J. Eur. Opt. Society-Rapid Publ. 22, 14( 2026) 135
Fig. 4. Schematic illustrations of different implementations of the lensless holographic microscope in transmission mode( a + c) and reflection mode( b + d + e). a) and b) show the main components of the setup and the light paths similar to that of Fourier holography. The reference source point is off-axis to allow for spatial phase shifting. In reflection mode b) the object illumination( and potentially the reference wave) are generated by a lens, placed directly in front of the beam splitter. c-e) show concrete technical implementations of the lensless microscope. c) is a Mach – Zehnder interferometer implementation for transmission measurements. The polarization in both arms is aligned by a k / 2 plate and polarizers. d) shows a Mach – Zehnder interferometer implementation in reflection mode, while e) shows a Michelson interferometer implementation for reflection mode where the reference wave is generated by the same lens as the object illumination being reflected by a reference mirror.
While this procedure will result in a poor, blurry reconstruction which does not achieve full diffraction-limited resolution, the energy will still be confined close to the object support. Consequently, since the cross-term is band-limited the interpolated M does not feature any spurious defects such as aliasing. The result in Figure 3d will be an interpolated representation of M( x, y) across a sampling grid having the size of the sensor but the pitch of the object plane.
In the second step, we can now modulate the densely sampled M with the high frequent but known analytical structure of the reference wave R. This yields M R = U H R * R = U H( assuming that the amplitude of the reference wave does not vary too much, so we can set | R | 1), thus digitally fully recreating the complex wave field U H in the sensor domain across the dense grid shown in Figure 3e. Now, we can simply employ the angular spectrum method [ 25 ] to yield an exact reconstruction of the complex wave field U O shown in Figure 3f.
To conclude, the two reconstruction steps do not include any approximations but are mathematically precise applications of the Rayleigh – Sommerfeld diffraction integral. Firstly, the sinc-interpolation of the complex hologram in Fourier space can not introduce any artifacts such as aliasing, since the complex hologram is band-limited, due to the Fourier holography setup. Secondly, the interpolated hologram is modulated with the analytical structure of the reference, while the subsequent propagation of the interpolated hologram via the angular spectrum method is mathematically equivalent to the application of the Rayleigh – Sommerfeld diffraction integral. This can exemplarily be seen in the exact reconstruction of the Siemens star shown in Figure 3f.
Because of the interpolation in the first reconstruction step, the computational complexity and runtime greatly depend on the desired feature size that shall be evaluated. For a N times N sized image and an interpolation ratio of q the reconstruction process consists of 1 FFT of a N N matrix, 3 FFT of a q N q N matrix and at least 2 multiplications of two q N q N matrices. The algorithm is implemented in MATLAB to run on a CPU.