1 536 J. Opt. Soc. Am. A/ Vol. 4, No. / February 007 Keith A. Nugent X-ray noninterferometric phase imaging: a unified picture Keith A. Nugent School of Physics, The University of Melbourne, Victoria 3010, Australia Received July 10, 006; accepted August 8, 006; posted September 1, 006 (Doc. ID 7800); published January 10, 007 A unified theory of noninterferometric phase recovery based on the so-called ambiguity function is introduced. The theory is used to analyze previously published techniques and unify them with the methods of phase-space tomography applicable to partially coherent data. The theory is then used to propose some new approaches to noninterferometric phase recovery. 007 Optical Society of America OCIS codes: , , , INTRODUCTION Noninterferometric phase imaging has its conceptual roots in the methods of astronomical adaptive optics. The twinkling of stars arises from the refraction of light passing through an inhomogeneous atmosphere. From a coherent optics perspective, a time-varying inhomogeneous atmosphere produces a time-dependent phase change. The correction of the phase distortions using phase sensing coupled with adaptive optics will, in principle, allow for the production of diffraction-limited images. Given that the sensing is a phase-measurement problem, interferometry is at first sight a natural option. However, twinkling can be observed using light that is manifestly partially spatially coherent and very highly temporally incoherent, and so it seems there should be methods for sensing wavefront distortion that do not require the complexity of interferometry. The astronomical community has proposed a number of sensors, the most famous of which is the Hartmann Shack. 1 This consists of an array of small lenses focusing light onto a detector array. As the wavefront slope changes at the lens, the lateral position of the focus shifts thus enabling the wavefront slope to be sensed. The spatial resolution is limited by the number of lenses in the array, and so these sensors do not permit high-resolution phase imaging. Other approaches have been proposed. One of these is curvature sensing. The physics of this method may be understood by contemplating the bottom of a rippling swimming pool in sunshine. Refraction forms the familiar bright lines across the bottom of the pool. These lines are produced by regions of the pool surface with a curvature such that the light is focused at the bottom; they therefore represent contours of constant surface curvature. The light at the entrance pupil of a telescope can be assumed to have a constant intensity distribution. If an image of the entrance pupil is formed and then defocused a little, the change in intensity is proportional to the Laplacian, or curvature, of the phase of the incident light. This signal may be used to provide feedback to a deformable mirror to correct for the atmospheric wavefront errors. The curvature sensing methodology enables a greater level of spatial resolution than the Hartmann Shack sensor and so can be developed for phase imaging. 3 Note that the effects of the refraction by the atmosphere are quite small yet are readily observable. Methods that use refraction and propagation should therefore open the way to a very sensitive approach to phase-sensitive imaging. The development of third-generation synchrotrons has created x-ray sources that are small and placed a long distance from the experimental station, resulting in a relatively high level of spatial coherence. It was soon noticed that small imperfections in the optics and windows of the beamline, such as variations in thickness, produced significant modulations in the observed intensity 4 analogous to the swimming pool effect just discussed. The resulting phase contrast is now used for imaging in both two and three dimensions. 5 As in adaptive optics, the coherence requirements are very modest and, indeed, it was subsequently shown that this mechanism yields phase contrast with conventional laboratory x-ray sources 6 and even in highly noncoherent environments such as neutron radiography. 7 Phase-contrast x-ray imaging is now a standard method for synchrotron-based radiography. The aim of the present paper is to provide an overview of x-ray phase-contrast imaging and to present a synthesis of the various quantitative phase imaging methods. The approach will elucidate how they relate to each other and identify their regimes of applicability. In particular, the phase-contrast methods are also related to the partially coherent method of wave-field recovery known as phase-space tomography. Section discusses wave-field propagation and intensity measurement from a partially coherent phase-space perspective. It then narrows the discussion to the case of a fully spatially coherent field. Section 3 introduces an important uniqueness test for phase-measurement techniques. Section 4 describes direct phase recovery methods, and Section 5 looks at nondirect (iterative) approaches. Section 6 considers new approaches that are suggested by the analysis, and Section 7 relates the for /07/ /$ Optical Society of America
