Laboratory demonstrations on a pyramid wavefront sensor without modulation for closed-loop adaptive optics system

Shengqian Wang; Changhui Rao; Hao Xian; Jianlin Zhang; Jianxin Wang; Zheng Liu

doi:10.1364/OE.19.008135

1. Introduction

The pyramid wavefront sensor (PWFS) is an innovative device proposed by Ragazzoni [1] with the special characteristics of variable gain and adjustable sampling in real time to enable an optimum match of the system performance, which make it an attractive option for next generation adaptive optics system compared with the Shack-Hartmann [2,3]. The Telescopio Nazionale Galileo is the first astronomical telescope that implements the PWFS in its adaptive optics system [4] and the Large Binocular Telescope also used this sensor as a key component [5]. Moreover, the PWFS which operates at near-infrared wavelengths is designed on the 3.5m telescope on the Calar Alto observatory in southern Spain [6].Furthermore, the PWFS is acted as a new and promising approach in the MCAO (multi-layer conjugate adaptive optics) to solve the problems related with the limited field of view [7], and it has also demonstrated that the PWFS has the ability to phase and align segmented mirrors [8].

At present most of the PWFS are used with modulation based on oscillating optical component in order to give a linear measurement of the local tilt, and it has been studied in the laboratory and successfully operated on the sky based on the modulation mode. Modulation plays a central role in the geometrical optical approximation to improve the linear range of the PWFS, which is very small in the nonmodulated case. Burvall discussed the approximate linear relationship between the wavefront derivative and the sensor response with modulation [9], and Vérinaud’s analysis explained the effects of modulation according to the diffraction optics theory [10]. However, the sensitivity of the PWFS will decrease with the modulation amplitude and the moving parts increase the complexity of the system. There is an attractive idea that rise with the use of the PWFS is how to work without any dynamic modulation, which would greatly simplify the optical and mechanical design of the adaptive optics system and also give highest sensitivity as expected to be achieved. Costa presented a study showing that the higher order uncompensated residuals of the atmospheric turbulence are equivalent to a modulation for the lower compensated modes [11,12]. Ragazzoni introduced a static modulation method based on a light diffusing element, which causes a blur of the spot on the pyramid pin, in a similar way to the dynamic modulation [13], and then LeDue tested this idea using a holographic diffuser [14]. Korkiakoski has found an iterative reconstruction method that can be used to evaluate the significance of the nonlinearities in the nonmodulated PWFS [15].

The aim of this paper is to clearly present the choices that can be made to design the PWFS without modulation using in the closed-loop adaptive optics system. Firstly, the special behavior of the PWFS without modulation is described with theoretical analysis and numerical simulations. The detection trend of the PWFS is studied to illustrate that the correction orientation of the nonmodulated PWFS is right to have the successive iteration process for the closed-loop adaptive optics system. (According to this paper, the detection trend is the normalized multiplication of the wavefront gradient with the PWFS signal. If the signal of the detection trend is a positive number, the polarity of the PWFS signal is accord with the wavefront slope and the closed-loop correction along this orientation will be effective.) At the same time, the behavior of the small linear range for the PWFS without modulation is considered in this paper. Finally the laboratory experimental setup based on this sensor and the liquid-crystal spatial light modulator (LC SLM) has been developed and the experimental results demonstrate the feasibility of the PWFS without modulation used in the closed-loop adaptive optics system.

2. Theory and simulation of the PWFS without modulation

The principle of the PWFS has been well described in many other papers so that we just refer the conclusions important to this work. When the PWFS is used as a wavefront sensor, four images of the pupil are created on the detector and the detection signals of the PWFS are calculated with the following formula [1]:.

S_{x} (x, y) = \frac{I_{1} (x, y) + I_{4} (x, y) - I_{2} (x, y) - I_{3} (x, y)}{I_{1} (x, y) + I_{2} (x, y) + I_{3} (x, y) + I_{4} (x, y)}

S_{y} (x, y) = \frac{I_{1} (x, y) + I_{2} (x, y) - I_{3} (x, y) - I_{4} (x, y)}{I_{1} (x, y) + I_{2} (x, y) + I_{3} (x, y) + I_{4} (x, y)}

where

I_{i} (x, y)

is the intensity distribution corresponding to the quadrant i of the detector. For circular modulation with angular amplitude θ (assuming the angular amplitude is bigger than the gradient of the wavefront aberration), it follows that the detection signals are proportional to the phase derivatives from the simple geometrical optics considerations.

