored while the state of polarization is the same at all point of a field, and (ii) vector case when the state of polarization of *partially coherent* field changes from point to point that requires explicit taking into account of vector nature of light. The first (scalar) case became the subject of intense investigations in last years [67, 73 - 76]. As a result of these investigations, new phase singularities of spatial and temporal correlation functions of quasi-monochromatic light fields have been revealed, as well as singularities of spectral components of polychromatic ("white-light") radiation [68, 77 - 82]. For that, vector singularities of partially coherent light fields have been revealed just recently [61 - 65]. Such singularities are elaborated in this

Let us consider vector singularities in partially coherent optical beams by giving the following simple instructive example. Mutually incoherent and orthogonally polarized Laguerre-Gaussian mode LG01 and a plane wave are coaxially mixed. Such components can be obtained from one laser (using a computer-generated hologram for forming LG01 mode) in interferometric arrangement with optical delay, *l* , considerably exceeding a coherence length of the used laser, *l* , or using two different lasers. Intensity of a plane wave is set

Thus, we consider two-component mixture co-directional orthogonally polarized beams, one of which contains a common phase singularity, *viz.* optical vortex. Interference between such beams with forming common interference fringes is excluded by two reasons: (i) specified mutual incoherence of the components: (ii) polarization orthogonality of them. Note, even only the second condition *per si* determines that, independently on the degree of mutual coherence of two beams over whole interval from zero (for optical path difference exceeding the coherence length) to unity (for zero optical path difference) *visually* observed and *photometrically* measured pattern remains unchangeable. However, more delicate polarization analysis of the combined beam enables to differentiate two limiting cases, *viz.* completely coherent and completely incoherent mixing of orthogonally polarized

**4.1 U and P singularities in partially spatially coherent combined beams** 

deliberately to be less than the peak intensity of the mode, see Fig. 12.

(about 6 Å for He-

quantum core is rather small. Its linear size is of order of magnitude <sup>3</sup>

Ne laser).

section.

components.

beginning of the Third Millenium [3].

Fig. 12. Mixing of vortex-supporting LG01 mode and plane wave with intensity less than peak intensity of a mode.

Let us firstly consider the limiting case when two components are completely mutually coherent. For the sake of distinctness (and for substantiveness of further consideration), we consider coherent mixing of orthogonally *circularly* polarized LG01 mode and a plane wave. Beside of all, choice of circular polarization basis possesses the advantage that it is invariant in respect to rotation of the coordinates, in contrast to linear or elliptical basis, which are relative ones [83].

Fig. 13. The lines of equal intensities of orthogonally polarized beams at Stokes space: equator of the Poincare sphere for circular polarization basis, coherent mixing (a); 45°-meridian including the poles for linear polarization basis, coherent mixing (b); diameter of the Poincare sphere connecting the poles for circular polarization basis, incoherent mixing (c).

In general, combined beam, everywhere with the unit polarization degree ( <sup>222</sup> <sup>123</sup> *sss* 1 , where 123 *sss* , , are the normalized second, third and fourth Stokes parameters, respectively [17, 83, 84], is elliptically polarized. But at the center of vortex of LG01 mode the field is circularly polarized with the state of polarization of a plane wave. A common phase singularity (vortex) of orthogonally polarized component of the combined beam lies at the bottom of this circular polarization. At the same time, the resulting field is polarized linearly at two contours where amplitudes of two components become equal to each other, see Fig. 13 a.

For that, owing to helicoidal structure of a wave front of LG01 mode, azimuth of linear polarization changes with changing phase difference of a mode and a plane wave. Such topological structure can be considered as elementary experimental model of the assemblage of C point and surrounding it L contour of conventional singular optics of vector fields. Really, similarly to the case of random vector fields, crossing L line where handedness is undetermined is accompanied by step-like changing handedness into opposite one, corresponding to predominant in intensity component with unchangeable azimuth of polarization. For comparison, Fig. 13 b illustrates the line of equal intensities of coherently mixed components in linear polarization basis.

It is of interest that the elementary structure shown in Fig. 13 a is directly related with description of polarized light at the circular complex polarization plane that is a stereographic projection of the Poincare sphere [84]. So, C point and L contours correspond to the pole of the Poincare sphere and its equator, see Fig. 14.

Fig. 14. The complex circular polarization plane. The center of coordinates corresponds to left-circular polarization (C point); the circle of unite radius separating red and blue areas corresponds to linear polarizations with changeable azimuth of polarization (L contours), its contours separate the area of the beam with left handedness (red) and right handedness (blue); right-circular polarization point lies at infinity.

