Abstract:
Disclosed are methods and apparatus for reconstructing a wave, including interpolation and extrapolation of the phase and amplitude distributions, with application to imaging apparatus, such as microscopes.

Description:
[0001]    This application claims the benefit of U.S. Provisional Application No. 60/838,228, filed Aug. 16, 2006. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    The most simple and therefore the most inclusive description of the method that I have discovered to achieve both phase and amplitude reconstruction along with super resolution is based primarily on three mainstays: the Rayleigh-Sommerfeld scalar wave diffraction equation, the mathematical law of Parseval and the principle of reconstruction through “iterative error reduction” 1 . 
         [0003]    Physically, the problem involves a three dimensional instrument having transverse Cartesian coordinates X &amp; Y and an orthogonal Z coordinate along the direction of propagation of the wave front of the scattered radiation. I define two conjugate transverse planes at X o , Y o  and X d , Y d . The subscripted o (output) plane is upstream of the subscripted d (diffraction) plane with the scattered wave front propagating from output to diffraction plane along the Z axis. To meet the strict requirements of the scalar diffraction equation, the wave front is considered to be null or zero at the output plane except within a moderately small hole say at the origin (X o =0, Y o =0). Still more physically, the output plane is a total occluding surface not allowing the scattered wave to exist in that plane at any point that is not within the area of the small hole. The diffraction plane is downstream from the output plane by a distance Z d  sufficiently large so that the two dimensional wave front there can be expressed as the two dimensional Fourier transform of that same wave front at the output plane upstream. Each point within the volume of space between the two conjugate planes can be located by its unique X, Y and Z coordinates. The wave function at each point has a complex value consisting of two rather than one number. The two numbers are called the amplitude and the phase. In sum then, one has a two dimensional amplitude distribution and a two dimensional phase distribution describing the scalar output wave everywhere in the space from Z o  to Z d , and in particular in the output plane. This wave propagates to the diffraction plane at Z d . The distribution in the diffraction plane of the amplitude and phase and therefore of the scattered wave is very different from what it was in the output plane. Mathematically the scattered wave in the output plane has been morphed into a new scattered wave in the diffraction plane by having traveled the distance Z d  between the two planes. The output wave has been Fourier transformed to yield the new diffracted wave. Now the essence of the first problem, the Fourier inverse problem becomes clear. 
         [0004]    Ordinarily, the Fourier Transform would take the complete (two dimensional complex) wave form in the output plane and transform it to the complete wave form in the diffraction plane. But current technology can only provide a way to measure the amplitude distribution (without the phase distribution) in the diffraction plane easily and with considerably more difficulty, the amplitude distribution (without the phase distribution) in the output plane. The phase information is not measurable and hence the scattered wave has lost a large percentage of the information it carries. Some estimate as much as 80% of the information in the scattered wave is lost here. However, assuming that we know the amplitude distribution alone in both planes, it is possible via the Gerchberg-Saxton 2  algorithm to infer the complete wave function in most cases. Now, assuming that we have the more likely situation, in which only the amplitude distribution is in hand for the diffraction plane, the phase distribution is again able to be calculated by knowing certain other constraints which exist naturally or which have been artificially placed on the wave function in the conjugate output plane. One of these useful constraints might be the boundary of the hole in the occluding screen through which the wave propagates. Mathematically, this boundary defines a so called area of support of the wave function at the output plane. Another constraint might place a known range of values, within which the amplitude and/or the wave phase must be. In fact, the investigator may make his own set of constraints by placing various phase plates (or partially occluding plates) whose specifications are known, within the area of support, and measuring the different diffracted wave amplitudes (without the phase distribution) corresponding to each perturbing filter. An arrangement such as this was studied by me. The computational algorithm (a modified form of the Gerchberg-Saxton algorithm) is given in “A new approach to phase retrieval of a wave front” by R. W. Gerchberg. 3  The contents of this article are incorporated by reference herein. The article shows that the method is always successful in finding the correct wave function with its phase function in both the output plane and the diffraction plane. However there is more. 
       SUMMARY OF THE INVENTION 
       [0005]    I have discovered that a modification of the algorithm given in the above-referenced Gerchberg article will not only solve the inverse phase problem, it will also extrapolate and/or interpolate the diffracted wave. This means that if there are doubtful or missed data points in the diffraction plane, the algorithm will generate the complex values that must be there. Moreover, if the diffraction pattern amplitude distribution measured is smaller than the full distribution, the modified algorithm will generate (extrapolate) the values that are missing and in so doing achieve super resolution in the recovered wave function. 
