Optical: systems and elements – Optical modulator
Reexamination Certificate
2000-11-08
2002-04-09
Epps, Georgia (Department: 2873)
Optical: systems and elements
Optical modulator
C359S279000, C359S299000, C250S550000
Reexamination Certificate
active
06369932
ABSTRACT:
FIELD OF THE INVENTION
The present invention is generally directed to a system and method for recovering wave front phase information and using the recovered information for display, and more particularly to a system and method for determining the phase function from measured intensity information.
BACKGROUND OF THE INVENTION
In a coherent monochromatic imaging system the problem of extracting phase information from a detection medium which records only intensity information remains a problem without a consistent solution. Several have proposed experimental methods for determining the phase function across the wave front. One such method disclosed in Gabor, D. “A New Microscope Principle”, Nature 161, 777 (1948) involves the addition of a reference wave to the wave of interest in the recording plane. The resulting hologram records a series of intensity fringes, on a photographic plate, which contain enough information to reconstruct the complete wave function of interest. However, in most practical applications this method is cumbersome and impractical to employ.
Other methods, which did not employ reference waves, have been proposed for inferring the complete wave function from intensity recordings. See, e.g., Erickson, H. & Klug, A. “The Fourier Transform of an Electron Micrograph: . . . ”, Berichte der Bunsen Gesellschaft, 74, 1129 (1970). For the most part these methods have involved linear approximation and thus are only valid for small phase and/or amplitude deviations across the wave front of interest. In general, these methods also suffer from the drawback of requiring intensive computational resources.
A further method proposed that intensity recordings of wave fronts can be made conveniently in both the imaging and diffraction planes. Gerchberg, R. & Saxton, W. “Phase Determination in the Electron Microscope,” Optik, 34, 275 (1971). The method uses sets of quadratic equations that define the wave function across the wave in terms of its intensity in the image and diffraction planes. This method of analysis is not limited by the above-described deficiency of small phase deviations, but again, it requires a large amount of computational resources.
In 1971 the present inventor co-authored a paper describing a computational method for determining the complete wave function (amplitudes and phases) from intensity recordings in the imaging and diffraction planes Gerchberg, R. & Saxton, W. “A Practical Algorithm for the Determination of Phase . . . ,” Optik, 35, 237 (1972). The method depends on there being a Fourier Transform relation between the complex wave functions in these two planes. This method has proven to have useful applications in electron microscopy, ordinary light photography and crystallography where only an x-ray diffraction pattern may be measured.
The so-called Gerchberg-Saxton solution is depicted in a block diagram form in FIG.
1
. The input data to the algorithm are the square roots of the physically sampled wave function intensities in the image
100
and diffraction
110
planes. Although instruments can only physically measure intensities, the amplitudes of the complex wave functions are directly proportional to the square roots of the measured intensities. A random number generator is used to generate an array of random numbers
120
between &pgr; and −&pgr;, which serve as the initial estimates of the phases corresponding to the sampled imaged amplitudes. If a better phase estimate is in hand a priori, that may be used instead. In step
130
of the algorithm, the estimated phases
120
(represented as unit amplitude “phasors”) are then multiplied by the corresponding sampled image amplitudes from the image plane, and the Discrete Fourier Transform of the synthesized complex discrete function is accomplished in step
140
by means of the Fast Fourier Transform (FFT) algorithm. The phases of the discrete complex function resulting from this transformation are retained as unit amplitude “phasors” (step
150
), which are then multiplied by the true corresponding sampled diffraction plane amplitudes in step
160
. This discrete complex function (an estimate of the complex diffraction plane wave) is then inverse Fast Fourier transformed in step
170
. Again the phases of the discrete complex function generated are retained as unit amplitude “phasors” (step
180
), which are then multiplied by the corresponding measured image amplitudes to form the new estimate of the complex wave function in the image plane
130
. The sequence of steps
130
-
180
is then repeated until the computed amplitudes of the wave forms match the measured amplitudes sufficiently closely. This can be measured by using a fraction whose numerator is the sum over all sample points in either plane of the difference between the measured and computed amplitudes of the complex discrete wave function squared and whose denominator is the sum over all points in the plane of the measured amplitudes squared. When this fraction is less than 0.01 the function is usually well in hand. This fraction is often described as the sum of the squared error (SSE) divided by the measured energy of the wave function: SSE/Energy. The fraction is known as the Fractional Error.
A theoretical constraint on the above described Gerchberg-Saxton process is that the sum squared error (SSE), and hence the Fractional Error, must decrease or at worst remain constant with each iteration of the process.
Although the Gerchberg-Saxton solution has been widely used in many different contexts, a major problem has been that the algorithm can “lock” rather than decrease to a sum square error (SSE) of zero. That is to say, the error could remain constant and the wave function, which normally develops with each iteration, would cease to change. The fact that the SSE cannot increase may in this way trap the algorithm's progress in an “error well.” See Gerchberg, R. “The Lock Problem in the Gerchberg Saxton Algorithm for Phase Retrieval,” Optik, 74, 91 (1986), and Fienup, J. & Wackermnan, C. “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am.A, 3, 1897 (1986). Another problem with the method became apparent in one dimensional pictures where non-unique solutions appeared. Furthermore, the algorithm suffers from slow convergence. To date, there are no alternative satisfactory solutions to these problems with the Gerchbcrg-Saxton method. Accordingly, there is a need for a system and method that can recover wave front phase information without the drawbacks associated with the prior art.
SUMMARY OF THE INVENTION
The method of the present invention is driven by an “error reduction” principle and requires a plurality of samples of the wave front from the object being observed. The method relies on the fact that the back focal plane of a convergent lens on which the scattered wave from the object impinges contains a wave function, which is directly proportional to the Fourier Transform of the object and is therefore directly proportional to the Fourier Transform of the image plane wave function of the object. In the case where the phase difference from one pixel to any of its neighboring pixels only changes slightly, prior art methods were computationally intensive in trying to distinguish between these slight phase differences. Since the actual back focal plane (BFP) wave transforms to the true image in the Image Plane, by the intervention of the drift space between these two planes (mathematically causing the BFP wave to undergo Fourier Transformation yielding the Image Plane wave), we have one very useful relationship between the measurements in these two conjugate planes. However, other relationships between the waves in these two planes are achievable by changing the phase distribution only (not the amplitude distribution) in the BFP. This can be accomplished by using known but physically different phase filters, in the BFP, whose effects on the BFP phase distribution are known. It is noted that there are other physical methods of effectively changing the phase in the BFP (e.g., the use of defocus).
Epps Georgia
Pennie & Edmonds LLP
Thompson Tim
Wavefront Analysis Inc.
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