Light-sheet fluorescence microscopy (LSM) has become an essential imaging modality in the life sciences, providing rapid volumetric imaging with minimal photodamage.1 Although this makes it an ideal candidate for in vivo imaging, such as endomicroscopy2^,3 and neural imaging in freely moving animals,4^,5 translation to miniaturized optical systems has been limited to date.6^,7 A critical technology for incorporating LSM into these applications is high-performance micro-objectives that can match their macroscopic counterparts in numerical aperture (NA) and field of view (FOV).8 Their performance comes at the cost of field curvature,9^,10 which presents an additional challenge for LSM. With the generally planar geometry of a Gaussian beam, only a limited portion of the illuminated area is brought into perfect focus in the image plane, limiting the effective FOV, imposing limits on the detection NA, or leading to loss in image contrast. Although the incorporation of scanning micromirrors could address this challenge to some extent,11 this further increases the system complexity and limits the achievable degree of miniaturization. Using Airy beams12 has been proposed to match field curvature due to its uniaxial acceleration when implemented as a static light-sheet.13 An Airy light-sheet can be generated by imposing a one-dimensional (1D) cubic phase profile on a Gaussian beam, which is then focused by a cylindrical lens. Similar to other configurations using propagation-invariant beams,14^,15 it has been used to excite a larger FOV compared to a Gaussian sheet, while maintaining the same sectioning ability.16^,17 However, incorporating the static Airy light-sheet only partially solves the curvature problem, as the planar Gaussian profile along its inactive axis still introduces an FOV reduction, limiting the performance required for micro-optical applications.

We have recently shown that this challenge can be addressed by an additional phase modulation on top of the cubic profile needed for the static Airy sheet.18 This phase profile depends linearly on the active coordinate and quadratically on the inactive coordinate and modifies the Gaussian profile of the light-sheet to achieve the same radius of curvature as the Airy beam. The result is a static, biaxially accelerating light-sheet that is unrestricted in its ability to section, thus increasing the effective FOV of the micro-objective. In this paper, we provide a detailed analytical derivation of the phase-plate parameterization using typical micro-objective specifications, allowing the light-sheet geometry to be tailored for maximized in-focus excitation. To verify the validity of our analysis, we perform simulations in the paraxial regime using ray-tracing and beam-propagation methods (BPMs). By designing, implementing, and characterizing a micro-optical excitation system that incorporates a 3D-printed millimeter-scale phase plate, we experimentally demonstrate our ability to generate such a light-sheet in a fully miniaturized fashion.

Figure 1(a) shows the optical architecture of a miniaturized light-sheet microscope with a high NA micro-objective. The illumination arm consists of a collimator (not shown) and a cylindrical lens to form a Gaussian light-sheet within the FOV of the micro-objective that is imaging the excited fluorescence onto a flat image sensor. The field curvature of the detection objective zd can be characterized by a radius rd≫FOV so that the detection field follows: zd(x,y)≈x2+y22rd.(1)

As common in LSM, we use the coordinate system of the detection objective for the light-sheet, in which

Figure 1(b) extends the concept to the generation of a static Airy light-sheet, resulting from a 1D cubic phase modulation of the collimated beam incident on the cylindrical lens, which allows field curvature matching in x. This already increases the effective FOV in this coordinate compared with the Gaussian light-sheet and improves the sectioning capability due to a thinner light-sheet. A secondary phase modulation of the proposed phase profile aims at bending the Airy light-sheet along its inactive axis, thus generating a static and yet biaxially accelerating sheet [Fig. 1(c)]. In the next two sections, we discuss the design of the active and inactive phase profiles to achieve biaxial field curvature matching.

An Airy light-sheet can be generated by modulating the collimated Gaussian beam with a beam waist w0 by a 1D cubic phase ϕAi at the back focal plane (BFP) of the cylindrical lens13 along its active direction z, such that the electric field follows, EAiBFP(y,z)=exp(−z2+y2w02)·exp[i2πα(zw0)3],(2)where α is the amplitude of cubic modulation. A schematic depiction of this configuration is shown in Fig. 2, where the approximate intensity profile around the focal plane can be obtained in the paraxial domain as19^,20IAi(x,y,z)≈|Ai(zz0−x22rAiz0+iaxkz02)|2·exp[2a(zz0−x2rAiz0)−2y2w02],(3)where Ai(·) denotes the Airy function, x is the axial coordinate, z0=f(6πα)13/(w0k) is a characteristic length, a=(6πα)−2/3 is an apodization factor, and k=2πn/λ is the wave vector in a medium of refractive index n.21 It has been demonstrated by numerous research groups19^,22^–25 that this function is highly asymmetrical with prominent sidelobes, and its main lobe follows a parabola with a radius of curvature rAi of rAi=6λn2f3w03α.(4)

