HiLo microscopy with caustic illumination

HiLo microscopy is an optical sectioning structured illumination microscopy technique based on computationally combining two images: one with uniform illumination and the other with structured illumination. The most widely used structured illumination in HiLo microscopy is random speckle patterns, due to their simplicity and resilience to tissue scattering. Here, we present a novel HiLo microscopy strategy based on random caustic patterns. Building on an off-the-shelf diffuser and a low-coherence LED source, we demonstrate that caustic HiLo can achieve 4.5 µm optical sectioning capability with a 20× 0.75 NA objective. In addition, with the distinct intensity statistical properties of caustic patterns, we show that our caustic HiLo outperforms speckle HiLo, achieving enhanced optical sectioning capability and preservation of fine features by imaging scattering fixed brain sections of 100 µm, 300 µm, and 500 µm thicknesses. We anticipate that this new structured illumination technique may find various biomedical imaging applications.


Introduction
Widefield fluorescence microscopy is indispensable for studying biological structures and dynamics.However, this technique is inherently limited by out-of-focus fluorescence background, stemming from the "missing cone" problem in the 3D optical transfer function (OTF) [1].Optical sectioning (OS) techniques [2,3] have been developed to overcome this limitation, enabling the removal of fluorescence background, enhancing image contrast, and facilitating high-fidelity 3D imaging.HiLo microscopy belongs to the class of OS-structured illumination microscopy (SIM) and works by computationally fusing two images: one illuminated uniformly and the other with structured illumination [4].Notably, random speckle patterns have emerged as the preferred choice for structured illumination in HiLo microscopy, attributed to their straightforward implementation and robust performance in the presence of tissue scattering, and have been applied to various in vivo applications [5,6].
Our focus is on further advancing OS-SIM techniques, especially HiLo microscopy, by exploring alternative structured illumination strategies that maintain system simplicity.Traditional HiLo implementations rely on laser sources [4,7] or intensity modulation with digital micro-mirror devices (DMDs) [8], both of which add to the complexity and cost of the system.In contrast, we propose the use of caustic patterns as a simple form of structured illumination.Caustics [9], or randomly focusing patterns characterized by wave-like intensity structures, offer a promising alternative due to their distinct intensity statistical properties, which differ from the Rayleigh statistics of fully developed speckle patterns [10].This distinction is advantageous for both illumination efficiency and minimizing photobleaching risks [11].Also, owing to the focusing mechanism, caustics not only provide much more versatility in controlling pattern density compared to speckles, but also make the use of incoherent light sources feasible.As we show in our experiments, the former aspect facilitates improved robustness against severe tissue scattering.The latter aspect is beneficial for OS, as under identical pattern densities, incoherent light leads to a quicker blurring of out-of-focus structures than what is observed with coherent illumination [1].
Here, we introduce a novel random illumination strategy for HiLo microscopy using caustic patterns.This approach simplifies the hardware requirements traditionally associated with HiLo techniques, allowing for implementation with an off-the-shelf holographic diffuser and an LED source.Our experimental prototype achieves 4.5 µm optical sectioning capability with a 20× 0.75 NA objective.Notably, we show that caustic HiLo offers enhanced optical sectioning capability and better preservation of fine features when imaging highly scattering fixed brain sections of 100 µm, 300 µm, and 500 µm thicknesses, outperforming speckle HiLo.We anticipate that this new OS-SIM technique may find various applications in biomedical imaging.

