Non-recursive transport of intensity phase retrieval with the transport of phase

Abstract

The transport of intensity equation (TIE) is a non-interferometric phase retrieval method that originates from the imaginary part of the Helmholtz equation and is equivalent to the law of conservation of energy. From the real part of the Helmholtz equation, the transport of phase equation (TPE), which represents the Eikonal equation in the presence of diffraction, can be derived. The amplitude and phase for an arbitrary optical field should satisfy these coupled equations simultaneously during propagation. In this work, the coupling between the TIE and TPE is exploited to improve the phase retrieval solutions from the TIE. Specifically, a non-recursive fast Fourier transform (FFT)-based phase retrieval method using both the TIE and TPE is demonstrated. Based on the FFT-based TIE solution, a correction factor calculated by the TPE is introduced to improve the phase retrieval results.

Retrieval of phase and three-dimensional topography using modified transport of intensity and phase equations with electrically programmable optical path lengths

“…under the assumption of near-uniform intensity, which typically is true for a phase object near the image plane. Equation ( 1) can be solved using the FFT to obtain the solution of the phase as 14,16 φ…”

“…Similarly, the TPE can be derived from the real part of the Helmholtz equation or paraxial wave equation and incorporates the effect of diffraction. 8 It is expressed as 8,14,16 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 1 1 7 ; 6 5 4…”

“…14 Recently, it has been shown that the TIE solution can be improved by incorporating a correction factor μðx; yÞ from the TPE without the need for a recursive solution. The non-recursive TIE+TPE can be expressed as 15,16 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 1 1 7 ; 5 5 5…”

“…An excellent review can be found in Zuo et al 13 Basunia et al 14 proposed recursively solving the TIE with the TPE to retrieve the (unwrapped) phase information. A non-recursive method incorporating TPE with TIE was introduced by Zhou et al, 15,16 which reduces computational time.…”

“…2, we first review the conventional TIE, TPE, and non-recursive TIE+TPE techniques to recover the unwrapped image phase. 15,16 In Sec. 3, we introduce the TIE, TPE and non-recursive TIE+TPE modified using a LC to provide electrically controllable OPLs needed for the implementation of TI.…”