Resolving power of diffraction imaging with an objective: a numerical study

Diffraction imaging in far-field can detect 3D morphological features of an object for its coherent nature. We describe methods for accurate calculation and analysis of diffraction images of scatterers of single and double spheres by an imaging unit based on microscope objective at non-conjugate positions. Quantitative study of the calculated diffraction imaging in spectral domain has been performed to assess the resolving power of diffraction imaging. It has been shown numerically that with coherent illumination of 532nm in wavelength the imaging unit can resolve single spheres of 2μm or larger in diameters and double spheres separated by less than 300nm between their centers. © 2017 Optical Society of America OCIS codes: (110.0180) Microscopy, (110.1650) Coherence imaging, (100.2960) Image analysis. References and links 1. G. C. Salzman, S. B. Singham, R. G. Johnston, and C. F. Bohren, "Light scattering and cytometry," in Flow Cytometry and Sorting, M. R. Melamed, T. Lindmo, and M. L. Mendelsohn, eds. (Wiley, New York, 1990), Ch. 5. 2. A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, "Cellular organization and substructure measured using angle-resolved low-coherence interferometry," Biophys. J. 82, 2256-2264 (2002). 3. M. M. Hanczyc, S. M. Fujikawa, and J. W. Szostak, "Experimental models of primitive cellular compartments: encapsulation, growth, and division," Science 302, 618-622 (2003). 4. K. V. Gilev, M. A. Yurkin, E. S. Chernyshova, D. I. Strokotov, A. V. Chernyshev, and V. P. Maltsev, "Mature red blood cells: from optical model to inverse light-scattering problem," Biomed. Opt. Express 7, 1305-1310 (2016). 5. M. Bessis, and N. Mohandas, "A diffractometric method for the measurement of cellular deformability," Blood cells 1, 307-313 (1975). 6. S. Holler, Y. Pan, R. K. Chang, J. R. Bottiger, S. C. Hill, and D. B. Hillis, "Two-dimensional angular optical scattering for the characterization of airborne microparticles," Opt. Lett. 23, 1489-1491 (1998). 7. J. Neukammer, C. Gohlke, A. Hope, T. Wessel, and H. Rinneberg, "Angular distribution of light scattered by single biological cells and oriented particle agglomerates," Appl. Opt. 42, 6388-6397 (2003). 8. X. Su, S. E. Kirkwood, M. Gupta, L. Marquez-Curtis, Y. Qiu, A. Janowska-Wieczorek, W. Rozmus, and Y. Y. Tsui, "Microscope-based label-free microfluidic cytometry," Opt. Express 19, 387-398 (2011). 9. K. M. Jacobs, L. V. Yang, J. Ding, A. E. Ekpenyong, R. Castellone, J. Q. Lu, and X. H. Hu, "Diffraction imaging of spheres and melanoma cells with a microscope objective," J. Biophotonics 2, 521–527 (2009). 10. K. M. Jacobs, J. Q. Lu, and X. H. Hu, "Development of a diffraction imaging flow cytometer," Opt. Lett. 34, 2985-2987 (2009). 11. K. Dong, Y. Feng, K. M. Jacobs, J. Q. Lu, R. S. Brock, L. V. Yang, F. E. Bertrand, M. A. Farwell, and X. H. Hu, "Label-free classification of cultured cells through diffraction imaging," Biomed. Opt. Express 2, 17171726 (2011). 12. Y. Sa, J. Zhang, M. S. Moran, J. Q. Lu, Y. Feng, and X. H. Hu, "A novel method of diffraction imaging flow cytometry for sizing microspheres," Opt. Express 20, 22245–22251 (2012). 13. Y. Feng, N. Zhang, K. M. Jacobs, W. Jiang, L. V. Yang, Z. Li, J. Zhang, J. Q. Lu, and X. H. Hu, "Polarization imaging and classification of Jurkat T and Ramos B cells using a flow cytometer," Cytometry A 85, 817-826 (2014). 14. R. Pan, Y. Feng, Y. Sa, J. Q. Lu, K. M. Jacobs, and X. H. Hu, "Analysis of diffraction imaging in nonconjugate configurations," Opt. Express 22, 31568–31574 (2014). 15. J. Zhang, Y. Feng, W. Jiang, J. Q. Lu, Y. Sa, J. Ding, and X. H. Hu, "Realistic optical cell modeling and diffraction imaging simulation for study of optical and morphological parameters of nucleus," Opt. Express 24, 366-377 (2016). 16. M. Portnoff, "Time-frequency representation of digital signals and systems based on short-time Fourier analysis," IEEE T. Acous. Speech Signal Proces. 28, 55-69 (1980). 17. M. A. Yurkin, and A. G. Hoekstra, "The discrete-dipole-approximation code ADDA: capabilities and known limitations," J. Quant. Spectrosc. Radiat. Transfer 112, 2234-2247 (2011). 18. H. Wang, Y. Feng, Y. Sa, Y. Ma, R. Pan, J. Q. Lu, and X. H. Hu, "Acquisition of cross-polarized diffraction images and study of blurring effect by one time-delay-integration camera," Appl. Opt. 54, 5223-5228 (2015). 19. X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003). 20. P. Zhang, P. M. Goodwin, and J. H. Werner, "Fast, super resolution imaging via Bessel-beam stimulated emission depletion microscopy," Opt. Express 22, 12398-12409 (2014).


