A new diffractive imaging technique called Imaging By Integrating Stitched Spectrograms (IBISS) is presented. Both the data collection and phase retrieval algorithm used in IBISS are direct extensions of frequency resolved optical gating to higher dimensions. Data collection involves capturing an array of diffraction patterns generated by scanning a sample across a coherent beam of light. Phase retrieval is performed using the Principal Components Generalized Projections Algorithm (PCGPA) by reducing the four dimensional data set of images to two remapped two-dimensional sets. The technique is successfully demonstrated using a Helium Neon laser to image a 350μm wide sample with 12μm resolution.
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The concept of phase retrieval from an oversampled, coherent scatter pattern was first suggested by Sayre in 1952 , and demonstrated experimentally for the first time in 1999 by Miao et. al.  in the original coherent diffraction imaging (CDI) implementation. In this version of CDI, a plane wave scatters from an isolated sample and is recorded in the far-field using a square law detector. In this regime, the diffraction pattern is proportional to the Fourier transform of the scattering electronic potential; however, because complex wave-propagation information is lost during this measurement, an iterative algorithm must be used to solve for the missing diffracted phases while concurrently reconstructing an image of the specimen. The original CDI algorithms proceed by making use of two constraints: 1) the measured diffracted amplitudes, and 2), the condition that the specimen is completely isolated (i.e. the entire finite size object is illuminated by the wave). The strength of this approach to CDI lies in the fact that it uses the amplitude of just one diffraction pattern, reducing data collection times and radiation dose. In most cases, the diffraction pattern is digitally sampled at a frequency very near the minimum necessary for unique phase retrieval. Because of this, conventional phase retrieval algorithms typically require a very large number of iterations and unique convergence is not always guaranteed.
In recent years, CDI techniques have greatly improved, enabling very high resolution 2D and 3D imaging of biological and materials samples using coherent light sources [3–11]. In order to alleviate constraints and enhance convergence properties of CDI, several new variants have been implemented with differing strengths and weaknesses. Major advances include keyhole-CDI and other curved wave-front variants [12–16], where the light illumination (rather than the sample) is truncated to satisfy the isolation constraint, or apertured illumination-CDI (AI-CDI), where an aperture is imaged directly onto the sample [17–19]. Another very successful technique that isolates the impinging wave rather than the specimen is ptychography [15,20,21]. In this approach, a well-characterized and localized coherent beam scatters from the specimen and several far-field patterns are collected from overlapping regions of the sample. The reconstruction algorithm then attempts to produce an object whose Fourier modulus is consistent with all of the collected scatter patterns. In ptychography, the wave truncation is achieved by means of a pinhole placed very near the sample being scanned.
Here we present Imaging by Integrating Stitched Spectrograms (IBISS) as a novel solution to phase retrieval. IBISS is similar to ptychography in that several diffraction patterns are collected containing overlapping, redundant information, which helps ensure unique and rapid convergence. However, in IBISS, the extra, overlapping data also provides enough information to simultaneously reconstruct the amplitude and phase of the beam, in addition to the sample, without the need for a priori information.
IBISS is a direct extension of conventional Frequency Resolved Optical Gating (FROG) [22,23] techniques to diffractive imaging. In a FROG measurement, the amplitude and phase of an ultrashort pulse are recovered by converting an unsolvable one-dimensional deconvolution into a solvable two-dimensional deconvolution. To do this, an ultrashort pulse is split into two paths, which are delayed with respect to each other. These two paths are then recombined in a nonlinear material that acts as a multiplier to gate portions of the pulse by the delayed pulse replica. The spectrum of the resulting signal is then recorded for a series of time delays. If a second harmonic generating (SHG) crystal is used, then the resulting nonlinear signal will be directly proportional to the product of the pulse with the delayed pulse. The PCGP algorithm can be used to retrieve the amplitude and the phase of the pulse using the spectrogram data. The PCGP algorithm is not constrained to inverting spectrograms generated using the pulse to gate itself; it is capable of inverting spectrograms generated by an unknown pulse and an unknown gating function. However, because the PCGP algorithm is a generalized projections algorithm, it does not form a projection onto a convex set, and is not guaranteed to converge. In spite of this, generalized projections algorithms are still expected to work well . For a general discussion and broad overview of FROG, please see  and the references therein.
