Generating dark and antidark beams using the genuine cross-spectral density function criterion

Milo W. Hyde; Svetlana Avramov-Zamurovic

doi:10.1364/JOSAA.36.001058

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

W (ρ_{1}, ρ_{2}) \propto J_{0} (β | ρ_{1} - ρ_{2} |) + α J_{0} (β | ρ_{1} + ρ_{2} |),

where

ρ = \hat{x} x + \hat{y} y

,

β

is a real constant,

J_{0}

is a zeroth-order Bessel function of the first kind, and

α

is a constant subject to the constraints

α^{*} = α

and

| α | \leq 1

. The spectral density

S

for dark and antidark beams is

S (ρ) = W (ρ, ρ) \propto 1 + α J_{0} (2 β ρ) .

For

α = - 1

, the spectral density has a “dark center,” and hence, when

- 1 \leq α < 0

, the beam is considered dark. The beam is called antidark when

0 \leq α \leq 1

.

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

J_{0} (β | ρ_{1} \mp ρ_{2} |) = \sum_{m = - \infty}^{\infty} {(\pm 1)}^{m} \exp [j m (ϕ_{2} - ϕ_{1})] J_{m} (β ρ_{1}) J_{m} (β ρ_{2}),

substituting this expression into Eq. (1), and simplifying produces

W (ρ_{1}, ρ_{2}) \propto \sum_{m = - \infty}^{\infty} [1 + α {(- 1)}^{m}] J_{m} (β ρ_{1}) \exp (- j m ϕ_{1}) J_{m} (β ρ_{2}) \exp (j m ϕ_{2}) .

Comparing Eq. (4) to the coherent-mode expansion or representation of

W

[12,16], namely,

W (ρ_{1}, ρ_{2}) = \sum_{m} λ_{m} ψ_{m}^{*} (ρ_{1}) ψ_{m} (ρ_{2}),

we see at once that

λ_{m} = 1 + α {(- 1)}^{m}, ψ_{m} (ρ) = J_{m} (β ρ) \exp (j m ϕ) .

Thus, a dark or antidark source can be generated by incoherently summing Bessel beams

ψ_{m}

weighted by

\sqrt{λ_{m}}

.

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

W (ρ_{1}, ρ_{2}) = \iint_{- \infty}^{\infty} p (v) H (ρ_{1}, v) H^{*} (ρ_{2}, v) d^{2} v,

where

*

is the complex conjugate,

p

is a non-negative function, and

H

is an arbitrary kernel [20,21].

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

H (ρ, v) = τ (ρ) [a \exp (j β v \cdot ρ) + b \exp (- j β v \cdot ρ)],

where

a

and

b

are constants,

τ

is in general a complex function,

v = \hat{x} v_{x} + \hat{y} v_{y}

, and

v_{x}

and

v_{y}

are random numbers drawn from the joint probability density function (PDF)

p

. Note that

H

is the weighted sum of “forward” and “reverse” traveling plane waves [“forward” and “reverse” referring to the sign of the exponential, analogous to

\exp (j k z)

and

\exp (- j k z)

being forward and reverse traveling waves, respectively], and the vector

v

determines the plane waves’ direction. Equivalently, this

H

is the weighted sum of inverse and forward Fourier transform kernels and is similar to the

H

, originally discussed in Ref. [20] and then subsequently by numerous others, for producing Schell-model sources [12,16]. This latter statement is not surprising considering that the dark and antidark CSD function in Eq. (1) is the sum of Schell-model and non-Schell-model sources.

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

W (ρ_{1}, ρ_{2}) = τ (ρ_{1}) τ^{*} (ρ_{2}) [a^{2} {\tilde{p}}^{*} (β ρ_{d}) + b^{2} \tilde{p} (β ρ_{d})] + τ (ρ_{1}) τ^{*} (ρ_{2}) a b [{\tilde{p}}^{*} (β ρ_{a}) + \tilde{p} (β ρ_{a})],

where

ρ_{d} = \hat{x} x_{d} + \hat{y} y_{d} = \hat{x} (x_{1} - x_{2}) + \hat{y} (y_{1} - y_{2})

,

ρ_{a} = \hat{x} x_{a} + \hat{y} y_{a} = \hat{x} (x_{1} + x_{2}) + \hat{y} (y_{1} + y_{2})

,

\tilde{p} (f) = \iint_{- \infty}^{\infty} p (v) \exp (- j v \cdot f) d^{2} v .

