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ORIGINAL RESEARCH article

Front. Phys., 09 January 2023
Sec. Optics and Photonics
This article is part of the Research Topic Optical Wave Propagation and Communication in Turbulent Media View all 7 articles

The behavior of partially coherent twisted space-time beams in atmospheric turbulence

Updated
  • Department of Engineering Physics, Air Force Institute of Technology, Dayton, OH, United States

We study how atmospheric turbulence affects twisted space-time beams, which are non-stationary random optical fields whose space and time dimensions are coupled with a stochastic twist. Applying the extended Huygens–Fresnel principle, we derive the mutual coherence function of a twisted space-time beam after propagating a distance z through atmospheric turbulence of arbitrary strength. We specialize the result to derive the ensemble-averaged irradiance and discuss how turbulence affects the beam’s spatial size, pulse width, and space-time twist. Lastly, we generate, in simulation, twisted space-time beam field realizations and propagate them through atmospheric phase screens to validate our analysis.

1 Introduction

New approaches in beam control include light with engineered space-time or spatiotemporal coupling. Recent papers have demonstrated space-time-coupled light which exhibits anomalous diffractive and refractive behaviors [14] as well as carries transverse (to the direction of propagation) orbital angular momentum in the form of spatiotemporal optical vortices (STOVs) [511]. These novel developments hold promise for exciting advancements in applications such as optical communications, optical tweezing, and quantum optics [2, 4, 1216].

Most of the space-time-coupled beam research manipulates coherent light, although this has begun to change with the development of partially coherent STOV and twisted space-time (and space-frequency) beams [1721]. The latter are non-stationary random fields with the beams’ spatial and temporal (or spectral) dimensions coupled in a stochastic twist. They are the spatiotemporal counterparts of traditional, spatially twisted Gaussian Schell-model beams [2227].

Spatially twisted partially coherent fields have been extensively studied since being introduced in 1993. This research includes beam synthesis [2833]; coherent modes/pseudo-modes [23, 26, 27, 3437]; angular momentum [3841]; and propagation behaviors in free-space, ABCD optical systems, and turbulence [35, 4251]. This stands in contrast to twisted space-time beams (and STOV beams more generally), where only their angular momentum and free-space propagation behaviors have been investigated [6, 8, 9, 11, 19, 20, 52, 53].

In this paper, we undertake, to our knowledge, the first study on the effects of atmospheric turbulence on twisted space-time beams. Using the extended Huygens–Fresnel principle, we derive an approximate expression for the mutual coherence function (MCF) of a twisted space-time beam after propagating through atmospheric turbulence of any strength. We then specialize the MCF to obtain the ensemble-averaged irradiance and discuss how turbulence affects the beam’s size, pulse width, and space-time twist. To validate our analysis, we compare the theoretical irradiance to the results of Monte Carlo wave-optics simulations. Lastly, we conclude with a brief summary of our findings.

2 Theory

2.1 Extended Huygens–Fresnel principle

Let us begin with the extended Huygens–Fresnel principle/integral:

Uρ,z,ω=kj2πzexpjkzUρ,0,ωexpjk2zρρ2expΨρ,0;ρ,z;ωd2ρ,(1)

where j=1, ω is the radian frequency, k = ω/c is the wavenumber, c is the speed of light, ρ=x̂x+ŷy is the source vector, and ρ=x̂x+ŷy is the observation vector. The optical field U in the integrand is a stochastic (frequency-domain) realization of a twisted space-time beam, and Ψ is a random complex function which models the phase and amplitude fluctuations of a point source propagating through atmospheric turbulence from ρ,0 to ρ,z at frequency ω [5457].

