1. INTRODUCTION

Digital-holographic detection, in practice, provides us with an estimate of the complex-optical field. As such, it can be applied to a variety of applications, which require the propagation of coherent light through distributed-volume atmospheric aberrations or “deep turbulence.” These applications include atmospheric characterization [1,2], adaptive-optics phase compensation [3,4], remote sensing [5,6], and free-space laser communications [7,8] (to name a few). With these applications in mind, this paper varies the parameters associated with digital-holographic detection in the off-axis pupil plane recording geometry (PPRG) to investigate its use as a deep-turbulence wavefront sensor (WFS). The goal throughout is to examine the parameters needed to produce worthwhile estimates of the amplitude and wrapped phase associated with the complex-optical field.

In practice, several different digital-holographic recording geometries exist, such as the off-axis image plane and on-axis phase shifting recording geometries [9–12]; however, this paper focuses on the off-axis PPRG. As shown in Fig. 1(a), the off-axis PPRG gets its namesake from placing the focal-plane array (FPA) in a pupil plane. Recall that in order to obtain an estimate of the complex-optical field, we need to first interfere a signal beam with a reference beam, so that we can record a digital hologram with the FPA. Using signal-processing techniques, the off-axis PPRG ultimately allows for the isolation of the desired complex-optical field, all from a single digital hologram. These advantages make the off-axis PPRG an ideal candidate for use in deep-turbulence applications, which involve high-resolution, low-latency wavefront sensing.

 

Fig. 1. Above is a diagram of digital-holographic detection in the off-axis PPRG. In (a), the coherent light from a master-oscillator (MO) laser actively illuminates a distant cooperative object (e.g., an unresolved ball bearing) creating a point-source beacon that propagates through deep turbulence to the entrance-pupil plane of an imaging system. Through collimation, the entrance-pupil plane creates a signal beam, which we interfere with a reference beam in the exit-pupil plane of the imaging system. Note that the reference beam manifests itself from an off-axis local oscillator (LO), which we derive from the MO laser. Also note that the exit-pupil plane is where we place the FPA, so that we can record a digital hologram. In (b), we can then use signal-processing techniques to manipulate the digital hologram into a wrapped-phase estimate.

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Current wavefront-sensing techniques suffer from degraded performance when in the presence of deep turbulence. For example, if the turbulence is strong enough and the propagation distance is long enough, the distributed-volume phase aberrations can cause the coherent light from a point-source beacon to interfere with itself [cf. Fig. 1]. Known as scintillation, this self interference is one cause of low signal-to-noise ratios (SNRs) over the field-of-view of the WFS, as well as branch points, which result from total-destructive interference. Branch points, in practice, also cause branch cuts to arise in the phase function. These $2\pi$ phase discontinuities get mapped to the null space of least-squares phase unwrapping and reconstruction algorithms [13]; thus, traditional wavefront-sensing techniques (e.g., the Shack–Hartmann WFS [14]) do not estimate the correct phase function when in the presence of deep turbulence. It is important to note that branch-point tolerant phase unwrapping and reconstruction algorithms do exist within the open literature [3,15]; however, the performance of these algorithms needs to be quantified with hardware solutions [16].

With the harmful effects of scintillation in mind, another cause of low SNRs is the extinction of light as it propagates through the atmosphere. Extinction, in general, results from molecular and aerosol absorption and scattering and ultimately leads to transmission losses through Beer’s law [17,18]. Together, the low-SNR effects of scintillation and extinction plague our ability to perform deep-turbulence wavefront sensing using traditional wavefront-sensing techniques. This outcome reinforces the need for a more robust WFS when in the presence of deep turbulence (i.e., scintillation and extinction).

With the use of a strong reference beam (e.g., one that is a large fraction of the FPA’s pixel-well depth), digital-holographic detection overcomes the low-SNR challenges caused by scintillation and extinction. In practice, a strong reference beam can boost the overall power of the signal beam to be well above the read-noise floor of the FPA. When faced with low SNRs, this boost allows us to approach the shot-noise limit (depending on the parameters of the FPA), which is a fundamental limit that does not allow for any more sensitivity with respect to detection. In regards to branch points, digital-holographic detection also grants access to a direct estimate of the wrapped phase; thus, we can directly sense the correct phase function when in the presence of deep turbulence.