Therefore, one must solve Eq. (4) for α with rAi=rd to obtain an Airy light-sheet with an acceleration matching the field curvature of the detection objective. Using Eq. (2), we obtain ϕAi(z)=2πrd6λn2f3·z3.(5)

To finalize the geometry of the light-sheet, the beam needs to be matched to the FOV of the micro-objective, which we define as the area around the focal point within which the intensity remains larger than the half of the maximum value.21 Since Eq. (5) does not depend on w0, we can control the geometry of the light-sheet in the propagation coordinate and in the inactive coordinate independently of each other via the focal length and the beam waist. Explicitly, the FOV along the active direction is obtained from FOVx=w0rd2ln2nf,(6)and along the inactive direction from FOVy=2ln2w0.(7)

To obtain a symmetric FOV, the condition rd=nf must be satisfied, which means that the focal length in the imaging medium must be equal to the radius of curvature.

To induce a parabolic axial shift of the focal point along the inactive coordinate y as well, we modify Eq. (3) by convolving it with a delta distribution δ(·) with the required shift in the detection coordinate. This operation can be expressed as Ibiaxial(x,y,z)=IAi(x,y,z)⊛δ(z−y22rd),(8)≈|Ai(zz0−x2+y22rdz0+iaxkz02)|2·exp[2a(zz0−2x2+y22rdz0)−2y2w02],(9)where ⊛ denotes the convolution operation. Details of this derivation can be found in Appendix A. By taking the Fourier transform of the electric field at the focal point x=0 along z, we can again obtain the required electric field at the BFP, which can be approximated as EbiaxialBFP(y,z)≈exp(−z2+y2w02)·exp[i2πα(zw0)3]·exp(−iπrdλf·y2z).(10)

The arguments of the first two exponential functions in Eq. (10) represent the conventional profile to generate an Airy beam as per Eq. (2), while the last term is an additional, phase-only term that we define as ϕinactive(y,z)=−πrdλf·y2z.(11)

In ray optical terms, this phase profile adds an additional launch angle from the phase plate to the cylindrical lens in its active coordinate, the magnitude of which varies as the square of the inactive coordinate. Because of the introduction of this additional angle in the BFP, each 1D Airy ray will be shifted in z at the focal plane of the cylindrical lens.

A combined phase plate with a sag height of hPP, which generates both the 1D Airy beam along the propagation coordinate, as well as the focus-shifted Gaussian beam along the inactive coordinate of the beam, can be obtained as hPP(y,z)=λ·(ϕAi+ϕinactive)2π(nm−1),(12)=rd6n2f3(nm−1)z3−12rdf(nm−1)y2z,(13)where nm is the refractive index of the phase plate. The individual profiles are plotted alongside the combined one in Fig. 3. An important point about Eq. (13) is that the Airy-related modulation scales linearly with rd, while the modulation related to the inactive profile is inversely proportional to it. This imposes a limit on the range of rd that can be realized. For lower values of rd, the α parameter is smaller, and the assumptions made in the derivation of the apodized Airy beam become invalid.19^,25 On the other hand, the modulation generated by the inactive profile dominates the phase plate. This leads to larger launch angles, which may violate the paraxial approximation of the optical element. At the upper end of the limit, it is useful to evaluate whether field curvature remains a problem as the beam becomes flatter and thicker, bypassing the initial problem.

To assess that curvature-matching in the paraxial regime, we performed ray-tracing simulations using commercial software (OpticStudio 22.1, Zemax LLC). The simulation model implemented the configuration shown in Fig. 2, with the phase profile described in Sec. 2.3. We calculated the curvature of the beam in the active direction by evaluating the caustic26 of 51 rays displaced in the active z-coordinate (XZ-section in Fig. 2). For the inactive direction, we obtained the displacement of the rays in the image plane by tracing 51 rays that were initially displaced along the y-direction. To further investigate the validity of Eq. (9), we propagated the field defined in Eq. (10) using a 2D fast Fourier transform (FFT) based BPM in the Fresnel regime.21 Propagation was performed to match the ray-tracing simulations, which included the same beam waist, phase plate profile, and focal length of the cylindrical lens. 3D stacks were calculated within a range of ±300μm around the focal point with a step size of Δz=5μm, and a grid size of (nx×ny)=(212×212)pixels with a width of Δx=Δy=0.2μm. We set the input parameters with an excitation wavelength λ=850nm, an input beam waist w0=240μm, an effective focal length of the cylindrical lens fz=1.4mm, and a refractive index of the phase plate nm=1.50927 to match the experimental setup described below. Two radii of curvature were chosen in a typical range for micro-objectives of 1.5 and 2 mm.8^,9^,28