2 Keith A. Nugent Vol. 4, No. /February 007 / J. Opt. Soc. Am. A 537 malism developed here to the partially coherent method of phase-space tomography. Finally, the paper is summarized and concluded in Section 8.. PARTIALLY COHERENT PROPAGATION Consider quasi-monochromatic partially coherent light under the paraxial approximation. Such light is characterized by the mutual optical intensity, which is related to the electric field by the following ensemble average: J z r 1,r = E z r 1 E z * r, where subscripts to indicate position along the optical axis are used. The observed intensity of the light is given by I z r = J z r,r. In the coherent case, the requirement to take the ensemble average in Eq. (1) may be dropped. An intuitive way to understand the flow of optical energy is via the generalized radiance function: B z r,u k J z r + x,r x u dx. exp ikx In this case, the propagation of the field and the intensity are obtained from the expressions 8 B z+ z r,u = B z r zu,u, I z r = B z r,u du. It is apparent that the effect of propagation is to shear the phase-space distribution, and the act of measuring the intensity is to take a projection across the variable u. Note that the generalized radiance is always real and may be interpreted as the quasi-probability distribution of the field in position and momentum, where the lateral component of the photon momentum is p=ku. The function is termed a quasi-probability because the distribution may not always be positive and so cannot be unambiguously interpreted as a physical probability. Experimental measurements of negative regions of this function for x-ray beams have recently been reported. 9 For these purposes, it is noted that an intensity measurement is a projection across momentum and that, as the process of free-space propagation is a shearing operation in phase space [Eq. (4)], the manner in which that projection is taken varies with the distance of propagation. Some insight into the physical situation can be gained by introducing the so-called ambiguity function, 10 which is simply the Fourier transform of the generalized radiance, and can be written in terms of the mutual optical intensity as follows: A z q,x = J z r + x,r x exp ikq r dr. The coherent ambiguity function can therefore be written in terms of the field via the expression, A q,x = E r + x E * r x exp ikq r dr. It is straightforward to show that the Fourier transform of the intensity distribution can be written in terms of the ambiguity function as Î q = A q,zq. This expression is represented schematically in Fig. 1. The intensity information is contained on a twodimensional surface through the four-dimensional q,x space. As z, the slope of the curve shown in Fig. 1 increases so that the far field corresponds to a slice along the q axis at x=0. Note that the schematics used in this paper, such as Fig. 1, are not intended to convey information about the appearance of the ambiguity function. Inspection of Eq. (7) will reveal that the maximum possible value of x corresponds to twice the maximum width of the scattering object, and so, if the object is finite so also is the ambiguity function in this dimension. The other dimension of the ambiguity function, q, relates to the Fourier transform of the object and so, for a finite object, is infinite in extent in this dimension. Note also that the ambiguity function is four dimensional, and that the propagation-induced rotation of the projection only occurs around a single axis. The limited information resulting from a small number of projections will often allow for the recovery of a coherent wave field, but it is not possible to retrieve the entire ambiguity function in this way, as would be required for the characterization of a partially coherent field. This is briefly discussed in Section 7. Using Eq. (8), we can write Î z q = J z r + zq,r zq exp ikq r dr. Clearly, in the coherent limit, one has Î z q = E r + zq E * r zq exp ikq r dr In this paper, Eq. (8) or equivalently, Eq. (10), is used to classify noninterferometric phase measurements. An in- Fig. 1. Fourier transform of the measured intensity corresponds to the plane through the four-dimensional ambiguity function. The sloped line is described by the equation x=z 0 q, where the symbols are defined in the text. The slice measured obviously depends on the location at which the intensity distribution is measured, and the projections for contact images z=0 and the Fraunhofer diffraction z are also shown.
3 538 J. Opt. Soc. Am. A/ Vol. 4, No. / February 007 Keith A. Nugent tensity measurement corresponds to a particular way of slicing through the ambiguity function, and in this paper, it is shown that the different forms of intensity-based phase measurement correspond to different ways of sampling the ambiguity function. 3. COUNTEREXAMPLE In the following sections, comments will be made on the uniqueness of the phase solution. Formal proofs will not be offered, but extensive use will be made of a particular case, the phase vortex, that can serve as a simple counterexample as a test of the uniqueness of the solution yielded by a given phase recovery method. A. Optical Vortex The phase vortex is a field that contains a phase singularity at its center and has a radially symmetric intensity distribution. These fields are known to carry orbital angular momentum, and this property has been demonstrated experimentally. 11 The existence of the phase singularity brings with it a number of complications for phase recovery and so these fields serve as a stringent test case for a test of uniqueness. 1 These fields have been observed experimentally in the x-ray region. 13 Fresnel diffraction may be written in polar coordinates as E z r, k exp ikr z f, exp ik z exp ik r z cos d d. If we assume that we may write the field in the form, E 0, = A exp im, 11 1 where m is known as the topological charge, then the Fresnel diffraction integral takes the form E z r, C r A exp k ik z d exp i m k r z cos d, 13 where C r exp ikr /z. The term in curly brackets can be rewritten in terms of a Bessel function so that Eq. (13) becomes E z r, C r exp k im A J m kr exp ik d, 14 z z where the prime with C r indicates that a further phase term has been absorbed into the prefactor. Since J m = 1 m J m, the intensity distribution is unable to distinguish the sign of the topological charge. Propagationbased phase measurement techniques must therefore fail. Note that a field of the form described by Eq. (1) must contain an intensity zero at =0, and the phase is not defined at this point. B. Implications for Practical Imaging The use of an optical vortex as a test case provides insight into the mathematical uniqueness of the phase recovery. It is apparent that the perfect rotational symmetry of a vortex is an important factor in whether the sign of its topological charge may be determined. One would speculate that any capacity to break the symmetry would allow for the charge to be determined, in much the same way that a blemish on an otherwise perfect spinning top will immediately reveal the sign of its angular momentum. 