In what follows, this paper quantifies the abilities of digital-holographic detection in the off-axis PPRG to accurately estimate the complex-optical field when in the presence of deep turbulence and detection noise. With that said, Section 2 gives an overview of the wave-optics and signal-to-noise models (both analytical and numerical) that we will use within the analysis. It also develops several parameters that we will use in an associated trade study. Section 3 then provides the results from this trade study with appropriate discussion. Following that, Section 4 provides a conclusion for this paper.

Before moving on to the next section, it is worth mentioning that this paper builds upon the analysis contained in a recent conference proceeding by Banet et al. [19] and is a “companion paper” to an analysis presented by Spencer et al. [10] with respect to digital-holographic detection in the off-axis image plane recording geometry (IPRG). Specifically, this paper formulates an analytical wave-optics model, which describes digital-holographic detection in the off-axis PPRG. It also includes detection noise in the numerical wave-optics model. In so doing, this paper validates the use of a closed-form expression for the SNR and formulates a method for optimal windowing when in the presence of deep turbulence and detection noise. Thus, this paper significantly expands upon the analysis presented in [19], which did not develop an analytical wave-optics model nor include the effects of detection noise in its numerical wave-optics model. The analysis also shows that, given a point-source beacon, the off-axis PPRG is less prone to pixel saturation because of focal-plane hot spots caused by deep-turbulence scintillation. This outcome results from the use of a stronger reference beam (e.g., one that is a larger fraction of the FPA’s pixel-well depth when compared with digital-holographic detection in the off-axis IPRG [10]). In comparison with the off-axis IPRG, we can better approach the shot-noise limit when using digital-holographic detection in the off-axis PPRG. This last point is important to remember when faced with low SNRs, for example, when investigating the branch-point problem with hardware solutions [16]. As such, it is our belief that the trade-space analysis contained in this paper will better enable future research efforts to design and implement digital-holographic detection in the off-axis PPRG for the purpose of exploring the branch-point problem.

2. WAVE-OPTICS AND SIGNAL-TO-NOISE MODELS

In this section, we develop wave-optics and signal-to-noise models (both analytical and numerical) for digital-holographic detection in the off-axis PPRG [cf. Fig. 1]. It is important to note that, throughout this paper, the computational simulations use MATLAB along with WaveProp [20] and AOTools [21], which are MATLAB toolboxes written by the Optical Sciences Company. For additional insight into these wave-optics and signal-to-noise models, we encourage the reader to review [9,10,19].

A. Analytical Wave-Optics Model

For the analytical wave-optics model, let’s start at the exit-pupil plane described in Fig. 1(a). Here, we can describe the signal complex-optical field $U_S(x_2,y_2)$ or “signal beam” that is incident on the FPA as

From the signal and reference beams [cf. Eqs. (1) and (3)], we can represent the real-valued hologram irradiance $I_H(x_2,y_2)$ that is incident on the FPA as

Provided Eqs. (5)–(8), the continuous photoelectron density $d_H(x_2,y_2)$, in units of photoelectrons (pe) per square meter follows as [22,23]

To go to the Fourier plane, we must use a 2D Fourier transform, such that

Moving forward, let us convert from spatial frequencies to spatial coordinates, so that ${\nu }_x=x_0/(M_T\lambda z)$ and ${\nu }_y=y_0/(M_T\lambda z)$. In effect, this choice scales the Fourier plane so that it is equivalent to the complex-optical field after propagation from the entrance-pupil plane back to the object plane. Let us also assume a uniform pixel pitch, so that $p\equiv x_s=y_s=w_x=w_y$. Thus, $N_P\equiv \mathrm{N}=\mathrm{M}$ is the number of FPA pixels across the exit-pupil diameter $D_2$.

From Eqs. (15) and (17), we arrive at the following relationship:

In units of pe, we can see that the terms within the curly braces in Eq. (21) repeat themselves at regular intervals of the Fourier-plane side length $L_F$ in both the $x$ and $y$ directions. This outcome is the result of 2D convolution with the separable comb functions [cf. Eq. (11)]. The second 2D convolution with the separable $\mathrm{sinc}$ functions serves to smooth out these repeated terms, whereas the envelope function $\tilde{\alpha }({\nu }_x,{\nu }_y)$ [cf. Eq. (16)] serves to amplitude modulate these repeated terms. Before moving on in the analysis, the reader should note that this result is physically relevant, as the sampling theorem dictates that a sampled function [e.g., the continuous photoelectron density $d_H(x_2,y_2)$ in Eq. (9)] becomes periodic upon finding its spectrum [22,23].

