Answer to Question #93248 in Optics for Joy

Question

In a double slit experiment, the wavelength of the light source is 405nm, the slit separation d is 19.44*10^-6m and the slits width a is 4.050*10^-6m. Consider the interference of the light from the two paths and also the diffraction of the light through h slits.
1. How many bright interference fringes are within the central peak of the diffraction envelope
2. How many bright fringes are within either of the first side peaks of the diffraction envelope

Answer 1

1. We can write for single slit

$a \sin θ = mλ (1)$

where a is the slit of width, m=1

Using (1) we got

$λ=a \sin θ (2)$

We can write for bright interference fringes

$d \sin θ = kλ (3)$

Using (3) we got

$k=\frac{ d \sin θ }{λ} (4)$

We put (2) in (4)

$k=\frac{ d \sin θ }{ a \sin θ }= \frac{ d }{ a } (5)$

Using (5) we got

k=4

The number of bright interference fringes is equal to

0, ±1, ±2, ±3, ±4

Answer

9

2. The first side diffraction maximum can be located by setting m=1.5.

Using (1) we got

$λ=\frac{ a \sin θ }{1.5} (6)$

We put (6) in (4)

$k=\frac{ 1.5 d \sin θ }{ a \sin θ }= \frac{ 1.5 d }{ a } (7)$

Using (7) we got

k=7

The number of bright interference fringes is equal to

0, ±1, ±2, ±3, ±4, ±5, ±6, ±7

Answer

15