If no modulation is applied, the PWFS only yields the sign of the wavefront slope based on the knife-edge test, that is to say the linear range of the signal is very small according to the geometrical optics theory. Using the diffraction theory of the Foucault knife-edge test, the relationship between the wavefront phase $ϕ (x, y)$ and the PWFS signal $S_{x} (x, y)$ is given by the following expression [9] (here the PWFS is simply replaced by a roof prism and interferences between the two pupils are neglected, and the same formula to the PWFS signal $S_{y} (x, y)$ )

S_{x} (x, y) = \frac{1}{π} \int_{- B (y)}^{B (y)} p . v . \frac{\sin (ϕ (x^{'}, y) - ϕ (x, y))}{x^{'} - x} d x^{'}

where

B (y)

is the edge point of the chord passing the point

(x, y)

in the pupil. In the integral the contribution of the sine-term is weighted with the distance between the integration point

(x^{'}, y)

and the measured point

(x, y)

. Obviously, there is not a linear relation between the PWFS signal and the phase differences according to this formula.

It is well known that the system can achieve successful closed-loop operation if the system satisfies two important conditions. The first one is that the detection trend of signal must be right during the closed-loop process, which means that the estimated correction is in the right direction to cause the residual variance of the closed-loop error to decrease continuously by successive iterations. The other one is that a small linear range is exist when the loop is closed successfully, which makes the closed-loop system to retain such a status in the closed-loop process. Due to the complexity of the theoretical formula (especially considering other impact factors, for example the interferences between the four pupils of the PWFS, the theoretical formula could become more complicated and hard to analyze), hence we adopt simulation method in the following discussion and present the preliminary simulation results to explain that it is possible to use the PWFS without modulation in the closed-loop adaptive system.

Firstly, to make a qualitative description of the detection trend of the PWFS without modulation, we derive a modal for the non-modulated PWFS measurement signal using diffraction optics and the elements for our simulation is consistent with the laboratory setup in the following text. Figure 1 shows the schematic diagram of this modal. The incoming wavefront is $W (x, y)$ , and a Fourier transform (labeled $F T$ in the figure) related the $W (x, y)$ to the $T (u, v)$ in the focal plane, where it is masked by the function representing the pyramid (in this case $H_{i} (u, v)$ is the spatial filter effect introduced by the i-th PWFS facet in the focal plane). Finally an optical relay lens projects four pupil images on the detector, that is to say the intensity distribution $I_{1} (x, y)$ , $I_{2} (x, y)$ , $I_{3} (x, y)$ and $I_{4} (x, y)$ can be accepted, and then the PWFS measurement signal $S_{x} (x, y)$ and $S_{y} (x, y)$ are formed from these four intensity patterns. The signal of $T r e n d X$ represents the normalized multiplication of the wavefront local tilt $W_{x}$ with the PWFS signal $S_{x} (x, y)$ (the same to $T r e n d Y$ ). Here the signal $T r e n d X$ is a quantitative representation of the detection trend and it has been normalized according to the maximum value in the simulation signal (because the detection signal amplitude is different for the different aberrations and we mainly want to explain that the actual slope and the PWFS signal have the same trend, we conducted a normalized processing to obtain the images with certain unity and intuitive). If the signal $T r e n d X$ is bigger than zero, this is in agreement with the fact that the detection trend of the phase difference is right when using $S_{x} (x, y)$ as the local wave-front derivative and the system will have the successive iteration process.

Fig. 1 Schematic diagram of the modal for the PWFS without modulation.

Abstract

1. Introduction

2. Theory and simulation of the PWFS without modulation

3. Experiment results

4. More considerations for the PWFS used in the practical astronomical AO system

5. Conclusions and future work

Acknowledgments

References and links

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Figures (21)

Equations (3)

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