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Polarization and coherence properties in self-healing propagation of a partially coherent radially polarized twisted beam

Yunqin Zhou, Zhiwei Cui, and Yiping Han

Open Access Open Access

Abstract

With the help of generalized Huygens–Fresnel integral, an analytical expression for the self-healing of a partially coherent radially polarized twisted (PCRPT) beam is derived. The coherence and polarization properties of the PCRPT beam in self-healing propagation are studied in detail. It shows that the existence of the twist phase is a double-edged sword for the self-healing properties of the beam. With the increase of the twist factor, the self-healing ability of beam intensity distribution decreases. However, the anti-disturbance performance of beam polarization improves at the same time. Besides, the polarization and coherence distribution of the beam are proved that own a slight self-healing ability when the obstacle is small. Our results will be helpful to the fields of optical tweezers, microscopy, optical communication, and so on.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The self-healing property of laser beams is that some laser beams can gradually reconstruct their profile in propagation after being disturbed by the obstacle. This special property was first found in Bessel beams [1–8]. Later, this property was found in other diffraction-free beams such as Airy beams [9–11], caustic beams [12] and Mathieu beams [13,14]. Gutierrez-Vega et al. pointed out that this feature should be common to all the diffraction-free beams [14]. However, it was subsequently shown that some diffracting beams like vector Laguerre–Gaussian beams [15], scalar and vector Bessel–Gauss beams [16], radially polarized Bessel–Gauss beams [17], and optical ring lattices [18] can also self-heal. All the beams mentioned above are completely coherent, but the beams that can be generated in the experiment are partially coherent. Based on this consideration, Wang et al. have successfully generalized the self-healing property to partially coherent beams [19]. The discussions about the self-healing properties of different partially coherent beams are also increasing [20–23].

On the other hand, the concept of self-healing properties is developing. The changes in the beam’s other degrees of freedom (like phase, orbital angular momentum, etc) after being disturbed also make people interested. Vyas et al. showed that the polarization distribution of partially obstructed vector Bessel-Gauss beam will recover at the focal plane [16]. Meanwhile, they showed that the vector Laguerre–Gaussian beam does not have this ability [15]. Liu et al. have explored the self-healing properties of a partially coherent vortex (PCV) beam and they demonstrated that the degree of coherence (DoC) distribution of a partially blocked PCV beam can self-reconstruct in the far field [24]. Moreover, Zhou et al. considered that a Gaussian Schell-model (GSM) beam was disturbed by the transparent turbid medium rather than an opaque obstacle, and found that the DoC of the GSM beam also owns self-healing properties in this situation [25].

In recent years, with the help of the necessary and sufficient conditions for devising correct cross-spectral densities [26,27], many new types of partially coherent beams have been proposed theoretically and produced experimentally. This greatly enriches the manipulations and applications of partially coherent beams. Mei and Korotkova introduced Laguerre–Gaussian Correlated Schell-model (LGCSM) beam [28]. Zhou et al. investigated the radiation forces and the trap stiffness induced by a focused LGCSM [29]. Chen et al. introduced Hermite-Gaussian correlated Schell-model (HGCSM) beam and found the HGCSM beam exhibits self-splitting properties and combining properties in different propagation situations [30]. Further, they have presented a method for generating optical coherence lattices and showed that information can be encoded into and recovered from the DoC of lattice in theory [31]. Ping et al. introduced radially polarized multi-Gaussian Schell-model (MGSM) beam [32] as an extension of scalar MGSM beam [33], and Song et al. investigated its propagation properties in oceanic turbulence [34]. Gao et al. presented a new class of broadband electromagnetic GSM beam whose polarization properties showed different features between the source and the far-zone [35]. Li et al. introduced vortex Gaussian cosine pseudo-Schell-model (GCPSM) source, which can produce high-quality tunable intensity distributions [36]. They also observed a remarkable anomalous self-focusing phenomenon in this beam. In addition to the contents above, there are many excellent works have been published [37–42]. For the reason of the length, we will not introduce them in detail here.

Due to the introduction of various new types of beams, the physics of the laser beam becomes richer, and the factors which can be used to manipulate the beam also increase. So a curious question arises: how will these newly introduced factors affect the self-healing properties of the beams? Here we take the twist phase as an example. It is a unique phase structure that only exists in partially coherent beams, and was first introduced by Simon and Mukunda in 1993 [43]. The twist phase can induce beams carrying orbital angular momentum (OAM) as same as the helical phase, and it was widely used in radiation force [44], imaging [45], optical communication [46], and quantum entanglement [47]. Its successful applications in these fields naturally arouse our interest in its function and performance in self-healing. Therefore, we will discuss the effect of the twist phase on self-healing properties in this paper. We select the partially coherent radially polarized twisted (PCRPT) beam as the object of our study. This beam was introduced by Wu, and he investigated the propagation properties of the beam in free space [48]. The statistical properties of PCRPT beam in a uniaxial crystal [49,50], underwater turbulent medium [51] and turbulent atmosphere [52] are investigated, respectively. Besides, the scattering properties of polychromatic PCRPT beam upon a deterministic medium were studied two years ago [53]. We will pay attention not only to the intensity but also to the polarization and coherence properties of the beam in disturbed propagation.

