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TIE is a classic phase retrieval algorithm. Suppose that a sample is illuminated by a monochromatic plane wave with constant intensity along the axis
$z$ , the sample’s intensity$I(x,y,{z_0})$ and phase$\varphi (x,y,{z_0})$ of the field at the focal plane satisfy the following equation [13]$$ - \nabla \cdot \left[ {I(x,y,{z_0})\nabla \varphi (x,y,{z_0})} \right] = {\left. {k\frac{{\partial I(x,y,z)}}{{\partial z}}} \right|_{z = {z_0}}}$$ (1) where
$k$ is wave number,$k = 2\pi /\lambda $ ,$\lambda $ is wavelength,${z_0}$ denotes the distance propagate along the optical axis$z$ .$\partial I(x,y,z)/\partial z$ is intensity derivative, which may be estimated by finite differences taken between the in-focus intensity image and the defocus intensity image[14], as shown in Fig.1.$${\left. {\frac{{\partial I(x,y,z)}}{{\partial z}}} \right|_{z = {z_0}}} \approx \frac{{I(x,y,{z_0}) - I(x,y,{z_0} - \Delta z)}}{{\Delta z}}$$ (2) where
$\Delta z$ is defocus distance. Substituting (2) into (1) and yielding the phase by use of Fourier methods [15]:$$\varphi (x,y,{z_0}) = k{\Im ^{ - 1}}\left[ {{{\left[ {2{\pi ^2}(f_x^2 + f_y^2)} \right]}^{ - 1}}\Im \left[ {{{\partial I(x,y,z)} / {\partial z}}} \right]} \right]$$ (3) where
$\Im $ and${\Im ^{ - 1}}$ denote the Fourier and inverse Fourier transform respectively,${f_x},{f_y}$ are the spatial frequencies in Fourier domain.$\varphi (x,y,{z_0})$ is equal to the product of the optical path length(OPL) through the sample with the wavenumber of the illumination. If a sample consisting of multiple materials with different refractive indices, the physical thickness$h(x,y,{z_0})$ of the sample is related to the OPL by$$h(x,y,{z_0}) = \frac{\lambda }{{{n_o} - {n_m}}}\frac{{\phi (x,y,{z_0})}}{{2\pi }}$$ (4) Where,
${n_o}$ is the refractive index of the object to be measured and${n_m}$ is the refractive index of the surrounding medium. The medium is usually air, so${n_m}$ is about 1.In addition, angular spectrum iteration is another classic phase retrieval method. The principle of angular spectrum iterative algorithm is shown in Fig.2. Among them,
${U_1} = \sqrt {{I_1}} \exp [j\phi ]$ and${U_2} = \sqrt {{I_2}} \exp [j{\phi '}]$ are the complex amplitudes at two different positions in the propagation direction of the light field, and${I_1}$ and${I_2}$ are the amplitudes captured at two positions above respectively. Firstly, a guess value is taken as the phase of${U_1}$ , and then the phase and the amplitude${I_1}$ of${U_1}$ are synthesized into the complex amplitude$U'_1$ . Then the angular spectrum propagation is used to get the complex amplitude${U_2}$ by the complex amplitude$U'_1$ , the real amplitude${I_2}$ is also used to replace the amplitude of the complex amplitude${U_2}$ to obtain$U'_2$ . Finally, the${U_1}$ is obtained by the angular spectrum propagation of$U'_2$ . Repeating this operation until the phase convergence is restored or reaching the preset counts of iteration. In this paper, the real amplitudes${I_1}$ and${I_2}$ are the amplitudes of the focal planes under green and blue illumination, respectively.In this paper, due to dispersion, the defocusing distance z in formula 2 is large, so the intensity difference method which is used to approximate the strength differential will cause a large error. The angular spectrum iterative algorithm is more dependent on the initial value. If the initial value is not selected properly, it is easy to converge to the local minimum. To solve these problems, a hybrid phase retrieval algorithm is used. The recovery result of TIE is taken as the initial value of the angular spectrum iteration algorithm, and then the angular spectrum iteration is used to iterate between the focusing plane and the defocusing plane until the final phase is obtained by convergence.
In the phase retrieval method based TIE and GS iterative, the accurate acquisition of intensity images is very important [16]. However, the acquisition of intensity images is usually realized by translating the CCD or the object manually or mechanically, which inevitably leads to the problems of slow speed and low accuracy. The CD-HPR method proposed in this paper has a good effect in solving this problem, which is mainly applied to single-lens system to retrieve phase.
