_{n}

^{2}= 10−16 to 10−14 m−2/3 and for various horizontal paths ranging from 1 to 10 km. Although the extended analytic theory stems from a study of coherent beams, the simulation results show good agreement with the analytical results for PCBs in fluctuation regimes ranging from weak to intermediate.

## 1.

## Introduction

Beam wander refers to the gross displacement of an optical beam intensity pattern relative to the pattern for an ideal (vacuum) propagation. Beam wander behavior and related attributes, such as root mean square (RMS) centroid and scintillation index, are important performance indicators of optical beams propagating through turbulence.^{1}2.^{–}^{3} These beam characteristics have been studied extensively by many researchers over the last five decades through different approaches and for a variety of beam types.^{4}5.6.7.^{–}^{8} The beam wander of a single ray was examined by Beckmann and Chernov using a geometrical optics (GO) approximation.^{4}^{,}^{9} An expression for the wander of a Gaussian beam was first developed using a Huygens–Fresnel approach,^{5} and later was derived by applying a Markovian random process approximation and Ehrenfest’s theorem from quantum mechanics.^{10}^{,}^{11} Eyyuboglu, Cil, and Baykal evaluated the beam wander behavior for several different beam types such as dark-hollow, flat-topped, annular, cos, and cosh-Gaussian beams.^{12}^{,}^{13} Beam wander effects for a spatially PCB were investigated by Berman et al., who introduced a photon distribution function method.^{14} To the best of our understanding, the effect of an average phase curvature applied to the PCB, for example “focus,” was not included in the study.

Recently, Andrews and Phillips developed expressions for the beam wander of a focused coherent Gaussian beam in a weak fluctuation turbulence regime by applying Rytov theory.^{15} Subsequently, Recolons et al. compared the theoretical models with simulations, and satisfactory agreement was observed.^{16} In terms of free space optical (FSO) applications, a convergent PCB or a PCB with some other specified wavefront curvature can greatly improve link performance.^{17} However, a convergent beam can still have a significant wander component. This brings us to the question: How does a focused PCB wander when propagating through turbulence?

In this paper, we extend the beam wander theory of Andrews and Phillips to include the focused PCB case. We examine the beam wander (RMS beam centroid) behavior and, in addition, investigate the beam size, the scintillation index, and the mean intensity patterns for both tracked and untracked beams. For validation purposes, a numerical wave optics simulation (WOS) is implemented to create focused PCBs and model their propagation through turbulence. Results are presented and compared with the analytic models for scenarios with turbulence strength varying from weak to strong and propagation distances ranging from 1 to 10 km.

## 2.

## Beam Wander Theory

## 2.1.

### Beam Wander Variance

Beam wander can be characterized by the transverse movement ${r}_{c}$ of the “hot spot” within the beam profile. The general expression of the beam wander variance, $\langle {r}_{c}^{2}\rangle $, was modeled by Andrews and Phillips as:^{15}

## (1)

$$\langle {r}_{c}^{2}\rangle =4{\pi}^{2}{k}^{2}{W}^{2}(Z){\int}_{0}^{Z}{\int}_{0}^{\infty}\kappa {\mathrm{\Phi}}_{n}(\kappa ){H}_{\mathrm{LS}}(\kappa ,z)\phantom{\rule{0ex}{0ex}}[1-{e}^{-{\mathrm{\Lambda}}_{P}Z{\kappa}^{2}{(1-z/Z)}^{2}/k}]\mathrm{d}\kappa \mathrm{d}z,$$^{18}

^{,}

^{19}

## (3)

$$W(z)={W}_{0}{[{(1-\frac{z}{F})}^{2}+(1+\frac{2{W}_{0}^{2}}{{l}_{c}^{2}}\left){\left(\frac{2z}{k{W}_{0}^{2}}\right)}^{2}\right]}^{1/2}\mathrm{.}$$Returning to Eq. (1), we now define the nondimensional output beam parameter ${\mathrm{\Lambda}}_{\stackrel{}{P}}$ of a PCB, and it can be written as

