High resolution imaging with differential infrared absorption micro-spectroscopy

Although confocal infrared (IR) absorption micro-spectroscopy is well established for far-field chemical imaging, its scope remains restricted since diffraction limits the spatial resolution to values a little above half the radiation wavelength. Yet, the successful implementations of below-the-diffraction limit far-field fluorescence microscopies using saturated irradiation patterns for example for stimulated-emission depletion and saturated structured-illumination suggest the possibility of using a similar optical patterning strategy for infrared absorption mapping at high resolution. Simulations are used to show that the simple mapping of the difference in transmitted/reflected IR energy between a saturated vortexshaped beam and a Gaussian reference with a confocal microscope affords the generation of high-resolution vibrational absorption images. On the basis of experimentally relevant parameters, the simulations of the differential absorption scheme reveal a spatial resolution better than a tenth of the wavelength for incident energies about a decade above the saturation threshold. The saturated structured illumination concepts are thus expected to be compatible with the establishment of point-like point-spread functions for measuring the absorbance of samples with a scanning confocal microscope recording the differential transmission/reflection. ©2013 Optical Society of America OCIS codes: (170.0180) Microscopy; (300.1030) Absorption; (300.6340) Spectroscopy, infrared; (300.6360) Spectroscopy, laser; (300.6390) Spectroscopy, molecular. References and links 1. W. Min, C. W. Freudiger, S. Lu, and X. S. Xie, “Coherent nonlinear optical imaging: beyond fluorescence microscopy,” Annu. Rev. Phys. Chem. 62(1), 507–530 (2011). 2. A. Volkmer, L. D. Book, and X. S. 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However, in the far-field, diffraction limits the resolution to values between λ and λ/2, where λ is the wavelength of the radiation used for imaging, and imposes thus a severe limitation in resolution especially for IRAS since in the mid-infrared the wavelengths span from 2 to 20 μm.Moreover, only synchrotrons provide broadband mid-IR sources of sufficient brightness to map efficiently the IR absorption at the diffraction limit using confocal microscopes, and IRAS table-top microscopes using thermal sources must be used at degraded resolution to achieve a realistic signal-to-noise ratio (SNR) [10][11][12][13][14][15][16][17][18][19][20].For a given wavelength, pulsed tunable IR laser sources match or exceed the brightness of synchrotron and can in principle be readily used for diffraction limited IRAS, although one trade-off is that only a single wavelength can be probed at a time.Today IRAS imaging with spatial resolution beyond the diffraction limit is achieved using near-field approaches (50-100 nm) with either broadband thermal sources or IR lasers [21][22][23].However, these techniques are limited to probing surfaces since they involve the scanning of the samples by a material probe.Developing far-field and table-top approaches for high resolution IRAS imaging that can overcome the above limitations remains thus highly desirable.
Pushed by the strong demand in life science, several schemes for achieving subdiffraction and, as a matter of fact, true nanoscale imaging, have been successfully developed with far-field fluorescence microscopy [24].These include reversible optical fluorescence transition (RESOLFT) microscopies such as for example stimulated emission depletion (STED) microscopy [25][26][27][28][29][30][31][32], non-linear saturated structured illumination microscopy (SSIM) [33,34], and single molecule localization microscopies such as stochastic optical reconstruction microscopy (STORM) [35,36].Several image subtraction methods have also been developed to improve the lateral resolution of fluorescence confocal microscopy, using different detection pinhole geometry [37][38][39] or vortex-shaped illumination [40,41].However, all these methods generally require the introduction of fluorescent markers in the samples which are the species that is truly measured in the samples.
Chemically sensitive label-free imaging with sub-diffraction resolution is on the other hand deemed possible based on the vibrational spectroscopies CARS [42,43] and IRAS [44].It is envisaged to control the spatial extension of the CARS emission using a vortex-shaped beam to suppress the vibrational Raman coherence through an incoherent process involving the excitation of a neighboring vibration or to periodically deplete the ground-state through Rabi oscillations.The scheme for IRAS involves a vortex-shaped IR pulse for saturating the vibrational mode of interest and subsequent probing with a second, delayed IR pulse [44].The expected improvement in spatial resolution is down to ca. 65 nm for CARS [43], and down to about λ/13 for IRAS using the vibrational depletion pump-probe scheme [44].In both cases these values would represent a breakthrough.