Let us support this intuitive consideration by formal description. Let us proceed from Jones vectors of two components, right-circularly polarized LG01 mode and left-circularly polarized plane wave,

$$\mathbf{E}\_{LG} = c(w/\rho) \exp(i\Lambda) \begin{bmatrix} \exp(i\rho) \\ \exp\left[i\left(\rho + \pi/2\right)\right] \end{bmatrix}, \mathbf{E}\_P = \begin{bmatrix} \exp(i\rho) \\ \exp\left[i\left(\rho - \pi/2\right)\right] \end{bmatrix} \tag{5}$$

where *c* is the amplitude factor corresponding to inhomogeneous amplitude distribution of a mode as a function of dimensionless radial coordinate, and *<sup>i</sup> e* is associated with helicoidal change of a phase of a mode under circumference of the central vortex (its explicit form for Laguerre-Gaussian mode is well known but is not relevant here). There is Jones vector of the combined beam:

$$\mathbf{E}\_{\text{Total}} = \mathbf{E}\_{LG} + \mathbf{E}\_{P} = \begin{bmatrix} E\_{\text{x}} \\ E\_{\text{y}} \end{bmatrix} = \begin{bmatrix} c \exp(i\Lambda) + 1 \\ c \exp\left[i\left(\Lambda + \pi/2\right)\right] + \exp\left(-i\pi/2\right) \end{bmatrix} \exp(i\phi) \tag{6}$$

General coherency matrix of the beam is found as

$$\mathbf{E}\{\mathbf{J}\} = \mathbf{E}\_{Total} \cdot \mathbf{E}\_{Total}^\* = \begin{bmatrix} E\_x \\ E\_y \end{bmatrix} \begin{bmatrix} E\_x^\* & E\_y^\* \end{bmatrix} = \begin{Vmatrix} I\_{xx} & I\_{xy} \\ I\_{yx} & I\_{yy} \end{Vmatrix} \tag{7}$$

or in explicit form:

284 Modern Metrology Concerns

where 123 *sss* , , are the normalized second, third and fourth Stokes parameters, respectively [17, 83, 84], is elliptically polarized. But at the center of vortex of LG01 mode the field is circularly polarized with the state of polarization of a plane wave. A common phase singularity (vortex) of orthogonally polarized component of the combined beam lies at the bottom of this circular polarization. At the same time, the resulting field is polarized linearly at two contours where amplitudes of two components become equal to each other, see Fig.

For that, owing to helicoidal structure of a wave front of LG01 mode, azimuth of linear polarization changes with changing phase difference of a mode and a plane wave. Such topological structure can be considered as elementary experimental model of the assemblage of C point and surrounding it L contour of conventional singular optics of vector fields. Really, similarly to the case of random vector fields, crossing L line where handedness is undetermined is accompanied by step-like changing handedness into opposite one, corresponding to predominant in intensity component with unchangeable azimuth of polarization. For comparison, Fig. 13 b illustrates the line of equal intensities of

It is of interest that the elementary structure shown in Fig. 13 a is directly related with description of polarized light at the circular complex polarization plane that is a stereographic projection of the Poincare sphere [84]. So, C point and L contours correspond

Fig. 14. The complex circular polarization plane. The center of coordinates corresponds to left-circular polarization (C point); the circle of unite radius separating red and blue areas corresponds to linear polarizations with changeable azimuth of polarization (L contours), its contours separate the area of the beam with left handedness (red) and right handedness

coherently mixed components in linear polarization basis.

to the pole of the Poincare sphere and its equator, see Fig. 14.

(blue); right-circular polarization point lies at infinity.

<sup>123</sup> *sss* 1 ,

In general, combined beam, everywhere with the unit polarization degree ( <sup>222</sup>

13 a.

$$\langle \mathbf{J} \rangle = \begin{vmatrix} c^2 + 2c\cos\Lambda + 1 & c^2 \exp(-i\pi/2) + 2c\sin\Lambda + \exp(i\pi/2) \\ c^2 \exp(i\pi/2) + 2\sin\Lambda + \exp(-i\pi/2) & c^2 - 2c\cos\Lambda + 1 \end{vmatrix} \tag{8}$$

Combining the elements of coherency matrix, one can find *full* Stokes parameters:

$$\begin{aligned} S\_0 &= f\_{xx} + f\_{yy} = 2(c^2 + 1); \qquad S\_1 = f\_{xx} - f\_{yy} = 4c \cos \Delta; \\\\ S\_2 &= f\_{xy} + f\_{yx} = 4c \sin \Delta; \qquad S\_3 = i \left(f\_{xy} - f\_{yx}\right) = 2(c^2 - 1). \end{aligned} \tag{9}$$

