Question about the derivation of the intensity formula of a diffraction gratingSimulating the Interference Pattern of Fraunhofer Diffraction by a Single SlitDiffraction Grating Spectrometry QuestionCovering centeremost slit of a N slit diffraction grating - what happens?Resolving power of a diffraction grating?Optics Diffraction Grating PlotTotal number of primary maxima in diffraction gratingDeriving formula for effect of slit width and multiplicity for multi-slit diffraction patternIntensity at $theta = 0$ in slit diffractionBasic, intuitive explanation for a diffraction gratingIntensity function of diffraction confusion
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Question about the derivation of the intensity formula of a diffraction grating
Simulating the Interference Pattern of Fraunhofer Diffraction by a Single SlitDiffraction Grating Spectrometry QuestionCovering centeremost slit of a N slit diffraction grating - what happens?Resolving power of a diffraction grating?Optics Diffraction Grating PlotTotal number of primary maxima in diffraction gratingDeriving formula for effect of slit width and multiplicity for multi-slit diffraction patternIntensity at $theta = 0$ in slit diffractionBasic, intuitive explanation for a diffraction gratingIntensity function of diffraction confusion
$begingroup$
In the notes I have, they have a diffraction grating with $2N + 1$ slits, a slit width of $2a$ and a slit spacing of $d$. They then say that the equation for the diffraction intensity pattern is given by:
$$I = I_0
left( fracsin((N+frac12)kdsintheta)sin(frac12kdsintheta) right)^2
left( fracsin(kasintheta)kasintheta right)^2
$$
They don't, however, give any proof or reason why this is the formula. I have been looking online for a way to justify this formula but I cant find anything. Anybody have a nice proof for this?
waves diffraction
edited 5 hours ago
Thomas Fritsch
1,546515
asked 7 hours ago
A. PavlenkoA. Pavlenko
324
$endgroup$
add a comment |
$begingroup$
In the notes I have, they have a diffraction grating with $2N + 1$ slits, a slit width of $2a$ and a slit spacing of $d$. They then say that the equation for the diffraction intensity pattern is given by:
$$I = I_0
left( fracsin((N+frac12)kdsintheta)sin(frac12kdsintheta) right)^2
left( fracsin(kasintheta)kasintheta right)^2
$$
They don't, however, give any proof or reason why this is the formula. I have been looking online for a way to justify this formula but I cant find anything. Anybody have a nice proof for this?
waves diffraction
edited 5 hours ago
Thomas Fritsch
1,546515
asked 7 hours ago
A. PavlenkoA. Pavlenko
324
$endgroup$
add a comment |
$begingroup$
In the notes I have, they have a diffraction grating with $2N + 1$ slits, a slit width of $2a$ and a slit spacing of $d$. They then say that the equation for the diffraction intensity pattern is given by:
$$I = I_0
left( fracsin((N+frac12)kdsintheta)sin(frac12kdsintheta) right)^2
left( fracsin(kasintheta)kasintheta right)^2
$$
They don't, however, give any proof or reason why this is the formula. I have been looking online for a way to justify this formula but I cant find anything. Anybody have a nice proof for this?
waves diffraction
edited 5 hours ago
Thomas Fritsch
1,546515
asked 7 hours ago
A. PavlenkoA. Pavlenko
324
$endgroup$
In the notes I have, they have a diffraction grating with $2N + 1$ slits, a slit width of $2a$ and a slit spacing of $d$. They then say that the equation for the diffraction intensity pattern is given by:
$$I = I_0
left( fracsin((N+frac12)kdsintheta)sin(frac12kdsintheta) right)^2
left( fracsin(kasintheta)kasintheta right)^2
$$
They don't, however, give any proof or reason why this is the formula. I have been looking online for a way to justify this formula but I cant find anything. Anybody have a nice proof for this?
waves diffraction
waves diffraction
edited 5 hours ago
Thomas Fritsch
1,546515
asked 7 hours ago
A. PavlenkoA. Pavlenko
324
edited 5 hours ago
Thomas Fritsch
1,546515
asked 7 hours ago
A. PavlenkoA. Pavlenko
324
edited 5 hours ago
Thomas Fritsch
1,546515
edited 5 hours ago
Thomas Fritsch
1,546515
edited 5 hours ago
Thomas Fritsch
1,546515
1,546515
asked 7 hours ago
A. PavlenkoA. Pavlenko
324
asked 7 hours ago
A. PavlenkoA. Pavlenko
324
asked 7 hours ago
A. PavlenkoA. Pavlenko
324
324
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
(image from Antonine education)
The light amplitude $E(theta)$ into direction $theta$ can be calculated
straight-forward by summing the contributions
- of all the slits ($n$ from $-N$ to $+N$)
- and of the parts of each individual slit ($x$ from $-a$ to $+a$)
The path difference of each contributing ray
(compared to the path length of the ray originating from the center of the grating)
is $(nd+x)sintheta$.
And hence its phase is $k(nd+x)sintheta$.
