The optics of Gaussian laser beams

PHYS 3330: Experiments in OpticsDepartment of Physics and Astronomy,

University of Georgia, Athens, Georgia 30602

(Dated: Revised July 2012)

In this lab you will test three predictions of classical electrodynamics as applied to propagatingGaussian laser beams.

I. INTRODUCTION

This lab will give you an opportunity to familiarizeyourself with the optical equipment available in the Mod-ern Optics Lab, to review some basic principles of ge-ometrical optics, and to learn about the properties ofGaussian laser beams. Most importantly, however, thislab will be your first opportunity to use model fitting totest theories of nature, and to communicate your resultsusing prose, figures, tables, and equations in a journal-style manuscript.

You will need to complete some background readingbefore your first meeting for this lab. Please carefullystudy the following sections of the “Newport Projects inOptics” document (found in the “Reference Materials”section of the course website): 0.1 “Geometrical Optics”,0.2 “Thin Lens Equation” and 0.6 “Lasers”. Also, readthrough the Melles Griot tutorial, all sections through2.7. Finally read section 11.7 of your textbook, “Physicsof Light and Optics,” by Peatross and Ware. Your pre-lab quiz cover concepts presented in these materials ANDin the body of this write-up. Don’t worry about memo-rizing equations – the quiz should be elementary IF youread these materials carefully. Please note that “taking aquick look at” these materials 5 minutes before lab beginswill likely NOT be adequate to do well on the quiz.

II. THE TRANSVERSE PROFILE OFGAUSSIAN BEAMS

We wish to make quantitative measurements of thetransverse intensity pattern of the laser beam emittedby a HeNe laser, and to compare these to a model pre-dicted by laser resonator theory, so as to confirm or refutethe model. One way to measure the transverse intensitypattern is to shine the beam onto a CCD camera andsimply capture a 2-D image. However, if a camera is notavailable, one can quite effectively employ the so-called“knife edge” technique. In fact, because knife edge datacan be acquired and processed very rapidly, this tech-nique is often favored over CCD cameras for high-endcommercial beam analyzers, such as the Coherent BeamAnalyzer. Here you will use the knife edge technique tomeasure the horizontal cross-sectional profile of a HeNelaser beam and extract from the data an estimate of thethe so-called “1/e2 radius” of the beam.

FIG. 1: The components for a knife-edge measurement of thebeam radius. The razor is glued to a kinematic mirror mount.Use the lower knob on the mount to push the razor bladethrough the beam. The gold colored device is the photodiode.

A. Procedure

1. Insert the photodiode into the laser beam. Centerthe beam on the black square in the middle of thegold colored can.

2. Connect the photodiode to the voltage measure-ment terminals of the Keithley digital multimeter.Make sure the voltage reading is less than 200mV;if it is not, ask your instructor to help you changeyour load resistor or to insert an attenuator intothe beam.

3. Position a mirror mount with a razor blade epoxiedonto it a few inches in front of the photodiode, sothat the razor blade is perpendicular to the beampropagation direction, and is about one millimeteraway from “cutting” into the beam.

4. The adjuster screws on the mirror mount have apitch of 80 (threads per inch). Calculate how manymillimeters the razor is displaced by one completerevolution of such a screw located directly behindthe razor. Note the dot painted on screw knob thatwill allow you to keep track of its rotational angle.If you are careful, you can reproducibly advancethe screw by 1/4 turn. Check that you will be able

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to cut across the entire beam before you startingtaking your data.

5. Using the knob directly behind the laser (the lowerof the two knobs), and in steps of 1/4 turns of theknob, cut through the entire beam with the razorblade, recording the voltage displayed at each step.

Decide yourself how to best estimate your uncertaintiesin voltage and razor blade position; discuss your ideaswith your instructor if you are unsure.