The paper is organized as follows: In section 2, we analytically derived the formulas about self-healing of the PCRPT beam based on the generalized Huygens–Fresnel integral [54]. In section 3, with the help of the unified theory of coherence and polarization [55–57], the distributions of average intensity, degree of polarization (DoP), state of polarization (SoP) and DoC are examined through numerical examples. The conclusion is summarized in section 4.

2. Analytical derivation

Let us consider a PCRPT beam, partially blocked by an opaque obstacle, propagating close to the $z$ direction into an ABCD optical system. The source plane is defined at $z=0$, as shown in Fig. 1. The statistical properties of a partially coherent vectorial beam in the spatial-frequency domain can be described by a $2\times 2$ cross-spectral density (CSD) matrix [55]

$$\boldsymbol{W}_{\alpha\beta}= \left[ \begin{array}{cc} W_{xx} & W_{xy}\\ W_{yx} & W_{yy}\\ \end{array} \right],$$
whose elements are defined as
$$W_{\alpha\beta}(\boldsymbol{r_{1}}, \boldsymbol{r_{2}})=\left\langle E_{\alpha}^{*}(\boldsymbol{r_{1}})E_{\beta}(\boldsymbol{r_{2}})\right\rangle,\qquad\alpha,\beta=x,y$$
where $\boldsymbol {r_{i}}=(x_{i},y_{i})$ is the transverse position vector, angular brackets denote the ensemble average, $E_{\alpha }, E_{\beta }$ denote the $\alpha$ and $\beta$ component of electric field, and the asterisk indicates complex conjugation.

 figure: Fig. 1.

Fig. 1. Illustrating the notation.

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For a PCRPT beam, its specific expression in the source plane is given as [48]

$$\begin{aligned} W_{\alpha\beta}(\boldsymbol{r_{1}}, \boldsymbol{r_{2}},0)= & \frac{\alpha_{1}\beta_{2}}{\omega_{0}^{2}}\exp\left(-\frac{x_{1}^{2}+y_{1}^{2}+x_{2}^{2}+y_{2}^{2}}{\omega_{0}^{2}}\right)\\ & \times\exp\left[-\frac{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}{2\delta_{0}^{2}}\right]\\ & \times\exp({-}ik\mu x_{1}y_{2}+ik\mu x_{2}y_{1}),\qquad\alpha,\beta=x,y \end{aligned}$$
where $\omega _{0}$ is the beam waist size, $\delta _{0}$ denotes the transverse coherence length in the source plane, $k=2\pi /\lambda$ is the wavenumber with the wavelength $\lambda$. $\mu$ is the twist factor, which indicates the degree of twist. The twist factor must satisfy inequality $|\mu |\leqslant 1/k\delta _{0}^{2}$ to keep the elements of CSD matrix non-negative.

With the help of the paraxial approximation, the propagation properties of the beam blocked by an opaque obstacle can be obtained by using the generalized Huygens–Fresnel integral, which is also called the Collins formula [54]

$$\begin{aligned} W_{\alpha\beta}(\boldsymbol{\rho_{1}}, \boldsymbol{\rho_{2}},z)= & \frac{1}{(\lambda B)^{2}}\int\int d^{2}\boldsymbol{r_{1}}d^{2}\boldsymbol{r_{2}}\times T^{*}(\boldsymbol{r_{1}})T(\boldsymbol{r_{2}}) W_{\alpha\beta}(\boldsymbol{r_{1}}, \boldsymbol{r_{2}},0)\\ & \times\exp\left\lbrace -\frac{ik}{2B}[A(r_{1}^{2}-r_{2}^{2})-2(\boldsymbol{r_{1}}\cdot\boldsymbol{\rho_{1}}-\boldsymbol{r_{2}}\cdot\boldsymbol{\rho_{2}})+D(\rho_{1}^{2}-\rho_{2}^{2})]\right\rbrace, \end{aligned}$$
where $\boldsymbol {\rho _{i}}=(u_{i},v_{i})$ is the transverse position vector in the receiver plane, $A, B, C, D$ are the transfer matrix elements of optical system and $T(\boldsymbol {r_{i}})$ is the transmittance function of the opaque obstacle. For convenience, we assume that this obstacle is circular and it has gaussian absorption efficiency. Therefore, $T(\boldsymbol {r_{i}})$ could be written as [7,58]
$$T(\boldsymbol{r_{i}})=1-\exp\left[ -\frac{(\boldsymbol{r_{i}}-\boldsymbol{r_{0}})^{2}}{\omega_{d}^{2}}\right],$$
in which $\boldsymbol {r_{0}}=(x_{0},y_{0})$ denotes the transverse position of the opaque obstacle in the source plane and $\omega _{d}$ is the size of obstacle. Substituting Eq. (3) and (5) into Eq. (4), after tedious integration and simplification [59], we obtain the analytical expressions of CSD matrix elements of the beam in the receiver plane. Since this result is too complex, we put it in Supplement 1. Based on these formulas, we could study the self-healing properties numerically.

