X-ray Image Reconstruction using the Transport of ... - CiteSeerX

Proc. 8th Int. Conf. X-ray Microscopy IPAP Conf. Series 7 pp.399-402

X-ray Image Reconstruction using the Transport of Intensity Equation KA Nugent 6FKRRORI3K\VLFV7KH8QLYHUVLW\RI0HOERXUQH9LF$8675$/,$ Recent years have seen a rapid growth in the development of techniques for the extraction of phase information from intensity measurements. In this paper the techniques based on observation of the propagation of light are reviewed and the applications described. The paper then goes on to discuss the application of these ideas to the non-crystallographic phase problem

KEYWORDS: Phase imaging, phase contrast, transport of intensity

G G 2π 6 ( U ) G ∇Φ ( U ) ≡ G , λ , (U )

1. Introduction With the advent of third-generation sources it was observed that the light leaving the synchrotron failed to have the expected intensity distribution. It was soon realized that the observed intensity distribution had in fact been strongly modified through refraction from slight non-uniformities in the synchrotron exit windows1). This was quite unanticipated and is, in a sense, a form of the speckle observed with visible lasers. It also demonstrated that very sensitive x-ray phase contrast was possible with these sources. The realization that the high coherence has major consequences, and offers major opportunities, has led to a rapid increase in the interest in coherent x-ray optics and x-ray phase imaging.

(3)

and it can be explicitly seen that this phase is also not defined at points of zero intensity. Furthermore, as an arbitrary vector field, the (average) Poynting vector may contain vorticity so that, in analogy with the scalar and vector potentials of electromagnetic theory, we may write2)

G G G G G G 6 ( U ) = , ( U ) [∇Φ 6 ( U ) + ∇ × Φ9 ( U )] , where

2. Phase and Partial Coherence

(4)

G Φ V and Φ9 are appropriately defined quantities

that have been termed the scalar and vector phase compo-

 3KDVHDQGWKH3R\QWLQJ9HFWRU Consider light from a star as it passes through the atmosphere. The density of the atmosphere fluctuates due to small thermal currents and this causes the refractive index of the air to change with time. The changes in refractive index bend the light into and out of the eye causing the apparent brightening and darkening that we know as twinkling. The essence of x-ray phase imaging can be understood from precisely this perspective using the theoretical device of the Poynting vector. The Poynting vector describes the direction and magnitude of the energy flow in the wave. In the case of coherent G G light with intensity , ( U ) , phase ĭ ( U ) and wavelength λ , the Poynting vector is given by

G G 6 (U )

G G Ȝ , ( U ) ∇ĭ ( U ) . ʌ

(1)

We immediately see that the phase gradient directly influences the flow of energy. In the case of quasimonochromatic partially coherent light, the Poynting vector fluctuates with time, however its average remains welldefined. We may therefore introduce a partial coherent phase which is GHILQHG via the expression G G 6 U

( )

where

=

λ 2π

G G , U ∇Φ U

( )

( ),

Figure 1: Interferogram showing an x-ray optical phase vortex. This data was obtained by observing the diffraction around a wire to produce a division of amplitude interferometer. The vortex is located at the point at which the fringe splits in two, indicating a vortex with topological charge 1. In other words, the vortex is carrying 1ƫof orbital angular momentum.

nents In order to remove any ambiguity in the definitions, we G G G require ∇ × ∇Φ 6 ( U ) = 0 and ∇ • ∇ × Φ9 ( U ) = 0 . Thought

(2)

of in this way, we see that phase can be regarded as having a vector as well as a scalar component. The scalar component is the familiar idea we encounter in our undergraduate curriculum. The vector component, as with electromagnetism

denotes a time average over a period much

longer than the coherence time. That is, the partially coherent phase gradient is given by

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and fluid flow, can be associated with vorticity, or angular momentum, in the field. The fact that electromagnetic waves can carry orbital angular momentum is now well-established and it has been shown that it is possible to transfer the angular momentum to trapped particles in the case of visible light. Such structures have also been observed in the x-ray region3). Interferometry is rather more challenging in the x-ray regime than in the visible light regime and so there are considerable advantages in developing non-interferometric phase recovery methods. My colleagues and I have therefore been developing non-interferometric phase recovery methods based on the so-called transport of intensity equation4). This is simply an expression for the conservation of energy on propagation. In general, energy conservation requires

G G ∇ • 6 (U ) = 0 .

