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., $mathrm{FT}{t_g(x)}$.
fourier-transform convolution dirac-delta
asked Jan 14 at 17:32
CDTCDT
84
$endgroup$
add a comment |
$begingroup$
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., $mathrm{FT}{t_g(x)}$.
fourier-transform convolution dirac-delta
asked Jan 14 at 17:32
CDTCDT
84
$endgroup$
add a comment |
$begingroup$
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., $mathrm{FT}{t_g(x)}$.
fourier-transform convolution dirac-delta
asked Jan 14 at 17:32
CDTCDT
84
$endgroup$
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
fourier-transform convolution dirac-delta
asked Jan 14 at 17:32
CDTCDT
84
asked Jan 14 at 17:32
CDTCDT
84
asked Jan 14 at 17:32
CDTCDT
84
asked Jan 14 at 17:32
CDTCDT
84
asked Jan 14 at 17:32
CDTCDT
84
84
add a comment |
add a comment |
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