1. INTRODUCTION

Holography has a rich history in applications involving long-range imaging [1] and wavefront reconstruction [2]. In practice, we can use these applications in concert to overcome atmospheric distortions [3–5]. With the advent of robust focal-plane arrays (FPAs), researchers began to measure and store holograms digitally [6]. In an effort to characterize the atmosphere [7–9], this early research provided the framework needed for deep-turbulence wavefront sensing using digital holography.

Many applications, such as free-space laser communications, involve propagation paths which experience deep-turbulence conditions. Also known as distributed-volume turbulence or strong turbulence, deep turbulence arises from atmospheric aberrations being distributed along the propagation path. Given spatially coherent light, this outcome gives rise to time-varying constructive and destructive interference. Known as scintillation, this phenomenon typically hinders wavefront-sensing performance. While all atmospheric-optical paths experience scintillation to some degree, the effects of scintillation are often negligible for the vertical-propagation paths (e.g., those associated with ground-based telescopes [10], which experience weak-turbulence conditions) and often appreciable for the horizontal-propagation paths (e.g., those associated with long-range imaging systems [11], which experience deep-turbulence conditions).

Traditional wavefront-sensing methods use localized irradiance measurements to estimate phase gradients [e.g., the Shack–Hartmann wavefront sensor (SHWFS)]. Note that these traditional methods enable near-diffraction-limited optical systems [10]; however, performance degrades substantially in the presence of strong scintillation. For all intents and purposes, strong scintillation occurs when, for example, the log-amplitude variance, which gives a measure for the amount of scintillation, becomes greater than 0.2 [12,13]. Given strong scintillation, branch points arise in the continuous-phase function, in particular, where there are amplitude nulls in the real and imaginary parts of the complex-optical field [14]. For gradient-based wavefront sensors, like the SHWFS, these amplitude nulls cause the wavefront sensor to measure and reconstruct noise. The branch points also add a rotational component to the phase function that gets mapped to the null space of a least-squares phase reconstructor, which manipulates the estimated phase gradients into a continuous-phase function. In turn, traditional wavefront-sensing methods do not perform well in deep-turbulence conditions.

We can alternatively use interferometric wavefront-sensing methods, such as the point-diffraction interferometer [15] and self-referencing interferometer (SRI) [16], to obtain an estimate of the complex-optical field. This estimate gives us access to the wrapped-phase function which contains both the irrotational and rotational phase components [14]. As such, we can use a branch-point-tolerant phase reconstructor to buy back performance when in the presence of strong scintillation [17]; however, this approach has yet to be demonstrated beyond a scaled-laboratory environment [18]. This last detail is most likely due to additional constraints caused by deep-turbulence conditions. For example, with an SRI, the received light is split to create a spatially filtered reference beam. Typically, researchers perform this spatial filtering by coupling the split received light into a single-mode optical fiber. When in the presence of strong scintillation, which arises with deep-turbulence conditions, this coupling results in efficiency losses and creates low signal-to-noise ratios (SNRs) that quickly lead to performance degradations [19].

To overcome the performance degradations caused by deep-turbulence conditions, we can instead use digital holography which is another interferometric wavefront-sensing method. In practice, digital holography is able to resolve the branch points associated with strong scintillation, since it provides us with an estimate of complex-optical field and access to the wrapped-phase function, which contains both the rotational and irrotational phase components [14]. Furthermore, digital holography is robust against the amplitude nulls caused by strong scintillation. The use of a strong reference beam from a local oscillator (LO) allows us to approach the shot-noise limit, since the signal beam is boosted above the read-noise floor of the FPA [20]. With these benefits in mind, this paper evaluates the performance of digital holography in the on-axis phase-shifting recording geometry (PSRG), as shown in Figs. 1 and 2. This paper, in turn, provides the necessary analysis needed to design and conduct future deep-turbulence experiments using digital holography in the on-axis PSRG. These experiments shall investigate both open- and closed-loop performance in scaled-laboratory and field environments. Such experiments shall also include additional factors not investigated in this paper, such as reference-beam nonuniformity, detector nonlinearities, laser practicalities, and vibration resistance.

 

Fig. 1. Illustration of digital holography in the on-axis PSRG. Note that we need phase-shifting optics (PSO) to implement our phase-shifting strategy (cf. Fig. 2).

