Holographic-based Zernike modal wavefront sensing

In Holography-based modal wavefront sensing, a multiplexed diffractive optical element is illuminated by the input wavefront, producing a series of pairs of spots in the image plane, one pair of spots for each Zernike mode. The amplitude of each Zernike mode can be measured based on the ratio of intensity of the corresponding pair of spots. A complete evaluation of the wavefront is achieved without the need for computation, which means that the principle sensor bandwidth is only limited by the readout speed of the photo-detectors (up to MHz). However, the intermodal crosstalk effect significantly decreases the signal to noise ratio, which finally limits the performance of the sensor. As part of a DFG project (in cooperation with the Institut für Systemdynamik, ISYS) the sensor response is studied statistically for random atmospheric aberrations and the sensor is optimized by modifying the system parameters to obtain better sensor sensitivity and accuracy.

A complex wavefront can be described as a superposition of a set of orthogonal aberration modes (e.g. Zernike polynomials) with varying amplitude. In HMWS, the amplitude of each mode can be measured by simply comparing the intensity difference of the two spots generated by a specially designed DOE. The basic principle for sensing one Zernike mode (Zk) is depicted in Fig.1. An incoming beam with an aberrated wavefront illuminates the hologram. The modulated beam passes through a convergent lens which performs the Fourier transform and forms a pair of spots, which are measured separately by two detectors. The transmissive hologram (transmittance function H(k_x,y)=cos(2πf k_x x+2πf k_y y- b_k Z_k(x,y)) is designed such that an input beam with aberration b_k Z_k leads to a strong intensity at detector 1 and an input beam with aberration -b_k Z_k leads to a strong intensity at detector 2. Based on the intensities at the two detectors (I_k1and I_k2), the magnitude of the kth Zernike component of the input wavefront can be calculated with a_k'=b_k I_k1-I_k2 I_k1+I_k2. Multiple aberration modes can be coded into a multiplexed hologram to simultaneously measure the amplitude of each Zernike mode, allowing a full reconstruction of the complex wavefront.

The main difficulty in this technique is the intermodal crosstalk problem. Changes of number and amplitude of other Zernike modes in the beam result in a change of output intensity for detecting the encoded mode Zk. To reduce the inaccuracy caused by the intermodal crosstalk, the system parameters (e.g. detector radius and phase bias encoded in the CGH) can be optimized considering the strength of the aberration. Calibrated response curves of low-order Zernike modes are further utilized to improve the sensor accuracy.

The system optimization is first implemented by simulation using random aberration data from the Hufnagel-Valley Boundary (HVB) atmospheric turbulence model. Fig. 2(a) shows an exemplary result for sensor with detector radius r=37.5 µm and phase bias bk=0.7 radians for each Zernike mode. Two hundred random aberrations were generated to find out the averaged necessary number of iterations for reducing the residual wavefront error to 0.1 λ RMS. The inset in Fig.2 (a) means that 139 out of 200 (about 70%) aberrations need 18 or less system correction loops to achieve the desired residual error. By optimizing the system parameters and employing the calibrated response curves for low-order modes, the system accuracy can be improved. Fig. 2(b) shows the optimized result where detector radius r=87.5 μm and bk=1.5 radians. The calibrated response curve is only used for Z4 to Z10. The cumulative histogram shows that the average number of iterations for obtaining a residual RMS wavefront error of 0.1 λ is reduced from 18 to 3. To validate the method that we used for optimizing the HMWS, we implemented a closed-loop adaptive optics system. The detailed system parameters can be found in [2]. Liquid crystal SLM is used as the aberration generator as well as wavefront correcting device. Random aberration is generated and displayed onto the SLM. The improvement of the system PSF as the correction loops progressed can be seen in Fig. 3. This optimization process can be easily adapted to other imaging systems (e.g. microscopy, retina imaging) with respect to their specified aberrations.

Fig. 3. The PSF of the system (a) before correction (b) after one iteration (c) after two iterations (d) after three iterations.

Microscopy, retinal imaging, ground-to-space laser communication, astronomy (laser guide star) etc.