Comparing Eq. (9) to Eq. (1), we see at once that

τ (ρ) = 1,

a^{2} {\tilde{p}}^{*} (β ρ_{d}) + b^{2} \tilde{p} (β ρ_{d}) = J_{0} (β ρ_{d}),

a b [{\tilde{p}}^{*} (β ρ_{a}) + \tilde{p} (β ρ_{a})] = α J_{0} (β ρ_{a}) .

Our goal now is to find the values of

a

and

b

and the positive, real function

p

that satisfies Eqs. (12) and (13).

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

α J_{0} (β ρ_{a}) = 2 a b \iint_{- \infty}^{\infty} p (v) \exp (- j β v \cdot ρ_{a}) d^{2} v .

We then apply the inverse Fourier transform to both sides of Eq. (14) yielding

α = 2 a b, p (v) = \frac{1}{2 π} δ (v - 1),

where

δ (x)

is the Dirac delta function.

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

a^{2} + b^{2} = 1 .

Substituting in

b = α / (2 a)

from Eq. (15), multiplying both sides of the resulting equation by

a^{2}

, and applying the quadratic formula produces

a = \pm \frac{1}{\sqrt{2}} \sqrt{1 \pm \sqrt{1 - α^{2}}} .

When

α = 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

U (ρ) = \sqrt{\frac{1 + \sqrt{1 - α^{2}}}{2}} [\exp (j β v \cdot ρ) + \frac{α}{1 + \sqrt{1 - α^{2}}} \exp (- j β v \cdot ρ)],

where

v

is a random wavefront slope, or gradient drawn from the joint PDF

p (v) = \frac{1}{2 π} δ (v - 1) .

Note that

p

is separable in magnitude and angle, i.e.,

p (v) = p (v, θ) = p (v) p (θ)

, where

p (v)

and

p (θ)

are the magnitude and angle marginal PDFs, respectively. This, of course, means that the magnitude and angle of

v

are independent random variables.

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.,

p (v) = δ (v - 1), p (θ) = \frac{1}{π (1 + δ_{0 n})} \cos^{2} (n θ / 2),

where

n

is an integer and

δ_{i j}

is the Kronecker delta function. Separable

p

, like the one above, was first considered by Wang and Korotkova [25] to create Schell-model sources that radiated beams with azimuthally varying far-zone intensity patterns. In Eq. (20), the

n = 0

case corresponds to the dark and antidark source discussed in Section 2.B (hereafter referred to as a zeroth-order dark and antidark beam). For a reason that will become apparent, we call the

n \neq 0

cases higher-order dark and antidark beams.

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

\tilde{p} (f, ψ) = \frac{1}{1 + δ_{0 n}} [J_{0} (f) + j^{- n} \cos (n ψ) J_{n} (f)] .

Substituting

\tilde{p}

into Eq. (9) and assuming the same values for

τ

,

a

, and

b

as above produces

W (ρ_{1}, ρ c_{2}) = \frac{1}{1 + δ_{0 n}} {J_{0} (β ρ_{d}) + j^{n} \cos (n ϕ_{d}) J_{n} (β ρ_{d}) - j^{n} \frac{[1 - {(- 1)}^{n}] α^{2}}{2 (1 + \sqrt{1 - α^{2}})} \cos (n ϕ_{d}) J_{n} (β ρ_{d})} + \frac{α}{1 + δ_{0 n}} [J_{0} (β ρ_{a}) + \cos (n π / 2) \cos (n ϕ_{a}) J_{n} (β ρ_{a})],

where

ϕ_{a, d} = \arctan (y_{a, d} / x_{a, d})

. The spectral density

S

of these new dark and antidark beams is

S (ρ) = 1 + α J_{0} (2 β ρ) + α (1 - δ_{n 0}) \cos (n π / 2) \cos (n ϕ) J_{n} (2 β ρ),

which when

n = 0, 1, 3, 5, \dots

simplifies to Eq. (2). When

n = 2, 4, 6, 8, \dots

, higher-order dark and antidark beams are produced. Figure 1 shows the

S

with

α = - 1

,

β = 1

, and

n = 0

, 2, 4, 6, 8, and 10. The figure clearly shows the higher-order modal behavior of these new sources. We note that, like all diffraction-free beams, higher-order dark and antidark beams possess infinite energy; therefore, they cannot be synthesized exactly, and an approximate dark (or antidark) source must suffice. Based upon the large number of papers and books that have been published discussing how to generate diffraction-free beams [9,17,26–32], we do not anticipate this important physical detail posing a challenge to realizing high-quality, approximate dark (or antidark) sources.

