Question

2. Laser light with wavelength 1 = 600 nm is sent through a screen with N narrow slits spaced a apart from each other (e.g. if there are 3 slits, they are at 0, a, 2a). You observe the resulting diffraction pattern on a screen placed far (meters) away. The screen is curved, so that the distance of the screen to the slits is constant with angle. This simplification means you don't have to take into account longer paths for larger angles or other geometric concerns. a) If N = 2, and a = 1200nm, write down a formula and draw a graph of the intensity vs. angle for -90 <0 <90. How many bright spots (points of maximal intensity) are there? b) If N = 2, what is the smallest value of a for which there is more than 1 bright spot? c) If N = 2, what is the smallest value of a for which there is at least 1 completely dark spot? d) If N = 100, what is the smallest value of a for which there is more than 1 bright spot, and what is the smallest value of a for which there is at least 1 completely dark spot?

Answer 1

Please verify my answer with solutions given or with a teacher for any errors or typos that may have crept in. I have assumed the student is acquainted with the basic principles, rules and formulas of wave optics.

a) we have for bright fringes,

where N is the number of slits, a is the space between adjacent slits, lambda is the wavelength

asing = NA

The graph for Intensity vs angle is as follows

A Intensity NVIVW N/a 21/a -2N/a N/a Diffraction angle

for three slits, there will be 3 maxima or bright points.

b) the maximum angle can be

006 = 0

and for very small angle

sino e

therefore $\theta = \frac{2\lambda}{a}$

and number of fringes

n = sin90) sino = 1/6

n=a/2

for Maxima, number of bright fringes is 2[n] + 1, therefore for more than 1 bright spot, n>=1/2

therefore

a/2x > 0.5 rzo

therefore minimum value of a has to be a= 600nm

c) for minima,

asint = (N+1/2)

N + 1/2 = 2 + 1/2 = 5/2

as previous again,

2.52 A =

number of fringes

n=a/2.54

therefore for 1 complete dark spot, 2[n]-1, i.e. 0<n<1 {since [n] is the greatest integer function i.e. if n=0.2, [n]=1}

0<a/2.5X < 1 0<a < 2.51 0<a < 1500nm

d) for N=100

for bright spot

urtog <D YOOID 90 < KOOTD

for dark spot

0 <α/100.5λ <1 0 <α< 100.5λ Ο <α < 60.3μm

I have tried to keep the answer as detailed and to the point as possible. Please provide an UPVOTE if it clears your query/question.