Abstract
We establish a general form of the cross-spectral density of statistical sources that generate vortex preserving partially coherent beams on propagation through any linear ABCD optical system. We illustrate our results by introducing a class of partially coherent vortex beams with a closed form cross-spectral density at the source and demonstrating the beam vortex structure preservation on free space propagation and imaging by a thin lens. We also show the capacity of such vortex preserving beams of any state of spatial coherence to trap nanoparticles with the refractive index smaller than that of a surrounding medium.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The concept of vortices was first introduced into optics by Coullet in 1989 [1] by analogy with the naturally encountered vortices, such as tornados and ocean whirlpools. This work was followed by a rigorous introduction of vortex-carrying optical beams with helicoidal wavefronts by Allen and co-workers [2]. The introduced vortex beams were shown in [2] to carry orbital angular momenta of $m\hbar$ per photon where $m$ is an integer known as a topological charge of the vortex. This pioneering work has established a link between an optical vortex on the one hand and the beam orbital angular momentum on the other, thereby having triggered a flurry of research on vortex beams [3,4]. A multitude of noteworthy features of optical vortex beams have been discovered to date. The latter include the phase singularity behavior of vortex beams; their ability to transfer the orbital angular momentum to neutral particles, thereby enabling trapping and tweezing the said particles, as well as an angular Doppler effect manifestation with vortex endowed beams [5,6]. These remarkable characteristics of optical vortex beams serve as a basis for their widespread applications to high-security and high-capacity quantum and classical optical communications, quantum storage, nanostructure processing, and superresolution imaging [7–9], to mention but a few relevant examples.
On the other hand, as the source coherence is reduced, fully spatially coherent vortex beams can be converted into partially coherent vortex beams (PCVBs) [10,11]. In general, partially coherent beams have been demonstrated to better resist ambient turbulence fluctuations than do their fully coherent counterparts [12], and are useful for diffractive imaging [13]. In this connection, PCVBs have received much attention, in part due to their interesting propagation properties such as self-shaping, self-reconstruction, and reduced turbulence-induced degradation and scintillations [10,11]. However, traditional partially coherent vortices, embedded within Schell-model beams [14–16], lose their vortex structure on free space propagation, especially in the low-coherence limit where their structural stability to random environment fluctuations is the greatest. Hence a natural question arises: Is it possible to combine the advantages of low spatial coherence, and hence the structural stability to ambient turbulence, and optical vortex structure by generating PCVBs whose vortex structure is immune to beam evolution through a linear ABCD system, including free space, at any spatial coherence level? Several groups have proposed particular classes of PCVBs satisfying this requirement [17–19]. To the best of our knowledge, however, no general approach to designing vortex preserving partially coherent beams (PCBs) has been put forward to date. Nor have any applications of such beams been explicitly discussed.
In this work, we establish, for the first time to our knowledge, a general form of the cross-spectral density of PCVBs that maintain their vortex structure on propagation though any linear ABCD optical system. In particular, we demonstrate how any such beam can be constructed using coherent pseudo-modes. This, in general pseudo-mode, decomposition offers a direct path to the laboratory realization of such beams [18]. We also introduce a class of vortex preserving partially coherent beams described by the cross-spectral density in a closed form. We then calculate the radiation forces due to the optical fields of such beams that enable nanoparticle trapping in the beam cores.
2. General form of vortex preserving partially coherent beams
In the space-frequency domain, the cross-spectral density (CSD) is used to characterize the second-order statistical properties of a PCB. To establish a generic form of vortex preserving PCBs, we introduce the CSD of a random ensemble $\{ U({\boldsymbol \rho },\omega )\}$ at the source as
We have now established the general form of the CSD, given by Eqs. (6), (9) and (10), of a statistical source generating vortex endowed PCBs that maintain their vortex structure on propagation through a generic ABCD optical system. Next, we proceed to introduce a class of such beams with a closed form CSD at the source to illustrate our general results.
3. New class of vortex preserving statistical beams
Consider a class of radially sinc-correlated vortex (RSCV) beams with the cross-spectral density in a closed form as
In the remainder of the paper, we adopt the modified free-space geometry whereby the beam, focused by a thin lens with focal length $f$=600mm, propagates toward a receiver plane. The distances from the lens to the source and receiver planes are $f$ and $z$, respectively. For this optical system, the corresponding elements of the transfer ABCD matrix can be expressed as
Applying Eqs. (6), (9), (14) and (15), we can obtain the intensity distributions $I\left ( {\textbf {r},z} \right ) = W\left ( {\textbf {r},\textbf {r},z} \right )$ of RSCV beams at the different propagation distances from the source plane. In Fig. 1, we exhibit intensity evolution of RSCV beams with the carrier wavelength of 632.8 nm and different topological charges $m$ at several propagation distances from the source, $z=0.6f$, $z=0.8f$ and $z=f$. We infer from the figure that the hollow dark cores of RSCV beams are maintained on the beam propagation all the way to the focal plane of the lens located in the far-zone of the source. It follows that the RSCV beam vortex structure remains intact as predicted by Eqs. (6), (9), and (10). Importantly, our conclusions can be extended to RSCV beams with any topological charge $m$ as is illustrated by different l rows of Fig. 1.To further illustrate the vortex structure preservation of RSCV beams, we study the focal plane ($z=f$) intensity distributions of such beams with different topological charges $m$ generated by sources of different states of coherence that are characterized by the source coherence width ${\sigma _c}$. We display the relevant numerical results in Fig. 2. To show the effect of the source coherence width ${\sigma _c}$ on the vortex structure, we plot the corresponding cross lines ${I\left ( {x,y = 0} \right )/{I_{\max }}\left ( {x,y = 0} \right )}$ of normalized RSCV beam intensity distributions in three cases: ${\sigma _c} = 1\textrm {mm}$ (black), ${\sigma _c} = 0.5\textrm {mm}$ (red), and ${\sigma _c} = 0.1\textrm {mm}$ (green) as well. We stress that our results indicate the vortex structure preservation in the focal plane for the RSCV beams with any topological charges $m$ even in the low coherence limit, $\sigma _c\ll \sigma _I$. We can see from the right l column in the figure that the source coherence has little effect on the central portion of the beams for all topological charges $m$, even though it strongly influences the power distribution in the beam tails. Thus, the vortex structure resilience on propagation of nearly incoherent RSCV beams through any linear ABCD system makes such beams quite different from typical PCVBs [10].
