W1A.5.pdf Digital Holography and 3-D Imaging 2017 © OSA 2017 1 Finite Difference Approach to Transport of Intensity Hongbo Zhang a , Wenjing Zhou a,b , Ying Liu c , Tian Tian a , Partha Banerjee d,e ,Ting-Chung Poon a,c a. Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA b. Department of Precision Mechanical Engineering, Shanghai University, Shanghai, 200072 China c. Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA d. Department of Electro-Optics and Photonics, University of Dayton, Dayton, OH 45469, USA e. Department Electrical and Computer Engineering, University of Dayton, Dayton, OH 45469, USA hbzhang@vt.edu, wjzz1331@vt.edu, yliu4@vt.edu, ttian2@vt.edu, pbanerjee1@udayton.edu, tcpoon@vt.edu Abstract A finite difference method is proposed for solving the transport of intensity equation (TIE). Simulation shows that the proposed numerical method is able to reconstruct the phase with good accuracy as compared with FFT-based TIE methods. 1. Introduction Phase retrieval can be achieved through either interferometric or non-interferometric techniques. Non- interferometric techniques for phase retrieval have been well studied [1, 2], and the use of the transport of intensity equation (TIE) as a non-interferometric method for phase retrieval has been proposed by Teague [1] and developed through continuous research efforts [3]. FFT-based numerical methods have been used for solving the TIE [4, 5]. Most of the FFT-based methods for solving the TIE are based on assumptions such as uniform intensity or that the curl of the transverse gradients of the intensity and phase is zero, and as such it may cause accuracy problems when these assumptions are not met. While numerical methods have been attempted to solve the TIE directly, a comparison between all these different methods is not available [6]. 2. Finite Difference Setup The imaginary part of the Helmholtz equation, also called the transport of intensity equation (TIE), is ∇ ⊥ ∙ ∇ ⊥  = − ! !" !" , (1) where (, , ) and (, , ) are the intensity and phase distributions, respectively, of the optical field from the object as a function of transverse coordinates ,  at a distance of propagation  , respectively,  ! is the wave number of light in free space, and ∇ ⊥ represents the transverse gradient. Writing out Eq. (1) explicitly, ∇ ⊥  ∙ ∇ ⊥  + ∇ ⊥ !  +  ! !" !" = 0. (2) Since ∇ ⊥  = !" !"  ! + !" !"  ! and ∇ ⊥  = !" !"  ! + !" !"  ! , where  ! and  ! are the orthogonal unit vectors in the  and  coordinate directions, Eq. (2) becomes !" !" !" !" + !" !" !" !" +  ! ! ! !" ! + ! ! ! !" ! +  ! !" !" = 0. (3) For numerical simulations, Eq. (3) is now discretized using the central finite difference scheme to give ! !!!,!,! !! !!!,!,! !∆! ! !!!,!,! !! !!!,!,! !∆! + ! !,!!!,! !! !,!!!,! !∆! ! !,!!!,! !! !,!!!,! !∆! + !,!,! ! !!!,!,! !!! !,!,! !! !!!,!,! ∆! ! + ! !,!!!,! !!! !,!,! !! !,!!!,! ∆! ! + ! ! !,!,!!! !! !,!,! ∆! = 0 (4)