B. Analysis

Plot the data with error bars.The predicted transverse intensity distribution of a

(round, TEM00) Gaussian laser beam can be written incartesian coordinates as,

I(x, y) = I0e−(x−x0)

2/2w02

e−(y−y0)2/2w0

2

(1)

where I0 = I(0, 0). Your photodiode doesn’t collect lightfarther than ± 4.5 mm from the center of the beam.However, this will not affect your ability to measure theshape of your beam, which is significantly smaller. Asa razor cuts into the beam to a horizontal distance x0from the center of the beam, it blocks all optical powerfor x > x0. The photodiode voltage V for a given razorposition x0, is then proportional to an integral over Eq(1):

V (x0) ∝∫ x0

−∞dx

∫ ∞−∞

dy I(x, y) ∝∫ x0

−∞dx I(x, 0) (2)

You will test the hypothesis that the beam profile inthe x (horizontal) direction is well modeled by Eq. (1),and estimate the w0 parameter. Work out the analyticalform of the integral Eq. (2). (Hint: it is a scaled andtranslated version of the normal cumulative distributionfunction). You must take this result and devise a modelto which to fit your data, deciding for yourself what freeparameters to use. Report and discuss the best fit pa-rameter values and their uncertainties. Also, report anddiscuss the value of chi-squared returned by the fit, anddiscuss the likelihood that your model is a good one (i.e,by considering the probability statistic (python functionchi2.sf())for that value of chi-square for the degrees offreedom in your fit.) If your chi-square is particularlyhigh, it may be that you need add parameters to youmodel. Make sure, as always, to provide informative,well labeled, and well captioned graphs of your data andfitted models, including error bars where appropriate.

III. THE LONGITUDINAL DIVERGENCE OFGAUSSIAN LASER BEAMS

Any Gaussian laser beam inevitably changes its trans-verse size as it propagates. Record the size of your laser

beam at various distances from the laser output port, upto 5m away if possible. Use the gauge card at your sta-tion to make the size measurements, and a tape measureto make the distance measurements. For a laser beamsufficiently dim as to not dazzle your eyes when viewedon the gauge card, the apparent beam diameter (i.e. thediameter at which the beam seems to disappear) is ap-proximately the 1/e2 diameter d ≡ 2w. As you will finddiscussed in any of the Gaussian laser beam referencesgiven above, the 1/e2 beam diameter d varies with dis-tance according to

d(z) =√d20 + θ2z2. (3)

Here d0 ≡ 2w0 is the 1/e2 beam diameter at the laseroutput, and z is the distance from the laser output. De-termine the value for θ based upon your measurements ofd(z) by fitting your data to the model of Eq. (3). Try fitswhere d0 is fixed by your measurements, and also whereyou let it be a free parameter. Discuss the relative qual-ity of two attempts at fitting in terms of the uncertaintyin the various fit parameters (i.e., do their uncertain-ties grow or shrink), in the apparent quality of the fit asjudged by a graphical inspection of the residuals, and inconsideration of their chi-square values.

IV. BEAM EXPANDERS

Many times when a laser is used in an optical system,there is a requirement for the beam to have a small diver-gence angle. For example, when measuring the distancefrom the Earth to the Moon via laser range finding, abeam one meter in diameter when it leaves earth expandsto several hundreds of meters when it reaches the moon,and is several kilometers in diameter when it comes backagain to earth.

As pointed out in section 0.6 “Lasers” of the NewportProjects in Optics reference, the product of the beamwaist and the divergence of a lens is a constant (for agiven wavelength):

d0θ =4λ

π(4)

Therefore, if we want a more collimated beam (i.e. onefor which the divergence angle θ is small), the beam di-ameter must be increased. A standard optical device forchanging the diameter of a laser beam is called a “beamexpander,” and utilizes two lenses. See Figure 2. Thefirst lens causes the beam to strongly diverge; the sec-ond lens, placed the correct distance away from the first,collimates the expanding beam. The output beam has alarger waist parameter, and thus a smaller divergence an-gle, than the input beam. The arrangement of the lensesis that of the telescope turned “the wrong way around.”The change in the beam radius is given by the ratio ofthe focal lengths of the two lenses; therefore, the finaldivergence angle should be equal to the divergence angleof the input beam divided by the power of the telescope.

