Before using adaptive optics (AO) systems, performance estimation and verification tests are essential [1-6]. In addition, there are temporal and spatial limitations to performing performance verification on-sky [7-9]. At the laboratory level, atmospheric turbulence simulations avoid these limitations. The phase plate is one of the laboratory level methods used for atmospheric turbulence simulation [10–15]. Other methods include atmospheric turbulence simulations using air heating [16], a deformable mirror (DM) [15, 17, 18], and a liquid crystal (LC) [19, 20].

Among these approaches, the air heating method is the most cost-effective and easy to implement. However, it is difficult to control the atmospheric turbulence parameters, and it can undesirably heat nearby optical elements [16]. LC and DM are popular for simulating various atmospheric conditions since they have high spatial and temporal capability, to modulate the optical path lengths [15–18]. But they are still expensive, and it remains difficult to simulate atmospheric turbulence with a small fried parameter without the use of beam reducing optics.

The phase plate does not have these disadvantages and can produce very high resolution and low-cost atmospheric turbulence simulations [10–15]. In addition, it is possible to change the effective turbulence characteristics in a limited way by overlapping multiple phase plates along the beam path [11, 15].

We recently developed an optical test bench with an atmospheric simulator to evaluate a recently developed AO system for 1.6 m telescopes with a reduction ratio of 71.4, as shown in Fig. 1. The AO system first uses a newly developed DM [21], so the optical test bench implements an AO system almost identical to an optical bench to confirm the performance predictions estimated with statistical analysis and computer simulation [6]. Specifically, the atmospheric simulator successfully mimicked the atmospheric turbulence at Bohyunsan Observatory, one of the candidate sites for the AO system, by rotating two phase plates in opposite directions. In this paper, we report on the fabrication and evaluation of the atmospheric simulator.

Figure 1. Schematic diagram of the adaptive optics (AO) under development [6].

We have organized this paper as follows: in the following section, we report the fabrication of the phase plates used in the atmospheric simulator, which was done using etching and spray techniques on glass substrates to generate realistic atmospheric-like atmospheric turbulence, i.e. Kolmogorov turbulences. Then we report the emulation of the atmospheric disturbance at Bohyunsan Observatory in terms of spatial and temporal behaviors. In Section 3, we analyze how closely the atmospheric disturbance simulation emulated reality with an investigation of the measured wavefronts on the optical table.

As far as we are aware [22, 23], this is the first report on an analysis of how the rotating phase plates dynamically emulate atmospheric disturbance in terms of wind speed and directions. The analysis is important because it prevents the beam from rotating as the plate rotates, which is unnatural, but could occur due to the residual wedge angles of the phase plates.

2.1. Overview

We are currently developing an AO system for a 1.6-m ground telescope using a DM that was recently developed in South Korea [21]. The telescope focuses the incoming beam to its Nasmyth focus, which is then collimated to the AO system (called an AO module) with a beam reduction ratio (BR) of 1/71.4 regarding the telescope pupil diameter, per Eq. (1). Table 1 summarizes its system specifications.

TABLE 1. System specifications of the adaptive optics (AO) system.

Parameters		Value
Beam Diameter	Telescope Pupil	1,600 mm
	AO Module	22.4 mm
Shack-Hartmann Wavefront Sensor	Diameter	22.4 mm
	Sub-aperture	23 × 23

(1)$$BR=\frac{Reducedbeamdiameter}{Telescopediameter}$$

where the telescope diameter is the entrance pupil diameter of the telescope, i.e. 1.6 m, and the reduced beam diameter is the collimated beam diameter entering the AO module.

The optical bench test implements an almost identical AO system (or module) on an optical table, to confirm the predicted system performance [6]. Figure 2 shows a schematic layout of the testbed composed of an atmospheric simulator and an AO simulator. Between the collimating optics and the tip/tilt mirror is the atmospheric simulator. The atmospheric simulator produces the atmospheric disturbance, by rotating two phase plates in the same direction or in different directions.

Figure 2. Schematic layout of the adaptive optics (AO) testbed.

The diameter of the entire phase-plate was 200 mm, and the actual beam diameter was 22.4 mm, located 70 mm away from the center. The two rotating phase plates can rotate in the same direction or in different directions, and the rotation speed is also adjustable. Figure 3 shows a schematic diagram of the phase plates and Fig. 4 shows a picture of two rotating plates.

Figure 3. Schematic diagram of the phase plate: (a) major dimensions, (b) effective sub-aperture of the phase plate as it rotates.