Fig. 1. Spectral densities $S$ of higher-order dark beams with $α = - 1$ and $β = 1$ . (a) $n = 0$ , (b) $n = 2$ , (c) $n = 4$ , (d) $n = 6$ , (e) $n = 8$ , and (f) $n = 10$ .

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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 $v$ 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 $β = 1$ and $α = - 1$ 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 $β = 1$ , $α = - 0.5$ , 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.,

W (ρ_{1}, ρ_{2}) \approx \sum_{m = - 50}^{50} [1 + α {(- 1)}^{m}] J_{m} (β ρ_{1}) \exp (- j m ϕ_{1}) J_{m} (β ρ_{2}) \exp (j m ϕ_{2}),

to approximate the infinite series in Eq. (4). For both the zeroth-order and

n = 4

higher-order dark sources realized using the genuine CSD criterion, we generated 100,000 optical field instances to form the dark beams. From these 50 coherent modes and 100,000 field realizations, we computed the spectral densities

S

and CSD functions

W (x_{1}, 0, x_{2}, 0)

.

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 $v$ 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 $ρ$ , i.e.,

RMSE = \sqrt{\frac{1}{N^{2}} \sum_{k = 1}^{N^{2}} {(S^{thy} [k] - S^{sim} [k])}^{2}}, ρ = \frac{\sum_{k = 1}^{N^{2}} (S^{sim} [k] - {\bar{S}}^{sim}) (S^{thy} [k] - {\bar{S}}^{thy})}{\sqrt{\sum_{k = 1}^{N^{2}} {(S^{sim} [k] - {\bar{S}}^{sim})}^{2}} \sqrt{\sum_{k = 1}^{N^{2}} {(S^{thy} [k] - {\bar{S}}^{thy})}^{2}}},

where

N^{2} = 512^{2}

was the number of pixels in an image,

k

was a discrete pixel index, and

\bar{S}

was the average value of the spectral density, versus the Monte Carlo trial number or coherent-mode number (whichever was applicable).

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 $ρ$ 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.

Fig. 2. Zeroth-order dark beam results with $α = - 1$ and $β = 1$ . (a) $S$ theory [Eq. (2)], (b) $S$ 50 coherent modes [Eq. (24)], (c) $S$ genuine CSD criterion 100,000 field realizations [Eqs. (18) and (19)], (d) $W (x_{1}, 0, x_{2}, 0)$ theory [Eq. (1)], (e) $W (x_{1}, 0, x_{2}, 0)$ 50 coherent modes [Eq. (24)], (f) $W (x_{1}, 0, x_{2}, 0)$ genuine CSD criterion 100,000 field realizations [Eqs. (18) and (19)], (g) two-dimensional root-mean-square error (RMSE) and correlation coefficient $ρ$ for the coherent modes $S$ computed against $S$ theory versus mode number, and (h) two-dimensional RMSE and correlation coefficient $ρ$ for the genuine CSD criterion $S$ computed against $S$ theory versus trial number.

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Fig. 3. Higher-order dark beam results with $α = - 0.5$ , $β = 1$ , and $n = 4$ . (a) $S$ theory [Eq. (23)], (b) $S$ genuine CSD criterion 100,000 field realizations [Eqs. (18) and (20)], (c) $W (x_{1}, 0, x_{2}, 0)$ theory [Eq. (22)], (d) $W (x_{1}, 0, x_{2}, 0)$ genuine CSD criterion 100,000 field realizations [Eqs. (18) and (20)], and (e) two-dimensional root-mean-square error (RMSE) and correlation coefficient $ρ$ for the genuine CSD criterion $S$ computed against $S$ theory versus trial number.

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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 $ρ$ ) than the genuine CSD method. The “knees” in the $ρ$ 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 $ρ$ results [Fig. 2(g)] is due to the fact that when $α = - 1$ , the even $m$ eigenvalues are zero; therefore, those terms of the series do not contribute to reducing the RMSE or increasing $ρ$ .

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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Generating dark and antidark beams using the genuine cross-spectral density function criterion

Abstract

1. INTRODUCTION

2. THEORY

A. Coherent-Mode Representation of Dark and Antidark Beams

B. Genuine CSD Function Criterion for Dark and Antidark Beams

C. Higher-Order Dark and Antidark Beams

3. SIMULATION

A. Setup

B. Results and Discussion

4. CONCLUSION

Acknowledgment

REFERENCES

Supplementary Material (1)

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Journal of the Optical Society of America A