Optics Question (I think about Fresnel-Kirchoff)

[rabbit44]

Homework Statement

A laser produces a beam of coherent light of wavelength lambda with plane
wave-fronts traveling along z. The amplitude of light transmitted by a trans-
parency in the x; y plane is independent of y and varies with x as

A[1 + cos(2*pi*x/d)]

where A is a constant and d > lambda. Show that three beams emerge
from the transparency, and find their angles to the z-axis.

Homework Equations

I think:

A(B) ~ int[u(x) exp(-iBx)]

Where B=k*sin(theta), u is the amplitude distribution given and A(B) is the amplitude at some theta.

The Attempt at a Solution

So I tried using that equation, hoping that only 3 values of theta would be valid or something, but it just didn't work at all. Am I completely off?

Thanks

 

[Jhero]

for your question and attempt at a solution. Let's work through this problem together.

First, let's start with the given amplitude distribution:

A[1 + cos(2*pi*x/d)]

We can rewrite this as:

A*cos^2(pi*x/d)

This tells us that the amplitude of the light is varying sinusoidally with x, with a maximum amplitude of A and a minimum amplitude of 0. This makes sense, as the cosine function varies between 1 and -1.

Next, let's consider the equation you provided:

A(B) ~ int[u(x) exp(-iBx)]

This equation represents the Fourier transform of the amplitude distribution, where u(x) is the amplitude distribution and B is the spatial frequency. We can use this equation to find the angles at which the three beams will emerge from the transparency.

First, we need to determine the spatial frequency, B. We know that B=k*sin(theta), where k=2*pi/lambda and theta is the angle of the beam with respect to the z-axis. So we can rewrite the equation as:

A(B) ~ int[u(x) exp(-2*pi*i*sin(theta)*x/lambda)]

Now, we can use the inverse Fourier transform to find the angle at which the three beams will emerge. The inverse Fourier transform is given by:

u(x) = (1/2*pi) * int[A(B) * exp(2*pi*i*sin(theta)*x/lambda) * d(theta)]

So we can plug in our values and solve for theta:

u(x) = (1/2*pi) * int[A*cos^2(pi*x/d) * exp(2*pi*i*sin(theta)*x/lambda) * d(theta)]

We can simplify this equation by using the trigonometric identity cos^2(x) = (1+cos(2x))/2:

u(x) = (1/4*pi) * int[A*(1+cos(2*pi*x/d)) * exp(2*pi*i*sin(theta)*x/lambda) * d(theta)]

Now, we can use the following property of the Fourier transform:

exp(2*pi*i*sin(theta)*x/lambda) = delta(x-(lambda/2*pi)*sin(theta))

Where delta(x) is the Dirac delta function. This tells us that the amplitude at a certain angle theta will only contribute at a certain value of x, which is given by (lambda/2*pi)*sin(theta).