## Abstract

In this work, we demonstrate how to generate dark and antidark beams—diffraction-free partially coherent sources—using the genuine cross-spectral density function criterion. These beams have been realized in prior work using the source’s coherent-mode representation and by transforming a ${J}_{0}$-Bessel correlated partially coherent source using a wavefront-folding interferometer. We generalize the traditional dark and antidark beams to produce higher-order sources, which have not been realized. We simulate the generation of these beams and compare the results to the corresponding theoretical predictions. The simulated results are found to be in excellent agreement with theory, thus validating our analysis. We discuss the pros and cons of our synthesis approach vis-à-vis the prior coherent modes work. Lastly, we conclude this paper with a brief summary, and a discussion of how to physically realize these beams and potential applications.

## 1. INTRODUCTION

Dark (or antidark) waves are characterized by a center dark (or bright)
notch in intensity that asymptotes to a constant value as one moves away
from the origin [1–6]. These waves were first observed as
optical solitons [1,2]. Ponomarenko
*et al.* [3]
showed that similar dark and antidark waves exist in linear media, are
diffraction-free beams [7–11], and are necessarily spatially partially coherent. Shortly
thereafter, Borghi *et al.* [4] generalized Ponomarenko
*et al.*’s scalar dark and antidark source,
producing an electromagnetic, or vector, version of the beam.

Being diffraction-free and possessing a dark center, dark (and antidark)
beams have many potential uses including in atomic optics, optical
trapping, and medicine [12–15]. Physically realizing these sources
is therefore important, and Ponomarenko *et al.*
[3] spend a significant amount of
time in their short paper discussing how to synthesize these beams using
the source’s coherent-mode representation [12,16], which has
very recently been demonstrated [17]. It is also worth mentioning that Turunen
*et al.* [5]
and Partanen *et al.* [6], while studying so-called specular and antispecular
beams, described how to realize a dark and antidark source by transforming
a ${J}_{0}$-Bessel correlated partially coherent field
[12,16,18,19] using a wavefront-folding interferometer.

Here, we present a new way to realize dark and antidark beams using the genuine cross-spectral density (CSD) function criterion derived in Refs. [20,21]. We show that these sources can be generated from an optical field consisting of the weighted sum of randomly tilted plane waves. Using this simple stochastic field realization, we generalize dark and antidark sources to produce new, higher-order dark and antidark beams.

This paper is organized as follows: in the next section, we present the
statistical optics theory necessary to realize dark and antidark beams
using the genuine CSD function criterion. We also show how to generalize
this result to produce higher-order dark and antidark sources. In
Section 3, we simulate the
generation of these beams and compare the results to the corresponding
theoretical expressions to validate the analysis in Section 2. We compare and contrast our genuine
CSD criterion approach to Ponomarenko
*et al.*’s coherent modes research. Lastly,
we conclude with a brief summary of our analysis and findings, and
potential applications of our work.

## 2. THEORY

#### A. Coherent-Mode Representation of Dark and Antidark Beams

We begin with a brief review of the prior work regarding dark and antidark partially coherent diffraction-free beams. As introduced in Ref. [3], the CSD function $W$ for dark and antidark beams is

Continuing to follow Ponomarenko *et al.* [3], using the summation theorem for
Bessel functions [22], i.e.,

#### B. Genuine CSD Function Criterion for Dark and Antidark Beams

As we will shortly demonstrate, dark and antidark beams can also be generated using the genuine CSD function criterion derived in Refs. [20,21]. The necessary and sufficient condition for a genuine CSD function $W$ is

Although $H$ originally introduced in Ref. [20] was purely a mathematical construct, we physically interpret it as a realization of a stochastic optical field drawn from a random process [23,24]. Here, we choose

Continuing with the analysis, substituting Eq. (8) into Eq. (7) and simplifying produces

We begin with Eq. (13), and note that since the right-hand side is rotationally invariant, the left-hand side must be as well. This implies, via Eq. (10), that $p$ is rotationally invariant, and when combined with the fact that $p$ is real, ${\tilde{p}}^{*}=\tilde{p}$. Putting this all together simplifies Eq. (13) to

Turning our attention to Eq. (12), we see, after substituting in the above $p$ and evaluating the rather trivial integrals, that

Substituting in $b=\alpha /(2a)$ from Eq. (15), multiplying both sides of the resulting equation by ${a}^{2}$, and applying the quadratic formula produces When $\alpha =0$, $a=\pm 1,0$. To keep $b$ finite, we choose the “$+$” root under the radical. The root choice on the outside of the radical in Eq. (17) is arbitrary; i.e., both end up producing the CSD function in Eq. (1). Without loss of generality, we choose the “$+$” root on the outside of the radical as well.Returning to Eq. (8), the optical field instance that produces a dark or antidark beam is

#### C. Higher-Order Dark and Antidark Beams

The $p$ in Eq. (19) is required to generate a dark or antidark beam as defined in Ref. [3]. We can generate new dark and antidark sources by choosing a different $p$. As an example, here we choose a $p$ that is again separable in magnitude and angle, but rotationally varies; i.e.,

Proceeding with the analysis, the Fourier transform of $p$, defined in Eq. (10), becomes

The stochastic field realization that generates higher-order dark and
antidark beams is the same as the one that produces Ponomarenko
*et al.*’s zeroth-order source [3], namely, Eq. (18). This time, however, the
$\mathit{v}$ is drawn from the joint PDF—in
this case, equal to the product of the two marginal PDFs—given
in Eq. (20). In
the next section, we validate the above analysis with Monte Carlo
wave-optics simulations where we generate the above partially coherent
beams.

