a) Convince yourself that the brightness pattern of light on the screen is a cosine function.
b) Now imagine a second set of slits placed just inward of the first set. How does the second set of slits modify the brightness pattern on the screen?
c) Imagine a continuous set of slit pairs with ever decreasing separation. What is the resulting brightness pattern?
d) Notice that this continuous set of slits forms a “top hat” transmission function. What is the Fourier transform of a top hat, and how does this compare to your sum from the previous step?
e) For the top hat function’s FT, what is the relationship between the distance between the first nulls and the width of the top hat? Express your result as a proportionality in terms of only the wavelenght of light λ and the diameter of the top hat D.
f) Take a step back and think about what I’m trying to teach you with this activity, and how it relates to a telescope primary mirror.
a) Convince yourself that the brightness pattern of light on the screen is a cosine function.
Firstly, let's consider what causes constructive or destructive interference. Constructive interference will occur when two light rays overlap while in phase. To be infuse, the parks and troughs of each wave must line up such that the peaks of the two waves strike the surface at the same time. In a similar condition, destructive interference occurs when the two waves are out of phase, or the peaks of wave 1 align with the troughs of wave 2, effectively canceling out both waves.
Double slit interference occurs because when the light rays from each slit strike the screen together, they have traveled different distances, so they may no longer be in phase with one another.
Let's quantify this. For constructive interference:
\(n\lambda=dsin\theta\)
The makes sense because the distance \(dsin\theta\) is approximately the difference in distances that the two light rays travelled, so it must be equal to a full number "n" of wavelengths \(\lambda\).
Brightness is at a maximum when the interference is completely constructive, or when \(sin\theta=0\) This occurs when \(sin\theta\) equals zero, which occurs at \(\theta=0,\pi\). We want brightness to be at a maximum at these situations, so a Cosine function makes sense, since \(cos\theta=1\) when \(\theta=0\).
b) Now imagine a second set of slits placed just inward of the first set. How does the second set of slits modify the brightness pattern on the screen?
This will cause even more diffraction and interference effects, effectively causing each maxima (bright spot) to be made of several smaller maxima, as each of the 4 light rays falls in and out of sync with its neighbors.
c) Imagine a continuous set of slit pairs with ever decreasing separation. What is the resulting brightness pattern?
This will cause the central maxima to be turned into an ever growing series of smaller and smaller maxima that get closer and closer to together, forming what is known as a top hat function.
d) Notice that this continuous set of slits forms a “top hat” transmission function. What is the Fourier transform of a top hat, and how does this compare to your sum from the previous step?
The Fourier Transform of a top hat is the \(sinc\) or \(\frac{sinx}{x}\) function. This makes sense because it resembles the spectra of a single slit (which all of the now infinitesimally tiny silts add up to). The tiny maxima will form a somewhat even light band in the waveform of a top hat or sinc function.
e) For the top hat function’s FT, what is the relationship between the distance between the first nulls and the width of the top hat? Express your result as a proportionality in terms of only the wavelenght of light λ and the diameter of the top hat D.
Since greater wavelengths of light exhibit greater bending or diffraction, a smaller slit induces greater diffraction, and the top hat diameter is approximately the width of the aperture, the relationship should be characterized by:
\[D\propto d_{null}\lambda\]
f) Take a step back and think about what I’m trying to teach you with this activity, and how it relates to a telescope primary mirror.
A telescope's primary mirror is on e the primary determining factors of the telescope's resolution. In this case, a larger primary mirror is similar to having a larger aperture, more distinct maxima and minima will be produced, thereby creating a more detailed top hat wave with a greater resolution and more detail.
Solved in colaberation with G. Grell.
