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Abstract
Determining the complete optical behavior of anisotropic samples using ptychography is always a difficult problem. We propose a novel birefringence measurement method based on mixed-state ptychography that can simultaneously obtain the azimuth angle and retardation of anisotropic samples in a single scan. By using a reference system transformation, the two mutually orthogonal object states are unambiguously retrieved, and their errors are greatly reduced. The normalized root mean square errors of the obtained azimuth angle and retardation are 0.0011 and 0.0041, respectively. This method offers more rapid data acquisition; compared to interferometric based methods, it has the advantages of unlimited field of view and a simpler, more stable setup. Further, it opens a new possibility for investigating anisotropic samples by means of mixed-state ptychography.
© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ptychography is a high-resolution coherent diffractive imaging (CDI) technique that uses an iterative retrieval algorithm instead of a physical image-forming optic. It reconstructs the amplitude and phase of both the probe and the object by using multiple diffraction patterns with overlapping illuminated regions [1–3]. Among the various CDI techniques, ptychography is particularly robust owing to redundancy in the data; it has the advantages of a wide field of view and fast convergence [4–7]. In the last few years, it has become a competitive method for visible light [8–10] and short-wavelength radiation including electron beams [11,12] and X-rays [13–15]. Except various kinds of samples, many valuable parameters related to the phase can be measured by ptychography. However, for anisotropic samples, such as biological tissues (molecular anisotropy) [16] and anisotropic materials (under an external load or in a relaxed state) [17,18], the presence of strong birefringence limits the applications of ptychography. Determining the complete optical behavior of these anisotropic samples using ptychography is always a difficult problem. Ferrand et al. [19] recently proposed a measurement scheme called vector formalism that uses a set of linearly polarized probes and polarization analyzers; simulations showed that this scheme has the ptychographic capability to retrieve the full anisotropy map of a sample [19,20]. Anthony et al. [21] obtained quantitative stress information (isochromatic, isoclinic, and isopachic) for a disc under diametrical compression using photoelastic ptychography in an experiment. Both methods, however, require multiple ptychographic scans and reconstructions under different optical polarization configurations. The data collection is a lengthy process, and the requirements for sample stability and matching during these reconstructions are high. Consequently, the prospect of using ptychography for birefringent objects is challenging.
Thibault and Menzel [22] showed in 2013 multiple probe and object modes could be reconstructed using a mixed-state ptychography algorithm. Mixed-state ptychography has already been applied to partially coherent illumination, imaging of vibrating samples, detector point spread, etc [23–26]. For anisotropic samples, the essence of optical birefringence is to decompose the incident beam into two orthogonally polarized beams with different refractive indeces [27,28]. Thus, mixed-state ptychography is a good choice for birefringent objects, although its use for this purpose has not been reported in the literature. In this paper, we propose a straightforward method based on mixed-state ptychography that can obtain quantitative birefringence maps of anisotropic samples from only one measurement. The method not only retains all the advantages of ptychography compared to other interferometric based approaches, which include an adjustable field of view, diffraction-limited resolution, and small nonlinearity errors, but also has a simpler optical setup and more rapid detection. The validity of our proposed method is verified. Furthermore, the reconstruction ambiguities and measurement performance are investigated.
2. Method
A mixed-state ptychographic iterative engine (PIE) algorithm allows simultaneous reconstruction of multiple probe and object states that are mutually incoherent. The data collection process is similar to that in ptychography. The anisotropic sample is scanned by a localized illumination, or probe, and a sequence of diffraction patterns is recorded in the far field. Considering its birefringent nature, the entire propagation process is described in combination with Jones matrix [29]. The probe needs to be linearly polarized light with a polarization angle α with respect to reference x-axis, which can be given as
where r is the object-space position coordinate, and p(r) is the illumination function. The sample birefringence property R(r) with phase retardation δ(r) and azimuth angle θ(r) of the fast axis may be expressed asInteraction of the probe and sample can be described by matrix multiplication and produces the exit wave ψ(r):where sj is the j-th sample translation shift, and the components ψj,x(r) and ψj,y(r) refer to the x and y directions, respectively. According to Eqs. (1) and (2), the two components have the formWe may form an object O(r) containing two orthogonal states Ox and Oy, which are responsive to different probe polarization states, so the exit wave becomes
