Twisted sinc-correlation Schell-model beams

Yujie Zhou; Weiting Zhu; Weiting Zhu; Weiting Zhu; Daomu Zhao; Daomu Zhao

doi:10.1364/OE.450254

1. Introduction

As we all know, partially coherent light not only has the advantages of high directivity and energy concentration [1–3] but also is insensitive to medium disturbances [4]. Therefore, such a light beam has great research value. An earlier partially coherent Gaussian Schell model (GSM) beam was proposed by Collett and Wolf, in 1978 [5]. Subsequently, theoretical and experimental researches on this great directional beam were studied in detail [6–13]. And beams with various shapes were also continuously raised, such as flat-topped beams [14,15], hollow beams [16,17], vortex beams [18–20], and Airy beams [21], etc. These beams have great potential applications in laser focusing, particle capture, nuclear fusion, and biomedicine. Meanwhile, a theory to devise bona fide spatial correlation function [22] of the light field was established in 2007. This provides a theoretical basis to explore various correlation functions with some unexpected properties. The unified theory of coherent polarization [23,24] closely links coherence and polarization of light field together. This offers a new approach for an in-depth understanding of optical statistical properties. Spatial modeling of the phase [25,26] of complex degree of coherence (CDC) is often regarded as an effective method to realize the light field adjustment.

Partially coherent beams endowed with various phases have been studied extensively. For example, linear phase induces GSM beam presenting off-axis effect upon propagation [27,28]; source beam with cubic phase would perform an airy-like distribution in the far field [21,26]; high-order indivisible phase can make the beam show self-focusing effect [29], and the twist phase that causes the beam to rotate [30]. Among them, a twisted beam [31–36] has been widely investigated due to its broad application potential [37]. And twisted beams have also been experimentally verified [38–40].

In this paper, we would introduce a new class of twisted sinc-correlation partially coherent sources to generate rotating profiles. Properties of such beam propagating in free space are studied in detail. We provide a few typical examples to specifically study the twist effect of light intensity and degree of coherence (DOC). In addition, adjustment of twist factor, beam width, and coherence length on intensity distribution are also explored.

2. TSCSM sources

Suppose a random statistically stationary scale source, locating at plane $z=0$ . The second-order correlation property of this light field, at angular frequency $\varpi$, can be described in terms of the cross spectral density (CSD) function as:

(1)$${{W}^{0}}(\mathbf{{{\rho }_{1}}},\mathbf{{{\rho }_{2}}},\varpi )={<}{{E}^{*}}(\mathbf{{{\rho }_{1}}},\varpi )E(\mathbf{{{\rho }_{2}}},\varpi )>,$$

where $\mathbf {\rho _{1}}=(x_{1}\text {, }y_{1})$ and $\mathbf {\rho _{2}}=(x_{2}\text {, }y_{2})$ are a pair of arbitrary points at transverse position vectors; $E(\mathbf {\rho } ,\varpi )$ denotes a component of an electric vector. The angular brackets present the ensemble average and the asterisk means the complex conjugate. For brevity, we omit the dependence on angular frequency. The genuine CSD function ${{W}^{0}}(\mathbf {{{\rho }_{1}}},\mathbf {{{\rho }_{2}}})$, which satisfied the sufficient non-negative definiteness, can be given as a superposition integral as [22]:

(2)$${{W}^{0}}(\mathbf{{{\rho }_{1}}},\mathbf{{{\rho }_{2}}})\text{=}\iint{p(\mathbf{\nu} ){{H}_{0}}^{*}}(\mathbf{{{\rho }_{1}}},\mathbf{\nu} ){{H}_{0}}(\mathbf{{{\rho }_{2}}},\mathbf{\nu} ){{d}^{2}}\mathbf{\nu} .$$

Here ${{H}_{0}}(\mathbf {\rho } ,\mathbf {\nu } )$ is an arbitrary kernel with a non-negative weight function $p(\mathbf {\nu } )$. $\mathbf {\nu }=({{\nu }_{x}},{{\nu }_{y}})$ is a 2D vector in the Fourier plane. In order to generate the sources with rotating intensity distribution, we consider the kernel as [32]:

(3)$$H(\mathbf{\rho} ,\mathbf{\nu})=\exp (-\frac{{{x}^{2}}}{2{{\sigma }_{x}}^{2}}-\frac{{{y}^{2}}}{2{{\sigma }_{y}}^{2}})\exp [-(uy+ix){{\nu}_{x}}+(ux-iy){{\nu}_{y}}],$$

here ${{\sigma }_{x}}$ and ${{\sigma }_{y}}$ are beam width and $u$ denotes the twist factor. Then, weight function is given as [41,42]:

(4)$$p(\mathbf{\nu})=\frac{{{\delta }_{x}}{{\delta }_{y}}}{{{\pi }^{2}}}\mathrm{rect}(\frac{{{\delta }_{x}}{{\nu}_{x}}}{\pi })\mathrm{rect}(\frac{{{\delta }_{y}}{{\nu}_{y}}}{\pi }),$$

where ${{\delta }_{x}}$ and ${{\delta }_{y}}$ are coherence length.