(5)

In the paraxial regime, which assumes that the energy propagates at a small angle to some direction (the optical axis), eq(5) becomes

G 2π ∂, ( U⊥ )

λ

∂]

G G = ∇ • [ , ( U⊥ ) ∇Φ ( U )] .

(6)

This is the transport of intensity equation and is an elliptic partial differential equation. Standard results of partial difG ∂, ( U⊥ ) ferential equation theory tell us that if we measure ∂]

and use these to solve eq(6) for the phase. 3. Solving for the phase The details of how the phase may be recovered will not be gone into here, but a simple example will provide the spirit of the methods. Consider a pure phase object, which is to say one that imparts only a phase shift to the wave, illuminated by a uniform plane wave with intensity , 0 . In this limit, the transport of intensity equation may be written

G 2π ∂, ( U )

λ

∂]

G 2 = , 0∇ Φ ( U ) .

(9)

G G Ö ( N ) , then It is well established that if Φ ( U ) ⇔ Φ G G 2 2Ö ∇ Φ (U ) ⇔ N Φ ( N ) , where ⇔ denotes a Fourier transform relationship. Thus, 2π 1 G −1 Φ (U ) = )7 λ ,0

1  N 2

)7

{ } G ∂, ( U ) ∂]

.

(10)

The matter is made rather more complex when the intensity contains a spatial variation, however it can be seen that the transport of intensity equation lends itself, in some cases at least, to a simple determination of the phase distribution. Note that elementary Fourier theory tells us that the singularity at N = 0 indicates an indeterminate absolute phase. This

Figure 2: Quantitative phase tomographic image of the complex x-ray refractive index. The image on the left shows the real (phase) component of the refractive index and shows a low-Z boron coating. The image on the right shows the imaginary (absorptive) part of the index and reveals a high-Z tungsten wire.

G G and , ( U⊥ ) , and , ( U⊥ ) > 0 over a simply connected region, then the phase is uniquely specified to within an additive constant5). If we make an intensity measurement at two planes separated by a small distance δ ] , then we can make the approximations:

G ∂, ( U⊥ ) ∂]

≈

G G , ( U⊥ ) +δ ] − , ( U⊥ ) −δ ] ] [ 2δ ] 1

(7)

and

1 G G G , ( U⊥ ) 0 ≈ , ( U⊥ ) +δ ] + , ( U⊥ ) −δ ] ; 2

[

]

(8)

is an inevitable consequence of phase measurements that do not use a reference wave.  4. Applications The refractive index of x-rays is complex, showing both an absorptive (imaginary) and a phase shifting (real) component. As the energy of the x-ray photon increases the magnitude of both components decreases, however the rate of decrease with energy is much slower for the real part and so the relative importance of the phase shift increases rapidly with energy. There has been a considerable amount of work exploring the applications of phase contrast imaging at third-generation synchrotron sources, and it has now been clearly demon-

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Proc. 8th Int. Conf. X-ray Microscopy

strated that phase visualization is possible using both synchrotron and laboratory-based x-ray sources6). In this paper we are concerned with the quantitative recovery of phase and we note that, in addition to the approach using the transport of intensity equation, methods adapted from electron microscopy have also been applied7). Also, some very interesting work on phase imaging using interferometry has been demonstrated8). Quantitative phase recovery using laboratory sources has been the subject of active research. Paganin and co-workers9) have shown that, where it is possible to assume that the object consists of a substance with uniform and known refractive index, then reliable quantitative phase images may be recovered, even using laboratory sources. However more complex approaches are required in the case where arbitrary complex refractive indices are required. McMahon et al10) have used the transport of intensity equation to obtain tomographic complex images of the complex refractive index and showed that the results obtained were quantitatively accurate (figure 1). Resolutions of better than one micron have been demonstrated11). 5. Non-periodic Phase Recoverty The methods described in the previous section have enabled the extraction of phase information from intensity measurements demonstrating that this is a powerful and flexible method. However the phase-problem of more general interest is the one related to crystallography. The observation that x-rays diffracted from a crystal are able to carry important structural information has a very long and distinguished history. When a plane wave strikes a diffracting structure, the far-field of the diffracted wave may be described as the Fourier transform of the wave exiting the sample. The detection process is only sensitive to the magnitude (i.e. the intensity) of the diffracted field, and the phase of the Fourier transform is lost. The magnitude of the Fourier transform can correspond to a number of different diffracting structures with the result that the structure may not be uniquely recovered. Efforts to solve this phase problem for crystals (that is, structures that are spatially periodic) have continued for many years and a number of very powerful techniques are now available. These methods have had an extraordinary impact in the field of protein crystallography. However, as light sources become more coherent and the biomolecules of interest become harder, or impossible, to crystallize, there is a need to develop methods that will allow the structure of non-crystalline samples to be recovered from diffraction measurements. The most challenging problem here is whether it will be possible to illuminate a single molecule using the pulse from an x-ray free-electron laser, obtain a measurable scattered signal and then use the measurement to determine the structure of the molecule. This general area, as applied to molecules as well as larger structures, has come to be known as coherent diffractive imaging (CDI). The first question is whether the diffraction pattern of an isolated non-periodic sample uniquely determines its structure. This theoretical problem was first considered in detail by Bates12) who showed that, apart from some trivial ambiguities, an isolated (which is to say finite) structure produces a diffraction pattern that is “almost” unique. That is, if the diffraction pattern of an isolated structure is measured, it should be possible to recover the structure.

