[Special Report] Wave front Reconstruction Using Small Lenses III

Mode and significance of Zernike polynomial equation

In modal sensing ofwavefrontreconstruction, the phase is calculated from the coefficient of Zernike polynomial equation using gradient information obtained from wavefront sensor.<Figure 5> shows phase reconstruction process using modal sensing of a given wavefront. <Figure 5> a) and b) show displacement vector of central point of reference wavefront and Hartmann image point of wavefront distortion. The vector length and direction vary based on the degree of distortion. <Figure 5> c) and d)show x axis and y axis of gradients obtained from the measured center point displacement. <Figure 5> e) shows the aberration mode values of coefficient ofZernike polynomial equation using gradient information. The astigmatism modes, Z04 and Z05, have large values. Lastly, <Figure 5> f) shows the phase obtained using coefficient of Zernike polynomial equation.

Shack-Hartmann style wavefront sensor was developed to perform high-speed sensing of distortedwavefront and computewavefrontinformation. Shack-Hartmann wavefront sensor is equipped witharray lens, beam narrower, image relay lensand CCD camera, etc. and has convenience of mobility. We applied an algorithm for real-time digital image processing through a computer.

We established a wavefront reconstruction algorithm suitable for closed-circuitadaptive optics using zonal sensing and modal sensing. zonal sensing was performed using reconstruction matrix suitable for each model andmodal sensing was performed using Zernike polynomial equation. An experiment was conducted using measured gradientinformation to assess the validity results.Southwellmodelyielded appropriate phase values. Hudgin and Fried modelshowed errors due to the algorithm’s inability to use the entire gradient information at the edge.

Figure 5.Wavefront reconstruction using modal sensing.a) Image point of reference wavefront, b) Image point of wavefront distortion, c) gradient information of the x-axis, d) gradient information of the y-axis, e) Zernike mode values, f) wavefront reconstruction using modal sensing.

In modal sensing, the number of conditions of Zernike partial differentiationpolynomial equationarray increased along with the difference of wavefront aberration in proportion to the number of modes regardless of the number of arrays. The error messenger drastically increased along with the number of modes and showed only slight increase after Z08, the fifth aberration. In an experiment, we found similar results from wave reconstruction through increasing the number of modes based on the gradientinformation of reference wavefront. Analyzing Zernike partial differentiationpolynomial equationarray showed that the appropriate number of modes can be determined using obtained gradient information.

Wavefront reconstruction through zonal sensing andmodal sensing of the samewavefront showed that the reconstruction phase from modal sensing was similar to Southwell’s results but the values of least square method were greater than those obtained fromzonal sensing. That is, modal sensing is suitable for analyzing aberration but not suitable in determining thephase value. Figure 6 shows wavefront reconstruction software that controlswavefront sensor.