## 3. SIMULATION

Here, we present simulations in which we generate zeroth-order and higher-order dark beams. We compare the simulated results to the theoretical expressions to validate our analysis and synthesis approach. Before presenting the results, we discuss the details of the simulation so that the interested reader can reproduce our results if desired.

#### A. Setup

For these simulations, we generated a zeroth-order dark beam with $\beta =1$ and $\alpha =-1$ using both the source’s coherent-mode representation [3] and our genuine CSD criterion method. To validate the higher-order beam analysis in Section 2.C, we also generated a higher-order dark beam with $\beta =1$, $\alpha =-0.5$, and $n=4$. We discretized these sources using $512\times 512$ computational grids with grid spacings equal to 78.125 mm as we show in Code 1, Ref. [33].

When generating the zeroth-order beam using the source’s coherent-mode representation, we used 50 coherent modes, i.e.,

The theoretical $S$ and $W$ for zeroth-order and higher-order dark and antidark beams are given in Eqs. (2) and (23) (for $S$) and Eqs. (15) and (20) (for $W$), respectively. The stochastic field realization that produces both zeroth-order and higher-order beams is given in Eq. (18), with the $\mathit{v}$ drawn from Eq. (15) for zeroth-order beams and Eq. (20) for higher-order beams.

To quantify the convergence and performance of our approach, we computed the root-mean-square errors (RMSEs) and correlation coefficients $\rho $, i.e.,

We performed these simulations using MATLAB R2017a. The MATLAB scripts (.m files) can be found in Code 1, Ref. [33].

#### B. Results and Discussion

Figures 2 and 3 show the zeroth-order and $n=4$ higher-order dark beam results, respectively. Figures 2(a)–2(c) show the theoretical, coherent modes, and genuine CSD criterion spectral densities $S$, respectively. Figures 2(d)–2(f) show the same results for $W({x}_{1},0,{x}_{2},0)$. Lastly, Figs. 2(g) and 2(h) show the RMSE and $\rho $ results versus mode and Monte Carlo trial numbers, respectively. Figure 3 is organized in the same way as Fig. 2, except there are no coherent modes results. Figures 2(a)–2(c), 2(d)–2(f), 3(a)–3(b), and 3(c)–3(d) are plotted on the same false color scales represented by the color bars above the respective subfigure groupings.

As evidenced by Figs. 2(a)–2(f) and 3(a)–3(d), the simulated results are qualitatively in excellent agreement with the theoretical predictions. The quantitative results in Figs. 2(g) and 2(h) show unequivocally that the coherent modes approach converges much faster and to a much smaller residual error (and higher $\rho $) than the genuine CSD method. The “knees” in the $\rho $ curves are at approximately 25 modes for the zeroth-order coherent modes result [Fig. 2(g)], and at approximately 250 trials for both the zeroth-order and higher-order genuine CSD results [Figs. 2(h) and 3(e), respectively]. The “stair-step” behavior of the coherent modes RMSE and $\rho $ results [Fig. 2(g)] is due to the fact that when $\alpha =-1$, the even $m$ eigenvalues are zero; therefore, those terms of the series do not contribute to reducing the RMSE or increasing $\rho $.

The observation that coherent modes is in many respects superior to the genuine CSD approach, insofar as it pertains to dark and antidark beams, is not surprising. The coherent modes—Bessel beams—look much more like the dark and antidark CSD function than tilted plane waves. This generally explains the convergence and error results.

We note that the benefit of the genuine CSD approach is its simplicity. The generation of higher-order dark and antidark beams is a good example of this. The same field realization that produces a zeroth-order beam also produces a higher-order dark and antidark beam. This is not the case for the coherent modes approach, where only the coherent-mode representation for the zeroth-order dark and antidark beam is known.

## 4. CONCLUSION

In this paper, we presented a method to generate dark and antidark beams using the genuine CSD function criterion. To date, these partially coherent sources have been generated using the source’s coherent-mode representation and by transforming a ${J}_{0}$-Bessel correlated source using a wavefront-folding interferometer. We generalized these sources, producing higher-order dark and antidark beams. The stochastic field instance that produced these partially coherent sources was simply a weighted sum of randomly tilted “forward” and “reverse” propagating plane waves.

To validate our analysis, we performed wave-optics simulations in which we generated traditional, zeroth-order, and higher-order dark beams. We compared the simulated results to the corresponding theoretical predictions and found them to be in excellent agreement. In addition, we found, not surprisingly, that generating zeroth-order dark and antidark beams using the source’s coherent-mode representation was superior to our genuine CSD criterion approach. Although the coherent modes method converged faster and to a smaller residual error, the benefit of our approach was its simplicity—the same field instance that produced a zeroth-order beam also produced a higher-order dark and antidark source. This was not the case for the coherent modes approach.

Dark and antidark beams, being the incoherent weighted sum of randomly tilted plane waves, can easily be synthesized in practice using a spatial light modulator or, simpler still, two tip-tilt, fast steering mirrors. The dark and antidark beam synthesis approach presented here will be useful in any application that uses these sources. These applications include, but are not limited to, optical trapping of atoms, particle manipulation, and medicine.

## Acknowledgment

The views expressed in this paper are those of the authors and do not reflect the official policy or position of the U.S. Air Force, the Department of Defense, or the U.S. government.

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