Image Quality Specification for Solar Telescopes

We used an un-obscured 2 dimensional pupil function $W({\bf x})$ described by

to model the telescope, where ${\bf x}$ represents the spatial coordinates at the pupil and $D$ is the aperture size. Several distinct values of pupil diameters were used starting from 30 cm to 200 cm, the values were chosen to represent the aperture size of the solar telescopes available in India and elsewhere in the world.

We implemented the AO correction in an idealistic way. We decomposed the instantaneous phase-perturbations over the pupil into a given number, $N_{Z}$, of Zernike Polynomials (Noll, 1976) using a least-square solution method (Stewart, 1993). A model phase front (phase perturbations over the pupil) was then synthesized from the Zernike coefficients obtained from the least square method and subtracted from the initial perturbed phase-front to obtain the residual phase perturbations after “AO correction”. As we sampled the phase screen with a sampling of 2 cm, and made a pixel-wise correction, it is an ideal correction up to 2 cm. Perturbations on spatial scales less than 2 cm are neither generated nor compensated in our approach. The number of Zernike polynomials used to model a given phase perturbation increased in steps of two radial orders at a time (2, 9, 20, 35, and so on) until the variance of the residual phase perturbations over the pupil becomes less than 1 radian². This criterion essentially enables us to terminate the simulation at a particular $r_{0}^{\prime}$ (that could be any value between 6 and 21) for a given $D$.

As we perform the Fourier transform using Fast Fourier Transform routines that keep the number of pixels same in either domain, we constrain the simulation window size to be at least twice that of the aperture size, with the aperture centered on the window, so that the transfer function is not truncated. As our simulations involved different aperture sizes, we choose a window size of 256 $\times$ 256 pixels so that apertures of up to 128 pixels could be simulated. This leaves us with the feasibility of performing simulations up to 2.56 m diameter aperture, for a 2 cm pixel sampling. However, we have performed simulations only up to $D=$ 2 m. As a result of the Fourier transformation relation between the pupil plane and the image plane, the field-of-view (in the image plane), determined by the pupil-plane pixel sampling, is $[\lambda/0.02]^{2}\approx 4.4^{\prime\prime}\times 4.4^{\prime\prime}$ arcsec² at 430.5 nm. As the atmospherically induced pupil-plane phase perturbations are independent of the wavelength, $\lambda$ becomes an independent parameter in our simulations. However, for the purpose of choosing the field-of-view, we have chosen the wavelength to be 430.5 nm. This also means that the $r_{0}$ values used in our simulations correspond to 430.5 nm.

Figure 3 shows the rms contrast as a function of $r_{0}$ for different telescope sizes under seeing-limited imaging. We find that it varies between 1.8 and 7.6%. It increases with $r_{0}$ as expected. It has a slight dependence on the telescope diameter as well; larger the diameter, higher the rms contrast, for a given $r_{0}$.

Figure 4 shows a semi-logarithmic plot of rms contrast as a function of Strehl ratio. The Strehl ratio spans over three orders of magnitude as $D/r_{0}$ changes from 1 to 33. However, the rms contrast changes by less than an order of magnitude for the same range of $D/r_{0}$. This is perhaps due to the intrinsically low contrast nature of the solar granulation. This plot helps us to specify the contrast of the solar granulation as metric for solar telescopes against the traditional Strehl ratio (which cannot be measured) for seeing-limited imaging. Conversely, it could also be used to estimate the efficiency of the telescope under seeing-limited imaging conditions (by comparing the observed contrast with the theoretical upper limit presented here).

The first order adaptive optics compensation is to stabilize the image by arresting or mitigating its random motion at kHz rate. It is equivalent to removing the 2D tilt in the atmospherically induced phase perturbations. The top and bottom panels of Figure 5 indicate the rms contrast and Strehl ratio after compensating for the fast varying wavefront tilt as a function of $D/r_{0}$. We observe that:

1. a)

   The Strehl ratio is a monotonically decreasing function of $D/r_{0}$. It rapidly decreases when $D/r_{0}$$\leq$10 but decreases relatively slowly when $D/r_{0}$$\ >$10.