Here we are especially interested in the case when 1 *c* . One just obtains for this case the *normalized* Stokes parameters:

$$\mathbf{s}\_0 = \mathbf{1}; \quad \mathbf{s}\_1 = \cos \boldsymbol{\Lambda}; \quad \mathbf{s}\_2 = \sin \boldsymbol{\Lambda}; \quad \mathbf{s}\_3 = \mathbf{0}. \tag{10}$$

Vanishing of the fourth Stokes parameter means that polarization at all points of the contour where intensities of the mixed components are equal to each other are equally distanced from the states of polarization of the components, i.e. neither right-circular nor left-circular predominate in intensity. It is in direct correspondence with Fig. 13 a. At all points of such L contour polarization is linear with the polarization azimuth <sup>1</sup> 2 1 0.5tan *s s* 2 , while the angle of ellipticity 3 0.5arcsin 0 *s* . Besides, the degree of polarization 2 2 1 2 *P ss* 1 . In correspondence with helicoidal structure of a wave front of LG01 mode, a phase difference of the components changes along the contour of equal intensities that results in changing azimuth of polarization. Thus, we obtain direct analog of L contour. Further, at the center of vortex of LG01 mode we have 0 *c* . Again, proceeding from Eq. (9) we find the normalized Stokes parameters 1, 0, 0, 1 , i.e. left-circular polarization of a plane wave. In the vicinity of such C point polarization is elliptical, with the azimuth of polarization changing with azimuthal coordinate and ellipticity decreasing from the vortex to L contour, Eq. (10), where handedness is undetermined an step-like changing by crossing this contour. It is all in quite correspondence with Fig. 14.

Thus, for circular polarization basis walking along contour of the combined beam "LG01 mode + plane wave" where intensities of the components become equal corresponds to moving along equator of the Poincare sphere that is determined only by the ratio of the second and third Stokes parameters. (For comparison, using linear polarization basis, to say 0° and 90°, one obtains by the same way the normalized Stokes parameters for the combined beam 1, 0, cos 0 90 , sin 0 90 that corresponds to points of 45° merdian of the Poincare sphere, see Fig. 13 b.)

Before consideration of the most general case of partial mutual coherence of the mixed orthogonally polarized components in the following section, let us consider other limiting case, *viz.* completely incoherent mixing of such components. There is no necessity to proceed now from Jones vectors and to form a coherency matrix of the combined beam. One can at once determine the Stokes parameters of mutually incoherent components and to sum directly them, without accounting phase relations that are irrelevant for incoherent summation. The normalized Stokes parameters of orthogonally polarized beams differ only in sign of the second, third and fourth parameters: 1, , , *sss* <sup>123</sup> and 1, , , *sss* <sup>123</sup> . It is clear that when two components become equal in intensities, the normalized Stokes parameters of the combined beam becomes1, 0, 0, 0 . The field at such elements of a field is completely unpolarized. There are just U singularities [61 - 63]. This case is shown in Fig. 13 c for the case of incoherent mixing of orthogonally circularly polarizer components. Trajectory of the imaging point for the combined beam in this case is the diameter of the Poincare sphere connecting two poles. U singularity is imaged by the center of this sphere, and all other points (beside the center and poles) image partially circularly polarized fields. For that, the length of a vector drawn from the center of the Poincare sphere to the imaging point inside it equals the degree of polarization. The point where the degree of polarization equals unity is referred to as P (completely polarized) point [61 - 63]. Its location is determined by the vortex of orthogonally polarized (scalar singular) component. The set of P points and U contours corresponding to extrema of the degree of polarization of a field are *the singularities of the degree of polarization* forming the vector skeleton of two-component mixture of orthogonally polarized beams. Note, in papers [61 - 64] consideration is carried out using the notion of the complex degree of polarization CDP, associated with orientation of the vector of polarization in the Stokes space and undergoing the phase singularity at the center of this space. So, U singularities can be considered just as vector singularities, *viz.* singularities of the vector of polarization, when its magnitude equals zero and a phase (orientation of the vector) is undetermined.

a phase difference of the components changes along the contour of equal intensities that results in changing azimuth of polarization. Thus, we obtain direct analog of L contour. Further, at the center of vortex of LG01 mode we have 0 *c* . Again, proceeding from Eq. (9) we find the normalized Stokes parameters 1, 0, 0, 1 , i.e. left-circular polarization of a plane wave. In the vicinity of such C point polarization is elliptical, with the azimuth of polarization changing with azimuthal coordinate and ellipticity decreasing from the vortex to L contour, Eq. (10), where handedness is undetermined an step-like changing by crossing

