Part I of this two-part paper uses wave-optics simulations to look at the Monte Carlo averages associated with turbulence and steady-state thermal blooming (SSTB). 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 and the distortion number to gauge the strength of the simulated turbulence and SSTB. These parameters simplify 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 SSTB. 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 I 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 SSTB. 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 SSTB. Part II of this two-part paper then investigates TTBI in the presence of turbulence and time-dependent thermal blooming (TDTB). 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 II of this two-part paper complements the analysis contained in this paper. In general, the steady-state assumptions contained in this paper allows us to explore the trade space in a computationally efficient manner, whereas the time-dependent assumptions contained in Part II allow us to examine the Monte Carlo averages with increased computational fidelity. For example, relative to the time-dependent results contained in Part II, the steady-state results contained in this paper 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 reduces the overall trade space and uses computationally efficient steady-state simulations 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 SSTB. For this purpose, we quickly review the details associated with the split-step beam propagation method (BPM), spherical-wave Rytov number, SSTB, distortion number, and parameters of interest in the following sections. Given a common setup, this paper shares most of these sections with Part II of this two-part paper on TTBI. The analysis in this paper specifically differs in the sections on SSTB 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.

### Steady-State Thermal Blooming

For SSTB, we assume that a transverse wind has cleared the source plane enough times so that the TDTB reaches a steady state with a fully formed crescent or half-moon irradiance pattern in the target plane (cf. Fig. 1). The heating and cooling of the air, as described by TDTB (cf. Part II), can reach a steady state if there are no variations in the exitance of the high-power laser source and ambient atmosphere. In turn, 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. (3)

$$\mathrm{\Delta}n(\mathbf{r})=\mathrm{\Delta}n(x,y,z)=-\frac{\mu (z)}{{v}_{\perp}(z)}{\int}_{-\infty}^{x}{i}_{\mathrm{HP}}(\xi ,y,z)\mathrm{d}\xi ,$$^{34}

^{–}

^{38}

## 2.4.

### 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. (3) into Eq. (5) results in the following relationship:## Eq. (6)

$$\varphi (x,y,Z)=-k{\int}_{0}^{Z}\frac{\mu (z)}{{v}_{\perp}(z)}{\int}_{-\infty}^{x}{i}_{\mathrm{HP}}(\xi ,y,z)\mathrm{d}\xi \text{\hspace{0.17em}}\mathrm{d}z.$$For focused high-power laser beams with initial power, ${P}_{0}$, and initial diameter, ${D}_{0}$, we can rewrite Eq. (6) in terms of a path-integral expression known as the distortion number, ${N}_{D}$. In particular,

## Eq. (7)

$$\varphi (x,y,Z)\approx -{N}_{D}\frac{{D}_{0}}{4\sqrt{2}{P}_{0}}{\int}_{-\infty}^{x}{i}_{\mathrm{HP}}(\xi ,y,0)\mathrm{d}\xi $$## Eq. (8)

$${N}_{D}=4\sqrt{2}{P}_{0}k{\int}_{0}^{Z}\frac{Z-z}{Z}\frac{\mu (z)\tau (z)}{D(z){v}_{\perp}(z)}\mathrm{d}z.$$Given propagation paths with constant atmospheric conditions, Eq. (8) simplifies, such that

## Eq. (11)

$${N}_{D}=\frac{4\sqrt{2}{P}_{0}k({n}_{0}-1)\alpha {\mathrm{e}}^{-\alpha Z}{\mathrm{e}}^{-\sigma Z}Z}{{C}_{P}{\rho}_{0}{T}_{0}{D}_{0}{v}_{\perp}}.$$^{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 SSTB.

## 2.5.

### 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) | ${N}_{D}$ | $8\sqrt{2},16\sqrt{2},32\sqrt{2}$ |

Initial power (kW) | ${P}_{0}$ | 62.72, 125.4, 250.9 |

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 ${r}_{0,\mathrm{sw}}$ (also known as 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. (12)

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

^{,}

^{51}

^{–}

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

## Eq. (13)

$${\theta}_{0}={[2.91{k}^{2}{\int}_{0}^{Z}{C}_{n}^{2}(z){(Z-z)}^{5/3}\mathrm{d}z]}^{-3/5}\Rightarrow {\theta}_{0}=0.314\frac{{r}_{0,\mathrm{sw}}}{Z}$$## Eq. (14)