Introduction
Angle-resolved study of elastically scattered light in far-field led to various tools for assay of micrometer-sized particles including biological cells by the contrast mechanism based on the 3D heterogeneity in refractive index (RI) [1][2][3][4].In comparison, imaging of coherent light scatter has been much less explored for the challenges to acquire and assess high-contrast images [5][6][7][8].In recent years, we have developed a diffraction imaging flow cytometry (DIFC) method for measurement of high-contrast images from micrometer-sized particles carried by a laminar flow through the focus of an incident laser beam [9][10][11][12][13].The essential design of DIFC imaging unit contains an infinity-corrected microscope objective of 0.55 in numerical aperture (NA), a tube lens and an imaging sensor placed at its focal plane Γ im as illustrated in Fig. 1(A).Previously we have developed and validated a method for accurate simulation of diffraction imaging process combining a vector wave model on light scattering and a geometric model for tracing the "rays" through the imaging unit [14,15].The new method can reproduce the diffraction images (DIs) measured at non-conjugate positions by varying angular cone of light detection for enhanced image contrast.This variability, however, makes it difficult to determine the resolving power of DIFC because no analytical relations exist between field-of-view (FOV) and scatterer's morphology at a non-conjugate position.In this report we describe and apply a method based on the short-time-Fouriertransform (STFT) algorithm [12,16] for analyzing DIs and finding the ability of the imaging unit to resolve small morphological variations in a scatterer made of two spheres, which is similar to the approach for deriving the Abbe and Rayleigh criterions on resolving two objects except that the spheres are connected in 3D space.