Summing these contributions you get
$$
beginalign
E(theta)
&= E_0 sum_n=-N^+N int_-a^+a e^ik(nd+x)sintheta textdx \
&= E_0 left( sum_n=-N^+N e^ikndsinthetaright)
left( int_-a^+a e^ikxsintheta textdx right) \
&= E_0 left( fracsin((N+frac12)kdsintheta)sin(frac12kdsintheta) right)
left( 2afracsin(kasintheta)kasintheta right)
endalign
$$
And finally you get the intensity by taking the absolute square of the amplitude
$$I(theta) = |E(theta)|^2$$
answered 7 hours ago
Thomas FritschThomas Fritsch
1,546515
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
(image from Antonine education)
The light amplitude $E(theta)$ into direction $theta$ can be calculated
straight-forward by summing the contributions
- of all the slits ($n$ from $-N$ to $+N$)
- and of the parts of each individual slit ($x$ from $-a$ to $+a$)
The path difference of each contributing ray
(compared to the path length of the ray originating from the center of the grating)
is $(nd+x)sintheta$.
And hence its phase is $k(nd+x)sintheta$.
Summing these contributions you get
$$
beginalign
E(theta)
&= E_0 sum_n=-N^+N int_-a^+a e^ik(nd+x)sintheta textdx \
&= E_0 left( sum_n=-N^+N e^ikndsinthetaright)
left( int_-a^+a e^ikxsintheta textdx right) \
&= E_0 left( fracsin((N+frac12)kdsintheta)sin(frac12kdsintheta) right)
left( 2afracsin(kasintheta)kasintheta right)
endalign
$$
And finally you get the intensity by taking the absolute square of the amplitude
$$I(theta) = |E(theta)|^2$$
answered 7 hours ago
Thomas FritschThomas Fritsch
1,546515
$endgroup$
add a comment |
$begingroup$
(image from Antonine education)
The light amplitude $E(theta)$ into direction $theta$ can be calculated
straight-forward by summing the contributions
- of all the slits ($n$ from $-N$ to $+N$)
- and of the parts of each individual slit ($x$ from $-a$ to $+a$)
The path difference of each contributing ray
(compared to the path length of the ray originating from the center of the grating)
is $(nd+x)sintheta$.
And hence its phase is $k(nd+x)sintheta$.
Summing these contributions you get
$$
beginalign
E(theta)
&= E_0 sum_n=-N^+N int_-a^+a e^ik(nd+x)sintheta textdx \
&= E_0 left( sum_n=-N^+N e^ikndsinthetaright)
left( int_-a^+a e^ikxsintheta textdx right) \
&= E_0 left( fracsin((N+frac12)kdsintheta)sin(frac12kdsintheta) right)
left( 2afracsin(kasintheta)kasintheta right)
endalign
$$
And finally you get the intensity by taking the absolute square of the amplitude
$$I(theta) = |E(theta)|^2$$
answered 7 hours ago
Thomas FritschThomas Fritsch
1,546515
$endgroup$
add a comment |
$begingroup$
(image from Antonine education)
The light amplitude $E(theta)$ into direction $theta$ can be calculated
straight-forward by summing the contributions
- of all the slits ($n$ from $-N$ to $+N$)
- and of the parts of each individual slit ($x$ from $-a$ to $+a$)
The path difference of each contributing ray
(compared to the path length of the ray originating from the center of the grating)
is $(nd+x)sintheta$.
And hence its phase is $k(nd+x)sintheta$.
Summing these contributions you get
$$
beginalign
E(theta)
&= E_0 sum_n=-N^+N int_-a^+a e^ik(nd+x)sintheta textdx \
&= E_0 left( sum_n=-N^+N e^ikndsinthetaright)
left( int_-a^+a e^ikxsintheta textdx right) \
&= E_0 left( fracsin((N+frac12)kdsintheta)sin(frac12kdsintheta) right)
left( 2afracsin(kasintheta)kasintheta right)
endalign
$$
And finally you get the intensity by taking the absolute square of the amplitude
$$I(theta) = |E(theta)|^2$$
answered 7 hours ago
Thomas FritschThomas Fritsch
1,546515
$endgroup$
(image from Antonine education)
The light amplitude $E(theta)$ into direction $theta$ can be calculated
straight-forward by summing the contributions
- of all the slits ($n$ from $-N$ to $+N$)
- and of the parts of each individual slit ($x$ from $-a$ to $+a$)
The path difference of each contributing ray
(compared to the path length of the ray originating from the center of the grating)
is $(nd+x)sintheta$.
And hence its phase is $k(nd+x)sintheta$.
Summing these contributions you get
$$
beginalign
E(theta)
&= E_0 sum_n=-N^+N int_-a^+a e^ik(nd+x)sintheta textdx \
&= E_0 left( sum_n=-N^+N e^ikndsinthetaright)
left( int_-a^+a e^ikxsintheta textdx right) \
&= E_0 left( fracsin((N+frac12)kdsintheta)sin(frac12kdsintheta) right)
left( 2afracsin(kasintheta)kasintheta right)
endalign
$$
And finally you get the intensity by taking the absolute square of the amplitude
$$I(theta) = |E(theta)|^2$$
answered 7 hours ago
Thomas FritschThomas Fritsch
1,546515
edited 1 hour ago
answered 7 hours ago
Thomas FritschThomas Fritsch
1,546515
answered 7 hours ago
Thomas FritschThomas Fritsch
1,546515
answered 7 hours ago
Thomas FritschThomas Fritsch
1,546515
1,546515
add a comment |
add a comment |
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