LC-1: Diffraction and Interference

OBJECTIVES:

To observe diffraction and interference and to measure the wavelength, $\lambda$, of laser light.

APPARATUS:

Optical bench; He-Ne laser; slit of variable width; thickness gauge; slide of 4 pair of electroformed double slits; short focal length lens; small opaque ball tipped pin; small circular holes in thin brass; screen; Pasco Interface with light and rotation sensors, mounting bracket, light aperature module; traveling microscope; interference demonstrations: optical flats (2), Newton's rings, 18 mm gauge blocks (2), interferometer & Na lamp.

INTRODUCTION:

The He-Ne laser provides plane light waves of wavelength 633 nm. The waves in a cross section of the beam are in phase (i.e. the light is ``coherent''). Any finite plane wave will spread by diffraction. The spreading is rapid if the beam is narrow, e.g. after passing the narrow slit in Part I below. If we illuminate two closely spaced narrow slits (narrow relative to the slit to slit spacing) by the same laser beam, the two spreading beams will overlap and interfere: Part II.

Experiment I: Single Slit Diffraction:

Figure 1: Schematic of single slit diffraction

THEORY:

The angular separation in radians of the first minimum from the center of the pattern is

$$\theta = \lambda /a$$

where $a$ is the width of the slit. (For your derivation you may refer to the text and remember that for small $\theta$, sin  $\theta  \cong  \theta$).

SUGGESTED PROCEDURE:

1.

Mount the laser and variable width slit on the optical bench as shown in Fig. 1. Observe the pattern on the supplied white screen and qualitatively explain in your lab book that the first minimum occurs as shown in Fig. 1 at $m=1$.

QUESTIONS:

Q1.

Qualitatively, how does the pattern vary as the slit is closed?

Q2.

In ``real'' life when would you begin to be concerned about the appearence of diffraction? Identify at least one (if not more) example of diffraction in your daily life.

Q3.

Qualitatively, how does the pattern vary as $\lambda$ decreases (from red to blue-violet)?
Click on the Single Slit Diffraction link immediately below (web-version) to test your prediction.

2.

To enable more quantitative observations the PASCO interface module has been configured to provide an Intensity vs. Linear position of the light sensor plot and table. Launch the experiment by CLICKing on the telescope icon below (web-version only). After the PASCO experiment window pops up, start the data acquisition by CLICKing once (or twice) on the REC button and move the combined light and rotary motion sensor in the lateral direction gently and smoothly by hand. Pratice starting on one side of the diffraction pattern and move smoothly towards the other.

3.

Configuring and aligning the optical components is very important! In particular the laser alignment and apperature settings are VERY important for getting good results.
There are two aspects for configuring the light sensor: aperature size and detector gain. Using a larger aperature lets more light reach the detector. How does this affect the image profile? A larger gain increases the sensor output but can peg the output at maximum and increase the noise. Adjust and record the setting which give you reasonably good results. Ask your instructor for a brief demostration if you are at all uncertain.

4.

Use the thickness gauge to set a slit width, $a \cong$ 0.1 mm. It is important that your slit is parallel. What would happen if the slits were not parallel to one another?

5.

Using the cross-hair feature of the PASCO graph display, measure $\theta$ (from $y/D$) for $m$=2, 1, 0, -1, -2 plot $m$ vs $\theta$. Does this form a straight line? From the slope and known value of $a$, calculate $\lambda$.

6.

Estimate uncertainties and then compare with the accepted value of 633 nm. Are there any obvious sources of systematic error?

Experiment II: Multiple Slit Interference

For double slit interference, shown in Fig. 2, the distance $y$ to the m$^{th}$ bright fringe from the midpoint 0 is

$$y = D \tan \theta \cong D \theta \mbox{\hspace*{.3in}and \hspace*{.3in}} \theta \cong m\lambda /d$$

Hence

$$\lambda  = yd/mD,$$

(For more information refer to the text and recall that for small $\theta$, $\theta \cong \sin\theta \cong \tan \theta$.)

Figure 2: Schematic of double (or multiple) slit interference.

SUGGESTED PROCEDURE:

1.

Mount the slide containing the four electroformed slits in the slit holder and illuminate one of the slit pairs with the laser. Two of the slit pairs have a center to center slit spacing of 0.25 mm and the other two have a 0.50 mm spacing. The slit pairs with the same spacing differ only in the width of each slit: one has slit widths of 0.04 mm and the other has slit widths of 0.08 mm.

2.

Note the difference in pattern and spacing of the resulting interference fringes on the wall. Neither slit width is negligible compared to the wavelength so both diffraction and interference effects are apparent in all patterns, and the interference fringes are contained within the diffraction envelope. Observe the interference pattern for all four electroformed slit pairs.

3.

Choose one pattern for careful measurements and measure the fringe separations at a number maxima ($m$= -1, 0, 1, etc.) and calculate $\lambda$ using the same general procedure as experiment I. The separations marked on the slits are only nominal. If time permits use the travelling microscope on page  to measure the slit separations.

4.

Estimate quantitatively your uncertainties. Compare with the accepted value.

Experiment III: Fresnel Bright Spot and Other Interference Patterns

INTRODUCTION:

The Fresnel bright spot (also called Poisson's bright spot) is a bright spot in the center of the shadow cast by every circular obstacle in the path of a plane wave. The effect is implicit in Fresnel's representation of a coherent plane wave by half-period zones (e.g. see Shortley and Williams, ``Elements of Physics'' 5th ed. p. 737 or, if this is the web-version, see this example ), but was first pointed out by Poisson (a disbeliever in the wave theory) in an attempt to ridicule Fresnel's wave theory. Arago subsequently showed experimentally that the spot existed. You, too, may demonstrate the spot:

1.

Replace the slit support rod with the rod containing the PASCO Slit Accessory wheel. Rotate and adjust the assembly to illuminate the circular obstacle in the laser beam.

2.

To obtain a large enough coherent plane wave it may be necessary to diverge the laser beam by placing a short focal length lens directly in front of the laser.

3.

Carefully align laser plus lens and obstacle to center the shadow along the beam axis. The coherently illuminated area (the plane wave) should be several times the area of the obstacle. Proper alignment may help significantly in ``cleaning up'' the image.

OPTIONAL:

Diffraction from a small circular opening: By analogy with the Fresnel bright spot in a circular shadow, you might expect a dark spot centered in the image of a circular opening. In this case the situation is more complex: the on-axis Fresnel bright spot in the shadow results from superposition of a large number of higher Fresnel half-period zones whereas the light from a small circular aperture comes only from a few low Fresnel half-period zones whose superposition on axis may result in either a dark or bright spot: dark for an even number of zones, bright for an odd.

Test this using the 2 circular aperatures on the PASCO Slit Accessory wheel. Illuminate the hole with the diverged laser beam in the same manner as described above. By varying the hole size and/or distance from hole to screen you may change the small number of half-period zones contributing and hence see the central point on axis as either a bright or a dark spot, e.g. see Fig. 13 Shortley and Williams, p. 740. Again careful alignment is important to obtain a clean symmetric image.

If time permits you may find it interesting to view the four addition two-dimensional diffraction patterns.

DEMONSTRATIONS ON THE DISPLAY TABLE:

A. Interference between optical flats, gauge blocks, etc.
B. Newton's rings
C. Michelson's interferometer