4 Optimum imaging wavelength of the telescope with partial correction adaptive optics

The effect of wavefront residual error of AOS on an optical image is to remove energy from the central peak, distributing it into a surrounding halo. The halo is absent for a perfect wavefront, but dominates the image structure in the presence of a wavefront error typical of uncompensated turbulence.

For the astronomical telescope with AOS, the image resolution is most affected. To investigate the image quality of AOS, it is useful to begin with the extended Marechal approximation, which provides a relationship between the figure error and the Strehl ratio under weak-turbulence conditions

$${\it Strehl}\approx \exp\left(-\sigma_{\rm fig}^2\right).$$ (20)

This is only valid for small residual phase error. For most partial correction AOS, this condition is generally not satisfied.

Assuming all of the energy lost from the central core of the beam is scattered over a region of width $\lambda/r_0(\lambda)$, a composite short-exposure image Strehl ratio (Parenti & Sasiela 1994) for partial correction AOS can be shown as

$${\it Strehl}_{\rm SE}\approx \exp\left(-\sigma_{\rm fig}^2\right)+\frac{1-\exp\left(-\sigma_{\rm fig}^2\right)}{1+[D/r_0(\lambda)]^2}\cdot$$ (21)

Accordingly, the short-exposure image resolution (Parenti & Sasiela 1994) for partial correction AOS can be expressed as

$$Resolution_{\rm SE}\approx 1.22(\lambda/D)\left(1\big/\sqrt{{\it Strehl}_{\rm SE}}\right).$$ (22)

Similarly, for the long-exposure imaging, the Strehl ratio and the resolution (Parenti & Sasiela 1994) for AOS can be approximated to

$${\it Strehl}_{\rm LE}\approx\frac{\exp\left(-\sigma_{\rm fig}^2\r... ...rm tilt}^2}+\frac{1-\exp\left(-\sigma_{\rm fig}^2\right)}{1+[D/r_0(\lambda)]^2}$$ (23)

and

$$Resolution_{\rm LE}\approx 1.22(\lambda/D)\left(1\big/\sqrt{{\it Strehl}_{\rm LE}}\right)$$ (24)

respectively. Note that the first term of Eqs. (21) and (23) is the relative peak intensity of the central core of the imaging beam and the second term is that of the surrounding halo. Generally, the peak intensity of the central core is much larger than that of the halo for imaging with AOS. As the figure residual error is very small, the short-exposure Strehl ratio can be simplified as mentioned above. For partial correction adaptive optical systems, the imaging performance as shown in Eqs. (21-24) is by far superior.

Figure 1 presents the short-exposure imaging resolution of AOS versus the observing wavelength for different figure residual root mean square errors. It can be seen from Fig. 1 that for a certain figure residual error, the imaging resolution improves with increasing observing wavelength as the observing wavelength is relatively shorter. When the observing wavelength is longer, the imaging performance gradually becomes worse with the lengthening of the wavelength. Note that there exists an optimum observing wavelength in which the imaging resolution is highest for a certain figure residual error. The larger the figure residual error, the longer the optimum wavelength. So in astronomical telescopes with adaptive optics, the appropriate imaging observation wavelength need to be selected based on the system capability and its mission in order to exploit fully the compensation function of AOS for atmospheric turbulence.

Figure 1: The imaging resolution vs. observing wavelength for adaptive optical systems.

The optimum wavelength would maximize the imaging resolution of the astronomical telescopes with adaptive optics. For convenience of analysis, a new function $F(\lambda)$ is defined to used to optimize the imaging observation wavelength of AOS,

$$F(\lambda)=(D/\lambda)^2 {\it Strehl} .$$ (25)

Evidently, the maximizations of the function $F(\lambda)$ and the imaging resolution are both equivalent.

