Spatial-domain filter enhanced subtraction microscopy and application to midIR imaging

We have experimentally investigated the enhancement in spatial resolution by image subtraction in mid-infrared central solid-immersion lens (c-SIL) microscopy. The subtraction exploits a first image measured with the c-SIL point-spread function (PSF) realized with a Gaussian beam and a second image measured with the beam optically patterned by a silicon π-step phase plate, to realize a centrally hollow PSF. The intense sides lobes in both PSFs that are intrinsic to the SIL make the conventional weighted subtraction methods inadequate. A spatial-domain filter with a kernel optimized to match both experimental PSFs in their periphery was thus developed to modify the first image prior to subtraction, and this resulted in greatly improved performance, with polystyrene beads 1.4 ± 0.1 μm apart optically resolved with a mid-IR wavelength of 3.4 μm in water. Spatial-domain filtering is applicable to other PSF pairs, and simulations show that it also outperforms conventional subtraction methods for the Gaussian and doughnut beams widely used in visible and near-IR microscopy. © 2017 Optical Society of America OCIS codes: (180.0180) Microscopy; (120.0120) Instrumentation, measurement, and metrology; (290.0290) Scattering; (170.5810) Scanning microscopy. References and Links 1. M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000). 2. Y. Fang, C. Kuang, Y. Ma, Y. Wang, and X. Liu, “Resolution and contrast enhancements of optical microscope based on point spread function engineering,” Front. Optoelectron. 8(2), 152–162 (2015). 3. O. Haeberlé and B. Simon, “Saturated structured confocal microscopy with theoretically unlimited resolution,” Opt. Commun. 282(18), 3657–3664 (2009). 4. C. Kuang, S. Li, W. Liu, X. Hao, Z. Gu, Y. Wang, J. Ge, H. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013). 5. H. Dehez, M. Piché, and Y. De Koninck, “Resolution and contrast enhancement in laser scanning microscopy using dark beam imaging,” Opt. Express 21(13), 15912–15925 (2013). 6. N. Liu, M. Kumbham, I. Pita, Y. Guo, P. Bianchini, A. Diaspro, S. A. Tofail, A. Peremans, and C. Silien, “Farfield subdiffraction imaging of semiconductors using nonlinear transient absorption differential microscopy,” ACS Photonics 3(3), 478–485 (2016). 7. A. Gasecka, A. Daradich, H. Dehez, M. Piché, and D. Côté, “Resolution and contrast enhancement in coherent anti-Stokes Raman-scattering microscopy,” Opt. Lett. 38(21), 4510–4513 (2013). 8. N. Tian, L. Fu, and M. Gu, “Resolution and contrast enhancement of subtractive second harmonic generation microscopy with a circularly polarized vortex beam,” Sci. Rep. 5(1), 13580 (2015). 9. S. You, C. Kuang, Z. Rong, and X. Liu, “Eliminating deformations in fluorescence emission difference microscopy,” Opt. Express 22(21), 26375–26385 (2014). 10. S. Segawa, Y. Kozawa, and S. Sato, “Resolution enhancement of confocal microscopy by subtraction method with vector beams,” Opt. Lett. 39(11), 3118–3121 (2014). 11. S. Segawa, Y. Kozawa, and S. Sato, “Demonstration of subtraction imaging in confocal microscopy with vector beams,” Opt. Lett. 39(15), 4529–4532 (2014). 12. Z. Rong, C. Kuang, Y. Fang, G. Zhao, Y. Xu, and X. Liu, “Super-resolution microscopy based on fluorescence emission difference of cylindrical vector beams,” Opt. Commun. 354, 71–78 (2015). Vol. 25, No. 12 | 12 Jun 2017 | OPTICS EXPRESS 13145 #285981 https://doi.org/10.1364/OE.25.013145 Journal © 2017 Received 8 Feb 2017; revised 13 Apr 2017; accepted 18 Apr 2017; published 31 May 2017 13. K. Korobchevskaya, C. Peres, Z. Li, A. Antipov, C. J. Sheppard, A. Diaspro, and P. Bianchini, “Intensity Weighted Subtraction Microscopy Approach for Image Contrast and Resolution Enhancement,” Sci. Rep. 6(1), 25816 (2016). 14. I. Pita, N. Hendaoui, N. Liu, M. Kumbham, S. A. Tofail, A. Peremans, and C. Silien, “High resolution imaging with differential infrared absorption micro-spectroscopy,” Opt. Express 21(22), 25632–25642 (2013). 15. C. Silien, N. Liu, N. Hendaoui, S. A. Tofail, and A. Peremans, “A framework for far-field infrared absorption microscopy beyond the diffraction limit,” Opt. Express 20(28), 29694–29704 (2012). 16. G. Carr, “Resolution limits for infrared microspectroscopy explored with synchrotron radiation,” Rev. Sci. Instrum. 72(3), 1613–1619 (2001). 17. M. Kumbham, S. Daly, K. O’Dwyer, R. Mouras, N. Liu, A. Mani, A. Peremans, S. M. Tofail, and C. Silien, “Doubling the far-field resolution in mid-infrared microscopy,” Opt. Express 24(21), 24377–24389 (2016). 18. S. M. Mansfield and G. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57(24), 2615–2616 (1990). 19. B. Birge, “PSOt-a particle swarm optimization toolbox for use with Matlab,”in Proceeding of the 3rd IEEE Swarm Intelligence Symposium (IEEE, 2003), pp. 973–990. 20. A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11(5), 455–459 (2012).