14 However, the creation of a direct mathematical inversion of the equations will generally require the vortices to be completely absent for the formalism to be valid, and so their presence may cast doubt on the validity of the reconstruction. Iterative techniques make no such assumption. In practice, any vortices that might be present will not have perfect symmetry, and in this case the vortices will be recovered. One cannot mathematically guarantee uniqueness, but the likelihood of any ambiguity will be vanishingly small. This conclusion is supported by simulation work in the context of transmission electron microscopy. 15 Vortices seem to develop very easily with propagation, and phase discontinuities will be present almost without exception in the far field. However, the methods discussed in this paper largely aim to recover the phase in the plane of a scattering object. If multiple scattering is absent and the object is thin, vortices will not usually develop within the object. Vortices, and other phase discontinuities, are therefore unlikely to present a significant practical limitation for weakly scattering objects. We now have the basic ideas in place to permit the consideration of the various approaches to quantitative phase measurement from the perspective of the ambiguity function. 4. DIRECT PHASE RECOVERY METHODS The aim of phase-measurement techniques is to recover the real and imaginary parts of a complex transmission function. Intrinsically, two independent data sets are required so as to provide a plane of information for each of the real and imaginary parts of the object. It will be seen that this requirement is generally met by the techniques that follow. The exceptions are methods that use a priori information to obviate one plane, and the multiplane technique that uses data with a great deal of redundancy. The methods in this section apply to the recovery of phase information in the image plane and so will apply to methods for which it is already possible to obtain a direct image either through a projection system or a lens-based imaging system. A. Method Based on Guigay Equations An object is illuminated with a coherent wave, and the intensity is measured over two planes displaced from one another along the optical axis. The spacing between the two planes can be quite wide but must be known. The ob-
4 Keith A. Nugent Vol. 4, No. /February 007 / J. Opt. Soc. Am. A 539 B. Transport of Intensity Equation This method is based on recovering phases from intensity measurements taken a differential distance apart (Fig. 3). In this case, the intensities are given by Î z q = A q,zq, Î z+ z q = A q, z + z q. 18 we relate these expressions via the expansion Fig.. Method of Guigay uses two quite widely spaced intensity measurements and so uses projections through the ambiguity function that has a relatively wide angular separation. ject is assumed to only weakly interact with the radiation, which permits a model for the propagation to be created that leads to a direct solution for the complex field. In terms of the ambiguity function, the data acquired are of the form (Fig. ) A q, z + z q A q,zq + z A q,zq. 19 z For generality, use the partially coherent form, z A q = q q E r + zq E * r zq exp ikq r dr. 0 Î z1 q = A q,z 1 q, Î z q = A q,z q. 15 One will, without loss of generality, take the case around z=0 and so insert this limit into Eq. (0) and then use the expression for the Poynting vector for energy flow in the field: Guigay 16 developed an analysis of Fresnel diffraction patterns that allows for the recovery of the phase directly under the assumption that the field is only slowly varying. The expression of this assumption is that the complex field in Eq. (10) may be expanded into the form E r = E 0 exp r + i r E 0 1 r + i r. 16 Substituting Eq. (16) into Eq. (10) yields Î z q = I 0 q cos z q F r + sin z q F r, 17 where F denotes the Fourier-transform operation. Two independent measurements of the intensity at different propagation distances allow this pair of equations to be solved for both the absorption and the phase. 17 The assumption of a weakly varying field allows for the linearization of the problem, which allows an equivalent transfer function to be written for free space. These ideas are adapted from methods in electron microscopy and have also been adapted for the examination of laboratorysource x-ray phase radiography. 18 To the best of the author s knowledge, the uniqueness of this recovery has not been explored in detail, but it is clear that it will fail to distinguish two vortices with charges of opposite sign. The formal regime of applicability has recently been broadened. 19 The phase part of the object function is described in Eq. (17) by the third term in the brackets, which is, for small z, linear in z and quadratic in spatial frequency. One can anticipate that if the displacement from the object is too small, there may be some noise sensitivity in the low spatial frequency components of the phase reconstruction. j r = ik lim x E r + x x 0 E * r x. With this, Eq. (0) may be written z A q,0 = i k q j r exp ikq r dr. Inverting the Fourier transform gives 1 I r z = 1 j r, 3 k which is the partially coherent transport of intensity equation reported elsewhere. 0 In the limit of a coherent field, we have Fig. 3. Transport of intensity approach uses two intensity measurements that are separated by a differential distance, and so the projections through the ambiguity function are also separated by a differential angle. This diagram also applies to the multiple wavelength approach, where the difference in slope arises from a change in the wavelength rather than a change in the measurement plane.