Provided Eqs. (19)–(21), we can use a window function $w(x_0,y_0)$ to isolate the turbulence-limited ASF [i.e., the third term within the curly braces of Eq. (21)]. In general,

In using Eqs. (21) and (22), we must satisfy Nyquist sampling with the FPA pixels, so that the repeated terms within the Fourier plane do not overlap and cause significant aliasing. Here, the Nyquist rate is $L_F$ and the Nyquist interval is $1/L_F$ [22]. If we also notice that we can express the Fourier-plane side length $L_F$ as

Provided Eqs. (22)–(26), we can use Eq. (21) to obtain an estimate of the turbulence-limited ASF. The reader should note once more that the final 2D convolution with the separable sinc functions in Eq. (21) acts to smooth out the Fourier plane. With that said, this diffraction-limited smoothing is often negligible when in the presence of deep turbulence. In the limit that ${\rho }_I\to 0$ and $Q_P\ge 4.88$, we can make use of the convolution-sifting property of the Dirac-delta function and neglect the 2D convolutions in Eq. (21). As a result, we can shift the Fourier plane, so that the origin is located at

To obtain an estimate of the normalized signal beam ${\widehat{\psi }}_S(M_T{x}_1,M_T{y}_1)$, which is incident on the entrance-pupil plane, we need to use a 2D inverse Fourier transform, such that

B. Numerical Wave-Optics Model

For the numerical wave-optics model or “model” for short, we used MATLAB along with WaveProp and AOTools to simulate complex-optical fields via discretized $\mathrm{N}\times \mathrm{N}$ grids. Starting at the simulated object plane [cf. Fig. 1(a)], the model used a narrow sinc function modulated by a raised-cosine envelope in a $4096\times 4096$ grid to emulate a point-source beacon of wavelength $\lambda =1\mathrm{\mu m}$. In order to avoid the detrimental effects of aliasing, the sampling of this function and the object-plane side length was set, so that, after propagation from the object plane to the entrance-pupil plane, the illuminated region of interest was half the user-defined, entrance-pupil plane side length (i.e., the simulations satisfied Fresnel scaling, such that $\mathrm{N}=S_0{S}_1/(\lambda Z)$, where $S_0$ and $S_1=16D_1$ are the object and entrance-pupil plane side lengths, respectively). The model then used the split-step beam propagation method to propagate the light from the point-source beacon a distance of $Z=7.5\mathrm{km}$ through 10 equally spaced Kolmogorov phase screens, which simulated the distributed-volume phase aberrations indicative of deep turbulence [26]. With 10 phase screens, the continuous and discrete calculations for the turbulence scenarios presented in Table 1 were within 1% error.

Table 1. Turbulence Scenarios Used in the Trade-Space Analysis

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With Table 1 in mind, a great deal of work has been done to characterize regimes in which deep-turbulence conditions exist. For example, in terms of optics, we often characterize turbulence with an index-of-refraction structure parameter $C_n^2$, which varies both seasonally and diurnally due to temperature fluctuations in the atmosphere. Given $C_n^2$, we can determine the Fried coherence diameter $r_0$, which gives us a metric for “seeing” [18,26–29]. This is said because increasing the size of the entrance-pupil diameter $D_1$ beyond $r_0$ yields diminishing returns in terms of imaging resolution. For horizontal-propagation paths (like those studied in this paper), the Fried coherence diameters for a plane wave and a spherical wave follow as

We can also determine the log-amplitude variance ${\sigma }_{\chi }^2$ from the index-of-refraction structure parameter $C_n^2$. In practice, ${\sigma }_{\chi }^2$ gives us a gauge for the amount of scintillation that the propagated light has experienced [18,26–29]; as such, we can also use it as a gauge for the density of branch points. For horizontal-propagation paths, the log-amplitude variances for a plane wave and a spherical wave follow as

