1.

Introduction

A 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 $L=0$ is the fundamental transverse mode (${\mathrm{TEM}}_{00}$) 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 $x$, $y$, $z$ directions: beam spot size, depth of focus (DOF), intensity profile in radial $(r,x,y)$ directions, and propagation in $z$ 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 Methods

2.1.

Gaussian Beam Model

The intensity profile $I_{\mathrm{G}}$ of a G beam is given by

Eq. (1)

$$I_{\mathrm{G}}(r,z)=I_0{\left[\frac{w_0}{w(z)}\right]}^2\exp [-\frac{2r^2}{w^2(z)}],$$

where $I_0$ is the incident intensity, $r$ is the transverse radius, $z$ is the path length, $w_0$ is the narrowest spot size (also known as beam waist), which equals $1/\mathrm{e}$ of Gaussian electric field ($E_{\mathrm{G}}$), and $w(z)$ is the spot size given by^1,2

Eq. (2)

$$w^2(z)=w_0^2[1+{\left(\frac{z}{z_0}\right)}^2],$$

Eq. (3)

$$z_0=\frac{\pi w_0^2}{\lambda },$$

where $z_0$ is the Rayleigh length (Fig. 1) and $\lambda$ is the wavelength. Length $2z_0$ is the DOF. At the beam waist ($r=w_0$, $z=0$, Fig. 1), the Gaussian beam electric field drops to $1/\mathrm{e}$ of $E_0$ (the maximum electric field) and the intensity drops to $1/{\mathrm{e}}^2$ of $I_0$. From Eq. (1), the intensity along $z$ axis ($r=0$) is

Eq. (4)

$$I_{\mathrm{G}}(0,z)=I_0\frac{1}{1+{(z/z_0)}^2},$$

which is an oval spanning $2z_0$ and with the maximum intensity at $z=0$. The intensity reduces to $1/2I_0$ at $z_0$ [Fig. 2(a)]. The DOF is the critical distance for G beam intensity in the $z$ direction, which depends on the magnification ($40\times$, $20\times$, etc.) and numerical aperture (NA) of the microscope objective lens used. The NA is given by

Eq. (5)

$$\mathrm{NA}=n\frac{D}{2f},$$

where $n$ is the index of refraction, $D$ is the diameter of the lens, and $f$ is the focal length. The DOF ($2z_0$) is given by

Eq. (6)

$$2z_0=\frac{2\pi w_0^2}{\lambda },$$

where

Eq. (7)

$$w_0=\frac{\lambda }{\pi \mathrm{NA}}\mathrm{.}$$

Fig. 1

Illustration of the key spatial electric field profile of a Gaussian beam. $E_0$, the maximum incident electric field at the origin; $r$, the transverse radius; $z$, path length; $z_0$, the Rayleigh length; $w(\mathrm{z})$, spot size; $w_0$, the narrowest spot size (beam waist). The electric field amplitude of Gaussian beam is given by $|E_{\mathrm{G}}(r,z)|=E_0[w_0/w(z)]\exp [(-r^2)/w{(z)}^2]$.

Fig. 2

Axial intensity of a Gaussian beam as a function of $z$ at $r=0$. (a) At $z=0$, intensity decreases to $1/2$ at $z_0$. (b) Two-photon fluorescent (2PF) intensity profile in which the intensity decreases to $1/4$ at $z_0$.

Thus, the relationship between the DOF and NA of the objective lens is

Eq. (8)

$$2z_0=\frac{2\lambda }{\pi {\mathrm{NA}}^2}\mathrm{.}$$

The beam waist ($w_0$) and DOF ($2z_0$) 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 $xy$ and $z$ 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 $w_0$.

The intensity of a G beam in 2PF ($I_{2\mathrm{p}}$) is proportional to the square of the intensity $I_{\mathrm{G}}$, i.e.,

Eq. (9)

$$I_{2\mathrm{P}}\propto \eta {\sigma }_2{I_{\mathrm{G}}}^2(r,z),$$

where $\eta$ is the fluorescence efficiency, and ${\mathit{ơ}}_2$ is the two-photon dye cross section with the unit GM and $1\mathrm{GM}={10}^{-50}{\mathrm{cm}}^4\mathrm{s}/\text{photon}$.³ The intensity $I_{2\mathrm{p}}$ is used to locate the excitation value of 2PF. It spreads over $2z_0$ at the $z$ axis, reduces to $1/4I_0$ at $z_0$ [Fig. 2(b)], and falls off fast to $e^{-4}{I}_0$ at the beam waist ($r=w_0$ and $z=0$).

