How to Use Harmonic Analysis to Determine g(x,y) in Different Cases?

[Amirjahi]

Homework Statement

The complex amplitudes of a monochromatic wave of wavelength λ in the z=0 and z=d planes are f(x,y) and g(x,y), resprctively. Assume d=10⁴λ, use harmonic analysis to determine g(x,y) in the following cases:

(a) f(x,y) = exp{(-jπ/λ)(x+y)}
(b) f(x,y) = cos(πx/2λ)
(c) f(x,y)=cos²(πy/2λ)

can anyone help me to solve this problem?
thank you

 

[blue_leaf77]

The other two sections of the template you have deleted are compulsory before you can hope to receive help. What equations in harmonic analysis, and additionally about the propagation of light, do you already know?

 

[Amirjahi]

Fourier Transform Equations:
F(ν_x,ν_y)=∫^∞_−∞f(x,y)exp(−i2π(ν_xx+ν_yy))dxdy

Transfer Function of Free Space (Fraunhofer Approximation):
g(x,y)=h₀exp(iπ(x²+y²)/λd)F(x/λd,y/λd)
3. The Attempt at a Solution
f(x,y)=1
F(ν_x,ν_y)=∫^∞_−∞exp(−i2π(ν_xx+ν_yy)dxdy=δ(ν_x−0)δ(ν_y−0)
g(x,y)=∫^∞_−∞F(ν_x,ν_y)H(ν_x,ν_y)exp(+i2π(ν_xx+ν_yy))dνxdνy
g(x,y)=h₀exp(iπ(x²+y²)/λd)δ(x/λd)δ(y/λd)

 

[blue_leaf77]

Good start. Unfortunately, the objects (the fields in the input plane) are defined for all spaces, i.e. both $x$ and $y$ run from $-\infty$ to $\infty$. In this case, the condition necessary for Fraunhofer approximation cannot be satisfied. Instead start with the Helmholtz equation in free space
$$
\nabla^2 E(x,y,z) + k^2 E(x,y,z) = 0
$$
with $k = 2\pi/\lambda$. Now apply Fourier transform on $E(x,y,z)$ to $x$ and $y$ only, so that in the end you will obtain a function $\tilde{E}(k_x,k_y,z)$. Hint: use the rule $\textrm{FT}[df(x)/dx] = ik_x\textrm{FT}[f(x)]$.

 

[Amirjahi]

Good start. Unfortunately, the objects (the fields in the input plane) are defined for all spaces, i.e. both $x$ and $y$ run from $-\infty$ to $\infty$. In this case, the condition necessary for Fraunhofer approximation cannot be satisfied. Start with the Helmholtz equation in free space
$$
\nabla^2 E(x,y,z) + k^2 E(x,y,z) = 0
$$
with $k = 2\pi/\lambda$. Now apply Fourier transform on $E(x,y,z)$ to $x$ and $y$ only, so that in the end you will obtain a function $\tilde{E}(k_x,k_y,z)$. Hint: use the rule $\textrm{FT}[df(x)/dx] = ik_x\textrm{FT}[f(x)]$.

yes i know that, but i want a solution of one of the cases above for better understand

 

[blue_leaf77]

Do you mean one of the (a), (b), and (c)? All of $f(x,y)$'s given in each case extends from minus infinity to plus infinity in both $x$ and $y$ directions, Fraunhofer approximation is not applicable here as it requires that $a^2/(z\lambda) << 0.1$ with $a$ the size of the input field. For instance, look at your previous work, in the input plane you have plane wave $f(x,y) = 1$, but at $z=d$ you got a point as your calculation resulted in an expression proportional a delta function. Does it look make sense to you?

 

[Amirjahi]

yes i mean one of a , b or c .
the same problem post recently , in the link below
https://www.physicsforums.com/threads/optical-fourier-transform-for-propagation.836793/
thank you

 

[blue_leaf77]

The right way to solve it is by the use of angular spectrum method. In case you are not familiar with it, start with the suggestion I pointed out in post #4. I just want to stress again, that Fraunhofer approximation will not work here.

 

[Amirjahi]

ok thank you for your help, I will try to solve it if possible

 

[Amirjahi]

The right way to solve it is by the use of angular spectrum method. In case you are not familiar with it, start with the suggestion I pointed out in post #4. I just want to stress again, that Fraunhofer approximation will not work here.

ok thank you for your help, I will try to solve it if possible

 

[Amirjahi]

ok thank you for your help, I will try to solve it if possible

hi

can i ask you the solution of FT[ f(x,y) = exp{(-jπ/λ)(x+y)}] ??

 

[blue_leaf77]

Take a look at this http://mathworld.wolfram.com/FourierTransformDeltaFunction.html.