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

Turbulence scenarios 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 | 0.1 | $0.156\times {10}^{-14}$ | 5.17 | 3.04 | 39.6 |

2 | 0.2 | $0.313\times {10}^{-14}$ | 7.83 | 2.00 | 60.0 |

3 | 0.3 | $0.469\times {10}^{-14}$ | 9.99 | 1.57 | 76.5 |

4 | 0.4 | $0.625\times {10}^{-14}$ | 11.9 | 1.32 | 90.9 |

5 | 0.5 | $0.781\times {10}^{-14}$ | 13.6 | 1.16 | 104 |

6 | 0.6 | $0.938\times {10}^{-14}$ | 15.1 | 1.04 | 116 |

7 | 0.7 | $1.09\times {10}^{-14}$ | 16.6 | 0.945 | 127 |

8 | 0.8 | $1.25\times {10}^{-14}$ | 18.0 | 0.872 | 138 |

9 | 0.9 | $1.42\times {10}^{-14}$ | 19.3 | 0.813 | 148 |

10 | 1.0 | $1.56\times {10}^{-14}$ | 20.6 | 0.763 | 157 |

## 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. (15)

$${P}_{N}=\frac{{\int}_{-\infty}^{\infty}\mathrm{cyl}\left(\frac{\sqrt{{x}^{2}+{y}^{2}}}{{D}_{Z}}\right){i}_{\mathrm{HP}}(x,y,Z)\mathrm{d}x\text{\hspace{0.17em}}\mathrm{d}y}{{P}_{0}},$$## Eq. (16)

$${S}_{P}=\frac{\mathrm{max}\{{i}_{\mathrm{HP}}(x,y,Z)\}}{{i}_{\mathrm{DL}}(\mathrm{0,0},Z)},$$## Eq. (19)

$$\mathrm{cyl}(\rho )=\{\begin{array}{ll}1& 0\le \rho <1/2\\ 1/2& \rho =1/2\\ 0& \rho >1/2\end{array};$$To calculate the number of branch points ${N}_{\mathrm{BP}}$ in Eq. (18), 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. (20) 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. (15)–(18) to visualize the following topics:

1. the number of phase screens needed to simulate the effects of SSTB,

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 Phase Screens Needed

To determine the number of equally spaced phase screens, ${N}_{\mathrm{PS}}$, needed to accurately simulate SSTB using the split-step BPM, we calculated both the normalized PIB, ${P}_{N}$, and the peak Strehl ratio, ${S}_{P}$, as a function of ${N}_{\mathrm{PS}}$. In addition, we calculated the rate of change of ${P}_{N}$ and ${S}_{P}$ as a function of ${N}_{\mathrm{PS}}$ via an operator, $\mathrm{diff}\{\odot \}$, that determines the running difference. Figure 2 shows the outcomes of these calculations. As shown, we can see that the rate of change of ${P}_{N}$ and ${S}_{P}$ go to zero (accurate to the third decimal place) when ${N}_{\mathrm{PS}}\ge 50$. Thus, in the analysis that follows, we used ${N}_{\mathrm{PS}}=50$ to simulate the effects of SSTB.

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 SSTB (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 SSTB 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 SSTB-only cases. Figure 3 demonstrates this hypothesis to be true for one Monte Carlo realization, where ${\mathcal{R}}_{\mathrm{sw}}=0.25$ and ${N}_{D}={N}_{C}=16\sqrt{2}\approx 22.6$ [cf. Eqs. (2) and (11), 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 II using time-dependent simulations (cf. Fig. 3 in Part II) 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 SSTB 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 SSTB-only cases. Figures 4 and 5 demonstrate this hypothesis to be true for one Monte Carlo realization, where ${\mathcal{R}}_{\mathrm{sw}}=0.25$ and ${N}_{D}={N}_{C}=16\sqrt{2}\approx 22.6$ [cf. Eqs. (2) and (11), 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 II using time-dependent simulations (cf. Figs. 4 and 5 in Part II) 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. (21)

$${\widehat{\varphi}}_{\mathrm{irr}}(x,y,0)=\mathrm{LS}\{\varphi (x,y,0)\}={\varphi}_{\mathrm{LS}}(x,y,0)$$## Eq. (22)

