Wavelength Measurement with a Machinist Ruler

Isaac AguilarCalifornia State University, Chico

(Dated: December 12, 2012)

The goal of this experiment was to calculate the wavelength of a laser using a machinist ruler.Light from the laser was shinned onto the laser and the reflected rays create a diffraction patternwhich can be used to calculate the wavelength. Since the laser is not oriented perpendicular tothe object creating the diffraction pattern the usual mλ = sin θ doesn’t work to calculating thewavelength. For this case the wavelength can be calculated using the incident angle, and reflectedangles of the beam in conjunction with the millimeter spacing on the ruler. The calculated valuesare around 650 nm - 680 nm which intuitively makes sense.

INTRODUCTION

The task of measuring the wavelength of alaser when equipped with a diffraction gratingstraight forward. But when equipped with amachinist ruler instead of the grating the taskbecomes a bit harder. The idea that a pathlength difference traveled by the incident wavescauses them to interfere with one another holds,but the functional form mλ = d sin θ for theinterference due to the path length difference isdoes not.

THEORY

A diffraction pattern is created when wavesinterfere with one another. An interference pat-tern is created when one wave take a shorteror longer path then another wave to the samepoint. This path length difference causes thewaves to interfere with one another creating adiffraction pattern. The task now becomes howto create a path length difference with a machin-ist ruler? The machinist ruler is made of metaland its scale is in millimeters. The millime-ter markings are grooved into the ruler and arepainted black which absorb any light incident onit while reflective surfaces reflect the light. Theabsorption of light between the grooves sepa-rates the single beam into two beams and cre-ates a path length difference between the inci-dent light. In figure 1 the path length differenceδ is created by two different paths δ1 and δ2. δ1

can be calculated using the triangle that con-tains the sides d and A.

FIG. 1. A and B are the path length differences δ1and δ2

δ1 = d cosα (1)

δ2 is calculated in a similar fashion

δ2 = d cos θn (2)

where θn is the angle at which the light reflectsoff the ruler and α is the angle of the incidentlight. The path length difference δ is the differ-ence between δ2 and δ1.

2

δ = δ2 − δ1 (3)

δ = d(cosα− cos θn) (4)

For constructive interference the path lengthdifference is equal to integer multiples of thewavelength

mλ = d(cosα− cos θn) (5)

EXPERIMENT

The experimental set up consisted of a reddiode laser, a power supply and a machinistruler placed on a table. The ruler was adjustedso that the reflected rays were projected ontothe wall in front of the laser. Masking tape wasplaced on the wall so that the bright fringe po-sitions could be recorded. Since the laser wasincident on the ruler at an angle α, the diffrac-tion pattern was projected onto the wall at thethat angle. Since the diffraction pattern extendsbeyond the angle α, θn is introduced. θn is mea-sured from the m = 0 fringe to the next brightfringe. The angle β is measured from wherethe undeflected laser hits the wall to the brightfringe position. β can be calculated using L1and L2.

β = tan−1 L1

L2(6)

The relationship between β, α and θn is givenby

β = α+ θn (7)

Since θn equals α at m = 0, α can be calcu-lated using equation (7).

α =β

2(8)

Now that the incident angle α is known cos θncan be calculated for each subsequent brightfringes.

FIG. 2. Basic set up of the lab

RESULTS

The data gathered from the experiment wasfit using a linear regression routine in gnuplot.cos θn was plotted against fringe number m.Solving equation (5) for cos θn gives this linearform

cos θn = −λdm+ cosα. (9)

Equation (9) shows that the slope is a func-tion of λ and the millimeter spacing d.

FIG. 3. Sample fit of a set of data.

3

Figure 3 shows a set of data fit against equa-tion (9). Multiplying the slope of the fit by thespacing d gives the wavelength of the laser. Ta-ble 1 shows the calculated values of λ.

TABLE I. Wavelength of laser

Data Set Wavelength (nm) Error ± (nm)

1 681 10

2 653 20

ERROR PROPAGATION

The general equation for error propagation is

δq =

√(∂q

∂x1δx1

)2

+

(∂q

∂x2δx2

)2

+ ... (10)

The error reported in Table 1 propagatesfrom the measurement of L1 and L2. The errorin λ is directly proportional to the error in θso using equation (6) in conjunction with equa-tion(10) will give the error in θ.

δθ =

√(∂θ

∂L1δL1

)2

+

(∂θ

∂L2δL2

)2

(11)

The error in λ can be found by taking equation(9) and substituting it into equation (10).

δλ =

√(∂λ

∂θδθ

)2

+

(∂λ

∂αδα

)2

(12)

Because the the incident angle α is constant,its derivative drops out and are left with thefollowing.

δλ = sin θδθ (13)

CONCLUSION

Calculating the wavelength of a laser usinga machinist ruler can be done. The reportedvalues of the wavelengths overlap within oneanother’s uncertainty. The wavelength of thered diode laser is between the range of 671nm to 673 nm. There are concerns that arisefrom this reported value. Has enough data beencollected to average out systematic errors thatarisen from the experimentation? Taken onlytwo sets of data is not enough to safely reportthis wavelength with confidence that the sys-tematic error has been minimized. The moredata set that can be taken and analyzed willhelp pinpoint whether this range is accurate.

[1] Taylor, John R. Introduction to Error Analysis:The Study of Uncertainties in Physical Measure-

ments University Science Books, Sausalito CA,1997