The exit wave propagates freely to the detector plane located in the far field. Because its two components, ψj,x and ψj,y, have mutually orthogonal polarization states, the measured intensity onto which they are simultaneously projected iswhere q is the reciprocal space coordinate, F{•} represents the Fourier transform, J is the total number of diffraction patterns, and Ψj,x and Ψj,y are the exit waves in x- and y-polarized directions at the detector plane, respectively. Consequently, the entire process can be thought of as the interaction of two probe states and the corresponding object states, which produces two wavefronts that sum incoherently to give the measured intensity. The relative weighting of the two probe states is related to the polarization angle α.The existing mixed-state PIE reconstruction algorithm is adopted to solve for the object states Ox and Oy. The ptychographic reconstruction must satisfy the modulus constraint, which holds that the sum of the intensities of all the diffraction waves equals the measured intensity. Thus, the corrected wave function at the detector plane is
where the superscript n represents the n-th iteration. The corrected exit wave function at the object plane can be obtained by the inverse Fourier transform, . Further, an update rule similar to that of the PIE is applied for each object state:where * denotes the complex conjugate, ε is a constant parameter used to avoid a zero value of . The illumination function p(r) can be updated using a similar approach in parallel or measured in advance. In either case, the polarization angle α is a known quantity. Finally, the amplitude and phase of both Ox and Oy are reconstructed.Because the function of each object state is associated with the phase retardation δ(r) and azimuth angle θ(r), the complex amplitudes retrieved via ptychography can be used to reveal the birefringence property of a sample under investigation. Using Eqs. (4) and (5), we can determine θ(r) and δ(r) by employing trigonometric operations:
where φr,x and φr,y are phase functions of the reconstructed object states Or,x and Or,y, respectively, and φr,x-y is the phase function of the complex amplitude Or,x-Or,y. Because each state reconstruction has an arbitrary constant phase offset, to circumvent the need for a priori knowledge of the investigated sample, an empty area is created artificially in which only the diffraction pattern of the probe is used, and the resulting reconstructions of object states show no phase shift. By enforcing this condition in a reconstruction together with the actual data, this problem is solved globally. In addition, the value of δ(r) from Eq. (9) lies in the range [−π, π], so we need to unwrap it in the presence of a larger stress [30]. The 2D quality-guided unwrapping algorithm can be adopted to generate smoothly varying phase maps [31]. Note that if the anisotropic sample has absorption property A(r) (A is the real number between 0 and 1), the above analysis is still valid, and A(r) can be obtained by the ratio of the total intensity of the exit wave to the probe intensity.3. Simulation result
The performance of our approach is first evaluated using a set of simulated data. The phase retardation and azimuth angle distributions of the birefringent specimen used for calculation are shown in Fig. 1(a). The retardation is inversely proportional to the radius, and the fast axis is along the radial direction. The parameters are chosen to realistically model a visible-light experiment with a wavelength of 633 nm and a probe approximately 1 mm in diameter, which is illustrated in Fig. 1(b). The probe is known and its polarization angle α is π/8. The corresponding modulus and phase of the two object states are given in Fig. 2 and clearly show different distributions owing to their different polarization states. The detector has 512 × 512 pixels (pixel size, 8.3 μm) and is placed 150 mm downstream from the object. The diffraction patterns are generated from a 10 × 10 regular grid with a random offset to avoid the raster grid pathology. The grid interval and the maximum random offset are 249 and 83 μm, respectively. The recorded diffraction data are discretized and in the range of [0, 255] (detector bit depth, 8 bit). We add Poisson noise to mimic the shot noise of the signal. Figure 1(c) shows one of the diffraction patterns.
Fig. 1 (a) Simulated specimen. Color scale indicates the retardation in radians. Superimposed white lines indicate the direction of the fast axis for the birefringent area. (b) Probe with modulus encoded as brightness and phase encoded as hue. (c) One of the diffraction patterns.
The reconstruction is repeated 10 times with random guesses of all the components of O(r). The modulus varies within the range of [0, 1]. All reconstructions are completed within 1000 iterations. Figure 3 shows the results of two reconstruction runs, each of which used different initial object guesses. For simplicity, only the phase distributions are shown in Fig. 3. Clearly the reconstructed object states are different in each case. In light of these results, in the following section the relationship between the various probe and object reconstructions will be further explored.
Fig. 3 Two different sets of reconstructions from the same data set using different initial guesses. (a) The first set of raw reconstructions. (b) The second set of raw reconstructions. The color scale applies to all the images.