Thus, for circular polarization basis walking along contour of the combined beam "LG01 mode + plane wave" where intensities of the components become equal corresponds to moving along equator of the Poincare sphere that is determined only by the ratio of the second and third Stokes parameters. (For comparison, using linear polarization basis, to say 0° and 90°, one obtains by the same way the normalized Stokes parameters for the combined

> 

Before consideration of the most general case of partial mutual coherence of the mixed orthogonally polarized components in the following section, let us consider other limiting case, *viz.* completely incoherent mixing of such components. There is no necessity to proceed now from Jones vectors and to form a coherency matrix of the combined beam. One can at once determine the Stokes parameters of mutually incoherent components and to sum directly them, without accounting phase relations that are irrelevant for incoherent summation. The normalized Stokes parameters of orthogonally polarized beams differ only in sign of the second, third and fourth parameters: 1, , , *sss* <sup>123</sup> and 1, , , *sss* <sup>123</sup> . It is clear that when two components become equal in intensities, the normalized Stokes parameters of the combined beam becomes1, 0, 0, 0 . The field at such elements of a field is completely unpolarized. There are just U singularities [61 - 63]. This case is shown in Fig. 13 c for the case of incoherent mixing of orthogonally circularly polarizer components. Trajectory of the imaging point for the combined beam in this case is the diameter of the Poincare sphere connecting two poles. U singularity is imaged by the center of this sphere, and all other points (beside the center and poles) image partially circularly polarized fields. For that, the length of a vector drawn from the center of the Poincare sphere to the imaging point inside it equals the degree of polarization. The point where the degree of polarization equals unity is referred to as P (completely polarized) point [61 - 63]. Its location is determined by the vortex of orthogonally polarized (scalar singular) component. The set of P points and U contours corresponding to extrema of the degree of polarization of a field are *the singularities of the degree of polarization* forming the vector skeleton of two-component mixture of orthogonally polarized beams. Note, in papers [61 - 64] consideration is carried out using the notion of the complex degree of polarization CDP, associated with orientation of the vector of polarization in the Stokes space and undergoing the phase singularity at the center of this space. So, U singularities can be considered just as vector singularities, *viz.* singularities of the vector of polarization, when its magnitude equals zero

0 90 that corresponds to points of 45°-

this contour. It is all in quite correspondence with Fig. 14.

0 90 , sin

and a phase (orientation of the vector) is undetermined.

beam 1, 0, cos

 

merdian of the Poincare sphere, see Fig. 13 b.)

Let us emphasize that the condition of occurring U singularity is equal to the condition of occurring L contour in completely coherent limit. It means that loci of C and L singularities in completely coherent fields and P and U singularities in partially coherent fields arising from completely incoherent orthogonally polarized components, correspondingly, coincide.

Displacement from U singularity results in predomination of one of two orthogonal components in intensity. The state of (partial) polarization is just determined by the predominant component. That is why, the degree of polarization can be determined in similar form as visibility:

$$P = \left| \frac{I\_1 - I\_2}{I\_1 + I\_2} \right|. \tag{11}$$

In other words, at each point of the combined beam equal in intensities parts of orthogonal components form unpolarized background, at which manifests itself completely polarized part corresponding to predominant in intensity component. This is in a complete agreement with classical decomposition of partially polarized beam into completely coherent and completely incoherent parts, which are added on intensities, without accounting phase relations [17, 84]. Note, there are no any device providing such decomposition in practice. However, share of completely polarized part can be determined experimentally through the Stokes polarimetric experiment, <sup>222</sup> *P sss* <sup>123</sup> or, equivalently, following Eq. (11). In theory one put in correspondence to such beams the set of two coherency matrices – for completely polarized and completely unpolarized parts of a beam [1, 17].

Thus, only two orthogonal states of polarization take place in combined beams of considered kind, which are separated by U singularities where the state of polarization is undetermined.

So, the considered limiting cases show the same location of C and P singularities and L and U singularities for the same set of components. However, vicinities of such singularities are essentially different. Only two orthogonal states of polarization are present in spatially partially coherent combined beams, and only the degree of polarization changes from point to point within the areas separated by U singularities.

#### **4.2 Vector singularities for partially mutually coherent mixed components**

Let us consider now the most general case, when two mixed components shown in Fig. 12 are orthogonally (circularly) polarized and are partially mutually coherent, so that the degree of mutual coherence of the components can be gradually changed from unity to zero. It can be implemented in the arrangement of the Mach-Zehnder interferometer with controllable optical pass difference between the legs of an interferometer, as shown in Fig. 15.

A half-wave plate at the interferometer input serves for fine balancing of intensity ratio between the legs of an interferometer without changing total intensity at its output. Two polarizers inside an interferometer are controllers setting orthogonal linear polarizations. LG01 mode is generated by a computer synthesized hologram. A quarter-wave plate at the output of an interferometer transforms orthogonal linear polarization into orthogonal circular ones.

