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1.IntroductionA Gaussian (G) beam is an ideal spatial form of a laser beam and one of a larger class of Laguerre–Gaussian solutions (LG) of the wave equation with orbital angular momentum L. The G beam at is the fundamental transverse mode () of the laser and is used to focus into material with a lens which takes a Fourier transform (FT). It is used in optical physics for propagating electro-magnetic energy in space with light-matter interactions. The profile of the G beam is a key element for imaging in multiphoton microscopy, including two-photon (2P) and three-photon (3P) applications, where a 140-fs pulsed G beam is used in the near-infrared to image objects in two-dimensional (2-D) and three-dimensional (3-D), particularly for cellular structures in the brain and other biological systems. There are several salient properties of G beam in the nonlinear optical process of two-photon fluorescence (2PF) as the beam propagates and scans in the , , directions: beam spot size, depth of focus (DOF), intensity profile in radial directions, and propagation in direction. The propagation for G beam in media has been reviewed by Saleh and Teich 1 and Yariv,2 but other features for small-scale emitters, such as dyes or quantum dots (QDs) in microscale biological tissues or media, have not been examined. This technical note investigates the important optical parameters and features of G beam in 2P microscopy for small nanometer-scale emitters in media, and examines the influence of these parameters on imaging microvessels in rat brain in vivo. 2.Materials and Methods2.1.Gaussian Beam ModelThe intensity profile of a G beam is given by where is the incident intensity, is the transverse radius, is the path length, is the narrowest spot size (also known as beam waist), which equals of Gaussian electric field (), and is the spot size given by1,2 where is the Rayleigh length (Fig. 1) and is the wavelength. Length is the DOF. At the beam waist (, , Fig. 1), the Gaussian beam electric field drops to of (the maximum electric field) and the intensity drops to of . From Eq. (1), the intensity along axis () is which is an oval spanning and with the maximum intensity at . The intensity reduces to at [Fig. 2(a)]. The DOF is the critical distance for G beam intensity in the direction, which depends on the magnification (, , etc.) and numerical aperture (NA) of the microscope objective lens used. The NA is given by where is the index of refraction, is the diameter of the lens, and is the focal length. The DOF () is given by whereThus, the relationship between the DOF and NA of the objective lens is The beam waist () and DOF () are two of the most salient features used to locate dyes or QDs in tissue compartments (e.g., inside and outside a blood vessel). A smaller beam waist and a smaller DOF lead to better image resolution in the and directions, clearer structures that can be observed, and a smaller point spread function. At the DOF, the focal spot area is blurred to twice of that at the beam waist . The intensity of a G beam in 2PF () is proportional to the square of the intensity , i.e., where is the fluorescence efficiency, and is the two-photon dye cross section with the unit GM and .3 The intensity is used to locate the excitation value of 2PF. It spreads over at the axis, reduces to at [Fig. 2(b)], and falls off fast to at the beam waist ( and ).The size of the 2PF image pattern follows in the plane. Figure 3(a) illustrates the schematic of a microvessel with diameter in brain tissue, where and , and the axis is along the vessel axial direction. The size of the 2PF oval in the plane [Fig. 3(b)] is the spatial convolution of the G beam and the emitting particle of diameter . Since the indices of tissue refraction and solvent are close, the reflection coefficient with index at the boundaries4 matches the little change in intensity due to the reflection of the vessel wall. Figure 4(a) shows the convolution of a G beam and the emission of a particle at size , where and . The 2PF image pattern is an ellipse with the major axis (DOF)5 and the minor axis . Figure 4(b) from Chaigneau et al.6 shows an example of a 2PF image of green beads (200 nm) on brain tissue layer; the image was distorted since the size of the particle was smaller than and , and corrections for the wavefront distortion were made. 2.2.Animal PreparationAdult female Sprague–Dawley rats (250 to 300 g, age 3 to 4 months, Hilltop Laboratory Animals Inc., Scottdale, Pennsylvania) were used for in vivo experiments. All procedures and the animals use were approved by the Institutional Animal Care and Use Committee of the City College of New York. The detailed procedure was similar to that described in Ref. 7. Briefly, the rat was subcutaneously anesthetized with pentobarbital sodium, and then its skull was exposed by shaving off the hair and cutting away the skin and connective tissue. The frontoparietal bone (left or right) was carefully thinned with a high speed microgrinder (0 to 50,000 rpm, DLT 50KBU, Brasseler, GA) until a portion of it () became soft and translucent. After skull grinding, the left (or right) carotid artery was cannulated with a PE50 tubing. The rat was then put on a stereotaxic instrument (David Kopf Instruments) with its head fixed with two ear bars and a mouth clamp. Cerebral microvessels (diameter 20 to ) were observed with a two-photon microscope (, , water immersion, Olympus) through the thinned part of the skull. FITC-dextran-70k (FD70s, Sigma; MW 70,000, Stokes radius ; emits at 520 nm) solution, concentration in 1% bovine serum albumin Ringer, was then injected into the cerebral circulation via the tubing at a perfusion rate of . 