# Fundamentals of photonics

1. Jan 14, 2016

### Amirjahi

1. The problem statement, all variables and given/known data

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 any one help me to solve this problem?
thank you

Last edited: Jan 14, 2016
2. Jan 14, 2016

### 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?

3. Jan 14, 2016

### Amirjahi

Fourier Transform Equations:
F(νxy)=∫−∞f(x,y)exp(−i2π(νxx+νyy))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(νxy)=∫−∞exp(−i2π(νxx+νyy)dxdy=δ(νx−0)δ(νy−0)
g(x,y)=∫−∞F(νxy)H(νxy)exp(+i2π(νxx+νyy))dνxdνy
g(x,y)=h0exp(iπ(x2+y2)/λd)δ(x/λd)δ(y/λd)

Last edited: Jan 14, 2016
4. Jan 14, 2016

### 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 $\textrm{FT}[df(x)/dx] = ik_x\textrm{FT}[f(x)]$.

5. Jan 14, 2016

### Amirjahi

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

6. Jan 14, 2016

### 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?

Last edited: Jan 14, 2016
7. Jan 14, 2016

### Amirjahi

8. Jan 14, 2016

### 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.

9. Jan 14, 2016

### Amirjahi

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

10. Jan 14, 2016

### Amirjahi

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

11. Jan 20, 2016

### Amirjahi

hi

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

12. Jan 20, 2016