Upon substituting from Eqs. (3) and (4) into Eq. (2), we can easily obtain the expression:

(5)$$\begin{aligned}{{W}^{0}}(\mathbf{{{\rho }_{1}}},\mathbf{{{\rho }_{{2}}}})&=\exp (-\frac{{{x}_{1}}^{2}+{{x}_{2}}^{2}}{2{{\sigma }_{x}}^{2}}-\frac{{{y}_{1}}^{2}+{{y}_{2}}^{2}}{2{{\sigma }_{y}}^{2}})\\ &\mathrm{sinc}[\frac{({{x}_{2}}-{{x}_{1}})-iu({{y}_{2}}+{{y}_{1}})}{{}^{{{\delta }_{x}}}/{}_{2}}]\mathrm{sinc}[\frac{({{y}_{2}}-{{y}_{1}})+iu({{x}_{1}}+{{x}_{2}})}{{}^{{{\delta }_{y}}}/{}_{2}}]. \end{aligned}$$

Partially coherent sources defined by the above equation can be called twisted sinc-correlation Schell-model (TSCSM) beams.

To have an in-depth view of this TSCSM beam, let us take a closer look at its orbital angular momentum (OAM). OAM density along the optical axis can be given as [43]:

(6)$${{M}_{\mathrm{orbit}}}(\mathbf{\rho} )={-}\frac{{{\varepsilon }_{0}}}{k}\operatorname{Im}{{\{{{y}_{1}}{{\partial }_{{{x}_{2}}}}W(\mathbf{{{\rho }_{{1}}}},\mathbf{{{\rho }_{2}}})-{{x}_{1}}{{\partial }_{{{y}_{2}}}}W(\mathbf{{{\rho }_{{1}}}},\mathbf{{{\rho }_{2}}})\}}_{\mathbf{{{\rho }_{{1}}}}=\mathbf{{{\rho }_{2}}}}},$$

with ${{\partial }_{j}}$ representing the partial derivation, ${{\varepsilon }_{0}}$ is the dielectric constant in vacuum and $\operatorname {Im}$ denotes the imaginary part. On substituting from Eq. (5) into Eq. (6), it yields

(7)$$\begin{aligned}{{M}_{\mathrm{orbit}}}(\mathbf{\rho} )&=\frac{{{\varepsilon }_{0}}}{2ku}\exp (-\frac{{{x}^{2}}}{{{\sigma }_{x}}^{2}}-\frac{{{y}^{2}}}{{{\sigma }_{y}}^{2}})\\ & \quad \{\mathrm{sinc}(\frac{4iux}{{{\delta }_{y}}})[\cos (\frac{4iu\pi y}{{{\delta }_{x}}})-\mathrm{sinc}(\frac{4iu\pi y}{{{\delta }_{x}}})]\\ &\quad +\mathrm{sinc}(\frac{4iuy}{{{\delta }_{x}}})[\cos (\frac{4iu\pi x}{{{\delta }_{y}}})-\mathrm{sinc}(\frac{4iu\pi x}{{{\delta }_{y}}})]\}. \end{aligned}$$

And a normalized OAM ${{m}_\mathrm {orbit}}$- roughly the angular momentum density per photon is:

(8)$${{m}_{\mathrm{orbit}}}=\frac{\hbar }{2u}[2-\frac{4\pi uy}{{{\delta }_{x}}}\coth (\frac{4\pi uy}{{{\delta }_{x}}})-\frac{4\pi ux}{{{\delta }_{y}}}\coth (\frac{4\pi ux}{{{\delta }_{y}}})].$$

When the twist factor is 0, particles do not carry OAM.(This can be calculated by using the l’Hôpital’s rule). At this time, spectral density no longer presents the twist effect upon propagation.

