gaussian

Optics Lab (Short Report)

Gaussian Beams using a single lens as aninterferometer

Brian Bannon(08332321)April 2013

Abstract

In this experiment the student is exposed visually to Gaussian beams in the un-dergraduate optics lab.Using a single lens as an interferometer, the student takeselectromagnetic radiation in the form of a laser beam and views it visually using acamera is able find and examine the Gaussian beam that is being induced in thelens and internal reflection is happening within the lens which produces the Gaus-sian Beam.The student will set up the experiment and try to discover the Gaussianbeams for different fringes and and different distances from the lens and camera,using the Fresnel diffraction method.In this experiment the student will show howa single lens can be used to produce a superposition of a collimated and a focusinglaser beam.This simple and inexpensive interferometer has essentially no path-delayjitter like other interferometer setups, while the phase between the two beams canbe controlled easily through subtle adjustments of the tilt of the lens.

Introduction

Weak surface reflections from a simple lens can be used to observe interferencesbetween collimated and focusing laser beams.The superimposed beams, the Gaussianbeam resulting from a double reflection within the lens, can be made to have similarintensities near the focus of the weaker beam by choosing the lens prescription andthe divergence of the incident laser. An advantage to this inexpensive setup is itsstability against vibrations.The relative phase between the two beams can be easilycontrolled through small adjustments to the lens alignment.Using this setup witha CCD camera the student can explore the amplitude and phase properties of aGaussian Beam in the vicinity of the focus.

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Data Analysis

The setup of the experiment is setup along a table with all the components all in linewith each other.The Laser passes through a polarised filter to dampen the intensityof the collimated beam on the CCD camera.next it passes through a converging lensto condense the beam so that the laser light can pass through a pin-hole were thelight hits a second lens which converges the focusing beam and were the internalreflection within the lens occurs and produces the Gaussian Beams which are seenon the CCD camera along with the collimated beam present through the lens.Firstly the student had to evaluate the distance between the second converging lensand the CCD camera. The distance from the first converging lens to the pin-holewas given to be 30cm, the distance from the pin-hole to the second converging lensis 100cm.The distance from the second converging lens to the camera is given by theformula,

di =n− 1

2nf (1)

In this equation the refractive index of glass is n=1.5, and f is the distance from thepin-hole to the second converging lens,f=100cm.

di =1.5− 1

2(1.5)(100) = 16.67cm

This is now the distance were the student should see the z0. Next a picture of thecollimated beam had to be taken to to use some analysis from the picture to find thedistance each pixel in the photo covered and to find the diameter and radius of theghost beam. The radius of the ghost beam is important for finding the z0 which willbe the distance of how far from the distance di the student must move the camerato produce more fringes in the Gaussian beam.

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Fig(1).Picture from the CCD camera of the collimated beam in the lower picturewith the Gaussian beam focus z0 just above it.

The format of this picture was saved by pixel size 476x316 pixels.Using MathCadsoftware to display the image as an array giving each pixel an numerical intensity,we could could graph the intensity of the light by the position of the beam.

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Fig(2).Graph of the intensity of Fig(1) with respect to position.

From the dimensions of the camera, given the student knew the hypotenuse being8mm, from the dimensions of the the picture in pixels we could find the distancecovered in one pixel.

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Fig(3).Dimension of the camera from using the pixels,the hypotenuse was 795pixels, the opposite 44 pixels and the adjacent 636 pixels.

In finding the size of 1 pixel the student took the length of the adjacent in mm anddivide it by the number of pixels across the adjacent.

pixel size =6.398x10−3

636= 1.006x10−5m

Now from Fig(1) the student noted the pixel length of the diameter of the collimatedbeam.

4diamter = 623− 168

Diameter = 4diamter x pixel size

Diameter = 4.577x10−3

With all the required information the student could now determine the radius ofthe ghost beam.

w0 =2λf ∗

π= 1.468x10−5m (2)

where f ∗ = diDiameter

The student now has to find z0 given by

z0 =kw0

2

2= 1.069x10−3m (3)

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where k = 2πλ

and λ = 633nm. With this now the camera will be moved eithertoward or away from the second converging lens a distance of z0, to see the differentfringes of the Gaussian Beam.Now the student took pictures of the different z0. The first being the focus of theGaussian beam.

Fig(4).Picture of z0 the Gaussian Beam above the collimated beam.

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Fig(5).Intensity profile of the z0 fringe.

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Fig(6).Picture of +z0 the Gaussian Beam above the collimated beam

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Fig(7).Picture of +2z0 the Gaussian Beam above the collimated beam

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Fig(8).Picture of +3z0 the Gaussian Beam above the collimated beam

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Fig(9).Picture of +4z0 the Gaussian Beam above the collimated beam

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Fig(10).Intensity profile of the different fringes ofTop:(l-r)+z0,+2z0,Bottom(l-r)+3z0,+4z0.

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Fig(11).Picture of −z0 the Gaussian Beam above the collimated beam

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Fig(12).Picture of −2z0 the Gaussian Beam above the collimated beam

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Fig(13).Picture of −3z0 the Gaussian Beam above the collimated beam

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Fig(14).Picture of −4z0 the Gaussian Beam above the collimated beam

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Fig(15).Intensity profile of the different fringes ofTop:(l-r)−z0,−2z0,Bottom(l-r)−3z0,−4z0.

Conclusion

In this experiment I completed everything I has set out to do and achieve with thisexperiment.The setup of the apparatus was quite difficult and hard to get a perfectimage on the CCD camera of the focusing beam. With some many componentsinvolved and each would have to aligned and adjust to capture the perfect image onthe camera.Dealing with the conversion from pixels to position was a bit tricky attime but once it was explained to me by the lecturer I was able to comprehend itand do some data analysis. With regard to the data itself, I tried to get the bestimages possible with the time I had in the lab.The graphs relate to the theoreticalgraphs given in the paper but don’t show a huge change when the camera was movedby z0 but a change between different fringes was seen.The theoretical predictions, Iwas unable to get out graphical I found the theoretical intensity formula to be tootricky and got bogged down with it and had not enough time to graph the theoreticalgraphs. Apart from that everything was fine and I achieved everything I set out todo from my abstract.

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References

[1] Visual introduction to Gaussian Beams using a Single lens as an interferometer,J.Peatross and M.V.Pack

Appendix

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