Part II of this two-part paper uses wave-optics simulations to look at the Monte Carlo averages associated with turbulence and time-dependent thermal blooming (TDTB). The goal is to investigate turbulence thermal blooming interaction (TTBI). At wavelengths near 1 μm, TTBI increases the amount of constructive and destructive interference (i.e., scintillation) that results from high-power laser beam propagation through distributed-volume atmospheric aberrations. As a result, we use the spherical-wave Rytov number, the number of wind-clearing periods, and the distortion number to gauge the strength of the simulated turbulence and TDTB. These parameters simply greatly given propagation paths with constant atmospheric conditions. In addition, we use the log-amplitude variance and the branch-point density to quantify the effects of TTBI. These metrics result from a point-source beacon being backpropagated from the target plane to the source plane through the simulated turbulence and TDTB. Overall, the results show that the log-amplitude variance and branch-point density increase significantly due to TTBI. This outcome poses a major problem for beam-control systems that perform phase compensation.

## 1.

## Introduction

At wavelengths near $1\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$, the effects of turbulence are often more dominant than the effects of thermal blooming. Thus, at short-exposure time scales, one does not typically see fully formed crescent or half-moon irradiance patterns, as described in Fig. 1, due to thermal blooming at wavelengths near $1\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$. From a historical perspective, however, fully formed crescent or half-moon irradiance patterns do repeatedly result from the dominant effects of thermal blooming at wavelengths in the mid-wave and long-wave infrared.^{1}^{–}^{5}

There is, nevertheless, an interaction that occurs between turbulence and thermal blooming at wavelengths near $1\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$.^{6}^{–}^{10} This so-called turbulence thermal blooming interaction (TTBI), in practice, results in an increased amount of scintillation, which is the constructive and destructive interference that results from propagating high-power laser beams through distributed-volume atmospheric aberrations or “deep turbulence.” In general, the scintillation caused by turbulence results in localized hot spots that cause localized heating of the atmosphere. This localized heating results in localized defocus-like optical effects (aka localized thermal blooming), which causes more constructive and destructive interference upon propagation through the atmosphere. Given weak-turbulence conditions, for example, the resulting scintillation caused by TTBI is often analogous to that experienced with deep-turbulence conditions. Therefore, thermal blooming can have a major impact on system performance (even at wavelengths near $1\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$).

Beam-control (BC) systems, in theory, can mitigate the nonlinear optical effects induced by thermal blooming.^{11} However, when one uses a single deformable mirror (DM) for phase-only compensation, analysis predicts the possibility of an instability due to positive feedback in the BC system.^{12}^{–}^{16} Appropriately termed phase compensation instability (PCI), the positive feedback arises with the time-dependent development of scintillation within the propagating high-power laser beam. Recall that the localized hot spots produce defocus-like optical effects in the atmosphere. A BC system corrects for the hot spots by applying focus-like phase compensations. In turn, these phase compensations increase the strength of the localized thermal blooming, which leads to a runaway condition until something mitigates the positive feedback (e.g., wind variations in the atmosphere^{17}^{–}^{21} or branch points in the phase function^{22}^{–}^{26}).

Whether from TTBI or PCI, an increase in the amount of scintillation leads to an increase in the amount of total destructive interference, which leads to amplitude nulls in both the real and imaginary parts of the complex optical field. These amplitude nulls cause branch points and branch cuts (i.e., $2\pi $ discontinuities) to arise in the collimated phase function of a backpropagated beacon.^{27}^{,}^{28} This outcome poses a major problem for BC systems that perform phase compensation. In practice, these branch points and branch cuts lead to fitting error in a BC system that uses a single continuous-face-sheet DM with a high-power coating for phase-only compensation.^{29}^{–}^{33} With this last point in mind, the atmospheric propagation research community needs to quantify the impacts of TTBI and PCI using the larger grid sizes allowed by modern-day, wave-optics simulations.

The following is Part II of a two-part paper on TTBI performed using AOTools and WaveProp—both of which are MATLAB toolboxes written by the Optical Sciences Company.^{34}^{–}^{38} In turn, this paper investigates TTBI in the presence of turbulence and time-dependent thermal blooming (TDTB). It does so via the Monte Carlo averages associated with the log-amplitude variance and the branch-point density. These metrics result from a point-source beacon being backpropagated from the target plane to the source plane through the simulated turbulence and TDTB. Part I of this two-part paper then investigates TTBI in the presence of turbulence and SSTB. Together, these papers will inform future wave-optics investigations.