2. b)

   The rms contrast is a non-monotonic function. It is highly dependent on the actual telescope diameter. The rate of enhancement of rms contrast with $r_{0}$ (seeing) is more rapid for small and intermediate size telescopes than that for large telescopes.

The four panels of Figure 6 show semi-logarithmic plots of rms contrast versus the Strehl ratio after AO correction. We note that the linear-log relation that existed before AO correction no longer exists. From the figure legends, we can identify N_z and D as the marker shape and color respectively. For example, cyan corresponds to a 200 cm telescope and triangles and squares correspond to 2 and 35 terms corrected, respectively. $r_{0}$ can be found by tracing the plots along a given D and N_Z. For example, the cyan triangle curve in the top left panel is made up of 16 points with the lowest data point corresponding to $r_{0}$ of 6 cm and the highest data point corresponding to $r_{0}$ of 21 cm. So, for each successive point, the value of Fried’s parameter increases in steps of unity. However, when we trace the cyan squares curve on the bottom right panel, we see that the maximum value of $r_{0}$ is only 19 cm. This is because we terminated the simulations for the cases where the phase variance was lesser than 1 radian² (Section 2.4.2). The “missing” $r_{0}$ = 20 and 21 cm points imply that the phase variance for that $D$ and $r_{0}$ in the previous value of N_Z (= 19) was less than 1 radian². The two red circles on the bottom left panel implies that for a 50 cm telescope, the mean square phase variance reduces to less than 1 radian² after correcting just 20 terms at $r_{0}$ = 7 cm; this yields corresponding Strehl and rms contrast as 0.6 and 6.5% respectively at $r_{0}$ = 7 cm. A 2 m class telescope will require compensation of up to 135 terms to bring the phase variance below 1 radian² at $r_{0}$ = 8 cm (blue stars on the bottom right panel). Other data points can be interpreted in a similar way.

In general, for small telescopes, both the Strehl ratio and the rms contrast increase (clustering near top right corner of the plots) with AO correction. However, for large telescopes, the increase is rather slow. Here again, this plot is quite useful to specify the granulation contrast as a metric after AO correction as against the traditional Strehl ratio.

The rms granulation contrast is a function of the solar granulation scene that is being observed. At large enough fields-of-view, this variation will not be high. We find a change in the intrinsic contrast with a change in region since we are using high-resolution images covering a very small field-of-view. So, we have repeated the entire simulation for 10 different solar regions following the method described in Section 2.5 and obtained the mean and standard deviation of rms granulation contrast.

Figure 7 is similar to Figure 5 showing the variation of rms contrast and Strehl ratio with D/$r_{0}$ for various telescope diameters. It can be seen from the top panel that the rms contrast can vary by up to 1 % above and below the mean value. However, the Strehl is independent of scene (it depends only the transfer function of the telescope). Similarly, Figures 6 and 8 are comparable but for the difference due to error bars arising from scene dependency.

As we already know the atmospheric path-length perturbations are achromatic. However, the wavelength dependency enters our simulations through the specifications of the Fried’s parameter $r_{0}$. Although we have used the shortest wavelength $\lambda$ = 430.5 nm, our results for a longer wavelength can be easily obtained by changing the input $r_{0}$ according to its wavelength dependency of $\lambda^{1.2}$. We have verified this through simulations as shown in Table 1. It can be seen that at long wavelengths, the Fried’s parameter is high and therefore the contrast is high (even though the intrinsic contrast of the granulation is lower at longer wavelength).