Given Eqs. (32)–(35), at the simulated entrance-pupil plane, a collimating lens of diameter $D_1=30\mathrm{cm}$ removed the parabolic phase from the propagated light, and the model created a “truth field” $U_S(x_1,y_1)$ by cropping out the center $256\times 256$ grid points. With that said, the model accounted for the demagnification between the simulated entrance- and exit-pupil planes via interpolation. As shown in Figs. 2(a) and 2(b), the model created a signal beam $U_S(x_2,y_2)$ by interpolating $U_S(x_1,y_1)$ to have the correct number of FPA pixels $N_P$ across the exit-pupil diameter $D_2$, resulting in an $N_P\times N_P$ grid. Upon interpolating, the model interfered $U_S(x_2,y_2)$ with a reference beam $U_R(x_2,y_2)$ to obtain a hologram irradiance $I_H(x_2,y_2)$ [cf. Eqs. (1), (3), and (5)]. Note that the model adjusted the LO coordinates $(x_R,y_R)$ in Eq. (3) to be 1/4 the simulated Fourier plane side length $L_F$ [cf. Eq. (20)]. In addition, the model set the amplitude of the reference beam $|A_R|$ to produce a per-pixel mean number of reference photoelectrons equal to 75% of the pixel-well depth, as opposed to 25% of the pixel-well depth in [10] (i.e., ${\overline{m}}_R=75,000\mathrm{pe}$ in this study for the off-axis PPRG and ${\overline{m}}_R=25,000\mathrm{pe}$ in a companion study for the off-axis IPRG). Also note that we observed no saturated pixels as a result of this choice. The model then scaled the amplitude of the signal beam $|A_S|$ to produce a per-pixel mean number of signal photoelectrons, such that ${\overline{m}}_S=0.283–28.3\mathrm{pe}$. By varying ${\overline{m}}_S$, the model allowed us to examine performance when in the presence of varying levels of detection noise.

 

Fig. 2. In (a) and (b), we show the normalized irradiance of the signal beam, whereas in (c) and (d), we show the resultant digital holograms for two different values of the number of detector pixels, $N_P$, across the exit-pupil diameter, $D_2$ (white circles). The black-dashed circles have diameters equal to the demagnified spherical-wave Fried coherence diameter, that is, $M_T{r}_{0-sw}$ for turbulence Scenario 5 in Table 1. It is important to note that we display the associated turbulence-limited sampling quotient for a spherical wave $Q_{P-sw}$ in the bottom-left corner of each subplot.

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With regard to detection noise, the model included both shot noise and read noise within the computational simulations. Shot noise followed a Poisson distribution, whereas read noise followed a Gaussian distribution. We set the read-noise standard deviation to 100 pe and the pixel-well depth to $100\times {10}^3\mathrm{pe}$. As such, the standard deviation of the shot noise varied within the computational simulations and was the dominate source of detection noise.

Following the above steps, we obtained our digital holograms [cf. Figs. 2(c) and 2(d)]. From here, the model performed signal-processing techniques to go to the simulated Fourier plane, isolate the turbulence-limited ASF, and obtain an estimate of the complex-optical field in the simulated entrance-pupil plane, that is, the “estimated field” ${\widehat{U}}_S(x_1,y_1)$. The reader can find a pictorial representation of these signal-processing techniques in Fig. 1(b). For all intents and purposes, these signal-processing techniques took place in five steps as follows.

Within these steps, we can infer several important conclusions about the number of FPA pixels, $N_P$, across the exit-pupil diameter, $D_2$, and the radius, ${\rho }_W$, of the window function in the simulated Fourier plane. First, as evidenced in Fig. 3, varying $N_P$ alters the side length of the simulated Fourier plane per Eq. (20), which shows that the Fourier-plane side length $L_F$ is directly proportional to $N_P$. While $N_P$ determines $L_F$, the value of ${\rho }_W$ actually determines the portion of the turbulence-limited point spread function (PSF) (i.e., the irradiance associated with the turbulence-limited ASF) to be inverse Fourier transformed. Thus, ${\rho }_W$ determines what spatial-frequency content makes it into the estimated field ${\widehat{U}}_S(x_1,y_1)$ and the sampling of ${\widehat{U}}_S(x_1,y_1)$.

 

Fig. 3. This figure displays the simulated Fourier plane for two different values of the number of detector pixels, $N_P$, across the exit-pupil diameter, $D_2$. From Eq. (21), the autocorrelation term and the Dirac-delta term appear at the center of each plot; however, the strengths of these terms are negligible because we used a strong reference beam and subtracted off the average photoelectron count from each digital hologram. With that said, the turbulence-limited PSF and its complex conjugate are readily visible in each plot, and we isolate the correct turbulence-limited PSF with a window function. Note that the white circles represent the window function, and the green-dotted circles demarcate the turbulence-limited bucket diameter for a spherical wave $D_{T-sw}$ for Scenario 5 in Table 1. Also note that the Fourier-plane side length, $L_F$, and radius, ${\rho }_W$, of the window function vary from subplot to subplot. Here, we express the size of ${\rho }_W$ and $L_F$ in terms of the approximate diffraction-limited bucket radius ${\rho }_I$.

