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Abstract
We derive analytical expressions for the length, thickness, and curvature of an Airy light sheet in terms of basic parameters of the cubic phase and the paraxially defined focusing optics that form the beam. The length and thickness are defined analogously to the Rayleigh range and beam waist of a Gaussian beam, hence providing a direct and quantitative comparison between the two beam types. The analytical results are confirmed via numerical Fresnel propagation simulations and discussed within the context of light-sheet microscopy, providing a comprehensive guide for the design of the illumination unit.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. INTRODUCTION
Light-sheet microscopy (LSM) is a state-of-the-art volumetric microscopy technique [1]. It is based on the illumination of a specimen perpendicular to the detection’s optical axis with a thin sheet-like beam to enhance image contrast and reduce photo damage and bleaching [2]. An inherent challenge of conventional LSM with Gaussian light-sheet (LS) illumination is the fundamental trade-off between the axial sectioning capability and the field of view (FOV), defined by the beam waist and depth of focus, respectively. This challenge can be overcome by using propagation-invariant beams [3], such as Airy [4] or Bessel beams [5], which provide significantly longer LSs for a given thickness at the cost of spurious energy in the beam’s sidelobes. However, the primary imaging parameters for illumination with propagation-invariant beams are not as well defined as for Gaussian beams and are usually estimated through beam propagation methods (BPMs) that require significant computational resources, especially when covering a large parameter space. Therefore, there is a need for a set of analytical expressions that quantitatively describes the geometry of these beams in terms of length and thickness, i.e., the FOV and axial sectioning, respectively. In addition, if these equations are expressed by means of the physical specifications of the illumination optics, for example, by the numerical aperture (${\rm NA}$) of the focusing optics and the phase mask optical thickness, the LSM design process would be significantly more intuitive.
The propagation characteristics of an Airy beam, which will be treated in this work, have been studied in detail from both theoretical [3,6,7] and experimental perspectives [8–10]. However, the emphasis of these earlier works was on the acceleration and self-reconstructing behavior of the beam. While analytical expressions for both acceleration and length exist, an expression for the beam’s thickness has, to the best of our knowledge, been limited to relative rather than explicit expressions [4,11].
In this paper, we use the analytical expression for an Airy beam’s electric field to derive the explicit dependency of the length, thickness, and curvature on the specifications of the illumination optics, as illustrated in Fig. 1. First, we relate the illumination specifications to the parameters of a ($1 + 1$)-dimensional Airy beam. Then, we employ those parameters to analytically deduce the length and thickness of an Airy beam, analogous to the beam waist and Rayleigh range of a Gaussian beam. Finally, to confirm the validity of the analytical derivations, we present numerical results obtained via a Fresnel-based BPM and compare them to those measurements.
Fig. 1. Schematic illustration of the illumination optics where a cubic phase is imposed onto a Gaussian beam with waist ${w_0}$. The propagation characteristics of the resulting Airy beam are plotted to the right of a lens with focal length $f$. Here, the beam’s length and thickness, which are the analytically derived parameters, are labeled. We denote the active direction of the lens by $x$ in the focal plane, the electric field by $E$, and the intensity by $I$. The same parameters in the back focal plane are denoted by a hat symbol ^.
2. ANALYTICAL MODEL
Since the Airy function is a solution of the paraxial wave equation, Airy beams can be classified as propagation-invariant beams of infinite energy [3,12]. An apodized and thus square-integrable form of the beam, which can be experimentally realized, has been demonstrated by Siviloglou et al. [13,14]. The electric field $E(x,z)$ of an apodized Airy beam with wavelength $\lambda$ can be described by
A. Definition of Beam Geometry
For a quantitative comparison of an Airy LS to a conventional one, we define the beam geometry analogous to that of a Gaussian beam. The thickness, as is for the beam waist of a Gaussian beam, is defined by the $1/{e^2}$ drop-off in intensity from the focal point along the lateral direction. The length is defined by the point where the maximum beam intensity diminishes by half along propagation direction $z$, similar to the Rayleigh range. It should be noted that the maximum intensity accelerates and is therefore not necessarily found on the optical axis [13]. Notwithstanding the out-of-axis acceleration, the maximum illumination intensity would not accelerate out of the detection plane of a collection objective that possesses sufficient collection FOV and depth of field. Thus, the beam parameter definitions for Airy LSM should take the collection path design into account but are beyond the scope of this paper.
These definitions can be applied to single-photon excited fluorescence (SPEF), where the signal is proportional to the illumination intensity, and multi-photon excited fluorescence (NPEF; $N$ is the number of absorbed photons), where the signal is proportional to the $N$th power of intensity [15].