3.1 Reconstruction ambiguities
In this section, we explain the ambiguous nature of the relationship between the reconstructed object states and the actual states. Our entire analysis in Section 2 is based on a fixed x–y coordinate system in the laboratory (Fig. 4). Further, the two orthogonal states are selected along the x and y directions. The corresponding two probe states are p(r) cosα and p(r) sinα, respectively, so their relative weighting is related to the polarization angle α. Because a linearly polarized probe can be decomposed into any two mutually orthogonal states, we assume that the physical probe is effectively modeled by two orthogonal probe states under a new coordinate system X–Y, which is formed by rotating the initial x–y system by an angle β. Using the vectors r and r′ to express the position coordinates (x, y) and (X, Y) of the same point in the two coordinate systems, according to the physical meaning of each parameter, we have
where ' denotes the parameters in the X–Y coordinates. Equation (11) shows that the polarization angle of the probe under the new reference frame, X–Y, changes to α + β, which means that the relative weighting has changed. Likewise, in combination with Eq. (4), Eq. (11) shows that the corresponding two independent object states have also changed. Note that the formed diffraction intensities do not change: I'j(u') = Ij(u). Thus, it is not sufficient to use only the modulus constraint, which holds that the sum of the intensities of all the diffraction states must match the intensity recorded by the detector. The polarization angle, as a known quantity, adds another constraint on the relative weighting of the probe states, thus ensuring that the ptychographic reconstruction is performed under the laboratory coordinates x–y.Noted that when β = π/2 − 2α, as shown in the coordinate system x'–y' in Fig. 4, the two orthogonal probe states are the same as that in the x–y coordinates, P'x' (r') = Py(r), P'y' (r') = Px(r). However, the corresponding two object states, O'x' (r') and O'y' (r'), are completely different from the states Oy(r) and Ox(r), and the relationships between them are complex. That is, O'y' (r') and O'x' (r') are conjugate solutions of Oy(r) and Ox(r), and hence cause reconstruction ambiguities. Certain linear combinations of the two sets of solutions also satisfy the relative weighting constraints. Note that the two sets of results in Fig. 3 have rotational symmetry with each other owing to the rotational symmetry of the phase retardation used in the simulation, which cannot represent a common situation.
3.2 Resolving the ambiguities
To resolve the ambiguity in the reconstructed object states, a further condition must be applied. If the reference frames x–y and x'–y' are simultaneously rotated by the same angle β (Fig. 4), the relative weightings between the probe states in the X–Y and X'–Y' coordinates are tan(α + β) and tan(α−β), respectively. Hence, a reference system transformation can be used to eliminate the ambiguities. We repeat our reconstruction with proposed method and the same parameters were used. As before, the standard mixed-state PIE algorithm on the fixed x–y coordinates is run for 200 iterations. Then the reference system is transformed to the X–Y coordinates (rotated by β). The initial object states are created according to Eqs. (4) and (11), and the retrieval algorithm is run for 200 iterations. Finally, the reference system is returned to the x–y coordinates. Six hundred iterations of the retrieval algorithm are required before the changes to the reconstructed object states between iterations become negligibly small. Figure 5 shows the complex reconstructed object images after the removal of the phase shift from all components of O(r) to ensure that their phase is zero in the surrounding scanned area. The quality of the reconstructed images is good, and the two object states are successfully reconstructed without any cross-talk.
Fig. 5 Object states reconstructed using the reference system transformation. The modulus (a) and phase (b) of Or,x, the modulus (c) and phase (d) of Or,y.
By using the recovered complex values Or,x and Or,y, the distributions of the phase retardation δ(r) and azimuth angle θ(r) are obtained using Eqs. (9) and (10), as shown in Fig. 6. The retrieved annular retardation and radial orientation of the fast axis agree well with the initial model. For a detailed comparison, Fig. 7 gives the retrieved and true values along the white dotted lines in Fig. 6, where the blue solid line represents the retrieved values, and the red dotted line represents the true values. The normalized root mean square (NRMS) errors of the phase retardation and azimuth angle on a 200 × 200 pixel central area of the reconstruction are 0.0041 and 0.0011, respectively. We find that the discontinuities and maximum deviation in Fig. 6(b) occur in those locations where the direction of the fast axis is perpendicular or parallel to the polarization direction of the probe. This is mainly because the denominator of Eq. (10), which can be reduced to according to Eq. (4), is 0 when θ is equal to α or α + 0.5π. The discontinuous data in Fig. 6(b) were interpolated to form the retrieved values in Fig. 7(b). As is the case in ptychography, the edge of the specimen contains some reconstruction errors because of fewer scan positions overlapped there, leading to a slight deviation from the ideal case [see Fig. 6(a)]. Considering these factors, the ptychographically determined data fit very well. In addition, affected by the discretized recorded data, higher resolution images and better results can be produced using a detector with higher dynamic range [32].
Fig. 6 Retrieved fast axis orientation (a) and retardation (b). The images have been cropped to show only the specimen.