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Fig. 2. Example of the PSO needed for the four-step method. This example also contains an illustration of the directional dependence of the $\pi$-phase shift upon reflection from a beam splitter (BS).

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It is worth mentioning that this paper builds upon the noiseless analysis contained in a recent conference proceeding by Thornton et al. [21]. Specifically, this paper develops and verifies the use of closed-form expressions for the SNR and field-estimated Strehl ratio with respect to the on-axis PSRG operating with three separate phase-shifting strategies—the four-, three-, and two-step methods [22,23]. Using detailed wave-optics simulations, which propagate a point-source beacon through deep-turbulence conditions, model digital holography with noise, and calculate the Monte Carlo averages associated with increasing turbulence strengths and decreasing focal-plane array sampling, the analysis shows the four-step method is the most efficient phase-shifting strategy. Furthermore, deep-turbulence conditions only degrade performance with respect to insufficient FPA sampling and low SNRs.

It is also worth mentioning that this paper is a “companion paper” to the analysis presented by Spencer et al. [24] and Banet et al. [25] with respect to digital holography in the off-axis image plane recording geometry (IPRG) and off-axis pupil plane recording geometry (PPRG), respectively. In practice, the off-axis IPRG and off-axis PPRG indirectly obtain an estimate of the complex-optical field from Fourier transformations and digital-signal-processing techniques (i.e., filtering the 2D spectrum of the digital hologram recorded with the FPA). Conversely, the on-axis PSRG directly obtains an estimate of the complex-optical field from multiple digital holograms being recorded on one or more FPA(s) and straightforward calculations [20].

In what follows, this paper develops closed-form expressions for two performance metrics (cf. Section 2), the SNR and field-estimated Strehl ratio, and verifies their use with detailed wave-optics simulations, which again, propagate a point-source beacon through deep-turbulence conditions, model digital holography with noise, and calculate the Monte Carlo averages associated with increasing turbulence strengths and decreasing focal-plane array sampling. (cf. Section 3). This outcome demonstrates the on-axis PSRG is another valid recording geometry for deep-turbulence wavefront sensing. Before moving onto the next section, it is important to note that digital holography in the on-axis PSRG and the SRI are similar in design, except for the origin of the reference beam. Therefore, this paper also includes a comparison between digital holography in the on-axis PSRG and the SRI to show the benefits of using a strong reference beam for applications involving deep-turbulence conditions (cf. Section 4).

2. DEVELOPMENT OF CLOSED-FORM EXPRESSIONS FOR TWO PERFORMANCE METRICS

This section provides an overview of the optical setup used for digital holography in the on-axis PSRG. It also develops estimate and noise models for three separate phase-shifting strategies—the four-, three-, and two-step methods. We then use the models to develop closed-form expressions for the SNR and field-estimated Strehl ratio. In the ensuing sections, we verify the use of these performance metrics via wave-optics simulations and then use them to compare the performance of digital holography in the on-axis PSRG to the performance of an SRI, both using the four-step method (i.e., the most efficient method).

A. Optical Setup

As shown in Fig. 1, to realize digital holography in the on-axis PSRG, we split a master oscillator laser into two optical legs. The first leg flood illuminates an unresolved, ball-bearing object creating a point-source beacon. Then, the reflected spherical wave propagates through deep-turbulence conditions and becomes the signal beam $U_S$ collimated in a pupil. The second leg creates a LO that gives rise to a reference beam $U_R$. After the reference and signal beams pass through the phase-shifting optics (PSO), we use the interference of light to create multiple holograms and record the resulting hologram irradiances $i_H^{(\delta )}$ on the FPA(s).

With Fig. 1 in mind, we show some of the details of the PSO in Fig. 2 for the four-step method. As shown in Fig. 2, the purpose of the PSO is to obtain the desired phase shift $\delta$ on the reference beam for the recorded holograms. Note that, in practice, a phase shift of $\pi$ occurs with reflections from the mirrors (M) and 50/50 beam splitters when the light is incident on the side favoring the black dot. The reflected light incident on the opposing side of the black dot does not incur a phase shift [26]. Additionally, we obtain a $\pi /2$-phase shift at the quarter-wave plate. Aggregating these phase shifts for the reference and signal beams results in the four holograms shown in Fig. 2 and has been demonstrated in hardware [27,28].

