### Experiment 8

#### Light as a Wave (Diffraction Grating)

Objective:

The objectives are to (a) verify the wave nature of light by measuring its wavelength in an interference phenomenon, (b) learn about diffraction grating, and (c) measure the wavelengths of red and violet colors.

Equipment:

A diffraction grating, a laser pointer of known wavelength, an optical bench, a target holder, a skew clamp, a tape measure, a white light bulb (40-60 watts) with a holder, a ruler, and a calculator

Theory:

Young's Double-Slit Experiment verifies that light is a wave simply because of the bright and dark fringes that appear on a screen.  It is the constructive and destructive interference of light waves that cause such fringes.

Constructive Interference of Waves

The following two waves (Fig. 1) that have the same wavelength and go to maximum and minimum together are called coherent waves.  Coherent waves help each other's effect, add constructively, and cause constructive interference.  They form a bright fringe.

Destructive Interference of Waves

In Fig. 2 however, the situation is different.  When the wave with amplitude A1 is at its maximum, the wave with amplitude A2 is at its minimum and they work completely against each other resulting in a wave with amplitude A2 - A1.  These two completely out of phase waves interfere destructively.   If A2 = A1, they form a dark fringe.

The bright and dark fringes in Young's experiment follow these formulas:

Bright Fringes:    d sinθk = k λ    where k = 0, 1, 2, 3, ...

Dark Fringes:      d sinθk = (k - 1/2 ) λ    where   k = 1, 2, 3, ...

The above formulas are based on the following figures:

Check the following statement for correctness based on the above figure.

Light rays going to D2 from S1 and S2 are 3(0.5λ) out of phase (same as being 0.5λ out of phase) and therefore form a dark fringe.

Light rays going to B1 from S1 and S2 are 2(0.5λ) out of phase (same as being in phase) and therefore form a bright fringe.

Note that SBo is the centerline

Going from a dark or bright fringe to its next fringe changes the distance difference by 0.5λ.

Diffraction Grating:

Diffraction grating is a thin film of clear glass or plastic that has a large number of lines per (mm) drawn on it.  A typical grating has density of 250 lines/mm.  Using more expensive laser techniques, it is possible to create line densities of 3000 lines/mm or higher.  When light from a bright and small source passes through a diffraction grating, it generates a large number of sources at the grating.  The very thin space between every two adjacent lines of the grating becomes an independent source.  These sources are coherent sources meaning that they emit in phase waves with the same wavelength  These sources act independently such that each source sends out waves in all directions.  On a screen a distance D away, points can be found whose distance differences from these sources are different multiples of λ causing bright fringes.  One difference between the interference of many slits (diffraction grating) and double-slit (Young's Experiment) is that a diffraction grating makes a number of principle maxima along with with lower intensity maxima in between.  The principal maxima occur on both sides of the central maximum for which a formula similar to Young's formula holds true.

D = the distance from the grating to the screen

d = the spacing between every two lines (same as every two sources)

If there are N lines per mm of the grating, then d, the space between every two adjacent lines or (every two adjacent sources) is

The diffraction grating formula for the principal maxima is:

d sin θk = k λ     where   k = 1, 2, 3, ...

Procedure:

1. Determination of (Lines/mm) of the Diffraction Grating:

a)      Fix a laser pointer and the diffraction grating (placed in a target holder) on an optical bench as shown.  Try to make a distance D (grating to wall) of about 1.5m.

b)      Make sure that the direction of the optical bench is normal (at right angle) to the wall and that you are measuring the perpendicular distance D from the grating to the wall.

c)      Measure y1 ,  y2 , and D with the precision of mm and record the values.

d)      Angles θ1 and θ2  may now be calculated from the measured values as follows:

e)      Use the  tan-1 function (built-in in your calculator) to calculate θ1 and θ2 .

f )      Use angles θ1 and θ2 along with the wavelength given on your laser pointer (in meters) and the diffraction grating formula to calculate d, the distance between adjacent spaces (sources) on the grating.  Find d once on the basis of k = 1  and once on the basis of k =2 .   Theoretically, the two values you obtain for d must be equal; however, due to measurement errors, they might be slightly different.  Find an average value for d in meters.

g)      From d, determine N, the number of lines per mm of the grating.

1.  Red and Violet Wavelengths:

a)      Hold a diffraction grating close to your eye and look at the objects around you.

You will see a continuous spectrum of rainbow colors around bright objects.  The diffraction grating separates the colors of white light similar to what a prism does.  White light coming from a bright object separates into its constituent colors as it passes thru the grating and reaches your eyes.  If you are looking through a grating at a bright spot such as the filament of a lit light bulb, you will be able to direct another person to move to the left or right and mark the ends of the spectrum you are observing.  By measuring the distance between each end of the spectrum and the bright filament Yviolet or Yred and D the distance from the filament to the grating (held by you), it is possible to calculate the angles θviolet and θred Then, by using the formula

d sin θk = k λ ,

the corresponding wavelengths for violet and red light can be determined.

Note that through the grating you will see more than one rainbow band.  You will see two or three bands on each side of the center.  If you use the 1st band to one side of the center, then k = 1.  For the 2nd band k = 2, and for the 3rd band k = 3.

b)      Place the optical bench near the board in your lab or class on a somewhat high table.

c)      Make sure that the optical bench stays at right angle to the board and mount a light-bulb so that it almost touches the board. Turn the light bulb on.

d)    Hold a diffraction grating at a fixed distance D from the lit bulb. When you look into the grating, your line of sight must be normal to the board.  A diagram of the set-up is shown below:

where V (in the diagram) is the Violet End of the spectrum, and R the Red end of it.  Also BV is the same as Y1V , the distance from the bulb to the violet end of the first fringe.  Similarly, BR is the same as Y1R, the distance from the bulb to the red end of the first fringe.

a)      While looking into the grating and observing the spectrum, guide your partner to the extreme ends of the spectrum so that he/she can mark those points on the board.  Your partner must have previously observed the same spectrum and have a good understanding of the experimental procedure.

b)      When those points are marked, double-check their precision and measure distances BV and BR to the nearest cm as shown in the figure.  Also measure D.

c)      From the data collected, calculate angles θviolet and θred and use each in the above-mentioned formula separately to find the corresponding wavelengths.

Data:

Given:

λ laser= As read from the laser pointer.

Measured:

Part A:                               Part B:

Yk =                                   Y1V =

D =                                    Y1R    =

N =

D =

Calculations

Show all calculations.

Comparison of the Results:

Calculate a % error on each measurement.

Conclusion:

To be explained by students.

Discussion:

To be explained by students.