Research ArticleAPPLIED SCIENCES AND ENGINEERING

Rapid laser solver for the phase retrieval problem

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Science Advances  04 Oct 2019:
Vol. 5, no. 10, eaax4530
DOI: 10.1126/sciadv.aax4530
  • Fig. 1 Basic DDCL arrangement for rapid phase retrieval.

    (A) Calculated scattered intensity distribution from the object (essentially the Fourier intensity distribution) is applied onto an SLM, which is incorporated into a ring degenerate cavity laser that can support up to 100,000 degenerate transverse modes. A mask shaped as the object boundaries (compact support) at the Fourier plane filters out extraneous modes that do not match the compact support. With this laser arrangement, the lasing process yields a self-consistent solution that satisfies both the scattered intensity distribution shown in (B) and the compact support constraint. (C) The reconstructed object intensity appears at the compact support mask and is imaged onto the camera. a.u., arbitrary units.

  • Fig. 2 Experimental results for real-valued centrosymmetric objects.

    Column (A) Intensity distributions of the actual objects. Column (B) Detected intensity distribution of the reconstructed objects, using a circular aperture as compact support. Column (C) Fourier intensity distributions at the SLM.

  • Fig. 3 Experimental results for complex-valued objects.

    Column (A) Intensity (brightness) and phase (hue) distributions of the actual objects. Column (B) Detected intensity distribution of the reconstructed objects, using mainly a circular aperture as compact support. Column (C) Fourier intensity distributions at the SLM. The first row shows an object with uniform phase distribution with a reconstruction fidelity of 0.91. The second (third) row shows the same object with arbitrary centrosymmetric (asymmetric) phase distribution with a reconstruction fidelity of 0.89 (0.81). The fourth row shows a noncentrosymmetric object with random asymmetric phase distribution and noncircular compact support with a reconstruction fidelity of 0.91.

  • Fig. 4 Experimental and quantitative results for fidelity as a function of object complexity.

    Top: Representative intensity distributions of objects with 4, 16, and 30 spots. Column (A) Intensity (brightness) and phase (hue) distributions of the actual objects. Column (B) Detected intensity distribution of the reconstructed objects, using a circular aperture as compact support. Column (C) Calculated Fourier intensity distributions applied to control the SLM. Column (D) Detected corresponding Fourier intensity distributions after modifications by SLM properties. Bottom: Quantitative fidelity values of the Fourier intensity distributions (blue) and the reconstructed object intensity distributions (red) as a function of the number of spots in the object (4 to 30). Inset: Fidelity values of the reconstructed object intensity distributions as a function of the fidelity values of the Fourier intensity distributions for all the measurements.

  • Fig. 5 Experimental results demonstrating the qualitative effect of tightness and asymmetry of compact supports.

    Column (A) Intensity distribution of the actual objects. Column (B) Detected intensity distribution of the reconstructed objects, using a circular aperture as compact support. Column (C) Detected intensity distribution of the reconstructed objects, using a square aperture as tight compact support (top row) and a circular aperture with a wedge as asymmetric compact support (bottom row). Column (D) Fourier intensity distributions at the SLM.

  • Fig. 6 Experimental quantitative results for reconstruction fidelity as a function of the compact support radius of the aperture normalized by the object size.

    Insets: Typical reconstructed object intensity distributions. (A) Compact support radius is 87% of the object radius. (B) Object radius is equal to compact support radius. (C) Compact support radius is 152% of the object radius.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/10/eaax4530/DC1

    Section S1. Detailed experimental arrangement

    Section S2. Convergence time to reach a solution

    Section S3. Simulation results

    Section S4. Runtime comparison to the RAAR phase retrieval algorithm

    Section S5. Phase measurement and reconstruction

    Fig. S1. Detailed experimental digital ring degenerate cavity laser arrangement.

    Fig. S2. Experimental arrangement with corresponding results for demonstrating rapid solutions.

    Fig. S3. Representative simulation results of inverse problem solutions with a DDCL.

    Fig. S4. Representative simulation results with phase aberrations.

    Fig. S5. Reconstruction fidelity in the RAAR algorithm.

    Fig. S6. Phase measurement methods.

    References (4144)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Detailed experimental arrangement
    • Section S2. Convergence time to reach a solution
    • Section S3. Simulation results
    • Section S4. Runtime comparison to the RAAR phase retrieval algorithm
    • Section S5. Phase measurement and reconstruction
    • Fig. S1. Detailed experimental digital ring degenerate cavity laser arrangement.
    • Fig. S2. Experimental arrangement with corresponding results for demonstrating rapid solutions.
    • Fig. S3. Representative simulation results of inverse problem solutions with a DDCL.
    • Fig. S4. Representative simulation results with phase aberrations.
    • Fig. S5. Reconstruction fidelity in the RAAR algorithm.
    • Fig. S6. Phase measurement methods.
    • References (4144)

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