We now consider the case for diffraction with slit number > 2, which we describe as a diffraction grating. Basically, we are dealing with the case of multiple slit interference patterns produced and it is of interest to examine the differences, e.g. from the single slit case, more closely. The diagram (Fig. 1) below) is instructive:
In terms of practical applications, note that any arrangement of elements which is equivalent in its action to a number of parallel equidistant slits of the same width is called a diffraction grating. Not surprisingly, this feature makes the grating a powerful instrument for the application of spectral analysis, an aspect my 2nd year physics students enjoyed both at Harrison College and the Univ. of Alaska- Fairbanks.
In this case of a diffraction grating each of the slits gives rise to a separate diffracted beam – such as depicted in Fig. 2, but which are so narrow that the diffracted beam from each spreads out over a sufficiently wide angle to interfere with all other beams.
Fig. 2
Now,
if the angle q
is increased slightly, wave fronts from the grating elements no longer arrive
at AA’ in phase with each other. Note here that even a relatively small change
results in almost complete destructive interference. Hence, the maximum at each
q is an extremely sharp
one. If q is increased enough we
find another maximum (2nd order) where ab = 2l, cd = 4l, ef = 6l etc. The wave fronts at
AA’ are again in phase. It follows from this that we can write:
ab = ml
where m = 0, +1,+ 2, +3,, …… where m is the order of the diffraction grating.
Then
the condition for a maximum is: d sin q = ml/d
Or:
sin q = ml/d
To see an example of such dispersion - how it works to separate different wavelengths, consider a grating which has 5 x 10 5 lines per m, and monochromatic light of wavelength 6 x 10 -7 m is incident. Then we first obtain d (the spacing) by taking:
d = 1m/ 5 x 10 5 = 2 x 10 -6 m
The various dispersed angles are than obtained by substituting for m, i.e. m = 0, m = 1. m = 2 etc.
Then: q o = 0 o.
q1 = 17 o (e.g. sin q1 = 1 ( 6 x 10 -7 m)/ 2 x 10 -6 m )
q2 = 37 o (e.g. sin q2 = 2 ( 6 x 10 -7 m)/ 2 x 10 -6 m )
q 3 = 64 o (e.g. sin q3 = 3 ( 6 x 10 -7 m)/ 2 x 10 -6 m )
And a simple sketch of the grating image can show the dispersed wavelengths. Also that there is no q4 given that the result is unphysical. (I.e. this is impossible since no sine value for any angle can exceed 1).
Applying this condition across different orders m, one obtains the intensity graph shown below:
Example
Problem:
A grating with 1,000 lines/mm is illuminated by monochromatic light of wavelength 400 nm. How many diffraction images are produced?
Solution:
We have: l = 4 x 10-7 m and d = 1 mm/1,000 =
10 -3 m/ 1,000 = 10 -6 m
Then:
(10 - 6 m) sin q = = 4 x 10-7
m
Therefore, the first diffraction image is at:
sin q1 = 4 x 10-7 m / (10 -6 m) » 0.4
whence: arc sin (0.4) = q1 = 23o.6
The second diffraction image is at: sin q2 = 2l/d
And: sin q2 = 2(4 x 10-7 m ) / (10 -6 m) » 0.8
q2 = arc sin (0.8) = 53o.1
For third order:
sin
q3 = 3l/d =
3(4 x 10-7 m ) / (10 -6 m) » 1.2
But, this is impossible since no sine value for any angle can exceed 1. Hence, only two images are produced, i.e. of orders 1, 2.
Intensity Distribution for an Ideal Grating:
To obtain the intensity distribution formula for an ideal grating we employ the method of adding complex amplitudes. Then, denote the number of slits by N, and the phase change - from one slit to the next by d . The resulting complex amplitude is the sum:
A exp(i q) = a [ 1 + e id + e i2d + e i3d +....... exp(i(N-1) d)
To find the intensity we must then divide by the complex conjugate, so that:
A 2 = a 2 {(1 - cos Nd )/ (1 - cos d)
Using: 1 - cos (x) = 2 sin 2 (x/ 2) we write:
A 2 = a 2 { sin 2 (Nd /2)/ sin 2 (d /2)} = a 2 sin 2 Ng /sin 2(g )
Where - as in the double slit case- we let g = d/ 2 = ( p/ l) d sin q
Then a 2 represents the intensity diffracted by a single slit. After inserting its value:
a 2 = A o 2 (sin 2 b/ b 2)
And we get:
I = A o 2 (sin 2 b/ b 2) (sin 2 Ng /sin 2(g )
The intensity in a Fraunhofer pattern for an ideal grating. To find the minima for: (sin 2 Ng /sin 2(g ) we just note when the numerator = 0. This occurs at: Ng = 0, p, 2p, 3p, .....pp. The condition for a minimum is in general:
g = pp / N
Excluding p-values for which p = mN.
The p-values correspond to path differences:
d sin q = l/N, 2l/N, 3l/N, ........(N-1) l/N, (N + 1) l/N,
Omitting values: 0, N l/N, 2N l/N,......
Which apply to principal maxima.
Angular dispersion:
The separation of any two colors (wavelengths): l1 and l2 , increases with the order number. We express this separation using a quantity called the angular dispersion. It is obtained by differentiating the grating equation with respect to l. Then: d cos q dq = m d l
Or: dq = m / d cos q
Or, using finite differences:
Dq /D l = m / d cos q
This equation shows for a given small differences in wavelength D l , the angular separation Dq is directly proportional to m, and inversely proportional to the grating spacing d.
Suggested Problems:
1) A diffraction grating has 600 lines/ mm. Find the angle q for the second order maximum if yellow light of l = 550 nm is used.
2) Two spectral lines in a mixture of hydrogen (H2) and deuterium (D2) gas have wavelengths of 656.30 nm and 656.48 nm, respectively. What is the minimum number of lines a diffraction grating must have to resolve these two wavelengths to first order (m= 1)?
3) Two spectral lines at 6200 Å. (doublet) are separated by 0.652 Å. Find the minimum number of lines a diffraction grating must have to just resolve this doublet in the 2nd order spectrum.
Remember 1 Å = 10- 8 cm
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