3. Numerical results and discussion

In the following examples, we set the parameters as $\lambda =632.8\rm ~nm$, $\omega _{0}=2\rm ~mm$, $\delta _{0}=0.1\rm ~mm$, $\mu =0.0005/{\rm mm}$ and $\omega _{d}=0.5\rm ~mm$, unless specified otherwise. Besides, in order to study the far field properties of the beam, we consider that the beam propagates in a focused system. A focusing lens is placed just behind the source plane. Hence, the elements of the transfer matrix are $A=1-z/f, B=z, C=-1/f, D=1$ with the focal length $f=450\rm ~mm$.

One of the most important properties of a laser beam is intensity. For a vectorial beam described by the CSD matrix, its ensemble average intensity can be obtained as [55]

$$\left\langle I(\boldsymbol{\rho},z)\right\rangle =W_{xx}(\boldsymbol{\rho},\boldsymbol{\rho},z)+W_{yy}(\boldsymbol{\rho},\boldsymbol{\rho},z).$$

Figure 2 shows the intensity distributions of a PCRPT beam at the transverse plane for different propagation distances. The center of the obstacle is set at the point $(1\rm ~mm, 1\rm ~mm)$. By comparing the first row and the second row of Fig. 2, it is found that the defect caused by the obstacle gradually disappears during the propagation from the source plane to the focal plane.

 figure: Fig. 2.

Fig. 2. The transverse intensity distributions of a PCRPT beam with an obstacle at different propagation distances. As a comparison, the first row shows the results without the obstacle. Each intensity distribution is normalized individually.

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We use the similarity degree [20] to characterize the degree of self-healing, and the detailed expression of the similarity degree is given as

$$\gamma(z)=\frac{\int\int\left\langle I(\boldsymbol{\rho},z)\right\rangle\left\langle I'(\boldsymbol{\rho},z)\right\rangle d^{2}\boldsymbol{\rho}}{\sqrt{\int\int\left\langle I(\boldsymbol{\rho},z)\right\rangle ^{2}d^{2}\boldsymbol{\rho}}\sqrt{\int\int\left\langle I'(\boldsymbol{\rho},z)\right\rangle ^{2}d^{2}\boldsymbol{\rho}}},$$
where $I(\boldsymbol {\rho },z)$ and $I'(\boldsymbol {\rho },z)$ denote the average intensity without and with obstacle, respectively. The range of $\gamma (z)$ is from 0 to 1. When $\gamma (z)=1$, it means that $I(\boldsymbol {\rho },z)=I'(\boldsymbol {\rho },z)$, i.e., the influence of the obstacle disappears completely. From the Fig. 3, one can find that when the beam propagates to near the focal plane, the value of the $\gamma (z)$ is very close to 1. Therefore, from the aspect of light intensity distribution, the PCRPT beam can self-heal in the far field.

 figure: Fig. 3.

Fig. 3. The variation of the similarity degree as a function of propagating distance.

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The effect of the twist factor on the intensity in self-healing is shown in Fig. 4. It can be seen that the change of the twist factor does not affect the blocking effect caused by the obstacle on the beam in the source plane. With the increase of the propagation distance, the beam with a smaller twist factor achieves self-healing earlier, which can be concluded by comparing (a-3), (b-3), and (c-3) of Fig. 4. The larger the twist factor, the greater the beam distortion. In the focal plane, the excessive twist factor even causes the beam to fail to self-heal perfectly, see Fig. 4(c-4). Meanwhile, we notice that the position of the disturbed part of the beam rotates rather than keeps unchanged. This is because the PCRPT beam rotates in propagation [48]. It is shown that even if the obstacle is fixed in a definite position, the defect caused by it still moves with the rotation of the beam.

 figure: Fig. 4.

Fig. 4. The transverse intensity distributions of a PCRPT beam at different propagation distances with selected values of the twist factor. First row, $\mu =0.0001/{\rm mm}$; second row, $\mu =0.0005/{\rm mm}$; third row, $\mu =0.001/{\rm mm}$. Each intensity distribution is normalized individually.