Method
Our setup is shown in Fig. 1(a).It includes a collimated LED (central wavelength λ = 470 nm, Thorlabs M470L5) as the light source, a backside-illuminated CMOS sensor (4000 × 3000 pixels, pixel size: 1.85 µm, The Imaging Source DFM 37UX226-ML) for image capture, and a fluorescence filter set (Thorlabs MF469-35, MF525-39, and MD498) to separate excitation and emission.Uniform illumination utilized in the excitation path is formed through the same manner as in standard epi-fluorescence microscopy, while structured illumination is generated by inserting a holographic diffuser (Edmund Optics 47-988) after the light source.The resulting caustic patterns are projected onto the sample plane using a relay system (f 1 = 300 mm, f 2 = 180 mm, Thorlabs ACT508-300-A, AC508-180-A) and a microscope (objective: Plan Apo 20× 0.75 NA, Nikon Eclipse TE2000-U).We measure a 370 × 278 µm 2 field of view (FOV) at the sample plane.Axial scanning is automated by rotating the microscope's fine focus knob with a step motor (Thorlabs MFC 1).The emitted fluorescence passes through the same microscope and another relay system (f 2 = f 3 = 180 mm, Thorlabs ACT508-180-A) before reaching the sensor.
The caustic HiLo algorithm is developed by adapting three existing HiLo algorithms [5,7,12], as illustrated in Fig. 1 where the cutoff frequency of the filter ⃗ k h is defined by HP( ⃗ k h ) = 1/2.On the other hand, I Lo ( ⃗ ρ) is extracted through several steps since a simple low-pass filtering to I u ( ⃗ ρ) does not reject out-of-focus background.We need an indicator to distinguish between in-focus low-frequency components and background.First, the normalized raw images are subtracted to remove variations from the object itself, retaining information proportional to the caustic illumination: where LP d [•] is a low-pass Gaussian filter with a cutoff frequency ⃗ k l defined similarly to ⃗ k h .As the caustic contrast decays as a function of defocus, the absolute value of I d ( ⃗ ρ) is able to resolve axial information.Additionally, the OS capability can be further tuned by applying an extra bandpass filter to I d ( ⃗ ρ) for pattern frequency selection: where ⃗ k ⊥ is the 2D frequency coordinate.Finally, the contrast map extracted from where F {•} and F −1 {•} denote the Fourier and inverse Fourier transforms, respectively.This local contrast is higher for in-focus components compared to out-of-focus components.However, two issues exist in C np ( ⃗ ρ).First, C np ( ⃗ ρ) includes the effects of noise, since shot noise and readout noise introduce additional variations in I u ( ⃗ ρ) and I s ( ⃗ ρ), resulting in a bias when quantifying the contrast.To account for this, the camera gain G (unit: e-/ADU) and readout noise level σ r (unit: e-) are calibrated, and the bias is calculated as ( The noise-corrected contrast map is then obtained as Second, C p ( ⃗ ρ) exhibits artifacts caused by the caustic patterns themselves.To overcome this issue, we apply a recently published HiLo-depatterning idea based on the non-local means (NLM) algorithm [12].For a given point ⃗ ρ * with a search window S, we calculate patch-based weights ω i * between ⃗ ρ * and all points within the search window in I u ( ⃗ ρ).These weights are then used to perform a weighted sum in C p ( ⃗ ρ), yielding the eventual C( ⃗ ρ): where β 2 is the patch size, and h is a hyper-parameter that determines the degree of the weighted average.By first multiplying C( ⃗ ρ) with I u ( ⃗ ρ), and then low-pass filtering using LP[•], the in-focus low-frequency components are obtained: where is the complementary low-pass filter.The HiLo image is synthesized with the combination of I Lo ( ⃗ ρ) and I Hi ( ⃗ ρ), followed by the NLM denoising algorithm: where , set as the ratio between the pair of spectra moduli at the cutoff frequency ⃗ k h , allowing for a smooth transition in the Fourier domain.For more details regarding the individual effects of the bandpass filter and the depatterning function, please refer to Fig. S1 in Supplement 1.

System analysis with a fluorescent layer and bead
Firstly, to quantitatively estimate the OS capability and determine the optimal filter parameters in our caustic HiLo system, we capture a z stack of caustic patterns using a thin fluorescent sample (Rhodamine 123 dissolved in water, Thermo Fisher Scientific).The sample is scanned axially over a range of 90 µm with a step size of 0.1 µm.Initially focusing on the focal plane to decide ⃗ k l , Fig. 2(a) shows the corresponding caustic pattern.Its 2D Fourier transform is displayed in Fig. 2(b), where the white dashed line represents the system bandwidth (corresponding to 2NA).Throughout this paper, all frequencies are normalized by k 0 , where k 0 = 2π/λ, unless stated otherwise.We compute the radial average of the frequency components, as shown in Fig. 2(c).By defining the threshold of the bandwidth as a 3 dB amplitude attenuation from the maximum, we determine that the pattern primarily occupies low frequencies with a peak at 0.048 and a bandwidth of 0.028 (lower bound: 0.032; upper bound: 0.06).As the goal of LP d [•] is to remove the caustic pattern while preserving the object information, we set | ⃗ k l | to be the lower bound of the pattern's bandwidth.
Next, we use the measured caustic stack to characterize the OS capability.We calculate the pattern contrast as a function of defocus z and apply an optional bandpass filter layer-wise to fine-tune the sectioning curve.According to Eq. (3), W( ⃗ k ⊥ ) reaches its maximum at the frequency 2 ln 2σ w .Therefore, we select six equally spaced values of σ w between 0.012 and 0.162 with a step of 0.03 to cover the entire range of spatial frequencies in the caustic pattern.Among these values, when σ w = 0.042, the maximum of W( ⃗ k ⊥ ) coincides with the primary frequency of the caustic pattern, indicating a minimal effect on the original pattern.In Fig. 2(d), three representative filters (σ w = 0.012, 0.042, 0.162) are illustrated alongside the simulated in-focus 2D OTF of the system.Figure 2(e) displays the corresponding sectioning curves with the σ w values shown in Fig. 2(d), as well as without σ w .Two observations can be made.First, all curves exhibit a secondary peak near the focus, attributable to the holographic diffuser's two