Subtraction microscopy has been also theoretically proposed for mid-IR imaging [14,15].These earlier proposals envisaged the exploitation of reflective objectives typically used for imaging in the mid-IR but for which the numerical apertures (NA) remain practically limited so that the PSF with a Gaussian beam exhibits a full-width at half maximum (FWHM) that is not narrower than ca.λ/1.3 in air [16,17].With a central solid immersion lens (c-SIL) [18,19], the mid-IR FWHM is reduced to λ/2 along the axis normal to the direction of polarization and the reconstruction of sparse specimens with a resolution of λ/2.6 was demonstrated when combining images recorded with crossed polarizations [17].
In this paper, we experimentally demonstrate subtraction microscopy in the mid-IR using a c-SIL by optically resolving polystyrene (PS) beads that are 1.4 µm apart (λ/2.4), using a silicon π-step phase plate to generate a one dimensional doughnut also called half-moon.Because the PSFs achieved with the c-SIL show strong side-lobes, a new subtraction method was also designed, where the Gaussian image is filtered in the spatial-domain prior to the subtraction, with a kernel matrix optimized so that the filtered Gaussian and doughnut PSFs match at their periphery.Because, it minimizes the side-lobes in the subtraction PSF, the spatial filtering of the Gaussian image leads to subtracted images that better preserve the relative intensities of the beads and that show higher spatial resolution, in comparison to the other subtraction methods [2][3][4][5][6][7][8][9][10][11][12][13][14][15].

Experimental methods
The mid-IR microscope used in these experiments has been described in [17] with the difference here that one of the two beam paths accommodates a π-step phase plate to generate the half-moon beam [Fig.1(a)].Briefly, the microscope is operated with a mid-IR PPLNbased synchronously-pumped optical parametric oscillator (LASERSPEC) pumped by a fibre laser (41 MHz, 40 ps, 1030 nm, 2.9 W, MULTITEL).The power at the objective was kept around 1 mW and the wavelength kept at 3.4 µm.The π-step phase plate was made from a double-side polished 1 cm 2 silicon plate with a single step of height 580 nm to operate at 3.4 µm.The fabrication process was based on a combination of UV lithography (SUSS MA6/BA6 mask aligner) and Reactive Ion Etching (ICP-RIE SENTECH SI 500).A cleaned i-type c-Si(100) wafer was first spin-coated (3000 rpm for 60s) with positive tone resist (SHIPLEY S1813) and the substrate was transferred to a hot plate at 115 °C and soft baked for 60s.UV lithography was then employed to define a sharp edge on the optical resist, and after resist development in a conventional MF319 solution, a Bosch reactive ion etching process was utilized to transfer the polymer pattern into the silicon substrate.An ultrasonic bath of acetone was employed for removing the organic residues.The phase plate was aligned using a mid-IR thermopile array (HEIMANN) placed in the beam path, with the step kept parallel to the linear polarization axis.We used a reflective objective of 0.4 NA (EDMUND OPTICS) with a 4 mm thick silicon c-SIL of radius 5 mm (ISP OPTICS) placed on the back side of a 1mm thick doubly polished silicon substrate (UNIVERSITY WAFER).The substrate frontside was sparsely covered with PS beads (1 µm, POLYSCIENCE) and immersed in distilled water for the measurements.The specimen reflection was measured with a MCT (HAMAMATSU) and normalized to a reference signal measured using a second MCT (THORLABS).