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Next, we will investigate the coherence and polarization properties of the beam in this disturbed propagation. The spectral degree of polarization (DoP) is an important indicator of the polarization, which could be written by the CSD matrix as [60]

$$P(\boldsymbol{\rho})=\sqrt{1-\frac{4~\rm{Det}~\boldsymbol{W}(\boldsymbol{\rho},\boldsymbol{\rho})}{[\rm{Tr}~\boldsymbol{W}(\boldsymbol{\rho},\boldsymbol{\rho})]^2}},$$
where $\rm {Det}$ and $\rm {Tr}$ denote the determinant and the trace of matrix, respectively. The DoP is bounded by 0 and 1, i.e., $0\leqslant P\leqslant 1$. The lower bound $P=0$ means that the beam field in this point is completely unpolarized. The upper bound $P=1$ means the field is fully polarized.

Figure 5 shows the DoP distributions of the PCRPT beam for different propagation distances. The first row is without the obstacle, one can see that the completely polarized beam in the source plane gradually depolarizes in propagation [61]. The closer to the center of the beam, the more the DoP decreases, and the DoP distribution is negative gaussian in the focal plane. The second row is the result with an obstacle. Different from the self-healing process of the beam intensity, the DoP distribution in the source plane does not seem to be affected by the obstacle. However, with the increase of the propagation distance, the DoP decreases faster in the position where intensity is disturbed. Since the position of intensity defect rotates, the disturbed area of the DoP also rotates. In the focal plane, the influence caused by the obstacle in DoP distribution is very conspicuous. Thus, it can be confirmed that whether there exists a defect at the light source by observing the DoP distribution of the beam in the far field.

 figure: Fig. 5.

Fig. 5. The transverse DoP distributions of the PCRPT beam with an obstacle at different propagation distances. As a comparison, the first row shows the results without the obstacle.

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In order to study the influence of the twist factor on the DoP, we plot the DoP distributions in the focal plane with and without an obstacle in Fig. 6. It is found that the existence and increase of the twist factor make the DoP decrease weaker. The larger the value of the twist factor, the smaller the size of the central negative Gaussian distributions. As long as the twist phase exists, the disturbed part rotates the same angle ($90^{\circ }$ clockwise) regardless of the $\mu$ value. The first column of Fig. 6 is without the twist phase (i.e., $\mu =0$), so the disturbed part of (b-1) keeps in the $(1\rm ~mm, 1\rm ~mm)$ direction.

 figure: Fig. 6.

Fig. 6. The transverse DoP distributions of the PCRPT beam with different twist factors in the focal plane. The first row shows the results without the obstacle, and the second row is the corresponding results with an obstacle.

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Following the work of Wolf and Korotkova [55,62], the CSD matrix of a partially coherent vectorial beam could be represented as a sum of the completely unpolarized part and the completely polarized part

$$\boldsymbol{W}(\boldsymbol{\rho},\boldsymbol{\rho},z)=\boldsymbol{W}_{\rm U}(\boldsymbol{\rho},\boldsymbol{\rho},z)+\boldsymbol{W}_{\rm P}(\boldsymbol{\rho},\boldsymbol{\rho},z),$$
where
$$\begin{aligned} \boldsymbol{W}_{\rm U}(\boldsymbol{\rho},\boldsymbol{\rho},z) & = \left[ \begin{array}{cc} A_{1}(\boldsymbol{\rho},\boldsymbol{\rho},z) & 0\\ 0 & A_{1}(\boldsymbol{\rho},\boldsymbol{\rho},z)\\ \end{array} \right],\\ \boldsymbol{W}_{\rm P}(\boldsymbol{\rho},\boldsymbol{\rho},z) & = \left[ \begin{array}{cc} B_{1}(\boldsymbol{\rho},\boldsymbol{\rho},z) & D_{1}(\boldsymbol{\rho},\boldsymbol{\rho},z)\\ D_{1}^{*}(\boldsymbol{\rho},\boldsymbol{\rho},z) & C_{1}(\boldsymbol{\rho},\boldsymbol{\rho},z)\\ \end{array} \right], \end{aligned}$$
with
$$\begin{aligned} A_{1} & =\frac{1}{2}\left[W_{xx}+W_{yy}-\sqrt{\left(W_{xx}-W_{yy}\right)^{2}+4|W_{xy}|^{2}}\right],\\ B_{1} & =\frac{1}{2}\left[W_{xx}-W_{yy}+\sqrt{\left(W_{xx}-W_{yy}\right)^{2}+4|W_{xy}|^{2}}\right],\\ C_{1} & =\frac{1}{2}\left[W_{yy}-W_{xx}+\sqrt{\left(W_{xx}-W_{yy}\right)^{2}+4|W_{xy}|^{2}}\right],\\ D_{1} & =W_{xy}. \end{aligned}$$
Combined with the Eq. (6), the average intensity of the completely polarized part and the completely unpolarized part are respectively expressed as
$$\begin{aligned} \left\langle I_{P}(\boldsymbol{\rho},z)\right\rangle =B_{1}(\boldsymbol{\rho},\boldsymbol{\rho},z)+C_{1}(\boldsymbol{\rho},\boldsymbol{\rho},z),\\ \left\langle I_{U}(\boldsymbol{\rho},z)\right\rangle =A_{1}(\boldsymbol{\rho},\boldsymbol{\rho},z)+A_{1}(\boldsymbol{\rho},\boldsymbol{\rho},z). \end{aligned}$$