Phase plates with a diameter of 0.78 mm were fabricated using a commercial two-photon polymerization based 3D printer (PPGT+, Nanoscribe GmbH & Co KG) with a commercial resin (IPS, NanoScribe GmbH & Co KG). The height profiles were pre-processed and subsequently fabricated on ITO-coated glass substrates using an in-house developed slicing and printing strategy, previously described.29 After development, the height profiles of the phase plates were measured by white-light interferometry (WLI; NewView 9000, Zygo Corporation). A 0.3-NA objective (I100384, Zygo Cooperation) was used to obtain the surface profile of the entire plate and then fitted with standard Zernike polynomials to quantify the shape deviation. Since the tilt of the printing field in this system can only be controlled to within a few micrometers from print to print, the tilt Zernike coefficients were not included in the deviation analysis. The surface roughness was estimated by measuring a 25μm×25μm area of the phase plate with a 0.8-NA objective (I190053, Zygo Cooperation). A high-pass Gaussian spline filter with a cutoff frequency of 25μm was applied and the resulting root mean square (RMS) error was calculated.30

A cylindrical gradient-index (GRIN) lens with an effective focal length of 1.4 mm (Grintech GmbH) was bonded to the back of the substrate after fabrication so the phase plate was located at its BFP. The entire assembly was illuminated with a fiber-based collimator (GRINTECH GmbH) using an SLED with a wavelength of 850 nm as the light source (Fig. 4). The beam waist of the illumination beam after the collimator was w0=240μm. In order to obtain the 3D beam profile in the vicinity of the focal plane of the cylindrical lens, the beam was imaged onto a CCD (UI-1240SE-NIR, IDS GmbH) using a 0.28-NA objective (Plan Apo 10x, Mitutoyo AC) and a relay lens, while the entire system was translated along the propagation direction of the light-sheet.

To compare the analytical analysis, simulations, and experiments, 2D cross sections of the beam intensity along its propagation direction at (y=0) and along the inactive direction at the focal plane (x=0) were measured. The thickness Δz and the FOVx,y were determined by finding the full width at half-maximum of the beam intensity in the respective direction. The radius of curvature in x and y was determined by finding the location of the maximum intensity in the respective coordinate and fitting Eq. (1) to these data points using a least-squares algorithm.31

Figure 5 shows the characterization of the surface profile obtained with the WLI. The surface errors become more pronounced toward the edge, where lift-off from the substrate is expected, but were found to be <1μm over a diameter of 780μm. With an RMS surface roughness below 10 nm, the phase plate can be considered an optical quality surface.

Table 1 compares the radii of curvature determined by Eq. (9), ray-tracing and BPM simulations with the experimental measurements. The ray-tracing simulations yielded radii of curvature that matched the design radii within numerical accuracy. Furthermore, the simulated beam profiles shown in Fig. 6 agree with those generated by evaluating Eq. (9). Deviations can be narrowed down to an asymmetric intensity distribution around the focus below 10μm, and predicted offsets in z are consistent with both design and numerical predictions. The radii of curvature obtained from measured data matched the design values in most cases. For rd=1.5mm, the radii in the propagation coordinate and in the inactive coordinate were found to be (0.2±0.1)mm and (0.1±0.1)mm smaller than the design value, respectively. For rd=2.0mm, the radius of the active coordinate is equal to the designed radius, while the inactive coordinate is (0.2±0.1)mm smaller than the designed radius. The beam profiles themselves show the typical Airy profiles, although a reduced sidelobe intensity is observed with an underestimation of the FOV in the propagation direction from 20% to 36% and 5% to 10% in the inactive coordinate.

The excellent agreement between the results of our analytical treatment and the ray-traced simulations is a good indication of the validity of the design approach within the paraxial regime. Limitations of this approximation may arise in different scenarios. As discussed in Sec. 2.3, lower values of rd will lead to a reduced modulation of the Airy profile. Therefore, we expect the radius to be underestimated in the propagation coordinate, as observed in the experimental data. In the inactive direction, we also expect to see an underestimated radius, but for a different reason. The higher additional deflection leads to a gradual violation of the paraxial approximation, resulting in aberrations and thus lower radii. On the other hand, we observed the Airy radius to be the same as the design radius for larger values of rd, whereas the radius along the inactive direction is slightly smaller. We attribute this to a manufacturing effect as for smaller values of rd, the sag height decreases and therefore requires more accurate manufacturing capabilities. Both effects can lead to the deviations reflected in the experimental results. Nevertheless, we have been able to adjust both radii to a degree that does not affect the improvement in effective FOV, which was the underlying design goal.

