1. Introduction

Temporal focusing (TF) nonlinear microscopy enables to simultaneously illuminate optically sectioned planes [1], lines [2, 3], or flexible two-dimensional patterns [4] without the need of tight spatial focusing of the laser beam. TF changes the relation between the lateral and axial dimensions of the illuminated spot (which depends on the objective’s numerical aperture, NA⁻¹, and NA⁻² respectively for Gaussian beam spatial focusing) to a more complex dependence, which also includes the system’s magnification (M), laser pulse duration (τ) and the laser beam cross-section dimensions [5]. This unique capability makes TF based systems useful in several optics related fields, such as standard [1, 6–9] and super-resolution [10, 11] fluorescence microscopy, harmonic generation microscopy [12], microfabrication and lithography [13–16], and optogenetic neural stimulation systems [17, 18].

Although TF is often implemented in a widefield temporal-focusing (WITEF) configuration, the implementation of a video-rate TF multiphoton microscope was based on scanning a temporally-focused line perpendicular to its long dimension [3, 8]. Line temporal focusing (LITEF) has two main advantages over WITEF. First, since multiphoton processes have a power-law dependence on light intensity, decreasing the illuminated area by a factor N, enhances two-photon excitation efficiency by N². Second, the optical sectioning of LITEF is expected to be tighter than that of WITEF, because the addition of spatial focusing contributes to the temporal focusing in the perpendicular plane [3]. Therefore, LITEF may be an attractive alternative to WITEF applications, where a stronger signal and tighter optical sectioning are required. The cost of using line focusing is an additional lateral scanning needed to illuminate a plane, however, in various applications including imaging and microfabrication, such (millisecond-timescale) scanning is acceptable. Two alternative LITEF optical setups were presented. The first design uses a cylindrical lens to focus a laser beam to a line on a diffraction grating (perpendicular to the grooves direction), and tube and objective lenses in a 4f configuration to image the grating surface onto the objective's front focal plane (Fig. 1(a)) [3]. Alternatively, the laser beam hits the grating surface directly, and a 4f configuration of a cylindrical and objective lenses is used to image the grating's surface onto the objective lens front focal plane [2]. In both options, the diffraction grating separates the incoming laser beam to its spectral components (in the x axis), and they re-unite in the objective focal plane where the sample is located and the grating surface is imaged. The spectral separation (in the xz plane, see Fig. 1(a)) results in pulse temporal stretching, which is compressed back to its original duration in the focal plane and re-stretched after it. Since multiphoton processes are sensitive to pulse duration, effective excitation is achieved only near the focal plane and optical sectioning without spatial focusing of the beam is attained. In the perpendicular (yz) plane the beam reaches the objective back aperture collimated and is focused to a line in the objective focal plane.

 

Fig. 1 Experimental system outline. (a) LITEF optical setup and inverted detection setup. Laser beam is focused by a cylindrical lens to a line (y axis) on the DPG transmission grating surface; the DPG is designed to diffract the laser beam and maintain the laser’s central wavelength in the same propagation direction. The tube and objective lenses image the grating surface onto the objective focal plane, where the pulse duration is minimal. The detection microscope uses a second objective and another lens to image the fluorescence on a CCD. (b) Detailed view of the sample region. Scattering samples were set over a 5µm layer of fluorescein. Measurements were obtained by axially moving objective 2 and the sample. (c) xz and yz projections of images taken at different distances from the TF focal plane using Nikon 40x NA = 0.8 objective (beam waist 0.75µm, line length 125µm). (d) Measurements (dots) of axial optical sectioning of the data shown in (c).

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The interactions of the illumination light with the medium in which it propagates (e.g. scattering) were recently shown to strongly affect the performance of WITEF [5], causing the axial sectioning to deteriorate much faster than in spatially focused two photon laser scanning microscopy (TPLSM). However, although central potential applications of LITEF involve interactions with strongly scattering biological media, its previous analyses [2, 19] did not consider scattering effects. Moreover, these studies focused on evaluating the temporal stretching of a propagating pulse and not on the dependence of LITEF performance on the optical system and media’s characteristics. In practice, LITEF light propagation is expected to have a non-trivial dependence on these characteristics, which will be different from that of WITEF, because of the addition of spatial focusing in the yz plane. In this paper we generalize our previous work on numerically modeling WITEF light propagation [5] to analyze LITEF light propagation in both transparent and scattering media: we validate the model by comparing its predictions to experimental measurements, use it to study the dependence of LITEF on the microscope and media’s parameters, and present an analytical formula that approximates this dependence. Finally we show that LITEF is more robust to scattering effects than WITEF and discuss the possibility of establishing a LITEF-based imaging modality for deep tissue imaging.