5 540 J. Opt. Soc. Am. A/ Vol. 4, No. / February 007 Keith A. Nugent Î z, q = A q,zq, Î z,k+ k q = A q,z 1+ k k q, 6 which is identical if we identify an effective small displacement z eff =z k/k. Following the analysis of the previous section through, one can therefore obtain a change in intensity between the two planes given by Fig. 4. Transport of the intensity equation measures the phase and intensity of the propagating field and so may be applied at any point in the wave field. That is, the projections need not take place around the plane of the object at z=0. I r = z k I r r. k 7 j r = 1 I r r, k 4 and the familiar transport of the intensity equation 3 can be recovered as I r z = 1 I r r. 5 k This expression has been used quite extensively in the recovery of phase information. 7,1, The uniqueness of the phase recovery has also been thoroughly explored, and it has been demonstrated to provide reliable phase reconstructions for a wide range of radiation types. It has been shown that the transport of intensity equation has a unique solution provided the intensity is strictly positive. The vortex field discussed in Section 3 fails this requirement, and so again, this method will fail to distinguish vortices of opposite charge. The data sets obtained must be closely spaced in z for Eq. (5) to be valid. This implies that slowly varying components may not be accurately recovered. The transport of the intensity approach therefore suffers from a propensity to amplify noise at very low spatial frequencies. Note that this formalism applies to the field, not to the object. It is thus possible to perform phase recovery at any plane in space (Fig. 4) even those well displaced from a diffracting object. This observation has implications for the discussion of far-field diffraction in Subsection 5.B. C. Multiple Wavelength Approach This approach 3 recognizes that wavelength,, and propagation distance, z, always appear in the diffraction integral in the combination z. Thus, assuming the optical properties of the sample are wavelength independent over the range used, a small change in wavelength can have the same effect on the propagation equation as a small change in propagation distance. A small change is made to the wavelength so that k k+ k. Let us redefine the values of q such that k + k q=k q+ q, which means that q= k/k q= / q. With this replacement, one can write expressions analogous to Eq. (18) as follows: This can be solved using the same methods as for the transport of the intensity equation. The relevant diagrams for this method are also Figs. 3 and 4, but the change in slope of the line is obtained by the rescaling of the q axis, as indicated in Eq. (6). A propensity to amplify noise at low spatial frequencies will also apply here, as for the transport of the intensity equation approach. D. Homogeneous Sample The homogeneous sample approximation assumes that the object consists of a single material with a known complex refractive index. Phase imaging, however, provides for greater contrast. Moreover, only one intensity data set is required. In terms of the ambiguity function, the data set required is (Fig. 5), Î z q = A q,zq, 8 augmented, as stated, with the a priori knowledge that the object consists of a single material. It is also assumed that its thickness varies slowly. The development of this method requires that the interaction of the sample with the incident field is sufficiently weak that an assumption of linearity can be made. The expression for the transmitted field can be used as Fig. 5. Homogeneous object approach makes a priori assumptions about the composition of the sample, and this assumption eliminates the need for a second measurement plane. The additional assumption that the object only weakly interacts with the sample allows for the measurement plane to be a nondifferential distance from the sample, as indicated by the large angle between the projection and the q axis.
6 Keith A. Nugent Vol. 4, No. /February 007 / J. Opt. Soc. Am. A 541 E r = E 0 exp ik 0 i t r, 9 where we have implicitly assumed that the object is made of a single known material. The ambiguity function can be used to write Î z q = I 0 exp k 0 t r + zq + t r zq ik 0 t r + zq t r zq exp ikq r dr. 30 We now use the slowly varying assumption to make a Taylor expansion of the thickness so that we obtain t r ± zq t r ± zq t r, Î z q = I 0 exp k 0 t r ik 0 zq t r exp ik 0 q r dr. We now use the Fourier derivative theorem, and that F t r = iq F t r, exp k 0 t r = k 0 t r exp k 0 t r, to reduce Eq. (30) to the form Î z r = I 0 F exp k 0 t r 1+ z q Inverting this, we can recover the thickness distribution from the expression t r = 1 k 0 ln F 1 + z q Î z q I This is the expression for a homogeneous object as first proposed by Paganin et al. 4 It can be noted that a very similar idea, based on the phase-attenuation duality of the interactions of high energy x rays with matter, has also been proposed for medical phase-contrast imaging. 5 It is possible to produce a vortex field with a homogeneous mask, 13 and so again, this method fails the test of true uniqueness. Note that the denominator in the brackets of Eq. (36) does not vanish as q 0, and so, this method is rather more stable to noise than the pure transport of intensity equation methods. E. Phase-Only Object This approach 6 uses intensity data obtained over a plane, a short distance from the object (Fig. 6). It is assumed that the object variation is sufficiently slow that the measurement differs only slightly from the sample free measurement, and that sample absorption is negligible. The data set therefore has the form Fig. 6. Phase-only approach makes an assumption about the wave field that there is a negligible amplitude modulation by the sample. Thus, only one measurement plane is required. The phase modulation can be quite high, in which case the measurement plane should be a differential distance from the sample. Î z q = A q, zq. 37 To obtain the phase result, the fact that Î 0 q =I 0 q and Eq. (10) may be used so that A q,0 = 1 z z Î z q I 0 q, 38 and Eq. () to conclude that A q,0 = 1 q F r, 39 z k 0 and r = k 0 z F 1 1 q Î z q I 0 q. 40 The assumption of a phase-only distribution requires that the intensity be uniform in the plane of the object. The intensity beyond the phase mask will be uniform as there will be a period of propagation before the vortex develops. It can therefore be concluded that this method fails the test of complete uniqueness. Note that the denominator in Eq. (40) does vanish as q 0. This method will therefore be rather unstable for low spatial frequencies. 