A method based on the transport of intensity equation (TIE) for phase retrieval is presented, which can retrieve the optical phase from intensity measurements in multiple unequally-spaced planes in the near-field region. In this method, the intensity derivative in the TIE is represented by a linear combination of intensity measurements, and the coefficient of the combination can be expressed by explicitly analytical form related to the defocused distances. The proposed formula is a generalization of the TIE with high order intensity derivatives. The numerical experiments demonstrate that the proposed method can improve the accuracy of phase retrieval with higher-order intensity derivatives and is more convenient for practical application.
©2011 Optical Society of America
The TIE can relate the object-plane phase to the axial derivative of the intensity distribution in the Fresnel region , and if the intensity distribution has no zeros, the TIE has a unique solution (within an additive constant) for coherent and partially coherent fields . Many effective numerical methods have been proposed for solving the TIE. Therefore, the TIE has been widely used for the recovery of optical phase from multiple intensity measurements, and phase recovery has been an essential imaging technique in many fields, e.g., optics , electron , neutrons  and x-ray microscopy .
In the numerical methods for solving the TIE, such as Fast Fourier Transform (FFT) [7,8], the multigrid (MG) [9,10], the Zernike polynomial expansion methods [11,12], and the Green’s function based method , the axial intensity derivative is commonly approximated by the finite difference method from two intensity measurements. However, two intensity measurements only achieve a maximum of second-order precision with respect to the separation of the adjacent intensity measurements if the intensity distributions do not contain noise . To improve the accuracy of the retrieved phase, multiple intensity measurements can be used to minimize the effect of the noise on the retrieved phase  or estimate the higher order axial intensity derivatives in the TIE [15,16]. The previous work mostly assumed that the intensity measurements are taken in the equally-spaced planes. However, the measurement errors exist objectively in practical application. Therefore, the intensity measurements are not really in the equally-spaced planes.
In this study, a generalization of the TIE is proposed, which uses multiple unevenly sampled intensity measurements for phase retrieval. Based on the higher-order Taylor expansions of the measured intensities, the axial intensity derivative in the TIE is expressed by a linear combination of the multiple unevenly sampled intensity measurements. Numerical simulations are conducted to test the method.
2. Unequally-spaced intensity measurements for phase retrieval
An optical plane wave in the x-y plane propagating along the axis can be expressed as:1]:
Assume that intensities measured in planes are available for solving the TIE. Expanding with Taylor series at as:Eq. (3) is ignored, which produce a linear Taylor approximation respect to. To improve the accuracy, we should estimate and compensate the higher-order intensity derivative in Eq. (3). This can be done by constructing a linear system from Eq. (3) as:
When the intensities are equally-spaced and symmetrically distributed relative to the focal plane, Eq. (4) is equivalent to the equation in  (Eq. (7) in ). When, Eq. (4) is reduced to the Eq. (7) in .
The solution to Eq. (4) can be obtained by inverting the matrix:Eq. (6) into Eq. (3), the order approximation to the intensity derivatives can be expressed with the form:Eq. (7) into Eq. (2), we can obtain a new generalization of the TIE with high order intensity derivatives:
In this study, the forward formula is used to estimate the intensity derivative. Table 1 shows the difference between the weight coefficients with unequally-spaced planes and that with equally-spaced planes . If the intensity measurement planes are equally-spaced, i.e., the proposed expressions of the coefficients are reduced to that in .
Numerical simulation experiments are conducted to demonstrate the feasibility of the proposed phase retrieval approach using intensity measurements in multiple unequally-spaced planes. A simulated optical fiber is chosen as the true phase profile (see Fig. 1 ).The dimensions of the image are 512 × 512 pixels with an assigned pixel size of 4. The phase value is between 0 and 0.5 radians. The radiation wavelength is chosen to be 550nm. The intensity distribution in the input plane is uniform. The root-mean-square-error (RMSE) is employed to quantify the accuracy of the retrieved phase:Figure 2 shows an example of the simulated diffraction images, which can be used to solve the TIE for phase retrieval. It should be mentioned that the diffraction images are sampled in unequally-spaced planes.
We carry out the phase retrieval simulations by solving the TIE with the FFT method . The first simulation assumes that the intensities are measured in the planes, is a random disturbance between 0 and , which denotes the measurement errors of the defocused distance. This numerical experiment includes the following two cases:
Case1: Ignoring, and the equally-spaced formula  is used to solve the TIE.
Case2: Considering, and the unequally-spaced formula is used to solve the TIE.
Figure 3 shows the recovered phases and the corresponding phase errors for case 2. In this numerical experiment, and, the defocused distances for the six intensities are respectively assigned to be and. The visual impression of the reconstructed phases is in good agreement with the true phase, and from the phase error images we can see that the visual differences between the reconstructed phases and the true phase tend to be smaller with the increase of the number of intensity measurements.
In the other simulation experiment, the intensities are measured in random planes between. Here we choose and. The intervals between the adjacent defocused planes are randomly chosen. In this simulation, the defocused distances of the six intensities are respectively set to be and. We name the second simulation as case 3.
Figure 4 shows the RMSE as a function of on a logarithmic scale. Figure 4(a) shows that the defocused distance errors can effect the oscillation of the RMSE curve, which corresponds to case 1. In this case, is ignored and the equally-spaced formula  is used to solve the TIE. Because the defocused distance errors are ignored, the accuracy of phase reconstruction cannot be improved by including higher order intensity derivatives. In case 2, we have considered the measurement errors, and the unequally-spaced formulas are used for phase retrieval. Consequently, the accuracy of the retrieved phases can be improved with the increase of the number of intensity measurements. The difference of the RMSE between case 2 and case 3 shown in Fig. 4(b) is due to the defocused distances. Since the noise is not considered, the smaller the defocused distances, the lower the RMSE are. Therefore, the RMSE in case 2 is larger than that in case 3.
We demonstrate a phase retrieval method based on the TIE from multiple unequally-spaced intensities, which helps relax the limitation of equally-spaced intensity measurements. Therefore, the proposed phase imaging method is more convenient for practical application. The numerical simulation results illustrate that the measurement error of the defocused distances effect the accuracy of phase retrieval and the proposed method can eliminate this influence. It can also improve the accuracy of the phase reconstruction with higher-order intensity derivative components just as the method with equally-spaced intensity measurements. The method has a potential use in the optical measurement system  for retrieving the phase with high accuracy in real time. However, the method presented in this paper does not consider the noise effect. More work will be done to simultaneously minimize the effects of the noise and the higher-order intensity derivatives on the retrieved phase from multiple unequally-spaced intensity measurements in the future.
This work is partly supported by a grant from the National Natural Science Foundation of China (No.60502018, No.61171005), the Innovation Fund and the Scholarship Award of BUAA for PhD Students, the Fundamental Research Fund for the Central Universities and the Aeronautical Special Fund of AVIC (No.CXY2010BH02).
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