It is important to note that this paper stands on its own as an independent article. However, it is also important to note that Part I of this two-part paper complements the analysis contained in this paper. In general, the steady-state assumptions contained in Part I allows us to explore the trade space in a computationally efficient manner, whereas the time-dependent assumptions contained in this paper allow us to examine the Monte Carlo averages with increased computational fidelity. For example, relative to the time-dependent results contained in this paper, the steady-state results contained in Part I provide an upper bound on both the increase in the log-amplitude variance and the branch-point density due to TTBI. Such results will hopefully prove fruitful in the development of next-generation scaling laws that account for the effects of TTBI.^{39}^{–}^{45}

In what follows, Sec. 2 contains the setup for this paper, whereas Sec. 3 explores the trade space. Results and discussion then follow in Sec. 4, with a conclusion in Sec. 5. Before moving on to the next section, it is worth mentioning that this paper builds upon the preliminary analysis presented by Murphy and Spencer in a recent conference proceeding.^{46} In particular, this paper develops time-dependent simulations with increased computational fidelity to clearly show an increase in both the log-amplitude variance and branch-point density due to TTBI. These results serve as a novel contribution to the atmospheric propagation research community.

## 2.

## Setup for the Wave-Optics Simulations

The desired setup is as follows. We wish to propagate a focused high-power laser beam with a wavelength of $1\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ along a propagation path with constant atmospheric conditions. Afterward, we wish to propagate a point-source beacon of the same wavelength back along the same path through the simulated turbulence and TDTB. For this purpose, we quickly review the details associated with the split-step beam propagation method (BPM), spherical-wave Rytov number, TDTB, number of wind-clearing periods, distortion number, and parameters of interest in the following sections. Given a common setup, this paper shares most of these sections with Part I of this two-part paper on TTBI. The analysis in this paper specifically differs in the sections on TDTB, number of wind-clearing periods, and parameters of interest. With that said, we include the shared sections in both papers for reading independence.

## 2.1.

### Split-Step Beam Propagation Method

In this paper, we make use of the commonly used split-step BPM, which simulates the propagation of monochromatic and polychromatic light through the atmosphere.^{47}^{–}^{50} As described by Schmidt,^{51} the split-step BPM divides the propagation path into independent volumes, with the atmospheric aberrations in each volume being represented by a single phase screen. The split-step BPM makes use of angular spectrum or plane-wave spectrum propagation to vacuum-propagate the light from a source plane to the first phase screen, applies the phase screen, and repeats this process until the monochromatic light reaches a target plane. Before moving on in the analysis, it is worth mentioning that AOTools and WaveProp make use of the split-step BPM. It is also worth mentioning that AOTools and WaveProp generate the phase screens associated with turbulence similar to the approach presented in Chapter 9 of Ref. 51. In this paper, we specifically use a Kolmogorov power-spectral density and do not add subharmonics or additional tilt to the generated phase screens.

## 2.2.

### Spherical-Wave Rytov Number

Provided the Rytov approximation, the propagation of a spherical wave through turbulence has an associated path-integral expression that serves as a gauge for the amount of scintillation.^{41}^{,}^{51}^{–}^{54} Known as the spherical-wave Rytov number ${\mathcal{R}}_{\mathrm{sw}}$ (aka the spherical-wave Rytov parameter or spherical-wave, log-amplitude variance), this path-integral expression takes the following form:

## Eq. (1)

$${\mathcal{R}}_{\mathrm{sw}}=0.563{k}^{7/6}{\int}_{0}^{Z}{C}_{n}^{2}(z){z}^{5/6}{(1-\frac{z}{Z})}^{5/6}\mathrm{d}z,$$^{52}

^{,}

^{53}and branch points and branch cuts (i.e., $2\pi $ discontinuities) readily show up in the phase function.

^{27}

^{,}

^{28}Thus, in the analysis that follows, we will use ${\mathcal{R}}_{\mathrm{sw}}$ as a gauge for the strength of the simulated turbulence.

## 2.3.

### Time-Dependent Thermal Blooming

The heating of the atmosphere due to an absorbed irradiance is balanced by the cooling of the atmosphere due to a transverse wind blowing across the high-power laser beam. Mathematically, we realize this heating and cooling via a forced-advection equation, such that

## Eq. (3)

$$\frac{\partial}{\partial t}\mathrm{\Delta}n(\mathbf{r},t)+{\mathbf{v}}_{\perp}(z)\xb7{\nabla}_{\perp}\mathrm{\Delta}n(\mathbf{r},t)=-\mu (z){i}_{\mathrm{HP}}(\mathbf{r},t).$$It is informative to determine the impulse response to the energy-balance equation. For this purpose, we rewrite Eq. (3) in operator form as

where $\mathcal{L}\{\odot \}=[\partial /\partial t+{\mathbf{v}}_{\perp}(z)\xb7{\nabla}_{\perp}]\odot $ is a linear operator that represents cooling of the atmosphere and $f(x,y,z,t)=\mu (z){i}_{\mathrm{HP}}(\mathbf{r},t)$ is a forcing function that represents heating of the atmosphere. Now, we can make use of a Green’s function analysis, such that## Eq. (6)