5. INDIRECT METHODS Indirect methods become important when it is not possible to write an analytic and invertible relationship between the wave field and the measurement. In this case, the phase recovery must be performed using other approaches, typically iterative, in which a solution is found that is consistent with the measured intensity measurements. This approach allows for the incorporation of additional a priori information and the inclusion of the consideration of experimental uncertainties, such as noise. A. Multiplane Intensity Measurements This approach has its origins in the through-focal series methods developed for transmission electron microscopy 7 and has been applied to develop quantitative threedimensional x-ray images. 8 Intensity measurements are obtained over planes at a range of distances along the op-
7 54 J. Opt. Soc. Am. A/ Vol. 4, No. / February 007 Keith A. Nugent tical axis. This method is also clearly applicable to methods for which an image can be obtained, such as in an electron microscope or in projection imaging using an x-ray source. The data used are of the form (Fig. 7), Î zn q = A q,z n q, n =1,,...,N. 41 The data are recovered by using the known laws of optical propagation to find a complex wave that is consistent with all of the measured data. The particular way in which the consistent solution is obtained has a number of possible implementations, the details of which will not be discussed in the present paper. Numerous intensity measurements will allow for a very robust phase recovery due to the significant redundancy in the data. However, as shown in Section 3, the diffracted intensity for opposite charged vortices will again be indistinguishable, and so the phase recovery is not necessarily unique. The use of multiple planes allows for a range of spatial frequencies to be probed at different sensitivities, and so the large amount of data used and their form tend to make this approach rather stable to the effects of noise. B. Far-Field Methods Far-field methods apply to imaging situations in which it is not possible to obtain a direct image due, for example, to the need for very high spatial resolution. These methods therefore typically apply to problems in which there are no lenses or where it is desired to image with a resolution far greater than is possible using conventional optical systems. The nature of the errors in the reconstructions in these approaches is different from methods that recover the phase in the plane of the image. In the cases that follow, the principal uncertainty in the image arises from the possibility that the iterative reconstruction has failed to converge onto a valid solution. 1. Gerchberg Saxton Algorithm The Gerchberg Saxton algorithm 9 uses an iteration between an intensity measurement in the far field and another in the plane in which the phase is to be recovered. That is, it uses the measurements (Fig. 8), Î 0 q = A q,0, Î q = A 0,q. 4 Fig. 7. Multiple plane method obtains many data sets at different distances from the sample and then undergoes a fitting procedure to find the wave field consistent with all of the data. Fig. 8. Gerchberg Saxton approach assumes that an intensity measurement of the sample is obtained (corresponding to a projection along the q axis), and a far-field diffraction measurement is obtained (corresponding to a projection along the x axis). An iterative method is then used to find a consistent complex amplitude. Fig. 9. Gerchberg Saxton Fienup approach uses a single set of far-field data and planar sample illumination and knowledge about the support (size and shape) of the object. The data are therefore described using the Fraunhofer formalism. The a priori size and shape knowledge serves to eliminate the need for a second data set. An iterative method is then used to find a consistent solution. This algorithm was also first developed in the field of electron microscopy. The basic idea is to measure the intensity distribution in one plane and guess the phase. Fourier transform the resulting field to obtain the corresponding diffraction pattern. Retain the resulting far-field phase distribution and correct the amplitudes to the measured values, inverse Fourier transform to obtain an updated near-field result, return the amplitudes to measured values, retain the phase, Fourier transform to the far field, and so on. It is found that this algorithm converges to recover the correct phase. The method clearly needs a high-resolution image in both planes and has found little application in the field of x-ray imaging. Opposite charged vortices will not be able to be distinguished, and so the resulting phase is not unique.. Gerchberg Saxton Fienup The Gerchberg Saxton Fienup algorithm 30 is a single data set algorithm that incorporates a priori knowledge about the shape of the scattering object. The algorithm uses only a far-field measurement (Fig. 9), Î q = A 0,q, 43