Figure 4. A picture of two rotating phase plates.

2.2. Atmospheric Disturbance at the Telescope Site

We mostly describe the optical effects of atmospheric disturbance using the Fried parameter (r_o), and the coherence time or critical time constant (τ_o), as given below [1-3].

(2)$$r_0(\lambda )=0.185{\lambda }^{6/5}{\cos }^{3/5}\varsigma {\left({\displaystyle \int C_n^2(h)dh}\right)}^{-3/5},$$

(3)$${\tau }_0=0.314\frac{r_0}{V_0}$$

where $C_n^2\left(h\right)$ is the refractive index structure function at altitude h. The observed wavelength is λ in microns, and the observed angle is ξ, and the average velocity of the turbulence is V_o. We can convert the Fried parameter at one wavelength to another wavelength using the equation below:

(4)$$r_0\left(\lambda \right)=r_0\left(0.5\right){(\lambda /0.5)}^{6/5}.$$

From previous studies [5, 24, 25], Bohyunsan Observatory was measured to have a Fried parameter (r_o) with a mean value of 85 mm and standard deviation of 13 mm at a wavelength of 0.5 μm, often simply expressed as 85 ± 13 mm. Table 2 lists the principal parameters of the atmospheric disturbance used for the design of the AO system. Table 3 lists the Fried parameters at four principal wavelengths.

TABLE 2. Atmospheric disturbance conditions at 0.5 μm.

Parameters	Symbol	Values
Fried Parameter	r0	85 ± 13 mm
Wind Speed	V0	8.3 ± 1.4 m/s

TABLE 3. Fried parameters at four principal wavelengths.

Condition	Wavelength (μm)	Values (mm)
Reference (SLODARa)	0.50	85 ± 13
Wavefront Sensing (Rayleigh LGSb)	0.53	92 ± 13
Wavefront Sensing (Sodium LGS)	0.59	103 ± 14
Science Imaging	0.63	114 ± 16

2.3. Fabrication of the Phase Plates

We fabricated the phase plates on glass (BK7) substrates using etching and spray techniques to generate realistic atmospheric-like atmospheric turbulence, i.e. Kolmogorov turbulences [13, 14]. We etched the glass substrate with hydrofluoric acid to produce an overall turbulence profile shape, then applied a Krylon transparent acrylic paint to fabricate high-frequency patterns. The detailed process parameters, which will be reported in another publication, including etching time, spraying position/distance, and number of spraying injections, were investigated prior to fabrication.

Figure 5 shows interferograms of the four phase plates. In operation, a beam passes through a small area, a 22.4 mm diameter aperture, as drawn in Fig. 3. Figure 6 shows phase maps of a phase plate as it rotates with 30° increments.

Figure 5. Interferograms of the fabricated phase plates (unit: wave).

Figure 6. Sub-aperture phase maps of a phase plate as it rotates.

2.4. Spatial Behavior: Fried Parameters of the Fabricated Phase Plates

In Kolmogorov’s theory, we express the phase structure function of the phase ϕ at the entrance pupil of the telescope, with the piston and tip/tilt terms removed, as expressed below [1-3].

(5)$$D_{\varphi }\left(r\right)=〈|\varphi \left(x\right)-\varphi \left(x\text{'}\right){|}^2〉=1.03{\left(\frac{r}{r_0}\right)}^{5/3}$$

(6)$$r=\left|x-x\prime \right|,$$

where x, x’ are position vector coordinates, and r is the distance between them. <> refers to the ensemble average.

We can rewrite Eq. (5) regarding the Fried parameter at the telescope pupil as below:

(7)$$r_0=\frac{1}{BR}\frac{0.95}{{〈|\varphi \left(x\right)-\varphi \left(x\text{'}\right){|}^2〉}^{5/3}}r.$$

Using Eq. (7) in a least square regression manner, we can estimate the Fried parameter of a phase plate from its measured phase ϕ_A over a small aperture of 22.4 mm diameter as it rotates, as shown in Fig. 4. Table 4 summarizes the Fried parameters of the four phase plates at the reference wavelength, i.e. 0.5 μm, named plates 1, 2, 3, and 4.

TABLE 4. Fried parameters of the four fabricated phase plates.

Plate No.	Mean (mm)	Standard Deviation (mm)
Plate 1	109	27
Plate 2	116	27
Plate 3	138	13
Plate 4	156	45

With two phase plates in use, the total wavefront deformation ϕ_t is a sum of the wavefront deformation due to each plate, designated ϕ₁ and ϕ₂.