The two-frequency cross-spectral density (CSD) function [55, 5862] can be obtained by taking the ensemble-averaged auto-correlation of Eq. 1, namely,

Wρ1,z,ω1,ρ2,z,ω2=k1k22π2z2expjk1k2zWρ1,0,ω1,ρ2,0,ω2expjk12zρ1ρ12expjk22zρ2ρ22expΨρ1,0;ρ1,z;ω1+Ψ*ρ2,0;ρ2,z;ω2d2ρ1d2ρ2,(2)

where we have assumed that the source field is statistically independent of the atmospheric turbulence fluctuations. The moment involving Ψ is related to the two-point, spherical wave structure function (WSF) [5557, 61, 62], and equals

expΨρ1,0;ρ1,z;ω1+Ψ*ρ2,0;ρ2,z;ω2=exp12Dρ1ρ2,0;ρ1ρ2,z;ω1,ω2exp2π20z0κΦnκ,ζk12+k222k1k2expjβκ2J0κRdκdζ,(3)

where Φn is the index of refraction power spectrum (assumed to be statistically isotropic) and β and R equal

β=ζzζ2z1k11k2R=ζzρ1ρ2+1ζzρ1ρ2.(4)

The approximate expression on the second line of Eq. 3 is derived using the method of smooth perturbations (also known as the Rytov approximation) and further assuming that Ψ is Gaussian distributed [5557, 6163]. We will return to Eq. 3 shortly.

The ultimate goal is to find the “two-time” MCF of a twisted space-time beam after propagating through turbulence. To do this, we must inverse Fourier transform Eq. 2, i.e.,

Γρ1,z,t1,ρ2,z,t2=Wρ1,z,ω1,ρ2,z,ω2expjω1t1expjω2t2dω1dω2.(5)

Applying Eqs 25 and interchanging the order of the integrals yields

Γρ1,z,t1,ρ2,z,t2=12π2c2z2ω1ω2Wρ1,0,ω1,ρ2,0,ω2exp12Dρ1ρ2,0;ρ1ρ2,z;ω1,ω2expjω1t1zcρ1ρ122zcexpjω2t2zcρ2ρ222zcdω1dω2d2ρ1d2ρ2.(6)

Assuming that the twisted space-time beam has a relatively narrow linewidth (or bandwidth) around mean or carrier frequency ωc (i.e., Δω/ωc ≪ 1), we can approximate Eq. 6 as

Γρ1,z,t1,ρ2,z,t21λc2z2expjωct1t2ρ1ρ122zc+ρ2ρ222zcWρ1,0,ω̄1,ρ2,0,ω̄2exp12Dρ1ρ2,0;ρ1ρ2,z;ω̄1+ωc,ω̄2+ωcexpjω̄1t1zcexpjω̄2t2zcdω̄1dω̄2d2ρ1d2ρ2,(7)

and, by evaluating Eq. 7, obtain a closed-form expression for the MCF. Before doing this, we need to discuss the functions D and W in the integrand.

2.2 Approximate two-point, spherical WSF D

Let us return to Eq. 3. By virtue of the source being narrowband, β ≈ 0. Letting Φn equal the von Kármán spectrum—namely,

Φnκ,ζ=0.033Cn2ζexpκ2/κm2κ+κ0211/6,(8)

where κm = 5.92/l0 and κ0 = 2π/L0 (Cn2, l0, and L0 are the index of refraction structure constant, the inner scale, and outer scale of turbulence, respectively)—the integral over κ evaluates to

Dρ1ρ2,0;ρ1ρ2,z;ω1,ω2=0.0332π2κ05/3U1;16;κ02κm2×k12+k220zCn2ζdζ0.0334π2κ05/3k1k20zCn2ζ×n=01nn!κ02R24nUn+1;n+16;κ02κm2dζ,(9)

where Ua;c;z is a confluent hypergeometric function of the second kind [6466]. In most physical scenarios, L0l0, and therefore, we can estimate Eq. 9 using the small argument relation for Ua;c;z. The result, after much analysis, is

Dρ1ρ2,0;ρ1ρ2,z;ω1,ω20.7817κ05/3k1k220zCn2ζdζ+8.7021κm5/3k1k20zCn2ζF1156;1;κm2R241dζ2.3450κ01/3k1k20zCn2ζR2dζ.(10)

