Model based optimization for refractive index mismatched light sheet imaging

Abstract

Selective plane illumination microscopy (SPIM) is an optical sectioning imaging approach based on orthogonal light pathways for excitation and detection. The excitation pathway has an inverse relation between the optical sectioning strength and the effective field of view (FOV). Multiple approaches exist to extend the effective FOV, and here we focus on remote focusing to axially scan the light sheet, synchronized with a CMOS camera’s rolling shutter. A typical axially scanned SPIM configuration for imaging large samples utilizes a tunable optic for remote focusing, paired with air objectives focused into higher refractive index media. To quantitatively explore the effect of remote focus choices and sample space refractive index mismatch on light sheet intensity distributions, we developed an open-source computational approach for integrating ray tracing and field propagation. We validate our model’s performance against experimental light sheet profiles for various SPIM configurations. Our findings indicate that optimizing the position of the sample chamber relative to the excitation optics can enhance image quality by balancing aberrations induced by refractive index mismatch. We validate this prediction using a home-built, large sample axially scanned SPIM configuration and calibration samples. Our open-source, extensible modeling software can easily extend to explore optimal imaging configurations in diverse light sheet imaging settings.

1. Introduction

Selective plane illumination microscopy (SPIM), or light sheet fluorescence microscopy (LSFM), encompasses techniques that enable rapid volumetric imaging of samples. SPIM approaches commonly involve two orthogonal objectives, one delivering light to the sample (excitation pathway) and one collecting emitted fluorescence (emission pathway) [1–3]. Shaping the excitation laser into a “light sheet” that is thin along the detection axis enables optical sectioning, leading to high-contrast images, even in densely labeled samples. Successful applications of SPIM span a wide range, including imaging of entire zebrafish embryos [2], tracking single cell dynamics in early mouse development [4], observing neural activity in the brains of small organisms [5], imaging of the central nervous system of mice [6], and extracting the hydrodynamic properties of flagella by observing their free diffusion [7]. These applications demonstrate SPIM’s capability to provide detailed, high-resolution images while minimizing photodamage and photobleaching.

All light sheet generation strategies have an intrinsic coupling of the width at the focus and the axial extent of the focus. One strategy to extend the axial extent for a given focus width is to use propagation-invariant beams, such as Bessel or Airy beams [8–10]. Gaussian beam propagation defines the beam waist ${\omega }_0=\frac{2\lambda }{\pi N{A}_{exc}}$ and confocal parameter (twice the Rayleigh length) $2\times z_R=2\times \frac{2\lambda }{\pi N{A}_{exc}^2}$ to parameterize the focus width and propagation length. Recently, both simulations and experiments have settled that Gaussian and sinc (given by a flat top profile) beams have larger $z_R$ than the Gaussian prediction, allowing for high-contrast imaging volumes similar to certain propagation-invariant beams [11,12]. The theoretical discrepancy arises because Gaussian predictions rely on the paraxial wave equation, which assumes small angles and does not account for diffraction effects at larger angles. Predominantly, prior simulations characterized various static and scanned light sheet modalities by propagating analytic electric field amplitude and phase distributions using beam propagation methods to generate the 3D intensity distribution about the focal plane [11–14]. Experimentally, deviation from a Gaussian focus results from overfilling the pupil, leading to a more uniform laser beam profile in the pupil and a sinc rather than a Gaussian distribution at the focus [15].

For samples that extend beyond the ${\omega }_0,z_R$ for a given optical configuration, there will be a spatially variant resolution and contrast, Fig. 1. Two common strategies to achieve a spatially invariant imaging system are to (1) limit the camera field of view (FOV) to the invariant region of the excitation or (2) increase the confocal parameter by increasing the excitation beam’s ${\omega }_0$ . The first strategy limits the effective imaging rate, and the second limits the optical sectioning, contrast, and potential resolution gains.

Fig. 1.

Spatially variant image formation and dependence on light sheet focus quality. Top: light sheet ZX focus profiles. Bottom: 200 µm z-stack (dz=0.5 µm) Z′X′ maximum intensity projections (gamma=0.75) of 1 µm beads embedded in agar. Unprimed coordinates denote excitation pathway and primed coordinates denote emission pathway. (A) Acquired using a static unaberrated light sheet focus. A tight region of high-quality imaging has a symmetric spatial variance around the light sheet focus. (B) Acquired using a static, aberrated, light sheet focus. An extended region of reasonable quality imaging has an asymmetric spatial variance around the aberrated light sheet focus.

