Computational Method for Wavefront Sensing Based on Transport-of-Intensity Equation

Abstract

Recently the transport-of-intensity equation as a phase imaging method turned out as an effective microscopy method that does not require the use of high-resolution optical systems and a priori information about the object. In this paper we propose a mathematical model that adapts the transport-of-intensity equation for the purpose of wavefront sensing of the given light wave. The analysis of the influence of the longitudinal displacement z and the step between intensity distributions measurements on the error in determining the wavefront radius of curvature of a spherical wave is carried out. The proposed method is compared with the traditional Shack–Hartmann method and the method based on computer-generated Fourier holograms. Numerical simulation showed that the proposed method allows measurement of the wavefront radius of curvature with radius of 40 mm and with accuracy of ~200 μm.

Reconstructing the Spatial Parameters of a Laser Beam Using the Transport-of-Intensity Equation

“…In turn, the calculation of the curvature radius of the wavefront can be based on the solution of the TIE, in which the field phase function φ(r ⊥ , z) is related to the intensity distributions I(r ⊥ , z) by the following equation [19,24]:…”

“…At the first stage, two intensity distributions I(r ⊥ , z 1 ) and I(r ⊥ , z 3 ) are recorded using a camera. Then, at the second stage, the phase φ(r ⊥ , z) is calculated using the registered intensity distributions by solving the TIE, and then the wavefront curvature radius R z 2 is calculated using the geometric method based on the phase data [19]. At the last stage, by the least-squares method the spatial parameters of the laser beam are calculated from the values of R z 2 according to the relations given in Table 1.…”

“…At the second stage, the phase function of the field φ(r ⊥ , z 2 ) was calculated using TIE for the intensity distributions in the two extreme planes z 1 and z 3 (the distance between the planes ∆z 1−3 was chosen arbitrarily and amounted to 30 mm). The resulting phase function of the field φ(r ⊥ , z 2 ) in the intermediate plane z 2 was used for the approximation and calculation of the wavefront curvature radius R(z 2 ) [19]. The radius of curvature was calculated for the different pairs of planes z 1 and z 3 from the obtained dataset.…”

“…At the same time, a method of real-time reconstruction of the complex amplitude exists for measuring the M 2 parameter by means of transport-of-intensity equation [17][18][19]. The transport-of-intensity equation (TIE) provides a non-interferometric way to obtain quantitative information on the phase of light beams by recording transverse field intensity distributions in two or more cross-sections [20,21].…”

Measurement of the radius of curvature of a spherical surface based on the transport-of-intensity equation

“…Стоит отметить, что соотношение (7) было получено с допущением о справедливости параксиального приближения. Это накладывает определенное ограничение, в силу которого минимальное значение радиуса кривизны волнового фронта, при котором точность предлагаемого метода будет сохраняться, составляет около 40 мм [27].…”