(8)$${\varphi }_t\left(x\right)={\varphi }_1\left(x\right)+{\varphi }_2\left(x\right).$$

Similarly, we can equate the structure function of the total wavefront deformation as below.

(9)$$D_{{\varphi }_t}\left(r\right)=〈|\varphi \left(x\right)-\varphi \left(x\text{'}\right){|}^2〉=1.03{\left(\frac{r}{r_{0,t}}\right)}^{5/3},$$

where the total or effective Fried parameter is r_0,t . Combining Eq. (5)–(9), we can estimate the total Fried parameter r_0,t from the two Fried parameters of the phase plates in use, namely r_0,1 and r_0,2 equated as below:

(10)$$r_{0,t}=r_{0,1}{r}_{0,2}{\left(r_{0,1}{}^{5/3}+r_{0,2}{}^{5/3}\right)}^{-3/5}.$$

Table 5 summarizes the estimated Fried parameters of all pairs. The pair of plate 2 and plate 3, expressed as Plate 2 + 3, provided values most similar to the Fried parameters at the site.

TABLE 5. Fried parameters of the overlapped phase plates.

Pair	Mean (mm)	Standard Deviation (mm)
Plate 1 + 2	74	18
Plate 1 + 3	80	11
Plate 1 + 4	84	22
Plate 2 + 3	83	11
Plate 2 + 4	87	22
Plate 3 + 4	96	12

2.5. Dynamic Behavior: Velocity of the Turbulence

The average velocity of the turbulence (V_o) corresponds to the duration over which the standard deviation of the phase fluctuations at a given point is of the order of 1 rad [1–3], equated as below:

(11)$$V_o={\left(\frac{{\displaystyle \int V{\left(h\right)}^{\frac{5}{3}}{C}_n^2\left(h\right)dh}}{{\displaystyle \int C_n^2\left(h\right)dh}}\right)}^{3/5}.$$

With two phase plates in use, the total average velocity of the turbulence (V_o,t) can be approximated as the weighted sum of the velocities of each plate, named V_o,1 and V_o,2, respectively, as below:

(12)$$V_{o,t}={\left(\frac{V_{o,1}{}^{5/3}{r}_{0,1}{}^{-5/3}+V_{o,2}{}^{5/3}{r}_{0,2}{}^{-5/3}}{r_{0,1}{}^{-5/3}+r_{0,2}{}^{-5/3}}\right)}^{3/5}.$$

To emulate the effect of wind at Bohyunsan Observatory, we rotated the phase plates at an angular speed to match the wind speed (V_o) of the site. Considering that the beam is reduced by the factor of the beam reduction ratio (BR) and the reduced beam is located 70 mm away from the center, as in Fig. 3, the simulated wind speed V_S in m/s is formulated as below:

(13)$$V_S=\frac{1}{BR}•\left(2\pi \frac{w}{60}\right)•r,$$

where r is the offset distance in meters from the center, i.e. 70 mm, and w is the rotational speed of the phase plate in revolutions per minute (rpm).

The phase plates in use have strengths similar to the atmospheric disturbance as shown in Table 4. In addition to that, since the phase plates are statistically uncorrelated, we can determine the rotational speed of the phase plates, given below, which is approximately 15.9 rpm for the simulation at an average wind velocity (V_o) of 8.3 m/s.

(14)$$w=BR•\left(\frac{60}{2\pi }\right)\left(\frac{V_o}{r}\right),$$

3.1. Set-up

Figure 7 shows a picture of the optical test bench. It consists of an atmospheric simulator and an AO simulator. A wavefront sensor (Shack-Hartmann sensor) within the AO simulator records the deformed wavefront of the collimated beams that have passed through the atmospheric simulator. Table 6 summarizes the specifications of wavefront sensing.

TABLE 6. Specifications of wavefront sensing.

Parameters	Value	Symbols
Sensor Type	Shack-Hartmann Sensor [26]	-
Wavelength	532 nm	-
Format (Lens Array)	23 × 23	N × N
Pitch	112 μm	P
Frame Rates	1 KHz	F

Figure 7. The optical test bench with the adaptive optics (AO) simulator and atmospheric simulator.