Method
It has been shown experimentally and numerically that the imaging unit for DI acquisition can record similar patterns of fringes or speckle distribution with the unit translated to an offfocus position by Δx > 0 [9,10,14].Positive Δx corresponds to moving the unit, including its sensor at Γ im , towards scatterer from a focusing position, which is defined as the location with Γ im conjugate to the plane of scatterer at the flow chamber center.Changing Δx varies the angular cone for DI acquisition and allows for contrast and FOV optimization [14].Examples of measured and calculated DIs of 640x480 pixels are presented in Fig. 1(b), 1(c), Fig. 3 and Fig. 4 with Δx set to 150 μm.One can match easily the measured DIs with calculated ones exhibiting high resemblance for scatterers of single and double spheres.We ignored the effect of CCD noise in consideration of resolution here since the measured DIs are of high contrast with dark current noise less than 0.2% of full scale in sensor's 12-bit pixel output.
Calculation of DIs consists of three steps as elucidated by Fig. 1(a).First, a scatterer is defined by its 3D distribution of complex RI as n s (r, λ) = n sr (r, λ) + in si (r, λ) with λ as the incident light wavelength.Elastic light scattering is modeled by a method of discrete dipole approximation (DDA) using a plane wavefield for the incident beam.We chose an opensource software ADDA which imports n s (r, λ) in a host medium of n h to obtain the Mueller matrix of elements S ij (θ s , φ s ) for calculation of scattered light intensities of different polarization attributes [4,17], where i or j = 1, 2, 3 4 and θ s and φ s are respectively the polar and azimuthal angles of scattered light "rays".For simplicity, we limit analysis here to calculation of unpolarized DIs by S 11 (θ s , φ s ).Extension to polarized DIs is straightforward using other elements [15,18].In the second step, the scattered light intensity proportional to S 11 (θ s , φ s ) is projected to an input plane Γ in defined in Fig. 1(a).Γ in is in the host medium of water with θ wm as the maximum cone angle from the x-axis for light detection and details of projecting S 11 to Γ in were described elsewhere [11].Finally, DI is calculated by ray-tracing each pixel in the input image at Γ in along a direction defined by (θ s , φ s ) through a virtual imaging unit with the same design as the one in our DIFC system to produce I(z, y) at Γ im using a commercial ray-tracing software (Zemax, 2009).The ADDA based DI calculation for single spheres has been validated against Mie based results as described in [14] and measured DIs acquired with our DIFC system.All computations except ADDA and Zemax were performed with in-house codes built on MATLAB ((MathWorks, 2013a).The ADDA simulations were carried out with value of dpl (dipoles per wavelength) set to 20 and the values of RI for spheres and host medium chose as those of polystyrene and water for λ = 532nm [19].For this study, we set out to quantitatively analyze the ability of DI on resolving morphological changes in scatterers of high symmetry or I(z, y) with oscillation in brightness.The STFT algorithm has been extended to transform I(z, y) into S(f, θ) in a 2D domain of frequency and angle.First, a pixel line I θ is sampled from I(z, y) at an angle θ from the z-axis as illustrated in Fig. 1(c) followed by Fourier transform on I θ after multiplication by a Gaussian window of width w.These operations can be described in continuous form as [16] 1) provides an effective way to identify a single sideband peak for characterizing local oscillations of different orientations.

Single spheres
With 2θ wm = 46.4°as determined by the choice of Δx = 150mm [14], DIs of single sphere were calculated with d ranging from 1.0 to 8.5μm with step of 0.   After DI simulations, STFT analysis was performed to transform these images into the 2D space of f and θ with the line sampling angle θ varied from 0 to 179° with a step of 1°.To further compress the spectral data for efficient analysis, we reduced the θ dependence of sideband parameters f s and M sm for 180 angles to 36 angular bands of 5° width for each bands' centers, marked as θ b , that ranges from 0 to 175°.The parameters were averaged over 3° for each θ b to minimize fluctuation.Figure 5 presents paired contour plots of f s (θ b , C) and M sm (θ b , C) with different C. One can clearly see that the DIs of double spheres produce distinct contour structures as C increases to above 0.30μm in the paired plots.Combined with the DIs shown in Fig. 4, these results indicate the resolving power of the diffraction imaging unit with λ = 532nm should be less than 0.3μm or 300nm as a scatter is "stretched" in all six directions in 3D space.