The results of our simulations, particularly those with AO corrections, correspond to ideal conditions. We have ignored the finite size of the wavefront sensor and corrector elements. We have also ignored the finite temporal delay that occurs in real systems. Thus, our results are only indicative of an upper limit on the contrast and the Strehl ratios. In what follows (Section 3.2.4), we compare the Strehl ratio and rms contrast obtained through our simulations with that obtained with real solar adaptive optics systems and thus derive an efficiency parameter. We then propose to use this efficiency parameter along with the upper limits obtained through our idealistic simulations, to specify the expected Strehl ratio and hence the rms granulation contrast measured by future solar telescopes. A caveat in this argument is that a certain degree of efficiency in the domain of Strehl ratio need not translate to the same degree of efficiency in the rms contrast, owing to the non-linear relationship between the Strehl ratio and rms contrast after AO correction. Scharmer et al. (2010) have reported an apparent efficiency factor of 54% in the rms solar granulation contrast after a low order ($\approx$ 30 modes) AO correction for a 1 m telescope (factor 1.85 mentioned in Figure 5). It implies that for a larger telescope, a similar efficiency might be achieved with a high order AO correction. It is clear that more data is required to get a better idea of if and how these parameters will affect the efficiency. Nevertheless, we can assume at least 50% efficiency in the rms contrast and use our results as a lower bound on the contrast to be expected.

Another limitation of our simulations is that we have assumed Kolmogorov type turbulence. In reality, the outer scale length could be finite and this would lead to a slightly better resolution (Tokovinin, 2002; Martinez et al., 2010). It is also known that the residual variances after compensation of a few low order Zernike terms is lower than that predicted by the Kolmogorov turbulence even when the outer scale 10 times larger than the aperture diameter (Winker, 1991). Thus, real systems could be better than what is predicted based on Kolmogorov turbulence. In the same vein, metrics like the Strehl ratio and rms contrast could also be higher and better respectively.

We could glean the Strehl ratio obtained with three practical solar AO systems. The first system was that of the 70 cm Vaccum Tower Telescope (VTT) (Berkefeld et al. (2012)). Here the residual variance of the corrected modes, uncorrected modes and wavefront sensor (WFS) errors are added to determine the Strehl (Table 2 - rows 1 to 6).

The second system was that the 76 cm Dunn Solar Telescope (DST). The Strehl values were estimated using three different methods. In the first method, a quasi long-exposure point spread function was estimated using the wavefront error (Marino,Rimmele, 2010). The corresponding optical transfer function was expressed as the product of three transfer functions and estimated appropriately. Finally, the Strehl ratio was estimated from the optical transfer function.

These values are listed in Table 2 rows 7 and 8. The efficiency here seems to be high (compared to the German VTT case) especially for the larger $r_{0}$ case. One possible explanation for this is the saturation of Strehl values with an increase in $N_{z}$. For $r_{0}$ = 17 cm, it was found through our simulations that correcting for 35 terms itself will result in a Strehl of 0.86. So we opine that such extreme cases should not be considered for calculating the efficiency. In the second method, the Strehl (Table 2 - row 10) was determined by extracting the wavefront error information after processing the AO corrected image using phase diversity method (Rimmele, 2000). Here again, the apparent high efficiency could be attributed to the combination of AO correction and image post processing and thus we exclude this case as well. Also, the values are taken from a figure in the paper. The figure only displays the best 20 out of every 100 frames. This could be another reason for high Strehl. In the third method, the residual errors from SHWFS were used to determine the Strehl (Table 2 - row 9). Here again, as stated in Rimmele (2000), the Strehl ratios are overestimated as the contribution of higher order modes (not detected by Shack-Hartmann wavefront sensor, SHWFS) is not taken into account.

The third system that was considered was the AO system of the 1 m New Vaccum Solar Telescope (NVST) at the Fuxian Solar Observatory in China (Rao et al. (2016)). They added the residual error of low-order corrected modes, high-order uncorrected modes, and aliasing error to estimate the total wavefront error. Following this, they used the expression from (Parenti and Sasiela, 1994) to estimate the short exposure Strehl ratio (see table 2 rows 11 - 15). The values predicted by our simulations are lower than the values reported by them. Understandably, it is not a fair comparison because we estimate Strehl ratios from long exposure images.

In summary, we find that the efficiency obtained from the VTT is likely to be unbiased and thus we can possibly conclude that the efficiency of practical solar AO systems is likely to be in the range of 40 to 55%.