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Abstract
The ability of a holographic optical spatial mode generator to reproduce a laser beam which has propagated through simulated atmospheric turbulence is characterized via wavefront sensor (WFS) measurements. The range of optical turbulence which can be effectively recreated is determined including its dependence on the pixel resolution of the spatial light modulator used as a hologram to modulate the beam. Optimal sampling of the hologram grating for producing turbulent spatial modes is studied analytically and experimentally. Mode fidelity above 90% is verified up to D/r0 = 10 with less than 3% error in the ratio D/r0. The Fried parameter is verified up to D/r0 = 50 demonstrating less than 9% error producing the targeted ratio D/r0.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Physically reproducing the effects of optical propagation through the atmosphere in a laboratory setting enables an evaluation of the coupling efficiency between optical devices for free-space optical applications. Examples of applications include communication over free-space channels, astronomy, and remote sensing [1–5]. One such technique employs a spatial light modulator (SLM) or digital micro-mirror device (DMD) as a hologram to modulate the phase and amplitude of a collimated input beam [6–9]. By displaying a computer-generated hologram (CGH) on the SLM, such a system can be used to produce a monochromatic beam in an arbitrary spatial mode. In particular, one can generate turbulent spatial modes simulating a beam which has propagated through multi-layer atmospheric turbulence with a compact optical setup using a single SLM.
This method offers a versatile laboratory simulation capability in comparison to traditional rotating phase plates. In addition to programmable implementation of a range of optical turbulence conditions—largely characterized by the ratio of aperture diameter to atmospheric coherence diameter $D/r_0$—one can also physically simulate other optical system features that affect coupling efficiency. For example, the CGH can be constructed to emulate a central obscuration in the telescope, tilt compensation from a fast-steering mirror, beam wander and pointing error, abberations associated to components in the optical train, and adaptive optics compensation. The same system can be used to generate fiber spatial modes, free-space modes, and Zernike abberations in a controlled manner. In this work, we study the capabilities of a holographic system to produce turbulent spatial modes simulating the light arriving at an optical receiver after propagation through the atmosphere. This receiver-side simulation capability is sufficient for many applications, including evaluating the coupling efficiency of receiver optical devices and in-hardware testing of adaptive optics systems in the presence of branch points and intensity nulls.
In this paper, we evaluate the fidelity of turbulent spatial modes generated holographically using a spatial light modulator to reproduce a laser beam which has propagated through multi-layer atmospheric turbulence up to $D/r_0=50$. In particular, we present direct measurements of the error in the targeted coherence diameter $r_0$. To our knowledge, this is the first precise measurement of $r_0$ reported in the literature for any atmospheric emulation system based on a liquid crystal SLM [9–20]; a notable gap considering the primary relationship between the ratio $D/r_0$ and receiver coupling efficiency [21,22]. Early measurements of the phase structure function found only coarse agreement with a 5/3 power law—insufficient for a measurement of $r_0$ [16,17]. Schmidt and Goda [18] showed improved power law behavior, albeit with a slightly lower exponent and relatively large, unquantified error in the coherence diameter $r_0$ attributed to underestimated tilt from the Shack-Hartmann WFS. Alternative measurements via the point spread function (PSF) presented in [19] found that phase screens with $D/r_0=8$ produced a long-exposure Strehl ratio expected for $D/r_0=4$. In contrast, the WFS measurements of the wave structure function presented in Section 5 yield a precise estimate of the coherence diameter $r_0$. These results reduce the uncertainty in adopting an atmospheric emulation system based on a liquid crystal SLM as a tool to characterize receiver optical device performance over free-space atmospheric channels.
