# Light sheet optics for PIV

> Shashikant Verma

## Objective

- To generate a green laser (532 nm) light sheet of $200$ mm height and $\sim 1.6$ mm thickness.

## Introduction

PIV is an optical diagnostics technique used for visualizing the flow
field by forming a thin light sheet of desired dimensions at the test
section. The light sheet can be obtained using a different combination
of spherical and cylindrical lenses as explained in [(book)](https://link.springer.com/book/10.1007/978-3-319-68852-7).
In order to make a thin light sheet [(Stopper et al.,)](https://www.sciencedirect.com/science/article/pii/S0894177709001575) have used two cylindrical lenses ($f_1 = 12.7$ mm and $f_2 = 200$ mm) and a spherical lens ($f = 1000$ mm).

## Following points should be considered while selecting the optics for PIV:

- Nd: YAG lasers need a combination of different lenses to generate
the thin light sheet of high intensity compared to argon-ion lasers.

- Cylindrical lens converges or diverges the beam only in one
direction (along their power direction) while spherical lens does
the same in both directions.

- Spherical lenses are easy to manufacture compared to the cylindrical
lens of the same focal length.

- There are chances that the high laser energy at the focal points may
result in ionization of air which affects the laser properties and
is not desired.

- Diverging lens should be placed first to avoid the ionization of air
near the focal point.

- It is very important to consider the energy per unit area for any
configuration. If a configuration results in high energy per unit
area, then the region close to the focal line should be covered to
avoid reflection by dust particles.

- Use of coated lens helps in preventing reflection from the lens
surface and it is important from a safety point of view.

- To have a minimum thickness for long distances, high focal length
lenses should be used.

- The height and thickness of the light sheet depend upon the
following parameters:

1. Radius of laser beam

2. Focal length of lenses

3. Distance between the lenses

4. Wavelength of laser beam

## Optics configuration

In the present study, we have used two spherical lenses (bi-concave
($f_1 = -50$ mm) and bi-convex ($f_2 = 150$ mm)), a plano-concave
cylindrical lens ($f_3 = 30$ mm) and a plano-convex cylindrical lens
($f_4 = 250$ mm) to generate the laser sheet of $200$ mm height and less
than $2$ mm thickness at the test section. The configuration is shown in
Figure 1.

|<img src="https://i.imgur.com/cHkQkX0.jpg">\|
|:--:|
|*Figure 1: Setup for obtaining thin light sheet* |

Lens 1 = Bi-concave spherical lens\
Lens 2 = Bi-convex spherical lens\
Lens 3 = Plano-concave cylindrical lens\
Lens 4 = Plano-convex cylindrical lens

## Detailed explanation for the selected optics

Our main objective is to generate the light sheet of desired height and
minimum thickness at the intended test section. After discussing with
different researchers and based on our requirement we have selected the
above configuration for forming the light sheet. It is recommended to
first collimate the laser beam then it becomes easy to control the
height and thickness of the sheet. The collimated beam is a parallel
beam with a constant diameter. The laser beam is first collimated by
using a diverging lens (Lens 1) and a converging lens (Lens 2). Then the
beam is expanded only in one direction by passing through the
cylindrical lens (Lens 3) without affecting its thickness. The sheet
thickness is controlled by another cylindrical lens (Lens 4). Both the
cylindrical lenses are placed in such a way that their axis is
perpendicular to each other. The distance in-between the lenses is
decided by focal lengths and desired sheet characteristics.

## Simulation and analytical calculations to find out focal length of lenses

The calculation for obtaining the desired sheet is done using
trigonometry.

- The focal point of both lens 1 and lens 2 coincide, so the beam
coming from the laser gets diverges from lens 1 in such a way as it
appears coming from the focal point of lens 2.

- The beam becomes parallel after passing through the lens 2.

- Then the beam diverges after passing through the lens 3 from where
its height increases while thickness remains unaffected. Based on
the distance of lens 3 from the test section and desired height of
the sheet, the focal length of lens 3 can be calculated using
trigonometry.

- The minimum thickness is obtained at the focal point of lens 4.

- Online simulation tools are also available to determine the sheet
characteristics. One such tool is 3DOptix which is used in the
present study to simulate the laser beam.

- Analytic calculation of sheet thickness can be performed by assuming
the beam to be a Gaussian beam. The following links will help in
calculating the sheet thickness [(1)](https://www.edmundoptics.com/knowledge-center/application-notes/lasers/gaussian-beam-propagation/),[(2)]()

## Calculation of laser sheet thickness

The thickness of a laser sheet can be calculated by assuming the laser beam as a Gaussian beam that undergoes converging and diverging during its flow. The actual laser has some deviation from the ideal Gaussian behavior. Due to diffraction, a Gaussian beam will converge and diverge from an area called the beam waist ($w_0$), where the beam diameter reaches a minimum value. A converging lens is used to control the thickness of the laser sheet. So when a Gaussian beam passed through a converging lens the minimum thickness is obtained at a focal point and the thickness can be calculated as follows:

$$w_{f}=\frac{\lambda f}{\pi w_{0}}$$

$$\omega_{0}=\frac{\lambda}{\pi \theta}$$ where,\
$\lambda$ = wavelength of laser beam, (532 nm)\
$\theta$ = half angle divergence (3.9 milliradian)\
$\omega_0$ = beam waist (86.84 $\mu$m)\
$f$ = focal length of converging lens (f = 250 mm)\
$\omega_f$ = half of beam thickness \
So the thickness of the beam obtained is two times of $\omega_f$ as 0.97 mm.