Eq. 10 includes both inner and outer scale effects. However, to evaluate Eq. 7 in closed form, we must let the inner scale l0 → 0 (κm). Using the large argument relation for 1F1, we obtain

Dρ1ρ2,0;ρ1ρ2,z;ω1,ω20.7817κ05/3k1k220zCn2ζdζ+2.9139k1k20zCn2ζR5/3dζ2.3450κ01/3k1k20zCn2ζR2dζ.(11)

We lastly assume that Cn2 is constant over the propagation path and set R5/3R2—an estimate known as the quadratic approximation [57, 62]. Substituting ω1=ω̄1+ωc and ω2=ω̄2+ωc as stipulated in Eq. 7 and noting that k1k2=k̄1+kck̄2+kckc2, we arrive at the final result

Dρ1ρ2,0;ρ1ρ2,z;ω̄1+ωc,ω̄2+ωc0.7817Cn2zκ05/3c2ω̄1ω̄22+1.0930Cn2z10.7152κ01/3kc2ρ1ρ22+ρ1ρ22+ρ1ρ2ρ1ρ2=2aωc2ω̄1ω̄22+2askc2ρ1ρ22+ρ1ρ22+ρ1ρ2ρ1ρ2.(12)

Equation 12 is very physical: The terms describe how atmospheric turbulence corrupts light’s spectral and spatial coherence. For traditional space-time separable beams, these two terms give rise to pulse and beam broadening, respectively [56, 57, 6773]. In our case, because of spatiotemporal coupling, both terms will affect the temporal and spatial beam sizes.

2.3 CSD function of a twisted space-time beam

With Eq. 12, we are one step closer to evaluating Eq. 7. We, of course, still need an expression for W. To find this expression, we begin with the MCF of a twisted space-time beam:

Γρ1,t1,ρ2,t2=A2expy12+y224Wy2expx12+x224Wx2expx1x222δx2expt12+t224Wt2expt1t222δt2expjμx1t2x2t1expjωct1t2;(13)

where A is the amplitude; Wx, Wy, and Wt are the spatial and temporal pulse widths; δx and δt are the spatial and temporal coherence widths; and μ is the space-time twist parameter [19]. The latter must satisfy μδtδx1 for the MCF in Eq. 13 to be genuine, i.e., square-integrable, Hermitian, and non-negative definite [58, 59]. Consequently, μ → 0 in the coherent beam limit δt, δx. When μδtδx=1, the twist in the beam is saturated [20, 25, 27]. We assume this condition for the simulations described in Section 3.

Note that Eq. 13 has the same general form as a twisted Gaussian Schell-model beam [22, 2527]; however, here, space and time are twisted. It is well known that the spectral density or average irradiance of a spatially twisted random beam rotates in the x-y plane as it propagates in the z direction [35, 40, 41, 74]. From Eq. 7, we see that t is linked paraxially to the propagation distance z; therefore, a twisted space-time beam rotates or tumbles in the x-z plane as it propagates. This behavior is described in Refs. [19, 20] for twisted space-time beams propagating in free space. What remains to be determined is how atmospheric turbulence affects the x-z plane rotation of twisted space-time beams.

We can find the CSD function W of a twisted space-time beam by Fourier transforming the MCF in Eq. 13, i.e.,

Wρ1,ω1,ρ2,ω2=12π2Γρ1,t1,ρ2,t2expjω1t1expjω2t2dt1dt2.(14)

Substituting Eq. 13 into Eq. 14 and evaluating the integrals yields

Wρ1,ω1,ρ2,ω2=A24πΩtexpy12+y224Wy2expx12+x224W̃x2expx1x222δ̃x2expω̄12+ω̄224Wω2expω̄1ω̄222δω2expμδω2ω̄21+12γt2ω̄1x1expμδω2ω̄11+12γt2ω̄2x2,(15)

where γt = Wt/δt, Wω = 2WtΩt, δω = 2δtΩt, and

Ωt2=14Wt2+12δt2212δt22,1W̃x2=1Wx2+μ2Wω2,1δ̃x2=1δx2+μ2δω2.(16)

With Eq. 15, we are now ready to evaluate the integrals in Eq. 7.

