Fourier transform of the convolution of a Dirac comb with the product of a complex exponential function and a...












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Straight to the math question: How can I calculate the following 1-dimensional spatial Fourier transform?



$frac{1}{2pi}int_{-infty}^{infty}left(e^{i(n-1)k_0frac{x^2}{2R}}mathrm{rect}(x/w)*sum_{l=-infty}^{infty}delta(x-lw)right)e^{-ik_xx}mathrm{d}x$



I will be very glad about an analytical solution, but also about any numerical implementation (Mathematica or python).



Parameters:



$n = 1.5$



$k_0 = 2pi/lambda = 2pi/(532times10^{-9}mathrm{m})$



$w = 43times10^{-6}mathrm{m}$



$R = 46.67times10^{-6}mathrm{m}$



Approach: I know that $mathrm{FT}{f*g} = mathrm{FT}{f}mathrm{FT}{g}$. Therefore I could calculate the Fourier transform of the Dirac comb, which gives me again a Dirac comb but of inverse frequency:



$mathrm{FT}{sum_{l=-infty}^{infty}delta(x-lw)} = frac{1}{w}sum_{l=-infty}^{infty}delta(k_{x}-frac{2pi l}{w})$



For the $e^{i(n-1)k_0frac{x^2}{2R}}mathrm{rect}(x/w)$ term I am not sure whether it is easier to evaluate the Fourier transform of the product or as the convolution of the Fourier transforms of each multiplicant. I know that the Fourier transform of a rect-function is a sinc:



$mathrm{FT}{mathrm{rect}(x/w)} = frac{w/2}{pi}mathrm{sinc}(k_xfrac{w}{2})$



I am not sure about how to evaluate the Fourier transform of the complex exponential term, let alone how to convolute the result with the sinc-function.



Background: My aim is to find the far-field diffraction pattern (Fraunhofer approximation) of a periodic grating structure consisting of thin planoconvex lenses (the grating lies in the xy-plane and is invariant along the y-direction). The intensity of the diffraction pattern will be proportional to the absolute square of the Fourier transform of the complex transmission function $t_g(x)$ of my grating.



The complex transmission function of a single grating element can be modelled as



$t(x) = e^{-ik_0(n-1)frac{x^2}{2R}} times mathrm{rect}(x/w)$,



where $k_0$ is the free space wave vector, $n$ the index of refraction of the grating, and $R$ is the radius of the circle that a single lens forms the cap of. The $mathrm{rect}(x/w)$-function restricts the complex exponential term to the width $w$ of a single grating element (i.e., a single thin lens). In order to account for the periodicity of the grating, we need to convolute $t(x)$ with a Dirac comb:



$t_g(x) = t(x)*sum_{l=-infty}^{+infty}delta(x-lw)$



What I want to find is now the Fourier transform of this transmission function, i.e., $mathrm{FT}{t_g(x)}$.










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$endgroup$

















    0












    $begingroup$


    Straight to the math question: How can I calculate the following 1-dimensional spatial Fourier transform?



    $frac{1}{2pi}int_{-infty}^{infty}left(e^{i(n-1)k_0frac{x^2}{2R}}mathrm{rect}(x/w)*sum_{l=-infty}^{infty}delta(x-lw)right)e^{-ik_xx}mathrm{d}x$



    I will be very glad about an analytical solution, but also about any numerical implementation (Mathematica or python).