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The imaging principle of obtaining intensity difference by chromatic dispersion in a single-lens system is shown in Fig.3. The object is placed in front of the lens,
${d_0}$ is the distance between the object and the lens,${d_i}$ is the distance between the image plane (CCD plane) and the lens.$U({x_0},{y_0})$ and$U({x_i},{y_i})$ are the complex amplitudes of the object plane and the image plane, respectively, which satisfy the following relationFigure 3. Schematic diagram of obtaining intensity difference by chromatic dispersion application in a single lens system
$$U({x_i},{y_i}) = \iint {h({x_i},{y_i};{x_0},{y_0})}U({x_0},{y_0}){\rm{d}}{x_0}{\rm{d}}{y_0}$$ (5) where
$h({x_i},{y_i};{x_0},{y_0})$ is the amplitude point-spread function.In the case of white light illumination, the preset green light with the center wavelength
${\lambda _{\rm{G}}}$ obtained by the green filter vertically irradiates the object, the measured intensity of image plane is$${I_{\rm{G}}}(x,y) = {\left| {{U_{\rm{G}}}({x_i},{y_i})} \right|^2}$$ (6) Then the blue light with the center wavelength
${\lambda _{\rm{B}}}$ is obtained by blue filter, the measured intensity of image plane is$${I_{\rm{B}}}(x,y) = {\left| {{U_{\rm{B}}}({x_i},{y_i})} \right|^2}$$ (7) ${I_{\rm{G}}}(x,y)$ and${I_{\rm{B}}}(x,y)$ are the in-focus intensities under different wavelengths, respectively. However, due to chromatic dispersion effect of an imaging system, in the same location, the intensity image obtained by green light is an in-focus image, and the intensity image obtained by blue light is a defocus image with respect to green light. Suppose the wavelength of green light is${\lambda _{\rm{G}}}$ , the refractive index through the lens is${n_{\rm{G}}}$ ,and the focal length through the lens is${f_{\rm{G}}}$ ; the wavelength of blue light is${\lambda _{\rm{B}}}$ , the refractive index through the lens is${n_{\rm{B}}}$ ,and the focal length through the lens is${f_{\rm{B}}}$ . By the Lens maker’s formula$$\frac{1}{{{f_{\rm{G}}}}} = ({n_{\rm{G}}} - 1)\left(\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}\right)$$ (8) $$\frac{1}{{{f_{\rm{B}}}}} = ({n_{\rm{B}}} - 1)\left(\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}\right)$$ (9) Where
${r_1}$ and${r_2}$ are respectively the radius of the left and right spheres of the lens. Because of${n_{\rm{G}}} \ne {n_{\rm{B}}}$ and${f_{\rm{G}}} \ne {f_{\rm{B}}}$ , the position of green and blue light imaging become different to produce an defocus distance$\Delta z$ . Below we derive the expression of the defocus distance$\Delta z$ from the imaging formula. The imaging formula is as follows$$\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$$ (10) where
$u$ ,$v$ and$f$ are object distance, image distance and focal length respectively. Considering the situation of green light firstly, assume that the object distance in the case of green light illumination is${u_{\rm{G}}}$ , image distance is${v_{\rm{G}}}$ , and${v_{\rm{G}}} = \dfrac{{{f_{\rm{G}}}{u_{\rm{G}}}}}{{{u_{\rm{G}}} - {f_{\rm{G}}}}}$ obtained by the imaging formula. Then change the green light to blue, the object distance${u_{\rm{B}}}$ will not change at this time, which is${u_{\rm{B}}}={u_{\rm{G}}}$ . By the imaging formula, the image distance${v_{\rm{B}}} = \dfrac{{{f_{\rm{B}}}{u_{\rm{G}}}}}{{{u_{\rm{G}}} - {f_{\rm{B}}}}}$ under the blue light. The defocus distance$\Delta z$ is the difference between the image distances under green and blue illumination, which is$$\Delta z = {v_{\rm{G}}} - {v_{\rm{B}}}{\rm{ = }}{u_{\rm{G}}}\left( {\frac{{{f_{\rm{G}}}}}{{{u_{\rm{G}}} - {f_{\rm{G}}}}}{\rm{ - }}\frac{{{f_{\rm{B}}}}}{{{u_{\rm{G}}} - {f_{\rm{B}}}}}} \right)$$ (11) Substitute (8) and (9) into (11), and get
$$\Delta z{\rm{ = }}{u_{\rm{G}}}\left( {\frac{1}{{{u_{\rm{G}}}({n_{\rm{G}}} - 1)(\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}) - 1}} - \frac{1}{{{u_{\rm{G}}}({n_{\rm{B}}} - 1)(\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}) - 1}}} \right)$$ (12) In particular, when