The data collection process for IBISS is directly analogous to that of FROG. Rather than gating a one-dimensional pulse in time, a spatial, two-dimensional beam is gated by a two dimensional sample which is translated in a plane perpendicular to the beam. In the same way that the signal in FROG is the product of the pulse and the delayed pulse, the signal immediately after the sample in IBISS is the product of the beam and translated sample. The two dimensional spectrum of this signal is recorded by placing a camera in the far-field of the sample or, more practically, at the focus of a Fourier transform lens, which is placed immediately after the sample. Instead of generating a two-dimensional data set composed of one dimensional vectors (representing the spectrum of the signal at different delay times), a four dimensional data set composed of two-dimensional spatial spectra at different scan positions is recorded.
The PCGP algorithm can be used to recover the amplitude and phase of the sample and beam in almost exactly the same way that it is used to recover the shape of a pulse and gate in FROG . When used to invert a FROG trace, the PCGP algorithm finds the best approximation for the two dimensional spectrogram that can be formed by remapping the outer product of a one dimensional pulse vector and a one dimensional gate vector. In IBISS, the PCGP algorithm finds the best approximation for the four-dimensional data that can be formed by mapping the outer product of two, two-dimensional matrices (images) representing the sample and the beam. To simplify programming of the PCGP algorithm for this imaging application, the four dimensional data set is remapped to a two-dimensional matrix, and the two-dimensional images of the sample and beam are remapped to vectors.
The beam and sample being measured in IBISS will be represented as follows: and where X and Y represent the displacement of the sample. The signal immediately after the sample isEq. (2)):Eqs. (1) and (3) as constraints. Equation (1), which is applied in the object domain, is used to obtain the next guess for the beam and sample as well as to construct the guess for the signal immediately after the sample. The constraint from Eq. (3), which is applied in the frequency domain, forces the amplitude of the guess for the spectrogram to be equal to the measured amplitude. Before this constraint is imposed, the error can be estimated in an RMS fashion, by summing the square of the difference between the two amplitudes and taking the square root of the sum.
In the most general case, the inversion algorithm begins with a random guess for the amplitude and phase of the beam and the sample and then uses Eq. (1) to generate a guess for the signal field, This guess is then Fourier transformed along the x and y directions to obtain a guess for the collection of spectra:Eq. (4) are replaced with the measured amplitudes, The result, is used to obtain the next guess for the beam and sample. The following is a more detailed description of how this is accomplished in the PCGP algorithm.
We will represent the discretized beam and sample as two-dimensional matrices:Eq. (5) to 1 x N2 vectors:Eq. (6) to the left by the row number minus one. The resulting matrix is shown below:Eq. (7) represents a linearized object domain signal at a particular scan position. The first N columns are the object domain signals which are formed by shifting the sample one through N steps in the Y direction and 0 steps in the X direction. The next N columns are the result of shifting the sample one through N steps in the Y direction and 1 step in the X direction and so on. These columns are reshaped into N by N sub-matrices composing the entire N2 x N2 matrix.
Now the first N x N elements in Eq. (8) are the beam multiplied by the un-shifted sample (X = 0 and Y = 0). The sub-matrix labeled X = -ΔX, Y = 0 is the beam multiplied by the sample shifted one step to the left in the X direction. The matrix labeled X = 0 and Y = -ΔX is the beam multiplied by the sample shifted one step up in the Y direction. Together the collection of sub-matrices represents the product for all combinations of Thus, there exists a straightforward transformation between the outer product and the object domain IBISS signal and vice versa. To obtain the IBISS spectrogram from the object domain signal, each sub-matrix is Fourier transformed.