Based on Eq. (11) and (12), the intensity distributions of the completely polarized and the completely unpolarized part of the disturbed PCRPT beam at different propagation distances are plotted in Fig. 7, respectively. As a comparison, the first row shows the whole intensity distributions of the PCRPT beam. From the first column of Fig. 7, one can see that the beam is fully polarized in the source plane, and the completely unpolarized part is negligible at this time. As the beam propagates, the completely unpolarized part appears and gradually becomes the main component. This is because a partially coherent fully polarized cylindrical vectorial beam will gradually depolarize as it propagates [61]. In the focal plane, the completely unpolarized part has become dominant. Although both these two parts have not gotten rid of the influence of the obstacle in the focal plane, the distortion of the completely unpolarized part is smaller. The intensity distribution of the completely unpolarized part is closer to the whole intensity distribution which finished self-healing (compare b-4 and c-4 to a-4). Therefore, the self-healing ability of the completely unpolarized light is stronger. This can explain the conclusion in the previous papers [19,20,58,63]: the shorter the coherence length, the better the performance of the self-healing. It is because that the beams with shorter coherence lengths are easier to depolarize and become completely unpolarized. This also explains the above results in this paper, i.e., the enhancement of the twist factor weakens the decrease of the DoP, which makes the beam maintain polarized more distance, so the self-healing ability decreases (in the aspect of intensity distribution).

 figure: Fig. 7.

Fig. 7. The transverse intensity distributions of the completely polarized part and the completely unpolarized part of the PCRPT beam with an obstacle at different propagation distances. The first row shows the whole intensity distributions of the beam. Each column is normalized with the maximum value of the whole intensity in this column.

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In addition to the DoP, the state of polarization (SoP) is another important indicator of polarization properties. The SoP of a partially coherent beam can be inferred from the generalized Stokes parameters (GSP) defined as [64]

$$\begin{aligned} S_{0}(\rho) & =W_{xx}(\rho,\rho)+W_{yy}(\rho,\rho),\\ S_{1}(\rho) & =W_{xx}(\rho,\rho)-W_{yy}(\rho,\rho),\\ S_{2}(\rho) & =W_{xy}(\rho,\rho)+W_{yx}(\rho,\rho),\\ S_{3}(\rho) & ={\rm i}[W_{yx}(\rho,\rho)-W_{xy}(\rho,\rho)], \end{aligned}$$
here $S_{0}(\rho )$ represents the average beam intensity. The polarization ellipse at one certain point of the beam can be obtained from the above parameters, as [17]
$$\begin{aligned} M_{1,2}(\rho) & =\frac{1}{\sqrt{2}}\left[\sqrt{S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}\pm\sqrt{S_{1}^{2}+S_{2}^{2}}\right]^{1/2},\\ \varepsilon & =\frac{M_{2}}{M_{1}}=\sqrt{\frac{\sqrt{S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}-\sqrt{S_{1}^{2}+S_{2}^{2}}}{\sqrt{S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}+\sqrt{S_{1}^{2}+S_{2}^{2}}}},\\ \psi & =\frac{1}{2}\arctan\frac{S_{2}}{S_{1}},\left( -\frac{\pi}{2}\leqslant\psi\leqslant\frac{\pi}{2}\right), \end{aligned}$$
where $M_{1}$ and $M_{2}$ represent the major and minor semiaxes, $\varepsilon$ is the degree of ellipticity, and $\psi$ denotes the orientation angle of the polarization ellipse, respectively.