There are two more items to consider with respect to Fig. 2: optical-path length and polarization. As shown, we do not draw the optical-path lengths to scale, and in a real system, we would want to match the optical-path lengths to ensure the proper phase shifts and minimize any losses in fringe visibility due to vibrations. Also note that we would need to image the signal and reference beams onto the FPA(s) to record the resulting hologram irradiances. To do so, we would need to employ relay optics (not shown here) to create conjugate pupil planes at the FPA(s), so that we conserve the phase of the signal and reference beams in forming our holograms. With respect to the polarization concerns, maximum fringe visibility only occurs when the reference and signal beam’s polarization states match. However, in a real system, the reference beam’s polarization state is probably different than the signal beam’s polarization state due to rough-surface scattering from the object. Therefore, also not shown here are the polarization optics we would need to use to maximize the fringe visibility in our holograms. Moving forward in the analysis, we simply assume that we match the polarization states and that we are only dealing with absolute phase shifts (e.g., there are no piston errors in the phase shifts).

With the above assumptions in mind, the on-axis PSRG can employ different phase-shifting strategies to calculate the complex-optical field [22,23]. In this paper, we analyze three methods, namely, the four-, three-, and two-step methods. As previously stated, we show the four-step method’s PSO in Fig. 2. The three- and two-step methods use similar PSO with some exceptions. In particular, we can modify the fraction of transmitted/reflected light off the first beam splitter encountered for the reference and signal beams to ensure equal amounts of light for each hologram. We can also exchange the final beam splitters with beam-combining optics to remove the unnecessary measurements while preserving the signal beam. In so doing, we only divide the signal beam by the number of holograms desired, and the holograms make use of corresponding strong reference beams to maximize SNR.

B. Estimate Model

Provided Figs. 1 and 2, the hologram irradiances $i_H^{(\delta )}$ take the following form:

Provided Eq. (2), we can perform algebraic manipulations to isolate the signal beam $U_S$ [22]. For each of the phase-shifting strategies considered in this paper, we arrive at the following relationships:

With Eqs. (1)–(6) in mind, we can record the hologram irradiances $i_H^{(\delta )}$ with a FPA, which performs a pixel-by-pixel integration [20]. Thus, for a FPA with $M\times N$ pixels,

Provided the right-hand sides of Eqs. (3)–(5), which depend on the measured hologram irradiances, $i_H^{(\delta )}$, we can use Eq. (9) to obtain a generic expression for the signal-beam estimate, ${\widehat{U}}_S^{(s)}$, in terms of the left-hand sides of Eqs. (3)–(5). In particular,

 

Fig. 3. Illustration of how digital holography in the on-axis PSRG allows us to access the wrapped-phase function. Here, we use the interference of light to create multiple holograms by mixing a phase-shifted reference beam with a signal beam. Note that the number of shifts or measurements required by the phase-shifting strategy is dependent on the phase-shifting method being used (here, we illustrate the four-step method). After we record the hologram irradiances with a FPA, we perform a straightforward calculation to obtain the wrapped-phase function from an estimate of complex-optical field [cf. Eq. (12) along with Eqs. (3)–(5)].

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C. Noise Model

Moving forward in the analysis, we would like to account for the effects of shot noise and read noise. For this purpose, we assume that the shot noise results from the random arrival times of the photons that are incident on the FPA, and that the read noise results from the read-out integrated circuitry of the FPA. We also assume that the shot noise follows a Poisson distribution, whereas read noise follows a Gaussian distribution.

For a Poisson-distributed random process, the mean is equal to the variance [26]. In this paper [cf. Eqs. (10) and (11)], 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. Here, we drop the FPA coordinates to denote the average over the entire detection area. Since we set ${\overline{m}}_R$ to 75% of pixel-well depth of the FPA, we assume that ${\overline{m}}_R\gg {\overline{m}}_S$ and that the mean number of hologram photoelectrons varies little from pixel to pixel because of the strong reference beam (${\overline{m}}_H\approx {\overline{m}}_R$). In turn, the Poisson-distributed shot noise follows a Gaussian distribution (to a good approximation when ${\overline{m}}_R\gg 1$) with variance ${\overline{m}}_R+{\overline{m}}_S\approx {\overline{m}}_R$. Armed with these assumptions, we can add the variances for each Gaussian-distributed random process and arrive at the noise variance ${\sigma }_n^2$, such that