Results and discussion
In line with [17], the PS beads are imaged at the water/silicon interface at 3.4 µm as asymmetric depressions with strong side-lobes due to the central obscuration of the reflective objective and the large refractive index difference between Si and water [Fig.1(b)].With the π-step phase plate aligned so that the step length matches with the beam polarization, the beads are imaged as two symmetric and elongated depressions as a result of the destructive interference along the phase plate step [Fig.1(c)].The FWHM of the half-moon node is ca.1.25 µm [Fig.1(d)].Although the FWHM in the Gaussian image varies (ca.1.4-1.8µm) with the lateral offset between the SIL and the objective generating aberration when imaging away from the SIL centre, we found that slight lateral adjustments of the π-step phase plate allowed maintaining the half-moon node FHWM to 1.3 ± 0.1 µm, so that the subtraction scheme is expected to provide a uniform and enhanced optical resolution.The optical resolution with the half-moon beam is verified by imaging two pairs of PS beads roughly aligned with the horizontal axis and that are respectively 2.3 ± 0.1 µm apart and unresolved with the Gaussian beam [Fig.2].With the half-moon beam, the first pair is imaged as 3 depressions.Thus, the half-moon image of the pair is in good approximation the incoherent addition of overlapping single bead images.Remarkably, although the second pair is unresolved with the Gaussian beam, 3 depressions are also measured with the half-moon beam which warrants that the half-moon beam affords indeed a higher spatial resolution along the horizontal axis, in keeping with the half-moon node FWHM being narrower than the Gaussian peak.The subtraction operation between Gaussian and half-moon image aims at taking advantage of this higher spatial resolution whilst maintaining a peaked PSF and thus readily interpretable images.
It is shown below that the weighted subtraction methods used earlier [2][3][4][5][6][7][8][9][10][11][12][13][14][15] induce significant intensity artefacts with the Gaussian and half-moon c-SIL images.Thus we are introducing a new subtraction method that aims at better matching the two PSFs at their periphery.Our proposition is to identify a suitable linear operation of the Gaussian image pixels that is optimized to achieve a subtracted PSF that is narrow and with little or no sidelobes.The method involves the identification of a kernel matrix K that minimizes the error function 2

