By using the beat frequency technique, the dual-wavelength digital holography (DWDH) can greatly increase the measurement range of the system. However, the beat frequency technique has a limitation in measurement range. The measurement range is not larger than a synthetic wavelength. Here, to break through this limitation, we propose a novel DWDH method based on the constrained underdetermined equations, which consists of three parts: (i) prove that the constrained underdetermined equation has a unique integer solution, (ii) design an algorithm to search for the unique integer solution, (iii) introduce a third wavelength into the DWDH system, and design a corresponding algorithm to enhance the anti-noise performance of DWDH. As far as we know, it is the first time that we have discovered that the problem of DWDH can belong in a problem of contained underdetermined equations, and it is also the first time that we have given the mathematical proof for breaking through the limitation of the measurement range. A series of results is shown to test the theory and the corresponding algorithms. More importantly, since the principle of proposed DWDH is based on basic mathematical principles, it can be further extended to various fields, such as dual-wavelength microwave imaging and dual-wavelength coherent diffraction imaging.
Digital holography is one of the most popular quantitative phase imaging techniques, but the refractive index and the thickness are always coupled in the phase. To solve the decoupling problem, multiple scanning methods such as tomography and total reflection are usually used, which is time-consuming. To increase the imaging speed and reduce the system cost, it is urgent to seek the decoupling method of scanning-free digital holography. In this paper, we find that the decoupling method of scanning-free digital holography can be transformed into a problem of solving constrained higher order equations. By introducing the Fresnel reflection formula, a six-degree equation about refractive index is constructed and the corresponding algorithm for solving the equation is given. By using the algorithm, the refractive index and thickness can be decoupled successfully. A series of results show that the proposed method is effective and has high anti-noise performance. This method provides a mathematical possibility for scanning-free digital holography to decouple the refractive index and complex pixel stepped thickness distributions. Therefore, it may provide a theoretical basis for the subsequent development of a real scanning-free digital holography system, which may have potential applications in the measurement of optical devices produced by the modern film deposition process and etching process.
It is known that phase ambiguity is always an inherent problem in digital holography. In this paper, a 2 π ambiguity-free digital holography method is proposed. The method naturally avoids phase ambiguity by a quasianalytic method. This quasianalytic method accurately calculates the true phase by constructing an equation and solving the solution of the equation. Thus, the inherent wrapping problem in digital holography is eliminated. For example, our experimental result shows that the true phase of the stepped specimen with the phase distributed in [0, 16 π ] can be obtained unambiguously. Since the proposed method naturally avoids the phase ambiguity problem, it may be beneficial to enlarge the application potential of the digital holography. The effectiveness and accuracy of the proposed method are verified by both numerical simulations and experimental results.
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