• Infrared and Laser Engineering
  • Vol. 49, Issue 10, 20200017 (2020)
Hong Cheng, Yong Liu, Jiajie Hu, Xiaolong Zhang, Huilong Deng, and Sui Wei
Author Affiliations
  • School of Electronics and Information Engineering, Anhui University, Hefei 230601, China
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    DOI: 10.3788/IRLA20200017 Cite this Article
    Hong Cheng, Yong Liu, Jiajie Hu, Xiaolong Zhang, Huilong Deng, Sui Wei. Hybrid phase retrieval with chromatic dispersion in single-lens system[J]. Infrared and Laser Engineering, 2020, 49(10): 20200017 Copy Citation Text show less

    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.
    $ - \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)

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    ${\left. {\frac{{\partial I(x,y,z)}}{{\partial z}}} \right|_{z = {z_0}}} \approx \frac{{I(x,y,{z_0}) - I(x,y,{z_0} - \Delta z)}}{{\Delta z}}$(2)

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    $\varphi (x,y,{z_0}) = k{\Im ^{ - 1}}\left[ {{{\left[ {2{\pi ^2}(f_x^2 + f_y^2)} \right]}^{ - 1}}\Im \left[ {{{\partial I(x,y,z)} / {\partial z}}} \right]} \right]$(3)

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    $h(x,y,{z_0}) = \frac{\lambda }{{{n_o} - {n_m}}}\frac{{\phi (x,y,{z_0})}}{{2\pi }}$(4)

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    $U({x_i},{y_i}) = \iint {h({x_i},{y_i};{x_0},{y_0})}U({x_0},{y_0}){\rm{d}}{x_0}{\rm{d}}{y_0}$(5)

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    ${I_{\rm{G}}}(x,y) = {\left| {{U_{\rm{G}}}({x_i},{y_i})} \right|^2}$(6)

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    ${I_{\rm{B}}}(x,y) = {\left| {{U_{\rm{B}}}({x_i},{y_i})} \right|^2}$(7)

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    $\frac{1}{{{f_{\rm{G}}}}} = ({n_{\rm{G}}} - 1)\left(\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}\right)$(8)

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    $\frac{1}{{{f_{\rm{B}}}}} = ({n_{\rm{B}}} - 1)\left(\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}\right)$(9)

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    $\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$(10)

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    $\Delta z = {v_{\rm{G}}} - {v_{\rm{B}}}{\rm{ = }}{u_{\rm{G}}}\left( {\frac{{{f_{\rm{G}}}}}{{{u_{\rm{G}}} - {f_{\rm{G}}}}}{\rm{ - }}\frac{{{f_{\rm{B}}}}}{{{u_{\rm{G}}} - {f_{\rm{B}}}}}} \right)$(11)

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    $\Delta z{\rm{ = }}{u_{\rm{G}}}\left( {\frac{1}{{{u_{\rm{G}}}({n_{\rm{G}}} - 1)(\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}) - 1}} - \frac{1}{{{u_{\rm{G}}}({n_{\rm{B}}} - 1)(\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}) - 1}}} \right)$(12)

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    $\Delta z{\rm{ = }}2{f_{\rm{G}}} - \frac{{2{f_{\rm{G}}}{f_{\rm{B}}}}}{{2{f_{\rm{G}}} - {f_{\rm{B}}}}}{\rm{ = }}4{f_{\rm{G}}}(\frac{{{f_{\rm{G}}} - {f_{\rm{B}}}}}{{2{f_{\rm{G}}} - {f_{\rm{B}}}}})$(13)

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    $\frac{{\partial I}}{{\partial z}} \approx \frac{{{I_{\rm{G}}}({x_i},{y_i}) - {I_B}({x_i},{y_i})}}{{\Delta z}}$(14)

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    ${\varphi _0}(x,y) = \frac{{2\pi }}{\lambda }{\Im ^{ - 1}}\left\{ {{{\left[ {2{\pi ^2}(f_x^2 + f_y^2)} \right]}^{ - 1}}\Im \left[ {{{\partial I} / {\partial z}}} \right]} \right\}$(15)

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    $\varPhi (m,n) = \exp\left[ {\frac{{ - i\pi }}{{\lambda D}}({m^2}\Delta {\xi ^2} + {n^2}\Delta {\eta ^2})} \right]$(16)

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    ${\varphi _i}(x,y) = {\varphi _o}(x,y) - C\varPhi ({{m}},{{n}})$(17)

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    $ RMSE = \sqrt {\dfrac{{\displaystyle\sum_{x,y} {{{\left[ {\varphi (x,y) - {\varphi _{ex}}(x,y)} \right]}^2}} }}{{M \times N}}} $ (18)

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    Hong Cheng, Yong Liu, Jiajie Hu, Xiaolong Zhang, Huilong Deng, Sui Wei. Hybrid phase retrieval with chromatic dispersion in single-lens system[J]. Infrared and Laser Engineering, 2020, 49(10): 20200017
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