    Parameters:



    $n = 1.5$



    $k_0 = 2pi/lambda = 2pi/(532times10^{-9}mathrm{m})$



    $w = 43times10^{-6}mathrm{m}$



    $R = 46.67times10^{-6}mathrm{m}$



    Approach: I know that $mathrm{FT}{f*g} = mathrm{FT}{f}mathrm{FT}{g}$. Therefore I could calculate the Fourier transform of the Dirac comb, which gives me again a Dirac comb but of inverse frequency:



    $mathrm{FT}{sum_{l=-infty}^{infty}delta(x-lw)} = frac{1}{w}sum_{l=-infty}^{infty}delta(k_{x}-frac{2pi l}{w})$



    For the $e^{i(n-1)k_0frac{x^2}{2R}}mathrm{rect}(x/w)$ term I am not sure whether it is easier to evaluate the Fourier transform of the product or as the convolution of the Fourier transforms of each multiplicant. I know that the Fourier transform of a rect-function is a sinc:



    $mathrm{FT}{mathrm{rect}(x/w)} = frac{w/2}{pi}mathrm{sinc}(k_xfrac{w}{2})$



    I am not sure about how to evaluate the Fourier transform of the complex exponential term, let alone how to convolute the result with the sinc-function.



    Background: My aim is to find the far-field diffraction pattern (Fraunhofer approximation) of a periodic grating structure consisting of thin planoconvex lenses (the grating lies in the xy-plane and is invariant along the y-direction). The intensity of the diffraction pattern will be proportional to the absolute square of the Fourier transform of the complex transmission function $t_g(x)$ of my grating.



    The complex transmission function of a single grating element can be modelled as



    $t(x) = e^{-ik_0(n-1)frac{x^2}{2R}} times mathrm{rect}(x/w)$,



    where $k_0$ is the free space wave vector, $n$ the index of refraction of the grating, and $R$ is the radius of the circle that a single lens forms the cap of. The $mathrm{rect}(x/w)$-function restricts the complex exponential term to the width $w$ of a single grating element (i.e., a single thin lens). In order to account for the periodicity of the grating, we need to convolute $t(x)$ with a Dirac comb:



    $t_g(x) = t(x)*sum_{l=-infty}^{+infty}delta(x-lw)$



    What I want to find is now the Fourier transform of this transmission function, i.e., $mathrm{FT}{t_g(x)}$.










    share|cite|improve this question









    $endgroup$















      0












      0








      0


      1



      $begingroup$


      Straight to the math question: How can I calculate the following 1-dimensional spatial Fourier transform?



      $frac{1}{2pi}int_{-infty}^{infty}left(e^{i(n-1)k_0frac{x^2}{2R}}mathrm{rect}(x/w)*sum_{l=-infty}^{infty}delta(x-lw)right)e^{-ik_xx}mathrm{d}x$



      I will be very glad about an analytical solution, but also about any numerical implementation (Mathematica or python).



      Parameters:



      $n = 1.5$



      $k_0 = 2pi/lambda = 2pi/(532times10^{-9}mathrm{m})$



      $w = 43times10^{-6}mathrm{m}$



      $R = 46.67times10^{-6}mathrm{m}$



      Approach: I know that $mathrm{FT}{f*g} = mathrm{FT}{f}mathrm{FT}{g}$. Therefore I could calculate the Fourier transform of the Dirac comb, which gives me again a Dirac comb but of inverse frequency:



      $mathrm{FT}{sum_{l=-infty}^{infty}delta(x-lw)} = frac{1}{w}sum_{l=-infty}^{infty}delta(k_{x}-frac{2pi l}{w})$



      For the $e^{i(n-1)k_0frac{x^2}{2R}}mathrm{rect}(x/w)$ term I am not sure whether it is easier to evaluate the Fourier transform of the product or as the convolution of the Fourier transforms of each multiplicant. I know that the Fourier transform of a rect-function is a sinc:



      $mathrm{FT}{mathrm{rect}(x/w)} = frac{w/2}{pi}mathrm{sinc}(k_xfrac{w}{2})$



      I am not sure about how to evaluate the Fourier transform of the complex exponential term, let alone how to convolute the result with the sinc-function.



      Background: My aim is to find the far-field diffraction pattern (Fraunhofer approximation) of a periodic grating structure consisting of thin planoconvex lenses (the grating lies in the xy-plane and is invariant along the y-direction). The intensity of the diffraction pattern will be proportional to the absolute square of the Fourier transform of the complex transmission function $t_g(x)$ of my grating.