To fully exploit SPIM’s potential space-time bandwidth, several innovative strategies have emerged to create a spatially invariant imaging system by extending the region of best excitation. Conceptually, these approaches synthesize an extended, high-quality FOV by imaging different sample areas with a translated light sheet. The differences between methods lie in the mode of extension, ranging from physically translating the sample through the light sheet focus [16], acquiring independent images of the light sheet in different positions [17], or continuously sweeping the light sheet position via remote focus [18,19], synchronously with the digital readout of a programmable CMOS detector, a technique known as axially scanned light sheet microscopy (ASLM) [20]. To achieve maximum performance, these methods rely on forming a high-quality light sheet focus at various points within the sample.

Inspired by recent insights into light sheet excitation profiles obtained by modeling and experiment, we undertook a more comprehensive effort to model light sheet profiles, subject to remote focusing, in refractive index (RI) mismatched samples [11,12,15]. Specifically, we aim to understand better the interplay between the specific remote focusing implementation and heterogeneous RI pathway on the excitation beam’s 3D intensity distribution. Heterogeneous RI can arise from mismatches in the excitation objective versus sample chamber, imaging media, sample, or some combination of all of these. To further explore the consequences of instrument and experimental design choices on the resulting image quality, we present a computational approach to simulate the 3D light sheet intensity distribution that integrates the remote focus pathway aberrations and the effects of a converging beam propagating through an index mismatch. While optical pathway aberrations have previously been integrated into optical modeling of focusing beams using a variety of computational approaches [21–24], the approach we provide here is open-source, compatible with other beam propagation methods, and extensible to any light sheet excitation pathway.

This work focuses on air immersion excitation objectives paired with a glass cuvette and an immersion medium, a combination commonly found when imaging large samples. Both our computational and experimental results confirm that low numerical aperture (NA) remote focusing with a tunable optic is a robust method to extend the FOV with uniform image contrast, even in an RI mismatch. However, even at a moderate NA of $\sim 0.3$ , we find significant changes to the excitation beam focus location and beam intensity profile for tunable optic remote focusing. In the presence of heterogeneous RI, our model predicts that there is a combination of applied light sheet displacement and excitation-lens-to-sample-chamber spacing that minimizes aberrations in the excitation path. We experimentally validated the model predictions by characterizing the point-spread function (PSF) across the FOV for static light sheets, demonstrating that there exist optimal combinations of light sheet position and objective-chamber spacing.

2. Methods

2.1. Optical modeling

We developed a model incorporating both ray tracing and field propagation to accurately model both aberrations caused by the light sheet optical train and diffraction during beam propagation. We defined the model coordinates with the propagation direction along the z-axis and the thin axis of the light sheet along the x-axis. The detection coordinates follow the typical SPIM convention, where the z’-axis is perpendicular to the light sheet propagation axis. The detection axes are distinguished using primed coordinates ( $x^{\prime },y^{\prime },z^{\prime }$ ). First, we launched a collection of collimated rays from a starting plane. We modeled the intensity distribution by stochastically sampling rays from either a Gaussian or uniform distribution to simulate Gaussian and flat-top laser profiles. Then, we ray traced through the optical system by applying Snell’s law of refraction at each surface while tracking each ray’s optical path length (OPL). Tracking each individual ray’s OPL allows us to account for aberrations induced by the optical system.

We modeled optical elements as flat or spherical surfaces with material refractive indices [25] according to the manufacturers specifications and a propagation wavelength of 561 nm (Fig. 4) or 488 nm (Fig. 5). Objectives are modeled as perfect lenses that transform a ray’s height and angle in the front focal plane (FFP) to the angle and height in the back focal plane (BFP) according to

$$\begin{array}{rl}\frac{h_2}{f}& =n_1\sin {\theta }_1\end{array}$$ (1)

$$\begin{array}{rl}{n}_2\sin {\theta }_2& =-\frac{h_1}{f}\end{array}$$ (2)

where $f$ is the vacuum focal length, we suppose the lens is immersed in media with refractive index $n_1$ and $n_2$ the left and right, respectively. The FFP is $n_{1}f$ to the left of the lens, and the BFP is $n_{2}f$ to the right. This construction ensures that an imaging system composed of two perfect lenses satisfies the Abbe sine condition but does not account for field-dependent or field-independent aberrations. Rays outside an optical element’s aperture are ignored for the remaining simulation. When ray tracing objectives, the rays are filtered in the BFP with an aperture equal to the pupil radius, $f\times \text{NA}$ .

Fig. 4.

