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Fundamentals of photonics

  1. Jan 14, 2016 #1
    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. jcsd
  3. Jan 14, 2016 #2

    blue_leaf77

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    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?
     
  4. Jan 14, 2016 #3
    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
  5. Jan 14, 2016 #4

    blue_leaf77

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    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)]##.
     
  6. Jan 14, 2016 #5
    yes i know that, but i want a solution of one of the cases above for better understand
     
  7. Jan 14, 2016 #6

    blue_leaf77

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    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
  8. Jan 14, 2016 #7
  9. Jan 14, 2016 #8

    blue_leaf77

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    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.
     
  10. Jan 14, 2016 #9
    ok thank you for your help, I will try to solve it if possible
     
  11. Jan 14, 2016 #10
    ok thank you for your help, I will try to solve it if possible
     
  12. Jan 20, 2016 #11
    hi

    can i ask you the solution of FT[ f(x,y) = exp{(-jπ/λ)(x+y)}] ??
     
  13. Jan 20, 2016 #12

    blue_leaf77

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