3. Free-space propagation of TSCSM beams

We now consider that the beam is transmitted through a thin lens, which locates at plane $z=0$, and propagates into a positive half-space $z>0$. The focal length is $400\mathrm {mm}$. And the wavelength is fixed at $632.8\mathrm {nm}$. Transfer matrix in such optical system is:

(9)$$\left[ \begin{matrix} A & B \\ C & D \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & z \\ 0 & 1 \\ \end{matrix} \right]\left[ \begin{matrix} 1 & 0 \\ -{}^{1}/{}_{f} & 1 \\ \end{matrix} \right]=\left[ \begin{matrix} 1-{}^{z}/{}_{f} & z \\ -{}^{1}/{}_{f} & 1 \\ \end{matrix} \right].$$

The propagation of paraxial beam through the ABCD optical system can be handled by the Collins formula [44]:

(10)$$W(\mathbf{{{r}_{1}}},\mathbf{{{r}_{2}}},z)=\frac{{{k}^{2}}}{4{{\pi }^{2}}{{B}^{2}}}\iint{p(\mathbf{\nu}){{H}^{*}}(\mathbf{{{r}_{1}}},\mathbf{\nu},z)H(\mathbf{{{r}_{2}}},\mathbf{\nu},z)d\nu_{x}d\nu_{y}},$$

where

(11)$$H(\mathbf{r},\mathbf{\nu},z)=\iint{H(\mathbf{\rho},\mathbf{\nu})}\exp \{\frac{ik}{2B}[A({{x}^{2}}+{{y}^{2}})-2(x{{x}^{'}}+y{{y}^{'}})+D({{x}^{'2}}+{{y}^{'2}})]\}{{d}^{2}}\mathbf{\rho} .$$

On substituting from Eqs. (3), (4) and (9) into Eq. (10), CSD of this novel focused beam at any propagation distance can be given as:

(12)$$\begin{aligned}W(\mathbf{{{r}_{1}}},\mathbf{{{r}_{2}}},z)&=\frac{{{k}^{2}}}{4{{B}^{2}}{{E}^{0.5}}}\exp [\frac{ikD({{x}^{'2}_{2}}+{{y}^{'2}_{2}}-{{x}^{'2}_{1}}-{{y}^{'2}_{1}})}{2B}]\\ & \exp [-\frac{{{k}^{2}}}{4{{B}^{2}}}(\frac{{x}^{'2}_{2}}{{{\varepsilon }_{x2}}}+\frac{{y}^{'2}_{2}}{{{\varepsilon }_{y2}}}+\frac{{x}^{'2}_{1}}{{{\varepsilon }_{x1}}}+\frac{{y}^{'2}_{1}}{{{\varepsilon }_{y1}}})]\iint{p({{\nu}_{x}},{{\nu}_{y}})F({{\nu}_{x}}},{{\nu}_{y}})d{{\nu}_{x}}d{{\nu}_{y}}.\\ \end{aligned}$$

With

(13)$$\begin{aligned}& {{\varepsilon }_{x1}}={{\varepsilon }_{x2}}^{*}=\frac{1}{2{{\sigma }_{x}}^{2}}+\frac{ikA}{2B};{{\varepsilon }_{y1}}={{\varepsilon }_{y2}}^{*}=\frac{1}{2{{\sigma }_{y}}^{2}}+\frac{ikA}{2B};\\ & E\text{=}{{\varepsilon }_{x1}}{{\varepsilon }_{x2}}{{\varepsilon }_{y1}}{{\varepsilon }_{y2}};\\ & {{\alpha }_{x}}=\frac{1}{4}(\frac{1}{{{\varepsilon }_{x1}}}+\frac{1}{{{\varepsilon }_{x2}}})-\frac{{{u}^{2}}}{4}(\frac{1}{{{\varepsilon }_{y1}}}+\frac{1}{{{\varepsilon }_{y2}}});\\ & {{\alpha }_{y}}=\frac{1}{4}(\frac{1}{{{\varepsilon }_{y1}}}+\frac{1}{{{\varepsilon }_{y2}}})-\frac{{{u}^{2}}}{4}(\frac{1}{{{\varepsilon }_{x1}}}+\frac{1}{{{\varepsilon }_{x2}}});\\ & {{\beta }_{x}}={-}\frac{k}{2B}(\frac{{{x}^{'}}_{1}}{{{\varepsilon }_{x1}}}+\frac{{{x}^{'}}_{2}}{{{\varepsilon }_{x2}}})+\frac{iku}{2B}(\frac{{{y}^{'}}_{2}}{{{\varepsilon }_{y2}}}-\frac{{{y}^{'}}_{1}}{{{\varepsilon }_{y1}}});\\ & {{\beta }_{y}}={-}\frac{k}{2B}(\frac{{{y}^{'}}_{1}}{{{\varepsilon }_{y1}}}+\frac{{{y}^{'}}_{2}}{{{\varepsilon }_{y2}}})+\frac{iku}{2B}(\frac{{{x}^{'}}_{1}}{{{\varepsilon }_{x1}}}-\frac{{{x}^{'}}_{2}}{{{\varepsilon }_{x2}}});\\ & \varepsilon =\frac{iu}{2}(\frac{1}{{{\varepsilon }_{x1}}}-\frac{1}{{{\varepsilon }_{x2}}})+\frac{iu}{2}(\frac{1}{{{\varepsilon }_{y2}}}-\frac{1}{{{\varepsilon }_{y1}}});\\ & F({{\nu}_{x}},{{\nu}_{y}})=\exp (-{{\alpha }_{x}}{{\nu}_{x}}^{2}-{{\alpha }_{y}}{{\nu}_{y}}^{2}+{{\beta }_{x}}{{\nu}_{x}}+{{\beta }_{y}}{{\nu}_{y}}+\varepsilon {{\nu}_{x}}{{\nu}_{y}}). \end{aligned}$$