|| ||
in a region of interest (ROI).Gaussian and half-moon (or doughnut) beams, and the value of each pixel in K are the parameters to be optimized.The operator ⊗ marks the discrete convolution.The ROI is defined in the periphery of the PSFs to exclude their centres, where it remains obviously advantageous to maintain a maximum intensity difference between the images.In this paper, ROI  was minimized using a particle swarm optimization (PSO) algorithm [19], although it is expected that other minimization methods can also be applied.Once, K is identified, the subtraction image is computed as Where g I and d I are respectively the Gaussian and half-moon (or doughnut) images.Notably the kernel must be computed only once for each pair of PSFs, and the method should be applicable to any implementation of subtraction microscopy.The spatial-domain filter enhanced subtraction method is first discussed for simulated Gaussian and doughnut PSFs [Fig.3(a)] as these are ubiquitous to confocal microscopy in the visible and near-infrared [2][3][4][5][6][7][8][9][10][11][12][13].All calculations were done in MATLAB®.The circular symmetry of the PSFs was imposed to the kernel matrix and the PSO converged rapidly [Fig.3(b)].The optimized kernel matrix exhibits pixel values that are zero or nearly zero for radius larger than the Gaussian FWHM, and although the kernel is not unique it systematically shows a ringed doughnut shape.The spatial-domain filtered Gaussian PSF matches very well with the doughnut PSF within the ROI whilst remaining peak-shaped [Fig.3(a)].
The same Gaussian and doughnut PSFs were used to simulate images (by convolution) of specimens made of four equivalent single pixel objects: one isolated to assess the subtraction PSF, and three aligned and closed-by to assess the intensity distortions.The subtraction was then computed from the simulated images according to Eq. ( 2), as well as according to the recently introduced intensity weighted subtraction [13] and to the constant weight scheme [2][3][4][5][6][7][8][9][10][11][12]14,17].For IWS, following [13], the simulated images were normalized to one and the subtraction performed according to 1 .
where the multiplication is applied pixel per pixel.For the constant weight subtraction, the images were used as generated with the PSFs defined in Fig. 3(a) and the subtraction computed according to where α was kept 1.The line profiles for the three subtraction methods and measured across the four identical single pixel objects are presented in Fig. 3(c).Line profiles computed for four similarly arranged single pixel wide objects but whose length were increased to simulate four identical thin nanowires are presented in Fig. 3(d).The effectiveness of the spatialdomain filter is demonstrated by observing that, for both single pixel objects and nanowires, K I shows the least negativities in the subtracted PSF, shows the highest uniformity in the intensity recorded over the four objects, and shows the highest visibility for the three closedby objects.Thus, it is clearly possible to generate a kernel that enhances the subtraction effectiveness for the Gaussian and doughnut pair presented in Fig. 3(a).The method must however be experimentally verified and this is done here with mid-IR c-SIL images recorded with Gaussian and half-moon beams.The half-moon enhances the spatial resolution along a single axis, crossed with the direction of polarization, and the subtraction scheme is validated using the Gaussian and half-moon images presented in Figs.4(a) and 4(b), where two isolated beads and two pairs of beads (closed-by and nearly aligned with the half-moon narrow profile) are found.The homogeneity of the beads and their arrangement on the surface were first verified using the specimen reconstruction method detailed in [17] and that applies to sparse specimens.Briefly, the reconstruction involves defining the specimen as a collection of point objects whose position and size are optimized so that the error between experimental and computed images is minimized.In [17], dense islands of 1 µm PS beads apart by λ/2.6 were resolved along both axes at 3.4 µm using cross-polarized Gaussian beams, with the same mid-IR c-SIL microscope as used here.The specimen sparsity is enforced using the iterative method of [20] and started with 11 randomly placed point objects of random size.The best reconstruction out of a series of 50 is presented in Fig. 4(c) and retains 8 objects.Their arrangement matches with our expectations that all the beads are of same size [with the exception of the two marked by an arrow in Fig. 4(c)] and that the upper half of the images show two pairs of beads, estimated to be apart by 1.5 ± 0.1 µm, and that the lower half include two single beads.
For computation of the subtraction schemes, after normalization to a background intensity of 1, the experimental c-SIL images were inverted and shifted to set their background at zero, so that the beads appear as protrusions within a background averaging to zero.The images of one of the single beads [marked by an arrow in Figs.4( The half-moon beam exploited in this paper is a first attempt at optical phase patterning in the mid-IR to enhance the spatial resolution and the resolution is optically improved down to ca. λ/2.4only along a single axis.The fabrication of phase plates of more complex design will afford the generation of doughnut beams and thus improve the resolution uniformly in the imaging plane, as this is currently possible in the visible and near-IR [2][3][4][5][6][7][8][9][10][11][12][13].Moreover, reducing the noise in the mid-IR experimental images by improving laser and microscope stability will further strengthen the advantages of our spatial-domain filter enhanced subtraction.