$${\widehat{\varphi}}_{\mathrm{rot}}(x,y,0)=\mathrm{arg}\{\mathrm{exp}(j[\varphi (x,y,0)-{\varphi}_{\mathrm{LS}}(x,y,0)])\},$$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 spherical-wave Rytov number, ${\mathcal{R}}_{\mathrm{sw}}$, and the distortion number, ${N}_{D}$. As both ${\mathcal{R}}_{\mathrm{sw}}$ and ${N}_{D}$ increase, both ${P}_{N}$ and ${S}_{P}$ typically decrease, which makes good sense. Note that the curves report the averages, and the error bars report the standard deviations associated with 100 Monte Carlo realizations. Also note that the widths of these error bars are reasonably small, and the curves themselves 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 spherical-wave Rytov number, ${\mathcal{R}}_{\mathrm{sw}}$, and the distortion number, ${N}_{D}$ [cf. Eqs. (17), (18), (2), and (11), respectively]. These metrics result from a point-source beacon being backpropagated from the target plane to the source plane through the simulated turbulence and SSTB. Here, we use ${\mathcal{R}}_{\mathrm{sw}}$ to gauge the strength of the simulated turbulence and ${N}_{D}$ to gauge the strength of the simulated SSTB. From Fig. 8(a), we can see that ${\sigma}_{\chi}^{2}$ deviates significantly from ${\mathcal{R}}_{\mathrm{sw}}$ when SSTB is present, even for the lowest ${N}_{D}$. Similarly, Fig. 8(b) shows that ${\mathcal{D}}_{\mathrm{BP}}$ increases significantly when SSTB increases in strength (i.e., as ${N}_{D}$ increases). Note that in Fig. 8 the curves report the averages and the error bars report the standard deviations associated with 100 Monte Carlo realizations. Also note that the widths of these error bars are reasonably small, and the curves themselves are reasonably smooth. Thus, we believe that 100 Monte Carlo realizations (i.e., the same number of realizations used in Part II 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 SSTB. For example, as shown in Fig. 8(a), the log-amplitude variance ${\sigma}_{\chi}^{2}$ increases monotonically as a function of the spherical-wave Rytov number ${\mathcal{R}}_{\mathrm{sw}}$. However, as the distortion number ${N}_{D}$ increases, this monotonic increase becomes increasingly saturated (i.e., it approaches an asymptotic value at a faster rate) due to TTBI. For all intents and purposes, this behavior is well documented in the atmospheric propagation literature^{52} and is a known shortcoming of the Rytov approximation (cf. the solid blue curve relative to the solid black curve). What is not well documented in the atmospheric propagation literature is the fact that TTBI causes this saturation to occur at a faster rate.

The branch-point density ${\mathcal{D}}_{\mathrm{BP}}$ also increases monotonically, as shown in Fig. 8(b), but in general, the number of branch points increases nonlinearly up until the strong scintillation regime (cf. the solid blue curve relative to the solid black curve). After this regime is met, the number of branch points grows linearly. This behavior is also well documented in the atmospheric propagation literature^{55} and poses a major problem for BC systems that perform phase compensation in the strong-scintillation regime. Again, what is not well documented is the fact that TTBI causes this nonlinear-to-linear transition to occur at a faster rate.

With the above behaviors in mind, the steady-state assumptions contained in this paper ultimately allowed us to explore the full trade space contained in Table 2 with computational efficiency. However, this computational efficiency came at the expense of computational fidelity. That is why we reserved the detailed examination of the Monte Carlo averages associated with the log-amplitude variance, ${\sigma}_{\chi}^{2}$, and the branch-point density, ${\mathcal{D}}_{\mathrm{BP}}$, to Part II of this two-part study. Relative to the time-dependent results contained in Part II, the steady-state results contained in this paper 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 SSTB. 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 SSTB, this paper made use of the following two parameters: the spherical-wave Rytov number 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 SSTB.

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 steady-state 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.