2.4 MCF of twisted space-time beam in atmospheric turbulence

Substituting Eqs 12, 15 into Eq. 7 and evaluating the integrals produces (after much analysis)

Γx1,0,z,t1,x2,0,z,t2=A2WtW̃tNFxNFyΔxΔyexpjωct̄1t̄2expNFx2Δxx12+x224Wx2expt̄12+t̄224Wteff2expμNFxΔxWt2W̃t2x1t̄2+x2t̄1expjkc2z1NFx2Δx+askc24Wx2Δxx12x22expjμ2NFx2ΔxWt2W̃t2t̄12t̄22expjμNFx2ΔxWt2W̃t2x1t̄2x2t̄1expjμNFx2Δxaωc2W̃t2x1+x2t̄1t̄2expjμ2Wx2ΔxWt2W̃t2askc2x1x2t̄1+t̄2expx1x222δxeff2expt̄1t̄222δteff2expx1x2t̄1t̄22δxteff2.(17)

Since the beam’s interesting behaviors occur in the x-t or x-z plane (the x and t dimensions are coupled), here, we present the MCF evaluated at y1 = y2 = 0. The undefined symbols in Eq. 17 are NFx,y=2kcWx,y2/z, which are the x and y Fresnel numbers for a fully coherent Gaussian beam; t̄=tz/c is the retarded time; W̃t2=Wt2+2aω/c2; and

Δx=1+4Wx2δx2+NFx2+8Wx2askc2+μ2aωc2Wt2W̃t2;Δy=1+NFy2+8Wy2askc2;14Wteff2=14W̃t2+μ2Wx2ΔxWt4W̃t4;12δteff2=12δt2+μ2Wx22ΔxΔxNFx2Wt4W̃t4+aωc214Wt2W̃t2;12δxeff2=NFx2Δx12δx2+askc21+2NFx2Δxaskc222Wx2Δx2+μ2aωc2NFx2ΔxWt2W̃t2;12δxteff2=μNFx2ΔxΔxNFx2Wt2W̃t2+askc24Wx2.(18)

Eq. 17 is organized so that the terms can be physically interpreted: Starting at the top and ignoring the carrier expjωct̄1t̄2, the amplitude term plus the first three exponentials comprise the ensemble-averaged irradiance (discussed in more detail below). The next (complex) exponentials on line 4 are the spatial and temporal chirps. These are followed by the space-time twist on lines 5 and 6. Lastly, the exponentials on line 7 model spatial and temporal coherence.

2.5 Average irradiance and physical discussion

The ensemble-averaged irradiance is found by evaluating Eq. 17 at equal space-time points, i.e.,

Ix,0,z,t=Γx,0,z,t,x,0,z,t=A2WtW̃tNFxNFyΔxΔyexpNFx2Δxx22Wx2exp1+μ24Wx2Wt4ΔxW̃t2t̄22W̃t2expμ2NFxΔxWt2W̃t2xt̄=Â2expx22Ŵx2expt̄22Ŵt2expμ̂xt̄.(19)

In order, the exponentials are the spatial beam shape, temporal beam (pulse) shape, and x-t plane rotation. The behavior of the beam can be understood by examining Ŵx, Ŵt, and μ̂ versus Fresnel number and turbulence strength. Figure 1 shows these curves: (A) plots Ŵx/Wx, Ŵt/Wt, and μ̂/μ over Fresnel numbers ranging from 100 (near field) to 0.01 (far field). The solid, dashed, dashed-dotted, and dotted traces show how these quantities evolve in free space (Cn2=0 m2/3) and atmospheric turbulence (Cn2=1014 m2/3 with L0 = 10 m, 50 m, and 100 m), respectively. For the latter, the (weak turbulence) spherical wave scintillation indices [57], i.e.,

σI2=0.5Cn2kc7/6z11/6,(20)

are annotated on the plot (centered on their corresponding Fresnel number) to show the strength of turbulence at that NFx. Figure 1B displays a zoomed-in view of Ŵx/Wx, Ŵt/Wt, and μ̂/μ over the boxed region in (A), viz., 15 ≥ NFx ≥ 0.5. Lastly, the results depicted in Figure 1 apply to a twisted space-time beam with λc = 1 μm, Wx = 2 cm, δx = 0.9Wx, Wt = 1 ps, δt = 0.9Wt, and μ=1/δxδt.

