A Fourier-based Solving Approach for the Transport of Intensity Equation without Typical Restrictions

The Transport-of-Intensity equation (TIE) has been proven as a standard approach for phase retrieval. Some high efficiency solving methods for the TIE, extensively used in many works, are based on a Fourier-Transform (FT). However, to solve the TIE by these methods several assumptions have to be made. A common assumption is that there are no zero values for the intensity distribution allowed. The two most widespread Fourier-based approaches have further restrictions. One of these requires the uniformity of the intensity distribution and the other assumes the collinearity of the intensity and phase gradients. In this paper, we present an approach, which does not need any of these assumptions and consequently extends the application domain of the TIE.


Introduction
Within the last decades, the number of applications for coherent diffraction has grown enormously. Among them are biomedical imaging, coherent tomography, digital holography, coherent x-ray imaging, holographic microscopy, optical metrology, wave sensing and many more. The information of the coherent illumination of an object is contained in a complex scalar field, composed of intensity and phase. Although, the intensity distribution can be measured by a CCD camera, the phase measurement is a rather challenging task. As a standard technique, interferometry has been used. However, it requires a cumbersome very sensible experimental setup. Due to powerful computer processors, available in recent years, phase retrieval has been established as the most widely used alternative to interferometry. The main advantage of phase retrieval is its ability to extract the phase information from the intensity measurements by appropriate numerical algorithms. Until now, several phase retrieval algorithms have been developed. However, each method has its drawbacks and limitations, which restrict its application. One of the algorithms in the paraxial domain is based on the Transport-of-Intensity equation (TIE) originally developed by Teague [1]. Contrary to iterative approaches like the Gerchberg-Saxton algorithm and its numerous modifications, the TIE is a deterministic approach. Thus, it avoids problems like non-convergence or stagnation, which are typical in many iterative approaches. Although, it is an elegant deterministic way for phase retrieval, its solution is not trivial. Two very successful fast solving approaches are based on the Fast Fourier Transform (FFT). However, they suffer from different assumptions related to the intensity behavior, which automatically restrict the application of these tools. One of these approaches proposed by [2] and applied in [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], for instance, assumes the uniformity of the intensity distribution (orthogonal to the propagation direction). Another approach [21][22][23][24][25][26][27][28][29][30][31][32], is also restricted to very special cases due to the assumption that the transverse gradient of the phase and the intensity must be collinear [33]. Moreover, zero values for the intensity must be strictly avoided in both approaches. In this work, we present an efficient algorithm without these restrictions, making the applicability of the TIE more general.

Iterative solution for the TIE
The TIE describes the coupling between the intensity I and the phase  in a plane orthogonal to the direction of the paraxial wave [1]   and z is the propagation direction of the field. If the complex amplitude is replaced by, exp( ) Ii  the TIE is the imaginary part of the paraxial Helmholtz equation [34]. In principle, the two coefficients k and I are known. The axial intensity derivative Iz  can be linearly approximated by the Finite Difference method using two intensity measurements in two parallel planes orthogonal to the propagation with the distance z  : 21 .

  
In order to remove the aforementioned restrictions of the standard solutions of the TIE, we propose the following change of Eq (1): Since it avoids the division by zero, the constant C enables zero values in the intensity distribution. It should be noted, that the Eqs. (1) and (3) The singularity at the frequency point   The initial phase is set to   0 ,0 xy   . In order to get the next estimation for the phase 1 n   , in each iteration the phase n  is introduced into Eq. (7). The iteration will be repeated, until the convergence is reached. The termination condition for the algorithm may be determined by a fixed iteration number, for instance.

Constraints on the phase changes under the paraxial assumption
The TIE is only valid in the paraxial domain and consequently its solution must satisfy the paraxial condition. However, in the proposed iterative method, the start phase distribution  (8), the following relationships can be derived: x y x y which will be used to replace the phase derivatives in Eq. (9) by the wave vector components. The triangle inequality [36] within the Eqs. (9) and (10)

Implementation of the paraxial constraints
To fulfill the paraxial condition, both constraints for the gradient and Laplace operator of the phase must be considered in the algorithm. These are the necessary constraints to make the set of possible solutions smaller and to prevent non-physical solutions for the phase distribution