Spectral density $S$ is equal to CSD, $W$, when $\mathbf {{r}_{1}}=\mathbf {{r}_{2}}$. And DOC has the form as:

(14)$$\mu (\mathbf{{{r}_{1}}},\mathbf{{{r}_{2}}},z)=\frac{W(\mathbf{{{r}_{1}}},\mathbf{{{r}_{2}}},z)}{\sqrt{S(\mathbf{{{r}_{1}}},\mathbf{{{r}_{1}}},z)}\sqrt{S(\mathbf{{{r}_{2}}},\mathbf{{{r}_{2}}},z)}}.$$

Applying Eqs. (12), (13) and (14), we can study the propagation through a focused optical system numerically in a convenient way.

4. Numerical results and discussions

4.1 Example of isotropic TSCSM beams

A stack of images taken in Fig. 1 illustrates the typical evolution of the transverse spectral density of a TSCSM beam upon propagation. There are two main effects on spectral density caused by twist factor. First, twist factor induced light intensity rotating to 90 degrees, during propagation. When $u>0$, light spot would rotate clockwise; otherwise, it would rotate anticlockwise. Secondly, twist factor can also adjust the intensity distribution. For sinc Schell-model beams (corresponding to the case where $u=0$), the Gaussian profile of source field would gradually transform into a flat profile with increasing propagation distance [39]. While light intensity in source plane with a larger value of twist factor, as shown in Fig. 1, is represented as a $2\times 2$ isotropic array. Upon propagation, spectral density maintains the rotational invariance. This novel beam does not possess a simple twist phase, yet has a twist associated with it.

Fig. 1. Average intensity generated by focused TSCSM beams at several distances in free space and setting ${{\sigma }_{x}}={{\sigma }_{y}}=2mm,{{\delta }_{x}}={{\delta }_{y}}=2mm;u=0.9$.

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Figure 2 presents the modulus of DOC of the field generated by the same source as in Fig. 1, with $x_{d}=x_{2}-x_{1},y_{d}=y_{2}-y_{1}$.DOC also exhibits a twist effect during propagation. With a positive value of twist factor, DOC rotates clockwise. Due to the disturbance of the medium, side lobes around the main part disappear gradually. At the same time, the main part of DOC would evolve from rectangular distribution to circle distribution. Then the twist effect could not be observed.

Fig. 2. Modulus of DOC of focused TSCSM beams at several propagation distances with parameters as in Fig. 1.

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4.2 Examples of anisotropic TSCSM beams

In this part, we first illustrate the statistical properties of anisotropic TSCSM beam with parameters ${}^{{{\sigma }_{x}}}/{}_{{{\sigma }_{y}}}={}^{{{\delta }_{x}}}/{}_{{{\delta }_{y}}}$. Transverse spectral density at several propagation distance has been shown in Fig. 3. It is shown that, spectral density only splits completely along $x$ direction in source plane, and divides into two parts. Comparing Fig. 1(a) and Fig. 1(h), one can see that, element of the array also shown its own tendency to split. During propagation, light intensity rotates counterclockwise to 90 degree.

Fig. 3. Average intensity generated by focused TSCSM beams at several distances in free space and setting:${{\sigma }_{x}}=1mm,{{\sigma }_{y}}=0.5mm,{{\delta }_{x}}=1mm,{{\delta }_{y}}=0.5mm;u=\text {-}0.9$.