In addition, we characterize the maximum ratio $D/r_0$ which can be produced with high fidelity based on the spatial resolution of the SLM. Related systems which employ an SLM directly as a phase screen are known to exhibit phase errors as the most severe phase gradients limit the SLM to several pixels per 2$\pi$ phase rotation [20]. Similar considerations for holographic systems (Section 2) yield a trade-off between the sampling of the hologram grating and the spatial bandwidth of the system—defining the maximum $D/r_0$ ratio one can generate. Optimizing the system for severe turbulence conditions thus requires specific attention to the sampling of the hologram grating. A numerical study of the effect of grating sampling on mode fidelity presented by Ando et al. found that the mode fidelity typically improves (with exceptions) as the grating pitch is increased from 2 to 10 pixels per period; however, the numerical study was limited to the shaping of Laguerre-Gaussian beams with only qualitative experimental verification from beam intensity profiles [23]. In this work, we study the effect of grating sampling on the mode fidelity of turbulent spatial modes produced by a holographic system both analytically (Section 2) and experimentally (Section 5). Measurements of the mode fidelity of a complex spatial mode require simultaneous phase and amplitude profiling and must account for the spatial resolution of the WFS—a procedure for this is given in Sections 4 and 5.
In Section 2 we present a theoretical analysis of the range of optical turbulence conditions which can be generated based on a hologram sampling analysis. The experimental setup is presented in Section 3 including techniques used for mitigating sources of error which can degrade the fidelity of the system. After describing the data analysis procedure in Section 4, the main results characterizing the performance of the system are presented in Section 5. Due to the slow response time of our SLM, we limit our study to simulation in a static context, i.e. producing static spatial modes corresponding to independent “snapshots" of a beam which has propagated through multi-layer atmospheric turbulence.
2. Producing optical turbulence with computer generated holograms
The system studied in this work produces a linearly-polarized, monochromatic beam with electric field $u_1(x,y;z)e^{i(kz-\omega t)}$ where $k=2\pi /\lambda$ is the wave number and $\omega = ck$ is the angular frequency. A phase-only hologram $e^{i\Phi (x,y)}$ is applied to a collimated beam to produce the prescribed beam $u_1(x,y)$ at the SLM plane $z=0$ in superposition with beams $u_m(x,y)$ which separate into a series of diffraction orders (Fig. 1)
with angular separation $p_1=(p_x^2+p_y^2)^{1/2}$ created by a phase grating of period $\Lambda = \lambda /p_1$ modulated by the hologram. The grating period $\Lambda$ determines the spatial bandwidth available to the first order beam $u_1$ and hence the range of optical turbulence which can be generated (Section 2.1). For a digital CGH—as implemented by an SLM—the grating period in each direction $\Lambda _{x,y} = \lambda /p_{x,y}$ is limited by sampling constraints (Section 2.2). The spatial modes $u_1$ are generated by numerical beam propagation through multiple random phase screens using the split-step beam propagation method (Section 2.4).
Fig. 1. Principle of a holographic spatial mode generator. A specified beam $u_1$ is created on the first-order diffraction from a hologram $e^{i\Phi }$ applied to a collimated beam.
2.1 Turbulence range constraint from hologram grating period
In order to extract the desired first order beam $u_1$, the angular separation $p_1$ of the various orders must be larger than the beam divergence of the diffracted beam. For turbulence-degraded beams, the divergence can be expressed in terms of Fried’s atmospheric coherence diameter $r_0$, with most of the power bounded within an angular diameter $3 \lambda /r_0$ (cf. Fig. 10 in [22]). The requirement $p_1 > 3 \lambda /r_0$ implies a hologram grating period
i.e., the period of the diffraction grating applied to the SLM should be less than one-third of the coherence diameter in order to spatially isolate the first order in the far field. Defining a pupil of diameter $D$ on the hologram, one obtains the upper limit where $P_g=D/\Lambda$ is the number of grating periods within the diameter of the hologram aperture.2.2 Grating constraints for pixelated holograms
In order to maximize the ratio $D/r_0$ supported by the system, it is necessary to separate the diffraction orders as much as possible to give maximum spatial bandwidth to the first order beam. The maximum angular separation $p_1$ is determined by the minimum grating period $\Lambda$, which can be analyzed based on the sampling theory for a discretely sampled spatial signal following the analysis of [24]. The range of spatial frequencies $\xi _x,\xi _y$ accessible for modulation by the hologram is limited by the spatial bandwidth of the device implementing the hologram $|\xi _x|,|\xi _y|\leq 1/2\delta$, where $\delta$ is the pixel pitch. To keep the first order beam within this modulation bandwidth, the grating period in each direction must satisfy $3/2\Lambda _{x,y}\leq 1/2\delta$, i.e. one requires at least $N_{x,y}=\Lambda _{x,y}/\delta = 3$ pixels per period in each direction. The angular separation $p_1$ is then maximized by using a diagonal grating $N_x=N_y$ to minimize the grating period $\Lambda$.
