Light sheets with extended length
Introduction
In a conventional wide-field fluorescence optical microscope, the excitation and imaging optics share the same objective lens and the excitation light passes through the specimen. The out-of-focus fluorescence thus adds background noise to the image. Confocal microscopy solves this problem by focusing the excitation light into a spot and rejecting the out-of-focus fluorescence light by a pinhole in the detection side. By scanning the focus point by point, the confocal microscopy allows for reconstructing the object’s three-dimensional (3D) structure. In such a scanning confocal microscope, although 3D high-resolution information can be obtained, illumination beam goes through the entire thickness of the specimen for many times, which causes serious phototoxicity to the sample and photobleaching to the fluorophore. Besides, confocal microscopy is slow because of its point-by-point scanning. Therefore, confocal microscopy by its nature is not suitable for 3D live-cell imaging though many efforts have been made to improve its performance [1].
Recent development of light-sheet microscopy (LSM) has demonstrated its capacity for live imaging of biological samples from single cells to tissues with high spatial and temporal resolutions [2], [3]. By illuminating the excitation light sheet from side perpendicular to the imaging objective axis, a light-sheet fluorescence optical microscope has the optical sectioning capability with ultra low phototoxicity. It is also much faster than a conventional confocal microscope because LSM unitizes plane-scanning rather than point-scanning. LSM is a powerful tool for four-dimensional (4D: 3D in space plus 1D in time) live-sample imaging.
In fluorescence LSM, the excitation light sheets is expected to be thin in axial dimension along the optical axis of detection objective (to eliminate off-focus excitation), and also be long in lateral dimension such that a large field of view can be covered. However, these two conditions contradict each other for the conventional LSM based on Gaussian beams because of the diffraction effect: a thinner Gaussian beam propagates a shorter distance. For a two-dimensional (2D) Gaussian-beam light sheet with a thickness approaching the diffraction limit of half a wavelength, the propagation length becomes shorter than a few wavelengths, in which the light sheet is too short to even cover a single cell. Non-diffracting Bessel beams have been utilized in order to produce longer light sheets. In these cases Bessel beams are scanned into a time-averaged sheet. While the scanning Bessel-beam light sheet indeed extends the propagation length significantly, its strong side lobes induce strong off-focus excitation and increases phototoxicity [4]: the Bessel beam has its energy evenly distributed among its rings [5], [6]. In reality, the propagation length is limited by the inverse of the finite annulus width of angular spectrum. Recently LSM using asymmetric Airy beams for extended field of view has been demonstrated [7], [8].
If the light energy of a Bessel beam is spread into multiple beams on the sheet, such a light sheet has much lower instantaneous intensity than a scanning Bessel-beam light sheet under the same illumination laser power, and thus leads to much lower nonlinear photodamage and phototoxicity to the specimen. This effect was demonstrated by Gao et al. using a linear array of Bessel beams [9]. It was later extended to 2D optical lattice light sheet microscopy (LLSM) for achieving higher resolution and lower phototoxicity [10]. However, LLSM is complicated to implement, has too many degrees of freedoms for optimization, and has limited access only within very a few groups. To produce a time-averaged uniform light sheet, LLSM requires dithering its 2D lattice pattern.
We recently demonstrated a simple and robust method to generate ultrathin line Bessel sheets (LBS) [11], [12]. Using this technique, we create smooth and long light sheets without dithering the beam. In this article, we reveal the theory and design principle of the LBS, as well as for producing more general 2D smooth light sheets with extended length with two-beam interference. Here we focus only the spatially-smooth light sheets without need of scanning or dithering the beams.
The article is organized as follows. In Section 2, we provide the general formalism of 2D light sheets. In Section 3, we describe the properties of single-spectrum-band light sheets. Then we derive and describe double-spectrum-band light sheets with extended length by two-beam interference in Section 4, where we also show the numerical comparison between these light sheets. We conclude in Section 5.
Section snippets
General formalism of 2D light sheets
We choose the coordinate system in which the coherent laser light, with wavelength , propagates along direction. The light sheet is confined in direction with a thickness and spread over direction with a width . Ideally, to achieve an absolute non-diffracting beam, its angular spectrum in – plane must be confined on a circular ring, i.e., is a constant where , so that the beam profile does not change along its propagation. The zeroth-order Bessel beam is a
Single-spectrum-band light sheets
We start from description of single-spectrum-band light sheets, because understanding the physics of the diffraction effect is helpful to design light sheets with extended length. For numerical computation, one can follow Eqs. (1)–(3). In order to obtain some insightful analytic expressions, we here take a 2D Gaussian beam as an example to illustrate the general relation between the light sheet parameters and the conclusions can be extended to general non-Gaussian sheets. Surprisingly, I have
Double-spectrum-band light sheets with extended length
Now we turn to production of light sheets with extended length. To ensure a constant propagation number in Eq. (1), the angular spectrum takes . That is, an ideal diffraction-free (or non-diffracting) light sheet is a superposition of two 2D plane waves with . In a real situation, the function has to be replaced by a spectrum distribution with a finite bandwidth, which limits the diffraction-free propagation range. Therefore, a double-spectrum-band
Summary
In summary, we have provided formalism for designing 2D light sheets with two symmetric bands in the space. For Gaussian-shape band structure, we provide analytic expressions under the paraxial approximation, which can be extended to rectangular spectrum with . As compared to the single-spectrum-band light sheet with the same thickness, the sheet propagation length is extended significantly. In the single-spectrum-band light sheet, the sheet propagation length is proportional to
Acknowledgements
The work was supported by Light Innovation Technology Ltd through a research contract with the Hong Kong University of Science and Technology (HKUST) R and D Corporation Ltd (Project code 16171510), and the Offices of the Provost, VPRD, and Dean of Science at HKUST (project no. VPRGO12SC02).
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