Typically,${\mathrm{\Phi}}_{n}(\kappa )$ in Eq. (1) is assumed to be the Kolmogorov spectrum, which is defined as where ${C}_{n}^{2}$ is the refractive index structure parameter and is assumed to be constant for horizontal propagation. Apply the geometrical optics (GO) approximation and the last term in Eq. (1) becomes:## (6)

$$1-{e}^{-{\mathrm{\Lambda}}_{P}Z{\kappa}^{2}{(1-z/Z)}^{2}/k}\approx \frac{{\mathrm{\Lambda}}_{P}Z{\kappa}^{2}{(1-z/Z)}^{2}}{k},\phantom{\rule[-0.0ex]{2em}{0.0ex}}\phantom{\rule{0ex}{0ex}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\mathrm{\Lambda}}_{P}Z{\kappa}^{2}}{k}\ll 1.$$^{15}

## (7)

$$\langle {r}_{c}^{2}\rangle =7.25{C}_{n}^{2}{Z}^{3}{W}_{0}^{-1/3}{\int}_{0}^{\stackrel{}{Z}}{(1-\frac{z}{Z})}^{2}{G}_{P}{(z)}^{-1/6}\mathrm{d}z,$$^{15}. For PCBs, this term could be significant and cannot be ignored. A closed-form analytic solution of Eq. (7) with Eq. (8) included is attainable and involves a hypergeometric function that is typically evaluated numerically. For the results in this paper, we bypass this step and simply evaluate Eq. (7) directly using numerical integration.

## 2.2.

### Mean Intensity Profiles

Consider a beam with the field $U(r,z)$, where $r$ is the transverse distance from the beam center in the plane perpendicular to the propagation direction. For a unit-amplitude field at the source plane ($z=0$), we have:

## (9)

$$U(r,0)=\mathrm{exp}(-\frac{{r}^{2}}{{W}_{0}^{2}}-i\frac{k{r}^{2}}{2F})\mathrm{exp}[i\xi (r,0)],$$^{20}

^{,}

^{21}the only free parameter that governs the beam profile is the receiving beam size/radius. Following Fante, who considered long-time (LT) and short-time (ST) averages, which correspond to untracked and tracked beams, respectively, the respective beam size values, ${W}_{\mathrm{LT}}(Z)$ and ${W}_{\mathrm{ST}}(z)$, are related by:

^{6}where $\langle {r}_{c}^{2}\rangle $ is the beam wander variance. Expressions for ${W}_{\mathrm{LT}}(Z)$ have been developed in several recent publications. We compared expressions presented by Andrews and Phillips, Ricklin and Davidson, and Korotkova et al. for the LT average beam size and irradiance distribution, and found that for a coherent Gaussian beam propagated over a relatively short distance and/or through weak turbulence, the theories from these authors

^{15}

^{,}

^{22}

^{,}

^{23}are almost identical and are consistent with our wave optics simulation results. However, for propagation through stronger turbulence, the theory put forward by Ricklin and Davidson provides a better fit to our simulation results. This expression for the LT average beam size is given by:

^{22}

## (11)

$${W}_{\mathrm{LT}}(Z)={W}_{0}{[{(1-\frac{Z}{F})}^{2}+(1+\frac{2{W}_{0}^{2}}{{l}_{c}^{2}}+\frac{2{W}_{0}^{2}}{{\rho}_{0}^{2}}\left){\left(\frac{2Z}{k{\stackrel{}{W}}_{0}^{2}}\right)}^{2}\right]}^{1/2},$$^{15}

^{,}

^{22}and

## 2.3.

### Scintillation Theory (On-Axis)

To be consistent with previous work on scintillation, we use the terms “tracked” for ST average results and “untracked” for LT average results in this section. The derivations of the on-axis scintillation indices of tracked and untracked coherent beams have been thoroughly discussed in several papers listed in Refs. 8 and 15. Corresponding analytical expressions for a focused PCB beam are found by incorporating the PCB beam size formulation. For a tracked beam, the on-axis scintillation index in weak turbulence is given by