Light sheet focus simulation results. (A) Light sheet widths and lengths for simulated and experimentally light sheets. Visual comparison of various simulated (B) and experimental (C) light sheet ZX profiles. The first two columns are example light sheets focusing in air with no offset. The last column is an example of displaced light sheets. The focal plane (dashed) and light sheet edges (solid) are highlighted in white. Data and simulations were acquired at a wavelength of 561 nm

Fig. 5.

Simulation and experimental results of optimizing optical system configuration. (A) The cuvette offset, $dc$ , is the distance between the objective and outer cuvette surface. The position of the objective surface is one working distance back from the FFP. The light sheet displacement, $df$ , is the distance between the inner cuvette surface and the midpoint focus. (B) Longitudinal spherical aberration (LSA) for various combinations of $dc$ and $df$ . (C)-(E) Wavefront decomposition, light sheet ZX planes, and experimental static light sheet PSF results with z standard deviation, ${\sigma }_z(y)$ , and focus extent (black lines) for positions highlighted in (B). The light sheet slice scale bar is 50 µm. Data and simulations use a wavelength of 488 nm.

After ray tracing through the final optical surface, we obtain the electric field on a grid and propagate it through the light sheet focus using the exact-transfer function (angular spectrum) method [26] that relates the electric field at axial position $z$ to the initial field,

$$E(x,y,z)={\mathcal{F}}^{-1}\left[\mathcal{F}[E(x,y,z=0)]e^{ız\sqrt{{\left(\frac{2\pi n}{\lambda }\right)}^2-k_x^2-k_y^2}}\right]$$ (3)

for a field propagating through a homogeneous refractive index medium. Here, $\mathcal{F}$ represents a 2D Fourier transform over the $x$ -, and $y$ -coordinates. We neglect polarization effects by making a scalar approximation for our field. The scalar approximation has been shown to be valid for $\text{NA}<0.2$ [27]. However, our approach could be extended to a full vectorial model by applying Eq. (3) to each polarization component.

We assumed radial symmetry for the optical pathways considered here, and only ray traced a one-dimensional slice through each optical pathway. This gives us a collection of points $\{(r_i,OP{L}_i){\}}_{i=1,\dots ,N}$ , as shown in Fig. 2(C). From these discrete points, we create an interpolating function for the optical path length, $OP{L}_{\text{int}}(r)$ . To handle the amplitude, we assume the density of rays is proportional to the radiant flux through the surface. First, we generate a histogram over radial positions, then create an interpolating function from this, ${\varphi }_{\text{int}}(r)$ . In order to satisfy energy conservation and accommodate a spatially varying Poynting vector, we create an interpolation function for the ray angles, ${\mathrm{\Theta }}_{\text{int}}(r)$ . Finally, we estimate the electric field on a 2D grid $(x_i,y_j)$ as Eq. (4). We choose grid spacing of 100 nm $\sim \lambda /(4n_{\text{water}})$ , which is Nyquist sampled with respect to the maximum propagating spatial frequency.

$$E(x_i,y_j)=\sqrt{\frac{{\varphi }_{\text{int}}\left(\sqrt{x_i^2+y_j^2}\right)}{\cos \left({\mathrm{\Theta }}_{\text{int}}\left(\sqrt{x_i^2+y_j^2}\right)\right)}}\exp \left[ikOP{L}_{\text{int}}\left(\sqrt{x_i^2+y_j^2}\right)\right]$$ (4)

Fig. 2.

Overview of modeling approach. (A) Light sheet model excitation pathway, where z is aligned with the propagation direction and x along the narrow axis of the light sheet. (B) The initial ray flux, $\varphi (r)$ , and path length difference, $\mathrm{\Delta }OPL(r)$ . (C) The ray $\varphi (r)$ and $\mathrm{\Delta }OPL(r)$ input for computing the electric field to be propagated using the exact-transfer function. The resulting intensity, $I(r)$ , and electric field phase amplitude, $\mathrm{\Phi }(r)$ are shown. (D) Focus quality was estimated by fitting the wavefront in a pseudo objective’s back focal plane.

