## Optical Setup and Image Formation Model  
### Multiwavelength Object and Image Modeling  
We present the problem of the absolute phase retrieval in the framework of thickness (profile) measurement for a transparent object. A coherent laser beam impinges on surface of the object of interest. A propagation of the coherent beam through the object results in a phase delay between the original input and the output beams. In general, this phase delay is spatially varying with variations proportional to variations of the thickness of the object. If the phase delay and the wavelength of the laser beam are known the thickness of the object can be calculated. However, only wrapped phase delays within the interval of \(2\pi\) radians can be measured in experiments with a single wavelength. It follows, that only very thin objects can be examined in this way. Use of multiwavelength beams allows to overcome this restriction and to reconstruct absolute phases, and in this way to produce surface measurements for much thicker objects. If the thickness is known the absolute phases for all wavelengths can be defined. Thus, the approach can be focused in two a bit different directions: measurement of the absolute phases for multiple wavelengths and measurement of the object surface. This task is similar to the problem of surface contouring of a reflective object [43]. The model developed below can be applied to both types of objects. Our aim to consider in details a more interesting case of the transparent object with dispersion of the refractive index, because ignoring this dependence can cause significant distortions in the profile of the object being reconstructed [53]. Let \(h_{o}(x)\), \(x\in R^{2}\), be a thickness variation of a transparent object to be reconstructed. A coherent multiwavelength light beam generates corresponding complex-valued wavefronts \(u_{o,\lambda}=b_{o,\lambda}\exp(j\varphi_{o,\lambda})\), \(\lambda\in\Lambda\), where \(\Lambda\) is a set of the wavelengths and \(\varphi_{o,\lambda}=\dfrac{2\pi}{\lambda}h_{o}(n_{\lambda}-1)\) are phase delays corresponding to \(h_{o}\), where \(n_{\lambda}\) is the refractive index. In the lensless phase retrieval, the problem at hand is a reconstruction of the profile \(h_{o}(x)\) from the noisy observations of diffractive patterns, registered at some distance from the object. For the high magnitude variation of \(h_{o}\) the corresponding absolute phases \(\varphi_{o,\lambda}\) take values beyond the basic phase interval \([-\pi,\pi]\), then only wrapped phases can be obtained from \(u_{o,\lambda}\). We demonstrate that the multiwavelength setup allows to reconstruct the profile of the objects which height variations are significantly greater than the used wavelengths as well as to reconstruct the corresponding multiwavelength absolute phases. In what follows, we apply the following convenient notation for object modeling:  
\[u_{o,\lambda}=b_{o,\lambda}\exp(j\mu_{\lambda}\varphi_{o}),\;\lambda\in\Lambda, \tag{1}\]where \(u_{o,\lambda}(x)\!\in\mathbb{C}^{2}\), \(\varphi_{o}(x)\in R^{2}\) is the object absolute phase in radians, \(\mu_{\lambda}>0\) are dimensionless relative frequencies and \(\Lambda=[\lambda_{o},\lambda_{1},...,\lambda_{n_{\lambda}-1}]\) is the set of the wavelengths. The link with the previous notation is obvious: \(\mu_{\lambda}=\dfrac{\lambda^{\prime}(n_{\lambda}-1)}{\lambda(n_{\lambda^{ \prime}}-1)}\) and  
\[\varphi_{o}=\dfrac{2\pi}{\lambda^{\prime}}h_{o}(n_{\lambda^{\prime}}-1). \tag{2}\]  
Here \(\lambda^{\prime}\in\Lambda\) is a reference wavelength and \(\varphi_{o}\) is an absolute phase corresponding to this wavelength. We are interested in two-dimensional imaging and assume that amplitude, phase and other variables, in what follows, are functions of the argument \(x\) given on a regular \(N\times N\) grid, \(X\subset\mathbb{Z}^{2}\). The parameter \(\mu_{\lambda}\) establishes a link between the absolute phase \(\varphi_{o}\) and the wrapped phase \(\psi_{o,\lambda}\) of \(u_{o,\lambda}\) which can be measured at the \(\lambda\)-channel.