Fig. 15. General arrangement for generation and detection of vector singularities in partially coherent beams: L – He-Ne laser; 2 and 4 – half-wave and quarter-wave plates, respectively; P – polarizers; A – linear analyzer; SGO – singularity generating object (computer synthesized hologram); BS – beam splitters; M – mirrors; CCD – CCD-camera; PC – personal computer. Insets show action 2 and 4 plates at the input and at the output of an interferometer; two prisms form optical path delay loop.

A quarter-wave plate and linear analyzer at the receiving end, together with CCD-camera matched with personal computer serve for Stokes-polarimetric analysis of combined beams. Two prisms at one of legs of an interferometer enable to control optical path difference and mutual coherence of the mixed components. Namely, one can control path delay *l* from zero to magnitude exceeding a coherence length (length of wave train) *l* of the used laser. Change of the ratio *l l* corresponds to change degree of mutual coherence of orthogonally polarized components. Thus, for 0 1 *l l* the combined beam is *simultaneously* partially spatially coherent (due to changing intensity ratio at cross-section of the resulting field) and partially temporally coherent (due to non-zero optical path difference between the components), one expects for increasing optical path difference the following.

As it has been mentioned above, the condition of arising of L contours and U contours in the limiting cases of mixing of orthogonally circularly polarized beams is the same: intensities of the components must be equal to each other. If the optical path difference increases from zero, field at the L contour remains linearly polarized, but the degree of polarization decreases. It follows from that the degree of polarization of a beam is determined by the degree of mutual coherence of its arbitrary orthogonal components [1, 17], here right-hand and left-hand circular components. It means that *U contour nucleates just at the bottom of L contour.*

The degree of polarization can be represented equivalently in terms of *measured* Stokes parameters (that will be used in the next section) or *theoretically*, *viz.* through the invariants

Fig. 15. General arrangement for generation and detection of vector singularities in partially

2 and

A quarter-wave plate and linear analyzer at the receiving end, together with CCD-camera matched with personal computer serve for Stokes-polarimetric analysis of combined beams. Two prisms at one of legs of an interferometer enable to control optical path difference and mutual coherence of the mixed components. Namely, one can control path delay *l* from zero to magnitude exceeding a coherence length (length of wave train) *l* of the used laser. Change of the ratio *l l* corresponds to change degree of mutual coherence of orthogonally polarized components. Thus, for 0 1 *l l* the combined beam is *simultaneously* partially spatially coherent (due to changing intensity ratio at cross-section of the resulting field) and partially temporally coherent (due to non-zero optical path difference between the components), one expects for increasing optical path difference the

As it has been mentioned above, the condition of arising of L contours and U contours in the limiting cases of mixing of orthogonally circularly polarized beams is the same: intensities of the components must be equal to each other. If the optical path difference increases from zero, field at the L contour remains linearly polarized, but the degree of polarization decreases. It follows from that the degree of polarization of a beam is determined by the degree of mutual coherence of its arbitrary orthogonal components [1, 17], here right-hand and left-hand circular components. It means that *U contour nucleates just at the bottom of L* 

The degree of polarization can be represented equivalently in terms of *measured* Stokes parameters (that will be used in the next section) or *theoretically*, *viz.* through the invariants

4 – half-wave and quarter-wave plates,

4 plates at the input and at the

respectively; P – polarizers; A – linear analyzer; SGO – singularity generating object (computer synthesized hologram); BS – beam splitters; M – mirrors; CCD – CCD-camera;

2 and

output of an interferometer; two prisms form optical path delay loop.

coherent beams: L – He-Ne laser;

following.

*contour.*

PC – personal computer. Insets show action

of the coherency matrix, which at the same time determine coherence properties of a field [17]:

$$P = \sqrt{1 - \frac{4 \det\{\mathbf{J}\}}{\mathbf{S} \mathbf{p}^2 \langle \mathbf{J} \rangle}}\,\tag{12}$$

For that, in general, the degree of polarization is always non less than modulo of the degree of mutual coherence of the components, for circularly polarized components

$$\left| \left| \mu\_{rl} \right| = \left| \frac{J\_{rl}}{\sqrt{J\_{rr} J\_{ll}}} \right|. \tag{13}$$

In general case, *P rl* , as the degree of coherence depends on the decomposition basis while the degree of polarization is invariant [5]. However, it has been shown [1, 17] that the degree of polarization is equal to the *maximal* degree of coherence, *rl* max *P* , in the case when the components are of equal intensities. This is just the case of L singularities and U singularities. It is of the most importance, that change of the optical path difference changes weights *rl* of completely coherent (and completely polarized) part of the combined beam and 1 *rl* of its completely incoherent part. Increasing *l l* difference corresponds to increasing of weight of U singularity against L singularity, so that one can follow gradual transformation of L contour into U contour.

