Experiment 2.15

SINGLE SLIT DIFFRACTION

AND

BABINET'S PRINCIPLE

PRELAB

PURPOSE

To use a diffraction grating to calibrate the wavelength of a laser. Using the known wavelength, the diffraction pattern of a hair will be measured and the hair diameter deduced.
 
 

MATERIALS optical bench, solid state diode laser, 3" X 5" index card with hole in center, tape, card holder with clip, diffraction grating (around 80 lines per mm) with slide holder, sheet of paper, half-meter stick, traveling microscope.

RELEVANT FORMULAS

Single slit diffraction:                                  

Angle Determination:


DISCUSSION

Determination of the Laser Wavelength:

A diffraction grating will be used to determine the wavelength of the helium-neon laser. For a diffraction grating that has a line spacing of d, the maxima are located by the following equation, called the "grating equation":
 
 

In this experiment, you will only need to use the first order maxima, so that the factor m = 1 in this case. The sine of the projection angle θ can be determined using the definition:

where s is the distance from the center to the maximum along the projection screen.

Diffraction by a Thin Opening:

As light passes through a narrow opening (a single slit) it "bends" and spreads out as it continues on the other side of the slit. This "spreading out" phenomenon is called diffraction and can be understood using the idea that each point across the slit is a source of light waves. Thus the slit can be thought of as an infinite number of infinitesimally closely spaced sources, all of which will interfere at some observation point distant from the slit.

The intensity pattern characteristic of a single slit has a central bright fringe and, on either side, a series of bright and dark regions. A bright region's intensity is lower the farther from the central fringe that it is located. A graphical representation of the intensity pattern is shown in Fig. 2.15-1.

Figure 2.15-1: Intensity pattern of a single slit. The distance s is measured along the screen from the central maximum. D is the distance from the slit to the screen and w is the slit width.
 
 
The location of the dark regions is the important thing about the single slit diffraction pattern. The relationship which gives the location of the minima (dark fringes) is:
 

                                                                                (1)

where w = slit width, θ  = angle measured from the slit center to the center of the dark fringe, λ = wavelength of the light source (if the source has many wavelengths present, we must consider each separately), and m = an integer that gives the order number. Note the integers along the horizontal axis of Fig. 2.15-1. The sine of the projection angle θ can be determined by measuring the distance from the slit to the screen D and the displacement of the fringes s, along the projection screen. You can apply the definition of the sine as the opposite side over the hypotenuse to find the factor in eqn. (1):
 

                                                                        (2)

If the angle θ is less than about 6 °, then the "small angle" approximation can be used. It involves approximating the sine of the angle with the tangent of the angle so that we can write:

Equation (1) for the location of the dark fringes then becomes:
 

where β is defined in Fig. 2.15.1.

Babinet's Principle states that the diffraction patterns of complementary objects are the same. The term complementary here means that opaque spaces in one object are transparent in the other object and vice versa. For example, a single slit and a hair of the same width are complementary.