2. Methods

2.1 Experimental setup

Our experimental setup is illustrated in Fig. 1(a). It is based on an upright LITEF microscope that illuminates a sample from above (optionally, the sample is located under a scattering medium), and an inverted microscope which images the sample from below without encountering scattering effects on the emitted light. The LITEF path uses a dual-prism grating (DPG) which consists of a transmission diffraction grating embedded between two prisms. The prisms angles (48°x42°x90°, BK7 glass) and the diffraction grating groove density (1200 lines/mm) are designed to refract and diffract the laser’s central wavelength (800nnm) toward the same direction of the incoming light propagation. The DPG based design simplifies the optical setup configuration, offers a high efficiency (85% measured efficiency vs. 87% predicted efficiency for both polarization states), and also enables to perform remote scanning of the focal plane [20]. The excitation source is an amplified ultrafast laser (RegA 9000, pumped and seeded by a Vitesse duo; Coherent), providing up to 200mW of average power at the sample plane at a 150KHz repetition rate (1.33 μJ/pulse). After passing through a beam expander, an electro-optic modulator (Conoptics), and a cylindrical lens (f = 75mm), the beam hits the DPG and reaches the grating tilted by an angle α’ = 18°. An f = 200mm tube lens (Nikon) was used together with three interchangeable objective lenses (Nikon 60x NA = 1, Nikon 40x NA = 0.8, and Zeiss 10x NA = 0.45. The latter combined with the Nikon tube lens had an actual magnification of 12; all objectives are water immersion) in a 4f configuration to image a temporally focused line onto the sample.

A scattering tissue phantom (prepared as described in ref [5].) was placed on top of a 5µm fluorescein layer near the objective's focal plane (see Fig. 1(b); the fluorescein layer thickness was measured using TPLSM axial scanning). This phantom mimics the scattering characteristics of cortical tissue with mean free path (MFP) of 200 µm and scattering anisotropy of g = 0.9 [5, 21]. To measure the fluorescence light intensity from the opposite side of the sample, as well as to estimate the illuminated line waist, we used a second objective lens (Olympus 20x NA = 0.5 water immersion, and Nikon 40x NA = 0.55 air), an imaging lens and a CCD camera (UEye 2220SE-M, IDS). The sample and the second objective lens were mounted on two micromanipulators (MP-285 and MP-225 respectively, Sutter), which were used to move the sample and the detection system to controlled distances from the TF plane with 1 μm steps. The thickness of the scattering medium above the fluorescein layer was measured by moving the sample from the scattering medium top to the fluorescein layer, measuring the distance, and subtracting the thickness of a cover slip (average thickness of 150 µm) that lies between them. Pulse duration of ~200fs was measured at the laser’s output using an autocorrelator (PulseCheck, APE). At the TF focal plane (after passing through all of the optical components) a similar pulse duration was estimated by fitting a WITEF optical sectioning measurements (i.e. by removing the cylindrical lens) to model predictions [5] for different pulse durations. Optical sectioning curves were calculated by integrating the fluorescence signal from an image acquired for each distance from the focal plane. All comparisons of model predictions to experimental measurements were compensated for the broadening introduced by the finite thickness of the fluorescein layer (see example in Fig. 2(d)).

 

Fig. 2 Numerical simulation of LITEF light propagation. (a) Schematic demonstration of light propagation in temporal and spatial focusing planes (xz and yz respectively), near the objective lens focal plane. Different colors in the xz planes represents different spectral components, each one is propagating in a different direction (β) and tilted in a different angle (α). (b) Snapshot of light propagation on the optical axis (in logarithmic scale), taken from the simulation. (c) Projections of simulated LITEF illumination of 5µm fluorescent layer (blurring by the imaging system was not simulated). (d) Optical sectioning curves for thin fluorescent layer (thickness$\to$0, blue line) and 5µm fluorescent layer (black line). Optical parameters: M = 40, NA = 0.8, w₀ = 0.75µm, l = 50µm.

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2.2 Computational model

In this section we present an adaptation of our WITEF light propagation model [5] to account for LITEF. The model assumes independent light propagation in the mutually-perpendicular spatial and temporal focusing planes (yz and xz planes, respectively). The original WITEF model geometry is two dimensional and describes light propagation in the optical axis and the spectral distribution axis (z and x axes, respectively). Here, we add an additional description for the propagation in the spatial focusing plane using a cylindrical Gaussian beam model in the y axis. In addition, our experimental setup now includes a DPG made of BK7 glass (see section 2.1 for details), which we incorporated into the model.