From a prior study using computer simulation [6], we found that the wavefront disturbance exhibits a run-to-run variation due to the random nature of atmospheric disturbance, which converges to values when the simulation time length is longer than 0.9 seconds, i.e. approximately 240 times the critical time constant of the applied atmospheric disturbance. Based on this finding, we recorded 1,000 wavefront data sampled for 1 second at 1 KHz. We present the n^th wavefront as ϕ_n where ϕ_n is a phase matrix of 23 × 23.

In addition to that, the wavefront was converted to the reference wavelength of 0.5 μm from the wavefront sensing wavelength, i.e. 0.532 μm, using Eq. (4) prior to application in the following analysis.

3.2. Statistical Behavior: Fried Parameter

We calculated the Fried parameters of the recorded 1,000 wavefront samples using Eq. (7). Figure 8 plots a histogram of the calculated Fried parameters with a bin size of 10 mm. Table 7 summarizes the Fried parameters of the site and the atmospheric simulator. The simulated mean value of the Fried parameter has a deviation of 2.4% from the design value, i.e. 85 mm.

TABLE 7. Fried parameters of the site and the simulator.

Condition	Minimum (1 sigma) (mm)	Mean (mm)	Maximum (1 sigma) (mm)
At the Site	72	85	98
Simulator(Design)	72	83	94
Simulator(Experiment)	72	87	102

Figure 8. Histogram of the measured Fried parameters of the atmospheric simulator.

3.3. Dynamic Behavior: Wind Velocity

We estimated the wind velocity and direction with an analysis of the temporal correlation between two temporally adjacent wavefront samples, namely: ϕ_n and ϕ_n+1. We first calculated the correlation matrix C between the two adjacent waterfronts as we shifted the two wavefronts with respect to each other until the two phases overlapped one pixel [27]. In this study, C was a matrix of 45 × 45. For a shift of (δi, δj), the value of C(δi, δj) is given as below:

(15)$$C\left(\delta i,\delta j\right)=\frac{{\displaystyle \sum _{ij}{\varphi }_n\left(i,j\right){\varphi }_{n+1}\left(i+\delta i,j+\delta j\right)}}{O\left(\delta i,\delta j\right)},$$

where (i, j) are the coordinates of a phase map. O(δi, δj) is the number of overlapped pixels with a shift of (δi, δj). Figures 9 and 10 show the overlapping matrix O and the correlation matrix C.

Figure 9. Overlapping matrix O.

Figure 10. Correlation matrix C.

Then we found the wind velocity (V) and direction (θ) by calculating the vector between the center point (i₀, j₀) and the peak point (i_p, j_p) in Fig. 10, as formulated below:

(16)$$V=\frac{P•F}{BR}•{\left({\left(i_p-i_0\right)}^2+{\left(j_p-j_0\right)}^2\right)}^{1/2},$$

(17)$$\theta ={\tan }^{-1}\left(\frac{j_p-j_0}{i_p-i_0}\right).$$

Figure 11 plots the calculated direction and intensity of the wind in polar coordinates. Table 8 summarizes the calculated wind velocities. The mean velocity of the wind was 7.5 m/sec, which was 90.4% of the mean wind velocity of the site, i.e. 8.3 m/sec. We estimated that the difference would come from the nonlinear angular rotation of the phase plates. In order to compensate for the effect, we needed to rotate the phase plate with an angular speed of 10% more than the expected value, based on a linear motion assumption, i.e. 17.5 rpm.

TABLE 8. Wind velocity at the site and from the atmospheric simulator.

Condition	Minimum (m/s)	Mean (m/s)	Maximum (m/s)
At the site	6.9	8.3	9.7
Simulator (Design)	8.3
Simulator (Experiment)	5.0	7.5	10.0

Figure 11. Wind velocity and direction of the atmospheric simulator.

We developed an atmospheric simulator that emulated the atmospheric turbulence at Bohyunsan Observatory by rotating two phase plates in opposite directions. The emulated atmospheric disturbance was tested to have a Fried parameter of 87 ± 15 mm and wind velocity of 7.5 ± 2.5 m/s. The Fried parameters were within 2.4% of the estimation, but the wind velocities were about 9.6% slower than expected, which was due to the nonlinear motion, i.e., the angular rotation of the phase plates. We found that we needed about 10% faster speed than the value expected from the linear wind motion.

In conclusion, we successfully mimicked the astronomical optical disturbance of Bohyunsan Observatory with an error of less than 10% in terms of Fried parameter and wind velocity.

The authors declare no conflict of interest.

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.

Supported by the Defense Industry Technology Center through a research program entitled “high-speed optical wavefront deformable mirror system performance analysis technology development research (UC180003D).”