## (15)

$${\sigma}_{I,\mathrm{tr},\text{weak}}^{2}(Z)=3.86{\sigma}_{R}^{2}\{0.4{[{(1+2{\mathrm{\Theta}}_{P})}^{2}+4{\mathrm{\Lambda}}_{P}^{2}]}^{5/12}\text{\hspace{0.17em}}\phantom{\rule{0ex}{0ex}}\mathrm{cos}[\frac{5}{6}\text{\hspace{0.17em}}{\mathrm{tan}}^{-1}\left(\frac{1+2{\mathrm{\Theta}}_{P}}{2{\mathrm{\Lambda}}_{P}}\right)]-\frac{11}{16}{\mathrm{\Lambda}}_{P}^{5/6}\},$$## (16)

$${\mathrm{\Theta}}_{\stackrel{}{P}}=\frac{1-\frac{Z}{F}}{{(1-\frac{Z}{F})}^{2}+{\left(\frac{2Z}{k{W}_{0}^{2}}\right)}^{2}(1+\frac{2{W}_{0}^{2}}{{l}_{c}^{2}})},$$## (18)

$${\sigma}_{I,\mathrm{un},\text{weak}}^{2}(Z)=4.42{\sigma}_{R}^{2}{\mathrm{\Lambda}}_{P}^{5/6}\frac{{\sigma}_{\mathrm{pe}}^{2}}{{W}^{2}(Z)}+{\sigma}_{I,\mathrm{tr}}^{2}(Z),$$## (19)

$${{\sigma}_{\mathrm{pe}}}^{2}=7.25{C}_{n}^{2}{Z}^{3}{W}_{0}^{-1/3}{\int}_{0}^{\stackrel{}{Z}}{(1-\frac{z}{Z})}^{2}\{{\left[\frac{1}{{G}_{P}(z)}\right]}^{1/6}-{\left[\frac{{K}_{r}^{2}{W}_{0}^{2}}{1+{K}_{r}^{2}{W}_{0}^{2}{G}_{P}(z)}\right]}^{1/6}\}\mathrm{d}z,$$^{15}which yields the general expression for the scintillation index for a tracked beam

## (21)

$${\sigma}_{I,\mathrm{tr}}^{2}(Z)=\mathrm{exp}\{\frac{0.49{\sigma}_{I,\mathrm{tr},\text{weak}}^{2}(Z)}{{[1+0.56(1+{\mathrm{\Theta}}_{P}){\sigma}_{I,\mathrm{tr},\text{weak}}^{12/5}(Z)]}^{7/6}}+\frac{0.51{\sigma}_{I,\mathrm{tr},\text{weak}}^{2}(Z)}{{[1+0.69{\sigma}_{I,\mathrm{tr},\text{weak}}^{12/5}(Z)]}^{5/6}}\}-1,$$## 3.

## Wave Optics Simulations

Comparison of analytic values with numerical WOS results helps validate both the theory and simulation modeling approaches. In addition, the numerical approach can help verify that the GO approximation used in the analytic result is acceptable. The WOS approach has been discussed in detail in various publications.^{24}25.^{–}^{26} For a focused PCB simulation we follow the approach developed for a collimated PCB but with a phase curvature term applied at the source plane.^{27}28.^{–}^{29} Specifically, the implementation procedure can be described as follows: (1) generate a random phase screen with appropriate spatial coherence length ${l}_{c}$;^{29} (2) apply the phase screen to a coherent beam in the source plane; (3) numerically “propagate” the beam to the observation plane^{28} through a separate set of phase screens that model atmospheric turbulence and compute the intensity; and (4) repeat steps 1 to 3 ${N}_{\mathrm{PS}}$ times, each time with a different realization of the spatial coherence screen (but without changing the turbulence screen realizations) and average the intensity at the observation plane. The average intensity is the PCB result. We typically limit ${N}_{\mathrm{PS}}$ to 30 to reduce computation time.