$$\mathcal{L}\{G(x,y,z,t;\xi ,\eta ,\zeta ,\tau )\}=-\delta (x-\xi ,y-\eta ,z-\zeta ,t-\tau ),$$## Eq. (7)

$$\mathrm{\Delta}n(x,y,z,t)={\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}{\int}_{-\infty}^{\infty}G(x,y,z,t;\xi ,\eta ,\zeta ,\tau )f(\xi ,\eta ,\zeta ,\tau )\mathrm{d}\xi \text{\hspace{0.17em}}\mathrm{d}\eta \text{\hspace{0.17em}}\mathrm{d}\zeta \text{\hspace{0.17em}}\mathrm{d}\tau .$$To determine the Green’s function $G(x,y,z,t;\xi ,\eta ,\zeta ,\tau )$ needed in Eq. (7), we can use a two-dimensional (2-D) Fourier transformation and a unilateral Laplace transformation to transform Eq. (6) into an algebraic expression. If we assume that there is no initial heating of the atmosphere, then we can easily solve for the Green’s function in the frequency domain. Afterward, we can use a 2-D inverse Fourier transformation and an inverse Laplace transformation to transform back into the spatial and temporal domains, so that

## Eq. (8)

$$G(x,y,z,t;\xi ,\eta ,\zeta ,\tau )=G(x-\xi ,y-\eta ,z-\zeta ,t-\tau )\phantom{\rule{0ex}{0ex}}=-\delta [x-\xi +{v}_{x}(z)(t-\tau ),y-\eta -{v}_{y}(z)(t-\tau ),z-\zeta ]\text{step}(t-\tau ).$$If we substitute Eq. (8) into Eq. (7), we can account for the effects of TDTB. With a little manipulation, we determine the change in the refractive index $\mathrm{\Delta}n(\mathbf{r},t)$ as

## Eq. (10)

$$\mathrm{\Delta}n(x,y,z,t)=-\mu (z){\int}_{0}^{t}{i}_{\mathrm{HP}}[x-{v}_{x}(z)(t-\tau ),y-{v}_{y}(z)(t-\tau ),z,\tau ]\mathrm{d}\tau .$$We can account for the effects of TDTB using AOTools and WaveProp, since these MATLAB toolboxes solve the integral expression found in Eq. (10) numerically.^{34}^{–}^{38} In particular, AOTools and WaveProp updates the change in the refractive index $\mathrm{\Delta}n(\mathbf{r},t)$ from a previous time step ${t}_{j}$ to the present time step ${t}_{j+1}$ with the time step $\mathrm{\Delta}t={t}_{j+1}-{t}_{j}$, viz.

## Eq. (11)

$$\mathrm{\Delta}n(x,y,z,{t}_{j+1})=-\mu (z){i}_{\mathrm{HP}}(x,y,z,{t}_{j+1})\mathrm{\Delta}t+\mathrm{\Delta}n[x-{v}_{x}(z)\mathrm{\Delta}t,y-{v}_{y}(z)\mathrm{\Delta}t,z,{t}_{j}].$$When using $N\times N$ grids to implement Eq. (11) into wave-optics simulations, it is important to note that the amount of spatial shift is often less than or greater than the grid sampling $\delta $. As a result, it is common practice to break the time development of Eq. (11) into subtime steps.^{35}^{,}^{36} Waveprop and AOTools subsequently uses linear interpolation for each subspatial shift. It is also important to note that when assuming Taylor’s frozen flow, time-dependent turbulence satisfies an unforced version of Eq. (3). As a result, WaveProp and AOTools uses Eq. (11), without the first term on the right-hand side of the equals sign, to simulate the effects of time-dependent turbulence.

## 2.4.

### Number of Wind-Clearing Periods

In this paper, we define the wind-clearing time, ${t}_{0}$, as the period of time needed for the initial transverse wind speed, $|{\mathbf{v}}_{\perp}(0)|={v}_{0}$ (in the source plane at $z=0$), to travel across the initial diameter, ${D}_{0}$, where

If there are no variations in the exitance of the focused high-power laser beam and ambient atmosphere, then TDTB can reach a steady state with a fully formed crescent or half-moon irradiance pattern in the target plane (cf. Fig. 1). Different parameters of interest (cf. the next section) require different multiples of ${t}_{0}$ for TDTB to reach a true steady state. As such, we define multiples of ${t}_{0}$ in terms of a parameter referred to here as the number of wind-clearing periods ${N}_{\mathrm{WCP}}$. Moving forward we will use ${N}_{\mathrm{WCP}}$ as a gauge for how much the wind transverses ${D}_{0}$.After the wind transverses the initial diameter ${D}_{0}$ at least once (when ${N}_{\mathrm{WCP}}\ge 1$), the conditions for SSTB are generally met. In practice, SSTB results in a static behavior of the change in the refractive index. This static behavior causes the time rate of change of the change in the refractive index to be zero. Consequently, we drop any time dependence, and the position-dependent change in the refractive index $\mathrm{\Delta}n(\mathbf{r})$ due to SSTB results as

## Eq. (13)

$$\mathrm{\Delta}n(\mathbf{r})=\mathrm{\Delta}n(x,y,z)=-\frac{\mu (z)}{{v}_{\perp}(z)}{\int}_{-\infty}^{x}{i}_{\mathrm{HP}}(\xi ,y,z)\mathrm{d}\xi .$$## 2.5.