When a delta pulse is focused into a line and impinges upon a diffraction grating (Fig. 1(a)), each spectral component is diffracted to a different direction and propagates a different optical path towards the focal plane. The propagation in the xz plane near the focal plane was previously described in detail [5]. Briefly, each spectral component propagates in a direction angle β as a tilted line, with tilting angle α (see Fig. 2(a)). All of the spectral components reunite in the focal plane and scan it together within picoseconds. The scanning speed depends on the angle α’ with which the incoming delta pulse phase front is tilted with respect to the diffraction grating, on the system’s magnification M, and on the DPG material (with refraction index n_DPG) and is given by [1] $\text{c/}\left({\text{n}}_{\text{DPG}}\cdot \text{M}\cdot \sin \alpha \prime \right)$. On the other hand, the focal plane is located in a medium with refractive index n_f, and is scanned by a line that propagates in direction β and is tilted by angle α with a scanning speed of$c\cdot \cos \left(\alpha -\beta \right)/\left({\text{n}}_{\text{f}}\cdot \sin \alpha \right)$. The focal plane is the image of the grating’s surface, and according to Fermat’s principle, the scanning time is equal. Therefore:

The propagation scheme in the yz plane is different. In this plane the cylindrical lens and the tube lens generate a telescope and the light reaches the objective lens nearly collimated. We modeled each spectral component as a cylindrical Gaussian beam in the yz plane, with an equal minimal waist (w₀) which is obtained in the focal plane (see Fig. 2(b)). The w₀ value was experimentally measured for each objective, and was corrected for the imaging PSF. The two-dimensional Gaussian beam formula is given by $I\left(y,z\right)=I_0\left(\frac{w_0}{w\left(z\right)}\right)\exp \left(\frac{-2y^2}{w^2\left(z\right)}\right)$. Therefore, each spectral component is characterized by its length, its tilting angle α, its propagation direction β, all in the xz plane, and its waist size w₀, in the yz plane.

In order to introduce tissue scattering effects into the model, we computed scattering kernels for various scattering depths, using a time-resolved Monte-Carlo simulation [22]. Medium parameters were: scattering MFP of 200 μm, g = 0.9 and negligible absorption. Upon entering the scattering medium, the different spectral elements’ intensity distributions are convolved with the matching scattering kernels. Since each spectral component has a different orientation as it propagates inside the scattering medium, we rotated the matching scattering kernel by the same angle to simulate the scattering directions.

3. Results

3.1 Model validation

To examine the model’s accuracy in a transparent medium under various optical configurations, we tested its predictions for TF’s main characteristic - the optical sectioning width. Optical sectioning was experimentally measured by axially scanning a 5µm layer of fluorescein solution across the focal plane. Results of these measurements and model predictions for three different optical setup parameters are shown in Fig. 3. The optical parameters were chosen to demonstrate LITEF capabilities for different applications: the first set of parameters (M = 40, NA = 0.8, line length = 125µm, beam waist = 0.75µm) represents commonly used system parameters for high resolution two-photon imaging, while the second set (M = 12, NA = 0.45, length = 500 µm, waist = 1 µm) is suitable for high resolution large field-of-view imaging. The third set (M = 60, NA = 1, length = 15µm, waist = 1.6µm) was chosen to explore the possibility of ultra-deep imaging that will be discussed in section 3.4.

 

Fig. 3 Model validation. Measured axial optical sectioning (dots) and model’s prediction (lines) for three sets of indicated optical parameters (200fsec pulses).

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3.2 Dependence on optical parameters in non-scattering media

Generalizing a similar result previously obtained for WITEF [5], we found an approximate formula that fits LITEF optical sectioning in transparent media. The sectioning profile of both our model predictions and the experimental measurements are consistently well fit with an analytical product of two square-roots of Lorentz-Cauchy functions given by:

Where F is the (peak-normalized) fluorescence signal and z is the axial distance from the TF focal plane. The optical sectioning parameters z_R1 and z_R2 depend only on the temporal and spatial focusing, respectively, highlighting the previously-noted separation of the two independent effects [2, 12, 19].