Figure 8 shows the GSP distributions of the PCRPT beam in the source plane and the focal plane. Part (a) and (b) are the results without the obstacle. Same as the distributions of the intensity and DoP, as long as the twist phase exists, the GSP distributions rotate $90^{\circ }$ clockwise from the source plane to the focal plane. It also can be seen that the distributions of $S_{1}/S_{0}$ are less affected by the obstacle whether the twist phase exists or not. In the focal plane, both c2 and d2 have returned to the state unaffected by the obstacle (compare with a2 and b2). However, the $S_{2}/S_{0}$ cannot eliminate the effect of the obstacle even in the focal plane (see c4 and d4, compare with a4 and b4). In this situation, the result with the twist phase is more closer to the distribution that without the obstacle. In other words, the SoP of the beam with a twist phase is more resistant to the disturbance caused by the obstacle.

 figure: Fig. 8.

Fig. 8. The transverse GSP distributions of the disturbed PCRPT beam in the source plane and the focal plane. As a comparison, part (a) and (b) show the results without the obstacle.

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To explore the effects of the obstacle size and source coherence length on the polarization properties of the beam, the DoP and GSP distributions in the focal plane under the selected parameters are plotted in Figs. 9 and 10. It can be seen from Fig. 9 that the polarization of the beam shows the self-healing effect when the obstacle is small ($\omega _{d}=0.2\rm ~mm$), both the DoP and GSP distributions in the focal plane recover to the undisturbed state. Nevertheless, this weak self-healing effect fails with the increase of the obstacle size. The larger the obstacle size, the more serious distortion of the DoP and GSP distributions. An interesting result appears in Fig. 10: when the source coherence length is large enough ($\delta _{0}=0.4\rm ~mm$), the center of the undisturbed DoP distribution appears a bright spot. The existence of twist factor still improves the anti-disturbance ability of beam polarization when the coherence length increases, we give the result when $\delta _{0}=0.2\rm ~mm$ in Supplement 1 to support that. However, the enhancement of the source coherence length cannot make the SoP self-heal perfectly, see the third row of Fig. 10. The distributions with the obstacle are still slightly distorted compared to the cases without the obstacle.

 figure: Fig. 9.

Fig. 9. The transverse DoP and GSP distributions of the disturbed PCRPT beam in the focal plane with selected values of the obstacle size.

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 figure: Fig. 10.

Fig. 10. The transverse DoP and GSP distributions of the disturbed PCRPT beam in the focal plane with selected values of the source coherence length.

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In addition to the polarization, coherence is another important correlation property of the beams. Wolf believes that polarization and coherence are the same physical phenomenon, i.e., correlations between fluctuations in light beams. Specifically, polarization comes from the fluctuation of the different field components at one point, while coherence arises from the fluctuation of the field between different points [55]. The quantity that describes the beam coherence is the degree of coherence (DoC), and the DoC between two points in one cross-section plane of partially coherent vectorial beams is given as [55]

$$\begin{aligned} \eta(\boldsymbol{\rho_{1}},\boldsymbol{\rho_{2}},z)=\frac{{\rm Tr}~W(\boldsymbol{\rho_{1}},\boldsymbol{\rho_{2}},z)}{\sqrt{{\rm Tr}~W(\boldsymbol{\rho_{1}},\boldsymbol{\rho_{1}},z)}\sqrt{{\rm Tr}~W(\boldsymbol{\rho_{2}},\boldsymbol{\rho_{2}},z)}}. \end{aligned}$$
The DoC is also bound by 0 and 1. The lower bound $|\eta |=0$ means that the beam fields between a pair of points are completely uncorrelated. The upper bound $|\eta |=1$ means that the fields between two points are completely correlated.

Figure 11 shows the $|\eta (\boldsymbol {\rho },\boldsymbol {-\rho })|$ distributions of the PCRPT beam for different propagation distances. The first row is absent of the obstacle, one can see that the Gaussian shape DoC distribution at the source plane gradually shrinks and a dark ring appears on its edge with the propagation of the beam. Same as the DoP, the effect of the obstacle on DoC distribution is weak in the near field but obvious in the far field. The result of (b-4) shows that the central Gaussian shape of the DoC is slightly flatter, and the direction of the flattening is the same as the position of the obstacle, i.e., the $(1\rm ~mm, 1\rm ~mm)$ direction.

 figure: Fig. 11.

Fig. 11. The transverse $|\eta (\boldsymbol {\rho },\boldsymbol {-\rho })|$ distributions of a PCRPT beam with an obstacle at different propagation distances. As a comparison, the first row shows the results without the obstacle.

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Figure 12 shows the $|\eta (\boldsymbol {\rho },\boldsymbol {-\rho })|$ distributions of the PCRPT beam in the focal plane with different twist factors. It can be seen that the existence and increase of the twist factor make the DoC distribution expand whether there exists an obstacle or not. All the DoC distributions change similar to those in (b-4) of Fig. 11 when the obstacle is present. It suggests that this type of change is independent of the twist phase.

 figure: Fig. 12.