      The complex transmission function of a single grating element can be modelled as



      $t(x) = e^{-ik_0(n-1)frac{x^2}{2R}} times mathrm{rect}(x/w)$,



      where $k_0$ is the free space wave vector, $n$ the index of refraction of the grating, and $R$ is the radius of the circle that a single lens forms the cap of. The $mathrm{rect}(x/w)$-function restricts the complex exponential term to the width $w$ of a single grating element (i.e., a single thin lens). In order to account for the periodicity of the grating, we need to convolute $t(x)$ with a Dirac comb:



      $t_g(x) = t(x)*sum_{l=-infty}^{+infty}delta(x-lw)$



      What I want to find is now the Fourier transform of this transmission function, i.e., $mathrm{FT}{t_g(x)}$.










      share|cite|improve this question









      $endgroup$




      Straight to the math question: How can I calculate the following 1-dimensional spatial Fourier transform?



      $frac{1}{2pi}int_{-infty}^{infty}left(e^{i(n-1)k_0frac{x^2}{2R}}mathrm{rect}(x/w)*sum_{l=-infty}^{infty}delta(x-lw)right)e^{-ik_xx}mathrm{d}x$



      I will be very glad about an analytical solution, but also about any numerical implementation (Mathematica or python).



      Parameters:



      $n = 1.5$



      $k_0 = 2pi/lambda = 2pi/(532times10^{-9}mathrm{m})$



      $w = 43times10^{-6}mathrm{m}$



      $R = 46.67times10^{-6}mathrm{m}$



      Approach: I know that $mathrm{FT}{f*g} = mathrm{FT}{f}mathrm{FT}{g}$. Therefore I could calculate the Fourier transform of the Dirac comb, which gives me again a Dirac comb but of inverse frequency:



      $mathrm{FT}{sum_{l=-infty}^{infty}delta(x-lw)} = frac{1}{w}sum_{l=-infty}^{infty}delta(k_{x}-frac{2pi l}{w})$



      For the $e^{i(n-1)k_0frac{x^2}{2R}}mathrm{rect}(x/w)$ term I am not sure whether it is easier to evaluate the Fourier transform of the product or as the convolution of the Fourier transforms of each multiplicant. I know that the Fourier transform of a rect-function is a sinc:



      $mathrm{FT}{mathrm{rect}(x/w)} = frac{w/2}{pi}mathrm{sinc}(k_xfrac{w}{2})$



      I am not sure about how to evaluate the Fourier transform of the complex exponential term, let alone how to convolute the result with the sinc-function.



      Background: My aim is to find the far-field diffraction pattern (Fraunhofer approximation) of a periodic grating structure consisting of thin planoconvex lenses (the grating lies in the xy-plane and is invariant along the y-direction). The intensity of the diffraction pattern will be proportional to the absolute square of the Fourier transform of the complex transmission function $t_g(x)$ of my grating.



      The complex transmission function of a single grating element can be modelled as



      $t(x) = e^{-ik_0(n-1)frac{x^2}{2R}} times mathrm{rect}(x/w)$,



      where $k_0$ is the free space wave vector, $n$ the index of refraction of the grating, and $R$ is the radius of the circle that a single lens forms the cap of. The $mathrm{rect}(x/w)$-function restricts the complex exponential term to the width $w$ of a single grating element (i.e., a single thin lens). In order to account for the periodicity of the grating, we need to convolute $t(x)$ with a Dirac comb:



      $t_g(x) = t(x)*sum_{l=-infty}^{+infty}delta(x-lw)$



      What I want to find is now the Fourier transform of this transmission function, i.e., $mathrm{FT}{t_g(x)}$.







      fourier-transform convolution dirac-delta






      share|cite|improve this question













      share|cite|improve this question











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      asked Jan 14 at 17:32









      CDTCDT

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