In the analysis that follows, we model the additive Gaussian noise [20], such that

D. Signal-to-Noise and Field-Estimated Strehl Ratios

In what follows, we formulate the SNR $S/N^{(s)}$ as the ratio of the mean signal power to the total noise variance. As such, we obtain the following relationship:

Another performance metric of interest in the analysis is the field-estimated Strehl ratio $S_F^{(s)}$ [24,25,31]. As shown in Appendix A and Appendix B, we can relate $S_F^{(s)}$ to the SNR $S/N^{(s)}$ via the following relationship [31]:

3. PERFORMANCE METRIC COMPARISON USING WAVE-OPTICS SIMULATIONS

This section develops the wave-optics simulations employed to verify the closed-form expressions developed above for the signal-to-noise and field-estimated Strehl ratios. We conduct this analysis entirely in MATLAB using the principles from Schmidt [32] with help from WaveProp [33] and AOTools [34], which are MATLAB toolboxes written by the Optical Sciences Company. For further insight on these wave-optics simulations, we list several references that include additional detail [21,24,25,35,36].

A. Numerical Model

With Fig. 1 in mind, we modeled the point-source beacon as a narrow sinc function modulated by a raised-cosine envelope on a $4096\times 4096$ numerical grid. Note that we set the physical side length of the numerical grid in the object plane, so that we met Fresnel scaling. We then propagated the point-source beacon to the pupil plane using the split-step beam propagation method. Here, we collimated the light and cropped the numerical grid to $256\times 256$, so that it had the same physical side length as the pupil diameter, $D$. We provide a list of simulation parameters in Table 1.

Table 1. Simulation Parameters Used in the Wave-Optics Simulations

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With Table 1 in mind, we modeled five distinct scenarios with increasing turbulence strengths. Table 2 lists the refractive-index structure parameter, $C_n^2$, spherical-wave log amplitude variance (Rytov number), ${\sigma }_{\chi -sw}^2$, and spherical-wave coherence length (Fried parameter), $r_{0-sw}$, for the various scenarios. For a given $C_n^2$, wavelength $\lambda$, and horizontal-path propagation distance $z$, we can calculate ${\sigma }_{\chi -sw}^2$ and $r_{0-sw}$, respectively, using the following formulas [37]:

Table 2. Turbulence Parameters Used for Five-Distinct Scenarios

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With respect to the split-step beam propagation method, we used 10 equally spaced Kolmogorov phase screens to achieve deep-turbulence conditions. By satisfying Fresnel scaling, we met all of the sampling requirements set forth by Schmidt [32], as discussed in Thornton et al. [21]. For model verification, the discrete calculations were within 1% error when compared to the continuous calculations [cf. Eq. (26) and (27)]. Additionally, we calculated the Monte Carlo averages associated with the magnitude of the complex degree of coherence in the pupil plane using 40 independent realizations for Scenarios 1–5 in Table 2, and the results closely matched theory [21]. We show an example of one realization of the irradiance and wrapped phase in Fig. 4 for the simulated signal-beam truth.

 

Fig. 4. (a) Irradiance and (b) wrapped phase for the simulated signal-beam truth for one realization of the turbulence (cf. Scenario 5 in Table 2).

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In addition to the turbulence strength, we varied the FPA sampling by changing the number of FPA pixels across a demagnified pupil image. For this objective, we interpolated the simulated signal beam in the pupil plane to match the size of the FPA. Note that we fixed the square-pixel width, so that the physical size of the FPA was proportional to the number of pixels across. Therefore, we demagnified the simulated signal beam, such that $M_T=W/D$, where $M_T$ is the transverse magnification, $W=N_p{w}_{x,y}$ is the side length of the FPA, $N_p$ is the FPA sampling, and $w_{x,y}$ is the square-pixel width. After this demagnification via interpolation, we scaled the mean number of signal photoelectrons ${\overline{m}}_S$ to vary the signal strength. For this purpose, we set the characteristics of the FPA, such that the quantum efficiency was 100%, as well as the pixel fill factor. Additionally, we assumed a uniform, linear pixel gain from zero to saturation (100,000 pe). As mentioned before, we then set the read-noise standard deviation, such that ${\sigma }_r=100\mathrm{pe}$, and the strength of the simulated strong reference beam, so that ${\overline{m}}_R=\mathrm{75,000}\mathrm{pe}$. Put another way, we set the mean number of reference photoelectrons ${\overline{m}}_R$ to 75% of pixel-well depth of the FPA to create a strong reference beam but avoid pixel saturation and excess shot noise.