Except for the single molecule localization techniques, the microscopies above achieve resolutions below the diffraction limit because of the spatial structuring of an incident radiation that exhibits either a single circular node (vortex) or a sinusoidal intensity profile, and that is sufficiently intense for saturating one optical process involved in the imaging.Thus, although more than one radiation is used depending on the optical information that is measured, only one is responsible for the resolution.For IRAS, and by extension for any optical absorption technique, we rationalize that only this one spatially structured radiation is required since the absorption can also be detected as a loss in its energy.This hypothesis is discussed and elaborated in this paper on the basis of simulations of the absorption of spatially defined intense IR beams in a mock organic sample.A single radiation scheme would indeed be experimentally attractive since for example, with respect to the vibrational depletion pump-probe scheme proposed earlier [44], pulse duration, pump-probe delay, and probe pump separation at the detector are then not relevant.
The simulations illustrate that a sensitive scanning scheme is possible by measuring with a confocal microscope the difference in transmitted/reflected energy between a vortex-shaped radiation and a reference taken here as Gaussian.The difference is proportional to the local absorbance.The spatial resolution is found to be down to less than λ/10 for a vortex about a decade above the saturation threshold.Although the sensitivity is a little less than the previously reported vibrational-depletion pump-probe scheme [44] for IRAS, the simulations reveal that, for the CH 2 stretches, 300 nm large mock organic alkane bulk specimens lead to a differential signal about 0.05% of the full transmitted/reflected vortex intensity, a value that is well within measurable range.

Scheme for differential absorption
We consider the irradiation of a sample incorporating two-level oscillators with an IR beam tuned to the wavelength of the vibrational transition, and with a pulsed temporal structure that, from an experimental point of view, will be necessary to achieve saturation of the transition, but that will not be required otherwise.We further consider that the IR beam is focused on the sample so that it exhibits a node along the optical axis, which is readily made experimentally with a vortex phase plate [45][46][47][48].A vortex absorption point-spread function (PSF) is then defined by 0, ( , ) ( , ) where ( , ) vortex x y Σ is the energy of the vortex pulse after absorption in the sample at the coordinates ( , ) x y defined in the sample plane, and where the indice 0 denotes the same quantity measured in the absence of absorption.The shape of the vortex PSF is reminiscent of the vortex geometry [Fig.1].However, because the sample will not absorb more photons than necessary to achieve saturation, that is, to excite half the local population of oscillators, the vortex PSF will plateau where the intensity on the sample is larger than the saturation threshold and will thus exhibit a node of width narrowing with the vortex pulse energy, and, as demonstrated experimentally with RESOLFT microscopy schemes, the width of the node can be much smaller than the diffraction limit [26][27][28][29]31,46].
The recovery of an absorption image confined to the region of that sub-diffraction node can be envisaged by subtraction from an image of reference defined with a PSF that ideally matches with that of the vortex one, except at the center where the node is absent.In an experiment, such a reference PSF can be readily approached by a Gaussian whose central part will be of uniform height for intensity above the saturation threshold.The absorption microscopy PSF for the Gaussian may be written as 0, ( , ) ( , ) where ( , ) Gauss x y Σ is the energy of the Gaussian pulse after absorption in the sample, and where the indice 0 denotes the same quantity in the absence of absorption.The difference Δ between the two PSFs gives a spatial profile illustrated in Fig. 1(c), and one readily sees that a desirably peakshaped differential IR (DIR) PSF will thus be achieved by directly measuring the difference between vortex and Gaussian beams after absorption in the sample: where C is a constant background for all ( , ) x y coordinates and scales simply with the difference in measured energy between vortex and Gaussian pulses in the absence of absorption.
