Wave-front Reconstruction For Shearing Inteferometry By Use Of The Fourier Transform

Lateral shearing interferometry employs interfering between an object wave-front and its sheared copy to test the wave-front. Without need of a reference wave, the shearing interferometry is more suitable in absence of available reference beam or the light source is low coherence. Using common-path system, the shearing interferomety has high insensitivity to circumstance disturbance than separate-path system. As a result, the shearing interferometry has become an important tool in optical testing and measurement and has a great value in applications. In this thesis, we present some new theoretical and experimental research work on the lateral shearing interferometry.In lateral shearing interferometry, inteferograms contain the information of the phase differences. The wave front reconstruction from phase differences is the most important step in lateral shearing interferometry. In various reconstruction algorithms, the two-dimensional wave-front reconstruction of the discrete Fourier transform is often used because of its fast compute, simpleness and accuracy. However, the algorithm requires the shear amount must be divided by the sampling dimensions, the spectrum leaking of the wave-front is inherent. The methods involved in this thesis are as follows:(1) Two-dimensional wave-front reconstruction based on single shear by use of the discrete Fourier transform. Firstly, the phase differences in orthogonal directions are processed by the discrete Fourier transform. Then the Fourier spectrum is obtained based on the least-square fitting. We replace the values of the coefficients at the leaking points by the average values of the adjacent points. Finally, the test wave-front is computed by the inverse Fourier transform of the expanding coefficients. The method is suitable for the continuous wave-front and is not suitable for the discontinuous wave-front.(2) Two-dimensional wave-front reconstruction based on two-shear by use of the discrete Fourier transform. The Fourier coefficients of the wave-front under test are computed from four phase differences according to two shears and two orthogonal directions. We solve the problem of the spectrum leaking by using the complementary frequency. However, the condition which must be satisfied is N=s1×s2, limiting the application in optical testing.(3) Two-dimensional wave-front reconstruction based on Fourier series and multi-shear interferometry. The expanding coefficients of the wave-front under test are computed from multi-group phase differences according to multi shears and two orthogonal directions. Finally, with the operation of inverse Fourier transform on the combined Fourier coefficient, the two-dimensional wave-front distribution can be reconstructed exactly in whole spatial frequency. The precision of the method is higher than the single-shear method and is suitable for any wave-front phase.(4) A novel method of solving the spectrum leaking. Firstly, derive a continuous Fourier transform from sampled data by means of the Whittaker-Shannon interpolation formula. Secondly, resample an appropriate lattice to avoid sampling of the leaking points of the expanding coefficients. Numerical and optical tests have confirmed that feasibility of the algorithm.(5) Wave-front reconstruction with tilt from multi-shear interferograms. When changing shear to test a wave-front, tilt error of the wave-front could vary because of the shearing method, environment vibration, or mechanical misalignment. The phase differences would have a bias when the experiment environment changes. When only one shear exists the bias will not influence the outcome. However, when the shears are two or more, the biases will not be matched and cause a big reconstruction error. In this thesis, we proposed a method which can remove automatically the tilt errors and improve the precision of reconstuction.Otherwise, because our multi-shear algorithm need multi-shears and multi group phase differences, the reconstruction error is easily influenced by the mechanical move. Therefore, we used the three-wave lateral shearing interferometer to overcome the difficulty. We can dynamically control the direction and amount of shear, as well as phase shift rely on the flexibility of Spatial Light Modulator (SLM). The setup needs not change the optics path and needs not change any experiment apparatus. We only need operate the computer and so it can alleviate the noise.We have performed numerical simulations for evaluating capability of wave front reconstruction methods mentioned in this thesis, and carried out experimental testing to verify the effectiveness of the methods proposed above. The results of numerical simulations and optical experiment confirm that the schemes proposed in this thesis have satisfactory reconstruction fidelity.