## References

**,” (1974). http://www.dtic.mil/dtic/tr/fulltext/u2/781763.pdf Google Scholar**

*Propagation of high-energy laser beams in the atmosphere***,” Proc. IEEE, 65 (12), 1679 –1714 (1977). https://doi.org/10.1109/PROC.1977.10809 IEEPAD 0018-9219 Google Scholar**

*High-power laser propagation: thermal blooming***,” Laser Beam Propagation in the Atmosphere, Springer-Verlag, Heidelberg, New York (1978). Google Scholar**

*Thermal blooming in the atmosphere***,” Proc. SPIE, 1221 1 –25 (1990). https://doi.org/10.1117/12.18326 PSISDG 0277-786X Google Scholar**

*Twenty-five years of thermal blooming: an overview***,” Phys. Rev. A, 4 (2), 763 –776 (1971). https://doi.org/10.1103/PhysRevA.4.763 Google Scholar**

*Temporal and spatial gain in stimulated light scattering***,” J. Opt. Soc. Am. A, 7 (6), 1103 –1124 (1990). https://doi.org/10.1364/JOSAB.7.001103 JOAOD6 0740-3232 Google Scholar**

*Perturbation growth by thermal blooming in turbulence***,” Phys. Rev. A, 41 (12), 6982 –6991 (1990). https://doi.org/10.1103/PhysRevA.41.6982 Google Scholar**

*Linear theory of uncompensated thermal blooming in turbulence***,” J. Opt. Soc. Am A, 8 (4), 637 –646 (1991). https://doi.org/10.1364/JOSAA.8.000637 Google Scholar**

*Perturbative approach to the small-scale physics of the interaction of thermal blooming and turbulence***,” Phys. Rev. A, 44 (10), 6862 –6876 (1991). https://doi.org/10.1103/PhysRevA.44.6862 Google Scholar**

*A linearized theory of transient laser heating in fluid***,” App. Opt., 13 (2), 331 –334 (1974). https://doi.org/10.1364/AO.13.000331 Google Scholar**

*Phase compensation for thermal blooming***,” J. Opt. Soc. Am., 67 (3), 290 –295 (1977). https://doi.org/10.1364/JOSA.67.000290 JOSAAH 0030-3941 Google Scholar**

*Properties of phase conjugate adaptive optical systems***,” J. Opt. Soc. Am. A, 6 (7), 1038 –1048 (1989). https://doi.org/10.1364/JOSAA.6.001038 JOAOD6 0740-3232 Google Scholar**

*Thermal blooming compensation instabilities***,” J. Opt. Soc. Am. A, 6 (12), 1859 –1862 (1989). https://doi.org/10.1364/JOSAA.6.001859 JOAOD6 0740-3232 Google Scholar**

*Scalar Green’s-function derivation of the thermal blooming compensation instability equations***,” Lincoln Lab. J., 5 (1), 151 –170 (1992). LLJOEJ 0896-4130 Google Scholar**

*Thermal-blooming laboratory experiments***,” J. Opt. Soc. Am. B, 9 (10), 1803 –1812 (1992). https://doi.org/10.1364/JOSAB.9.001803 JOBPDE 0740-3224 Google Scholar**

*Linearized theory of thermal-blooming phase compensation instability with realistic adaptive-optics geometry***,” Proc. SPIE, 1221 229 –240 (1990). https://doi.org/10.1117/12.18345 PSISDG 0277-786X Google Scholar**

*Effects of a random wind field on thermal blooming instabilities***,” Lincoln Lab. J., 5 (1), 131 –150 (1992). LLJOEJ 0896-4130 Google Scholar**

*The theory of compensated laser propagation through strong thermal blooming***,” Lincoln Lab. J., 5 (2), 273 –293 (1992). LLJOEJ 0896-4130 Google Scholar**

*Scaled atmospheric blooming experiments***,” J. Opt. Soc. Am. A., 15 (5), 1212 –1226 (1998). https://doi.org/10.1364/JOSAA.15.001212 JOAOD6 0740-3232 Google Scholar**

*Wind-shear induced stabilization of PCI***,” J. Opt. Soc. Am. A., 26 (7), 1638 –1653 (2009). https://doi.org/10.1364/JOSAA.26.001638 JOAOD6 0740-3232 Google Scholar**

*Linear analysis of thermal blooming compensation instabilities in laser propagation***,” Pure Appl. Opt., 6 (103), 256 –269 (1997). https://doi.org/10.1088/0963-9659/6/1/012 PAOAE3 0963-9659 Google Scholar**

*The influence of wavefront dislocations on phase conjugation instability with thermal blooming compensation***,” Proc. SPIE, 7816 781609 (2010). https://doi.org/10.1117/12.859936 PSISDG 0277-786X Google Scholar**

*Impact of spatial resolution on thermal blooming phase compensation instability***,” Proc. SPIE, 8165 816503 (2011). https://doi.org/10.1117/12.894079 PSISDG 0277-786X Google Scholar**