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If the diffracted intensity is directly reconstructed without phase it will recover the autocorrelation function of the sample, which is twice the size of the original sample. Viewed from this perspective, Bates’ result shows that the autocorrelation function of a finite structure (almost) uniquely determines the structure. Because an autocorrelation function is twice the size of the object, the diffraction pattern must be sampled at twice the density that would be needed for the object itself. For this reason, the method has also come to be known, perhaps misleadingly, as the “oversampling” method. Methods have been developed that are able to find the solution to this phase problem. These are all based on developments and improvements of the so-called GerchbergSaxton13) algorithm. This method iterates between real space and diffraction space by imposing the measured diffraction intensities in diffraction space, while letting the phase of the diffraction pattern float, and imposing the known extent of the sample (its support) in real space, while letting the structure within the support float. It can be shown that the iteration will converge and numerical tests indicate that the reconstruction is almost always the correct one. In the last few years, the “oversampling” method has been successfully demonstrated experimentally14). However the questions linger about reliable convergence due to the possibility that the iteration in the algorithm might stagnate at the wrong solution. Modern x-ray optics is now able to produce focal spots with a width of only a few tens of nanometres15). The nonperiodic phase problem is concerned with recovering structures, such as large proteins, with a size of some tens of nanometers. The assumption of an incident plane wave is implicit in the analysis underpinning crystallography, yet modern x-ray optics is able to create waves with a phase curvature that is significant over the scale of the diffracting structure. It is therefore appropriate to ask whether controlling the incident wave phase curvature offers a new handle on the crystallographic and non-crystallographic phase problems. If the incident wave is spherical, one gets the interesting result that the difference between the diffracted intensity and the diffraction of a planar wave obeys an equation that is formally identical to the transport of intensity equation16). Where it not for the presence of vortices, one could then directly apply the methods described in this paper to solve the phase problem. However, vortices are almost inevitably present in the far-field and so the phase cannot be reliably recovered. As discussed earlier, the ambiguity in the phase recovery is often principally a matter of symmetry. It can be shown that the use of two incident waves with orthogonal cylindrical curvature allows the phase to be recovered uniquely. Moreover, my group has shown that simply using a curved beam can improve convergence and yield more reliable results17). As an interesting aside, we note that a related observation was also made when related phase recovery techniques were used to diagnose the optical problems discovered immediately after the launch of the Hubble telescope18). The use of these methods as a scientific tool is in its infancy, though related methods have already been applied to the atomic scale electron imaging of a carbon nanotube19). 6. Conclusions In this paper we have taken a perspective on optical propagation that allows phase to be viewed in a very geometric man-