However, although a spatial filter ultimately selects only the first order beam, it is nevertheless important to sample the hologram well enough to create higher orders which carry substantial power relative to the first order, otherwise aliasing of the higher orders can corrupt the first order beam (Fig. 2). To keep the $m$-th order beam within the spatial bandwidth of the pixelated hologram requires $(2m+1)/2\Lambda _{x,y} \leq 1/2\delta$, i.e., at least $N_x,N_y= 2m+1$ pixels per period. Of course, increasing the number of pixels per period comes at the cost of reducing the spatial bandwidth (and spatial modes) available to the first-order beam as illustrated in Fig. 2.
Fig. 2. Spatial frequency content of a hologram $\Phi$ generated using the method described in Section 2.3 using a vertical grating with 3 pixels per period (top), and 5 pixels per period (bottom), as represented by the intensity in the focal plane of a thin lens of focal length $f$ within the spatial bandwidth $|x|/\lambda f \leq 1/2\delta$ of the hologram. The desired PSF is shown at right. Note the improved accuracy of the PSF obtained by increasing the grating sampling to modulate the second order comes at the cost of a reduced spatial bandwidth available to the first order. Images from numerical simulation of a 128$\times$128 pixel CGH with $D/r_0=2$.
Precisely how many orders should be kept within the spatial bandwidth of the pixelated hologram to prevent significant degradation of the first order beam depends on the method used to generate the hologram $\Phi$. Specifically, to minimize the number of pixels per period required to produce a spatial mode with high fidelity requires a hologram $\Phi$ for which the total power in the higher order beams in (1) is small.
2.3 Holograms for severe turbulence emulation
In this work, we study a holographic technique introduced by Arrizón et al. (Type 3 in [24]) for which the power in the higher orders decreases rapidly, making it well-suited for producing severe optical turbulence. For a specified optical field $u_1(x,y)=A(x,y)e^{i\phi (x,y)}$ the hologram $\Phi (x,y)$ is defined by identifying the first-order term in the Jacobi-Anger expansion
with that of (1). This produces a pair of equations for $\Phi (x,y)=a(x,y)\sin (\alpha (x,y))$ given by The amplitude $A$ is normalized to the range $0\leq A \leq 0.58$ so that the inverse for the first order Bessel function $J_1^{-1}$ is well-defined. We find that a convenient approximation is given byExperimental results presented in [24] use a diagonal grating with 5 pixels per period in each direction. The restriction on the achievable turbulence range Eq. (3) based on the grating period motivates a general assessment of the impact of the grating period on the performance of the system. Figure 3 shows a parametric plot of the $m$-th order Jacobi-Anger expansion Eq. (4) in the complex plane for $-\pi \leq \alpha \leq \pi$ with grating amplitude $a=1.84$ and $|m|\leq 4$. Observe that the partial expansion already lies very close to the unit circle with $|m|= 4$ orders. The deviation of the partial expansion from unit modulus decreases with the grating amplitude $a < 1.84$.
Fig. 3. Convergence of the Jacobi-Anger expansion to the unit circle in the complex plane shown via a parametric plot of the partial sum with $a=1.84$, $-\pi \leq \alpha \leq \pi$, and $|m|\leq 1,2,3,4$ (left). The total power in the $m$-th order relative to the first-order is bounded by $J_m^2(a)/J_1^2(a)$ and already becomes small for the third- and fourth-order beams (right).