To determine the asymmetric light sheet intensity from these radially symmetric simulations, we combined two simulations using excitation beams having Gaussian waists matching the two asymmetric light sheet waists, ${\omega }_0^{(x)}$ and ${\omega }_0^{(y)}$ . In analogy with the solution for Gaussian beam intensities (Eq. (5)), we combined the two solutions to obtain an asymmetric solution using ${\omega }_0^{(x)}$ and ${\omega }_0^{(y)}$ (Eqs. (6),(7)).

$$\begin{array}{rl}{I}_G(x,y,z;{\omega }_0^i)& ={\left(\frac{{\omega }_0^i}{{\omega }^i\left(z\right)}\right)}^2\exp \left[-2\frac{(x^2+y^2)}{{\omega }^i{\left(z\right)}^2}\right]\end{array}$$ (5)

$$\begin{array}{rl}{I}_{asy}(x,y,z;{\omega }_0^{(x)},{\omega }_0^{\left(y\right)})& =\frac{{\omega }_0^{\left(x\right)}{\omega }_0^{\left(y\right)}}{{\omega }^{\left(x\right)}\left(z\right){\omega }^{\left(y\right)}\left(z\right)}\exp [\frac{-2x^2}{{{\omega }^{\left(x\right)}\left(z\right)}^2}+\frac{-2y^2}{{{\omega }^{\left(y\right)}\left(z\right)}^2}]\end{array}$$ (6)

$$\begin{array}{rl}& =\sqrt{I_G(\sqrt{2}x,0,z;{\omega }_0^{\left(x\right)})}\sqrt{I_G(0,\sqrt{2}y,z;{\omega }_0^{\left(y\right)})}.\end{array}$$ (7)

We assume a similar relationship holds for our model profiles $I_M$ . In practice, we simulate the light sheet propagation along the axis with the tight light sheet focus, $x$ , and assume an exact Gaussian form along the tall axis of the light sheet, Eq. (8). We expect that optical aberrations only weakly affect this axis, and we find that simulations using this approach agree well with our experiments.

$$I_{\text{LS}}(x,y,z;{\omega }_0^{\left(x\right)},{\omega }_0^{\left(y\right)})\approx \frac{{\omega }_0^{\left(y\right)}}{{\omega }^{\left(y\right)}\left(z\right)}\exp [-\frac{2y^2}{{\omega }^{\left(y\right)}{\left(z\right)}^2}]\sqrt{I_M(\sqrt{2}x,0,z;{\omega }_0^{\left(x\right)})}$$ (8)

Before determining the light sheet intensity profile, we first characterize the focal plane using ray tracing. Determining the focal plane in the presence of aberrations is complicated by the spread in the optical axis intersection between paraxial and marginal rays. For small aberrations, the best-fitting Gaussian reference sphere is centered at the midpoint between the two ray intersections [28]. Beyond this, diffraction effects must be considered. Our optical model defines the focal plane as the midpoint between the paraxial and marginal ray intersections with the optical axis.

To characterize wavefront aberrations, we estimated the effective objective pupil wavefront by re-sampling the rays using an objective with matching NA and focal length defined by the focal plane. The action of the objective eliminated the curvature attributed to focusing, resulting in a flat pupil plane to characterize wavefront aberrations. We defined the wavefront $W(r)$ as the optical path length difference with respect to the the paraxial ray, Eq. (9). Wavefront distributions were fit using an $8^{th}$ order polynomial. Here we only include azimuthally symmetric aberrations, as our simulation is symmetric. For example, $C_4$ is a direct measure of spherical aberration while defocus is characterized by a combination of $C_2$ and $C_4$ [29]. To provide more insight, we convert the polynomial coefficents to Zernike coefficients describing aberrations up to tertiary spherical aberration. From this analysis, we find that primary spherical is the dominant aberration, as illustrated in Fig. 5.

To quantify wavefront aberrations, we defined an aberration function by subtracting the piston term (Eq. (10)). For a holistic interpretation of the focus quality, we calculated the wavefront Strehl ratio (Eq. (11)) and root-mean-square deviation (RMS) (Eq. (12)) [29,30].

$$\begin{array}{rl}W(\rho )& =-[\text{OPL}(\rho )-\text{OPL}(0)]=-\sum _{i=0}^8{C}_i{\rho }^i\end{array}$$ (9)

$$\begin{array}{rl}\mathrm{\Phi }(\rho )& =W(\rho )-C_0\end{array}$$ (10)

$$\begin{array}{rl}S& =\frac{1}{{\pi }^2}{\left|{\int }_0^{2\pi }{\int }_0^1{e}^{i{k}_o\mathrm{\Phi }(\rho )}\rho d\rho d\varphi \right|}^2\end{array}$$ (11)

$$\begin{array}{rl}\text{RMS}& =\sqrt{\frac{1}{\pi }{\int }_0^{2\pi }{\int }_0^1{|\mathrm{\Phi }(\rho )-\overline{\mathrm{\Phi }}|}^2\rho d\rho d\varphi }\end{array}$$ (12)

where $\rho$ and $\varphi$ describe the normalized pupil position, and $\overline{\mathrm{\Phi }}$ is the mean wavefront aberration.