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ner. In this way, we are able to develop a suite of techniques that allow phase to be recovered in a very direct manner in a variety of different contexts. The approach is now wellestablished for the measurement of the phase of images using x-rays, light, neutrons and electrons. However there are some residual ambiguities in the phase recovery that have origins in optical vortex structures. We have experimentally shown that these structures exist even for highly energetic x-ray photons. The ideas described for image plane phase recovery may also be adapted to phase recovery in the far-field, as is appropriate for crystallographic phase recovery methods. The resulting method of coherent diffractive imaging opens the way to very high resolution lensless x-ray imaging. Acknowledgment I would like to thank the many colleagues who have contributed to the work described in this paper. I would particularly mention Dr Andrew Peele now of La Trobe University; Dr; Dr David Paganin now of Monash University; Dr Anton Barty now of the Lawrence Livermore National Laboratory; Dr Henry Chapman of the Lawrence Livermore National Laboratory; and Dr Harry Quiney and Dr Ann Roberts of the University of Melbourne. References 1) Snigirev A, Snigireva I, Kohn V, Kuznetsov S, Schelokov I , On the possibilities of x-ray phase contrast microimaging by coherent highenergy synchrotron radiation Review Of Scientific Instruments 66, 54865492 (1995) 2) Paganin D.and Nugent K.A., Non-Interferometric Phase Imaging with Partially-Coherent Light, Physical Review Letters, 80, 2586-2589 (1998) 3) Peele A.G,. McMahon P.J., Paterson D., Tran C.Q., Mancuso A.P., Nugent K.A., Hayes J.P., Harvey E., Lai B. and McNulty I., Observation of an X-ray Vortex, Opt.Letts., 27, 1752-1754 (2002) 4) Teague MR, Deterministic Phase Retrieval - A Green-Function Solution, Journal Of The Optical Society Of America 73, 1434-1441 (1983) 5) Gureyev TE, Roberts A, Nugent KA, Partially Coherent Fields, The Transport-Of-Intensity Equation, And Phase Uniqueness, Journal Of The

IPAP Conf. Series 7 Optical Society Of America A-Optics Image Science And Vision 12, 1942-1946 (1995) 6) Wilkins SW, Gureyev TE, Gao D, Pogany A, Stevenson AW Phasecontrast imaging using polychromatic hard X-rays Nature 384, 335-338 (1996) 7) Cloetens P, Ludwig W, Baruchel J, Van Dyck D, Van Landuyt J, Guigay JP, Schlenker M Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays Applied Physics Letters 75, 2912-2914 (1999) 8) Momose A, Takeda T, Itai Y, Hirano K Phase-Contrast X-Ray Computed Tomography For Observing Biological Soft Tissues Nature Medicine 2, 473-475 (1996) 9) Paganin D, Mayo SC, Gureyev TE, Miller PR, Wilkins SW Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object Journal Of Microscopy-Oxford 206: 33-40 (2002) 10 ) McMahon PJ, Peele AG, Paterson D, Nugent KA, Snigirev A, Weitkamp T, Rau C, X-ray tomographic imaging of the complex refractive index, Applied Physics Letters, 83, 1480-1482 (2003) 11) McMahon PJ, Peele AG, Paterson D, Lin JJA, Irving THK, McNulty I, Nugent KA, Quantitative X-ray phase tomography with sub-micron resolution, Optics Communications 217, 53-58 (2003) 12 ) Bates RHT Fourier Phase Problems Are Uniquely Solvable In More Than One Dimension .1. Underlying Theory, Optik 61, 247-262 (1982) 13) Gerchberg.RW, Saxton WO Practical Algorithm For Determination Of Phase From Image And Diffraction Plane Pictures, Optik 35, 237-246 (1972) 14) Miao JW, Charalambous P, Kirz J, Sayre D Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized noncrystalline specimens, Nature 400, 342-344 (1999) 15) Bergemann C, Keymeulen H, van der Veen JF Focusing X-ray beams to nanometer dimensions Physical Review Letters 91. 204801 (2003) 16 ) Nugent K.A., Peele A.G., Chapman H.N.and Mancuso A.P., Unique phase recovery for nonperiodic objects, Physical Review Letters, 91, 203902 (2003) 17) Quiney HM, Nugent KA & Peele AG, Iterative Image Reconstruction Algorithms using Wavefront Intensity and Phase Variation, Optics Letters, in press. 18) Fienup JR, Marron JC, Schulz TJ, Seldin JH Hubble Space Telescope Characterized By Using Phase-Retrieval Algorithms Applied Optics 32, 1747-1767 (1993) 19) Zuo JM, Vartanyants I, Gao M, Zhang R, Nagahara LA, Atomic resolution imaging of a carbon nanotube from diffraction intensities, Science 300, 1419-1421 (2003)