Equation ( 4) suggests an experimental scheme to realize the method in which periodically alternating series of vortex and Gaussian pulses are focused onto the sample.The integrated intensity measured after absorption in the sample will then exhibit a component of the same period.The latter component is indeed proportional to the desired differential absorption signal and can be readily measured by synchronous detection, for example.The differential absorption image is then generated by mapping this signal, by varying the sample position in the beams.Noteworthy, although the differential PSF exhibits a central peak that clearly stands out of the background, an imperfect match between the vortex and reference spatial profiles away from the central axis will generate concentric rings around the peak [see side lobes in Fig. 1(d)], which are obviously undesirable artifacts.The precise shape of the reference beam, taken here as Gaussian, is thus crucial.However, as it will be discussed below, the use of a confocal arrangement for collecting the beam drastically improves the PSF by strongly reducing these side-lobes.

Model
The algorithm and Matlab TM scripts used here to model the differential absorption scheme have been adapted from those reported in [44].The scripts consider that the measurements are made in transmission, since the temporal evolution of the intensity is then simpler to compute.Noteworthy, the differential scheme is expected to perform the same in reflection, which will be simpler to implement experimentally as only one objective needs then to be aligned.The sample is modeled in space as an ensemble of voxels containing each a collection of independent oscillators and, for each sample voxel, the change in intensity due to stimulated absorption/emission [49], along the coordinate z defined as the axis of propagation of the light pulses, normal to the sample surface, is inferred from where k is the stimulated emission/absorption Einstein coefficient, ρ is the density of oscillators, N Δ is the difference in relative population density between the fundamental and excited vibrational state, ( , ) I t r is the local intensity at a given time t and sample position ( , , ) , with ( , ) x y the coordinates in the sample plane set as normal to the direction of propagation of the light, and h and c are respectively the Planck constant and the speed of light in vacuum.N Δ is determined by where Γ is the spontaneous de-excitation rate, and N the relative population density of excited oscillators at a given position in the sample.
The system of Eq. ( 5) and Eq. ( 6) is readily solved using the backward Euler approach and independently for each set of ( , )  x y coordinates.The solution is found by iteration with the boundary conditions 0 ( , ) 0 , where 0 t is a reference time before irradiation of the sample.0 ( , , , ) 0 I x y z t = describes the spatial and temporal structures of the vortex and reference pulses impinging the sample at 0 z : where r and θ are the polar coordinates in the plane normal to the direction of propagation, and where 0,vortex I and 0,Gauss I are constants adjusted to reproduce the desired pulse energies.
The pulse duration are then given by 0 2 ln(2)τ and the spatial extension of the pulses on the sample are defined by the waist 0 w , which corresponds to a full width at half maximum (FWHM) of 0 2 ln(2) w for the Gaussian profile and a FWHM ca.1.8 times larger for the vortex profile.
The energy Σ of the pulses after absorption in the sample is computed by integrating the intensity in the sample plane ( , )  x y and over time, at the coordinate z exceeding the sample thickness, for a given relative position of the sample in the beams.IR absorption and differential IR absorption images are then created by repeating the calculations whilst scanning the relative sample position.In line with Eq. (1), Eq. ( 2), and Eq. ( 4), the differential IR absorption is defined by 0, (%) 100 where the normalization to the incident energy 0,vortex Σ affords an assessment of the sensitivity of the method.
Aiming to a realistic prediction of the differential IR absorption microscopy performances, the FWHM of the Gaussians was set to values that are no smaller than those expected with an objective of numerical aperture (NA) 0.85, corresponding to commercially available aspherical lenses at 3.5 μm.At the diffraction limit, experiments have verified that an objective with NA 0.7 focuses a Gaussian beam with λ at 3.5 μm into a spot of FWHM ca.2.3 μm [10] and thus a waist of ca.1.4 μm (FWHM of ca.1.9 μm) is thus readily expected for a NA of 0.85 by interpolation.The waist 0 w of the vortex was adjusted to that value producing a pattern where the distance between two maxima along the diameter of the vortex spot is 2.2 μm, in agreement with the value that we have found using the vectorial diffraction theory detailed in [48].For the Gaussian reference, larger waists were also used to facilitate the compensation of the naturally broader vortex.Noteworthy, when the confocal spatial filtering is taken into account the compensation is facilitated and is adequate also when using Gaussian waist set to the diffraction limit.The pulse duration was arbitrarily set at 1.0 ps, although larger values may also be of experimental interest.