Additionally, we employed ray tracing to characterize the longitudinal spherical aberration (LSA) as the difference in optical axis intersection between paraxial and marginal rays Fig. 5 [31].

2.2. Simulation of light sheet illuminations

We simulated the remote focus pathway used in the light sheet acquisitions as shown in Fig. 2(A). Each model included a thick lens of variable focal length to simulate an electrotunable lens (ETL), two achromatic doublet lenses ( $L_1$ , $L_2$ ), and an objective ( ${\text{EXC}}_{\text{obj}}$ ). We assume the light sheet propagates through a cuvette, which we model as a flat (plano-plano) glass surface of thickness 1.25 mm and RI= $1.4585$ . The region inside the cuvette is filled with water, RI= $1.33$ . The RI mismatch at the cuvette surface reduces the rays' angles and shifts the focus.

The ETL was modeled using a flat first surface, a spherical second surface of variable curvature, and fixed edge thickness. The ETL was parameterized by the edge thickness and the lens power in diopters ( ${\text{m}}^{-1}$ ). The lens material RI was set to 1.3 to match the manufacturer’s specifications, and edge thickness was 10 mm. To map the ETL power or focal length to the second surface radius of curvature, $R_2$ , we applied the lens maker’s equation.

To align the lenses and ensure the ETL and objective are in conjugate planes, we deployed ABCD matrix methods to identify the lens’ principal and focal planes. After ray tracing each component, we determined the intensity distribution about the focus as described above and extracted the central slice, $I_M(x,0,z)$ , for comparison to experimental data.

All computation, except for data acquisition, was performed using a Linux Mint 20 server with two Xeon E5-2650 CPUs (Intel) with 16 cores each, 1 TB RAM, and two GeForce RTX 3090 Ti (Nvidia) with 24 GB of memory. It took 2 minutes to simulate one remote focus pathway and generate the 3D light sheet in a $249\times 2049\times 2049$ voxel volume using the CPU. The system GPUs were not used for the simulations presented here.

2.3. ETL remote focus optimization

We used ray tracing to characterize the axial aberrations and investigate the effects of using an ETL for remote focusing. The remote focus pathway includes the same components described above, utilizing a low NA, long working distance excitation objective (0.14 NA Mitutoyo 378-802-6). In each case, we tested all physically attainable configurations of cuvette position and ETL curvature. The cuvette offset (dc) positions spanned the space between the lens’s surface element and the lens’s working distance. Although the perfect-lens objective model only requires specifying the location of the principal plane, we additionally assume the objective has a working distance of 34 mm to match our Mitutoyo excitation objective. The position of the fictitious surface element is, therefore, 6 mm after the principal plane. In the presence of the cuvette, the ETL power is constrained to keep the focal plane within the cuvette bounds. The light sheet displacement (df) or focal shift is calculated with respect to the last cuvette surface.

2.4. Experimental setup

We modified an existing remote focus SPIM setup to capture light sheet images in transmission [32,33], as illustrated in Fig. 3(A). We imaged beads embedded in agar using the optical pathway in Fig. 3(B) to validate the ETL optimization model. A National Instruments DAQ (USB-6343) synchronized the ETL and illumination. Below, we detail the illumination pathway, followed by the detection optics.

Fig. 3.

Optical configurations for measurements. (A) Optical pathway used to image light sheets in transmission. (B) Optical pathway used to characterize PSF's to validate the ETL optimization model. $L_i$ are achromat doublets, Ap. is the adjustable aperture used to modulate the NA. $L_{cyl}$ is the cylindrical lens, TL is the tube lens. The dotted lines represent conjugate planes, red and blue indicating conjugate to the ${\text{BFP}}_{EO}$ and ${\text{FFP}}_{EO}$ respectively.

Both excitation optical pathways utilized an excitation source (Oxxius L4Cc) with four wavelengths (488 nm, 523 nm, 561 nm, 638 nm) combined into a single fiber output, with 561 nm used for light sheet measurements and 488 nm used to image beads. For all measurements, the fiber output was collimated using 250 mm focal length achromatic lens (Thorlabs AC508-250-A-ML), the effective NA was modulated using an adjustable rectangular slit (Ealing 74-1137-000) conjugate to the ${\text{BFP}}_{EO}$ . An ETL (Optotune EL-16-40-TC) was utilized for the remote focusing unit.