8 Keith A. Nugent Vol. 4, No. /February 007 / J. Opt. Soc. Am. A 543 and iterates so as to find the data that are consistent with both the measurement and the known shape (or support) of the object. As no measurement is required at the object plane, it can be used to obtain a high-resolution image from diffraction data. Bates 31 has shown that the resulting reconstruction is almost unique to within the ambiguities of constant phase offset, phase conjugation and lateral shift. Almost means that it is possible to create ambiguous structures, 3 but they are thought to be very rare and of no practical significance. The charge-conjugation ambiguity identified by Bates 31 includes the vortex example of Section 3. Knowledge of the support places a limit on the bandwidth of the variation in the x axis of the ambiguity function. Thus, it is related to the Gerchberg Saxton algorithm insofar as one has the knowledge about a property of the horizontal projection in Fig. 6 but does not have a full measurement of it. This approach to imaging has now been demonstrated experimentally by a number of workers and is hoped to be used as the basis for the interpretation of noncrystalline diffraction data. The method has become topical as a potential route to the determination of molecular structures using x-ray free-electron lasers Gerchberg Saxton Fienup in the Fresnel Region This previous two forms of phase recovery assume a Fourier-transform relationship between the near field and the far field. Physically, this implies that the field has negligible curvature in both planes. That this assumption can be relaxed even at the molecular scale in modern x-ray optics has been recently pointed out by Nugent et al. 37 It is possible to either measure in the Fresnel region with planar illumination 38 or to illuminate with a beam containing significant curvature. In either case, the data set is a Fresnel diffraction pattern and has the form (Fig. 10), Î q = A q, q, 44 where is a measure of the distance from the sample, the curvature of the incident beam, or a combination. Note that the slope in the line shown in Fig. 10 implies that there is a correspondence between the angle of diffraction and the position in the sample the diffraction information carries with it some spatial information. This additional spatial constraint allows for better convergence than is typically found with plane-wave illumination but has the disadvantage that the diffraction pattern is sensitive to sample motion, a sensitivity absent with planar illumination. Note that the resolution of the image is not determined by the resolution of the optic used to produce the curved illuminating beam. The properties of the Gerchberg Saxton Fienup algorithm in the Fresnel region have been examined, 39 although in a rather different context, and the result has been shown to be unique. The convergence properties have also been examined and shown to be far superior to the approach in the Fraunhofer limit, 40 and the method has very recently been applied to the characterization of precision x-ray optics 41 and to coherent diffractive imaging. 4 The method has been shown to be unique to within phase conjugation. Again, vortices point to a possible ambiguity in the phase recovery, but in this case, the ambiguity is identical to the insensitivity to phase conjugation. Note that it is also possible to obtain diffraction patterns in the far field, but for which the illuminating field contains a slightly different curvature. The differences in the intensity distributions have been shown to obey an equation entirely analogous to the transport of intensity equation, 37 a conclusion that is perhaps not surprising in the picture developed here (Fig. 4). 4. Ptychography The electron microscopy method known as ptychography 43,44 falls broadly into the present category. This method aims to recover an image with a resolution much greater than could be acquired using a practical lens. In this method, the sample is placed into a beam, typically diverging, and the diffraction pattern is recorded. The sample or beam is then displaced a number of times with the diffraction pattern recorded at each displacement. The data sets are then assembled into a single image, typically using iterative methods. 45,46 The illuminating fields are spherically curved, and so this can be regarded as an application of the ideas used in the Fresnel form of coherent diffractive imaging (Fig. 10) but to multiple sets of data 47 where the support is determined by the illuminating beam rather than the object and can be changed by moving the illumination. In ptychography, then, changing the position of the illumination, rather than the observation position, yields the phase information. Fig. 10. Gerchberg Saxton Fienup approach in the Fresnel region allows for some phase curvature on the sample or at the detector and the data are described by the Fresnel formalism. This curvature introduces a correlation between the position in the measurement plane and position in the sample. As a result, the convergence is much better than with the Fraunhofer diffracted data, but the method is now sensitive to sample movement. 5. Astigmatic Diffraction That it should be possible to control the phase structure of the illuminating wave allows for the possibility of using this as a way of determining the phase of the detected wave. A particular case of this was discussed by Nugent et al. 37 who looked at the consequences of illumination with two orthogonal cylindrically curved waves (Fig. 11),
9 544 J. Opt. Soc. Am. A/ Vol. 4, No. / February 007 Keith A. Nugent 6. NONPROPAGATION METHODS In this paper, the various forms of propagation-based phase imaging have been understood on the basis of how the intensity measurement samples the ambiguity function. Given that the process of propagation allows for an intensity measurement to sample along a line through the function, it seems that all possible combinations may have been used in one form or another, leading to an array of both direct and iterative methods. The potential for new intensity-based phase measurement schemes may depend on the use of either new sources of a priori information or in finding different ways of projecting across the ambiguity function. Other possible sources of a priori information will depend on the details of the imaging problem and are highly problem dependent. In this paper, knowledge of the composition of the sample, knowledge that absorption is negligible, and knowledge of the shape and size of the object have been mentioned. There is no doubt that many other possibilities will emerge from time to time. The development of astigmatic imaging opens up the possibility of sampling the ambiguity function in different ways. Suppose that, instead of allowing the radiation to propagate from a given sample plane, we obtain two measurements at the sample plane but