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Fig. 7. Holographic reconstructions of a) a USAF resolution chart and b) an MEMS chip measured in reflection mode. The smallest resolved elements( group 9, element 1) of the resolution target have a line width of 0.98 lm. The reconstruction of the MEMS achieves a resolution of at least 2 lm and shows that capacitors and springs in the single-digit micrometer range can be resolved across a field of view of around 2.5 mm. such as lensless inline holography using iterative phase retrieval have the same robustness as our proposed technique, computation times can also be high, in the second to minutes range [ 31, 32 ]. However, the main drawback of these techniques, is the limitation of resolution to the pixel pitch, which we can easily overcome. Multi-shot holography and pixel-super-resolution techniques instead are able to reach resolutions down to the diffraction limit and reach real time reconstruction using deep neural networks [ 33, 34, 35 ]. Since these techniques need multiple measurements to achieve sub-pixel resolution, they however do not have same robustness as single shot measurements, which limits their applicability to transient processes.
Fig
. 8. Modulation transfer function( MTF), representing the mean Michelson contrast, evaluated exemplarily for the reconstructions in transmission and reflection mode of the USAF resolution charts shown in Figures 6a and 7a. The MTF of the transmission measurement remain relatively high for all available resolution groups. The MTF in reflection mode is significantly lower, likely due to wavefront errors in the reference wave. The dip around 200 line pairs per mm in reflection mode can likely be attributed to degradations of elements close to the edge of the illuminated aperture and is otherwise not significant. The error bars represent the standard deviation across individual bars in Figures 6a and 7a respectively.
around 0.9 up to resolutions of 456 line pairs per mm( lp / mm) and drops only to 0.71 ± 0.04 for the smallest available line pairs( 645 lp / mm, 0.78 lm linewidth). This indicates that a higher resolution might be achievable if smaller elements were available. The MTF in reflection mode however is significantly lower with values of 0.22 ± 0.02 at 512 lp / mm. This is likely due to wavefront errors in the reference wave.
A comparison with the state of the art of lensless microscopic techniques shows, that, while single shot techniques
4 Conclusion and outlook
We have developed a novel reconstruction process for diffraction-limited lensless holographic microscopy utilizing the inherent sampling pitch transformation of the Fresnel propagation for an initial spatially limited reconstruction of the hologram. This initial reconstruction, however, will always be severely degraded by the Fresnel approximation and is consequently only used to interpolate the hologram by zero-padding the Fresnel reconstruction and propagating the zero-padded signal back into the hologram plane. The interpolated hologram with a pixel pitch corresponding to the diffraction limit can now be propagated into the object plane again using any exact wave field propagator, such as the angular spectrum method.
Using single-shot measurements, we achieve a resolution of approx. 1.5 lm across a field of view of approx. 7 7mm 2, corresponding to a space bandwidth-product of 20 million, surpassing many conventional microscope objectives. A resolution below 1 lm, close to the diffraction limit requires higher signal-to-noise ratios which we achieve by using temporal instead of spatial phase shifting in reflection mode and by limiting the illuminated object area and