The size of the 2PF image pattern follows ${I_{\mathrm{G}}}^2$ in the $xz$ plane. Figure 3(a) illustrates the schematic of a microvessel with diameter $d$ in brain tissue, where $d>2z_0$ and $d>2w_0$, and the $x$ axis is along the vessel axial direction. The size of the 2PF oval in the $yz$ plane [Fig. 3(b)] is the spatial convolution of the G beam and the emitting particle of diameter $d$. Since the indices of tissue refraction and solvent are close, the reflection coefficient with index $n=1.375$ at the boundaries⁴ matches the little change in intensity due to the reflection of the vessel wall.

Fig. 3

(a) Schematic of a microvessel in the brain. The $z$ direction is the laser beam scanning direction. (b) Convolution of a Gaussian beam profile with a vessel of diameter $d$ ($d>2w_0$ and $d>2z_0$) yields the 2PF pattern of scanning in yz plane.

Figure 4(a) shows the convolution of a G beam and the emission of a particle at size $d$, where $d<2z_0$ and $d<2w_0$. The 2PF image pattern is an ellipse with the major axis $2z_0$ (DOF)⁵ and the minor axis $2w_0$. 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 $2w_0$ and $2z_0$, and corrections for the wavefront distortion were made.

Fig. 4

(a) Convolution of a Gaussian beam profile with a vessel of diameter $d$ ($d<2w_0$ and $d<2z_0$) yields the 2PF pattern of scanning in $yz$ plane. (b) Image of 2PF of emitter particle (beads, $d=200\mathrm{nm}<2z_0$) [adapted from Fig. 2(a), Ref. 6].

3.

Results and Discussions

Figure 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 ($I$), 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 $y$ and $z$ 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 $40\mu \mathrm{m}$) was much larger than $2w_0$ and $2z_0$, 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.

Fig. 5

Distribution of 2PF intensity profiles in the cross-sectional view of a rat brain microvessel in $y$ and $z$ directions. Different spatial positions were digitized at vertical direction of the vessel lumen. Track line 1, the cross section at $1/4$ radius; track line 2, the cross section at $1/2$ radius; track line 3, the cross section in the middle. The size of vessel diameter $d>2w_0$ and $2z_0$ gives full resolution.

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 $xy$ plane and then along the $z$ direction. The resulting 2PF intensity is plotted in the $xy$ plane at $z=0$, and the intensity of the G beam ${I_{\mathrm{G}}}^2$ is plotted in the $yz$ (or $xz$) plane. The length of the $yz$ plot is $2z_0$ and the diameter of the vessel or particle is $d$. A small $2z_0$ is needed to obtain the actual images of a vessel in the $z$ direction without any image correction, which is typically $2\mu \mathrm{m}$ when using a $40\times$ 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 $8\times$ 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 $80\mu \mathrm{m}$ and the Ltr is approximately $480\mu \mathrm{m}$. The imaging depth of $500\mu \mathrm{m}$ in brain tissue can be achieved in theory.

The spatial resolution may be improved when the G beam has a smaller beam waist $w_0$ or Rayleigh length $z_0$, 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 $\propto 1/z^4$ when $z\gg z_0$. The beam waist of the 2P intensity profile is

For 3PF, the intensity ($I_{3\mathrm{P}}$) is obtained similarly as

Similarly, for 4PF, the intensity ($I_{4\mathrm{P}}$) is

The salient parameters of 2P, 3P, and 4P imaging are compared in Table 1. The reduction of the beam waist $w_0$ from 2PF to 4PF will lead to higher spatial resolutions in the $x\text{-}y$ plane. Equation (3) shows that a longer wavelength $\lambda$ will lead to a shorter Rayleigh length $z_0$; therefore, an increase of the wavelength from 2PF to 4PF will result in higher spatial resolution in the $z$ 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 1

Spatial 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 ($>1\mathrm{mm}$) 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 $z$-axis. Thus, a Gaussian beam overcomes this disadvantage of the Bessel beam, and is used to locate nanometer structures in $z$-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 ($w_0$ and $z_0$), 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.