     
    
     DETAILED DESCRIPTION 
       [0006]    To describe the new algorithm, I begin with the minimum physical essentials for collecting data from a transmission type specimen with a range of illumination absorption and phase change distributed over its extent. This specimen is mounted in a hole in a 2 dimensional occluding plane at Z o  which is located a distance Z d  upstream of the diffraction plane. The size and location of the hole as well as the distance Z d  are known. Illumination of the specimen will be by a coherent, uniform amplitude and phase wave front. Immediately upstream or downstream of the specimen (ideally at the specimen) the wave front phase will be changed by a phase filter so that the wave front at the specimen will be the sum of the unknown specimen amplitude and phase distribution and the known phase filter distribution. The filter phase distribution is known. The filter amplitude distribution is equal to a constant of one. Depending on the problem, there will be several different phase filters that will substitute for each other during the gathering of data about the specimen, as will be shown. Incidentally, different holey occluding filters may be used rather than phase filters, where “holey” filters are light-occluding barriers with known spatially distributed holes of possibly varying size. Combinations of holey filters and phase filters can also be used. The output wave (at the specimen) will propagate along the Z axis to the diffraction plane located at Z=Z d . There the intensity or rather the amplitude distribution of the wave front will be measured for each different phase filter successively. The number of diffraction patterns in hand will equal the number (say, N) of phase filters used. The diffraction patterns will be known. All the data that are necessary are now in hand. 
         [0007]    Computer processing all these data is done in an iterative manner. Successive cycles of the algorithm produce better and better estimates of the specimen amplitude and phase distributions. There is the possibility that successive estimates do not change. In this case additional filters will be required to generate additional diffraction patterns. However, the algorithm is guaranteed not to diverge from the correct estimate in a mean squared error sense. 
         [0008]    Remember now that the wave function in the diffraction plane is the Fourier transform of the filtered wave function in the specimen plane. For no particular reason let us begin the first iterative algorithm cycle in the diffraction plane corresponding to one particular filter. We have in hand the amplitude distribution of the wave which was measured and we will combine that with the best guess we can make for the phase distribution to yield our first estimate of the complete wave function for that particular filter in the diffraction plane. Put this estimate through an inverse Fourier transform to yield an estimate of the filtered specimen wave. In the computer use an inverse phase filter to cancel the effect of the actual filter. (If an element of the physical filter shifted the phase by say plus 37 degrees, the inverse filter would shift the phase of that element by minus 37 degrees). This yields the first raw estimate of the specimen phase and amplitude distribution. Save this specimen estimate. I usually use two two dimensional matrices each with X &amp; Y indices which cover the output plane. One matrix contains the Real part of the complex numbers which define the specimen wave function at each point on the specimen and the other part contains the Imaginary part. Now do the same procedure with each diffraction plane amplitude distribution adding the Real and Imaginary parts of the wave forms generated into the 2 corresponding Real and Imaginary matrices. Now divide each number in the 2 matrices by the number of diffraction patterns (N) which have been used. Also, since we know that the value of the true specimen wave is zero outside the hole in the occluding screen plane, we can set all values of elements outside the hole to zero. At this point we may be able to incorporate any data that we know about the true specimen wave into the estimated wave function that is contained in the two matrices, always taking care to make the correction as small as possible if there is a range of correction that will satisfy the known a priori constraint. Clearly, at this point our two matrices hold the first estimate of the wave function in the output or specimen plane!! Note that we have taken some number (say N) of recorded diffraction patterns in the diffraction plane to generate just one estimate of the wave function in the output plane before we apply any phase or occluding filters to it. The next step in the algorithm is to generate estimates of the N diffraction patterns that this estimate of the specimen wave function will produce after it has been modified by a phase (or occluding) filter. 
         [0009]    I take one of the phase filters and essentially apply it to the estimate of the specimen wave function in the output plane. Then I propagate the wave to the diffraction plane. In the computer, this is done by mathematically Fourier transforming the filtered estimate of the specimen wave function. The diffraction pattern amplitude distribution generated will not match that which was physically measured and corresponded to that filter. So, I replace the diffraction wave amplitude distribution with the measured distribution leaving all points on the wave at which I have no a priori measured data untouched. As the algorithm proceeds these points will assume the value that they must have. The points so developed may be said to be extrapolated or interpolated depending on their location in the diffracted wave. Note that I have not modified the phase distribution of the diffracted wave. Thus the second estimate of the diffracted wave corresponding to the phase filter chosen is in hand and this diffracted wave function is saved. I do the same procedure for the next N−1 remaining filtered output waves yielding a total of N diffracted wave second estimates. These are the new N diffracted wave estimates with which I begin the next cycle of the iterating algorithm. 