More quantitatively, one can use Eq. (4) to calculate the relative power $\eta _m/\eta _1$ in the $m$-th order beam, where
Note that this power depends only on the amplitude $A(x,y)$ of the desired beam through the grating amplitude $a(x,y)=J_1^{-1}(A(x,y))$. For $m>1$, $J_m(a)/J_1(a)$ increases monotonically in $a$ over the interval $0\leq a\leq 1.84$ [25]. For beams $u_1$ supported in a finite domain $(x,y)\in \Omega$, it follows that the power $\eta _m/\eta _1$ in the higher orders relative to the first-order is largest for a flat-top beam profile $A\equiv A_{\max }$ for $(x,y)\in \Omega$, and decreases with the maximum normalized amplitude $A_{\max }\leq 0.58$. Hence, the relative power in the higher orders is bounded by $\eta _m/\eta _1 \leq J_m^2(a_{\max })/J_1^2(a_{\max })$, shown in Fig. 3 for $m\leq 4$.Note that even for the maximum grating amplitude $a_{\max }=1.84$ radians the relative power becomes quite small already for the third and fourth order beams. Specifically, the expansion yields less than 4% relative power in the third order and less than 0.2% relative power in the fourth order, giving a rough bound on the noise in the first order one can expect using 5 pixels per period and 7 pixels per period, respectively. On the other hand, the total power in the second order can be up to 30% of that in the first order, suggesting more significant noise from the second order may impact performance for holograms using only 3 pixels per period. The results presented in Section 5 provide an examination of the impact of this in the context of generating turbulent spatial modes. It may be noted that other techniques for phase-amplitude modulation using phase-only holograms have been studied which require similar considerations [24,26–29].
2.4 Beam propagation simulations
In order to generate complex amplitude holograms, full wave optics simulations are employed to numerically simulate a beam propagating through the atmosphere. In this work, we consider space-to-ground paths simulated using the split-step beam propagation method [30]. The turbulence is represented by splitting the propagation path into segments $[z_i,z_{i+1}]$ and locating phase screens $e^{i\varphi (x,y)}$ at the center-of-mass of the refractive index structure constant $C_n^2(z)$ within each segment in order to accurately simulate both the phase and amplitude of the beam [31,32]. The phase screens are generated with an FFT-based method with spatial statistics determined by the modified von Kármán phase power spectral density
Table 1. To simulate a range of turbulence conditions, two sets of simulations were configured, with $r_0 = 20$ cm for $D/r_0 < 10$ and $r_0 = 10$ cm for $D/r_0 \geq 10$. Both scenarios model weak scintillation with Rytov variance $\sigma _\chi ^2=0.02$. The sampling was chosen to match the hologram sampling ($D/\Delta x = 840$).
For the purposes of this work we have restricted to higher elevation ($60^\circ$) propagation paths in order to avoid the formation of branch points in the simulated phase profile. Although the holographic technique is capable of producing branch points [7], they present a significant obstruction for the WFS phase reconstruction algorithm limiting our ability to verify phase profiles with branch point singularities.
3. Experimental setup
The experimental setup is shown in Fig. 4. The input light source is a linearly polarized laser coupled to a polarization-maintaining single mode fiber. To maximize the power of the light modulated by the SLM, the light is aligned to match its required input polarization using a half-wave plate (HWP). To decrease the intensity variation of the input light, an 80 mm focal length collimator is used to create a 14.5 mm diameter beam, overfilling the 6.72 mm hologram aperture. Remaining intensity variations of the input beam are carried over into the output beam in accord with Eq. (4).
Fig. 4. Optical layout of spatial mode generator. This diagram shows a ray trace of the system. The blue ray traces are the input light to the spatial light modulator. The red ray traces are the zero order reflection from the SLM, green/purple are the first order reflections, and yellow/cyan are the second order reflections
Next, the input light passes through a 50:50 beam splitter (BS) to the reflective SLM. The SLM supports a wavelength range of 1500-1600 nm and has a 15.36${\times }$9.6 mm surface area comprised of an array of 1920${\times }$1200 active liquid crystal elements with 8 $\mu$m pixel pitch. Each element has 10-bit depth (1024 phase levels) from 0 to 2$\pi$. The 250 ms response time of the SLM limited the refresh rate to about 2 Hz. This is well under the required refresh rate of around 100 Hz needed to recreate an evolving atmosphere in real-time. Therefore, the system is evaluated using 100 static samples, enough to capture statistics for each turbulence scenario $D/r_0$ given in Table 1.