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FIG. 2: Beam expanders. (a) With diverging lens; (b) withconverging lens.

A. Procedure: Constructing a beam expander

Construct a beam expander with whatever lenses areprovided at your station.

1. Mount a short focal length (25.4 mm) positive ornegative lens, whichever is provided at your station,in the beam.

2. Align the lens by the following procedure

• Direct the laser at a known reference point(such as a pencil dot on a white note card)located a few meters away.

• By: (1) raising or lowering the post in the postholder, (2) rotating the post in the postholder,and/or (3) sliding the entire mount on the ta-ble, adjust the lens position so that the out-going beam is still centered on the referencepoint.

• Check that your lens is normal to the direc-tion of propagation of the beam (i.e., not tiltedhorizontally or vertically). Do this by observ-ing the light reflected from the surfaces of thelens comes straight back on itself. There aretwo reflections, one from each surface. Checkthis by positioning a notecard with a smallhole (slightly larger than the laser beam) ap-proximately 50cm–100cm upstream of the firstlens. 1) Get the two reflections to be at thesame height as the hole by adjusting the lensheight. 2) Slide the lens mount side to side,and rotate the lens post, to get the two reflec-tions to overlap each other and to be centered

on the hole in the card – this assures that thebeam is at the center of the lens and that thelens is not tilted with respect to the beam.

3. Mount a positive lens of focal length (100-150 mm),provided at your station, one focal length awayfrom the real or virtual waist formed by the firstlens (see Figure 2). Find the focal lengths of yourlenses by reading the part number of the edge ofthe optic, and consulting the wooden cases in whichthe lenses are stored in the Room 213. Measure thepositions of your lenses with an accurate ruler.

4. Repeat the alignment procedure for this new lens.

5. Find tune the distance of the second lens from thefits. To do this, observe the beam as far down-stream as possible (i.e., until you hit the wall acrossthe room). Have your lab partner make small ad-justments of the distance that lens 2 is from lens 1,while you monitor the size of the beam. The lensspacing is correct when this distant spot is as smallas possible.

B. Measuring the divergence of an expanded beam

Measure the divergence of your expanded beam.Record the size of your laser beam at various distancesfrom the last lens of your telescope, up to 5m away ifpossible. Use the gauge card at your station to makethe size measurements, and a tape measure to make thedistance measurements. It may be difficult to see anydivergence at all. Plot the beam diameter vs distancefor the expanded beam, with error bars for both axes,as you did for the original, smaller beam. Fit your datato the model of Eq. (3), to estimate a new beam diver-gence angle. Calculate the magnification power of thebeam telescope from the known lens focal lengths, andcompare the angle you found to the angle predicted byEq. (4). Remember to propagate all uncertainties, andmake a quantitative statement as to the agreement be-tween the prediction and your measurement of the newdivergence angle.

V. HINTS AND TIPS

Alignment of components is made simpler by di-recting the optical paths along the rows of screw holeson the optical table.When handling optical components treat the op-

tical surfaces of lenses and mirrors with care. Never touchthe surfaces – handle the components by their edges. As-semble components over the optical table, not over thefloor.Laser safety. The HeNe lasers in this lab are “Class

IIIa” laser systems, which mean they are low power, vis-ible lasers for which the eye’s natural blinking reflex pro-

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tect the user from damage by momentary exposure to thebeam. Most laser pointers and firearm laser sights are inthis category.

However, observe these safety rules in the lab at alltimes.

• Do not intentionally look directly into the laserbeam.

• Clearly identify beam paths and ensure that theydo not cross populated areas or traffic paths.

• Terminate the beam at the end of its useful path

with a beam stop, preferably black in color. Blackaluminum foil is provided for this.

• Keep all beams at waist level.

• Watches and other jewelry that might enter the op-tical plane should must be removed while workingwith lasers.

• Use only diffuse (for example, white 3x5 card) re-flectors for viewing HeNe laser beam – never any-thing shiny.