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Provided the truth field $U_S(x_1,y_1)$ and the estimated field ${\widehat{U}}_S(x_1,y_1)$, the model used a metric dubbed the field-estimated Strehl ratio $S_F$ to quantify performance. In practice,

C. Signal-to-Noise Models

As previously stated, the numerical wave-optics model included both shot noise and read noise within the computational simulations. Here, we assumed that the shot noise resulted from the random arrival times of the photons that were incident on the FPA and that the read noise resulted from the read-out integrated circuitry of the FPA. The reader should note once more that the modeled shot noise followed a Poisson distribution, whereas read noise followed a Gaussian distribution.

For a Poisson-distributed random process, the mean is equal to the variance [25]. In this paper, the mean is equal to the sum of the per-pixel mean number of photoelectrons from the signal and reference beams, ${\overline{m}}_S$ and ${\overline{m}}_R$, respectively. Given that ${\overline{m}}_R\gg {\overline{m}}_S$, we can assume that the mean number of photoelectrons varies little from pixel to pixel because ${\overline{m}}_R$ is at 75% of the pixel-well depth. In turn, the Poisson-distributed shot noise follows a Gaussian distribution (to a good approximation) with variance ${\overline{m}}_S+{\overline{m}}_R$. Armed with these assumptions, we can add the variances for each Gaussian-distributed random process and arrive at the total noise variance ${\sigma }_n^2$, such that

In terms of obtaining a closed-form expression for our analytical SNR, we can determine the SNR, in general, as the expected value of the signal power in the Fourier plane divided by the variance of the noise in the Fourier plane [30]. This ratio gives the following relationship:

In terms of obtaining our numerical SNR, we first determined the power associated with the signal and noise in the windowed area of the simulated Fourier plane. Next, we determined the power associated with just the noise in the windowed area of the simulated Fourier plane. The reader should note that we determined this noise-only power by taking a 2D FFT of the reference beam that was incident of the FPA without the signal beam present. This step was only valid if ${\overline{m}}_R\gg {\overline{m}}_S$. Put another way,

With all of this in mind, we can determine the numerical SNR as the ratio of the expected value of the signal-only power to the variance of the noise in the Fourier plane. In mathematical form, our numerical SNR becomes

 

Fig. 4. This plot displays the percentage errors between the analytically and numerically calculated SNRs for 10 different SNRs as given by Eq. (39) for turbulence Scenarios 1 and 5 in Table 1. We calculated the SNRs by using the maximum allowed radius, ${\rho }_W$, of the window function in the Fourier plane (i.e., ${\rho }_W$ was set to one fourth of the Fourier-plane side length $L_F$). To vary the SNRs, we adjusted the per-pixel mean number of photoelectrons from the signal beam. Here, the error bars depict the standard-deviation offsets for 20 independent realizations of turbulence and 20 independent realizations of detection noise, that is, 400 realizations.

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3. RESULTS AND DISCUSSION

In this section, we present the results of a trade-space analysis that

In Fig. 5(a), we see that the performance remains stagnant for values of $N_P$ that exceed $\sim 32\mathrm{pixels}$ for each turbulence scenario. While this result might seem strange, recall that $N_P$ only affects the Fourier-plane side length $L_F$ and is completely decoupled from the size of the window function that encompasses the turbulence-limited PSF. Figure 5(b) shows that the performance increases as the SNR increases for each value of $N_P$, but that the performance curves approach a higher asymptotic value when $N_P=96$ and $N_P=192$ than when $N_P=16$. This outcome is most likely due to aliasing that occurs when $Q_P<4.88$, as discussed in Section 2.A.

 

Fig. 5. In this plot, we see the trade-space trends that arise from varying the number of FPA pixels, $N_P$, across the exit-pupil diameter, $D_2$, in (a) and the SNR in (b) for the turbulence scenarios in Table 1. All of the solid colored lines are averages of 20 independent realizations of turbulence and 20 independent realizations of detection noise, that is, 400 realizations. The dashed lines are standard-deviation offsets from these averages. In both plots, the radius, ${\rho }_W$, of the window function in the Fourier plane is always four times the approximate diffraction-limited bucket radius, ${\rho }_I$.

