Elsevier

Ultramicroscopy

Volume 103, Issue 2, May 2005, Pages 153-164
Ultramicroscopy

Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy

https://doi.org/10.1016/j.ultramic.2004.11.006Get rights and content

Abstract

This paper examines the behaviour of the new Ptychographical Iterative Engine (PIE) algorithm when part of the initial information it requires is inaccurately known. This could be the parameters describing the illuminating wavefunction, the precise location of the specimen relative to the illuminating wavefunction, or other information that is assumed about the physical system. The tolerance of the algorithm for unavoidable problems such as noise and source incoherence is also investigated, leading to the conclusion that this approach to phase retrieval is very robust. It can not only tolerate errors in the assumed parameters, but can often be used as a method of characterising the parameters more accurately.

Introduction

Iterative phase retrieval methods have been known for many years [1], [2], but have recently come to particular prominence in the context of high resolution microscopy. This is for two reasons. Firstly, it is increasingly important to be able to find the phase of a wavefunction in order to obtain high resolution information about the specimen being examined. Secondly, the iterative methods offer a particularly appealing approach to solving the phase problem. This is because they come in a variety of forms allowing many different experimental approaches to their implementation. When considering the iterative phase retrieval approach, particular attention must be paid to the popular Fienup algorithm [2] (a refinement of the Gerchberg Saxton algorithm [1]) which has been experimentally implemented by a number of researchers [3], [4], [5]. Although the Fienup method is experimentally possible, it suffers from several limitations. These include the fact that the specimen examined must be relatively small in order to satisfy the Nyquist rate information requirement, and also that one cannot examine any particular region of interest, being limited to the exact region exposed by the support function. It is also important to note that the Fienup approach, along with most methods of phase retrieval, does not handle the situation where the wave incident onto the specimen is highly curved [6].

Another approach to phase retrieval is the method known as ptychography [7], [8]. This method involves moving a scanning probe, across the specimen, to every position in an xy grid, and recording the diffraction patterns thus obtained. The result is a four-dimensional data set that can be processed to solve for the phase of the specimen transmission function. One advantage of ptychography is that the diffraction pattern information gathered can be measured out to high angles, allowing high resolution reconstruction of the specimen wavefunction. However ptychography is hampered by the difficulty of measuring the very large amount of data required to form the data set.

A new method of phase retrieval, which we call the PIE (Ptychographical Iterative Engine) algorithm has recently been described by the authors [9], [10]. The approach combines the ideas behind ptychography and iterative phase retrieval to produce a powerful new technique of retrieving the phase of a wavefunction in situations where the incident radiation can be moved laterally, relative to the specimen. The early version of this algorithm worked well only when the exit function could be passed through a sharp aperture, giving a wavefunction with clearly defined zero regions. However the algorithm has since been dramatically improved, and is now effective for a much wider range of experimental situations. This algorithm surmounts the problems faced by the Fienup approach while retaining the benefits of using an iterative approach. In particular the PIE method works very well with any incident radiation, including that which is highly focussed or curved, or any aberrated lens system. This means that the method is easily applied to the Scanning Transmission Electron Microscopy (STEM) configuration.

In the PIE approach, measured images or diffraction patterns taken at different positions of the incident radiation beam, are used to perform a phase retrieval. The result is a recovery of the complex transmission function of the specimen being examined, which can then be used to discover information about the structure of the specimen, such as its projected potential. The use of diffraction plane measurements means that high frequency information can be measured, and thus this method can be used to achieve super-resolution, or resolution beyond the information limit given by the transfer functions of the lenses in the microscope system. This is because an objective lens is not used to produce a focussed image of the post-specimen wavefield. The information is instead inferred from diffraction data.

The PIE algorithm requires accurate knowledge of the incident radiation, which in the STEM case is the probe, in order to retrieve the specimen transmission function. The relative positions the probe is solved to must also be known so that the algorithm can function correctly. However in experimental practice it is likely that the probe positions will not be known with complete accuracy. It is also likely that some of the parameters that characterise the probe will be inaccurately measured. Other experimental problems will also arise, such as the inevitable Poisson noise and incoherence caused by the finite size of the source. These will also impact the success of the algorithm. It is important to understand the effects of all of these variables on the algorithm's behaviour, in order to judge which parameters are the most important ones to control and characterise exactly.

Section snippets

Description of the algorithm

In what follows, the example of a probe incident onto a specimen (as in STEM, for example) is used to illustrate the experimental arrangement (Fig. 1), however it is important to note that this is only one of a great number of possible arrangements of apparatus. Let O(r) and P(r) represent two-dimensional complex functions. In what follows, O(r) will physically represent an exit wave that would emanate from an object function which is illuminated by a plane wave, or equivalently the

Analysis of algorithm when assumed input information is incorrect

We now begin investigating the effect of various problems that may occur in an experiment based on this algorithm. This is done by simulating the effect of several different sources of error in the recorded diffraction patterns (noise and incoherence) and in the probe (incorrect probe parameters).

Conclusions

This paper has demonstrated the operation and behaviour of the PIE algorithm in a number of different simulated situations, all of which mimic various problems that could occur in an experiment. The results allow us to conclude that the new algorithm is tolerant of both noise and incorrect initial assumptions about the probe parameters.

The first part of the investigation explored the behaviour of the PIE algorithm when statistical and random noise are introduced into the simulated input data.

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