Conclusion
PS beads (1 µm) were imaged with a silicon c-SIL using Gaussian and half-moon beams to demonstrate the enhancement of resolution by subtraction in mid-IR microscopy.The halfmoon beam was generated by introducing a silicon π-step phase plate optimized for a central wavelength of 3.4 µm.Along the axis normal to the direction of polarization, the width of the half-moon node is 1.25 µm, and we verified that an all optical spatial resolution <λ/2 is possible by resolving beads that are 1.4 ± 0.1 µm apart (λ/2.3).The weighted subtraction exploited in earlier studies when imaging with Gaussian and doughnut beams [2][3][4][5][6][7][8][9][10][11][12][13][14][15]

Fig. 1 .
Fig. 1.(a) Mid-IR c-SIL scanning microscope.M: mirror; F1 and F2: near-IR filters; L1 and L2: beam expander/collimation; ChW: mechanical chopper wheel; LP: linear polarizer; BS: pellicle beam splitter; S: beam shutter (blocking the Gaussian beam path as shown or used to block the half-moon beam path); TP: thin plate; π-PP: π-step phase plate; Pol: linear polarization axis; MCT: mid-IR MCT detectors; L: lens; OBJ: reflective objective; SIL: silicon central solid immersion lens; SPL: sample with specimen frontside immersed in water; NDF: neutral density filter.(b) Images of a single 1 µm PS bead recorded with the Gaussian beam.Sale bar 5 µm.The double-tipped arrow marks the direction of polarization.The image was normalized to a background of 1. (c) Same as (b) with the half-moon beam.(d) Line profiles extracted from (b) and (c).

Fig. 2 .
Fig. 2. (a) Gaussian image of 1 µm PS beads.Scale bar 10 µm.(b) Same with half-moon beam.(c) Line profiles (i) and (ii) extracted from (a) and (b) showing that 2 nearby beads are observed as 3 depressions with the half-moon beam (see arrows).(d) Line profiles (iii) and (iv) extracted from (a) and (b) showing that 2 beads unresolved with the Gaussian beam are also observed as 3 depressions with the half-moon beam (see arrows).
of a point-object recorded with the

Fig. 3 .
Fig. 3. (a) Line profiles of simulated Gaussian (blue, unbroken) and doughnut (orange) PSFs, with spatial-domain filtered Gaussian (blue, dashed) after optimization of the kernel.(b) Evolution of the error function with the number of PSO iterations, with the optimized kernel shown in inset.(c) Line profiles of Gaussian and subtracted images for a specimen made of one isolated pixel object and 3 adjacent pixel objects, all 4 objects being aligned (thick black dashed line): (i) shows Gaussian image (orange, dashed), (ii) shows K I (orange), (iii) shows

Fig. 4 .
Fig. 4. (a) Mid-IR c-SIL image of PS beads recorded with the Gaussian beam.Scale bar 5 µm.(b) Same as (a) with the half-moon beam.(c) Reconstruction of the sample using the method presented in [17].(d) Gaussian image after application of the spatial-domain filter.(e) Subtraction image I K .(f) Subtraction image I IWS .(g) Subtraction image I α .(h) Line profiles measured across the PS beads identified by the blue and orange arrows: (i) for Gaussian image (a) (ii) for I K image (e), (iii) for I IWS image (f), and (iv) for I α image (g).
a) and 4(b)] were then used respectively as Gaussian and half-moon PSFs to generate the kernel matrix, optimized along a single axis and with mirror symmetry.The resulting spatial-domain filtered Gaussian image is shown in Fig. 4(d), and the subtraction images K I , IWS I and I α are shown in Figs.4(e)-4(g).Line profiles extracted across the single bead marked by a blue arrow and the pair of beads marked by a yellow arrow are shown in Fig. 4(h), for the three subtraction schemes.In keeping with the isolated bead and the two beads in the pair being all three of same size, it is once again observed that K I overperforms IWS I and I α .The line profiles for K I show indeed the best agreement between the intensities of the three beads, the least negativities, and the best resolution of the two beads (1.4 ± 0.1 µm) in the pair, clearly unresolved in the Gaussian image.