FIGURE 1
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FIGURE 1. (A) Ŵx/Wx, Ŵt/Wt, and μ̂/μ from Eq. 19 versus Fresnel number NFx. (B) Zoomed-in view of Ŵx/Wx, Ŵt/Wt, and μ̂/μ over the boxed region in (A). The solid, dashed, dashed-dotted, and dotted traces show the results in free space (Cn2=0 m2/3) and atmospheric turbulence (Cn2=1014 m2/3 with L0=10 m, 50 m, and 100 m), respectively. The text annotations report the weak turbulence spherical wave scintillation index values σI2 at the corresponding NFx. These results apply to a twisted space-time beam with parameter values equal to λc =1 μm, Wx =2 cm, δx =0.9Wx, Wt =1 ps, δt =0.9Wt, and μ=1/δxδt.

Starting with the free-space (solid) curves in Figure 1, we see that for NFx > 10, the twisted space-time beam is effectively in the source plane, with ŴxWx, ŴtWt, and μ̂<μ/5. Things begin to change for 10 > NFx > 1: Most noticeably, the beam grows significantly larger due to diffraction. Indeed, over this range, the beam expands nearly three times its original size in the x direction. In addition to Ŵx, the pulse width also changes in this region because of spatiotemporal coupling. Beginning around NFx ≈ 10, Ŵt starts to contract (shorten) and continues to do so until NFx ≈ 1. This shortening of Ŵt is met by an increase in μ̂. When considered together, the result is a beam that rotates in the x-t (or x-z) plane—the beam effectively “trades” Ŵt to do so. Lastly, for NFx < 1, Ŵx continues to grow larger due to diffraction, Ŵt asymptotes (the pulse width stops contracting), and μ̂ falls rapidly toward zero. Physically, the twisted space-time beam is in the far zone, diffraction dominates, and the beam no longer rotates.

Examining the turbulence (dashed, dashed-dotted, and dotted) curves, we generally observe the same behavior; however, the beam’s evolution described above is effectively pushed to the left, i.e., toward higher Fresnel numbers. Where the separation between free-space (diffractive) and turbulence-induced behavior occurs (in other words, at what NFx), of course, depends on Cn2 and L0. Nevertheless, some general trends are evident and independent of turbulence strength:

1. The beam’s size Ŵx asymptotically expands much more rapidly in turbulence than in free space (z3 vice z2) [69, 7173].

2. After initially contracting, the pulse width Ŵt lengthens and continues to grow longer. While this can clearly be seen in Figure 1, more insight can be gained by examining the mathematical expression for Ŵt, namely,

1Ŵt=1W̃t1+μ24Wx2Wt4ΔxW̃t2.(21)

In atmospheric turbulence, the term containing the twist parameter μ tends to zero like z−2 (in free space, the term asymptotes to a constant value). For large z, the result is therefore ŴtW̃t=Wt+2aω/c2. The turbulence contribution to the pulse width grows linearly with z [57, 67, 68], thus explaining the increasing pulse width.

3. The x-t plane rotation μ̂ decays much more rapidly in turbulence than in free space. Examining the mathematical relation for μ̂ reveals that it approaches zero like μ̂z3 in turbulence (vice μ̂z1 in free space) as z.

3 Validation

In this section, we validate Eq. 19 by generating, in simulation, twisted space-time beam field realizations and propagating those realizations through atmospheric turbulence phase screens. Before presenting and analyzing the results, we discuss the simulation setup.