Discussions and summary
The resolving power of DI for morphological variations in scatterers is poorly understood due to the complex relations between the changes and image features.The 2D STFT method described in this report provides a resource for such investigations.Analysis on DIs by single spheres clearly shows that the STFT sideband parameters cannot be used to distinguish small spheres of d = 1 or 1.5μm.It should be pointed out that other image processing methods could be applied to extract pattern change among DIs by spheres of d < 2μm by, e.g., quantifying the size changes of the central bright spot.In contrast, the results with DIs of two spheres demonstrate the ability of DIs in 2D form to reflect a scatter's morphological changes in 3D space including the axial direction of imaging.Furthermore, each DI represented by a horizontal line in f s or M sm plot displays different angular variations (shown by colors) in Fig. 5 that could be used to detect small changes (stretched along 6 directions) in morphology.The results thus provide direct evidences that the objective based diffraction imaging has resolutions for morphological changes that are in par with the Abbe diffraction limit at about 0.5 μm on the lateral resolution of far-field imaging using conventional incoherent illumination and an objective of NA = 0.55.We also performed additional study with two spheres replaced by equally sized cylinders and similar results were obtained.Even though the resolving power of DIFC is less than those of super-resolution microscopy methods [20], the combined advantages of DIFC make it a valuable tool for assay of small particles and cells in terms of label-free acquisition, simplicity in design, fast throughput and automated image processing.
In summary, we applied and validated a method of diffraction imaging simulation to calculate DIs and developed a 2D STFT method for analysis of resolving power for an imaging unit using objective of NA = 0.55.The results show that the imaging unit can resolve morphological changes smaller than 300nm for a coherent illumination with wavelength at 532nm.
is the FOV over I θ (z) = I(z, y) with y = y c + (z−z c )tanθ and (y c , z c ) are the center coordinates of I.After tests with different window width, we fixed w at 3 times of the mean pixel distances of major peaks in I θ that yields optimized spatial and spectral resolutions.A clear sideband peak at frequency f s can be identified in |S(f, θ; z)| of DI containing oscillating patterns with this choice of w. Figure 2 presents two examples of |S|.Comparisons to STFT on parallel lines of I(z, y) and 2D Gabor transform show that Eq. ( 5 μm.These results serve as the baseline data for extraction of parameters from the STFT spectra to characterize the oscillating patterns with varying periods.As shown in Fig. 3, diffraction images of spheres with d = 1.0 and 1.5μm contain no oscillating patterns due to the finite value of θ wm and thus no sideband can be observed in |S|.In comparison, as d increases to 2.0 μm and above, a sideband appears clearly in |S(f, θ; z)| for estimating d value of the sphere from the peak frequency f s .Since the fringe patterns in DIs displayed in Fig. 3(a) are symmetric to the horizontal direction, we present in Figs.3(b) and 3(c) only the STFT spectra on lines sampled at θ = 0.

Fig. 3 .
Fig. 3. (A) Calculated DIs of single spheres with different diameter d as marked, bar = 10° in detection angle in water; (B) |S(f, 0°; z)| of calculated DIs in (A) at z of maximum M s with legends showing d and z (in unit of pixel size Δ) values, f v and f s are indicated by the arrows for the case of d = 5.0μm; (C) f s and M sm obtained from DIs versus d, the solid line is a straight line fitted to f s with M sm > 1.Other parameters are the same as those in Fig. 2. Due to the local nature of brightness oscillation in I θ , sidebands exist only in STFT spectra of sampled lines windowed at certain z's.As the Gaussian window slides on I θ , f s varies near the middle of the line and disappear at the ends.A figure of merit M s (θ,z) is defined for evaluation of existence and sharpness of a sideband in |S(f,θ; z)| as follows

Fig. 4 .
Fig. 4. Examples of calculated DIs of two spheres with d = 3.0μm, bar = 10°.The direction and magnitude of (C) are marked in each image.Other parameters are the same as those in Fig. 2.

Fig. 5 .
Fig. 5.The paired contour plots of f s (θ b , C) and M sm (θ b , C) obtained from calculated DIs of two spheres with d = 3.0μm, 0.075μm ≤ C ≤ 6.0 μm and (C) directions as marked in each image.The white dash lines indicate C = 0.30μm.Other parameters are the same as those in Fig. 2.