2.3.Two-Photon Microscopy and Image CollectionAn Ultima two-photon microscope (Prairie Tech. Inc., WI, excitation wavelengths 680 to 1080 nm) was used in the experiments. FITC-dextran-70k solute was excited with 140-fs laser pulses at 800 nm using Ti-Sapphire laser (Chameleon Ultra II laser, Coherent Inc.), and the fluorescence emitted from the brain specimen was collected by the photomultiplier tubes after passing the barrier filter (). For this imaging system is , is , and the DOF is . When the fluorescence solution was injected into the cerebral circulation, images were simultaneously recorded. The images were collected as image stacks in the direction every 5 s, and were reconstructed into a segment with the cross-sectional area of () and thickness of [Fig. 3(a)]. Collected images were analyzed offline for measuring the light intensity profile inside and outside the microvessel. 3.Results and DiscussionsFigure 5 demonstrates the cross-sectional light intensity profile of a cerebral microvessel filled with fluorescein dye (FITC-dextran-70 k) in rat brain tissue. The light intensity profiles (), at different cross-sectional distances from the center of the lumen were digitized to show variations on the 2P emission of fluorescein dye in the and directions, respectively. Comparison of the intensity plots for track lines 1, 2, and 3 demonstrated that the closer the cross section is to the center of the lumen, the higher the peak intensity and the wider the spectral peak width are. Since the diameter of the cerebral microvessel imaged (20 to ) was much larger than and , there was no distortion in the images and no correction was needed for vessel structure reconstruction. In addition, the light intensity profile can be used for the determination of microvessel permeability and solute diffusion coefficient in the brain tissue,7 since the fluorescein dye travels from the vessel lumen to the brain tissue passing through the vessel wall. An important feature of the 2PF imaging of G beam is that its emission occurs in regions that are occupied by G beam as the beam scans. The G beam scans first in the plane and then along the direction. The resulting 2PF intensity is plotted in the plane at , and the intensity of the G beam is plotted in the (or ) plane. The length of the plot is and the diameter of the vessel or particle is . A small is needed to obtain the actual images of a vessel in the direction without any image correction, which is typically when using a lens. Another important feature in 2PF imaging is the excitation wavelength. To attain the deepest penetration, due to linear absorption of the host media the wavelength should be within the range of 700 to 900 nm, typically around 800 nm. The image is blurred by multiscattering in the tissue where the limit of coherent penetration of the ballistic pulse intensity is governed by scattering length (Ls). The transport scattering length (Ltr) governs the distance to change the photon direction (typically 5 to 8 Ls). Typical Ltr used in optical coherence tomography is about 0.5 to 1 mm for tissues where the diffusion equation becomes valid for Ltr scattering length.8,9 The effect of multiscattering in a Gaussian profile is discussed in detail by Ying et al.5 The Ls for brain tissue is about and the Ltr is approximately . The imaging depth of in brain tissue can be achieved in theory. The spatial resolution may be improved when the G beam has a smaller beam waist or Rayleigh length , which introduces the application of 3PF or four-photon fluorescence (4PF). For example, by substituting Eq. (2) into Eq. (1) and applying Eq. (9), the intensity of 2PF is given as The beam intensity falls off to approximately when . The beam waist of the 2P intensity profile is For 3PF, the intensity () is obtained similarly as where is the particle 3P cross section with dye with the unit of .10 The beam intensity falls off to approximately when . The beam waist of 3P intensity profile isSimilarly, for 4PF, the intensity () is where is the particle 4P cross section with dye with the unit of .10 The beam intensity falls off to approximately when . The beam waist of 4P intensity profile isThe salient parameters of 2P, 3P, and 4P imaging are compared in Table 1. The reduction of the beam waist from 2PF to 4PF will lead to higher spatial resolutions in the plane. Equation (3) shows that a longer wavelength will lead to a shorter Rayleigh length ; therefore, an increase of the wavelength from 2PF to 4PF will result in higher spatial resolution in the axis. Using 3PF or 4PF other than 2PF shows a promising way to improve imaging resolution; however, in vitro or in vivo experiments are still needed to further investigate the improvement both qualitatively and quantitatively. Table 1Spatial parameters for multiphoton fluorescence.