3.1 Simulation setup

3.1.1 Numbers of grid points, spacings, trials, etc.

In these wave-optics simulations, we generated and propagated twisted space-time beam field realizations through independent instances of atmospheric turbulence. The Fresnel numbers for these simulations were NFx = 10, 5, 2.5, and 1. For each NFx, we computed the ensemble-averaged irradiance Ix,0,z,t from 1,000 independent field and turbulence realizations. The source and observation planes were discretized using three-dimensional grids that were Ny × Nx × Nt = 1, 200 × 1, 200 × 128 with spacings equal to Δsrc = 1.58 mm, Δobs = 2.5 mm, and Δt = 0.0781 ps.

3.1.2 Generating twisted space-time fields

We generated twisted space-time beam field realizations using the approach described in Ref. [31]. The technique utilizes Gori and Santarsiero’s integral criterion for genuine CSD functions and MCFs, colloquially known as the superposition rule [75, 76]. Specialized for our purposes, a thermal (or pseudo-thermal) twisted space-time beam field realization can be generated by evaluating the following superposition integral:

Uρ,t=rvx,vt12pvx,vtHρ,t;vx,vtdvxdvt,(22)

where r is a zero-mean, unit-variance, delta-correlated, complex Gaussian random function [31], and p and H are

pvx,vt=απexpαvx2βπexpβvt2Hρ,t;vx,vt=Aexpy24Wy2expσxx2expσtt2expjxjαμtvxexpjt+jβμxvt.(23)

The α, β, σx, and σt relate to the physical twisted space-time beam parameters in Eq. 13 via the relations [19, 35].

14Wx2=σxβμ22,14Wt2=σtαμ22,12δx2=βμ24+14α,12δt2=αμ24+14β.(24)

In the simulations, we produced twisted space-time beams with the following parameter values λc = 1 μm, Wx = Wy = 2 cm, δx = 0.9Wx, Wt = 1 ps, δt = 0.9Wt, and μ=1/δxδt—the same as in Figure 1. These parameter values corresponded to α = 3.24 cm2, β = 0.81 ps2, σx = 0.2168 cm−2, and σt = 0.8673 ps−2. We evaluated Eq. 22 as a matrix-vector product, where the vx and vt dimensions were discretized using 64 grid points each, with spacings equal to Δvx = 0.0645 cm−1 and Δvt = 0.1291 ps−1, respectively.

3.1.3 Atmospheric turbulence

The index of refraction structure constant and outer scale for the atmospheric turbulence was Cn2=1014 m2/3 and L0 = 10 m, corresponding to the dashed traces in Figure 1. We simulated propagation through this turbulence using the split-step algorithm described in Refs. [70, 7780]. For NFx = 10, 5, 2.5, and 1, we discretized the continuous propagation paths using 4, 5, 9, and 20 equally spaced, statistically independent phase screens generated using the Fourier transform (also known as the spectral) method and augmented with subharmonics [70, 78, 81, 82]. The strength of each phase screen (Cn2, Fried’s parameter r0, or coherence width ρ0) was selected such that the discrete-path spherical wave r0 and scintillation index σI2 matched those of the desired, continuous turbulent path. To capture the change in phase due to turbulence over the light source’s bandwidth, we divided each phase screen by kc to convert from radians to meters of optical path length (OPL).

Note that we did not simulate the other turbulence conditions reported in Figure 1 due to computational constraints. Accurately simulating turbulence with a given outer scale requires phase screens that have physical dimensions on the order of L0. Simulating the L0 = 50 m and 100 m atm would have required grids that were (approximately) 25 and 100 times larger (in numbers of points), respectively, than those used in the L0 = 10 m simulations (see Section 3.1.1).

3.1.4 Procedure

On each Monte Carlo trial,

1. We generated a twisted space-time beam realization and an instance of atmospheric turbulence as described above.

2. We then Fourier transformed the twisted space-time beam realization to the ω domain using a fast Fourier transform (FFT) computed along the third dimension of U.

3. We propagated U to each of the 4, 5, 9, or 20 (depending on NFx) planes using the convolution form of the Fresnel diffraction integral (also known as the angular spectrum propagation method [78, 80]), which we evaluated using FFTs computed along U’s spatial dimensions.

