WISE 187 Report for Amara Tazeem

Measuring the Wavelength of
Laser Light with an Ordinary Ruler

Amara Tazeem
WISE 187 Course
Stony Brook University
May 2003

Introduction

Light and its characteristics have baffled physicists for centuries. The question: is light a steady stream of particles or is it a wave that spreads from a source, was not completely answered until the early 20th century.The debate finally ended with the realization that light has both wave and particle properties. By 1830, most scientists had accepted the wave theory of light. It was not until 1905, when Einstein showed that light emits energy in packets or quanta, that the scientific community accepted the particle theory of light.

In this experiment the traditional wave theory of light and including its predictions about diffraction and interference were studied. The data gathered was used to calculate the wavelength of a beam of red laser light.

Diffraction is characteristic of the wave nature of light. It occurs when light comes across an obstacle in its path. The wave theory of light predicts that the light bends around the object in its path, into the region behind it, called the shadow region. As the light waves of light bend around the object, they cross each others path, hence interfering with one another.

There are two types of interference that can occur. Waves can interfere constructively or destructively. In constructive interference the crest or maximum of one wave interferes with the crest of a second wave producing a bright spot. This occurs when the waves are in phase with one another. This pattern is produced for an integer multiple of the wavelength of light used. The formula for this is d sin = m . If the crest of one wave interferes with the trough or minimum of a second wave the maximum and minimum cancel to give a dark spot. This is referred to as destructive interference. This occurs when the wave fronts are 180o out of phase with one another so that they differ by a factor of �. The formula for this is given by d sin = (1/2 +m). This is shown in the diagram below (taken from Ref. 2).

The type of interference pattern observed depends on the wavelength of light used. In order to observe an interference pattern at all, the light used must by coherent. Lights emitted from coherent sources have a specific phase relationship relative to one another. That is, the light waves could be fully in phase, fully out of phase, partially in phase or partially out of phase. What matters is that the waves have a specific orientation relative to one another. It has to be directed a certain way so that the waves from the two sources, or from a single source, can continuously interfere with one another. If the sources aren't coherent and the waves generated are in random phases relative to one another, no pattern is produced, and a fully illuminated projection screen will be seen.

The first full coherent source of light, the Light Amplification by Stimulated Emission of Radiation (laser), was not developed until the 1950s. It is this relatively new technology that was used in this experiment to produce an interference pattern.

One final important point with regard to this experiment is that visible light comes in a whole array of colors from violet at one end with a wavelength of 500 nm to red at the other end with a wavelength of 700 nm. Because interference patterns are sensitive to the wavelength of light, using "white" light, which consists of light with various wavelengths, could be a problem. Hence it is essential to isolate the different bands of light to get a source that is monochromatic, of one wavelength. In this experiment monochromatic red coherent laser light was used.

Procedure

A Helium Neon laser was used to emit red laser light. A beam of light was directed at the edge of a machinist ruler with 100 divisions to an inch, for the first group of data. The divisions were etched into the surface of the ruler, allowing them to be used as "reflection gratings." to create diffraction patterns. The pattern of spots was observed on the opposite wall. it was traced onto aced onto a piece of graph paper to be analyzed later. It was noted that the intensity of the spots decreased as their distance from the first spot increased.

The ruler was then flipped so that the grating size was increased from 100 to an inch to 64 to an inch. The interference pattern produced here was also traced. The previous mentioned calculations were conducted while the angle at which the laser beam hit the ruler was kept the same. In order to understand what effect varying the angle had in the pattern produced, for the next set of data, it was varied and the same two ruler gratings were used.The last two patterns were observed as the beam hit the ruler at the same angle to the horizontal.

Data

For Theta 1

As can be seen from the pictures above, the distance to the wall was calculated from the position of maximum brightness of the laser beam on the ruler. This distance is L = 74.4 +/- 0.2 cm for this set of data.

The graphs below exactly reproduce the diffraction patterns that were observed for angle 1 using the 100 divisions to an inch and the 64 divisions to an inch gratings. The numbers recorded next to each spot tell the distance of that spot from the first spot, which the image of the beam as directly misses the ruler.

For Theta 2

Distance to the wall from position of maximum brightness on ruler = L = 69 +/- 0.2 cm

Analysis

The lowest spot in the interference patterns shown above is that of the beam of light as grazes the edge of the ruler. The second lowest spot in the pattern above is from light being reflected without diffraction. This is called the zeroth order of diffraction, even though no diffraction is actually ocurring. The one above it is the first order, and so on. The origin is taken as the position directly between the spot that goes straight past the ruler and the zeroth order spot. This is illustrated in the figure below (see ref. 1).

Using the distance from the origin to the zeroth order spot, as x, and the distance to the wall, d, the angle between the horizontal and the zeroth order spot was calculated. This angle should be the same for the 100 marks per division grating and the 64 marks per division.

The data above shows the angle calculated using tan = (x/d). A small variation in the angles can be seen when they were supposed to be the same. The difference is relatively small and thus is not significant to the results of this experiment. Also this is to be expected because the instruments used are not highly precise.

When searching for a formula that related all the data gathered to the wavelength of light , an incorrect formula was found on one web site from the University of Melbourne Physics Department instructional labs (see Ref. 3). The formula defined x as the distance to the wall, y as the vertical distance to the spot, from the origin, and d as the distance between the markings on the ruler. The formula given is as follows:

This formula can be shown as incorrect by a simple dimensional analysis. On the left side n has no units and is a length. On the right side x, y and d are also units of length. So

[L] = [L][L³/L²]

[L] = [L][L]

[L] = [L²]

This is obviously wrong since a quantity with units of length can't equal a quantity with units of length squared. The correct formula should have units of [L²] in the numerator as well as the denominator. The correct formula is stated in A.L. Schawlow's 1965 paper on measuring the wavelength of light using a ruler (see Ref. 4).

The formula used in this experiment to calculate the wavelength of the laser light is slightly different than Schawlow's. The derivation of the formula, shown below, starts with the Bragg equation, and then uses a small-angle approximation.

The final equation gained from the derivation above resembles a regular linear equation of the form y = Ax + B. In the above, ² is the y variable, the expression 2[(1 - cos ₁)] is the B term, and the slope is 2/d, where d is the spacing of the ruler markings, and m is the diffraction order. The equation has been plotted in the graph below.

The above graphs are plotted as diffraction order vs. the angle squared. Here the angle is in degrees. These graphs were made for the 100 divisions per inch data that was gathered for the two different angles. The linear relationship proved above is clearly visible. Also it can be seen that the lines obtained for the two different angles are approximately parallel.This is also expected from the formula above since d, and are the same for both, and the only difference between the two sets of data is the incident angle of the laser beam with the ruler. This is only present in the B term in the equation and hence serves to only shift the two lines relative to one another.