Fig. 7 Comparison of true and retrieved azimuth angle (a) and retardation (b) along white dotted lines in Fig. 6(a) and (b), respectively
4. Performance analysis
4.1 Effect of polarization angle
As is well-known, the sensitivity of the birefringence property to linearly polarized light is related to its polarization angle α. Thus, the effect of α on the convergence rate is analyzed. The normalized sum of the squared deviations S is used to describe the deviations of the calculated intensities in iteration n from the recorded intensities. The differences between the retrieved object states and original object states Om are evaluated using the NRMS error metric Em. We compare only the areas of the object where five or more scan positions overlap. S and Em are calculated as follows:
For the initial x–y coordinates, the changes in S and Em with the number of iterations (n < 200) for different polarization angles are plotted in Fig. 8. The reconstructions are performed with random initial guesses for the object states Ox and Oy. The intensities converge very rapidly, and the deviation is reduced to less than 1% after 10 iterations for all polarization angles. However, the retrieved object states converge much more slowly than the intensity. Further, the convergence rates of Ox and Oy are different for the same polarization angle. The main reason is that the polarization angle α affects the weighting of the two probe states or the proportions of the object states in the recorded intensities. The further from π/4 the polarization angle is, the greater the difference in the convergence rate is. In other words, as the proportion of the object state Ox or Oy in the recorded intensities increases, Ox or Oy converges more rapidly. The convergence speed of the intensity is determined by Ox, which is dominant in the intensities. The errors of Ox and Oy decrease to some extent, and then stagnate, as a result of the reconstruction ambiguities described in Section 3.1.
Fig. 8 Effect of polarization angle α on convergence rate. Changes in intensity deviation S (a) and error Em (b) with number of iterations
4.2 Performance analysis
Figure 9 shows the changes in S and Em with the number of iterations using the proposed reference system transformation for different rotation angles β when α = π/8. Compared to the direct PIE retrieval without coordinate system rotation, our method converges more rapidly because the rotation effectively resolves the ambiguities, which also greatly reduces the intensity deviation and the errors of the reconstructed object states. Ex and Ey [bottom and top plots in Fig. 9(b)] are reduced by 18 and 13 times, respectively. Further, the final accuracies of Ox and Oy are independent of the rotation angle β, which affects only the convergence speed and error in the X–Y coordinates (n from 201 to 400). This result indicates that our method unambiguously retrieves the entire set of unknowns.
Fig. 9 Changes in intensity deviation (a) and object errors (b) with number of iterations for different rotation angles β. The top and bottom plots in (b) correspond to Oy and Ox, respectively.
Further, we perform simulations to compare the convergences of the proposed approach under different polarization angles α, as shown in Fig. 10. All the errors are reduced when our method is used, and the relationship between the final errors and α is still consistent with Fig. 8(b). Ey is always greater than Ex except at α = π/4, where Ey is equal to Ex. The errors Ex and Ey are reduced on average by 18.2 and 14.3 times, respectively. The NRMS errors of the obtained azimuth angle and retardation are shown in Table 1. The NRMS values for different polarization angles are essentially the same. On the basis of the error curves in Fig. 10, we employ a polarization angle of π/8 to measure anisotropic samples.
Fig. 10 Comparison of errors of proposed approach at β = π/12 and that of the approach without rotation for different polarization angles α. (a) Error Ex. (b) Error Ey
Table 1. NRMS errors of obtained azimuth angle and retardation at β = π/12 for different polarization angles
In addition, we execute two rotations in the process of reference system transformation. First, we rotate the reference system to the X–Y coordinates; then we return it to the x–y coordinates. Figure 11 shows that two rotations are sufficient for our purpose. More rotations are unhelpful for reducing the errors. These conclusions are verified for other anisotropic samples, and the same error curves are obtained, indicating that the above simulations have general validity.
Fig. 11 Relationship between reconstructed error and number of iterations using different rotation times
5. Conclusion
We proposed a birefringence measurement scheme based on mixed-state ptychography for the first time. The multi-state ptychographic algorithm based on the Jones matrix of birefringence property is derived. By decomposing the sample into two mutually orthogonal object states, where each object state responds to only one probe state, the azimuth angle and retardation can be simultaneously obtained by a single ptychography scan. This method offers a more environmentally stable setup and takes less time for data acquisition. A mathematical analysis reveals inherent ambiguities in the reconstructions and shows how these ambiguities can be resolved by a reference system transformation. The validity is verified by direct comparison to the approach without rotation. The errors of the reconstructed object states Ox and Oy are reduced by 18 and 13 times, respectively. The NRMS errors of the obtained azimuth angle and retardation are 0.0011 and 0.0041, respectively (α = π/8, n = 1000). The analysis shows that the convergence rate and errors are closely related to the relative proportions of the two object states in the recorded diffraction patterns, which is determined by the polarization angle. This conclusion also applies to other mixed-state ptychography techniques. In addition, this method has some limitations. There should be an empty scanning area to remove the phase shift. And it is very complicated to deal with anisotropic sample with a superimposed phase distribution, which need further analysis. The experiments is currently in preparation and some more detailed analysis such as the dynamic range of detector and measure accuracy need to be made. We believe that this method provides an effective means of quantitatively determining the anisotropic distribution of sample and a new potential to investigate anisotropic samples by means of mixed-state ptychography.
Funding
Natural Science Foundation of Shanghai, China (17ZR144820); Foundation of the Chinese Academy of Sciences, China (CXJJ-17S060)
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