As is well known, whenever vortex beams are scattered by a small obstacle, a neutral dielectric sphere, for example, the momentum of the beam and the sphere are exchanged, giving rise to radiation and scattering forces pushing the sphere to a stable equilibrium position within the beam field. This phenomenon lies at the heart of optical trapping of small particles [3]. In the following section, we discuss the trapping forces exerted by sinc-correlated vortex beams onto a small spherical particle.
4. Trapping dielectric nanoparticles with radially sinc-correlated vortex beams
We now consider a spherical dielectric nanoparticle of radius $a$ that we assume to be much smaller than the carrier wavelength of the beam, $a \ll \lambda$. In this approximation, we can adopt the Rayleigh scattering theory [36] to analyze the radiation forces experienced by the nanoparticle in the field of an RSCV beam. A detailed analysis [25,37] shows that the scattering and gradient forces are the leading forces experienced by the particle in the radiation filed, whereas the gravitational, buoyancy, drag, and Brownian forces are negligible in comparison. Hence, we only take the scattering and radiation forces into consideration in this work.
We assume the RSCV beam, focused by a thin lens with the focal length $f$, to propagate toward a receiver plane. Further, the distances from the lens to the source plane and to the receiver plane are assumed to be $l$ and $z$, respectively. The corresponding ABCD transfer matrix can then be expressed as
The RSCV beam intensity profiles have a circular symmetry in any transverse plane, resulting in the circularly symmetric radiation forces acting on a trapped nanoparticle. In Fig. 3, we display the density plots of the $x$-component of the gradient force $F_x(x,y=0,z)={\textbf {e}}_x\cdot \textbf {F}_{grad}( x,y=0,z)$ and the magnitude of the scattering force $F_{scat}(x,y=0,z)$ in the vicinity of the focal plane of the focusing lens. Further, we display in Fig. 4 the cross lines of $F_x(x,y=0,z=f)$ and $F_{scat}(x,y=0,z=f)$ of the RSCV beam in the focal plane. We consider three cases here: highly coherent RSCV source (${\sigma _c} = 20\textrm {mm}$), moderately coherent RSCV source (${\sigma _c} = 8\textrm {mm}$), and nearly incoherent RSCV source (${\sigma _c} = 2\textrm {mm}$). As the gradient force is nearly three orders of magnitude greater than the scattering force, the latter is virtually negligible in practice. With the coherence level of the beam decreasing from very high to very low, the gradient force changes rather insignificantly. Note that the positive value of $F_x$ corresponds to the $x$-component of the gradient force in the positive $x$ direction. It can be inferred from Fig. 4 that there exists only one stable equilibrium position, labelled by a solid black dot in Fig. 4, for any trapped particle in a transverse plane of any RSCV beam in the neighborhood of the focal plane. The said equilibrium positions are also denoted by dashed white lines in Fig. 3.
Our theoretical and numerical results imply that we can indeed combine the advantages of low spatial coherence of the source and a stable vortex structure of the generated beam to ensnare a small particle by an RSCV beam that will be relatively stable to inevitable fluctuations in a surrounding medium. Thus, RSCV beams of any spatial coherence level can trap nanoparticles, made of a material with the refractive index smaller than the one of the surrounding medium, in any transverse plane in the vicinity of the focal plane. Finally, we note that the magnitudes of the gradient and scattering forces drastically decrease for the vortex beams with topological charges $m>1$ in agreement with previously published results [37].
5. Summary
In this work, we have established, for the first time to our knowledge, a general form of the cross-spectral density function of partially coherent sources generating vortex beams that maintain their vortices on propagation through any linear ABCD optical system, including free space and a thin optical lens. To illustrate our general results, we introduced a new class of partially coherent vortex beams with a closed form cross-spectral density function at the source. We demonstrated through numerical simulations that each member of the new class of vortex endowed beams maintains its vortex structure on propagation through an ABCD system regardless of the source state of spatial coherence. As well we showed that the vortex preserving partially coherent beams can be used in particle trapping applications by determining the radiation and scattering forces exerted by any such beam onto a small Rayleigh particle. Instructively, the vortex structure preservation of the new beams implies that novel beams of a very low state of coherence, which are highly stable to intrinsic environment fluctuations, can be employed to trap Rayleigh nanoparticles with the refractive index smaller than that of a surrounding medium.
Funding
National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11525418, 11874046, 11904247, 11947239, 11974218, 91750201); Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-05497); Innovation group of Jinan (2018GXRC010); China Postdoctoral Science Foundation (2019M662424).
Disclosures
The authors declare no conflicts of interest.
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