${u_{\rm{G}}} = 2{f_{\rm{G}}}$ which means the object is placed at the double focal length of the left side of the single lens system$$\Delta z{\rm{ = }}2{f_{\rm{G}}} - \frac{{2{f_{\rm{G}}}{f_{\rm{B}}}}}{{2{f_{\rm{G}}} - {f_{\rm{B}}}}}{\rm{ = }}4{f_{\rm{G}}}(\frac{{{f_{\rm{G}}} - {f_{\rm{B}}}}}{{2{f_{\rm{G}}} - {f_{\rm{B}}}}})$$ (13) Substitution of (6) and (7) into (2) gives
$$\frac{{\partial I}}{{\partial z}} \approx \frac{{{I_{\rm{G}}}({x_i},{y_i}) - {I_B}({x_i},{y_i})}}{{\Delta z}}$$ (14) the phase
${\varphi _0}(x,y)$ of the object can be obtained by substituting (14) into (1)$${\varphi _0}(x,y) = \frac{{2\pi }}{\lambda }{\Im ^{ - 1}}\left\{ {{{\left[ {2{\pi ^2}(f_x^2 + f_y^2)} \right]}^{ - 1}}\Im \left[ {{{\partial I} / {\partial z}}} \right]} \right\}$$ (15) In addition, it is worth noting that the phase retrieval of single-lens imaging system introduces additional phase aberration of the quadric sphere. In the experiment, the phase
${\varphi _0}(x,y)$ is compensated by the following phase mask.$$\varPhi (m,n) = \exp\left[ {\frac{{ - i\pi }}{{\lambda D}}({m^2}\Delta {\xi ^2} + {n^2}\Delta {\eta ^2})} \right]$$ (16) where
$m \times n$ is size of phase mask,$\Delta \xi $ and$\Delta \eta $ are discretized sampling intervals D is an adjustable parameter that compensates for the wavefront curvature. The compensated phase${\varphi _i}(x,y)$ is as follows.$${\varphi _i}(x,y) = {\varphi _o}(x,y) - C\varPhi ({{m}},{{n}})$$ (17) where
$C$ is a constant and the value is adjusted according to the results in the experiment.Considering the long propagation distance of the light field during actual imaging and the large defocusing distance, the hybrid phase retrieval algorithm is used to get the improved retrieval phase. In this paper, the phase recovered by TIE is used as the initial phase of the angular spectrum iterative algorithm and the process of hybrid phase retrieval algorithm is shown in Fig.4. Then the angular spectrum propagation is continuously used between the focusing and defocusing planes, and the intensity of the complex amplitude obtained by the angular spectrum propagation is replaced by the true intensity of the focusing and defocusing planes each time, When the preset number of iterations is reached or the phase is converged, a better retrieval phase
$\varphi $ is obtained finally. -
The relevant simulation experiments are given to test the method according to the theory described above. It is assumed that a pure-phase object with phase shift ranging from 0 rad to 2
$\pi $ rad, is illuminated by a monochromatic plane wave, as shown in Fig.5(a). For simulation purposes that the following parameters are chosen: image size N×M = 256×256, pixel size dx×dy = 4 μm×4 μm the focal length of the lens is f = 150 mm, object distance and image distance under green illumination are 2f = 300 mm, and${n_{\rm{G}}}= 1.527$ ,${n_{\rm{B}}}=1.519$ , then the defocus distance$\Delta z =$ 9.2 mm by (12). The in-focus and the defocus intensity distributions shown in Fig. 5(b) and Fig. 5(c) are simulated under green light and blue light with the center wavelengths${\lambda _{\rm{G}}} =$ 532 nm and${\lambda _{\rm{B}}} =$ 470 nm respectively. The two intensity images are calculated by (15) to obtain the initial phase distribution in the image plane, then the mask for compensation in (16) is used for this phase shown in Fig.5(d). We can see that the compensated phase is still fuzzy due to the large propagation distance of the light field and long defocusing distance in the experiment. We use angular spectrum iterative algorithm to improve the phase, we can see that the phase is more clear shown in Fig. 5(e). The gray values of the Fig. 5(a) and (e) are selected for comparison, as shown in Fig.5(f), and its curve fitting is well.In order to further verify the accuracy of the retrieved results, here the RMSE defined in (18) is adopted.