introduce a phase variation at the sample, r. In this case, the second, phase-varied measurement has the form, A q,x = E r + x E * r x exp ik r + x r x exp ikq r dr. 46 Fig. 11. Astigmatic diffraction case illuminates the sample with waves that have orthogonal cylindrical curvature (the lighter curve in the figure indicates a wave that is curved out of the plane sketched in this two-dimensional representation). Although experimentally challenging, the convergence is excellent, and the solution is truly unique. Indeed, the formalism has a close relationship to that of the transport of the intensity equation, and it should be possible to perform a direct reconstruction, though this is probably unnecessary. We assume the phase variation may be written in the form, so that r + x r x = r,x, 47 A q,x = E r + x E * r x exp ik q r r,x dr. Î y q x,q y = A q x,0, q x,q y, Î x q x,q y = A 0,q y,q x, q y The Fourier transform of the modified intensity distribution is then given by This approach yields a unique solution and has excellent convergence properties, however, it will no doubt be extremely challenging to put into practice. The approach of astigmatic diffraction has also been extended to a preliminary investigation of crystalline diffraction 3 and to transmission electron microscopy. 48 The symmetry-breaking property acts to break the degeneracy of, for example, oppositely charged vortices and results in this method producing a truly unique solution, though at the price of significant experimental complexity. Î q = E r + zq E * r zq exp ik q r r,zq dr. 49 Given the condition Eq. (47), this is a rather general expression and could itself be used as the basis for an iterative noninterferometric phase recovery algorithm. If we further conjecture, and with some loss of generality, that the aberration introduced may be written in the form, r,zq = r zq, 50 then this allows for the inclusion of some of the preceding cases. Specifically, the Fresnel Gerchberg Saxton Fienup method corresponds to r,zq r, and the astigmatic diffraction corresponds to 1 x; y. Using Eq. (50), obtain an aberrated intensity distribution with the Fourier transform, Fig. 1. Figures 10 and 11 indicate methods that use modifications to the sample illumination. In principle, these approaches can be generalized to allow for arbitrary phase curved illumination, as indicated. It is conjectured that this opens up new approaches to phase recovery that may have significant experimental advantages in some circumstances.
10 Keith A. Nugent Vol. 4, No. /February 007 / J. Opt. Soc. Am. A 545 Î q = A q,z q zq. 51 An example of this is shown in Fig. 1. If we know r,zq to be small, then it follows that Î q Î 0 q + ik r,zq E r + zq E * r zq exp ikq r dr. 5 This reverts to the transport of the intensity equation for a spherical phase aberration, r,zq r. These aberration-based methods have not, to the best of the author s knowledge, been put into practice except where they correspond to the specific cases mentioned. This may represent an opportunity for further exploration. One would imagine that such a method could also incorporate a priori information and so permit phase recovery with a single data set. 7. IMAGING THE COMPLETE WAVE FIELD The discussion so far has concerned the recovery of phase and so implicitly assumes that the field is coherent. However, the description of an intensity measurement as a slice through the ambiguity function leads to the possibility of characterizing partially coherent fields using intensity measurements, 49 and for purposes of completeness, this method is briefly discussed. The scheme obtains many intensity measurements, which yield projections across the ambiguity function, rather as shown in Fig. 8. In practice, three-dimensional intensity measurements only yield a measurement over a three-dimensional subset of the four-dimensional ambiguity function; as discussed earlier, free-space propagation produces a rotation only about a single axis and so cannot fully map out the four-dimensional space. This problem is resolved by essentially extending the method of astigmatic diffraction to become a four-dimensional method by obtaining threedimensional intensity measurements for different cylindrical curvatures. 50 An implementation of the method has been discussed for visible light optics 51 but has yet to be experimentally implemented for light or any other form of radiation. Phase-space tomography has been experimentally implemented for x rays in a recent series of papers. It has been used to make a thorough measurement of the coherence, 5 to develop a more flexible form of x-ray microscopy 9 and to characterize and correct aberrations in the x-ray optical system. 53 The characterization of a full four-dimensional coherence functions has yet to be demonstrated, but if it can be practically achieved, then it will be possible to have a data set containing all of the information that can be extracted from the scattered field. 8. SUMMARY AND CONCLUSION This paper has presented an exploration of methods for noninterferometric phase retrieval as currently implemented in x-ray imaging. The methods have been viewed in terms of how the ambiguity function is sampled and has provided a unifying framework that also includes extensions to partially coherent wave-field determination. The broad overview is summarized in Table 1. In this table, a classification system is proposed in which three broad properties are introduced in the form XYZ. X is either I, F, or A, corresponding, respectively, to whether the method is applicable using a measurement in the intermediate (Fresnel) region, the Fraunhofer region or whether an aberrated field is used to illuminate the sample. The Fresnel region has clear advantages in terms of convergence and uniqueness but requires a rather stable experimental system and may not be practical for many applications in very high-resolution imaging given the need for illumination with curved wavefronts. Aberration-based methods have only recently been demonstrated and show good convergence but some sensitivity to experimental drift if the experimental exposure is very long. Y