Optical phases, as noted earlier, cannot be directly measured with electronic devices. But luckily optical waves do carry some innate properties that make their phases manageable with algorithmic manipulations, that is, Fourier transforming the near-field distribution of the wave field to be determined corresponds to their far-field appearance. With this mathematical struc-ture, algorithmic phase retrieval contributes a way for recovering the phase given the measurement of the optical far-field pattern and some other prior
5.2. Iterative Phase Retrieval
knowledge. In other words, there essentially exists a phase distribution that, when combined with the known Fourier modulus data, satisfy all the prior constraints [70].
The most dominant class of phase-retrieval methods is practicing an it-eration scheme that modifies the object in each cycle until the progressively improved Fourier moduli are good enough and all the prior constraints are satisfied. Before an acceptable retrieval is reached, many alternate projec-tions onto different constrained spaces should be performed to facilitate con-vergence.
5.2.1 Elementary Projectors
The phase problem can be generalized to a problem defined as [75]
Problem. Given two constraint sets S1 and S2, find x such that x ∈ S1∩ S2.
In almost all practical applications of the physical world we really live in, both S1 and S2 can be assumed to be subsets of a finite-dimensional Hilbert space H. The structure of the constraint sets decides whether the problem subject to these sets difficult to solve or not. For example, searching for the solution belonging to S1∩ S2 will not be thought a challenge provided that both S1 and S2 are linear constraint spaces. Next, let us move on to the definition of the basic operation – the projection onto a set.
Definition. Let S ∈ H be a closed constraint set. A projection PS from an x ∈ H onto S is a mapping that finds a point PS(x) ∈ H such that kx − PS(x)k is minimized.
Looking for an answer point in S1∩ S2 is undoubtedly equivalent to finding out the convergence point of the sequentially alternate projections onto S1 and S2 from a given initial guess point. Since x ∈ S if and only if x = PS(x), all solutions to Problem satisfy the condition
xsol= PS 1(xsol) = PS 2(xsol) . (5.1) Based on this notion, iterative phase-retrieval algorithms are all to reach a final convergence point on S1∩ S2 through successive alternate projections.
There are many operators that perform fundamental projections, referred to as elementary projectors [76]. Here, we just name a few for helping com-prehend their basic operational principles in the language of vector space projection. The first example is the nonnegativity mapping.
Projector 1 (Nonnegativity).
If ρ(n) is the value of pixel n in the object domain, then
ρ′(n) =
For real-valued objects one can impose nonnegativity constraint, setting all negative pixels values to zero and leaving the positive pixel values unchanged.
Objects of this kind never assume negative values. For instance, the inten-sity of a light wave, the instantaneous magnitude of a voltage or current signal, and the probability function of a quantum system, etc. are always nonnegative.
Projector 2 (Support).
Let B be the support. Support projection is then the mapping
ρ′(n) =
The assumption that the non-support values are 0 is not accurate all the way. In some applications [75], the exit wave emerging from a small isolated specimen immersed in a largely-extended incoming wave is certainly not zero outside the support. This situation should be noted before employing sup-port mapping.
Projector 3 (Modulus).
Let I be the measured intensity profile. Simple modulus projection is the mapping
ρ′(n) = F−1√
I Fρ (n)
|Fρ (n)|.
The modulus projector is a composite operator consisting of two Fourier transformations and one modulus replacement. During alternate Fourier projections, the newly derived phase information is preserved and fed into the next iteration. In our proposed phase-retrieval algorithm that will show up in the later content, only this projector is employed to retrieve the accurate phase footprint of the hologram.
5.2. Iterative Phase Retrieval
Figure 5.1: (a) Image reconstructed from a diffraction pattern with correct phases. (b) Inter-projections between two identical holograms obtained from Figure 3.5 (a) or Figure 3.5 (b).
5.2.2 Single-Hologram Inter-Projections Algorithm
The twin image problem in holography can be ascribed to the loss of the diffraction phase during recording a hologram. The intensity-only mea-surement provides two solutions in the conventional image reconstruction method. When the object is numerically reconstructed in digital Fourier holography, the obscured image in the focusing plane arises from the in-terference of the object field and the diffraction wave of the twin object in inversely symmetric form. In Gabor holography illustrated in Figure 3.5 (b), the reconstructed object is superimposed by the defocused image of an iden-tical twin located on the opposite side of the hologram and at twice the reconstruction distance away from the original object. By providing the cor-rect phase information to the measured hologram, only the original object is restored as it is the inverse process of object diffractions (Figure 5.1 (a)).
The object can also be reconstructed from the hologram on the other side by taking the complex conjugate of the diffraction field. As shown in Fig-ure 5.1 (b), the diffraction images on two opposite holographic planes have the same intensity pattern if the test object only modulates the amplitude of the incident wave field [77]. The measured intensity image can thus serve as the constraint in both holographic planes and can be applied to retrieve the diffraction phase.
To obtain the lost diffraction phase information in holograms, we itera-tively propagate the wavefield back and forth between the two holographic planes separated by twice the reconstruction distance and apply the field am-plitude of the measured hologram as the only constraint. Details of our phase retrieval algorithm between iterations are summarized in the following steps.
The first complex diffraction wavefield consists of the field amplitude from the recorded hologram and a randomly assigned phase distribution. The
syn-thesized wavefield numerically propagates to the opposite diffraction plane.
Since the intensity pattern on the both planes has to be identical, we replace the modulus of the resulting wavefield with the recorded field amplitude and keep the obtained phase to form the new complex diffraction pattern. The new wavefield then backpropagates to the original diffraction plane. The modulus of the computed wavefield is modified again by the amplitude con-straint. The inter-projection of the measured Fourier modulus between the two planes is performed by computing the Rayleigh-Sommerfeld diffraction integral with a simple double fast Fourier transform method, whose formalism takes the form of Eq.(4.22) as an implementation of the Fresnel convolution method in 4.2.2. The iterative procedure repeats until the error between the recorded hologram and the computed squared modulus is smaller than a designated threshold. Once the lost diffraction phase is recovered, the exact object can be reconstructed from the complex diffraction pattern through the inverse process of Eq.(3.18) or Eq.(3.19). It is worth noting that only one elementary projector, modulus mapping in Projector 3, involves in the entire phase retrieving process, nothing more.