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Then, behavior of DOC has been illustrated in Fig. 4. Results show that DOC rotates counterclockwise around the beam center upon Propagation. This is consistent with the direction of rotation of light intensity.

Fig. 4. Modulus of DOC of focused TSCSM beams at several propagation distances with parameters as in Fig. 3.

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Another case of source beam has been analyzed as following. We chose parameters with ${}^{{{\sigma }_{x}}}/{}_{{{\sigma }_{y}}}={}^{{{\delta }_{y}}}/{}_{{{\delta }_{x}}}$. Similar to Fig. 1, spectral density in Fig. 5 exhibits a rotating Gaussian array upon propagation. And Fig. 6 presents the transverse DOC of source beams in various propagating distances. Apparently, DOC performs a non-unidirectional rotation upon propagation. It is shown that DOC first rotates a small angle clockwise. After that, it would rotate counterclockwise. Compared with that in source plane, DOC would finally rotate to 90 degrees.

Fig. 5. Average intensity generated by focused TSCSM beams at several distances in free space and setting: ${{\sigma }_{x}}=1mm,{{\sigma }_{y}}=0.3mm,{{\delta }_{x}}=0.3mm,{{\delta }_{y}}=1mm;u=\text {-}0.9$.

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Fig. 6. Modulus of DOC of focused TSCSM beams at several propagation distances with parameters as in Fig. 5.

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By choosing an appropriate kernel function and nonnegative weight function, we construct this new kind of twisted beam. Due to the regulation of twist factor, light intensity presents a splitting effect. It is worth noting that transverse spectral density rotates as a whole around the light field center, instead of each lobe rotating around its respective lobe center.

4.3 Analysis of splitting effect of TSCSM beams

In this section, we conduct a detailed analysis of this splitting effect of intensity distribution.

As shown in Fig. 7, the effect of twist factor, beam width, and coherent length on the spectral density in source field and far field, respectively, are given. For simplicity, we choose the $x=0$ plane and always set $\sigma _{x}=\sigma _{y}$, $\delta _{x}=\delta _{y}$. For that case where $u=0$, spectral density presents a Gaussian profile in source plane (seen in Fig. 7(a1)), and a flat profile in far-field plane (seen in Fig. 7(a2)), respectively. With parameter u increasing, spectral density in central area would become weaker, even to zero. At this time, the light spot splits into two parts along the y-direction. Compared with the intensity distributions in source field and far field, it is also found that splitting effect of spectral density becomes stronger, with increasing propagation distance. Besides, the greater the value of beam width, the stronger the intensity attenuation in the central area of the light spot. And spectral density in central area would decline rapidly with a smaller value of coherence length.

Fig. 7. Illustrating the spectral density of focused TSCSM beams in source field (row 1 ) and far field (row 2) with parameters: ${{\sigma }_{x}}\text {=}{{\sigma }_{y}}=1mm,{{\delta }_{x}}={{\delta }_{y}}=1mm$ (column 1);${{\delta }_{x}}={{\delta }_{y}}=1mm,u=0.3$(column 2); ${{\sigma }_{x}}\text {=}{{\sigma }_{y}}=1mm,u=0.3$ (column 3).

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5. Conclusion

In summary, we have introduced a typical TSCSM beam. Statistical properties of such isotropic and anisotropic TSCSM beams propagating in free space have been well studied. Spectral density would rotate to 90 degrees upon propagation, for both isotropic and anisotropic light sources. Besides, with a positive value of twist factor, light spot would rotate clockwise; otherwise, it rotates counterclockwise. However, the twist effect of DOC with different parameters setting, would behavior variously. DOC could rotate to 90 degrees, and its rotation direction is the same as that of the spectral density. Through parameters adjustment, a non-unidirectional twist of DOC during propagation can also be achieved. In addition, twist factor causes attenuation of the light intensity in central area. Particularly, spectral density in central area would attenuate more, as the coherence length and twist factor increase, and the beam width decreases. Our result might be beneficial for communication, imaging , and particle capture.

Funding

National Natural Science Foundation of China (12174338, 11874321); Key Laboratory of Optical Field Manipulation of Zhejiang Province (ZJOFM2020004).

Disclosures

The authors declare no conflict of interest.

Data availability

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

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Twisted sinc-correlation Schell-model beams

Abstract

1. Introduction

2. TSCSM sources

3. Free-space propagation of TSCSM beams

4. Numerical results and discussions

4.1 Example of isotropic TSCSM beams

4.2 Examples of anisotropic TSCSM beams

4.3 Analysis of splitting effect of TSCSM beams

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (14)

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