### Distortion Number

It is useful to describe the refraction caused by SSTB in terms of a phase error $\varphi (x,y,Z)$ measured in radians. For this purpose,

where again $k=2\pi /\lambda $ is the angular wavenumber and $Z$ is the propagation distance. Substituting Eq. (13) into Eq. (14) results in the following relationship:## Eq. (15)

$$\varphi (x,y,Z)=-k{\int}_{0}^{Z}\frac{\mu (z)}{{v}_{\perp}(z)}{\int}_{-\infty}^{x}{i}_{\mathrm{HP}}(\xi ,y,z)\mathrm{d}\xi \text{\hspace{0.17em}}\mathrm{d}z.$$For focused high-power laser beams with initial power, ${P}_{0}$, and initial diameter, ${D}_{0}$, we can rewrite Eq. (15) in terms of a path-integral expression known as the distortion number, ${N}_{D}$. In particular,

## Eq. (16)

$$\varphi (x,y,Z)\approx -{N}_{D}\frac{{D}_{0}}{4\sqrt{2}{P}_{0}}{\int}_{-\infty}^{x}{i}_{\mathrm{HP}}(\xi ,y,0)\mathrm{d}\xi $$## Eq. (17)

$${N}_{D}=4\sqrt{2}{P}_{0}k{\int}_{0}^{Z}\frac{Z-z}{Z}\frac{\mu (z)\tau (z)}{D(z){v}_{\perp}(z)}\mathrm{d}z.$$Given propagation paths with constant atmospheric conditions, Eq. (17) simplifies, such that

## Eq. (20)

$${N}_{D}=\frac{4\sqrt{2}{P}_{0}k({n}_{0}-1)\alpha {\mathrm{e}}^{-\alpha Z}{\mathrm{e}}^{-\sigma Z}Z}{{C}_{P}{\rho}_{0}{T}_{0}{D}_{0}{v}_{\perp}}.$$^{34}This regime is where the radians of distortion induced by SSTB give rise to significant fluctuations in the fully formed crescent or half-moon irradiance pattern (cf. Fig. 1). Thus, in the analysis that follows, we will use ${N}_{D}$ as a gauge for the strength of the simulated TDTB.

## 2.6.

### Parameters of Interest

Table 1 contains all the parameters of interest in the wave-optics simulations. It is important to note that the wave-optics simulations used $N\times N$ grids. The side length $S$ was the same in both the source and target planes creating unity scaling within the wave-optics simulations. By choice, the wave-optics simulations also satisfied Fresnel scaling, such that $N={S}^{2}/(\lambda Z)$, where $\lambda $ is the wavelength and $Z$ is the propagation distance. The resulting $N\times N$ grid also minimized the effects of aliasing without making the wave-optics simulations too computationally expensive.

## Table 1

Parameters of interest in the wave-optics simulations.

Parameters (MKS units) | Symbol | Value(s) |
---|---|---|

Grid | $N\times N$ | $1024\times 1024$ |

Side length (m) | $S$ | 2.263 |

Wavelength (m) | $\lambda $ | $1\times {10}^{-6}$ |

Propagation distance (m) | $Z$ | 5000 |

Distortion number (rad) | ${N}_{D}$ | $16\sqrt{2}$ |

Initial power (kW) | ${P}_{0}$ | 125.4 |

Ambient refractive index difference | $({n}_{0}-1)$ | $2.602\times {10}^{-4}$ |

Absorption coefficient (${\mathrm{m}}^{-1}$) | $\alpha $ | $5\times {10}^{-6}$ |

Scattering coefficient (${\mathrm{m}}^{-1}$) | $\sigma $ | $5\times {10}^{-5}$ |

Specific heat at constant pressure (J/kg/K) | ${C}_{P}$ | 1004 |

Density of air at constant pressure ($\mathrm{kg}/{\mathrm{m}}^{3}$) | ${\rho}_{0}$ | 1.293 |

Ambient temperature (K) | ${T}_{0}$ | 300 |

Transverse wind speed (m/s) | ${v}_{\perp}$ | 5 |

In addition to Fresnel scaling, we ensured that the wave-optics simulations had at least 10 pixels across the spherical-wave Fried parameter ${r}_{0,\mathrm{sw}}$ (aka the spherical-wave coherence diameter or spherical-wave coherence length).^{41}^{,}^{51}^{–}^{54} Given a propagation path with constant atmospheric conditions, the associated path-integral expression simplifies into the following closed-form expression:

## Eq. (21)

$${r}_{0,\mathrm{sw}}={[0.423{k}^{2}{\int}_{0}^{Z}{C}_{n}^{2}(z){\left(\frac{z}{Z}\right)}^{5/3}\mathrm{d}z]}^{-3/5}\Rightarrow {r}_{0,sw}={(0.159{k}^{2}{C}_{n}^{2}Z)}^{-3/5}.$$^{41}

^{,}

^{51}

^{–}

^{54}it also makes use of the isoplanatic angle, ${\theta}_{0}$, and the Greenwood frequency, ${f}_{G}$, such that

## Eq. (22)

$${\theta}_{0}={[2.91{k}^{2}{\int}_{0}^{Z}{C}_{n}^{2}(z)(Z-z{)}^{5/3}\mathrm{d}z]}^{-3/5}\Rightarrow {\theta}_{0}=0.314\frac{{r}_{0,\mathrm{sw}}}{Z}$$## Eq. (23)

$${f}_{G}={[0.102{k}^{2}{\int}_{0}^{Z}{C}_{n}^{2}(z){v}_{\perp}{(z)}^{5/3}\mathrm{d}z]}^{3/5}\Rightarrow {f}_{G}={(0.102{k}^{2}{C}_{n}^{2}{v}^{5/3}Z)}^{3/5}.$$## Table 2

Turbulence scenario used in the wave-optics simulations. Recall that Cn2 is the refractive index structure coefficient, Rsw is the spherical-wave Rytov number, D0 is the aperture diameter, r0,sw is the spherical-wave Fried parameter, θ0 is the isoplanatic angle, λ/D0 is the diffraction-limited angle, and fG is the Greenwood frequency.

Scenario | Rsw | Cn2 (m−2/3) | D0/r0,sw | θ0/(λ/D0) | fG (Hz) |
---|---|---|---|---|---|

1 | 0.25 | $0.391\times {10}^{-14}$ | 8.96 | 1.75 | 68.6 |

Before moving on to the next section, it is important to note that we used the Greenwood frequency ${f}_{G}$ (rounded to one significant figure) to determine the time step $\mathrm{\Delta}t$ needed in the wave-optics simulations. In particular,

## Eq. (24)

$$\mathrm{\Delta}t=\frac{1}{20{f}_{G}}=0.714\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{ms}.$$## 3.

## Exploration Using the Wave-Optics Simulations

In this section, we make use of the wave-optics simulations setup in the previous section to explore the trade space. The goal is to investigate TTBI in terms of the normalized power in the bucket (PIB), ${P}_{N}$, and the peak Strehl ratio, ${S}_{P}$, associated with a focused high-power laser beam being propagated from the source plane at $z=0$ to the target plane at $z=Z$, and the log-amplitude variance, ${\sigma}_{\chi}^{2}$, and the branch-point density, ${\mathcal{D}}_{\mathrm{BP}}$, associated with a point-source beacon being backpropagated from the target plane at $z=Z$ to the source plane $z=0$. Here,

## Eq. (25)

$${P}_{N}=\frac{{\int}_{-\infty}^{\infty}\mathrm{cyl}\left(\frac{\sqrt{{x}^{2}+{y}^{2}}}{{D}_{Z}}\right){i}_{\mathrm{HP}}(x,y,Z)\mathrm{d}x\text{\hspace{0.17em}}\mathrm{d}y}{{P}_{0}},$$## Eq. (26)

$${S}_{P}=\frac{\mathrm{max}\{{i}_{\mathrm{HP}}(x,y,Z)\}}{{i}_{\mathrm{DL}}(\mathrm{0,0},Z)},$$## Eq. (29)

$$\mathrm{cyl}(\rho )=\{\begin{array}{ll}1& 0\le \rho <1/2\\ 1/2& \rho =1/2\\ 0& \rho >1/2\end{array};$$To calculate the number of branch points ${N}_{\mathrm{BP}}$ in Eq. (28), we used the following relationship:

where ${N}_{+}$ is the number of positively charged branch points and ${N}_{-}$ is the number of negatively charged branch points within the collimated phase function $\varphi (x,y,0)$. This relationship says that we can determine the location of a branch point when the line integral around the closed curve $C$ of the gradient of the collimated phase function $\nabla \varphi (x,y,0)$ does not equal zero, specifically, where $\nabla \varphi (x,y,0)$ is a nonconservative vector field. To account for this relationship numerically, AOTools and WaveProp discretely samples the continuous integral in Eq. (30) by breaking the $N\times N$ grid into a bunch of $2\times 2$ subgrids and summing up the phase derivative around each point. A positive $2\pi $ value results in a positively charged branch point and a negative $2\pi $ value results in a negatively charged branch point. In turn, to calculate ${N}_{\mathrm{BP}}$, we computed the sum of the total number of positive and negative branch points associated with the pixels found within $\varphi (x,y,0)$.In what follows, we will use the metrics defined in Eqs. (25)–(28) to visualize the following topics:

1. the number of wind-clearing periods needed to simulate the effects of TDTB,

2. the focused high-power laser beam in the target plane,

3. the backpropagated point-source beacon in the source plane, and

4. the overall trade space in terms of the normalized PIB, ${P}_{N}$, and the peak Strehl ratio, ${S}_{P}$.

