# Astigmatic Laser Mode Convertors

## 1. Introduction

The objective of my project is to investigate the intensity and phase structure of transverse laser modes. In general, transverse laser modes are best described by a product of Hermite polynomial and Gaussian(HG modes). And Laguerre-Gaussian laser modes can be obtained by the direct conversion of the Hermite-Gaussian laser output using a mode converter utilizing the Gouy phase shift in the region of an elliptical beam waist. However, a direct use of the mode converter requires a laser operating in a high-order HG mode. This can be readily achieved putting metallic wires inside a properly misaligned laser cavity.

## 2. Mode Converter & Orbital Angular Momentum

The wave front (top) and the intensity pattern (bottom) of the simplest Laguerre Gauss () or doughnut mode. The index l is referred to as the winding number and (p+1) is the number of radial nodes. The customary Gaussian mode can be viewed as LG mode with l=0. The handedness of the helical wave fronts of the LG modes is linked to the sign of the index l and can be chosen by convention. The azimuthal phase term of the LG modes results in helical wave fronts. The phase variation along a closed path around the beam center is . Therefore in order to fulfill the wave equation the intensity has to vanish in the center of the beam. For the paraxial wave equation
(1)
both the Hermite Gaussian
 (2)
and the Laguerre Gaussian
 (3)
modes are a possible set of basis vectors.
Therefore it is possible to transform HG modes into LG modes and vice versa:

Experimentally this was realized by astigmatic mode conversion. The mode converter consists of two cylindrical lenses canonically disposed with respect to one another. This system of lenses introduces a Guoy phase shift and converts the incoming HG10 mode into the LG01 mode

## 3. Alternative Method

The field distribution in the HG mode HG1,0 rotated 45o or 135º relative to the x-axis ( without the propagation factor ) is equivalent to the sum ( or difference ) of two HG 0,1 and HG1,0 modes,

(4)

Approximated field distribution : two out-of-phase fundamental Gaussian beams

(5)

The intensity distributions corresponding to the fields (4) and (5) are similar (see Fig.)

Intensity distribution for a Hermite-Gaussian beam HG1,0 rotated 45o relative
to the x-axis (a) and for two out-of-phase fundamental Gaussian beams (b).

## 4. Experiments

Beam splitter BS1 splits a beam from a He-Ne laser oscillating in the fundamental mode into two beams. The mirrors and the beam splitter BS2 are adjusted so that the beam from the two arms are slightly misaligned. Changing the relative phase between the beams by moving a prism we can get the required field distribution at the input of the mode convertor.

## 5. Further Study

• The design of the mode converter
• Study of the Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) mode.

## 6. Acknowledgements

I deeply appreciate Dr. John Noe for his instruction and generous help, both for the physics and the webpage. I would like to thank Prof. Harold Metcalf's group members for help and supply.

## References:

1. Dmitri V.Petrov, Fernando Canal, Lluis Torner, Opt. Comm. 143 (1997) 265-267
"A simple method to generate optical beams with a screw phase dislocation"

2. M.W.Beijerbergen, I.Allen, H.E.I.O. van der Veen, and J.P.Woeman, Opt.Comm.96 (1994) 123-132
"Astigmatic laser mode converters and transfer of orbital angular momentum"

3. M. Padgett, J. Arlt, and N. Simpson, Am. J. Phys. 64 (1), 1996
"An experiment to observe the intensuty and phase structure of Laguerre-Gaussian laser modes"

4. J.Courtial, M.J.Padgett, Opt.Comm. 159 (1999) 13-18
" Performance of cylindrical lens mode converter for producing Laguerre-Gaussian modes"

5. L.Allen,M.W.Beijersbergen,R.J.C.Spreeuw,and J.P.Woerdman, Physics Review A, 45 (1992)8185-8189
"Orbital angular momentum of light and transformation of Laguerre-Gaussian laser modes"

6. Chr.Tamm, Physics Review A, 38 (1988)5960-5963
"Frequency locking of two transverse optical modes of a laser"

7. Alipasha Vaziri, Alois Mair, Gregor Weihs, Anton Zeilinger
Institut für Experimentalphysik Universität Wien, Austria

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