In the excitation optical pathway used for light sheet characterization, Fig. 3(A), the beam was collimated and expanded by a factor of 4 before passing through the aperture (Thorlabs AC508-250-A-ML, Thorlabs AC254-050-A-ML, AC508-200-A-ML) and a truncated Gaussian line profile was generated by focusing a cylindrical lens (Thorlabs ACY254-200-A) on the ETL and relayed to the ${\text{BFP}}_{EO}$ (Thorlabs AC508-150-A-ML, AC508-100-A-ML) before being focused by a 0.3 NA air objective (Nikon MRH00105).

The excitation optical pathway used for standard SPIM operation used a cylindrical lens (ACY254-50-A) to generate a line profile conjugate to the imaging plane. The line profile is refocused on the ETL and relayed to the $\text{BFP}$ of a 0.14 NA air objective (Thorlabs AC254-050-A-ML, AC508-180-A-ML, AC508-150-A-ML, Mitutoyo 378-802-6).

For both measurements, the detection optics were mounted on an adjustable motorized stage (Thorlabs KST101) to record z-stacks for light sheet characterization or to perform auto-focusing during SPIM operation. Light sheets were imaged in transmission using a 0.5 NA long working distance objective (Mitutoyo 378-805-3) and a 200 mm tube lens (Thorlabs TTL-200) focused on a scientific CMOS camera (Teledyne Photometrics Iris15). The effective pixel size of 85 nm was adequate to Nyquist sample all the light sheets considered here. For imaging beads, the detection optics were replaced with a 0.5 NA 2x magnification objective and 1x tube lens (Olympus MV PLAPO 2XC, Olympus MV PLAPO 1X), forming an image on the same scientific CMOS camera. Fluorescence light was isolated using a 30 mm filter wheel (Finger Lakes Instrumentation HS0433417) and barrier filter (Semrock FF01-536/40-32-D).

The control computer for all experiments was a Dell Precision 3630 running Microsoft Windows 10 Enterprise. This computer has a Core i7-9700K CPU (Intel) with 8 cores, 64 GB RAM, and a GeForce RTX 3060 GPU (Nvidia) with 12 GB of memory. The navigate software package [34] was used to collect the data shown in Fig. 1(A),(B) and Fig. 5(C)–(E). Micromanager 2.0 [35] was used to acquire light sheets in transmission.

2.5. Light sheet profile acquisitions

We measured light sheets focusing in air and through an RI mismatch as expected during normal imaging conditions. The RI mismatch was created using a 10 mm $\times$ 20 mm $\times$ 45 mm (22.5 mm path length) glass cuvette filled with water. For each, we modulated the effective NA by adjusting the long-axis extent of the laser elliptical focus in the objective back focal plane and then acquired a z-stack for a range of ETL powers. The position of the index mismatch is fixed such that when the ETL is flat, the focus is centered in the cuvette. The remote focus distance is measured relative to the light sheet position when the ETL is flat. After adjusting the aperture, the ETL current was reset to zero, and the detection stage was adjusted to center the light sheet. To acquire data, we first moved the detection stage to align the FOV to the desired remote focus distance, then adjusted the ETL power to recenter the light sheet focus. We captured z-stacks for remote focus positions for each NA ranging from ±4.0 mm. Aberrations limited the remote focus range for the higher effective NAs. The z-stack range was adjusted to observe out-of-focus features of the light sheets.

The 3D raw data was processed to extract a light sheet slice in the ZX plane which we could compare to our simulation results. First, we identified the focal plane using the peak intensity. The ZX plane with the greatest average intensity was chosen to compare with our model results. The resulting 2D image is rotated to align the propagation and $z$ -axis, typically less than 1°. For light sheets imaged in water, stage displacements were scaled by $1.33$ to reflect the image plane shift imposed by the index mismatch.

2.6. Light sheet profile analysis

We quantified the simulated and experimental light sheets' radial confinement or width, $\omega (z)$ , and propagation length, $z_L$ . The width is characterized by the full width at half-maximum (FWHM) of the radial intensity, and the length is the range over which $\omega (z)\le \sqrt{2\ln 2}{\omega }_0$ .