Scaling law
Defocus z (µm) sets of axially displaced foci that are approximately symmetric about the diffuser plane.We can model the random surface of the diffuser as a series of convex and concave lenses, each having similar curvature.The convex lenses converge light and form the first focus, while the light passing through the concave lenses generates the second focus after the lens relay.In our setup, the separation between the two focal planes is measured to be 12.7 mm from the diffuser side, which corresponds to roughly 11.4 µm at the sample plane.Second, W( ⃗ k ⊥ ) not only reshapes the sectioning curves but also shifts them, especially for small σ w values.This occurs because out-of-focus images contain more low-frequency components compared to the in-focus image.
With a small σ w , W( ⃗ k ⊥ ) suppresses the majority of frequencies in the in-focus image while retaining most of them in the out-of-focus images.
To decide the value of ⃗ k h , we measure the full width at half maximum (FWHM) of all the sectioning curves and plot them in Fig. 2(f).The solid blue line represents the FWHM without σ w (FWHM = 7.5 µm).As σ w increases, the FWHM decreases, asymptotically approaching 4.5 µm (σ w = 0.162).The sectioning quantified in Fig. 2(f) is only applied to I Lo ( ⃗ ρ) through Eq. ( 8).To ensure the same sectioning range is obtained for I Hi ( ⃗ ρ), we select ⃗ k h such that the axial bandwidth of OTF( ⃗ k h , k z ) matches the ultimate sectioning range.Based on the Stokseth approximation [13], we find that | ⃗ k h | ≈ 0.06 without σ w and | ⃗ k h | ≈ 0.1 when σ w = 0.162.We further estimate the scaling law of the OS for our caustic HiLo technique.To accomplish this, the sectioning curves illustrated in Fig. 2(e) are rearranged.We align the maxima of all the curves and confine the range from 0 to −40 µm to mitigate the influence of the twin foci.In order to study the sectioning effect from the structured illumination alone, we capture another caustic stack by positioning the thin fluorescent sample at the front focal plane of the objective and moving the diffuser in 1-mm steps over a 100-mm range.This stack is then interpolated to the same z step as the fluorescent caustic stack.All the contrast decay curves are displayed on a log-log scale in Fig. 2(g), following the method in [14].The scaling lines are not strictly linear, and different σ w values change the slope as well.Despite this, we make basic estimates.As depicted by the gray curve, which represents the sectioning due to the illumination alone, the pattern contrast scales approximately as z −0.8 for large defocus distance z.After acceleration of the detection OTF, the contrast decay follows approximately a z −1.5 relationship, as shown by the blue curve.With the additional W( ⃗ k ⊥ ) (σ w = 0.162), the yellow line decays around z −2 .Secondly, we perform the point spread function (PSF) calibration using a 200-nm fluorescent bead (Phosphorex) over a range of 90 µm with a step size of 0.1 µm.The experimental widefield 3D OTF is obtained directly by applying a 3D Fourier transform to the widefield PSF stack.To compute the HiLo 3D OTF, we implement a slightly modified HiLo process to the widefield PSF stack as follows: 1. high-pass filtering the widefield PSF to obtain PSF Hi ( ⃗ ρ, z); 2. multiplying the widefield PSF with the sectioning curve (σ w = 0.162), followed by low-pass filtering, to result in PSF Lo ( ⃗ ρ, z); 3. combining PSF Hi ( ⃗ ρ, z) and PSF Lo ( ⃗ ρ, z) to generate PSF HiLo ( ⃗ ρ, z); 4. applying a 3D Fourier transform to obtain the HiLo 3D OTF.In Fig. 2(h), we present the overlay of the two OTFs.The widefield OTF shows good agreement with the theory.The HiLo OTF predominantly overlaps with the widefield OTF, with the exception of expanded frequency support around the zero frequency, effectively filling the missing cone and thereby providing OS.Finally, with PSF HiLo ( ⃗ ρ, z), we analyze the system resolution in Fig. 2(i) by Gaussian fitting of the experimental data.The lateral and axial resolutions are approximately 405 nm and 2.25 µm, respectively, agreeing well with the theoretical values of 350 nm ( λ 2NA ) and 1.87 µm ( 2λ NA 2 ).The slight discrepancies between the measurements and the formula predictions can be attributed to imperfections in the optics and mechanics of the system.