Each topic is the subject of the following sections. These sections will inform the results and discussion presented in Sec. 4.

## 3.1.

### Number of Wind-Clearing Periods Needed

To determine the number of wind-clearing periods, ${N}_{\mathrm{WCP}}$, needed to accurately simulate TDTB using the split-step BPM, we calculated both the normalized PIB, ${P}_{N}$, and the peak Strehl ratio, ${S}_{P}$, as a function of ${N}_{\mathrm{WCP}}$. In addition, we calculated ${P}_{N}$ and ${S}_{P}$ assuming SSTB conditions. Figure 2 shows the outcomes of these calculations. As shown, we can see that both ${P}_{N}$ and ${S}_{P}$ reach steady state (accurate to the third decimal place) when ${N}_{\mathrm{WCP}}\ge 1$. Thus, in the analysis that follows, we used ${N}_{\mathrm{WCP}}=2$ to simulate the effects of TDTB.

Note that we used 50 equally spaced phase screens to simulate the effects of both TDTB and SSTB in Fig. 2. In general, this choice was accurate to the third decimal place (cf. Fig. 2 in Part I). For convenience in the wave-optics simulations, we also used 50 equally spaced phase screens to simulate the effects of turbulence. Therefore, in the analysis that follows, the phase screens used for simulating turbulence and TDTB (using the split-step BPM) were collocated along the propagation path. This choice led to small percentage errors (less than a 10th of a percentage) between the continuous and discrete calculations of the parameters found in Table 2.^{51}

## 3.2.

### Focused High-Power Laser Beam

To create the focused high-power laser beam, we used a series of steps starting with the creation of a positive thin lens transmittance function of circular diameter ${D}_{0}$ and focus $Z$. Assuming plane-wave illumination, we then set the exitance ${i}_{0}$ of the focused high-power laser beam, such that ${i}_{0}=4{P}_{0}/(\pi {D}_{0}^{2})$. This series of steps created a top-hat or flat-top beam profile in the source plane with approximately 256 grid points across ${D}_{0}$. As discussed above, AOTools and WaveProp then used the split-step BPM to propagate the focused high-power laser beam from the source plane to the target plane.

Based on previously published theoretical explorations,^{6}^{–}^{10} we hypothesized that the simulated turbulence and TDTB would increase the amount of scintillation found in a propagated high-power laser beam due to TTBI. In turn, the normalized PIB ${P}_{N}$ and peak Strehl ratio ${S}_{P}$ for the TTBI case would typically decrease in comparison to the simulated diffraction-limited, turbulence-only, and TDTB-only cases. Figure 3 demonstrates this hypothesis to be true for one Monte Carlo realization, where ${\mathcal{R}}_{\mathrm{sw}}=0.25$, ${N}_{\mathrm{WCP}}=1$, and ${N}_{D}={N}_{C}=16\sqrt{2}\approx 22.6$ [cf. Eqs. (2), (12), and (20), respectively]. Notice that we report values for both ${P}_{N}$ and ${S}_{P}$ at the top of each normalized irradiance subplot in Fig. 3. Also notice that these irradiance subplots are similar but different to those reported in Part I using steady-dependent simulations (cf. Fig. 3 in Part I) since both papers use the same Monte Carlo realization of turbulence.

## 3.3.

### Backpropagated Point-Source Beacon

To create the backpropagated point-source beacon, AOTools and WaveProp used a series of steps starting with the creation of a positive thin lens transmittance function of square width $2{D}_{0}$ and focus $Z$. Assuming Fresnel scaling, AOTools and WaveProp vacuum-propagated this positive thin lens transmittance function from the source plane to the target plane using angular spectrum or plane-wave spectrum propagation. This series of steps created a sinc-like function in the target plane with three pixels across its central lobe. As discussed above, AOTools and WaveProp then used the split-step BPM to backpropagate the point-source beacon from the target plane to the source plane.

