## Laser Beam Propagation

### Overview

In my first few weeks here at the Laser Teaching Center, I developed an interest in transverse laser modes. To perform experiments with transverse laser modes, one needs to be knowledgeable of math specifically involving Gaussian wave equations. In attempt to familiarize myself with the kind of math involved, along with all the lab equipment, I decided to do this mini-project on measuring the spot size of a laser beam at various distances after propagation through a convex lens.

### Theory

The propagation of a laser beam is a widely understood topic. When the intensity is measured across the beam, the distribution follows that of a Gaussian curve, which has the appearance of a bell curve. The intensity is at a maximum at the center of the beam and decreases outward with the radius. The Gaussian equation is as follows:

`				I (x,z) = A*exp[-2(x-x0)^2/w(z)^2]		(1)`

where I is the intensity, A is the amplitude (an arbitrary number), x-x0 is the radial distance (x0 being the center), and w(z) is the radius of the beam z distance away from the actual laser. Furthermore, w(z) can also be calculated mathematically with the given equation:

```				w(z) = w0 [1+(z/z0)^2]^(1/2)			(2)
```

where w0 is the smallest possible width and z0 is the Rayleigh range. z0 is defined as the distance from the beam waist (the location of w0 on the axis of propagation) where the w(z) = √ 2*w0. Within this distance, the laser beam remains relatively parallel, thus allowing 2z0 to be called the "depth of focus." z0 is dependent on w0 with this given formula:

```				z0 = (π/λ)*w0^2					 (3)
```

When the laser beam propagates through a convex lens, the rays will converge at a certain distance, the focal point f, away from the lens and form a new w0. This focal point, which varies with the distance of the lens from the laser, is not to be mistaken for the focal length of a lens itself. To obtain the new w0, the focusing angle, θ, is needed:

```
θ = arctan (d/2f)				(4)
```

where d is the initial diameter of the beam. The following equation then relates θ to w0:

```				λ = (2π)*(λ/2w0)				(5)
```

Using this new w0, a new z0 can be found with equation (3) allowing then both quantities to be used to solve for w(z) of the laser beam in equation (2) after the lens.

When (x-x0) = w(z), the intensity at that point becomes exp(-2) of the maximum intensity (about 14%), allowing this distance to be defined as the end of the beam. Since the Gaussian curve is symmetrical, there is an identical point located on the other side of the center, which is also about 14% of the maximum intensity.

Using a pinhole is a very effective way of measuring the width of a laser beam, given that the pinhole is smaller than w0. However, if a small enough pinhole is unavailable, then an alternate method is needed. To solve this problem, a razor blade can be used in place of a pinhole to conveniently and accurately measure the diameter of any sized laser beam.

The intensity distribution of the razor blade method turns out to be equal to the double integral of the Guassian equation. Solving such an equation involves knowing the error function at a given point …

The goal of this project was to experimentally measure the radius of a laser beam, w(z), at various distances after propagation through a convex lens and then compare it to its theoretical prediction obtained by equation (2).

### Experimental Setup

The beam from a Melles-Griot HeNe laser (wavelength 632.8 nm) was centered on a Thorlabs DET110 photodetector approximately 40 inches away. A Thorlabs 1.0 inch diameter convex lens of 15 cm focal length was placed 20 inches in front of the laser. A razor blade (2 inches x 1 inch) was then placed at various 1 inch distances between the lens and photodetector at about 40 inches from the laser. It was attached upright on a stand which was then placed on two mils-driven (1/1000 of an inch) Daedal Inc. translators, one laid on top of the other, with one of the translator knobs pointing outwards and the other translator knob pointing horizontally. The outward pointing translator was then set to read at 800 mils, or just enough so that the razor blade just barely blocked the path of the beam and that a maximum intensity could be measured. The outward pointed translator knob was then turned at various intervals causing the razor blade to advance and block off part of the propagating beam, until it was fully covered. An intensity reading in the form of current was recorded at each interval.

To confirm accuracy of the razor blade method, the intensity distribution of an unaltered laser beam was measured using both the razor blade and pinhole methods. The data was then plotted and compared to its theoretical curve, (the integral of the original Gaussian curve). The width of the Gaussian curve was altered accordingly to produce the best fit graph. Pinhole and razor blade measurements of the laser beam width prior to the lens were also taken to confirm that the intensity distribution was indeed Gaussian.

The black solid line represents a theoretical Gaussian curve based on Equation (1) and the green solid line is its integral. The pink boxes and dots represent experimentally collected data points and they generally fit its theoretical counterpart. The razor blade method is an accurate way of measuring the diameter of a laser beam.

To do this for every 1 inch from the laser for 20 inches is rather tedious. However, because the Gaussian curve is symmetrical, technically only two points are needed to figure out the width. As mentioned earlier, the end of the beam is defined as I0*exp[-2] and I0*(1-exp[-2]) (I0 is the maximum intensity located at the center of the beam)and so these intensities were calculated and then matched experimentally using the razor blade method.

The pink solid lines represent the theoretical beam width as it gets farther away from the lens. Because the width is half the total distance across, all of the data and equations had to be negated to get the bottom portion of the hyperbola. The black cross-shaped dots represent data points collected during experimentation. However, what isn't shown on the graph is that all the data points had been multiplied by a factor of √ π. This was unexpected because originally the theoretical curve did not match the data points, with the exception of the beam diameter at the beam width at 6 inches.

 Victor WangSummer 2006 Home  Laser Teaching Center