The phase retrieval process uses this transformation to iterate back and forth between the object and frequency domains, imposing constraints in each. After beginning with a random guess for the phase and amplitude of both the sample and beam, these matrices are linearized and their outer product computed. Using the row shifting and sub-matrix reshaping procedure described previously, the object domain IBISS signal is produced. The spectrogram is formed by taking the two-dimensional Fourier transform of each sub-matrix of the object domain signal. At this point, the amplitude of the spectrogram is replaced with the square root of the measured spectrogram intensity data but the phase of the guess is retained. The result is inverse Fourier transformed back to the object domain. The row shifting procedure is reversed to obtain what will be called an outer product form matrix. The result is not a true outer product (Rank 1 matrix) unless the guess for the sample and beam were correct. The principal components of the outer product form matrix can be found by performing a singular value decomposition or using the power method in exactly the same way as is done using the PCGP algorithm for inverting a two dimensional FROG trace . These principal components represent the two vectors with the largest weighting factor when the outer product form matrix is decomposed into a superposition of outer products. The outer product of the principal component vectors is the best rank 1 approximation of the outer product form matrix in the least squares sense. The principal components, which form a generalized projection, are selected as the next best guess for the beam and the sample. This process is repeated until the error between the guess for the spectrogram amplitude and the measured spectrogram amplitude is sufficiently small. The possible ambiguities of the resulting reconstruction are the addition of a constant and linear phase to the sample and beam matrices (the same ambiguities that exist for FROG) [28,29].
This algorithm inherently requires that the sample either be isolated or periodic. This constraint arises because of the way the algorithm generates the guess for Esig (Eq. (8)) from the guess for the beam and sample matrices. From Eq. (8), it can be seen that in each sub-matrix, the sample matrix (Gsample) has been circularly shifted, that is, any elements that are shifted beyond the edge of the matrix are wrapped around to the other side. This method generates an accurate representation of Esig when the sample being imaged is isolated (surrounded by a completely opaque material) because the elements of Gsample that get wrapped around are either zeros or multiplied by zeros in Ebeam. This circular shifting also works for imaging periodic samples since the elements of the sample that are wrapped around exactly match the next part of the sample being illuminated as long as the scan distance is some multiple of the sample’s period.
Another limitation of this imaging technique arises due to the rapid increase of the computation time with input size. Since the pixel resolution is determined by the number of scan positions, it is desirable to record as many diffraction patterns as possible. However, the runtime of the algorithm iswhere N is the number of scan positions in one dimension.
In our experimental setup (see Fig. 1 ), a Helium-Neon laser was used to illuminate the sample and the diffraction pattern was collected by a CCD. A spatial filter was used to produce an ideal, Gaussian, laser beam intensity cross section. The spatial filter consisted of a 6 cm lens used to focus the laser beam through a 25µm circular pinhole (1.). A second 6 cm lens was placed after the pinhole to collimate the beam. Since the IBISS algorithm can reconstruct the amplitude and phase of both the sample and the beam, the exact shape of the beam incident on the sample is not critical. In order to emphasize this point, a 25µm by 25µm square pinhole (2.) was imaged onto the sample. The sample (3.) was placed on a motorized stage that moved in a plane perpendicular to the beam propagation direction (with 1 µm step size). A Fourier transform lens (L3), with a focal length of 10 cm, was placed immediately after the sample. The camera was placed at the focus of the Fourier transform lens (4.) and was mounted on a translation stage which moves in the direction of the beam path so that its position could be precisely controlled. The camera is a windowless, USB controlled, CMOS camera with 1024 by 1280 pixels, each of width 5.2µm.
The sample, shown in Fig. 1(b), was a thin (12.7um), opaque, stainless steel sheet with an L shaped hole that was cut using a focused laser. The longer leg of the L is about 350µm while the shorter leg is about 250µm. Data was acquired by performing a line-by-line scan of the sample across all X and Y positions. At each position, the program records a series of images at different exposure times, which are then combined to form a final, high dynamic range (HDR) image.
The bit depth of the camera is 8 bits (corresponding to 256 different intensity values). In order to take advantage of this entire range, it would be necessary to pick an exposure time such that the brightest pixel had the maximum value, but was still not saturated. To increase the effective bit depth, multiple pictures are captured at different exposure times. The smallest exposure time has no saturated data while larger exposure times have more and more saturated data. These images are then combined into one, HDR image by replacing the saturated data in one picture with a properly scaled set of unsaturated data from a picture taken at a shorter exposure time. This process has the effect of increasing the bit depth so that a much wider range of intensity values can be recorded and therefore finer details of the diffraction pattern can be used during reconstruction of the sample.