We accounted for each phase-shifting strategy by dividing the simulated signal beam by the number of measurements needed for the corresponding method [cf. the factor $s$ in Eq. (12)]. With this in mind, we then estimated the complex-optical field in the pupil plane with the corresponding pixel-by-pixel formulas [cf. Eq. (12) along with Eqs. (3)–(5)]. We show an example of one realization of the irradiance and wrapped phase in Fig. 5 for the simulated signal-beam estimate.

 

Fig. 5. (a) Irradiance and (b) wrapped phase for the simulated signal-beam estimate for one realization of turbulence (cf. Scenario 5 in Table 2).

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B. Numerical Results

The results of the wave-optics simulations presented here cover a threefold trade space with respect to the four-, three-, and two-step methods. In particular, we quantify performance by

As shown in Fig. 6, we verify the use of Eqs. (16)–(25). Here, we compare both the numerical field-estimated Strehl ratio $S_F$ and numerical SNR $S/N$ to theory. Note that the differences between the theoretical lines and data points are less than 1% for the Monte Carlo averages from 40 independent realizations of turbulence and 30 independent realizations of noise for all of the scenarios given in Table 2. To calculate $S_F$, we made use of the following relationship (cf. Appendix A):

 

Fig. 6. Numerical field-estimated Strehl ratio $S_F$ versus the numerical SNR $S/N$ with a comparison to theory. Shown here are the Monte Carlo averages from 40 independent realizations of turbulence and 30 independent realizations of noise for all of the scenarios given in Table 2 and the three separate phase-shifting strategies of interest in this paper. Note that $N_p=256$ for all of these results.

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With the results of Fig. 6 in mind, in Fig. 7 we compare the numerical field-estimated Strehl ratio $S_F$ as a function of signal strength ${\overline{m}}_S$ to theory [cf. Eqs. (23)–(25)]. Again, the observed error between theory and the numerical results is less than 1% for the Monte Carlo averages from 40 independent realizations of turbulence and 30 independent realizations of noise for Scenario 1 and 5 in Table 2. From the results, we see that the four-step method is the most efficient phase-shifting strategy, despite having the most required signal-beam splits. The complete sampling of the phase in $\pi /2$ steps, in practice, results in a more precise estimate of the complex-optical field. Put another way, the $\zeta$ term within the total noise variance $\zeta {\sigma }_n$ [cf. Eq. (16)] introduces less noise into the estimate for the four-step method.

 

Fig. 7. Numerical field-estimated Strehl ratio $S_F$ versus the signal strength ${\overline{m}}_S$ with a comparison to theory. Shown here are the Monte Carlo averages from 40 independent realizations of turbulence and 30 independent realizations of noise for Scenario 1 and Scenario 5 in Table 2 and the three separate phase-shifting strategies of interest in this paper. Note that $N_p=256$ for all of these results. Also note that the open circles represent the results from Scenario 1, whereas the plus signs represent the results from Scenario 5.

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It is important to note that the results contained in Figs. 6 and 7 made use of a FPA with 256 pixels across the simulated demagnified pupil image (i.e., $M_T=1$). With that said, Fig. 8 shows the relative percent difference between the theoretical and numerical field-estimated Strehl ratios for Scenarios 1 and 5 from Table 2 and the four-step method. Here, $M_T$ varied such that the modeled, square FPA size ranged from 16 to 256 pixels across ($N_p$), and as such, the signal-beam estimate’s grid size varied from $16\times 16$ to $256\times 256$. To compare the signal-beam estimate to the simulated signal-beam truth (256 pixels across), we upsampled via linear interpolation. We plot the results as a function of the FPA sampling $N_p$ ($y$ axis) and the signal strength ${\overline{m}}_S$ ($x$ axis). In Fig. 8, we calculate the relative percent difference as

 

Fig. 8. Relative percent difference ($\mathrm{\Delta }{S}_F$) between the theoretical and numerical field-estimated Strehl ratios for (a) Scenario 1 and (b) Scenario 5 from Table 2. We plot the results as a function of the FPA sampling $N_p$ and mean signal strength ${\overline{m}}_S$ for the four-step method [cf. Eq. (23)]. Shown here are the Monte Carlo averages from 40 independent realizations of turbulence and 30 independent realizations of noise.