Computer-generated complex amplitude phase holograms are applied to the SLM as discussed in Section 2. These holograms limit the phase range to $[-1.84,1.84]$ reducing the number of usable phase levels from 1024 to 600. A vertical hologram grating was used with parameters listed in Table 2. The holograms included a circular aperture by defining the hologram for a beam $u_1(x,y)$ truncated by a circular pupil with 6.72 mm diameter corresponding to 70% of the diameter of the central 1200${\times }$1200 pixel active area. This yields 840 addressable pixels across the diameter of the output beam. A hologram creating a flat-top $u_1$ with flat phase profile was used during system alignment.
After the SLM, a 4$f$ system is employed to relay the optical field at the SLM plane to an output plane that is accessible. As seen in Fig. 4, the 4$f$ system starts at the SLM. The modulated light is turned by the 50:50 BS, sending half of the remaining power into an $f = 79$ mm aspheric focusing lens (L1) placed one focal length from the SLM. The angle of the BS is rotated to direct the first order reflection through an adjustable-width slit placed at the focus of L1. The slit width is set to $w = f/\Lambda$ depending on the hologram grating period $\Lambda$. This places the blade edges half-way between the diffraction orders to allow transmission of as many spatial frequencies as possible around the first order while blocking all other SLM diffraction orders. The light is then collimated with a second 79 mm focal length aspheric lens (L2). As shown in Fig. 1, each diffraction order has two copies created on each side of the zero order which differ by a conjugation of the phase. To ensure that the unconjugated first order reflection is chosen by the slit a hologram generated to produce a focusing beam is applied to the SLM and the beam is verified to focus past the output plane.
There are several unwanted reflections depending on the alignment of the system that can overlap with the modulated first order output. These reflections were identified by placing a camera with a 3 $\mu$m pixel resolution at the focus of a 100 mm focal length lens. The lens is placed at the output plane. An image seen in Fig. 5 shows the modulated beam with 3 unwanted reflections measured at the focus. There are two reflections that originate from the internal surfaces of the BS. These reflections do not move when the BS or the SLM is rotated. A third reflection was found to originate at the surface of the SLM. This reflection is not modulated by the SLM. This reflection does not move with tip/tilt changes of the SLM surface but does move with the BS rotation. Using a combination of fine rotational control for the BS and tip/tilt for the SLM surface the modulated first order can be moved away from these reflections, and then the 4$f$ system with spatial filter is realigned.
Fig. 5. Image collected at the focus of a 100 mm focal length lens. Three unwanted reflections are seen near the first order reflection shown here modulated using a hologram with $D/r_0=10$. Two reflections originate from internal surfaces of the beam splitter. A third reflection originates from the SLM but is not modulated by the hologram.
Measurements of the mode fidelity and wave structure function were obtained using an InGaAs WFS based on quadriwave lateral shearing interferometry (QWLSI) [35] with a 9.6${\times }$7.68 mm sensor sensitive to 1550 nm light. The WFS reports phase and intensity profiles with 160${\times }$128 pixel sampling with 60 $\mathrm {\mu }$m pitch and phase resolution less than $\lambda /775$. The InGaAs sensor in the WFS was placed at the output plane in Fig. 4 (a propagation distance $f$ from L2). An analysis mask with 6.46 mm diameter centered on the intensity image of the flat-top beam is defined in the WFS software. To keep the local phase gradients within the range of the sensor the tip/tilt of the WFS was adjusted such that the analysis software reported a beam tip/tilt of less than 3 milliradians.