Atmospheric turbulence is simulated with a split-step approach involving a series of random screens evenly spaced along the propagation path.^{30} Our turbulence screens assume a Kolmogorov spectrum. The number of turbulence screens (${N}_{\mathrm{TS}}$) used for a particular scenario is determined following the criteria described in Ref. 24. For each data point presented, 500 propagations are simulated through different realizations of turbulence, which means a total of ${N}_{\mathrm{PS}}\times {N}_{\mathrm{TS}}=30\times 500=15,000$ propagations are required for each point. Both LT (untracked) and ST (tracked) cases are investigated. For ST (tracked) results, the beam centroid for each of the 500 turbulence realizations is found and each intensity pattern is translated to the center of the numerical grid before the measures are computed. The WOS program codes were implemented in the MATLAB environment.

The link parameters for results presented in this paper are listed in Table 1. The analysis and simulations can be applied to wide range of parameters, but the choices here are characteristic of moderate-length terrestrial links (1 to 10 km) in weak (${\sigma}_{R}^{2}<1$) to strong (${\sigma}_{R}^{2}>1$) intensity fluctuation regimes. The phase curvature parameter $F$ is set equal to the propagation distance $Z$ so the effect of focus over a range of distances can be investigated. PCB coherence lengths of 2 and 5 cm are typical of values that provide near-optimal link performance in applications like FSO communications.^{17}

## Table 1

WOS simulation parameters for link scenarios studied in this paper.

Scenario parameters | Cn2 | σR2 | NTS |
---|---|---|---|

W0=5 cm, λ=1 μm, Z=1 to 10 km, F=Z, lc=2 or 5 cm | 10−16 m−2/3 | 0.003 to 0.23 | 2 |

10−15 m−2/3 | 0.03 to 2.3 | 2 to 6 | |

10−14 m−2/3 | 0.3 to 22.7 | 2 to 20 |

Limiting the number of PCB phase screens (${N}_{\mathrm{PS}}=30$) when performing the turbulence simulations saves computation time but introduces additional fluctuations in the results, especially when ${l}_{c}$ is relatively small. To compensate for this effect, we simulate propagation through vacuum and calculate the beam wander variance ${\langle {r}_{c}^{2}\rangle}_{V}$ and scintillation index ${\sigma}_{c,V}^{2}$ that are purely caused by the limited number of PCB screens. To illustrate the magnitude of these errors, Fig. 1 presents vacuum results for ${l}_{c}=2$ and 5 cm. The false beam wander variance increases roughly in a linear fashion with propagation distance but the false scintillation index for both tracked and untracked beams tends to be independent of $Z$. The values are larger for the shorter coherence length. In the next section we define how corrections are implemented to compensate for the effects of using a limited number of PCB screens.

## 4.

## Analytic and WOS Results

In this section we present examples of analytic and WOS beam size, beam wander, and scintillation index results. However, a few example irradiance profiles are first presented to illustrate the characteristics of an average PCB propagated through turbulence. Figure 2 shows analytic tracked (ST) and untracked (LT) profiles created with Eqs. (13) and (14) and corresponding WOS results. In general, the profiles are Gaussian in shape and the tracked beams have higher peak intensities than the untracked beams. These examples also illustrate the close match between the analytic and WOS results. We reiterate that all the results in this section assume the beam is being focused for the indicated propagation distance.

## 4.1.

### Beam Size

The analytic expressions for LT and ST average beam sizes are given in Eqs. (10) and (11). For the WOS results, the first step in finding the LT beam size is averaging the two-dimensional intensity profiles produced by the 500 turbulence realizations and measuring the ${e}^{-2}$ irradiance radius. This gives an initial estimate of the beam size ${W}_{\mathrm{LT},S,0}$. The beam size corrected for effects of a limited number of PCB screens is given by

## (23)

$${W}_{\mathrm{LT},S}=\sqrt{{W}_{\mathrm{LT},S,0}^{2}-{\langle {r}_{c}^{2}\rangle}_{\stackrel{}{V}}}\mathrm{.}$$Analytic and WOS beam size results are presented in Fig. 3. As expected, the beam size increases with propagation/focus distance, stronger turbulence, and smaller coherence length. The analytic and WOS results are consistent in all cases for the turbulence strengths of ${C}_{n}^{2}={10}^{-16}$ and ${10}^{-15}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{m}}^{-2/3}$. For the stronger turbulence case of ${C}_{n}^{2}={10}^{-14}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{m}}^{-2/3}$ and with the tracked beam, the analytic prediction has somewhat higher values than the WOS results.

