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Under the scalar paraxial approximation, an optical wavefield is considered to be complex function dependent on position; i.e., at a given location in space the optical field is a complex value with an intensity and phase. The optical wavefield propagates through space and can be modeled using the Fresnel transform. Lenses, apertures, and other optical elements can be used to control and manipulate the wavefield and to perform different types of signal processing operations. Often these optical systems are described theoretically in terms of linear systems theory leading to a commonly used Fourier optics framework. This is the theoretical framework that we will assume in this manuscript. The problem which we consider is how to recover the phase of an optical wavefield over a plane in space. While today it is relatively straightforward to measure the intensity of the optical wavefield over a plane using CMOS or CCD sensors, recovering the phase information is more complicated. Here we specifically examine a variant of the problem of phase retrieval using two intensity measurements. The intensity of the optical wavefield is recorded in both the image plane and the Fourier plane. To make the analysis simpler, we make a series of important theoretical assumptions and describe how in principle the phase information can be recovered. Then, a deterministic but iterative algorithm is derived and we examine the characteristics and properties of this algorithm. Finally, we examine some of the theoretical assumptions we have made and how valid these assumptions are in practice. We then conclude with a brief discussion of the results.

Visible light is an electromagnetic field with a wavelength that ranges roughly from ultraviolet (300 nm) to infrared (800 nm). An electromagnetic field is vectorial in nature with three electric and magnetic orthogonal field components. When designing a microwave radio receiver (long wave wavelengths relative to visible light) one must consider how these different magnetic and electric field components interact with each other and with conducting strips of metal and dielectric substrates that are used for impedance matching, mixing, and waveguiding. The resulting equations (that follow from Maxwell’s equations) are quite complex and require intensive numerical calculation procedures since the different vectorial components of the EM field interact with each other forming coupled vectorial equations [

Fortunately when examining optical problems in the visible regime it is often possible to make the scalar approximation. This means that we can assume that the different vectorial components of the optical wavefield do not “see” each other and hence each vectorial component can be treated independently as a separate scalar problem. This approximation is very useful and quite accurate and can be used to analyze a wide range of important and practical optical problems.

Therefore in this manuscript we assume that a scalar description of light propagation is valid and use the Fresnel transform to relate an optical field in one plane to that in another plane, where the optical planes are separated from each other by an axial distance

We note that in this theoretical framework if we know the intensity and phase of the optical field in a give optical plane, it is then possible to calculate (numerically or under certain circumstances analytically) the field distribution in any other plane using the LCT [

In a practical sense then it is important to be able to measure experimentally both the intensity and the phase of an optical wavefield in a given plane. Nowadays high quality low noise CCD and CMOS sensors are commercially available and widely used. With these devices it is relatively straightforward to measure the intensity of an optical field in a particular plane. We will now briefly discuss several techniques for indirectly measuring the phase distribution of the field.

Digital holography (DH) was first proposed by Gabor himself in [

These approaches are quite different in nature and require the capture of two separate intensity distributions. Usually these distributions are captured in an image plane and in a slightly defocused plane. Using the so-called transport of intensity equation, see, for example, [

Finally we turn our attention to iterative phase retrieval approaches. Here usually two or more intensity distributions are captured in different LCT domains, or in physical optics terms the intensity is captured in several different optical planes. Often these different optical planes are separated from each other by sections of free-space and combinations of lenses, but also in the image and corresponding Fourier transform planes. The origin of these types of algorithms dates back to the iterative Gerschberg-Saxton (GS) approach first published in the 70s and was further developed Fienup et al. [

The image-Fourier plane (GS) algorithm works as follows:

One starts with an initial guess of the phase distribution in the image plane

We then numerically calculate the Fourier transform of this distribution and compare the calculated intensity in the Fourier plane to the experimentally measured result. The calculated intensity distribution will differ from the experimentally measured distribution because our initial guess at the phase distribution will not be correct

At this stage we replace the incorrect numerically calculated intensity distribution with the correct experimentally measured intensity, while retaining the phase values and numerically propagate the distribution back to the image plane

We do not expect the calculated image intensity to be the same as the measured intensity distribution. Again we replace the numerically calculated distribution with the measured distribution