Diagram from http://web.utk.edu/~cnattras/Phys250Fall2012/modules/module%201/diffraction_and_interference.htm
b) Now imagine a second set of slits placed just inward of the first set. How does the second set of slits modify the brightness pattern on the screen?
c) Imagine a continuous set of slit pairs with ever decreasing separation. What is the resulting brightness pattern?
d) Notice that this continuous set of slits forms a “top hat” transmission function. What is the Fourier transform of a top hat, and how does this compare to your sum from the previous step?
e) For the top hat function’s FT, what is the relationship between the distance between the first nulls and the width of the top hat? Express your result as a proportionality in terms of only the wavelenght of light λ and the diameter of the top hat D.
f) Take a step back and think about what I’m trying to teach you with this activity, and how it relates to a telescope primary mirror.
Firstly, let's consider what causes constructive or destructive interference. Constructive interference will occur when two light rays overlap while in phase. To be infuse, the parks and troughs of each wave must line up such that the peaks of the two waves strike the surface at the same time. In a similar condition, destructive interference occurs when the two waves are out of phase, or the peaks of wave 1 align with the troughs of wave 2, effectively canceling out both waves.
Double slit interference occurs because when the light rays from each slit strike the screen together, they have traveled different distances, so they may no longer be in phase with one another.
Let's quantify this. For constructive interference:
\(n\lambda=dsin\theta\)
The makes sense because the distance \(dsin\theta\) is approximately the difference in distances that the two light rays travelled, so it must be equal to a full number "n" of wavelengths \(\lambda\).
Brightness is at a maximum when the interference is completely constructive, or when \(sin\theta=0\) This occurs when \(sin\theta\) equals zero, which occurs at \(\theta=0,\pi\). We want brightness to be at a maximum at these situations, so a Cosine function makes sense, since \(cos\theta=1\) when \(\theta=0\).
b) Now imagine a second set of slits placed just inward of the first set. How does the second set of slits modify the brightness pattern on the screen?
This will cause even more diffraction and interference effects, effectively causing each maxima (bright spot) to be made of several smaller maxima, as each of the 4 light rays falls in and out of sync with its neighbors.
c) Imagine a continuous set of slit pairs with ever decreasing separation. What is the resulting brightness pattern?
This will cause the central maxima to be turned into an ever growing series of smaller and smaller maxima that get closer and closer to together, forming what is known as a top hat function.
d) Notice that this continuous set of slits forms a “top hat” transmission function. What is the Fourier transform of a top hat, and how does this compare to your sum from the previous step?
The Fourier Transform of a top hat is the \(sinc\) or \(\frac{sinx}{x}\) function. This makes sense because it resembles the spectra of a single slit (which all of the now infinitesimally tiny silts add up to). The tiny maxima will form a somewhat even light band in the waveform of a top hat or sinc function.
e) For the top hat function’s FT, what is the relationship between the distance between the first nulls and the width of the top hat? Express your result as a proportionality in terms of only the wavelenght of light λ and the diameter of the top hat D.
Since greater wavelengths of light exhibit greater bending or diffraction, a smaller slit induces greater diffraction, and the top hat diameter is approximately the width of the aperture, the relationship should be characterized by:
\[D\propto d_{null}\lambda\]
f) Take a step back and think about what I’m trying to teach you with this activity, and how it relates to a telescope primary mirror.
A telescope's primary mirror is on e the primary determining factors of the telescope's resolution. In this case, a larger primary mirror is similar to having a larger aperture, more distinct maxima and minima will be produced, thereby creating a more detailed top hat wave with a greater resolution and more detail.
Solved in colaberation with G. Grell.
Diagram from http://web.utk.edu/~cnattras/Phys250Fall2012/modules/module%201/diffraction_and_interference.htm
It looks like you may be confusing how the many slits combine and how the many cosines add up. What we're trying to ask in part c) is what happens when you add up all of these cosine waves. Do you really get a tophat function?
ReplyDeleteNow, in part d), what happens when you combine a bunch of slits. Imagine you have a piece of cardboard that you are carving slits out of. If you keep adding more and more slits inside your original ones, eventually you will have no cardboard remaining, and it will just be one big slit. This is what we mean by a top hat function! Does this make sense?