         [0010]    It is noted here that the computer processing of the data is slightly different if holey perturbing filters are used, rather than phase filters, to generate the different diffraction patterns. In this case, if a return falls on a pixel that is blocked by the holey filter, that return is not counted in the averaging to achieve the estimate of the output specimen. 
         [0011]    I define a figure of merit as the sum of the squares of the differences between the diffraction amplitude distributions measured minus those estimated. I call this the error energy for any particular cycle. It will be found that this error energy can not increase and that given sufficient numbers of filtering data, will always decrease approaching a limit of zero. Of course zero error will mean that not only has the phase inverse problem been solved but so too has the extrapolation and interpolation problems been solved in cases where this was required. 
         [0012]    The method just described lends itself to implementation in a number of physical embodiments, which in turn suggest some possibly useful variations of the method. For example, shown in  FIG. 1  is a light microscope  100 , comprising a coherent light source  110 ; a specimen  122  mounted on a glass slide  123  placed over a hole  121  in an occluding plane  120  located at position Z o  along the Z-axis; a rotating phase filter  140  comprising multiple sectors, each giving a different phase shift distribution; a detector  130  (for example, a charge-coupled-device detector) located at position Z d  along the Z-axis; and a computer  150  connected to detector  130 , the computer  150  being capable of performing digital signal-processing functions, such as two-dimensional Fast Fourier Transforms. The above-described method may be applied to the operation of microscope  100 , thereby achieving super-resolution. In order to do this, phase filter  140  is rotated N times, each time interposing a different phase-modifying sector into the path of light propagating from the sample to the diffraction plane. This then generates the N diffraction patterns at detector  130  required as inputs to the algorithm. The algorithm is run as described above, generating not only the phase distribution at the diffraction plane, but also the amplitude and phase distribution at points not measured by detector  130 , for example at points beyond the physical extent of detector  130  (extrapolation), and/or at omitted or doubtful points in between those measured by detector  130  (interpolation). 
         [0013]    In the apparatus just described, it is helpful if the occluding plane  120  is painted black, and if the hole  121  has well-defined edges, so that the area of support (the area in which the lightwave can exist at the plane Z 0 ) is well-defined. In the figure, hole  121  is shown as substantially circular, but it can have other shapes. 
         [0014]    While the figure shows light from source  110  being transmitted through specimen  122 , in other variations such an apparatus can work with reflected or scattered light, rather than transmitted light. 
         [0015]    Instead of rotating a multi-sector phase filter such as filter  140  through the light path, other techniques can be used to achieve the necessary phase variations to produce the required N diffraction patterns. For example, a mechanism that moves individual phase filters into and out of the light path by translation rather than rotation can be used. Spatial light modulators can be applied to modify the phase by sequentially varying amounts. Also, occluding or partially occluding filters can be used. See FIG. 4 of Ref. 3 for an example. 
         [0016]    The above variations have been based on modifying the phase of the lightwave sequentially in time. However, by the use of optical techniques such as beam splitting, multiple phase-altered copies of the output wave can be produced and applied to multiple detectors in parallel. If the resulting diffraction patterns are then inverse-Fourier-transformed in parallel, a considerable speed-up in the operation of the algorithm can be achieved. 
         [0017]    Note that the microscope of  FIG. 1  is a lensless microscope. It is thus free of the typical imperfections introduced by defects in lenses, etc. However, the methods and apparatus described herein can also be applied to lens-based systems, such as cameras. What is required in each instance is the identification of a pair of conjugate planes, in the sense that the wave functions at the respective planes are related via the Fourier transform (or, at lesser drift distances, via the Fresnel transform—see Ref. 3, eqn. 1). In the case of a camera, suitable conjugate planes are the back focal plane of a camera lens and the image plane. Applications to telescopes are also expressly contemplated. 
         [0018]    Much of the above discussion has been in terms of visible light. However, it will be appreciated that the method and apparatus can be applied to phase and amplitude reconstruction of other waves and wave-like phenomena, such as radio waves, X-rays, and electron waves. 
         [0019]    The following documents are incorporated in this disclosure by reference, in their entirety. Documents 3 and 4 are physically attached to and included in this disclosure as Appendices I and II.
   1. Gerchberg, R. W. and Saxton, W. O., 1972, Optik, 35, 237   2. Gerchberg, R. W., 1974, Optica Acta, v.21, n. 9, 709   3. Gerchberg, R. W., 2002, J. of Modern Optics, v. 49, n 7, 1185   4. U.S. Pat. No. 6,906,839, issued Jun. 14, 2005, to Ralph W. Gerchberg