4. In each plane, we converted the atmospheric phase screen in meters of OPL to radians using the ω values along the third dimension of U. We then applied the phase screen to the field and propagated U to the next plane.

5. Upon reaching the observation plane, we Fourier transformed the field back to the t domain using an FFT computed along U’s third dimension.

6. Lastly, we computed the trial irradiance Ix,0,z,t=Ux,0,z,t2.

We repeated this procedure 1,000 times.

3.2 Results

Figures 24 show the results of the twisted space-time beam simulations. Figures 2, 3—which report the ensemble-averaged irradiances Ix,0,z,t after propagating through free space (included as a reference) and atmospheric turbulence, respectively—are organized in the same manner: The top row shows the theoretical Ix,0,z,t given in Eq. 19 for Fresnel numbers NFx = , 10, 5, and 1, respectively. The bottom (second) row displays the same for the simulated Ix,0,z,t. The images in Figure 3 are encoded using the same color scales as the corresponding subfigures in Figure 2. Row and column headings have been added to both figures to aid the reader. Lastly, Figure 4 reports the theoretical and simulated Ŵx/Wx, Ŵt/Wt, and μ̂/μ versus Fresnel number NFx. The solid and dashed curves in the figure are the same as those shown in Figure 1B; however, here, we have added the simulated results denoted by the markers ◦ and ⊳. We obtained these results by fitting Gaussian functions to the simulated Ix,0,z,t.

FIGURE 2
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FIGURE 2. Ensemble-averaged irradiance Ix,0,z,t free-space results: (A) theory NFx = , (B) theory NFx = 10, (C) theory NFx = 5, (D) theory NFx = 1, (E) simulation NFx = , (F) simulation NFx = 10, (G) simulation NFx = 5, and (H) simulation NFx = 1.

FIGURE 3
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FIGURE 3. Ensemble-averaged irradiance Ix,0,z,t turbulence results: (A) theory NFx = , (B) theory NFx = 10, (C) theory NFx = 5, (D) theory NFx = 1, (E) simulation NFx = , (F) simulation NFx = 10, (G) simulation NFx = 5, and (H) simulation NFx = 1.

FIGURE 4
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FIGURE 4. Theory [Eq. 19] and simulation Ŵx/Wx, Ŵt/Wt, and μ̂/μ versus Fresnel number NFx. The symbols ◦ and ⊳ are the results of the simulation.

Inspection of Figure 3 reveals good agreement between simulation and theory in weak to moderately strong atmospheric turbulence [Figures 3B, C, F, and G]. In contrast, the agreement is rather poor in strong turbulence [Figures 3D, H]. This discrepancy is likely caused by the quadratic approximation we used to derive the MCF in Eq. 17 and subsequently Ix,0,z,t in Eq. 19. The validity of the quadratic approximation (and the extended Huygens–Fresnel principle, more generally) is suspect in strong turbulence [6163, 83]. Thus, the disagreement in Figures 3D, H is somewhat expected. The results in Figure 4 are consistent with those in Figure 3—we observe good agreement in weak-to-moderate turbulence and poor agreement in strong turbulence. Although the theoretical relations for Ŵx/Wx, Ŵt/Wt, and μ̂/μ generally underestimate the effects of turbulence on those parameters, they do accurately predict the trends versus Fresnel number and turbulence strength.

3.3 Experimental verification

Before concluding, we briefly discuss the process for experimentally verifying the theoretical and simulated results presented above. Twisted space-time beam field realizations can be physically synthesized using an apparatus known as a Fourier transform pulse shaper (FTPS) [1, 4, 9, 8487]. An FTPS consists of two identical gratings separated by a 4f cylindrical lens (CL) system. At the center of the 4f system is a spatial light modulator (SLM). Assuming a pulsed laser beam is input into the FTPS, the first grating-CL-2f system spreads and maps the input beam’s spectrum into physical space at the SLM plane. The SLM modifies the field in the space-frequency (x-ω) domain, which is then transformed back to the space-time domain by the second grating-CL-2f system. Partial coherence manifests by incoherently summing many independent twisted space-time beam realizations.