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With Fig. 8 in mind, a couple of features become apparent in the analysis. First, we reach steady-state differences between the theoretical and numerical results when ${\overline{m}}_S\gtrsim 10$, which is where the $S/N\gtrsim 10$ and $S_F\gtrsim 0.9$. As we increase the turbulence strength, the induced sampling errors also increase, and the results for Scenarios 2–4 also follow this trend. Thus, turbulence strength only affects the FPA sampling requirements. The largest differences occur with smaller values of $N_p$ and stronger turbulence strengths. Second, at low SNRs ($S/N<10$), the differences vary greatly with $N_p$. This outcome is due to the smoothing that occurs given the coarser FPA sampling [cf. the 2D convolution in Eq. (12)]. It also shows that when we properly sample the Fried parameter [cf. Eq. (27)], digital holography estimates the complex-optical field exceptionally well [38].

4. SRI COMPARISON

As discussed above, the SRI is an alternative interferometric wavefront-sensing method [16]. The primary difference between digital holography and the SRI is that the SRI splits the received signal beam to create a reference beam via spatial filtering, typically with a single-mode optical fiber. With that said, Rhoadarmer and Klein provide further discussion on the design of an SRI [28]. Various phase-shifting strategies also exist for the SRI, but similar to digital holography in the on-axis PSRG, the four-step method has the best performance [39].

In this section, we model deep-turbulence wavefront sensing using the SRI in the same fashion as the four-step method for digital holography in the on-axis PSRG (i.e., in an open-loop configuration) with a few exceptions. The first exception is that we set the beam-splitter ratio $\beta$, which splits the signal beam to create the reference beam, so that ${\overline{m}}_S={\overline{m}}_R$. This choice maximizes the SNR [40]. Additionally, we include the effects of a fiber-coupling efficiency ${\eta }_c$. Wheeler and Schmidt [19] showed that this efficiency depends on the spatial coherence radius ${\rho }_0$ or the coherence length $r_0$, since $r_0\approx 2.1{\rho }_0$ [41], relative to the pupil diameter $D$. From Table 2, the corresponding ${\eta }_c$ for Scenario 1 and Scenario 5 is 10% and 1%, respectively. Therefore, in the analysis that follows, we make use of the following three cases: ${\eta }_c=100\%$ for the ideal case, ${\eta }_c=10\%$ for Scenario 1, and ${\eta }_c=1\%$ for Scenario 5.

Because of the losses encountered with the fiber-coupling efficiency ${\eta }_c$, we need to introduce a new term: ${\overline{m}}_i$, which is the mean number of incident photoelectrons. For digital holography in the on-axis PSRG, ${\overline{m}}_i={\overline{m}}_S$, since we use 100% of the signal beam. On the other hand, for the SRI, ${\overline{m}}_S=\beta {\overline{m}}_i$, which is the percent of the incident light split for the signal beam, and ${\overline{m}}_R=(1-\beta ){\eta }_c{\overline{m}}_i$, which is the percent of the incident light both split and coupled into the single-mode optical fiber for the reference beam. In turn, ${\overline{m}}_L=(1-\beta )(1-{\eta }_c){\overline{m}}_i$ is the percent of the incident light lost due to fiber coupling, so that ${\overline{m}}_i={\overline{m}}_S+{\overline{m}}_R+{\overline{m}}_L$ for the SRI.

Similar to the analysis presented above (cf. Section 2), Rhoadarmer and Barchers formulated closed-form expressions for the SRI [31]. With respect to the four-step method, the SNR $S/N_{SRI}^{(4)}$ and field-estimated Strehl ratio $S_{F,SRI}^{(4)}$ follow as

