Ultramicroscopy of Nerve Fibers and Neurons: Fine-Tuning the Light Sheets

Abstract

Light sheet generation approaches employed in the ultramicroscopy imaging technique are presented, and three different designs are discussed. These include slit-based as well as more advanced slit-free setups, including those based on aspheric optics, that is, inherently aberration-free. The performance of the light sheets thus generated is briefly described theoretically, and experimentally compared. Images of a chemically cleared fruit fly (Drosophila melanogaster), nerve fibers in an entire mouse embryo, and individual neurons in intact mouse hippocampi are presented.

Appendix

The complex light intensity in the X direction (i.e., perpendicularly to the slit) of an axially symmetrical Gaussian beam propagating in vacuum along the z-axis is described by Eq. 1 [20]:

$$ E\left(x,z\right)={E}_0\frac{w_0}{w(z)}\exp \left(-\frac{x^2}{w^2(z)}\right)\exp \left[- ik z- ik\frac{x^2}{2R(z)}+ i\phi (z)\right] $$

(1)

where

E₀ is the amplitude of the illuminating light

w₀ is the radius of beam waist at the focus (z = 0)

w(z) is the radius of the beam at distance z from the focus

i is the complex unit, $\sqrt{-1}$

k is the wave number (2π/λ, where λ is the wavelength of the illuminating light)

R(z) is the radius of curvature of the beam at distance z

φ (z) is the longitudinal phase delay at distance z

$$ R(z)=z\left[1+{\left(\frac{z}{z_R}\right)}^2\right] $$

(2)

$$ w(z)={w}_0\sqrt{\left[1+{\left(\frac{z}{z_R}\right)}^2\right]} $$

(3)

$$ \phi (z)=\arctan \left(\frac{z}{z_R}\right) $$

(4)

where z_R is the Rayleigh range (Fig. 9),

$$ {z}_R=\frac{\pi {w}_0^2}{\lambda }. $$

(5)

Fig. 9

Detail of a light beam at focus. Rayleigh range (z_R) describes the beam expansion along the direction of propagation (Z), and is defined as the distance from the waist (diameter 2w₀) to the point where the cross section of the beam is approximately doubled. For a Gaussian beam, this occurs at a beam diameter of $\sqrt{2}{w}_0$. Adapted from Ref. [20] by permission of © Wiley Interscience

At the beam waist (z = 0), w(z) = w₀, R(z) = R₀, φ(z) = φ₀ and Eq. 1 becomes simplified as follows:

$$ E(x)={E}_0\exp \left(-\frac{x^2}{w_0^2}\right)\exp \left(- ik\frac{x^2}{2{R}_0}+i{\phi}_0\right) $$

(6)

E(x) is the complex amplitude of light while its (measurable) intensity is

$$ I(x)=\kern0.5em {\left|\mathrm{E}\left(\mathrm{x}\right)\right|}^2={E}_0^2\exp \left(-\frac{2{x}^2}{w_0^2}\right) $$

(7)

This is a well-known Gaussian distribution. As the beam is axially symmetrical the same type of dependence is obtained for the Y direction, that is, the intensity can be generalized as follows:

$$ I\left(x,y\right)=\kern0.5em {\left.\mid E\Big(x,y\Big)\right|}^2={E}_0^2\exp \frac{-2\left({x}^2+{y}^2\right)}{w_0^2} $$

(8)

However, in a slit-based system the Y direction (parallel to the slit) is of little relevance as the beam is truncated by the slit mainly in one direction only (X in our notation), and later also “flattened” (in the same direction) by the cylindrical lens.

The effect on the Gaussian beam of the slit (width = 2a in the X direction) is described by the Heavyside step function A_p(x) which equals 1 for −a ≤ x ≤ a, and 0 elsewhere. A generalized Huygens–Fresnel diffraction integral then describes the truncated (modified Gaussian) beam [20,21,22,23,24,25]:

$$ E(x)={E}_0\underset{-a}{\overset{a}{\int }}{A}_p\left({x}_0\right)E\left({x}_0\right)\exp \left[-\frac{ik}{2B}\left({Ax}_0^2-2{xx}_0^2+{Dx}^2\right)\right]{dx}_0 $$

(9)

where A–B and D are elements of the optical transfer matrix. Equation 9 is not dependent on C, the third element of the transfer matrix. In the Y direction, the beam is practically untruncated (assuming a relatively narrow slit), so the Heavyside step function is not applied.