1 Introduction

It is well known that atmospheric turbulence (Roddier 1981) severely limits the resolution of large ground-based telescopes in the visible and near infrared bands. Adaptive optics (Jiang et al. 1995) techniques can be used to correct random optical wavefront distortions induced by the atmospheric turbulence in real time.

The performance requirements for adaptive optics systems (AOS) depend on their mission. In the visible band, large telescopes require very complex AOS with hundreds of correction channels in order to produce high-resolution object images. But the large telescopes with low-order AOS perform well in the near infrared band. The imaging observation wavelength is one of the most significant factors in the requirements of AOS. For ground-based solar telescopes (Rimmele & Radick 1998), high-resolution monochromic imaging observations are usually done at a single wavelength with a very narrow band because of strong enough sunlight. For nighttime astronomical telescopes, a range of wavelengths is often used for imaging observation. For nighttime astronomical adaptive optical systems, the observing wavelength during setting up an observation is generally confined to the end portion of the red visible (R and I bands), the non-thermal near IR (J and H bands), or thermal IR (K band) according to the practical requirements and system capabilities.

In selecting an imaging observation wavelength, two competing effects must be considered. On the one hand, as the imaging wavelength is increased, the wavefront residual error decreases, resulting in a sharper point spread function. On the other hand, diffraction increases with wavelength, producing the opposite effect. For telescopes with partial correction AOS, the appropriate observing wavelength should be selected on the basis of the system capability and the working conditions in order to achieve the best angular resolution.

In previous publications, Tyler (Tyler & Fender 1994) derived an expression for the wavelength giving maximum resolution for the telescopes with AOS. In his study, the imaging Strehl ratio is simplified as $S=1-\sigma_{\rm fig}^2$, where $\sigma_{\rm fig}^2$ is the variance of the residual phase error. This is only valid for small residual phase error. For the partial correction AOS, this condition can commonly not be satisfied. In this paper, we derive the expressions of the optimum short-exposure and long-exposure imaging observation wavelength of astronomical telescopes with partial correction adaptive optics based on the analysis for the wavefront residual error of AOS. The paper is organized as follows. The parameters of atmospheric turbulence are briefly introduced in Sect. 2. In Sect. 3, the wavefront residual error of AOS is analyzed by considering the limitation factors of the practical AOS. Section 4 derives the optimum imaging observation wavelength on the basis of the relationship between the angular resolution and the wavefront residual error for the astronomical telescopes with partial correction AOS. In Sect. 5, the optimum imaging observation wavelength is computed for the 61-element adaptive optical system at the 1.2 m telescope of the Yunnan Observatory.