*Impact of branch points in adaptive optics compensation of thermal blooming and turbulence***,” AFIT/OSE/ENP/11-M02, Air Force Institute of Technology, Wright Patterson Air Force Base, Ohio, (2011). http://www.dtic.mil/dtic/tr/fulltext/u2/a538538.pdf Google Scholar**

*Branch point mitigation of thermal blooming phase compensation instability***,” J. Opt. Soc. Am. A, 31 (15), 2865 –2882 (1992). https://doi.org/10.1364/AO.31.002865 JOAOD6 0740-3232 Google Scholar**

*Branch cuts in the phase function***,” J. Opt. Soc Am. A, 15 (10), 2759 –2768 (1998). https://doi.org/10.1364/JOSAA.15.002759 Google Scholar**

*Branch point problem in adaptive optics***,” Opt. Commun., 200 43 –72 (2001). https://doi.org/10.1016/S0030-4018(01)01546-2 OPCOB8 0030-4018 Google Scholar**

*Adaptive optics wave function reconstruction and phase unwrapping when branch points are present***,” Opt. Exp., 16 (10), 6985 –6998 (2008). https://doi.org/10.1364/OE.16.006985 OPEXFF 1094-4087 Google Scholar**

*Optical phase unwrapping in the presence of branch points***,” App. Opt., 53 (18), 3821 –3831 (2014). https://doi.org/10.1364/AO.53.003821 Google Scholar**

*LSPV+7, a branch-point-tolerant reconstructor for strong turbulence adaptive optics***,” in Proc. OSA pcAOP, PTh2D.2 (2017). Google Scholar**

*Branch-cut accumulation using LSPV+7***,” Proc. SPIE, 10194 1019493 (2017). https://doi.org/10.1117/12.2258681 PSISDG 0277-786X Google Scholar**

*Compensation in the presence of deep turbulence using tiled-aperture architectures***,” (1986). Google Scholar**

*A discussion of some issues associated with the evaluation and compensation of thermal blooming***,” App. Opt., 18 (15), 2638 –2644 (1979). https://doi.org/10.1364/AO.18.002638 Google Scholar**

*Algebraic model for CW thermal-blooming effects***,” Proc. SPIE, 5792 76 –87 (2005). https://doi.org/10.1117/12.603384 PSISDG 0277-786X Google Scholar**

*Methodology for comparing worldwide performance of diverse weight-constrained high energy laser systems***,” Opt. Exp., 21 (12), 14789 –14798 (2013). https://doi.org/10.1364/OE.21.014789 OPEXFF 1094-4087 Google Scholar**

*Enhanced, fast-running scaling law model of thermal blooming and turbulence effects on high energy laser propagation***,” J. Opt. Soc. Am. A, 33 (12), 2477 –2484 (2016). https://doi.org/10.1364/JOSAA.33.002477 JOAOD6 0740-3232 Google Scholar**

*General wave optics propagation scaling law***,” Opt. Exp., 26 (12), 15609 –15622 (2018). https://doi.org/10.1364/OE.26.015609 OPEXFF 1094-4087 Google Scholar**

*Far-field propagation of partially coherent laser light in random mediums***,” Proc. SPIE, 10772 1077208 (2018). https://doi.org/10.1117/12.2319869 PSISDG 0277-786X Google Scholar**

*Investigation of turbulence thermal blooming interaction using the split-step beam propagation method***,” Appl. Phys., 10 129 –160 (1976). https://doi.org/10.1007/BF00896333 APPYEK 1080-9198 Google Scholar**

*Time-dependent propagation of high energy laser beams through the atmosphere***,” Appl. Phys., 14 99 –115 (1977). https://doi.org/10.1007/BF00882638 APPYEK 1080-9198 Google Scholar**

*Time-dependent propagation of high energy laser beams through the atmosphere: II***,” App. Opt., 57 (15), 4090 –4102 (2018). https://doi.org/10.1364/AO.57.004090 Google Scholar**

*Polychromatic wave-optics models for image-plane speckle. 1. Well-resolved objects***,” App. Opt., 57 (15), 4103 –4110 (2018). https://doi.org/10.1364/AO.57.004103 Google Scholar**

*Polychromatic wave-optics models for image-plane speckle. 2. Unresolved objects***,” App. Opt., 37 (21), 4525 –4535 (1998). https://doi.org/10.1364/AO.37.004525 Google Scholar**

*Density of turbulence-induced phase dislocations*## 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.