may be either D or I, corresponding to either a direct or an iterative solution. A direct solution will find a field that is entirely consistent with the measurement, including any noise or errors that exist in the data. That the direct methods have this clear relationship to the data can restrict their application only to data with a very high signal-to-noise ratio. Indirect methods will intrinsically have a metric for the optimization of the fit to the data, and so it is straightforward to incorporate a measure of the noise. The fit can also be constrained to include any a priori physical information, a feature that can be very difficult to include into direct methods. Finally, Z refers to the number of data sets used. Z=1 implies a method that must incorporate a priori sample information, Z= is the minimum required for a direct general solution, and Z=N implies many data sets, which must therefore incorporate redundancy in the data acquired. Methods that use a single data set must include a priori information in some form, such as object support in the iterative far-field methods or knowledge of some broad optical properties such as in the phase-only and homogeneous sample cases for direct methods. Note that such a priori information can assist in the stabilization of the solution against noise. Z= typically corresponds to a fully general direct method, and considerable care must be taken to ensure that the two measurements are precisely aligned even a slight lateral misalignment will be interpreted as a significant phase gradient by direct solution methods. Z=N will have considerable noise robustness due to a great deal of redundancy in the data set. Also, being typically iterative, it can also have a noise metric included in the fitting procedure. X-ray phase imaging is an active area of research with ideas emerging quite rapidly, and there will no doubt be many more approaches to add to this synthesis in coming years. Nonetheless, it is hoped that the discussion in this paper will help with the clarification of the applicability of various techniques, will allow a broader overview of how methods of phase recovery fit together and stimulate the invention of new methods. ACKNOWLEDGMENTS The author thanks many of his colleagues for stimulating discussions, including Chanh Tran, Andrew Peele, Garth Williams, and Harry Quiney. The support of the Austra-
11 546 J. Opt. Soc. Am. A/ Vol. 4, No. / February 007 Keith A. Nugent Table 1. Simple Classification System Proposed for the Approaches to Phase Recovery Discussed in This Paper a Class Method Measurement Conditions ID High measurement Î z1 q =A q,z 1 q, separated planes Î z q =A q,z q (method of Guigay) ID Closely separated Î z q =A q,zq, measurement planes (transport of intensity) Î z+ z q =A q, z+ z q ID Wavelength variation Î z, q =A q,zq, Î z,k+ k q =A q,z 1+ k q k Radiation must interact only weakly with the object No special conditions Sample must have properties that are precisely wavelength invariant ID1 Homogeneous sample Î z q =A q,zq Sample must have known and uniform composition Radiation must interact weakly with the sample ID1 Phase-only sample Î z q =A q, zq Measurement plane must be close to sample Interaction must be known to have negligible absorption IIN Multiple plane methods Î zn q =A q,z n,q, n=1,,...,n No special conditions FI Gerchberg Saxton Î 0 q =A q,0, No special conditions approach Î q =A 0,q FI1 Gerchberg Saxton Fienup Î q =A 0,q Object must have known support AI1 Fresnel Gerchberg Saxton Fienup Î q =A q, q Object must have known support AI Astigamtic diffraction Î y q x,q y =A q x,0, q x,q y, Î x q x,q y =A 0,q y,q x, q y No special conditions but experimentally challenging Unique solution AI/AI1 Aberration Slowly varying aberration a The classification is in terms of whether the recovery uses measurements in the Fresnel I, far fields F, or a nonplanar illumination A ; whether the recovery is direct D or iterative I, and the number of data sets required 1,, or many, indicated by N. lian Research Council is acknowledged through a Federation Fellowship and the ARC Centre of Excellence for Coherent X-ray Science. The author s address is REFERENCES 1. J. M. Beckers, Adaptive optics for astronomy principles, performance, and applications, Annu. Rev. Astron. Astrophys. 31, 13 6 (1993).. F. Roddier, Curvature sensing and compensation a new concept in adaptive optics, Appl. Opt. 7, (1988). 3. M. R. Teague, Deterministic phase retrieval: a Green s function solution, J. Opt. Soc. Am. 73, (1983). 4. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation, Rev. Sci. Instrum. 66, (1995). 5. P. Cloetens, W. Ludwig, J. Barucher, D. Van Dyck, J. Van Landuyr, J. P. Guigay, and M. Schlenker, Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays, Appl. Phys. Lett. 75, (1999). 6. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Phase-contrast imaging using polychromatic hard x-rays, Nature 384, (1996). 7. B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, Imaging phase radiography with neutrons, Nature 408, (000). 8. M. J. Bastiaans, Application of the Wigner function to partially coherent light, J. Opt. Soc. Am. A 3, (1986). 9. C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Faterson, and I. Mc Nulty, X-ray imaging: a generalized approach using phase-space tomography, J. Opt. Soc. Am. A, (005). 10. J. P. Guigay, The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams, Opt. Commun. 6, (1978). 11. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Direct observation of transfer of angular-momentum to absorptive particles from a laserbeam with a phase singularity, Phys. Rev. Lett. 75, (1995). 1. T. E. Gureyev, A. Roberts, and K. A. Nugent, Partially coherent fields, the transport-of-intensity equation, and phase uniqueness, J. Opt. Soc. Am. A 1, (1995). 13. A. G. Peele, P. J. McMahon, D. Paterson, C. Q. Tran, A. P. Mancuso, K. A. Nugent, J. P. Hayes, E. Harvey, B. Lai, and I. McNulty, Observation of an x-ray vortex, Opt. Lett. 7, (00). 14. K. A. Nugent, D. Paganin, and T. E. Gureyev, A phase odyssey, Phys. Today 54, 7 3 (001).
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