Assignment 5

Problem 1:

Calculate the intensity distribution of the interference pattern for up to four equally-spaced sources. Assume light shines on a series of equally spaced slits. The spacing between the slits is d. The diffraction pattern is observed on a screen a distance L away from the slits, L >> d.

If we view the slits as sources of electromagnetic waves, then these sources are coherent, the electric fields

E(x,t)=E_maxcos(kx-wt+f)

of all the sources are in phase. But if we observe the diffraction pattern on the screen a distance z away from the x-axis so that z/L = tanq, then the electric field of source n is out of phase with the electric field of source 1 (n-1)d, where

d = k d sinq.

The total electric field at z is the sum of the fields due to all of the sources. The intensity at z is proportional to the square of the amplitude of the resultant field.

The resultant field at z is given by

where a is the phase of the electric field of source one at position z on the screen and N is the number of sources.

The amplitude of this field at z is given by

The intensity distribution as a function of d is given by (E_res/E_max)².

Procedure:

Open a Microsoft Excel Spreadsheet.

Let column A contain the phase shift d, from -14 to +14 ins steps of 0.1, starting in row 3.

Let columns B, C, D, and E contain a*cos(0), b*cos(d), c*cos(2d), and d*cos(3d), respectively, starting in row 3.

The coefficients a, b, c, and d in row 2 are used to turn a source on or off. If a coefficient is 1 the source is turned on and if it is zero a source is turned off. Start with a = b = 1, c = d= 0.

Let column F contain the sum of columns B through E, starting in row 3.

Let columns G, H, I, and J contain a*sin(0), b*sin(d), c*sin(2d), and d*sin(3d), respectively, starting in row 3.

Let column K contain the sum of columns G through J, starting in row 3.

Let column L contain the square of column F plus the square of column K, starting in row 3. Column L contains the intensity distribution as a function of d, starting in row 3.

Construct a plot of the intensity distribution as a function of d (column L versus column A).

With a = b = 1 and c = d = 0 the plot shows the intensity distribution of two sources.

Copy your plot to a Microsoft Word document. For a single source the intensity is normalized to one. What is the intensity of the central maximum for two sources?

Turn on another source by setting c =1.

Copy your plot of the intensity distribution as a function of d for three sources to a Microsoft Word document. What is the intensity of the central maximum for three sources?

Turn on another source by setting d = 1.

	Copy your plot of the intensity distribution as a function of d for four sources to a Microsoft Word document. What is the intensity of the central maximum for four sources?
	In your own words, describe how the intensity distribution changes, as you add more equally-spaced sources.

Problem 2:.

Suppose light from a Helium-Neon laser (wavelength l = 633 nm) is expanded and collimated into a beam with a diameter of 2 cm, and then split into two beams that intersect as shown in the drawing below.

(a) If the full angle between the two overlapping beams is 10°, how many fringes appear in the overlapping region?
(b) When the angle between the overlapping beams is increased to 30°, does the fringe spacing in the overlap region increase or decrease? By how much?

Problem 3:. Explain!	Problem 4: Explain!
Problem 5: Explain!	Problem 6: Explain!
Problem 7: Explain!	Problem 8: Explain!