To process raw images into light sheet parameters, we first interpolated the axial intensity to identify the diffraction focal plane. Next, the focal plane intensity was fit using a 3-Gaussian intensity model, Eq. (13), to account for side lobes and more complex beam structure caused by diffraction. The width, ${\omega }_0$ was defined using the peak lobe width and reported using the FWHM. Next, we numerically evaluated each plane’s FWHM, including side lobes. After calculating the width for each z-plane, we interpolated the resulting distribution. In calculating the light sheet length, we found a numerical approach to define the radial extents (where $\omega (z)\ge \sqrt{2}{\omega }_0$ ) was more consistent for comparing Gaussian and non-Gaussian light sheets than fitting using a Gaussian function.

$$I(r)=I_{\text{offset}}+\sum _{i=0}^{i=2}{I}_i\exp \left[-2\frac{{(r-{\mu }_i)}^2}{{\omega }_i^2}\right]$$ (13)

2.7. Bead imaging to characterize imaging system quality

We prepared 1 µm beads embedded in 1 % agar (FluoSpheres F8823). Using the SPIM setup in Fig. 3(B), we performed imaging experiments at cuvette offsets positions from 10 mm to 35 mm, every 5 mm. For each cuvette position, the detection stage was translated to focus ∼4 mm from the leading surface, in the middle of the cuvette and ∼4 mm from the back surface. We captured 80 µm z-stacks, dz=1.0 µm, at each position. To quantify our results, we localized and fit the beads to a Gaussian model [36] and then applied a median and mean filter to the resulting axial standard deviation distribution along the propagation axis. To estimate the propagation length of the light sheet, we numerically identified the FWHM of the focus.

3. Results

3.1. Comparison of simulated and experimental light sheets

The model results (dashed lines) depicted in Fig. 4 represent light sheets with zero ETL defocus applied, agree with prior results [11,12]. The model results in water suggest an index mismatch is a limiting factor in obtaining tighter light sheets, but these light sheets also exhibit longer axial profiles. A comparison between the model results in air and in water supports prior theoretical work, which found that the spherical aberration induced by an index mismatch extends the propagation length compared to a Gaussian focus of equal width [11,14].

The experimental results with no shift applied closely follow the model expectations from Eq. (8) and Fig. 4. Our experimental results are consistent with previous observations of Gaussian beam and sinc light sheets that exhibit longer propagation lengths for a given ${\omega }_0$ [12,15]. Focusing in water, the tightest attainable light sheet is limited by aberrations. Both the model and experimental results show that focusing through an index mismatch is a limiting factor in achieving quality light sheets. In this initial experiment, we did not optimize the relative spacing of the excitation optics and cuvette for the RI mismatch case.

The visual comparison between the model and experimental data emphasizes our ability to accurately capture features of the light sheets, Fig. 4(B),(C). While the model replicates focal features such as side lobes effectively, as aberrations increase, we observe minor deviations for out-of-focus features. We suspected this discrepancy arises because these features are highly sensitive to the precise phase profile of the beam or to the objective lens aberrations, which we are not modeling. A visual comparison between experimental and acquisition light sheets with applied focus shows we can capture the aberration characteristics associated with correcting first-order defocus with an ETL.

3.2. Remote focus optimization for heterogeneous refractive index

Using an objective lens designed for a different RI will lead to aberrations and impact the excitation profile and detected image quality. In addition, many large sample SPIM experiments have an RI mismatch between the optics, sample immersion media, and sample itself, leading to a displacement of the focus. Traditionally, addressing this issue involves holding the sample chamber constant and displacing the light sheet (physically or using remote focusing) until it is within the sample. Leveraging our model, we explored optimal configurations for remote focusing in such environments. We primarily focus on aberration effects introduced by the RI mismatch between the optics and the sample immersion medium, which is often the dominant source of aberration in the system. In Supplement 1 ^{(1.4MB, pdf)} section 3, we further explore the effect of immersion media-sample RI mismatch.

There are up to two optimal light sheet positions for a given imaging configuration where the light sheet aberrations are minimized. These positions occur where the ETL induces an optical path length difference ( $\mathrm{\Delta }\text{OPL}$ ) that counterbalances the path length difference accumulated through the remaining optics (see Supplement 1 ^{(1.4MB, pdf)} section 2). The RI mismatch imposes an additional $\mathrm{\Delta }\text{OPL}$ , degrading focus quality and shifting the focal plane further from the objective.

$$\mathrm{\Delta }\text{OPL}\propto \sqrt{{\left(\frac{r}{d}\right)}^2+1}-1$$ (14)

Here, $d$ is the distance propagated in the medium, and $r$ is the distance from the optical axis. Consequently, adjustments in the ETL lens curvature are required to maintain the focal position within the FOV. Using the ETL remote focus model, we predicted the optimal spacing between the objective and the cuvette to minimize the beam waist ( ${\omega }_0$ ) and maintain these values when shifting the light sheet via remote focusing Fig. 5.