#### **4.3 Experimental reconstruction of "pure" and "mixed" polarization singularities**

Mixing of LG01 mode and plane wave was performed in the arrangement Fig. 15 [66]. Intensity of a plane wave was considerably (approximately by the order of magnitude) less the peak intensity of a mode, both circularly (orthogonally) polarized. The following results have been obtained under such conditions.

Fig. 16 shows the combined beam whose view, as was mentioned above, within experimental accuracy remains the same at arbitrary optical path delay set in the interferometer. This photo has been obtained for incoherent mixing of two components for *l l* 3 (under condition realized in paper [68]. We measured spatial distribution of the Stokes parameters and looked for the elements where 123 *sss* 0 (0 *P* , U contours), and 3*s* 1 (P point), see discussion after Eq. (10). In such a manner, we were in a position to reconstruct a vector skeleton of partially spatially coherent combined beam formed by completely mutually incoherent components, as in Refs [61 - 64]. Experimental error in determining the normalized Stokes parameters was at the level 7%; this determines reliability with which we reconstructed P point and U contours. P and U singularities for this case are shown in right fragment of Fig. 16. Two U contours separate the areas with right-circular and left-circular polarization shown by different colors. Within these areas 2 2 1 2 *s s* 0 , while 3*s* 1. Let us emphasize that the full Stokes-polarimetric experiment over combined beam cross-section is necessary in this case, as operating only with rotating linear analyzer does not provide differentiation partial circular polarization from complete elliptical polarization.

Fig. 16. The partially coherent combined beam (left) and its vector skeleton formed by P and U singularities (right) for completely incoherent mixing of circularly polarized components.

Separate maps of the Stokes parameters are less representative being only row material for finding out the degree of polarization, ellipsometric parameters of a field, and vector singularities. That is why, we demonstrate separately from 2D pattern shown in right fragment of Fig. 16 1D cross-section of the degree of polarization of this combined beam, see Fig. 17. Red curve shows two-lateral radial dependence of *P* computed following Eq. (11). Blue curve shows experimentally obtained distribution found as the combination of measured Stokes parameters, here *P s* <sup>3</sup> . Quantitative discrepancy of two curves (both in positions of zeroes and in heights of side-lobes) is obvious and is explained by anisotropy of the vortex. Nevertheless, behavior of the experimental dependence is in quite satisfactory qualitative agreement with the simulation results. Namely, one observes two zeroes of the degree of polarization at the each side of the central optical vortex that are the signs of two U contours. Moreover, experiment has proved typical conical vicinity of U contours recently predicted and observed in paper [61], which are reliable sign of true singularity of any kind, in contrast to local minimum.

Another limiting case (completely mutually coherent components) for *l l* 1 (approximately 0.05) is illustrated in Fig. 18. Again, spatial maps of the Stokes parameters were obtained and the elements 3*s* 0 and 2 2 1 2 *s s* 1 where selected. There are the lines of linear polarization. Than, in several selected points of such L lines we determined the azimuth of polarization, again, by two ways: firstly as <sup>1</sup> 2 1 tan *s s* , and, secondary, as direct measurement of the azimuth of polarization by rotating a linear analyzed up to complete extinction of a field at the specified point that corresponds to crossed azimuth of polarization of the combined beam and the axis of maximal transmittance of analyzer. Description between two results for determining of the azimuth of polarization do not exceeded 0.1 rad.

Perfect extinction of a beam at the specified points just shows that the degree of polarization *P* 1 (in contrast to the previous case of completely mutually incoherent components, where intensity at the analyzer output is independent on its orientation). Also, for certain orientations of a quarter-wave plate and analyzer, the field at each other point can be extinguished that shows that everywhere the degree of (elliptical) polarization equals unity. It is worth to compare Fig. 18 with a view of the circular complex polarization plane (Fig. 14) to be convincing of that, really, such polarization distribution over of a combined beam is close experimental analogue of the circular polarization plane.

Fig. 16. The partially coherent combined beam (left) and its vector skeleton formed by P and U singularities (right) for completely incoherent mixing of circularly polarized components.

