Physics 4510 Optics
Homework Assignment Problem Set #5
(Turn in before class by Dec 8th, no late homework will be accepted)
1. (15%) Prove that the secondary maxima of a single-slit diffraction pattern occur
when β = tan β .
2. (15%) A single-slit Fraunhofer diffraction pattern is formed using white light. For
what wavelength does the second minimum coincide with the third minimum
produced by a wavelength of λ = 400 nm.
3. (20%) For an apodized slit which has a transmission function of
⎡ ⎛ y ⎞2 ⎤
1 2
f ( y) =
exp ⎢− 2⎜ ⎟ ⎥ ,
a π
⎣⎢ ⎝ a ⎠ ⎦⎥
proof that the Fraunhofer diffraction pattern is
∞
k 2a2
)
E ∝ ∫ f ( y )e iky dy = exp(−
−∞
8
by using the relation
∫
∞
e −u du = π .
2
−∞
Note that the diffraction pattern has no sidebands.
4. (15%) Consider a double slits aperture, each slit has a width b, and are separated
by a distance h. Prove that the intensity of the Fraunhofer diffraction pattern is
sin β 2
I = I0 (
) cos 2 γ
β
where β =
1
1
kb sin θ and γ = kh sin θ .
2
2
5. (15%) Continue on problem #4, the first term of the diffraction pattern of a
sin β 2
) typically accounts for diffraction pattern of a single
double slits aperture (
β
slit, and the cos γ terms corresponds to the interference between the two slits.
Missing orders occur at those values of sin θ which satisfy, at the same time, the
condition for interference maxima and the condition for diffraction minima. Show
h
that this leads to the condition ( ) = integer.
b
2
6. (20%) Plot the diffraction pattern I (θ ) of the double slits aperture for θ from 180o to 180o in Problem #5, where I 0 = 1 , b = 1µm , h = 3µm and the optical
wavelength λ = 632nm .
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--
% Problem #6
% plotting the double slit diffraction pattern
clear all;
theta_angle = -180:.01:180;
theta = theta_angle/180*pi;
lambda = 632e-9;
b=1e-6;
h=3e-6;
k=2*pi/lambda
beta = .5*k*b*sin(theta);
gamma = .5*k*h*sin(theta);
I=(sin(beta)./beta.*cos(gamma)).^2;
plot(theta_angle, I);
xlabel('Angle');
ylabel('Diffraction Pattern');
1
0.9
0.8
Diffraction Pattern
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-200
-150
-100
-50
0
Angle
50
100
150
200