Results and discussion
The feasibility of the concept is first examined by computing the differential absorption PSF associated with the CH 2 stretch vibrational mode at ca. 3.5 μm on a sample defined as a single pixel of mock octadecyltrichlorosilane [Fig.2].The dimensions of the pixel are set at 25 × 25 nm 2 , which remains small in comparison to the expected resolution.The vortex and reference pulse energy is the same and is varied from 1 nJ to 1 μJ.The reference is set as a Gaussian of waist 1.7 μm, a value that was found to minimize the magnitude of uncompensated rings around the central PSF peak, although side lobes remain clearly visible for all the energies [Fig.2(a)].
The resolution is measured as the FWHM of the central PSF peak and its value is plotted in Fig. 2(b), along with the PSF value at the peak in Fig. 2(c).As expected, the FWHM decreases sharply with the energy of the vortex pulse and down to ca. 270 nm (λ/13) for 100 nJ, where the vortex maximum intensity is ca. 10 times above the saturation threshold [44].This vortex energy of 100 nJ was previously found to be a realistic optimum with respect to the IR irradiation damage threshold of samples at 3.5 μm [44].Noteworthy, the FWHM is also reduced to around 1.2 μm for values below the saturation threshold (1 nJ), which is an effect of structured illumination [61].The side lobes in the differential absorption PSF seen in Fig. 1(d) and Fig. 2(a), and due to the poor superposition of the vortex and reference away from the optical axis, are very undesirable for high resolution imaging.The side lobes can in principle be reduced by replacing the Gaussian reference by one with a profile that matches better with the vortex.Although possible, this approach is however not straightforward to envisage from an experimental point of view.A better approach for suppressing the side lobes is to make the method less sensitive to these differences between vortex and reference profiles, and it is shown below that a confocal spatial filtering will just do that.
In a confocal microscope, the volume of sample contributing to the detected signal is controlled by the introduction of a pinhole conjugated with a defined sample volume [62].The confocal spatial filtering effect was introduced in our simulation by weighting the intensities leaving the sample [Eq.( 5)] with a Gaussian distribution of diffraction limited FWHM adjusted to a collecting lens of NA 0.85, corresponding to the maximum numerical aperture that is commercially available at 3.5 μm.The differential absorption PSF computed in combination with the confocal spatial filtering are shown in Fig. 3(a) for vortex energies of 10 and 100 nJ.The curves were calculated with the same parameters as those used for the PSF presented in Fig. 2(a) where no spatial filtering of the detected intensity was then introduced.Obviously, in Fig. 3(a), the side lobes are significantly reduced and the PSF are essentially point-like as expected, with a FWHM also marginally reduced.
We further illustrate the differential absorption method by computing IR absorption and differential IR absorption images of a sample made of nine alkane cubes of 300 × 300 × 300 nm 3 with the same density of CH 2 as the 2 nm thick octadecyltrichlorosilane film [Fig.3(b)], and which allow us to assess the merit of the scheme when imaging organic nanoparticles.The pulse energy is set at 100 nJ and the images are computed taking into account of the confocal spatial filtering.The IR absorption image calculated for the reference Gaussian does not reveal any fine feature reminiscent of the nine organic domains, which of course results from the then limited resolution [Fig.3(c)].Conversely, for the vortex, the absorption image reveals the nine domains as depressions on a locally varying background consistent with the vortex PSF in Fig. 1(d).For the differential absorption, the domains are resolved as protrusions, marking the increased IR absorption, and the background is now perfectly corrected by the reference [Fig.3(e)].Obviously, in an experiment the differential absorption can be recorded with more sensitivity directly by synchronous detection than if both the vortex and Gaussian absorption image are recorded independently.At the center of a domain the differential signal is 0.05% of the energy reaching the detector [Fig.3(f)].The sensitivity of the differential absorption is however expected to be less than the one reported for the vibrational depletion pump-probe scheme, where the maximum signals