We evaluate our predictions by imaging and quantifying the point-spread function (PSF) across the camera FOV, using diffraction-limited fluorescent microspheres in agar for various light sheet shifts (df) and objective-to-cuvette spacing (dc). The PSF ${\sigma }_z$ fit results along the propagation direction reflect a combination of imaging system resolution and the optical sectioning attributed to the light sheet’s focal region. Quantifying the extent over which the PSF remains confined in the light sheet propagation direction, we find that our model well predicts the optical sectioning and propagation length in the presence of RI mismatch-induced aberrations, Fig. 5. For our remote focus pathway, the optimal position for placing the cuvette to achieve a uniform PSF ${\sigma }_z$ within the propagation length and maximizing the remote focus range was determined to be between 25 mm and 30 mm. Furthermore, we observed that when the cuvette is positioned at either extreme—either too close or too far from the objective—the quality of the light sheet improves with displacements away from these extreme positions.

4. Discussion

In this work, we have explored the effect of remote focusing and refractive index mismatch on light sheet intensity distributions for large FOV SPIM. We confirmed through numerical modeling and experimental validation that RI mismatches lead to aberrated light sheet intensity profiles. Our modeling tools demonstrated that some recovery is possible by jointly optimizing the applied light sheet shift and the objective-to-sample-chamber spacing, which we then experimentally verified using test samples of fluorescent microspheres embedded in agar. We further validated our conclusions by calculating the third-order Seidel aberration contributions of each optical element to show how they balanced each other to yield optimal configurations. We also extended our modeling approach to demonstrate that immersion media to sample RI mismatch has a negligible contribution to aberrations compared to the air-objective-to-immersion-media RI mismatch. One significant benefit of optimizing the light sheet pathway is that the focus is optimized to enhance image contrast and correctly positioned within the sample. By incorporating remote focus pathway aberrations in the light sheet simulation, we demonstrated that considering both diffraction effects and non-paraxial aberrations is crucial for accurately characterizing light sheets in remote focus and RI mismatch applications.

Our findings are enabled by an open-source software package that combines ray tracing and beam propagation to generate accurate simulations of the light sheet focus [37]. By carefully accounting for all of the optical elements and various refractive index media in the optical pathway, we demonstrate that it is possible to simulate remote-focus light sheet pathways using efficient, CPU-only software. For large samples, such as cleared tissue, zebrafish, or organoids, an air immersion objective is common to generate the light sheet focus. Here, we use ray tracing to account for the remote focus pathway aberrations in one such air-immersion objective remote focus configuration. Our developed tools enabled us to estimate the remote focus range and determine the optimal RI mismatch position to maximize optical sectioning for this specific configuration. We provide examples with our software package on how to customize and define arbitrary light sheet optical configuration, allowing users to explore the best configurations for their particular setup.

Moving forward, it may be possible to integrate our approach into an on-the-fly adaptive framework to improve light sheet image quality [23]. Existing adaptive frameworks rely on heuristic metrics or guide stars to calculate the required corrections on deformable or configurable elements to account for RI mismatches and other image-degrading aberrations [32,38–46]. These approaches significantly improve obtainable image quality across multiple SPIM optical configurations at the cost of more complicated optical configurations and potentially expensive modifications. Our results suggest that simply starting with an optimized imaging configuration will enhance image contrast across the FOV. Our findings also apply to extended FOV methods, such as ASLM, where optimizing both the RI mismatch position and the light sheet position will further improve image quality, independent of additional corrective measures. Future work could extend these methods to account for invariant illumination in reconstructing light sheet acquisitions of large samples, where the cuvette position and remote focus parameters should be updated to maintain the best quality light sheet. Additionally, extending the model to the entire image formation framework would enable the exploration of adaptive optics informed by modeling to assist the correction of excitation and emission pathway aberrations dynamically.

Our findings on the optimal configurations for light sheet optical pathway configurations, especially in the presence of RI mismatches, provide valuable guidelines for improving imaging quality in SPIM. This study offers a practical strategy for enhancing SPIM performance across a range of biological samples and imaging configurations and a community resource software package to model, predict, and characterize light sheet quality.

Funding

National Heart, Lung, and Blood Institute10.13039/100000050 (R01HL068702).

Disclosures

The authors declare no conflicts of interest.

Data availability

Example datasets for the imaging of the light sheet in transmission and bead measurements are available [47]. All imaging data is available upon request. Code to run analysis is available on our Github repository [37].

Supplemental document

See Supplement 1 ^{(1.4MB, pdf)} for supporting content.