Steps (2)-(4) are repeated until the algorithm converges, i.e., when there is little difference between the measured and calculated intensity distributions. The similarity between the measured and numerically calculated distributions can be estimated with a root mean square error measure [

In [

In the iterative phase retrieval approach that we have just been discussing the intensity distributions are captured using CCD/CMOS arrays. And so in practice the intensity distribution is only recorded over a finite spatial extent and with a finite number of pixels which limits the spatial resolution and also introduces numerical sampling effects [

Therefore in this manuscript we deliberately take a different approach that emphasizes a theoretical analysis using a relatively simple model. Similar to the iterative phase retrieval approach we analyze how the phase may be estimated from two intensity measurements: one made in the image plane and one made in the Fourier plane. We do however make an important simplification. We assume that we can measure the intensity distribution in the Fourier plane over an infinite extent and ignore any sampling or discretization process that occurs due to the finite extent of the pixels in the CCD/CMOS array. Furthermore we assume that the intensity distribution in the image plane consists of a finite set of discrete point sources. With these two simplifications we can derive an analytical solution to the phase retrieval problem.

The manuscript is organized in the following manner: In Section

We begin our analysis by examining the idealized optical system depicted in Figure

Depiction of phase retrieval setup for an idealized phase retrieval experiment, OFT: optical Fourier transform. A set of

We imagine that we can measure the intensity in the Fourier domain over an infinite extent and with a sampling step size that is extremely small, so that any sampling effects can be ignored. We also imagine that only a finite number of perfect point sources in the input plane (with known intensity but unknown phase) contribute to the Fourier plane intensity distribution. Hence in this analysis we exclude any physical imposed limiting factors on the ability of our system to make a measurement. This will allow us to significantly simplify the analysis and to concentrate on another and arguable more important effect: the existence of a very high number of alternative solutions to the phase retrieval problem. As we shall see it is possible that many different combinations of phase values can produce identical intensity distributions. Hence this phase retrieval problem is said to be ill-posed.

We refer back to Figure

We thus begin by noting that the Fourier transform of a point source is given by

We also note that the larger the value of

Now setting

We set

Once

If we can find values for

We repeat this iterative process

To proceed further we must therefore find a solution to the following coupled set of equations

In this section we will examine some of the implications of the theoretical analysis we have just derived. First we will examine how we may reduce the number of possible solutions at each level of the iteration, reducing them from eight to two. To find a phase value for each contributing point source the algorithm must move through

We note that it is possible to rewrite the coupled equations, (

This is where the difficulties with our phase retrieval approach begin in earnest. Although it is possible to find an analytical solution to the coupled equations (

Having decided on a particular choice of

Fortunately, we have found from numerical simulations that although there are eight solutions possible in principal, including solutions with complex values, in practice when we substitute real physical parameters into our system of equations several of these solutions yield the same answer. We have found that at each step in the algorithm only two solutions are produced. This is the case for all simulations we have run. We do not pursue this question further here, concluding that at each step in an iterative algorithm we are able to find two solutions that are physically valid and satisfy (

From the previous section we have found that it is possible at each stage in the iterative procedure to reduce the number of solutions to two possible answers. At each stage in the algorithm then a “binary” choice has to be made about which solution to choose. If we have

Decision tree illustrating the choices made by the algorithm as it steps through the unknown phase values. At each stage of the algorithm, a choice between two paths, Path 0 or Path 1, is made. A particular path through the solution space is then uniquely identified with a bit sequence. For example, if the dashed red boxes are chosen at each stage of the algorithm being chosen, this corresponds to the path

We have run numerous numerical simulations of the equations derived in Section

The correct phase solution is always found among the other different solutions

The overwhelming majority of solutions have complex numbers for the phase and hence can be excluded as real physical solutions to the phase retrieval problem

There are often several other real valued solutions in addition to the actual real solution

We have run several numerical simulations with

We conclude that if we had unlimited computing resources and time it would in principle be possible to calculate all possible solutions even when

From the preliminary analysis we have reported in the previous section, we find that the overwhelming majority of the solutions yield phase values that are complex and hence “unphysical” in nature and can be discarded. This opens up the opportunity of pruning the solution space. We will now outline a strategy that can be employed to greatly reduce the search space. This solution depends on unique binary number that identifies each particular path through the search space and we note that it will be a