4.1 Optimum wavelength for short-exposure imaging

For short-exposure imaging, the imaging resolution depends mainly on the figure residual error of AOS, so that $F(\lambda)$ can be written as

$$F(\lambda) (D/\lambda)^2 {\it Strehl}_{\rm SE}$$ =

$$\approx (D/\lambda)^2 \left[\exp\left(-\sigma_{\rm fig}^2\right)+\frac{1-\exp\left(-\sigma_{\rm fig}^2\right)}{1+\left(D/r_0(\lambda)\right)^2}\right]\cdot$$ (26)

For astronomical imaging through adaptive optics correction for atmospheric turbulence, the imaging Strehl ratio and the resolution mostly lies on the intensity distribution of the central core of the beam. Note that generally the diameter of the system is much larger than the coherence length, namely $D\gg r_0(\lambda)$, so the function $F(\lambda)$ can be simplified as

$$F(\lambda)\approx(D/\lambda)^2\exp\left(-\sigma_{\rm fig}^2\right).$$ (27)

Substituting Eqs. (16) into Eq. (27), one can obtain

$$F(\lambda)\approx(D/\lambda)^2\exp[-A(\lambda_0/\lambda)^2]$$ (28)

where A is the figure residual error of AOS at the wavelength $\lambda_0=0.55~\mu$m,

$$[f_{\rm G}(\lambda_0)/f_{\rm 3dB}]^{5/3}[ 1+2.2156(2\pi\tau f_{\rm 3dB})^{2/3}$$

$$+1.3292(2\pi\tau f_{\rm 3dB})^{5/3}]$$

$$+(\lambda_{\rm W}/\lambda_0)^2\pi f_{\rm 3dB} \sigma_{\rm fn}^2/f_{\rm s}+0.34[d_{\rm s}/r_0(\lambda_0)]^{5/3}$$

$$+(\theta/\theta_0(\lambda_0))^{5/3}+(D/d_0(\lambda_0))^{5/3}.$$ (29)

In order to verify the existence of the optimum imaging observation wavelength, one can obtain, by taking the second order derivative for the function $F(\lambda)$ and letting it be less than zero, the following inequation:

$$\frac{{\rm d}^2F(\lambda)}{{\rm d}\lambda^2} \left(\frac{6D^2}{\lambda^4}-\frac{14AD^2\lambda_0^2}{\lambda^6}+\frac{4A^2D^2\lambda_0^4}{\lambda^8}\right)$$

$$\times\exp\left[-A(\lambda_0/\lambda)^2\right]<0.$$ (30)

Solving the above inequation, one can obtain

$$\sqrt{A/3}\lambda_0<\lambda<\sqrt{2A}\lambda_0.$$ (31)

That is to say that there exists an optimum imaging observation wavelength for AOS only as $\sqrt{A/3}\lambda_0<\lambda<\sqrt{2A}\lambda_0$.

Figure 2: Optimum short exposure observing wavelength and the corresponding image resolution.

Figure 3: Optimum long exposure observing wavelength and the corresponding image resolution.

Taking the first order derivative for the function $F(\lambda)$ and letting it be equal to zero, we can write:

$$\frac{{\rm d}F(\lambda)}{{\rm d}\lambda}=\left(\frac{-2D^2}{\lamb... ...{2AD^2\lambda_0^2}{\lambda^5}\right)\exp\left[-A(\lambda_0/\lambda)^2\right]=0.$$ (32)

For short-exposure imaging, the optimum imaging observation wavelength for AOS, taking the solution of Eq. (32), can be derived as

$$\lambda_{\rm SE\_opt}=\sqrt{A}\lambda_0.$$ (33)

According to Tyler (Tyler & Fender 1994), the optimal imaging observation wavelength for AOS should be

$$\lambda_{\rm SE\_opt}^*=\sqrt{2A}\lambda_0.$$ (34)

Subsituting Eq. (33) into Eq. (22), the corresponding resolution for short-exposure imaging at the optimum wavelength can be expressed approximately as

$$Resolution_{\rm SE\_opt} \approx 1.22\sqrt{eA}(\lambda_0/D)$$

$$1.22\sqrt{e}\left(\lambda_{\rm SE\_opt}/D\right).$$ = (35)