Fig. 12. The transverse $|\eta (\boldsymbol {\rho },\boldsymbol {-\rho })|$ distributions of the PCRPT beam with different twist factors in the focal plane. The first row shows the results without the obstacle, and the second row is the corresponding results with an obstacle.

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Finally, the $|\eta (\boldsymbol {\rho },\boldsymbol {-\rho })|$ distributions of the PCRPT beam in the focal plane with selected values of the source coherence length and obstacle size are shown in Fig. 13. It can be concluded that the increase of the source coherence length broadens the DoC distributions, which is similar to the result caused by the increase of the twist factor. The DoC of the beam shows the self-healing effect when the obstacle is small ($\omega _{d}=0.2\rm ~mm$), but this weak effect disappears when the obstacle size is larger. The larger the obstacle size, the more serious distortion of the DoC distributions. The central shape of the DoC becomes more flatter and the dark ring becomes more uneven with a larger obstacle. All the directions of the DoC flattening in Figs. 12 and 13 are the same, so the direction of this flattening is only related to the position of the obstacle.

 figure: Fig. 13.

Fig. 13. The transverse $|\eta (\boldsymbol {\rho },\boldsymbol {-\rho })|$ distributions of the disturbed PCRPT beam in the focal plane with selected values of the source coherence length and obstacle size.

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4. Conclusion

In summary, we discussed the disturbed propagation of the PCRPT beam analytically. The results showed that the PCRPT beam owns the self-healing properties. We focused on the influence of the twist phase on these properties and found that the intensity distribution of the beam can self-heal when the twist factor is not too large. However, an excessive twist factor will make the beam fail to self-heal perfectly. The polarization and coherence properties of the beam in this disturbed propagation were investigated. It is found that the existence of the obstacle accelerates the decrease of the DoP, but the increase of the twist factor leads the DoP to drop slower. Besides, it was shown that the polarization and coherence distribution of the PCRPT beam own a weak self-healing effect when the obstacle is small. Nevertheless, neither the polarization nor coherence distributions at the focal plane can perfectly eliminate the influence of the obstacle with a larger size. The effect of the twist phase on the beam propagation is similar to that of the source coherence length: its increase will cause the beam to maintain polarized a longer distance, i.e., the negative Gaussian distribution of the DoP will shrink and the DoC distribution will broaden. Of course, it also owns the function that the source coherent length does not have: it can make the beam rotate in propagation and its existence improves the anti-disturbance ability of the beam polarization. Our results will be helpful to the fields of optical tweezers, microscopy, optical communication, and so on.

Funding

National Key Research and Development Program of China (6142411203109).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

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Figures (13)
Fig. 1.
Fig. 1. Illustrating the notation.

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Fig. 2.
Fig. 2. The transverse intensity distributions of a PCRPT beam with an obstacle at different propagation distances. As a comparison, the first row shows the results without the obstacle. Each intensity distribution is normalized individually.

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Fig. 3.
Fig. 3. The variation of the similarity degree as a function of propagating distance.

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Fig. 4.
Fig. 4. The transverse intensity distributions of a PCRPT beam at different propagation distances with selected values of the twist factor. First row, $\mu =0.0001/{\rm mm}$; second row, $\mu =0.0005/{\rm mm}$; third row, $\mu =0.001/{\rm mm}$. Each intensity distribution is normalized individually.

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Fig. 5.
Fig. 5. The transverse DoP distributions of the PCRPT beam with an obstacle at different propagation distances. As a comparison, the first row shows the results without the obstacle.

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Fig. 6.
Fig. 6. The transverse DoP distributions of the PCRPT beam with different twist factors in the focal plane. The first row shows the results without the obstacle, and the second row is the corresponding results with an obstacle.

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Fig. 7.
Fig. 7. The transverse intensity distributions of the completely polarized part and the completely unpolarized part of the PCRPT beam with an obstacle at different propagation distances. The first row shows the whole intensity distributions of the beam. Each column is normalized with the maximum value of the whole intensity in this column.

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Fig. 8.
Fig. 8. The transverse GSP distributions of the disturbed PCRPT beam in the source plane and the focal plane. As a comparison, part (a) and (b) show the results without the obstacle.

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Fig. 9.
Fig. 9. The transverse DoP and GSP distributions of the disturbed PCRPT beam in the focal plane with selected values of the obstacle size.

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Fig. 10.
Fig. 10. The transverse DoP and GSP distributions of the disturbed PCRPT beam in the focal plane with selected values of the source coherence length.

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Fig. 11.
Fig. 11. The transverse $|\eta (\boldsymbol {\rho },\boldsymbol {-\rho })|$ distributions of a PCRPT beam with an obstacle at different propagation distances. As a comparison, the first row shows the results without the obstacle.