B. Airy Beams in Terms of Illumination Specifications
The following equations emphasize the results of our derivations. Please find the details of the derivations in Supplement 1, Section 1. To express the geometry parameters defined above in terms of Eq. (1), Airy beam parameters ${x_0}$ and $a$ have to be related to the input Gaussian beam and the illumination optics. By taking the Fourier transform $\hat E(\hat x) = {\cal F}\{E(x,z = 0)\}$ of Eq. (1) at the focal plane, one finds
1. Curvature of an Airy Beam
The acceleration $x(z)$, also known as curvature, can be deduced by equating the real part of the argument of the Airy function in Eq. (1) to zero and solving for $x$ [13]. This offset can be expressed by the radius $r$ of a circle that is estimated by a parabola:
The comparison yields the explicit radius of curvature [8] describing its linear dependency on $\alpha$.2. Thickness of an Airy Beam
To find the thickness $\Delta x$ of an apodized Airy beam, one needs to express the thickness $\Delta s$ of the Airy function ${\rm Ai}(s)$ in normalized coordinates $s = x/{x_0}$ as a function of $a$, i.e., $\Delta s = f(a)$. Assuming a linear relationship between $\Delta s$ and $a$, we find the thickness to be
3. Length of an Airy Beam
Since the maximum of the intensity $I$ accelerates, it makes sense to evaluate the length along the beam’s main lobe, i.e., inserting Eq. (7) into Eq. (1). For small values of $a$ (Supplement 1, Section 2.C), the main lobe intensity is proportional to
In LSM, the length of the illumination beam can be related to the area in which fluorescence is excited [18]. As for a certain intensity threshold no or a negligible amount of excitation can be expected [15], it makes sense to relate a drop-off in intensity along the propagation axis, i.e., the length of the beam, to the definition of the FOV. For an Airy beam, this depends on the implementation of the LSM setup: if the LS accelerates within the detection plane of the objective, it is sensible to relate the FOV to the length of the beam, i.e., ${\rm FOV} = 2{z_{{\rm Ai}}}$ (Fig. 2, left). As a result, an optimal $\alpha$ for a desired FOV can be found, which is
This configuration can be realized for a $(2+1)$-dimensional Airy beam for which the phase modulation is tilted by 45° as compared to the detection plane [19].Fig. 2. Schematic implementations of Airy light sheets and their respective fields of view (FOV) for the detection objective. The coordinate system refers to the collection path, where $z$ is the optical axis of the objective. In the in-plane configuration, the beam accelerates along $x$ within the detection plane and the FOV corresponds to the length of the beam. In the off-plane configuration, the beam accelerates along $y$, which means that it accelerates out of the detection plane, and only the “cap” of the beam is imaged. For a proper definition of FOV, the detection objective’s properties must be taken into account. Illumination in the other direction can be achieved via scanning for both in- and off-plane, or using a cylindrical lens (off-plane only).
In a configuration where the beam accelerates out of the detection plane, which is the case for a $(1+1)$-dimensional Airy beam focused through a cylindrical lens creating a static LS [20], the problem becomes more involved. Since the main lobe accelerates out of the detection plane, the detection optics have to be taken into account. Therefore, the FOV is not only a function of the beam’s length, but also of the depth of field of the detection optics. To reduce off-plane illumination, an increase in $\alpha$ would be an intuitive way to increase both the length of the beam and the radius of curvature. As in this case more power is distributed through the sample, the total incident power has to be increased to achieve the same amount of excitation. However, this increases thickness and therefore decreases contrast, and also extends the photodamage to more parts of the tissue. For a mathematically solid estimation of the interplay among illumination, collection, and sectioning, multiplication of the detection point-spread function with the estimated excitation is necessary. However, this optimization problem is beyond the scope of this discussion.
Fig. 3. Comparison of analytically derived expressions (solid lines) for the beam’s length, thickness $\Delta x$, and radius of curvature $r$ with BPM simulations for three different numerical apertures.
3. NUMERICAL SIMULATIONS
To verify our analytical calculations, we performed numerical simulations by means of a Fresnel-based, one-dimensional FFT-BPM implemented in Python ([21], Numpy Version 1.21.3). The input parameters were assumed to be a Gaussian beam with a waist of ${w_0} = 255\;\unicode{x00B5}{\rm m}$ in air ($n = 1$), modulated by a cubic phase with $\alpha$ parameters in the range of one to five. The wavelength was set to $\lambda = 750\;{\rm nm}$. This beam was propagated from the back focal plane of a lens to the lens with focal length $f$, which was determined according to the ${\rm NA}$ in the range of 0.1–0.3 used for the respective sweep. At the lens plane, a phase $\phi$ simulating the focusing of the lens was modulated on the propagated beam. To model the lens, we used a hyperbolic phase profile [22] with a $\kappa = - 2$ to compensate for spherical aberrations when simulating values of ${\rm NA} \gt 0.2$. Then, the beam was propagated a distance of $2f$ in steps of $\Delta z$ to obtain the propagation profile after the lens. A spatial sampling of $\Delta x = 15\;{\rm nm}$ and $\Delta z = f{/2^{10}}$ ensured convergence on a total grid size of ${2^{15}} \times {2^{10}}$ (${n_x} \times {n_z}$). The length and thickness were determined by numerically evaluating a volume around the focal plane using linear interpolation between the sampled points. The radius of curvature was determined by fitting a parabola to the lateral position of the maximum intensity along $z$ in the range $|z| \lt 0.8{z_{{\rm Ai}}}$.