$$ RMSE = \sqrt {\dfrac{{\displaystyle\sum_{x,y} {{{\left[ {\varphi (x,y) - {\varphi _{ex}}(x,y)} \right]}^2}} }}{{M \times N}}} $$ (18) Where
$\varphi (x,y)$ and${\varphi _{ex}}(x,y)$ represent the recovered phase and the original phase, respectively, and the value of RMSE between them is 0.107 6.In order to compare the accuracy of TIE and hybrid algorithms, the experimental results of TIE and hybrid algorithms based on dispersion are presented in Fig.6. Fig.6(a) is the simulated original phase, Fig.6(b) and Fig.6(c) are the retrieval phases obtained by TIE and hybrid algorithm respectively, and their mean square error with the original phase is 0.267 5 and 0.098 7 respectively.It can be seen that the hybrid algorithm significantly improves the accuracy.The numerical experimental results are sufficient for the correctness and effectiveness of CD-HPR in a single-lens system.
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The experimental arrangement used to test the CD–HPR is illustrated in Fig.7. LED white light (GCI-060411, Daheng optics, China) as a light source is used. A bandpass filter of the known central wavelength is placed before the white LED, a variable aperture is placed between the two in order to control the range of the light field and to make the light field strictly symmetrical about the optical axis. A plane wave is obtained by collimating lens (f = 150 mm). The sample is a micro-lens array, which is consist of some single lens made of silicone oil with refractive index of 1.579, the filling material surrounding the single lens is the PDMS with refractive index of 1.403, the maximal thickness of the lens is 1.15 mm. The focal length of a lens in a single-lens system is f = 150 mm, and a CCD (1 280 pixel × 1 024 pixel, pixel size 5.2 μm × 5.2 μm) was built in the image plane.
First, the preset illumination wavelength (green light) is obtained by bandpass filter with the central wavelength at 532 nm and the full width at half maximum of 22 nm. In this case, the in-focus image can be captured at image plane as shown in Fig.8(a). Then the filter is replaced by the filter with the central wavelength at 470 nm to get blue light, the defocus image is acquired by CCD in the same place as shown in Fig.8(b). Finally, the phase recovered by CD-HPR is shown in Fig. 8(c). The red line in Fig.8(c) is converted from phase to thickness by (4), as shown in Fig.8(d). And 3D display can also be obtained as shown in Fig.8(e). The maximum thickness of the lens measured by the CD-HPR method is approximately 1.19 mm, close to the actual thickness, and the entire experiment process only needs to replace different filters, which verifies the effectiveness of CD-HPR in a single lens system.
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摘要: 相位恢复是利用能观测到的强度信息恢复原始相位信息。强度传输方程(TIE)作为一种传统的非干涉相位恢复技术,只需通过测量至少两个相近平面的强度信息即可计算出相位信息。这种方法通常需要通过移动被测物体或摄像机来获取强度图像,不可避免地会产生机械误差。提出了一种新的相位恢复方法:与色散融合的混合相位恢复算法(CD-HPR)。通过设置不同波长的光通过单透镜系统得到物体在同一位置成的像,这样不需要机械运动就能获得聚焦和散焦图像强度,然后结合散焦量与波长之间的关系计算出散焦量,再用强度传输方程计算初始相位信息。角谱迭代算法的使用较好地改进了初始相位值。在仿真实验中,该方法恢复的相位与原始相位之间的均方差为0.1076;同时,通过实验恢复了透镜阵列的相位,实验结果与实际参数的误差为3.4%,证明了该方法的正确性和有效性。该方法扩展了传统方法要求光源为单色的局限性,提高了计算精度。Abstract: Phase retrieval is to recover the original phase information by using the intensity information obtained from observation. Transport of intensity equation (TIE), as a traditional non-interference phase retrieval technique, can compute the losing phase information from only a minimum of two intensity measurements at closely spaced planes by solving the equation. This method usually requires the acquisition of intensity images by moving the object to be tested or CCD, which inevitably results in mechanical errors. A new phase retrieval method called chromatic dispersion-hybrid phase retrieval (CD-HPR) was proposed. The object was imaged at the same position by setting different wavelengths of light after passing through the single-lens system, in-focus and defocus intensity images were obtained without mechanical movement, and the initial phase information of an object was calculated from the phase retrieval technique based on TIE by combining the relationship between the defocus amount and the wavelength. Next angular spectrum iteration was used to improve the initial phase information. In this simulation, the RMSE between the phase recovered by this method and the original phase was 0.1076. At the same time, the phase of the lens array was restored by experiment. The error between the experimental result and the real parameter is 3.4%, which proves the correctness and effectiveness of the proposed method. This method extends the limitation of the traditional method that requires the light source to be monochromatic and improves the calculation accuracy.
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