Again, based on previously published theoretical explorations,^{6}^{–}^{10} we hypothesized that the simulated turbulence and TDTB would increase the amount of scintillation found in a backpropagated point-source beacon due to TTBI. As a result, the log-amplitude variance ${\sigma}_{\chi}^{2}$ and branch-point density ${\mathcal{D}}_{\mathrm{BP}}$ for the TTBI case would typically increase in comparison to the simulated diffraction-limited, turbulence-only, and TDTB-only cases. Figures 4 and 5 demonstrate this hypothesis to be true for one Monte Carlo realization, where ${\mathcal{R}}_{\mathrm{sw}}=0.25$, ${N}_{\mathrm{WCP}}=1$, and ${N}_{D}={N}_{C}=16\sqrt{2}\approx 22.6$ [cf. Eqs. (2), (12), and (20), respectively]. Notice that we report values for both ${\sigma}_{\chi}^{2}$ and ${\mathcal{D}}_{\mathrm{BP}}$ at the top of each normalized irradiance subplot in Fig. 4 and each wrapped phase subplot in Fig. 5. Also notice that these irradiance subplots are similar but different to those reported in Part I using steady-state simulations (cf. Figs. 4 and 5 in Part I) since both papers use the same Monte Carlo realization of turbulence.

To further explore the effects of TTBI, we also calculated the irrotational phase estimate, ${\widehat{\varphi}}_{\mathrm{irr}}(x,y,0)$, and rotational phase estimate, ${\widehat{\varphi}}_{\mathrm{rot}}(x,y,0)$. Figure 6 shows these estimates. In practice, these estimates originated from the collimated phase functions $\varphi (x,y,0)$ found in Figs. 5(b) and 5(d), for the turbulence-only and TTBI cases, respectively. To calculate ${\widehat{\varphi}}_{\mathrm{irr}}(x,y,0)$ and ${\widehat{\varphi}}_{\mathrm{rot}}(x,y,0)$ from $\varphi (x,y,0)$, we used the following relationships:^{27}^{–}^{31}

## Eq. (31)

$${\widehat{\varphi}}_{\mathrm{irr}}(x,y,0)=\mathrm{LS}\{\varphi (x,y,0)\}={\varphi}_{\mathrm{LS}}(x,y,0)$$## Eq. (32)

$${\widehat{\varphi}}_{\mathrm{rot}}(x,y,0)=\mathrm{arg}\{\mathrm{exp}(j[\varphi (x,y,0)-{\varphi}_{\mathrm{LS}}(x,y,0)])\},$$Figure 6 shows that both the irrotational phase estimate, ${\widehat{\varphi}}_{\mathrm{irr}}(x,y,0)$, and the rotational phase estimate, ${\widehat{\varphi}}_{\mathrm{rot}}(x,y,0)$, contain more phase features due to the TTBI case [cf. Figs. 6(b) and 6(d)] compared to the turbulence-only case [cf. Figs. 6(a) and 6(c)]. This outcome says that a BC system, in general, will need more resolution and stroke to correct for ${\widehat{\varphi}}_{\mathrm{irr}}(x,y,0)$. It also says that ${\widehat{\varphi}}_{\mathrm{rot}}(x,y,0)$ will lead to an increased amount of fitting error because of the $2\pi $ discontinuities associated with branch points and branch cuts, especially when using a single continuous-face-sheet DM with a high-power coating for phase-only compensation.^{29}^{–}^{33}

## 3.4.

### Overall Trade Space

Figure 7 shows the overall trade space in terms of the normalized PIB, ${P}_{N}$, and the peak Strehl ratio, ${S}_{P}$. Here, we calculated both metrics as a function of the number of wind-clearing periods ${N}_{\mathrm{WCP}}$. As ${N}_{\mathrm{WCP}}$ increases, both ${P}_{N}$ and ${S}_{P}$ decrease, which makes good sense. Note that the bold curves report the averages and the light curves report the standard deviations associated with 100 Monte Carlo realizations. Also note that the widths of these light curves are reasonably small, and the bold curves are reasonably smooth. Thus, we will use the same number of Monte Carlo realizations in the analysis that follows.

## 4.

## Results and Discussion

This section contains results for the trade space setup in Sec. 2 and explored in Sec. 3. In particular, Fig. 8 shows results for the log-amplitude variance, ${\sigma}_{\chi}^{2}$, and the branch-point density, ${\mathcal{D}}_{\mathrm{BP}}$, both as a function of the number of wind-clearing periods ${N}_{\mathrm{WCP}}$ [cf. Eqs. (27), (28), and (12), respectively]. These metrics result from a point-source beacon being backpropagated from the target plane to the source plane through the simulated turbulence and TDTB. Here, we use ${N}_{\mathrm{WCP}}$ to help gauge when the time-dependent simulations reach steady state. From Fig. 8, we can see that both ${\sigma}_{\chi}^{2}$ and ${\mathcal{D}}_{\mathrm{BP}}$ overshoot and settle into steady state when ${N}_{\mathrm{WCP}}\ge 0.5$. This outcome is consistent with Fig. 7. Note that in Fig. 8 the bold curves report the averages and the light curves report the standard deviations associated with 100 Monte Carlo realizations. Also note that the widths of these light curves are reasonably small, and the bold curves are reasonably smooth. Thus, we believe that 100 Monte Carlo realizations (i.e., the same number of realizations used in Part I of this two-part study) are adequate in quantifying the effects of TTBI.