4. Data analysis
The wave structure function $\mathcal {D}(r)=\mathcal {D}_\phi (r) + \mathcal {D}_\chi (r)$ was calculated and compared to that of the Rytov theory for a plane wave in slant-path turbulence [36]
The intensity statistics were also verified by comparing the log-amplitude variance $\sigma _\chi ^2$ to the expression
expected from the Rytov theory for plane wave propagation in weak turbulence [33].Finally, we also use the phase and amplitude profiles measured by the wavefront sensor to calculate the fidelity
In order to analyze the fidelity at increasing turbulence levels, it is necessary quantify the impact of the fixed finite resolution of the WFS on the accuracy of the measurement. As an estimate of the fidelity measurable with our setup at a given coherence diameter $r_0$, we use as reference the squared modulus of the complex degree of coherence
Finally, we note that the numerical simulations were sampled at the spatial resolution of the SLM which is roughly 7 times that of the WFS. In order to identify potential artifacts of the WFS measurements, we numerically model the behavior of a QWLSI phase reconstruction [35,39] by calculating the least squares phase gradients of the simulated phase over a moving window with 120${\times }$120 $\mu$m$^2$ area—corresponding to the effective integration area for a single sample of the phase gradient—and reconstruct the phase using a simple FFT-based phase reconstruction algorithm. The simulated intensity is averaged over a 1 pixel (60${\times }$60 $\mu$m$^2$) area. The reconstructed simulations are used as a point of reference for comparison to the measured spatial statistics and fidelities reported in Section 5.
5. Results and discussion
This section presents results characterizing the performance of the spatial mode generator using a range of hologram grating periods $\Lambda$ listed in Table 2. First, baseline measurements of the peak-to-valley (PtV) and rms wavefront error (WFE) are given for each grating using a hologram generating a flat-top beam profile. Next, the spatial structure and coherence diameter $r_0$ of the optical turbulence produced by the SLM is verified via the wave structure function. Finally, the fidelity of the turbulent spatial modes is presented and compared for each grating period.
Table 2. Listing of the grating periods $\Lambda$ tested and corresponding measurements of a flat-top modulated beam.
Measurements of the flat-top beam are given in Table 2. The diameter of the flat-top modulated beam was measured at 6.57$\pm$0.1 mm FWHM and 6.86$\pm$0.1 mm at $1/e^2$ of peak intensity. Thus, although the SLM hologram does not produce a hard aperture at the pupil diameter $D=6.72$ mm in the output plane, an intensity roll-off of 300 $\mu$m from FWHM to $1/e^2$ was produced centered on the intended aperture diameter. The rms WFE was close to $\lambda /40$ for all grating periods, though a slightly higher WFE was observed using a hologram grating with only $N_x=3$ pixels per period. The PtV WFE was within $\lambda /2$ with the largest departures near the edge of the WFS analysis pupil.
Phase and intensity profiles were collected and analyzed for a range of simulated turbulence conditions described in Table 1 with $2\leq D/r_0 \leq 50$ using 100 holograms for each propagation scenario. Above $D/r_0=50$, the WFS phase reconstruction became unstable and we no longer obtain reliable measurements. This instability corresponds to $r_0 = 130$ $\mu$m, or roughly 2 WFS pixels per coherence diameter. Examples of the phase and intensity measurements are shown in Fig. 6 against the simulated profile used to generate the corresponding holograms.
Fig. 6. Example phase and intensity profile with comparison to the simulated profile used to generate phase holograms with $P_g=93$ and $D/r_0 =8$.
In Fig. 7, the measured wave structure function is plotted from the WFS phase and amplitude profiles obtained from a beam generated using a hologram grating with $N_x=5$ pixels per period and $P_g=168$ periods across the hologram pupil. The wave structure function of the emulated beam matches theory and simulation well up to $D/r_0=50$. In particular, the emulation clearly exhibits the 5/3 power law behavior characteristic of propagation through atmospheric turbulence. The large relative deviation seen at $D/r_0 = 2$ occurs as the rms phase difference $[\mathcal {D}_\phi (r)]^{1/2}$ drops below $\sim 0.16$ radians, corresponding to the residual rms wavefront error $\lambda /40$ of our reference beam (cf. Table 2).