The first function in the product describes the sectioning due to the temporal focusing, and depends on the microscope’s NA in the TF plane (and on the objective’s filling in the x axis), magnification, the illuminated line length and the laser's pulse duration. The second function describes the sectioning due to the spatial focusing and depends only on the beam waist, i.e. on the objective’s NA in the spatial focusing plane (and on the objective’s filling in the y axis). We note that choices of different combinations of light source, diffraction grating, cylindrical lens, and tube lens will affect the filling in the x and y axes. Examples of fitting Eq. (2) to the model’s results are shown in Fig. 4. Next, we found that for a wide range of parameters (magnification 10-60, pulse duration 100-400 fsec, numerical apertures 0.45-1, line length 5-200 μm, and beam waist 0.5-1.5 μm), the dependence of z_R1 and z_R2 on the optics can be well-approximated by the following expression:

 

Fig. 4 Comparison of calculated axial optical sectioning for different beam waists (dots) and best-fit products of two square roots of Lorentz-Cauchy functions (lines). Optical parameters: M = 20, NA = 1, l = 50µm, tau = 100fsec.

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Fig. 5 Comparison of LITEF calculated optical sectioning and its analytical approximation. (a) Scatter plot of the estimated Lorentz-Cauchy parameters, according to Eqs. (2) and (3), and their calculation from fitting the functions to the numerical model results. Error bars in right panel indicates standard deviation. (b) Comparison of model calculated optical sectioning (dots) vs. Equations (2) and (3) (lines). Optical parameters are indicated next to each graph.

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3.3 Scattering effects

The use of an amplified laser source enabled the measurement of light penetrating through more than 1mm of the scattering phantom - these measurements and model predictions were compared for two different optical setups (Fig. 6(a)). Interestingly, according to both the theoretical and experimental results, LITEF exhibits a relatively slow deterioration of the optical sectioning with scattering depth: no significant broadening was measured for the high magnification setup, and a broadening by a factor less than 1.5 was measured in the low magnification configuration at a depth of 6 scattering MFPs. For comparison, WITEF with similar optical parameters to the high magnification setup exhibits a 5-fold broadening of the axial sectioning over a much smaller range of 2.5 MFPs (500µm) [5]. This highlights the relative robustness of LITEF over WITEF for tissue scattering effect on sectioning (Fig. 6c).

 

Fig. 6 Scattering effects. (a) Optical sectioning of two optical setups at different scattering depths. Dots represent experimental measurements; rectangles are model calculation results (connected by a dotted line). Insets show model’s prediction vs. experimental measurements for sectioning profile, and xy/xz projection images taken at specific points in the graph. Optical parameters: 1) M = 12, NA = 0.45, l = 500µm, w = 1 µm, tau = 200fsec. 2) M = 40, NA = 0.8, l = 125 µm, w = 0.75 µm, tau = 200fsec. (b) Measured attenuation of the LITEF signal (logarithmic scale) and exponential fit as a function of scattering phantom thickness. (c) Comparison of broadening of optical sectioning FWHM through 500µm of the scattering phantom for the two LITEF setups from (a, b) vs. WITEF broadening for setup 2 (ref [5].) and vs. expected optical sectioning of a spatially-focused beam (TPLSM, ref [23].). (d) TPLSM excitation decay constant [23] vs. decay constant ranges measured in LITEF (panel b), and WITEF [5] in scattering phantoms.

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The Fluorescence signal power as a function of depth in scattering media was also measured (Fig. 6(b)). The fluorescence signal exponential attenuation fit corresponds to a 1/e decay constant of 127 µm for the x12 NA = 0.45 setup and 105 µm for the x40 NA = 0.8 setup. Interestingly, these measured decay constants are intermediate between the spatial focusing MFP (100 µm independent of optical parameters, e.g., ref [23], box 1), and the 190 µm for the x60 NA = 1 simulated WITEF imaging setup [5] (Fig. 6(d)).

3.4 Deep tissue penetration

The previous section's results (Fig. 6) suggest the possibility of penetrating deeper than 6 scattering MFPs into a scattering phantom. Moreover, LITEF appears significantly more robust to scattering effects than WITEF. Interestingly, an attempt to penetrate into a similar depth using a standard two-photon laser scanning microscope (spatial focusing) will result in creation of out-of-focus excitation on the tissue’s surface due to the high flux of photons [21]. Temporal focusing microscopy, however, isn’t expected to excite fluorescence on the tissue's surface as efficiently because the pulse is temporally stretched far from the focal plane and two-photon absorption is reduced. Therefore, it may be possible to establish an imaging method analogous to scanning two-photon microscopy that will be more suitable for deep tissue imaging, by illuminating a temporally-focused line (that is as short as possible) and raster scanning it over a region of interest. In order to investigate the feasibility of such imaging, we removed a beam expander from our setup and by using a high magnification objective (x60, NA = 1) illuminated a 15 µm-long temporally focused line onto the 5µm fluorescein layer under scattering phantoms. Removing the beam expander resulted in poor filling of the objective (worse effective NA), and a beam waist of 1.6 µm was measured. We measured penetration of more than 9 scattering MFPs into the scattering phantom without significant loss of optical sectioning (Fig. 7). We note that an optical setup that will further decrease the line length to 5 µm and optimize the objective’s filling is expected to achieve optical sectioning of ~2 µm. Therefore, the suggested method may be used to penetrate very deep into tissue, beyond what is possible with standard imaging methods.