4.2 Optimum wavelength for long-exposure imaging

For long-exposure imaging, the imaging resolution depends not only on the figure residual error but also on the tilt residual error of AOS, Thus $F(\lambda)$ can be simplified as

$$F(\lambda)\approx(D/\lambda)^2\frac{\exp[-A(\lambda_0/\lambda)^2]}{1+5B(\lambda_0/\lambda)^2}$$ (36)

where B is the tilt residual error of AOS at the wavelength $\lambda_0=0.55~\mu$m,

$$B=[f_{\rm T}(\lambda_0)/f_{\rm c}]^2+(\lambda_{\rm T}/\lambda_0)^2\pi f_{\rm c}\sigma_{\rm tn}^2/f_{\rm st}.$$ (37)

Similarly, one can derive the optimum imaging observation wavelength for long-exposure imaging as

$$\lambda_{\rm LE\_opt}=\left[\sqrt{\left(A+\sqrt{A^2+20AB}\right)\big/2}\right]\lambda_0.$$ (38)

Subsituting Eq. (38) into Eq. (24), the corresponding resolution for long-exposure imaging at the optimum wavelength can be expressed approximately as

$$Resolution_{\rm LE\_opt} \approx 1.22\left\{\left[\left(A+\sqrt{A^2+20AB}\right)\big/2+5B\right]\right.$$

$$\left. \times \exp\left[2\big/\left(1+\sqrt{1+20B/A}\right)\right]\right\}^{1/2}(\lambda_0/D)$$

$$1.22\Bigg\{\left[\left(A+\sqrt{A^2+20AB}\right)+10B\right]$$ =

$$\times\frac{\exp\left[2\big/\left(1+\sqrt{1+20B/A}\right)\right]}{A+\sqrt{A^2+20AB}}\Bigg\}$$

$$\times(\lambda_{\rm LE\_opt}/D).$$ (39)

Figures 2 and 3 present optimum imaging observation wavelength and the corresponding image resolution for short-exposure imaging and long-exposure imaging respectively. It can be seen from Figs. 2 and 3 that the optimum imaging observation wavelength of AOS depends directly on the wavefront residual error of AOS. The larger the wavefront residual error, the longer the optimum imaging observation wavelength is. Especially, the optimum imaging observation wavelength for short-exposure imaging, see Eq. (33), can be explained as the corresponding wavelength under the condition that the figure residual error is 1 rad².

Furthermore, the optimal observing wavelength and the corresponding short-exposure imaging resolution according to Tyler and Fender's (Tyler & Fender 1994) result, see Eq. (34), are also presented in Fig. 2. Note that the optimal wavelength derived by Tyler and Fender (Tyler & Fender 1994) is larger than that in this paper. The imaging performance according to Tyler and Fender's result is inferior to the result of this paper. The interpretation is that the imaging short-exposure Strehl ratio in Tyler and Fender's derivation (Tyler & Fender 1994) is approximated as $S=\exp(-\sigma^2_{\rm fig})\approx 1-\sigma^2_{\rm fig}$. If the required precision is 10%, the figure residual error should be less than 0.14 rad². When the figure residual error is equal to 0.6 rad², the precision can only attain 27%. In contrast with the results of this paper, the percentages of the departure of the optimal wavelength derived by Tyler and Fender (Tyler & Fender 1994) and the corresponding imaging resolution are 41.4% and 10% respectively. Evidently, the imaging resolution based on the calculated optimal observing wavelength in this paper is superior to that according to the optimal wavelength derived by Tyler and Fender (Tyler & Fender 1994). This result shows also that the calculation in this paper is more precise and more effective than the rough estimate carried by Tyler and Fender (Tyler & Fender 1994).

For nighttime astronomical telescopes with adaptive optics, the imaging observations are mostly done at a range of wavelengths. Figure 4 presents the percentage of the short-exposure imaging resolution departure from the optimum resolution as function of the percentage of the observing wavelength departure from the optimum wavelength. As AOS is not operated in the optimum wavelength but just 10% longer or shorter, the resolution degrades only about 1%. This result shows that the derived optimum observing wavelength in this paper can be applied not only in the monochromic observation with very narrow band but also in the nighttime observation with a certain range of wavelength.

Figure 4: The percentage of the short-exposure imaging resolution departure from the optimum resolution as a function of the percentage of the observing wavelength departure from the optimum wavelength.