Fig. 7. The left panel shows the wave structure function calculated from WFS measurements of a beam generated using holograms modulating a phase grating with $P_g=168$ and $N_g=5$ pixels per grating period. The separation distance is normalized by the diameter of the aperture. The right panel shows a comparison of the simulated $D/r_0$ and the measured $D/r_0$ obtained from the structure function.
Based on the strong 5/3-law behavior, the coherence diameter $r_0$ is calculated for a range of grating periods and the ratio $D/r_0$ is compared to the numerical simulation used to generate the holograms in Fig. 7. Restricting to $P_g>3\cdot D/r_0$ and $N_g\geq 5$ the measured $D/r_0$ matches the raw simulated value within $3\%$ for $D/r_0\leq 10$ and within $9\%$ for $D/r_0\leq 50$.
The measured $D/r_0$ begins to lag the simulated values at higher $D/r_0$. However, this is apparently an artifact of the phase reconstruction affecting the spatial statistics for separation distances near the resolution of the WFS. Specifically, the QWLSI-reconstructed simulation exhibits the same subtle lag in the structure function at small separations as the WFS measurements—visible in the left panel of Fig. 7—and the apparent lag in the measured $D/r_0$ is also reproduced in the QWLSI-reconstructed simulation (dashed curve in right panel of Fig. 7).
As the number of grating periods $P_g$ across the diameter of the SLM aperture decreases, the measured $D/r_0$ begins to drop off. In particular, with $P_g=31$ this drop-off begins with $D/r_0$ above 10 to 15, and with $P_g = 80$ the drop-off can be seen for $D/r_0$ above 25 to 30, in general agreement with Eq. (3). Fig. 8 shows an example of the effects of decreasing the number of grating periods for a hologram with $D/r_0 = 50$. As predicted by Eq. (3), with $P_g=31$ filtering of the high spatial frequencies from the vertical slit is clearly visible in the phase profile, leading to a “stretched" appearance in the phase along the horizontal axis. This degradation of the phase profile is consistently accompanied by intensity voids which can be used to partially verify the optical turbulence range when a WFS is unavailable. However, note that spatial filtering is not visually apparent at $P_g=93$ despite violation of the bound given in Eq. (3).
Fig. 8. The simulated phase/intensity (left) used to generate a hologram with $D/r_0=50$ compared to the measured profile as the number of grating periods $P_g$ is decreased. The intensity images are individually normalized to the peak intensity.
The intensity statistics are also verified via the log-amplitude variance $\sigma _\chi ^2$ in Fig. 9. With at least 5 pixels per grating period $N_g\geq 5$ the measured log-amplitude variance is close to the simulated value for smaller $D/r_0$, but the variance is seen to increase with $D/r_0$ away from the simulated value $\sigma _\chi ^2 = 0.02$. This is likely due to a small deviation $\Delta z$ of the WFS from the 4$f$-plane along the optical axis. A small propagation distance $\Delta z$ past the 4$f$-plane becomes effectively equivalent to an additional propagation distance $M^2\Delta z$ in the full scale simulation, where $M=D_{sim}/D_{lab}$ represents the magnification of the simulated beam relative to the laboratory beam. The expected increase in scintillation can be estimated using (14) to calculate the adjusted log-amplitude variance assuming an additional vacuum propagation distance $M^2\Delta z$ at the end of the simulated propagation path. The adjusted Rytov variance is shown in Fig. 9 with displacements $\Delta z$ of 1 mm and 3 mm along the optical axis, with magnification determined by $D_{lab} = 6.72$ mm and $D_{sim}$ given in Table 1. Note that the log-amplitude variance is also seen to increase as the bound $D/r_0 < P_g/3$ is violated, as may be expected due to the appearance of intensity voids associated to spatial filtering seen in Fig. 8.