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Fig. 12.
Fig. 12. The transverse $|\eta (\boldsymbol {\rho },\boldsymbol {-\rho })|$ distributions of the PCRPT beam with different twist factors in the focal plane. The first row shows the results without the obstacle, and the second row is the corresponding results with an obstacle.

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Fig. 13.
Fig. 13. The transverse $|\eta (\boldsymbol {\rho },\boldsymbol {-\rho })|$ distributions of the disturbed PCRPT beam in the focal plane with selected values of the source coherence length and obstacle size.

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Equations (15)

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W α β = [ W x x W x y W y x W y y ] ,
W α β ( r 1 , r 2 ) = ⟨ E α ∗ ( r 1 ) E β ( r 2 ) ⟩ , α , β = x , y
W α β ( r 1 , r 2 , 0 ) = α 1 β 2 ω 0 2 exp ⁡ ( − x 1 2 + y 1 2 + x 2 2 + y 2 2 ω 0 2 ) × exp ⁡ [ − ( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 2 δ 0 2 ] × exp ⁡ ( − i k μ x 1 y 2 + i k μ x 2 y 1 ) , α , β = x , y
W α β ( ρ 1 , ρ 2 , z ) = 1 ( λ B ) 2 ∫ ∫ d 2 r 1 d 2 r 2 × T ∗ ( r 1 ) T ( r 2 ) W α β ( r 1 , r 2 , 0 ) × exp ⁡ { − i k 2 B [ A ( r 1 2 − r 2 2 ) − 2 ( r 1 ⋅ ρ 1 − r 2 ⋅ ρ 2 ) + D ( ρ 1 2 − ρ 2 2 ) ] } ,
T ( r i ) = 1 − exp ⁡ [ − ( r i − r 0 ) 2 ω d 2 ] ,
⟨ I ( ρ , z ) ⟩ = W x x ( ρ , ρ , z ) + W y y ( ρ , ρ , z ) .
γ ( z ) = ∫ ∫ ⟨ I ( ρ , z ) ⟩ ⟨ I ′ ( ρ , z ) ⟩ d 2 ρ ∫ ∫ ⟨ I ( ρ , z ) ⟩ 2 d 2 ρ ∫ ∫ ⟨ I ′ ( ρ , z ) ⟩ 2 d 2 ρ ,
P ( ρ ) = 1 − 4   D e t   W ( ρ , ρ ) [ T r   W ( ρ , ρ ) ] 2 ,
W ( ρ , ρ , z ) = W U ( ρ , ρ , z ) + W P ( ρ , ρ , z ) ,
W U ( ρ , ρ , z ) = [ A 1 ( ρ , ρ , z ) 0 0 A 1 ( ρ , ρ , z ) ] , W P ( ρ , ρ , z ) = [ B 1 ( ρ , ρ , z ) D 1 ( ρ , ρ , z ) D 1 ∗ ( ρ , ρ , z ) C 1 ( ρ , ρ , z ) ] ,
A 1 = 1 2 [ W x x + W y y − ( W x x − W y y ) 2 + 4 | W x y | 2 ] , B 1 = 1 2 [ W x x − W y y + ( W x x − W y y ) 2 + 4 | W x y | 2 ] , C 1 = 1 2 [ W y y − W x x + ( W x x − W y y ) 2 + 4 | W x y | 2 ] , D 1 = W x y .
⟨ I P ( ρ , z ) ⟩ = B 1 ( ρ , ρ , z ) + C 1 ( ρ , ρ , z ) , ⟨ I U ( ρ , z ) ⟩ = A 1 ( ρ , ρ , z ) + A 1 ( ρ , ρ , z ) .
S 0 ( ρ ) = W x x ( ρ , ρ ) + W y y ( ρ , ρ ) , S 1 ( ρ ) = W x x ( ρ , ρ ) − W y y ( ρ , ρ ) , S 2 ( ρ ) = W x y ( ρ , ρ ) + W y x ( ρ , ρ ) , S 3 ( ρ ) = i [ W y x ( ρ , ρ ) − W x y ( ρ , ρ ) ] ,
M 1 , 2 ( ρ ) = 1 2 [ S 1 2 + S 2 2 + S 3 2 ± S 1 2 + S 2 2 ] 1 / 2 , ε = M 2 M 1 = S 1 2 + S 2 2 + S 3 2 − S 1 2 + S 2 2 S 1 2 + S 2 2 + S 3 2 + S 1 2 + S 2 2 , ψ = 1 2 arctan ⁡ S 2 S 1 , ( − π 2 ⩽ ψ ⩽ π 2 ) ,
η ( ρ 1 , ρ 2 , z ) = T r   W ( ρ 1 , ρ 2 , z ) T r   W ( ρ 1 , ρ 1 , z ) T r   W ( ρ 2 , ρ 2 , z ) .
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