**P**

Separate maps of the Stokes parameters are less representative being only row material for finding out the degree of polarization, ellipsometric parameters of a field, and vector singularities. That is why, we demonstrate separately from 2D pattern shown in right fragment of Fig. 16 1D cross-section of the degree of polarization of this combined beam, see Fig. 17. Red curve shows two-lateral radial dependence of *P* computed following Eq. (11). Blue curve shows experimentally obtained distribution found as the combination of measured Stokes parameters, here *P s* <sup>3</sup> . Quantitative discrepancy of two curves (both in positions of zeroes and in heights of side-lobes) is obvious and is explained by anisotropy of the vortex. Nevertheless, behavior of the experimental dependence is in quite satisfactory qualitative agreement with the simulation results. Namely, one observes two zeroes of the degree of polarization at the each side of the central optical vortex that are the signs of two U contours. Moreover, experiment has proved typical conical vicinity of U contours recently predicted and observed in paper [61], which are reliable sign of true singularity of any kind,

Another limiting case (completely mutually coherent components) for *l l* 1 (approximately 0.05) is illustrated in Fig. 18. Again, spatial maps of the Stokes parameters were

polarization. Than, in several selected points of such L lines we determined the azimuth of

measurement of the azimuth of polarization by rotating a linear analyzed up to complete extinction of a field at the specified point that corresponds to crossed azimuth of polarization of the combined beam and the axis of maximal transmittance of analyzer. Description between

Perfect extinction of a beam at the specified points just shows that the degree of polarization *P* 1 (in contrast to the previous case of completely mutually incoherent components, where intensity at the analyzer output is independent on its orientation). Also, for certain orientations of a quarter-wave plate and analyzer, the field at each other point can be extinguished that shows that everywhere the degree of (elliptical) polarization equals unity. It is worth to compare Fig. 18 with a view of the circular complex polarization plane (Fig. 14) to be convincing of that, really, such polarization distribution over of a combined beam is

two results for determining of the azimuth of polarization do not exceeded 0.1 rad.

1 2 *s s* 1 where selected. There are the lines of linear

*U***<sup>1</sup>**

.

*U***<sup>2</sup>**

2 1 tan *s s* , and, secondary, as direct

in contrast to local minimum.

obtained and the elements 3*s* 0 and 2 2

polarization, again, by two ways: firstly as <sup>1</sup>

close experimental analogue of the circular polarization plane.

Fig. 17. 1D distribution of the degree of polarization of the combined beam formed by two mutually incoherent orthogonally polarized components defined in Fig.12 and shown in Fig. 16.

Fig. 18. C and L singularities in combined beam assembled from completely mutual coherent orthogonally (circularly) polarized LG01 mode and plane wave. At L lines, where intensities of two mixed component are equal, the azimuth of polarization changes in agreement of prediction illustrated in Fig. 14 Areas of different colors correspond to opposite handedness.

At last, we have elaborated experimentally intermediate case, when 0 1 *l l* , lying between ones considered above. For step-by-step increasing optical path difference between the same orthogonally (circularly) polarized components, we obtained spatial distributions for the Stokes components 0 90 45 45 , , , , , *r l II I I II* and found from them the Stokes parameters. Further, the degree of polarization and ellipsometric parameters of the combined beam were determined as the combinations of these parameters.

Before formulating the conclusions from our observations, let us represent one of row (intermediate) results undergoing following processing. Fig. 19 illustrates combined beams "LG01 mode + plane wave" (with large intensity ratio, so that one does not visualizes a plane wave) for relative optical path differences close to unity (coherent limit illustrated in Fig. 18) and slightly exceeding a half of the coherence length of used laser, left fragments of Fig. 19. Other fragments of this figure are the intensity distributions 45 *I* (central column) and 45 *I* (right column) used for forming the third Stokes parameters. (Other pairs of intensity distributions show the same tendency). Though two orthogonally polarized components do not interfere, their equal polarization projections selected by properly oriented polarizer *can* interfere depending on their mutual coherence. If the degree of mutual coherence of the components is not zero, their equally polarized projections interfere with forming typical patterns indicating phase singularity. To have enough spatial resolution for determining place of vortex, we set non-zero interference angle between the components (which as small enough to be no influencing on accuracy of polarization measurements). For that, instead of snail-like pattern typical for coaxial mixing of LG01 mode and a plane wave, we obtained interference forklets. Comparison of the central and left columns of Fig. 19 shows that spatial intensity distributions for orthogonal polarization projection of the combined beam are complementary in a sense that dark forklet is replaced by bright one.

