# Bessel beam simulation

## Theory

To create a Bessel beam we have to focus an annulus of light. The annulus of light can be created by masking a Gaussian beam with an appropriate mask, defined by the inner and outer radii. Fixing the focal length of the microscope objective to 1, the inner and outer radius defines the minimum and maximum angles, respectively. We have $R_1=\tan(\alpha_{MIN})$ and $R_2=\tan(\alpha_{MAX})$. The two angles $\alpha_{MIN}$ and $\alpha_{MAX}$ define the minimum and maximum numerical apertures of the system, respectively. If the refractive index of the medium in which we are focusing the annulus is n, we have

$\sin(\alpha_{MIN})=\frac{NA_{MIN}}{n}$

and

$\sin(\alpha_{MAX})=\frac{NA_{MAX}}{n}$

To describe an annulus, I use $NA_{MAX}$ and the ratio $\varepsilon=\frac{NA_{MIN}}{NA_{MAX}}$. Note that $\varepsilon$ goes from 0 to 1, 0 being a full circle (normal Gaussian beam) and 1 an infinitely thin annulus.

### Calculating the PSF

To calculate the illumination PSF of a system with a given $NA_{MAX}$and $\varepsilon$, we need to calculate the electromagnetic field in three dimensions. This is accomplished by following the approach of B. Richards and E. Wolf (B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II," Proc. R. Soc. London Ser. A 253, 358-379 (1959).). The calculation is performed by the Java code used in simulation (see also a C version of the the code).

## Simulations results

### Simulation parameters

• The refractive index n, choosen to be 1.333 (refractive index of water)
• The wavelength $\lambda$ (488 nm for single-photon excitation, 920 nm for two-photon)
• The $NA_{MAX}$, varying from 0.1 to 1.3
• The NA ratio $\varepsilon$, varying from 0 (Gaussian beam) to 0.9

Note that increasing $\varepsilon$ results in more elongated Bessel beams. However, the power transmitted by the mask decreases as $1-\varepsilon^{2}$, therefore more elongated Bessel beam requires a lot of energy to be produced, because of the low throughput of the mask.
Single photon excitation For the two-photon excitation, the PSF with the given values of $NA_{MAX}$ and $\varepsilon$ was calculated and then squared to account for the nonlianear effect.

How to read the graphs For both graphs, the black and colored lines represent the width of the PSF along the radial and axial direction, respectively. Values are in nm. For example, using an NA of 0.6 and an NA ratio of 0.7 results in a Bessel beam which is 200 nm wide and 5000 nm long (single-photon).

## Creating a Bessel beam

To create a Bessel beam, one needs to focus a ring of light. As shown above, such ring of light can be obtained by masking a gaussian beam. To get a thin and elongated beam, it is important to have a large max. NA and a small ratio between the outer and inner radii. However, a small ratio means a low throughput (i. e. most of the incoming power is reflected by the mask). When using two-photon excitation, using annular beams instead of gaussian beams before the mask would increase the throughput. To create an annular beam, one could use a lens in combination with an axicon. The spot diagram simulates an f=1000 mm lens before an axicon of 0.5 deg. The simulation was run in OSLO, using an f=1000 lens from the THORLABS catalog and an axicon corresponding to the THORLABS 0.5 deg axicon (thickness 5 mm, cone angle 179.5, UV Fused Silca). To model the conical surface in OSLO, the curvature was set to -0.01 mm and the conical constant to -13131, according to the formula k=-(1/(tan(theta))^2+1)