Experimental demonstration on non-scattering fluorescent fibers
To verify the performance of our technique, we image fluorescent fibers depicted in Fig. 3 utilizing three methods: caustic HiLo, speckle HiLo, and confocal microscopy (objective: UPLSAPO 20× 0.75 NA, Olympus FV3000).For the first two HiLo strategies, to ensure a fair evaluation, identical algorithm parameters are used, except for ⃗ k h , which is determined by their respective OS curves, and the same diffuser is employed (see Fig. S2 in Supplement 1 for more discussion related to pattern generation and comparison).We acquire three co-localized image stacks using uniform, caustic, and speckle illumination, with a range of 30 µm and a step size of 0.5 µm.Through cross-correlation registration, the same volume is obtained via confocal microscopy, under optimal voxel size (Nyquist sampling) and aperture size (1 AU), to serve as the ground truth.For quantitative assessment of the OS improvement, we calculate the signal-to-background ratio (SBR) as the average signal from the sample region divided by the average background fluorescence within a given volume.The signal area is selected by thresholding the optically sectioned image, while the background area is determined by thresholding both widefield and optically sectioned images and then applying an XOR operation to these thresholds.The SBR of the widefield image is 1.7.Upon implementing HiLo, the SBR improves to 7.2 for the caustic version and to 9.3 for the speckle version.In comparison, the confocal measurement yields an SBR of 5.9.The background rejection strengths of these three methods are further quantified by the sectioning curves plotted in Fig. S3 in Supplement 1.We further compare the depth-encoded maximum intensity projection (MIP) among the widefield, confocal and caustic HiLo stacks, in which hue represents depth and value represents intensity.We conclude that caustic HiLo is competitive in OS capability with confocal microscopy, but slightly less effective than speckle HiLo for this non-scattering sample, which is due to the higher pattern density in the fully developed speckles.

Experimental demonstration on scattering fixed mouse brain sections
To demonstrate the advantage of caustic HiLo in imaging scattering tissues, we conduct experiments on fixed mouse brain cortex slices, where a specific subset of inhibitory interneurons expresses GCaMP6f (detailed sample preparation information is available in Supplement 1).The samples, with thicknesses of 100 µm, 300 µm, and 500 µm, showcase varying degrees of scattering strength.Targeting 25-µm-thick volumes starting from the surfaces of the slices, we keep identical step sizes and acquisition procedures to those used for the fluorescent fibers.Figure 4(a) illustrates surface-layer images from all the illumination strategies for the 100-µm-thick and 300-µm-thick samples.As the thickness increases, the SBR of the widefield measurements decreases from 1.5 to 1.2.After processed by caustic HiLo and speckle HiLo, the corresponding results are shown in Fig. 4(b).For the 100-µm-thick sample, caustic HiLo enhances the SBR to 6.4, surpassing 5.8 achieved with speckle HiLo.Such a difference can be observed from the enlarged views, where caustic HiLo reveals more neuronal details.As the sample thickness reaches 300 µm, increased scattering reduces the SBR for caustic HiLo to 5.1 and for speckle HiLo more significantly to 3.5, thus accentuating the differences in their performance.These distinctions are visually verified by a more pronounced background in the speckle HiLo enlarged views.Progressing to the 500-µm-thick sample, while both approaches still enhance the SBR, the overall pattern contrast remains low with either method, leading to comparable reconstruction levels (as detailed in Fig. S4 of Supplement 1).This is indicative of the inherent limitations faced by the two techniques in highly scattering environments.In Fig. 4(c), we visualize the caustic HiLo stacks for both the 100-µm-thick and 300-µm-thick samples.We attribute the robustness of caustic HiLo under tissue scattering to its lower pattern density.Specifically, high-density patterns that are in focus directly contribute to the signal of interest, whereas high-density patterns that are out of focus lead to an accumulation of fluorescence background in structured illumination measurements [15].Assuming the contrasts of caustic and speckle patterns at the focal plane are nearly identical, the denser out-of-focus speckle patterns will exacerbate the contrast more severely.This interplay between in-focus and out-of-focus patterns underlies our findings, suggesting that the intensity statistics of caustic patterns (see Fig. S2 in Supplement 1) offer an advantageous balance between pattern contrast and OS capability.