 

Fig. 7 Ultra-deep penetration into scattering phantom. A 15µm line is illuminated through more than 9 scattering MFPs without significant loss of optical sectioning. Dots represent experimental measurements, rectangles - model calculations results (connected by a dotted line). Insets show optical sectioning measurements and their model predictions for specific depths.

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4. Discussion

In this study we combined theoretical and experimental work to expand the understanding of LITEF, and emphasized its unique characteristics and capabilities in comparison with WITEF and spatial focusing. The enhanced sectioning capability of LITEF allows the use of lower magnification objectives, which offer a larger field of view, while still maintaining microscopic-resolution imaging.

Related models for describing TF light propagation were recently published [16, 24], highlighting the growing importance of TF based applications. We note that the framework for our models (both WITEF and LITEF) differs from these models, since it is an imaging based framework, which describes light propagation in an optical setup similar to the one described in Fig. 1(a). Alternative configurations for TF applications, which do not image the grating surface onto the objective focal plane (see Fig. 1 in ref [16]. and in ref [24].), would require some adaptations of this model to analyze them. Moreover, in this model we assume that the configuration of the optical setup before the objective lens (laser beam bandwidth and diameter, grating groove density, and tube lens focal length) achieves an adequate filling of the objective lens' back aperture. Even though the model enables to refer to any level of objective filling, we preferred to analyze the scenario of a well-designed optical microscope in which there is an attempt to co-optimize the optical sectioning (which requires to overfill the objective back aperture) as well as the system’s efficiency (which requires to underfill the objective back aperture).

Two new analytical expressions were introduced to describe LITEF's sectioning characteristics. Equation (2) describes the axial sectioning profile as a product of two square-roots of Lorentz-Cauchy functions, and highlights the separable role of spatial and temporal focusing in creating this profile (the WITEF temporal-only profile is described by a square-root of a single Lorentz-Cauchy function [5, 19]). Equation (3), which has the same form as the equivalent expression we previously found for WITEF [5] (but has different parameter values, due to the change in system’s characteristics), may serve as a basis for approximating the optical sectioning for a given set of optical parameters. The dependence on five different parameters, compared to just one in spatial focusing, offers a relatively high level of flexibility in the optical design process.

LITEF is a hybrid method of spatial and temporal focusing, and it was found here to have intermediate robustness to scattering effects relative to the two “pure” methods (Fig. 6). We have shown that the measured effects of scattering on axial sectioning are relatively minor and enable the use of LITEF deep inside scattering media, for applications like imaging and single-cell excitation, without significant deterioration of the system’s performance. This is in contrast to the rapid deterioration of WITEF's optical sectioning [5] - the difference between the relative sectioning deterioration could be of an order of magnitude within less than 1mm of tissue propagation. Thus, LITEF appears to be generally more suitable for deep tissue applications than WITEF [for multiphoton imaging, tissue scattering will also affect the detected light propagation, an effect that can be partially mitigated using descattering algorithms [8] and customized detection schemes]. Secondly, the signal attenuation rate, shown in Fig. 6(b and d), gives an indication about the expected scattering effects. For standard spatial focusing the decay constant is expected to be 100 µm (MFP/2 [23]), since only ballistic (non-scattered) photons contribute to the nonlinear fluorescence signal. LITEF’s decay constants were slightly longer, which also means that scattered photons contribute to the fluorescence signal (this effect is even stronger for WITEF, where longer decay constants are observed). Interestingly, we note that our results on the relative resilience of TF axial sectioning to scattering are complementary to recent findings on the scattering-resilience of diffractive pattern TF illumination [25] that were based on a different model describing light wave propagation.

Finally, based on the observation of ultra-deep penetration by very short LITEF lines (Fig. 7), we put forward the idea of combining such micro-line illumination and raster scanning across a volume, as a method that in principle may allow imaging beyond the physical out-of-focus limits [21]. Although micro-LITEF is demonstrated to have impressive scattering resilience - the expected improvement in penetration depth of the proposed imaging method remains to be verified and the system’s implementation would require additional work.