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In Fig. 6(a), the performance reaches a different distinct maximum for each turbulence scenario, and there appears to be an optimal value of ${\rho }_W$. As the strength of turbulence increases, the high-spatial-frequency content contained in the turbulence-limited PSF increases and necessitates the use of a wider window function in the Fourier plane to better encompass this high-spatial-frequency content. Thus, the optimal value of ${\rho }_W$ increases for increasing turbulence strengths.

 

Fig. 6. In this plot, we see the trade-space trends that arise from varying the radius ${\rho }_W$ of the window function in the Fourier plane in (a) and the SNR in (b) for the turbulence scenarios in Table 1. All of the solid colored lines are averages of 20 independent realizations of turbulence and 20 independent realizations of detection noise, that is, 400 realizations. The dashed lines are standard-deviation offsets from these averages. It is important to note that the peaks in the curves in (a) are pushed out to higher values of ${\rho }_W/{\rho }_I$ as the strength of turbulence increases (${\rho }_W/{\rho }_I=31.5$ for Scenario 1 and ${\rho }_W/{\rho }_I=38.0$ for Scenario 5). Again, ${\rho }_I$ is the approximate diffraction-limited bucket radius. In both plots, the number of FPA pixels, $N_P$, across the diameter of the exit pupil, $D_2$, is 256.

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Figure 6(b) shows that smaller values of ${\rho }_W$ provide better results in low SNR scenarios and that larger values of ${\rho }_W$ provide better results as the SNR increases. This is said because larger values of ${\rho }_W$ (which correspond to larger window functions) cause more of the Fourier plane, and thus the noise in the Fourier plane, to be encompassed. Therefore, smaller window functions are more robust to noise in the Fourier plane because they encompass a tighter region of the turbulence-limited PSF and less of the regions where the signal floor is comparable with the noise floor in the Fourier plane.

Figures 7(a)–7(d) affirm the behaviors found in Fig. 6 by looking at the wrapped-phase estimate for varying values of ${\rho }_W$. The lowest value of ${\rho }_W$ gives a wrapped-phase estimate that is smooth when compared with the wrapped-phase truth because of the 2D convolution with the sombrero function in Eq. (30). As the value of ${\rho }_W$ increases, the wrapped-phase estimates look more like the wrapped-phase truth, as seen in Fig. 7(c), but eventually become degraded by high-frequency noise, as seen in Fig. 7(d).

 

Fig. 7. Here, we show the wrapped-phase truth in (a) and the wrapped-phase estimates in (b)–(d) determined via a single independent realization of digital-holographic detection in the off-axis PPRG. The number of FPA pixels, $N_P$, across the exit-pupil diameter, $D_2$, is 256 for these results, and the turbulence strength corresponds to Scenario 5 in Table 1. Note that the black-dashed circle in (a) has a diameter equal to the Fried coherence diameter for a spherical wave $r_{0-sw}$.

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The final point of discussion lies in the analysis of the performance peaks when varying the value of ${\rho }_W$ in Fig. 6(a). With that said, we can explain this behavior when the noise floor in the Fourier plane is compared with the turbulence-limited PSF without noise. Figure 8 displays the azimuthally averaged and normalized turbulence-limited PSFs, with and without noise, as well as the simulated and expected noise-only power, as per Eq. (37). Upon examination of the turbulence-limited PSFs without noise (blue curves in Fig. 8), the reader can see that the crossing points with the noise-only power (red curves in Fig. 8) occur at nearly the same radial values as do the peaks in $S_F$ in Fig. 6(a). The end result is that the optimal window radius occurs right as the turbulence-limited PSF dips below the noise floor in the Fourier plane. As seen in Fig. 7, this minimizes the smoothing caused by convolution with the sombrero function [cf. Eq. (30)] while omitting high-frequency noise. Figure 8 verifies the claim that increasing the turbulence strength increases the high-spatial-frequency content contained within the turbulence-limited PSF and necessitates a wider window function in the Fourier plane to achieve optimal performance.

