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
Observation diversity, where a specimen is imaged with a series of patterned illuminations and/or collection point-spread functions (PSFs), is frequently exploited in wide-field and scanning microscopy to enhance the spatial resolution [1–12]. Subtraction microscopy involves scanning with Gaussian and doughnut beams and the weighted subtraction of the two images, with the gain in resolution resulting from the doughnut node being narrower than the Gaussian peak [2–12]. The concept has been exploited for a variety of implementations including fluorescence [3–5], scattering , transient absorption , coherent Raman , and second harmonic generation . However, because Gaussian and doughnut PSFs do not match well at their periphery, subtraction microscopy typically generates subtraction PSFs with significant side-lobes and negative intensities, leading to severe intensity distortions in the reconstructed images which are particularly detrimental for closely packed and dense specimens. To reduce these artefacts, it is possible to exploit other PSFs that better match in their periphery [9–12]. Digital processing is however generally preferable due to Gaussian and doughnut beams being today readily prepared in the visible and near-infrared. Suppression of the negative intensities by setting these to zero has been used  but only benefits for very sparse specimens. Intensity weighted subtraction (IWS) has been recently proposed and demonstrated, where a matrix of subtraction weights is exploited  instead of the constant weight used otherwise [2–12].
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 .
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 [2–15].
2. Experimental methods
The mid-IR microscope used in these experiments has been described in  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 PPLN-based 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 cm2 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).
3. Results and discussion
In line with , 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. 1.4-1.8 µm) 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 [2–15] 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 side-lobes. The method involves the identification of a kernel matrix K that minimizes the error function19], although it is expected that other minimization methods can also be applied. Once, K is identified, the subtraction image is computed as
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 [2–13]. 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  and to the constant weight scheme [2–12,14,17]. For IWS, following , the simulated images were normalized to one and the subtraction performed according toFig. 3(a) and the subtraction computed according toFig. 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 spatial-domain filter is demonstrated by observing that, for both single pixel objects and nanowires, 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 closed-by 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  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 , 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  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(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 , and 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 overperforms and . The line profiles for 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.
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/2.4 only 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 [2–13]. 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.
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 half-moon 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 [2–15] generates significant intensity artefacts with the c-SIL PSFs. A new subtraction method is thus proposed where the Gaussian image is first filtered in the spatial-domain with a kernel optimized to match the Gaussian and half-moon PSFs at their periphery, thereby reducing the side-lobes and negativities responsible for these artefacts. The effectiveness of the method was experimentally demonstrated for the c-SIL and, from simulations it is predicted that the method will be equally efficient for the Gaussian and doughnut beams, ubiquitous to many implementations in the visible and near-IR.
European Commission (LANIR, FP7 280804 2012-2015); Science Foundation Ireland (SFI) (13/TIDA/I2613 and 13/CDA/2221); Integrated Nanoscience Platform for Ireland (INSPIRE) (PRTLI5, PhD Scholarship); Irish Research Council (IRC) (GOIPG/58/2013, PhD Scholarship).
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