For the comparison, we formulated results from two perspectives, as shown in Figs. 9 and 10. Here, we verified the use of the closed-form expressions for the SRI [cf. Eqs. (31) and (32)]. Note that the differences between the theoretical lines and data points are less than 1% for the Monte Carlo averages from 40 independent realizations of turbulence and 30 independent realizations of noise. Also, we made use of a FPA with 256 pixels across the simulated demagnified pupil image. With that said, the results show that digital holography in the on-axis PSRG outperforms the SRI with respect to the numerical field-estimated Strehl ratio $S_F$ (cf. Fig. 9) and the numerical SNR $S/N$ (cf. Fig. 10) by multiple orders of magnitude. As shown in Fig. 10, for example, there are notable differences in the incident mean number of incident photoelectrons ${\overline{m}}_i$ required to reach an $S/N=10$ with approximately 21 dB difference for the ideal case (green to yellow), 28 dB difference for Scenario 1 (green to orange), and 37 dB difference for Scenario 5 (green to red). These differences represent the necessary SRI signal amplification needed to achieve similar performance to digital holography in the on-axis PSRG due to the lack of a strong reference beam from the LO provided by digital holography.

 

Fig. 9. Numerical field-estimated Strehl ratio $S_F$ versus the mean number of incident photoelectrons ${\overline{m}}_i$ for digital holography in the on-axis PSRG and the SRI with 100%, 10%, and 1% fiber-coupling efficiency. Here, the solid lines represent the theoretical results for the four-step method [cf. Eq. (23) for digital holography in the on-axis PSRG and Eq. (32) for the SRI]. The open circles represent the numerical results for Scenario 1, whereas the x’s represent the numerical results for Scenario 5 (cf. Table 2). Shown here are the Monte Carlo averages from 40 independent realizations of turbulence and 30 independent realizations of noise.

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Fig. 10. Numerical SNR $S/N$ versus the mean number of incident photoelectrons ${\overline{m}}_i$ for digital holography in the on-axis PSRG and the SRI with 100%, 10%, and 1% fiber-coupling efficiency. Here, the solid lines represent the theoretical results for the four-step method [cf. Eq. (19) for digital holography in the on-axis PSRG and Eq. (31) for the SRI). The open circles represent the numerical results for Scenario 1, whereas the x’s represent the numerical results for Scenario 5 (cf. Table 2). Shown here are the Monte Carlo averages from 40 independent realizations of turbulence and 30 independent realizations of noise.

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5. CONCLUSION

The results presented here showcase the strengths of digital holography in the on-axis PSRG for the purposes of deep-turbulence wavefront sensing. Throughout this paper, we develop closed-form expressions for the field-estimated Strehl and signal-to-noise ratios for the two-, three-, and four-step methods. Using detailed wave-optics simulations, which propagate a point-source beacon through deep-turbulence conditions, model digital holography with noise, and calculate the Monte Carlo averages associated with increasing turbulence strengths and decreasing focal-plane array sampling, we also verify the use of these closed-form expressions. Overall, the results show the four-step method is the most efficient phase-shifting strategy and deep-turbulence conditions only degrade performance with respect to insufficient FPA sampling and low SNRs.

The first result is somewhat counterintuitive since the four-step method requires the most signal-beam splits. However, the results of the closed-form expressions and detailed wave-optics simulations show the four-step method is more concise with less noise in estimating the complex-optical field. Furthermore, when the FPA sampling and SNR is sufficient, the percent difference between the theoretical results and the numerical results is negligible, regardless of the turbulence strength. In general, the results show when the FPA sampling is greater than 32 pixels and the SNR is greater than 10, the field-estimated Strehl ratios are greater than 0.9 (with respect to the four-step method).

A comparison to the SRI also shows the benefits of using a strong reference beam to perform interferometric wavefront sensing. For this purpose, we modeled the SRI in an ideal way and included the effects of a fiber-coupling efficiency to provide more realistic SNRs with increasing turbulence strengths. In the low-SNR regime, the SRI needs tens of decibels more signal-beam power to achieve similar performance to digital holography in the on-axis PSRG. As such, this comparison provides a performance benchmark for applications involving deep-turbulence conditions.

In summary, this paper evaluates the performance of digital holography in the on-axis PSRG and enables the optimal design of such a deep-turbulence wavefront sensor. By employing four $\pi /2$ phase shifts, we minimized the total noise and improved system performance in terms of both the SNR and field-estimated Strehl ratio. System performance also approached theoretical limits when we sampled the four digital holograms with at least five pixels across the Fried parameter. Since digital holography provides a strong reference beam from a LO, we then showed it outperforms the SRI in low signal-to-noise conditions for deep-turbulence applications. In turn, this paper provides the necessary analysis needed to design and conduct future deep-turbulence experiments using digital holography in the on-axis PSRG.