# Creating Higher-order Mode Laser Beamswith an Open Cavity Laser

## Introduction

When I had my Lasers class (PHY452) last semester I started working in the Laser Teaching Center on a little project to get some extra credit. My first work was trying to figure out the profile of a Gaussian beam. After winter break, I thought I should do some other things based on this little project. Then I started to work on Higher Order Hermitian- Gaussian Modes (HG modes) and Laguarre Gaussian Modes (vortex modes). I had a really good time in the Laser Teaching Center this semester.

The next step of this project is to get optical vortex from HG modes by using a light cnoverter and get optical vortex.

## Background

Laser beams can only exist in certain specific spatial patterns called modes, which are the discrete solutions of the paraxial wave equation that describes light moving at small angles to some axis. Most lasers are designed to produce only the zero-order or "fundamental" mode. It has cylindrical symmetry and a cross-sectional intensity profile I(r) described by the Gaussian ("bell curve") function exp(-2r^2/w^2). Higher order modes with more complex patterns are also possible, as are superpositions of higher order modes. In the higher order modes of the Hermite-Gaussian (HG) type the Gaussian function of the fundamental mode is multiplied by two polynomial functions that depend on x only and y only, respectively - Hm(x) and Hn(y). These functions are labelled by two integers m and n, which correspond to the degree of the polynomial. A mode with order m (or n) has m (n) nodes or dark bands along the x (y) - direction. It can easily be seen that HG modes have rectangular symmetry. A second complete family of higher-order modes, the Laguerre Gaussian (LG) modes labelled by l and p, is the product of functions of the polar coordinates r and phi and has circular symmetry. Since the phase of an LG mode varies as exp(ilphi), for l > 0 there is a singularity (dark spot) in the center of the beam. These singular LG modes have a twisted wavefront and are often referred to as optical vortices.

## Creating Good Higher-order HG Modes

### Introduction

The first goal of this project is to learn how to create stable and good quality higher-order HG modes with an open-cavity helium neon (HeNe) laser. (In such a laser the low-pressure gas mixture is separated from the atmosphere by a Brewster window and there is an open space between this window and the output coupler mirror which allows objects to be placed in the intracavity beam of the laser.) The method used so far with our laser is to insert a human hair (diameter about 75 microns) into the exposed laser beam. While this method does work, the results are unpredictable and unreliable. We have recently obtained more stable higher-order modes by inserting a glass microscope slide with fine scratches into the cavity. The slide is tilted at Brewster's angle to suppress reflections, and the scratches (created by wiping the glass with sandpaper) are much smaller than the width of a hair, 10 microns or less. We plan to try very thin wires next and to improve the positioning mechanism.

### Set Up

I inserted this thin wire into the cavity and observed HG mdoes by putting CCD in front of the laser.

### Result

 HG10 mode HG20 mode HG11 mode

## Measuring the Relative Phase of HG Modes

### Introduction

If one uses a beam splitter to split HG beams and then overlap them together to observe the interference pattern, he would see the relative phase difference between lobes of HG modes.

### Set Up

I used one beam splitter infront of the laser and it splitted the beam into two branches. I used a couple of lens to amplifiy the size of one branch of beams to make it a reference beam. Then two branches need to be overlap together again. I overlaped the beam from the lower branch showed in the set up (which was not be amplified) with one lobe of reference beam. The result was showed below.

### Result

 HG10 mode interference of HG10 HG11 mode interference of HG11 HG20 interfence of HG20

I could see phase difference between two lobes clearly in the first two pictures. One can see that how a bright horizontal fringe in one lobe conresponding to a dark one in the other one. That was caused by the phase difference between two lobes.

## Optical Vortex

### Introduction

Light can be twisted by two cylindrical lens around its axis of propogation. If one shines a twisted light on a flat screen, he can see a light "ring". Because ligth with different phase cancel each other at the center and that makes a singularity point at the center of spot. Here I want to first introduce formulas for a Hermitian Gaussian(HG) and Laguerre-Gaussian mode. Here are equations for HG and LG which are propogate along z axis.

One can see from this equation that HG beam is living in Cartesian coordinates. It depends on x,y and propogation direction z. At the same time, an LG mode is depending on r (distance from the center of "ring"), \[Phi] (angle) and propogation direction z. LG modes only excist in cylindrical coordinates.

From equations above,we saw that LG mode can be decompsed into HG modes. A LG(1,0) mode can be decomposed into two parts. The first one is a HG01 mode and the second is HG10 mode which is half Pi out of phase with the first HG component. This decomposition can be illuminated by graph below.

### Set Up

The only changes I made in this setup compared to the last one is this converter in the path of lower beam. The converter was made up by two cylindrical lens with same focal length which were seperated by double focal length diatance. Because cylindrical lens only has one direction be curved, so input beam will undergo different changes in Gouy phase (1). That can make the phase whift between two components I want. For more details about how this converter works, one can check M.Pagget's paper(2).

### Result

 Input light Reference beam Vortex Inteference pattern

This time, I made vortex I got from this converter and overlaped it with reference HG beam. The interference pattern was showed in the last picture. In this last picture, we can see this "fork" clearly in the center. Why there is a "fork"?

### Explanation to the "fork"

We can see in this picture, when I overlap reference beam and vortex, the vortex inferenced with two different parts of reference beam. Due to the phase difference between lobes, there will be a "phase jump" at the center which is caused by discontinuation of phase.

### References

"An experiment to observe indensity and phase structure of Languerre-Gaussian laser modes", M.Pagett, J.Arlt and N.Simpson, American Journal of Physics, 3 Apr 1995