A Gaussian beam is superior to a Bessel beam for backward and forward multiphoton microscopes with applications in 3-D imaging. Since the focal spot of the Gaussian beam is well localized compared to the long line of a Bessel beam, Gaussian beam is commonly used for optical slicing and 3-D reconstruction. The Bessel beam is diffraction free and well collimated over long distances () with surrounding rings. It works well in multiphoton arrangement excitation in perpendicular viewing and scanning, and can be used in light sheet microscopy to generate 2-D and to some extent 3-D images of thick media.11–13 However, in the multiphoton excitation process, only approximately 5% of the Bessel beam is in the central spot while a large amount is lost in the rings, and since these rings are not important in the formation of images, the long collimated Bessel beam may not be used to locate nanometer-scale structures in the -axis. Thus, a Gaussian beam overcomes this disadvantage of the Bessel beam, and is used to locate nanometer structures in -stacks acquired with the backward transmission microscope. In conclusion, this study examines the critical properties of a Gaussian beam for biologists and biomedical engineers who are using 2P or multiphoton microscope. The spatial forms of Gaussian beam ( and ), with micrometer-scale resolution, provide the information needed for determining microvessel permeability and solute diffusion coefficients in tissue. With the application of 2P or multiphoton microscopy, high resolution imaging of tissue or vessels can be obtained. This work can be used to investigate neuronal or vascular structure in physiological or biological process in vivo or ex vivo. AcknowledgmentsWe thank Dr. Bingmei M. Fu for providing rats. This work was partially supported by ARO research grants, and by NIH (Grant Nos. G12-RR003060 (NCRR/RCMI) and SC1HD068129). ReferencesB. E. A. SalehM. C. Teich,
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BiographyLingyan Shi is a research associate at IUSL and Biology Department at the City College of New York (CCNY). She obtained her PhD degree in biomedical engineering from CCNY. Her research interests include in vivo deep tissue imaging by using multiphoton with femtosecond laser pulses to understand functional interactions between neural cells and brain vasculature, and quantification/regulation of blood–brain barrier permeability for drug delivery. Other research interests include biomedical optics and cardiovascular biomechanics. Adrián Rodríguez-Contreras is a biology assistant professor at the City College of New York. In 2008, he established the developmental neurobiology laboratory combining anatomical, electrophysiological, and two-photon microscopy tools to study brain development in rodents and barn owls. Other research interests include applying novel methods in fluorescence microscopy for in vivo studies, exploring the cellular mechanisms involved in the growth of brain tumors and understanding the functional interactions between neural cells and the brain vasculature. Robert R. Alfano is a distinguished professor of science and engineering at the City College of CUNY. He has pioneered many applications of light and photonics technologies to the study of biological, biomedical, and condensed matter systems, invented and used in his research supercontinuum and novel tunable solid-state lasers. He has received his PhD in physics from New York University and is a fellow of the American Physical Society, Optical Society of America, and IEEE. |