These results clearly show that TTBI results in an increased amount of scintillation when simulating turbulence and TDTB. For example, as shown in Fig. 8(a), the log-amplitude variance ${\sigma}_{\chi}^{2}$ had steady-state averages of 0.437 for the TTBI case (when ${N}_{\mathrm{WCP}}\ge 0.5$), and 0.291 for the turbulence-only case (when ${N}_{\mathrm{WCP}}\ge 0$). These steady-state averages agreed well with those obtained from simulating turbulence and SSTB [cf. the horizontal black lines in Fig. 8(a)], which had averages of 0.454 for the TTBI case (when ${\mathcal{R}}_{\mathrm{sw}}=0.25$ and ${N}_{D}={N}_{C}=16\sqrt{2}\approx 22.6$) and 0.297 for the turbulence-only case (when ${\mathcal{R}}_{\mathrm{sw}}=0.25$ and ${N}_{D}=0$), leading to percentage errors of 3.88% and 1.92%, respectively.

The branch-point density ${\mathcal{D}}_{\mathrm{BP}}$, as shown in Fig. 8(b), had steady-state averages of 390 for the TTBI case (when ${N}_{\mathrm{WCP}}\ge 0.5$), and 127 for the turbulence-only case (when ${N}_{\mathrm{WCP}}\ge 0$). These steady-state averages did not agree well with those obtained from simulating turbulence and SSTB [cf. the horizontal black lines in Fig. 8(b)], which had averages of 487 for the TTBI case (when ${\mathcal{R}}_{\mathrm{sw}}=0.25$ and ${N}_{D}={N}_{C}=16\sqrt{2}\approx 22.6$), and 164 for the turbulence-only case (when ${\mathcal{R}}_{\mathrm{sw}}=0.25$ and ${N}_{D}=0$), leading to percentage errors of 24.8% and 29.3%, respectively. This lack of agreement was probably due to two phenomena. The first being due to the fact that TTBI does not allow the radians of distortion caused by TDTB to equal those caused by SSTB, and the second being due to the sensitivity of ${\mathcal{D}}_{\mathrm{BP}}$ as a metric.^{55} Both phenomena warrant further investigation (in subsequent papers). Nonetheless, the results presented in this paper clearly show that TTBI results in an increased amount of scintillation when simulating turbulence and TDTB.

With the above percentage errors in mind, the time-dependent assumptions contained in this paper ultimately allowed us to examine the Monte Carlo averages associated with the log-amplitude variance, ${\sigma}_{\chi}^{2}$, and the branch-point density, ${\mathcal{D}}_{\mathrm{BP}}$, with increased computational fidelity. However, this increased computational fidelity came at the expense of computational time. That is why we reserved the full trade space exploration to Part I of this two-part study. Relative to the time-dependent results contained in this paper, the steady-state results contained in Part I provide an upper bound on both the increase in ${\sigma}_{\chi}^{2}$ and ${\mathcal{D}}_{\mathrm{BP}}$ due to TTBI (cf. Fig. 8 in both papers). It is our hope that such results will prove fruitful in the development of next-generation scaling laws that account for the effects of TTBI.^{39}^{–}^{45}

## 5.

## Conclusion

In this paper, we used wave-optics simulations to look at the Monte Carlo averages associated with turbulence and TDTB. The goal throughout was to investigate TTBI. At a wavelength near $1\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$, TTBI increases the amount of scintillation that results from high-power laser beam propagation through distributed-volume atmospheric aberrations. In turn, to help gauge the strength of the simulated turbulence and TDTB, this paper made use of the following three parameters: the spherical-wave Rytov number, the number of wind-clearing periods, and the distortion number. These parameters simplified greatly, given a propagation path with constant atmospheric conditions. In addition, to help quantify the effects of TTBI, this paper made use of the following two metrics: the log-amplitude variance and branch-point density. These metrics resulted from a point-source beacon being backpropagated from the target plane to the source plane through the simulated turbulence and TDTB.

Overall, the results showed that TTBI causes the log-amplitude variance and the branch-point density to increase. These results pose a major problem for BC systems that perform phase compensation. In turn, the time-dependent simulations presented in this paper will provide much needed insight into the design of future systems.

## Acknowledgments

The author of this paper would like to thank the Directed Energy Joint Transition Office for sponsoring this research, and C. E. Murphy for many insightful discussions regarding the results presented within.

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*Investigation of branch-point density using traditional wave-optics techniques*## Biography

**Mark F. Spencer** is a senior research physicist and the principal investigator for Aero Effects and Beam Control at the Air Force Research Laboratory, Directed Energy Directorate. In addition, he is an adjunct assistant professor of optical sciences and engineering at the Air Force Institute of Technology (AFIT), within the Department of Engineering Physics. He received his PhD in optical sciences and engineering from AFIT in 2014. He is a senior member of SPIE.