A. Results
Figure 3 shows the comparison between the analytical expressions for LS geometry and the simulated beam profiles. Values predicted by the analytical model for the length underestimated those obtained by BPM with a relative deviation of up to 9.5% for small values of $\alpha$ and $\rm NA$, and less than 1% for larger values of $\alpha$. The relative thickness deviations were below 0.9% for the entire parameter range. The radius of curvature showed increasing deviations with ${\rm NA}$ of up to ${-}14\%$ for small values of $\alpha$. For large values of $\alpha$ and small values of ${\rm NA}$, the deviations were below 1.7%.
4. DISCUSSION AND CONCLUSION
To simplify the design process of a LSM, we have derived explicit expressions for the radius of curvature $r$ [Eq. (8)], thickness $\Delta x$ [Eq. (9)], and length ${z_{{\rm Ai}}}$ [Eq. (11)] of an Airy beam similar to the beam waist and Rayleigh range of a Gaussian beam. While the general relations have been predicted before [4,8], our calculations allow a quantitative prediction and comparison of those with respect to the used optical components. For example, we can explicitly state that the Rayleigh range equivalent of an Airy beam is $3\alpha \sqrt {2\ln 2}$-fold larger than that of a Gaussian beam [Supplement 1 Eq. (S35)].
The numerically obtained values for the length of the beam show their highest deviations of up to 9.5% for small values of $\alpha$. This behavior is expected, as Eq. (10) underestimates the length for higher values of $a$ and, thus, lower values of $\alpha$ (Supplement 1 Section 2.C). The lower the NA, the higher these deviations appear when comparing simulated and predicted values. The closest match between simulations and theory was found for the thickness relations, which can even be extended to different drop-off definitions and to multi-photon excitation. The largest deviations were obtained for the prediction of curvature that may arise from the reciprocal calculation, translating small deviations into larger ones. Although being consistent with previous publications that investigate curvature [8,14,20], special attention should be paid when estimating curvatures for larger values of NA.
Furthermore, a comprehensive definition of the desired FOV including a relation of an optimal $\alpha$ was provided. We also found that Eq. (12) provides values for $\alpha$ that closely match the experimentally found ones in previous publications. In Ref. [4], where the authors optimized their $\alpha$ to be 7 for SPEF, we estimate ${\alpha _{{\rm opt}}} = 7.1$ (used parameters: ${\rm NA} = 0.42$, $\lambda = 532\;{\rm nm} $, ${\rm FOV} = 200\;\unicode{x00B5}{\rm m}$, $n = 1.33$). In Ref. [19], they found an optimized value for $\alpha$ of 10.3 for TPEF, which matches closely with our calculation of ${\alpha _{{\rm opt}}} = 9.1(4)$ (used parameters: ${\rm NA} = 0.3$, $\lambda = 930.9\;{\rm nm} $, ${\rm FOV} = 622(34)\;\unicode{x00B5}{\rm m}$, $n = 1.33$). The deviations in the optimized $\alpha$ value could be attributed to the difference in the definitions of NA.
In summary, this paper derived and discussed the Rayleigh range and beam waist equivalents for an Airy beam. Furthermore, it explicitly relates the optical specifications to the properties of the beam. This may significantly improve the work-flow for designing the LS microscope’s illumination as well as its collection path in the future.
Funding
Horizon 2020 Framework Programme (Grant no. 871212).
Acknowledgment
We thank Mingzhou Chen (University of St. Andrews) for advice on the definition of the $\alpha$ parameter and insights on the derivation of the characteristic length and apodization factor. We further express our gratitude to Fritz Bayer (ETH Zurich) for verifying the mathematical derivations, to Tim Winter (University of Freiburg) for working on the propagation code, and to Christian Eggeling (Friedrich Schiller University Jena) for his supervision of S. Schulz.
Disclosures
The authors declare no conflicts of interest.
Data availability
The simulation and evaluation code used to generate the results presented in this paper is available in Code 1, Ref. [23].
Supplemental document
See Supplement 1 for supporting content.
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