 

Fig. 8. This figure displays the azimuthally averaged and normalized [with respect to the maximum of (a)] turbulence-limited PSFs with and without noise as well as the simulated-noise power and expected-noise power ${\sigma }_n^2$ in the Fourier plane for Scenarios 1 and 5 [cf. Table 1] in (a) and (b), respectively. Note that the values of $\rho /{\rho }_I$ at the crossings of the blue and red curves for each turbulence scenario are roughly equal to the values of ${\rho }_W/{\rho }_I$ at the peaks in performance for Fig. 6(a) (${\rho }_W/{\rho }_I=31.5$ for Scenario 1 and ${\rho }_W/{\rho }_I=38.0$ for Scenario 5). Also note that the $y$ axes for both plots are normalized and displayed on a $\log$ scale to help visualize the high-spatial-frequency content of the turbulence-limited PSFs. Both plots show the averages of 20 independent realizations of turbulence and 20 independent realizations of detection noise, that is, 400 realizations.

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With all of this in mind, we extended the analysis of the optimal window radius to account for all of the turbulence scenarios in Table 1. Figure 9 ultimately shows that we have developed a method for optimal windowing of the turbulence-limited PSF. Here, the differences between the window radii associated with maximum performance (like those seen in Fig. 6) and the crossover radii where the turbulence-limited PSF dips below the noise floor in the Fourier plane (as seen in Fig. 8) are small. Across all scenarios, the differences waver around a mean value of $-0.0498$ with a standard deviation of 0.6449 in units of length normalized to the approximate diffraction-limited bucket radius ${\rho }_I$. These results indicate good agreement between the window radii (associated with maximum performance) and the crossover radii (where the turbulence-limited PSF dips below the noise floor in the Fourier plane).

 

Fig. 9. This figure displays the differences between the window radii associated with maximum performance and the crossover radii where the turbulence-limited PSF dips below the noise floor in the Fourier plane for all of the turbulence scenarios in Table 1. As a function of SNR, all of the points waver around a mean value of $-0.0498$ with a standard deviation of 0.6449 in units of length normalized to the approximate diffraction-limited bucket radius ${\rho }_I$ (which is a small quantity compared with the radius ${\rho }_W$ of the window functions in the Fourier plane). The crossing points of the turbulence-limited PSFs with the noise floor were determined by taking the average of the first 10 points that were within $0.0005{\mathrm{pe}}^2$ of the expected-noise power ${\sigma }_n^2$. The data show the averages of 20 independent realizations of turbulence and 20 independent realizations of detection noise.

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4. CONCLUSION

From the results presented in this paper, we find that, when using the appropriate parameters, digital-holographic detection in the off-axis PPRG is a robust and effective method for sensing the complex-optical field when in the presence of deep turbulence and detection noise. The results show that varying the number of FPA pixels across the exit-pupil diameter directly affects the side length of the Fourier plane, which contains the turbulence-limited PSF. Increasing the number of FPA pixels across the exit-pupil diameter past the point where aliasing becomes mitigated also offers no appreciable change in performance. It is important to note that the radius of the window function in the Fourier plane determines what spatial-frequency content makes it into the estimated field and the sampling of this estimated field. With this in mind, the results show that the window radii of maximum performance correspond to where the turbulence-limited PSF crosses the noise floor in the Fourier plane. This outcome provides us with a method for optimal windowing of the turbulence-limited PSF. In addition, our numerically calculated SNRs match well with our analytically calculated SNRs. This outcome means that future research efforts can calculate experimental SNRs with high accuracy via a closed-form SNR expression (provided that the strength of the reference beam is much, much greater than the strength of the signal beam and that these future research efforts have accurate equipment specifications).

Similar to the companion paper by Spencer et al. [10], this paper shows that digital-holographic detection offers a distinct way forward (with respect to wavefront sensing) when faced with the low SNRs brought about by deep-turbulence scintillation and extinction. We make this claim because digital-holographic detection uses a reference beam to boost the signal beam well above the read-noise floor that is created by the FPA. In practice, this strong reference beam allows us to reach a shot-noise-limited detection regime, and, in contrast with the companion paper by Spencer et al. [10], this paper shows that we are able to set the per-pixel reference beam strength to be much higher in the off-axis PPRG than in the off-axis IPRG (75% of the pixel-well depth as opposed to 25% in [10]) given a point-source beacon. Overall, the off-axis PPRG is less prone to pixel saturation because of focal-plane hot spots caused by deep-turbulence scintillation; thus, we can better approach the shot-noise limit when in the face of low SNRs. Given that digital-holographic detection in the off-axis PPRG also provides us with access to the complex-optical field directly, we can better explore the associated branch-point problem.