In summary, we have derived a phase profile to generate an Airy light-sheet that follows the curved image plane of a typical micro-objective, while incorporating the advantages of using an Airy beam to provide a larger FOV and better sectioning. Ray-tracing and BPM simulations in the paraxial regime validated the analytical analysis, with deviations between the two within numerical accuracy. A micro-optical implementation with milimeter-scale 3D-printed refractive phase plate was developed and combined with GRIN lens and experimentally demonstrated to achieve curvature matching in the design range.

To obtain the required phase profile, we need an expression of the biaxial electric field at the focal plane x=0, which we can approximate from the apodized Airy beam convoluted by the delta distribution δ as in Eq. (9) such that Ebiaxial(x=0,y,z)=Ai(zz0)·exp(azz0−y2w02)⊛δ(z−y22rd).(14)

The electric field at the BFP can be obtained by taking the Fourier transform along the active direction of the lens Fz{·}, i.e., EbiaxialBFP(y,z)=Fz{Ai(zz0)·exp(azz0−y2w02)⊛δ(z−y22rd)}.(15)

Since the Gaussian component is only dependent on y, it is neither affected by Fz, nor by the delta distribution and can thus be separated, EbiaxialBFP(y,z)=Fz{Ai(zz0)·exp(azz0)⊛δ(z−y22rd)}·exp(−y2w02).(16)

Using the convolution theorem, we can also simplify the convolution operation to a product, such that Fz operates individually on the apodized Airy function, whose identity is known from Eq. (2). The Fourier transform of a linear delta distribution is an exponential function, Fz{δ(z−y22rd)}∝exp(−ikzy22rd),(17)where kz≈2πz/(λf) is the wave vector in the active coordinate. Inserting those identities into Eq. (16) leaves EbiaxialBFP(y,z)=Fz{Ai(zz0)·exp(azz0)}·Fz{δ(z−y22rd)}·exp(−y2w02),(18)≈exp(−z2w02)·exp[i2πα(zw0)3]·exp(−iπrdλf·y2z)·exp(−y2w02),(19)=exp(−z2+y2w02)·exp[i2πα(zw0)3]·exp(−iπrdλf·y2z),(20)which is a Gaussian beam (the first exponential function) on which the cubic Airy phase (the second exponential argument) as well as the biaxial tilt (the third exponential argument) is imposed.

We would like to underline the fact that several assumptions were made in the derivation of Eq. (20) which can be summarized by the following points:

Any violation of those assumptions may result in aberrations of the Airy beam, which leads to a variation in length and thickness of the light-sheet, and in mismatch of the field curvature, as discussed in Section 5.

In general, the argument of the δ function in Eq. (20) can be modified to resemble an arbitrary focal shift in z depending on a function g(y), i.e., s(z,y)=δ[z−g(y)].(21)

An example for a benchtop application is the combination of a spatial light modulator (SLM) with a cylindrical lens to generate a 1D-Airy light-sheet. If the imaging arm is rolled with respect to the detection arm, the light-sheet will intersect the focal plane at an angle. By setting g(y) to match this tilt, one can compensate this misalignment without moving any mechanical components by employing a phase of ϕinactive=kz·g(y),(22)onto the SLM.

Table 2 accompanies the results presented in Table 1 by additionally stating the length and thickness of the light-sheet.

Yanis Taege received his BSc degree in physics from the University of Constance in 2017 and his MSc degree in applied physics from the University of Freiburg in 2019. Since 2020, he has been pursuing his PhD in the Laboratory for Micro-Optics at the University of Freiburg working on 3D micro- and nanofabrication of microendoscopic systems.

Tim Samuel Winter received his BSc degree in microsystems engineering from the University of Freiburg in 2020. He is currently pursuing his MSc degree in microsystems engineering at the University of Freiburg and Tampere University.

Sophia Laura Schulz received her BSc degree in optometry and her MEng degree in laser- and optotechnologies from Ernst Abbe University of Applied Sciences in Jena. She has been working for GRINTECH GmbH since 2020 and has been enrolled as a physics PhD student at Friedrich-Schiller University of Jena since 2021. Her main field of interest is the development and especially optical design of miniaturized systems for various biophotonics applications.

Bernhard Messerschmidt received his PhD in physics from Friedrich-Schiller University of Jena in 1998. Now, he is a managing director at GRINTECH in Germany and is also responsible for all R&D activities. His research focuses on biomedical optics, gradient index optics, and optical metrology.

Çağlar Ataman received his PhD in electrical engineering from Koç University, Istanbul, Turkey, in 2008, and worked as a postdoctoral researcher at École Polytechnique Fédérale de Lausanne, in Switzerland, between 2008 and 2012. From 2012 to 2021, he was a senior scientist at the Department of Microsystems Engineering of the University of Freiburg, Germany, where he has been an assistant professor since 2021, leading the Microsystems for Biomedical Imaging Laboratory.