## 4.2.

### Beam Wander

The analytic beam wander result is given in Eq. (7). The corrected WOS result ${\langle {r}_{c}^{2}\rangle}_{S}$ is obtained via

Figure 4 shows comparisons of the analytical and WOS results. The wander increases with propagation/focus distance and also with turbulence strength. Dependence on the coherence length ${l}_{c}$ is weak for the cases presented. The analytic and WOS results are generally consistent, although the WOS tends to produce slightly larger beam wander values. This trend is most obvious in the strong turbulence case (${C}_{n}^{2}={10}^{-14}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{m}}^{-2/3}$) at longer propagation/focus distances.## 4.3.

### Scintillation Index

The analytic scintillation index expressions (on-axis) for tracked and untracked beams are given in Eqs. (21) and (22). For the WOS, the scintillation index is found by computing the normalized variance of the irradiance at the center of numerical grid (optical axis) for the 500 turbulence realizations. Again, the tracked results include the translation of the beam to remove the centroid wander. To compensate for the effects of the finite number of PCB phase screens, the following is used:^{23}

## (25)

$${\sigma}_{I,S}^{2}=\frac{{\sigma}_{I,S,0}^{2}-{\sigma}_{I,V}^{2}}{1+{\sigma}_{I,V}^{\mathrm{2,}}},$$Figure 5 presents the scintillation results. In general, the index values increase with distance and turbulence strength. The WOS values generally follow the analytic results, although there is clearly more scatter in the WOS points for the ${l}_{c}=2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cm}$ cases. This is likely due to imperfect correction for the limited number of PCB screens. Saturation of the index value is evident at longer distances for the stronger turbulence cases. We note that for increasing distance beyond that shown in Fig. 5, and therefore increasing Rytov variances, the focused PCB index values appear to peak and then reduce to a saturation value in a way that is similar to results for coherent beams.^{31} We expect the saturation regime for a PCB to occur at larger Rytov variances than for a comparable coherent beam.^{18} However, further study of this regime would likely require: (1) inclusion of a finite inner scale in the turbulence spectrum, as the scintillation index is highly sensitive to the inner scale; and (2) the exploration of WOS approaches that accurately model the strong fluctuation regime, likely involving high densities of grid points and turbulent screens.

## 5.

## Conclusion

The extension of coherent beam wander theory to the case of a focused PCB essentially involves the substitution of the beam waist expression that includes the PCB parameters. Comparison of the resulting analytic expressions for beam size, beam wander, and on axis-scintillation index with WOS results indicates that the GO approximation used in the coherent theory appears reasonable for the PCB theory, at least for most of the link values we studied. The WOS results required corrections for errors caused by the use of a limited number of PCB phase screens. Differences between the analytic and WOS results were primarily seen in stronger turbulence situations. The differences could be due to the use of the Rytov approximation in the analytic derivations but also to numerical issues in the WOS—for example, incomplete correction of the error due to the limited number of PCB screens.

## Acknowledgments

This work was supported by the Air Force Office of Scientific Research under the Sensing, Surveillance, and Navigation program, grant FA9550-09-1-0616.

## References

## Biography

**Xifeng Xiao** received her BS and MS degrees in physics from Xiamen University, Fujian, China, in 1998 and 2001, respectively. She earned her MS and PhD degrees in electrical engineering in 2004 and 2008, respectively, from New Mexico State University, where she is currently a research assistant professor. Her main research includes simulation and modeling of free-space laser communication, liquid-crystal polarization, acousto-optic imaging spectrometer design, and polarimetric imaging.

**David G. Voelz** is a professor of electrical engineering at New Mexico State University. He received his BS in electrical engineering from New Mexico State University in 1981 and his MS and PhD degrees in electrical engineering from the University of Illinois in 1983 and 1987, respectively. From 1986 to 2001, he was with the Air Force Research Laboratory in Albuquerque, NM. His current research interests include spectral and polarimetric imaging, laser imaging and beam projection, laser communications, adaptive optics, and astronomical instrumentation development.