The PCGP algorithm requires that the number of different scan positions be equal to the number of pixels in each image. A huge number of pictures would be required if the full resolution of the camera (1280x1024) were used. Therefore the images are first cropped square and then binned to 32x32 or 64x64 pixels, since larger resolutions would result in extremely long reconstruction run times. Furthermore, the pixel resolution of the reconstructed images is equal to the number of scan positions since the collection of diffraction patterns is the same dimensionality as the outer product of the sample and beam matrices. So if 64x64 diffraction patterns are recorded, each with 64x64 pixels, then the full collection of diffraction patterns has a size of 642x642. Since this is also the dimensionality of the outer product of the sample and beam matrices, the size of those matrices must be 64x64.
The PCGP algorithm requires that the step size of the scanned sample (dX) be equal to the sampling period of the beam and sample (dx). Since x and y, the spatial coordinates of the beam and sample, are Fourier transform pairs with fx and fy, the spatial frequencies associated with the diffraction patterns, it must be the case that and In the paraxial limit, the value of fx can be estimated by where xc is the distance from the center of the diffraction pattern, λ is the wavelength, and z is the focal length of the Fourier transform lens. Combining these equations, we getEq. (9) gives a required step size of 11.88µm. The full scan distance, which limits the size of the sample to be imaged, is just the step size multiplied by the number of scan positions.
The diffraction pattern must be sampled at a frequency greater than or equal to the Nyquist frequency, which is twice the highest frequency component of the sample being illuminated. The period of the highest frequency component of the object will be the reciprocal of half of its largest feature size:Eq. (11) into Eq. (10) is:Eq. (12):Eq. (13) gives a value of 5.53x103 m−1 for the Nyquist frequency (fN). The actual sampling rate was 1/(2p) = 6.01x103m−1 which is greater than fN.
A data set was collected by imaging a 25µm by 25 µm square pinhole onto the “L- shaped” sample with a magnification of two. Images of the beam and sample, as well as the collection of diffraction patterns recorded during a full scan across the sample, are shown in Fig. 2 . The image of the beam shown in Fig. 2 was captured by removing the sample and placing another camera in exactly the same plane in order to record the shape of the beam at the sample. This camera was then removed, the sample was replaced, and the data set was recorded with a total of 64x64 diffraction patterns captured. Each image was taken with the sample at a different position relative to the beam and was binned down to a size of 64x64 pixels. Figure 2(a) and 2(b) show images of the sample and laser illumination beam respectively. Figure 2(c) shows a concatenation of all the diffraction patterns, while Fig. 2(d) shows a zoomed-in view of a few individual diffraction patterns.
Figure 3 plots the reconstructions of the amplitude and phase of both the sample and beam after running the algorithm for 100 iterations, which required approximately 30 minutes on a high powered cluster running Matlab code. The reconstructed amplitudes of the beam and sample were found to be in excellent agreement with the measured profiles (insets in Fig. 3(a) and 3(c)). Figure 4 shows the RMS error between the amplitude of the measured diffraction patterns and the guess for the diffraction patterns, at each iteration. After about sixty iterations the error no longer changes significantly, indicating that the algorithm has converged. As expected, the IBISS spectrogram resembles the object and can be used to generate a first guess for the amplitude in the algorithm. A smoothed guess for the amplitude is generated by binning each sub-matrix (diffraction pattern) to one pixel, and is shown in the inset of Fig. 2(c).
IBISS requires the collection of a large number of diffraction patterns which generally results in long data collection times. The higher the desired pixel resolution, the more diffraction patterns must be recorded. With such large amounts of data, running the PCGP algorithm on a desktop computer can become impractical and time consuming. The spatial resolution of this technique is the physical step size between sample scan positions which is constrained by the wavelength of light being used, the pixel size of the camera, the number of scan positions, and the distance from the sample to the camera. Despite these restrictions, IBISS has the following important advantages over conventional diffractive imaging techniques: the large set of overlapping diffraction patterns results in rapid convergence to the correct images of both illuminating beam and sample. Thus, the IBISS algorithm can reconstruct the shape and phase of the beam along with that of the sample without any a priori knowledge of the exact shape of the illuminating beam. This technique might find applications in electron microscopy and wavefront measurement .
The authors gratefully acknowledge support from a National Security Science and Engineering Faculty Fellowship and facilities from the National Science Foundation Engineering Research Center in EUV Science and Technology.
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