Turbulence (besides outdoor experiments which are generally uncontrolled) can be controllably generated in the laboratory using several different methods [88]. Of these, phase plate/wheel [8992] or hot-air [93, 94] techniques are the most germane, and systems employing those methods are easily capable of reproducing the turbulence conditions simulated above.

Lastly, to observe the beam’s behavior in x-t domain, we follow the procedure described in Refs. [1, 5]: The light at the output of the turbulence generator transits a grating-CL-2f system and then is measured by a detector. The detector measures the light’s spatially resolved spectrum averaged over many independent field and turbulence realizations, i.e.,

Sx,z,ω=Ux,z,ω2.(25)

Note that this quantity is also referred to as the spectral density [58, 59, 71, 72]. Using Eq. 14, the spectral density relates to the MCF via

Sx,z,ω=12π2Γx,z,t1,x,z,t2expjωt1t2dt1dt2,(26)

and consequently, the ensemble-averaged irradiance Ix,z,t is not directly recoverable. Likely, the easiest course of action is to compare the measured spectral density to its theoretical and simulated counterparts to validate the latter.

4 Conclusion

In this paper, we focused on a recently introduced, partially coherent, space-time-coupled field known as a twisted space-time beam. Twisted space-time beams are similar to traditional twisted Gaussian Schell-model beams; however, instead of being spatially twisted (like the latter), the former possess a stochastic twist which couples their space and time dimensions. Like STOV beams, this spatiotemporal twist imbues twisted space-time beams with transverse (to the direction of propagation) angular momentum.

Generalizing the original research presented in Refs. [19, 20], here, we studied how twisted space-time beams behave as they propagate through atmospheric turbulence. Applying the extended Huygens–Fresnel principle, we derived the MCF for twisted space-time beams after propagating a distance z through atmospheric turbulence of arbitrary strength. From the MCF, we obtained the ensemble-averaged irradiance and quantified the effects of turbulence on beam size, pulse width, and space-time twist. We then simulated twisted space-time beam propagation through atmospheric turbulence to validate our theoretical analysis. The simulated results were found to be in good agreement with theory in weak-to-moderate turbulence. On the other hand, we observed rather poor agreement in strong turbulence, where our theoretical expression for the ensemble-averaged irradiance underestimated the effects of turbulence on the beam size, pulse width, and space-time twist. It did, however, accurately predict the trends of those parameters versus Fresnel number and turbulence strength.

Light with engineered space-time or spatiotemporal coupling is a new and exciting aspect of beam control research, with potential revolutionary uses in optical communications, optical tweezing, and quantum optics. While the free-space propagation characteristics of space-time-coupled beams are generally understood, much less is known about how these beams behave in random media. The results in this paper are a first step toward this goal.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author contributions

MH performed the tasks of conceptualization, formal analysis, investigation, methodology, validation, visualization, and writing.

Acknowledgments

The views expressed in this paper are those of the authors and do not reflect the official policy or position of the US Air Force, the Department of Defense, or the US government. MH would like to thank the Air Force Office of Scientific Research (AFOSR) Physical and Biological Sciences Branch (RTB) for supporting this work.

Conflict of interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: atmospheric turbulence, coherence, random media, random fields, space-time coupling, spatiotemporal coupling, statistical optics

Citation: Hyde IV MW (2023) The behavior of partially coherent twisted space-time beams in atmospheric turbulence. Front. Phys. 10:1055401. doi: 10.3389/fphy.2022.1055401

Received: 27 September 2022; Accepted: 02 December 2022;
Published: 09 January 2023.

Edited by:

Muhsin Caner Gokce, TED University, Turkey

Reviewed by:

Serkan Sahin, TED University, Turkey
Guoquan Zhou, Zhejiang Agriculture and Forestry University, China

Copyright © 2023 Hyde IV. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Milo W. Hyde IV, milo.hyde@afit.edu

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.