Discussion
Since our work focuses on the proof-of-concept verification and comparison of different HiLo strategies, we currently switch illumination modules manually after each axial scanning round.Although this process inevitably introduces a long acquisition time, our setup remains relatively stable when imaging static objects.Additionally, we find that the algorithm parameters are tolerant of possible sample / stage displacement during the acquisition because they are derived from statistics in either the frequency or spatial domain.Regarding future improvements for caustic HiLo, we could implement a dual-LED illumination configuration-one path with a diffuser installed and the other without-to enhance system stability, similar to the one implemented in the recent speckle HiLo system [12].
Compared to the OS-SIM techniques [16], which use a regular grid pattern with an incoherent source, our method offers two advantages.First, for a single optically sectioned image, the number of necessary measurements is reduced from three to two.Most importantly, even though we apply HiLo to the OS-SIM setup, our use of random caustic illumination provides better resilience against system imperfections than regular patterns.With regular patterns, accidental mechanical misalignment or optical aberrations can result in a high likelihood that the low-pass filter used to generate I Lo ( ⃗ ρ) in the conventional HiLo algorithm fails to eliminate the applied patterns, leading to unwanted artifacts in the final images [17].Conversely, caustic HiLo, specifically designed for random patterns, avoids this issue, making it particularly beneficial for systems constrained by space and cost.

Conclusion
In summary, we have introduced a caustic illumination approach for HiLo microscopy and experimentally demonstrated its effectiveness on multiple samples, including fluorescent fibers and fixed brain sections of varying thicknesses.Our solution is compact and cost-effective, eliminating the need for a laser or light modulator.Most importantly, we show that caustic HiLo provides enhanced robustness against tissue scattering compared to speckle HiLo.Inspired by techniques for customizing caustic intensity statistics [11] and super-resolution microscopy with "delta" speckles [18], a promising future direction involves developing custom diffusers to tailor caustic patterns for specific imaging applications.
Funding.National Institutes of Health (R01NS126596, S10OD024993) and a grant from 5022-Chan Zuckerberg Initiative DAF, an advised fund of Silicon Valley Community Foundation.

Fig. 1 .
Fig. 1.Overview of caustic HiLo microscopy.(a) The caustic HiLo setup.The caustic patterns are generated by inserting a holographic diffuser into the excitation path.Insets: examples of uniform and caustic illumination imaged with a uniform fluorescent layer.(b) Illustration of the caustic HiLo algorithm with a simulated resolution target.

Fig. 2 .
Fig. 2. System characterization.(a) The caustic pattern captured at the focal plane.(b) 2D Fourier transform of (a).(c) The radial average of (b) reveals the diffuser's bandwidth and primary frequency.(d) Comparison between the in-focus OTF and bandpass filters having different σ w values.(e) Contrast decay calculated from a z stack of caustic patterns, with each layer prefiltered to tune the sectioning range.(f) The FWHM of the sectioning curves as a function of σ w .The sectioning range without filtering is 7.5 µm (blue solid line).(g) Contrast decay on a log-log scale within the range of 0 to −40 µm.(h) The experimentally measured HiLo 3D OTF expands the support around the zero frequency.The widefield OTF appears in red, the HiLo OTF in green, and their overlapping region produces a yellow hue.(i) Gaussian-fitted resolution analysis.In (h) and (i), the sample is a 200-nm fluorescent bead.

Fig. 3 .
Fig. 3. Imaging of fluorescent fibers with different modalities.(a)-(c) Example images from uniform, caustic, and speckle illumination.(d)-(f) The OS capability of caustic HiLo is comparable to that of confocal microscopy, but is slightly weaker than that of speckle HiLo for the non-scattering fiber sample.Images (a)-(f) are located at the same layer.(g)-(i) Depth-encoded MIPs from the widefield, confocal, and caustic HiLo stacks.The volume of the imaged area is 370 × 278 × 30 µm 3 .