for the same sample and with the same vortex at 100 nJ are expected to be at least an order of magnitude higher, for an optimized pump-probe delay [44].This is because the absorption is here measured superposed to ca. 100 nJ, while for the pump-probe scheme it is superposed to a much weaker probe pulse of ca. 1 nJ.Yet, the SNR required for measuring a cubic 300 nm alkane domains with the differential scheme presented here is thus of the order of 10 −4 -10 −5 (i.e., for a magnitude a decade above noise level).Since earlier works have highlighted that subtraction-based imaging are more noise-sensitive than schemes where imaging is achieve with a single measure [37][38][39][40][41], and that this may impact on the experimental resolution [40], it remains important to compare the SNR requirements of the differential absorption scheme with past experimental achievements in absorption spectroscopy and microscopy.Variations in transmission of the order of 10 −4 were recorded in a pump-probe absorption microscopy scheme [63] and a SNR of 16:1 rms was achieved when detecting IR absorption in a Fourier transform setup with a reference beam 10 4 more intense than the signal beam [21].Moreover, differential noise compensation schemes with a sensitivity of 10 −5 are possible using solid-state mid-IR laser sources [64] and are directly applicable to the differential scheme when the vortex and reference energies are the same at the detector.It appears thus that, for the example presented in Fig. 3(b)-3(f), the sensitivity required for the differential absorption scheme is well within actual experimental capabilities.Finally, although it exhibits a lower sensitivity than the vibrational depletion pump-probe scheme, the differential absorption approach is advantageously much simpler to implement: the careful compromise between pulse duration and spectral resolution, the need for achieving precise timing of the arrival of the pump and probe pulses at the sample, and the separation of the pump and probe beams at the detector are indeed not relevant to the differential scheme.

Conclusions
Simulations were used to show that simply mapping the difference in transmitted/reflected IR energy between a vortex-shaped beam and a Gaussian reference with a confocal microscope affords the generation of vibrational absorption images of spatial resolution well beyond the current state-of-the-art in IRAS far-field micro-spectroscopy.The simulations of the differential absorption reveal a spatial resolution better than λ/10 for incident energies about a decade above the saturation threshold for the CH 2 stretch in alkanes.The concept of saturated structured illumination is thus seen to be compatible with the establishment of below-thediffraction limit, point-like PSF for measuring the IR absorbance of samples with a scanning confocal microscope set to recording the differential transmission/reflection.The detector sensitivity for organic nanoparticles is seen to be well within actual experimental capabilities.

Fig. 1 .
Fig. 1.(a) Differential absorption microscopy scheme where reference Gaussian (G) and vortex-shaped (V) pulses are focused by a lens (L) alternatively on the sample (SPL) and where the transmitted intensity is integrated by a detector (D).A scanner is used for producing images (xy).(b) Same as (a) but with confocal spatial filtering of the transmitted beam using a pinhole (PH).(c) Illustration of the transmission and change in population of a system of twolevel oscillators for increasing incident intensity.(d) Difference Δ between vortex and reference PSF.

Fig. 2 .
Fig. 2. (a) Differential absorption PSF for vortex energies of 1 μJ (continuous), 100 nJ (shorter dash), 10 nJ (dash), and 1 nJ (dash-dot).The PSF were normalized with their maximum at 1 for clarity.(b) PSF FWHM for the data shown in (a).(c) PSF value at peak for the data shown in (a) without normalization.

Fig. 3 .
Fig. 3. (a) Differential absorption PSF with confocal spatial filtering for vortex energies of 100 nJ (shorter dash) and 10 nJ (dash) (pixel: 25 × 25 nm 2 ).The PSF were normalized with their maximum at 1 for clarity.(b) Sample geometry for an array of nine cubic domains of alkane (300 × 300 × 300 nm 3 ; 300 nm between domains), (c) corresponding Gaussian reference IR absorption image, (d) vortex IR absorption image, and (e) differential absorption image.(f) Line profiles extracted from the DIR and Gaussian reference images (dash).For (b-f) the pulse energy is 100 nJ and one pixel is 50 × 50 nm 2 .