J. Eur. Opt. Society-Rapid Publ. 22, 14( 2026) 133
Fig. 2. Wigner-space diagrams as well as amplitude and phase images of a wave field throughout the lensless imaging process. m is the bandwidth of the signal, while x is its lateral extent. a) Amplitude of wave field U O in the object plane, its support in phase space defines the space bandwidth-product( SBP) of the wave field. b) Phase of propagated wave field U H, the central box represents the SBP of a camera sensor. The propagation shears the support in phase space. High frequencies are recorded as badly sampled alias frequencies. c) Phase of propagated spherical reference wave R. d) Phase of the measured cross term M = U R *, the spherical reference wave aligns the SBP of the coherence function and the sensor.
The inverse Fresnel propagation formula has great advantages with respect to diffraction limited lensless imaging. It transforms a complex hologram sampled across the sensor grid, into an object wave field across a different sampling grid, whose pitch Dq = kz / L is approximately consistent with the diffraction limit d [ 1 ]. Hence, the Fresnel propagation formula inherently provides diffraction limited imaging. Unfortunately, since it is rooted in the Fresnel approximations, it only yields acceptable results for numerical apertures NA < 0.1, which makes it a mediocre choice for microscopy applications.
Precise reconstruction beyond this limit and without further pre-knowledge is not a simple task. The main problem is the high frequent chirp S 1( x, y) inequation( 5), which is substantially undersampled in the sensor domain. This, therefore, impedes the application of precise methods such as wave propagation by means of plane wave decomposition( or angular spectrum method). To realize this, we can write down the gradient of the phase /( x, y)= arg { S 1 }, e. g., in x- direction r x / ðx; yÞ ¼k x = z; ð7Þ
with k = 2p / k being the wave vector. If we insert x = L S / 2 to determine the highest phase gradient close to the sensors edges, we yield
r x / ðx; yÞj x¼LS = 2 ¼ 2pL S 2kz 2p 1
2d ¼ m N; ð8Þ
where we have used the Abbe diffraction limit equation( 1) in the last step. As a result, we obtain approximately the Nyquist frequency m N of the dense sampling grid in the object plane. Hence, S 1 is undersampled by a factor of approximately q in the sensor domain.
This has two major consequences which can be exemplified by the Wigner-space diagrams in Figure 2 that represent the space-bandwidth distribution of differently propagated wave fields( e. g., Fig. 2a shows the band-limited space-bandwidth of a wave field in the object plane): Firstly, as depicted in the Wigner-space diagram in Figure 2b, we cannot experimentally sample the wave field itself using the camera. The use of e. g., a plane reference wave would thus result in fringes too dense to be sampled by the finite-sized pixels. The pixels would average the high frequent parts of the interference pattern. Secondly, as shown in equation( 8), if we do not know the entire wave field across the dense sampling grid, we cannot precisely reconstruct it. However, the wave field can be approximated across such a dense sampling grid through interpolation of the sparse hologram [ 27 ].
2.2
Lensless Fourier holography
The solution to the first problem regarding sampling has been widely reported in the state of the art under the term Lensless Fourier Holography [ 28, 29 ]. The basic idea is to use a spherical reference wave R( x, y) which appears to originate from the center of the object, as shown in Figure 2c. This can be realized with a beam splitter for example. Since S 1 in equation( 6) can be regarded as a parabolic approximation of a point source located in the center of the object as well, we can assume R S 1 for small numerical apertures. The measured cross term becomes M ¼ U H R U H S
1 which can be sampled as shown in Figure 2d. In this situation, the reconstruction task reduces to a simple Fourier transform of M( x, y) as seen from equation( 5), hence the name of the technique. The origin of the reference wave can also be laterally shifted to realize off-axis holography due to modulation of the interference pattern with a spatial carrier frequency. In this case, no additional phase shifting is required( single shot operation) to separate real and conjugate image. Additionally, the source of the reference wave, e. g., a fiber tip, can be placed next to the object, so that the beam splitter can be omitted.