From Figure

The algorithm then runs as follows:

For a given

WHILE

Using Eq. (

IF

IF

ELSE:

IF

ELSE

Return

This algorithm will run through the search space checking the solutions. If a complex solution is found it will choose the alternate solution. For a given stage in the algorithm if both

When employing the “pruning” approach to search the solutions space we must still search a very large space; however the depth of the search per path will be greatly reduced; see, for example, [

In this manuscript we have examined the problem of measuring the phase of an optical wavefield. In the Introduction, we briefly reviewed several approaches for making this measurement before concentrating on iterative phase retrieval. In these phase retrieval problems an attempt is made to estimate the phase of wavefield using usually two or more intensity distributions captured in different Linear Canonical Transform planes. An iterative FFT based algorithm is employed that tunes/modifies the phase values iteratively so that the difference between a numerically calculated intensity distribution and an experimentally captured intensity distribution is minimized. Fienup [

Here we have deliberately chosen to examine a simplified version of the phase retrieval problem, assuming that we can make perfect measurements of the intensity distribution in an image and its corresponding Fourier plane. We assume that the distribution in the image plane consists of

It is possible in principal to find solutions to the phase retrieval problem that satisfy the physical constraints of the problem. The algorithm for finding these solutions works iteratively but deterministically and will find among the many solutions returned the physically correct solution

The number of solutions is enormous; if there are

We can modify the algorithm so that “unphysical” solutions are excluded which we refer to as pruning. Using a pruning approach we can significantly reduce the space of possible solutions. The performance of the pruning approach can be approximately estimated using statistical techniques. A critical factor determining whether the algorithm can be computed in a realistic time is the “branching” rate or depth of search before a given solution can be excluded

It is important to also highlight the significant practical shortcomings of this theoretical analysis.

We model the field whose phase values we wish to recover as a set of point sources (with known amplitude) in an image plane which are separated uniformly from each other by a fixed distance,

There is however a difference in the modeling in this paper for the intensity distribution recorded in the Fourier plane. In this paper we effectively assume that we can measure the continuous Fourier plane intensity; i.e., it is as if we can record and measure the intensity over an infinite plane in the Fourier domain and with an infinitely fine spatial resolution. Making these assumptions means that we can develop convenient analytical equations relating the unknown phase values directly to experimental measurements. These analytical equations are used then to develop the iterative algorithm so that the phase values can be estimated. This contrasts with real experimental systems in that the intensity in the Fourier plane is measured only at a finite number of locations, i.e., at each pixel, and is therefore sampled. And secondly the Fourier intensity distribution is only measured over the finite extent of the CCD/CMOS array. This summarizes the theoretical differences between the analysis presented in this manuscript and the standard theoretical treatment of the iterative phase retrieval problem.

We note however that it is possible to replicate experimentally the theoretical conditions we have assumed in the manuscript. We can achieve this by making several individual measurements of the Fourier plane intensity distribution where we move the CCD/CMOS array between measurements. Then using standard digital signal processing operations we synthetically increase both the spatial resolution and the spatial extent of the measurement by stitching together the single intensity measurements.

A significant shortcoming of this theoretical approach is the effect that even small amounts of measurement noise would have on the algorithm. In every experiment there will be measurement error, for example, electronic noise in the readout of a CCD or CMOS device. If a measurement is made on a variable that has a low signal level, the relative error tends to be larger [

So we can see that to turn this theoretical approach into a practical solution would require careful experimental measurements and the development of additional signal processing algorithms to overcome unavoidable noise in these measurements. Nevertheless we hope that by approaching the phase retrieval problem from a different direction we stimulate some new interesting ideas and insights and also shed some light on the character of the solution space for the phase distributions that are recovered from intensity measurements.

In this manuscript we deliberately focused on a simple 1D theoretical model (we only consider the

This is a theoretical paper: all signal processing data can be generated from equations therein.

The authors declare that they have no conflicts of interest.

The author gratefully acknowledges the EU commission (H2020-TWINN-2015 HOLO, project Nr. 687328), for supporting his attendance at Summer School “Digital Holography and Related Techniques” organized in Stuttgart, 21-23 June 2017. Part of the work described in this paper results from discussions that the author had during this meeting.