Fig. 9. Log-amplitude variance calculated from wavefront sensor measurements. The increase with $D/r_0$ is compared to the increase expected from the Rytov theory (14) adjusted for a 1 mm and 3 mm displacement of the WFS from the 4$f$-plane.
Finally, in Fig. 10 we plot the fidelity of the generated spatial modes referenced to the simulated profile used to create the holograms. The average and standard deviation for each set of 100 holograms at each turbulence level is shown. Restricting to $P_g>3\cdot D/r_0$ and $N_x\geq 5$ the fidelity exceeds 90% for $D/r_0 \leq 10$ and exceeds 80% for $D/r_0\leq 20$. It should be noted that as $D/r_0$ increases, the calculated fidelity becomes increasingly sensitive to small translational and rotational perturbations of the overlap against the reference. The fidelity of the QWLSI-reconstructed simulations agrees with the estimate provided by the complex degree of coherence $|\gamma (r_1)|^2$ in (16) and thus yields an estimate of the limitation of the fidelity measurement imposed by the spatial resolution of the WFS at a given turbulence level. The decrease in the measured fidelity tracks that of the QWLSI-reconstructed simulations as the turbulence level $D/r_0$ increases.
Fig. 10. Fidelity of the emulation referenced to the simulated spatial mode used to generate holograms. Calculated from WFS phase/intensity profiles.
Note that the fidelity does not improve beyond $N_x=9$ pixels per grating period and already approaches the maximum with only $N_x=5$ pixels per period. Furthermore, the mode fidelity is not severely degraded at the minimum of $N_x = 3$ pixels per grating period, though there is a visible improvement when at least two diffraction orders are kept within the spatial bandwidth of the SLM (i.e. $N_x\geq 5$).
6. Conclusion
In this paper we characterized the ability of a spatial light modulator to reproduce a laser beam which has propagated through simulated multi-layer turbulence using complex amplitude phase holograms. The measured value of $D/r_0$ matched the simulated value within 3% up to $D/r_0 = 10$ and within $9\%$ up to $D/r_0 = 50$, although the latter estimate appears to be limited by the WFS phase reconstruction (Fig. 7). Optimizing the grating period according to the constraints $P_g>3\cdot D/r_0$ and $N_x\geq 5$ derived in Section 2, the fidelity of the generated spatial modes was shown to exceed 90% for $D/r_0\leq 10$ and $80\%$ for $D/r_0\leq 20$. Although the fidelity was maximized with $N_x = 9$ pixels per period—in agreement with the analysis of Section 2.—a grating with $N_x=5$ pixels per period was sufficient to produce high mode fidelity, with reasonable mode fidelity still obtained using only $N_x=3$ pixels per period.
The precise reproduction of the desired wave structure function in Fig. 7 is the clearest demonstration of a 5/3 power law produced by a liquid crystal SLM of which the authors are aware. In contrast to non-holographic systems which employ the SLM directly as a phase screen, the spatial filtering and single SLM design of the holographic system minimizes diffractive phase errors from the SLM aperture [17]. The hologram grating further allows one to separate the modulated reflection off of the SLM from the unmodulated reflection and other stray beam cube reflections corrupting the PSF (cf. Fig. 5), which can lead to biasing of the Strehl ratio similar to that observed in [19]. Furthermore, the assessment of the measured log-amplitude variance shows that—in contrast to typical beam-shaping SLM applications—significant intensity variations are associated to very small displacements from the 4$f$-plane. Precisely locating this plane is thus important to prevent unintentional formation of intensity nulls and branch points when producing severe turbulence, and is required when characterizing the system with a WFS.
As mentioned in the introduction, the results presented in this paper are based on measurements obtained in a static context. Evaluation of the performance of a holographic spatial mode generator for real-time simulation of an evolving atmospheric channel is left for future consideration.
Funding
National Aeronautics and Space Administration.
Acknowledgments
The authors would like to thank Evan Katz and Benjamin Croop for their contributions to this project. This work was supported by the NASA Space Communications and Navigation (SCaN) Program and the Glenn Research Center Communications & Intelligent Systems Division.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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