The main conclusion follows from comparison of fragments b and e (c and f). Decreasing of the mutual coherence of the mixed components and decreasing of the degree of polarization of the combined beam are accompanied by decreasing of ability of equal polarization projections of the mixed components to interfere that manifests itself in decreasing of visibility of interference pattern. So, in fragments b and c of Fig. 19 ( *l l* 0.05 ) the measured visibility is 0.97, while in the fragments Fig. 19 e and f ( *l l* 0.56 ) visibility is 0.24 (with experimental error non exceeding 5%). It shows the feasibility allows determine *the degree of mutual coherence* of two orthogonally polarized beams by measuring *the degree of polarization* of the combined beam formed by such components found from Stokes parameters. Namely, in our experiment *rl* for *l l* 0.56 also equals 0.24. For the reasons discussed in Section 3, such measurements are preferably be performed at the elements of the combined beam where intensities of two beams are equal to each other (where L and U singularities co-exist in case of partial mutual coherence of the components), while at such singular elements of the combined beam *P rl* .

Thus, vector singularities occurring in light fields, which are simultaneously partially spatially and partially temporally coherent have been considered. It has been shown that in the case of partially coherent mixing of two orthogonally *circularly* polarized components conventional vector singularities, *viz.* C points and L lines submerged in a field of elliptical

At last, we have elaborated experimentally intermediate case, when 0 1 *l l* , lying between ones considered above. For step-by-step increasing optical path difference between the same orthogonally (circularly) polarized components, we obtained spatial distributions for the Stokes components 0 90 45 45 , , , , , *r l II I I II* and found from them the Stokes parameters. Further, the degree of polarization and ellipsometric parameters of the

Before formulating the conclusions from our observations, let us represent one of row (intermediate) results undergoing following processing. Fig. 19 illustrates combined beams "LG01 mode + plane wave" (with large intensity ratio, so that one does not visualizes a plane wave) for relative optical path differences close to unity (coherent limit illustrated in Fig. 18) and slightly exceeding a half of the coherence length of used laser, left fragments of Fig. 19. Other fragments of this figure are the intensity distributions 45 *I* (central column) and 45 *I* (right column) used for forming the third Stokes parameters. (Other pairs of intensity distributions show the same tendency). Though two orthogonally polarized components do not interfere, their equal polarization projections selected by properly oriented polarizer *can* interfere depending on their mutual coherence. If the degree of mutual coherence of the components is not zero, their equally polarized projections interfere with forming typical patterns indicating phase singularity. To have enough spatial resolution for determining place of vortex, we set non-zero interference angle between the components (which as small enough to be no influencing on accuracy of polarization measurements). For that, instead of snail-like pattern typical for coaxial mixing of LG01 mode and a plane wave, we obtained interference forklets. Comparison of the central and left columns of Fig. 19 shows that spatial intensity distributions for orthogonal polarization projection of the combined beam are complementary

The main conclusion follows from comparison of fragments b and e (c and f). Decreasing of the mutual coherence of the mixed components and decreasing of the degree of polarization of the combined beam are accompanied by decreasing of ability of equal polarization projections of the mixed components to interfere that manifests itself in decreasing of visibility of interference pattern. So, in fragments b and c of Fig. 19 ( *l l* 0.05 ) the measured visibility is 0.97, while in the fragments Fig. 19 e and f ( *l l* 0.56 ) visibility is 0.24 (with experimental error non exceeding 5%). It shows the feasibility allows determine *the degree of mutual coherence* of two orthogonally polarized beams by measuring *the degree of polarization* of the combined beam formed by such components found from Stokes

discussed in Section 3, such measurements are preferably be performed at the elements of the combined beam where intensities of two beams are equal to each other (where L and U singularities co-exist in case of partial mutual coherence of the components), while at such

*rl* . Thus, vector singularities occurring in light fields, which are simultaneously partially spatially and partially temporally coherent have been considered. It has been shown that in the case of partially coherent mixing of two orthogonally *circularly* polarized components conventional vector singularities, *viz.* C points and L lines submerged in a field of elliptical

*rl* for *l l* 0.56 also equals 0.24. For the reasons

combined beam were determined as the combinations of these parameters.

in a sense that dark forklet is replaced by bright one.

parameters. Namely, in our experiment

singular elements of the combined beam *P*

polarizations coexist with singularities arising just in partially coherent fields, such as U and P singularities as the extrema of the degree of polarization. Gradual transformation of C and L singularities into P and U singularities, respectively, accompanying decreasing degree of mutual coherence of the components has been experimentally shown. So, conventional polarization singularities of completely coherent fields (C points and L lines) are vanish in incoherent part of the combined beam, so that the only polarization of the component predominant in intensity remains in the vicinities of P points and U lines.

Fig. 19. The combined beams "LG01 mode + plane wave" with relative optical path differences *l l* 0.05 (a) and *l l* 0.56 (d); the corresponding intensity distributions behind a linear analyzer for determining the third Stokes parameters: +45° (b and e) and – 45° (c and f). Decreasing visibility of interference fringes in fragments e and f corresponds to decreasing in parallel the degree of mutual coherence of the mixed components and the degree of polarization of the combined beam.
