Understanding the Formula for Dark Fringe in Single-Slit Diffraction

In summary, the conversation discusses the formula for dark fringe in single-slit diffraction. The text explains that the path difference between two narrow strips at the top edge and center of the slit is (a/2)sinθ, and if this is equal to λ/2, cancellation of light occurs. The reason for choosing these specific points is questioned, but it is explained that the book is simplifying the method by considering two corresponding points on the half wavefronts. This allows for destructive interference from every point on the incident wavefront to be accounted for. Another method that does not divide the wavefront in half may not accurately account for the contribution from all points.
  • #1
KFC
488
4
Hi guys,
I have a doublt about deducing the formula of dark fringe for single-slit diffraction. In the text, it reads "Consider two narrowstrips, one just below the top edge of the drawing of the slit and one at its center. The difference in path length to point P is [tex](a/2)\sin\theta[/tex], where a is the slit width. Suppose this path difference happens to be equal to [tex]\lambda/2[/tex]; then light from the two strips arrives at point P with a half-cycle phase difference, and cancellation occurs."

Why it said "happens to be ..." ? Why it must be the top point and the center of the slit? It seems not that convincible! What about I pick the top point of the slit and a point apart the top edge of the slit by a/2.25 and said "Suppose the path difference (from these two points) happens to be [tex]\lambda/2[/tex] ..." then I will also get a similar formula but with different [tex]\theta[/tex]! So the text choose that specific points make sense?
 
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  • #2
A rough approximation with a wide slit is that light passing through the center of the slit does so unperturbed (think of it as in wavelets). However, the light that strikes the edge of the slit will be diffracted. This diffracted light at the edges will travel outward back towards the center of the slit and interfere with the light that originally passed through unperturbed.
 
  • #3
I think the book is referring to a simplified method of finding the positions of the first minima.The wavefront is divided in half and the book is considering two corresponding points on the two half wavefronts and calculating the angle such that the waves from these two points interfere destructively.At this angle there will also be destructive interference between the waves from every other set of corresponding points on the two half wavefronts,in other words, by dividing in half the contribution from every single point on the incident wavefront can be accounted for.With your method KFC you have not divided the wavefront in half and although from some sets of points there will be destructive interference you cannot extend this method to work out the contribution from the remaining points.
 

What is single-slit diffraction?

Single-slit diffraction is a phenomenon that occurs when a wave, such as light, passes through a narrow opening or slit. This causes the wave to spread out and interfere with itself, resulting in a diffraction pattern.

How does single-slit diffraction differ from other types of diffraction?

Single-slit diffraction is different from other types of diffraction, such as double-slit diffraction, because it only involves one narrow opening or slit. This results in a simpler and more distinct diffraction pattern.

What factors affect the diffraction pattern in single-slit diffraction?

The diffraction pattern in single-slit diffraction is affected by several factors, including the width of the slit, the wavelength of the wave, and the distance between the slit and the screen where the pattern is observed.

How is single-slit diffraction used in real-world applications?

Single-slit diffraction is used in various real-world applications, such as in optical instruments like telescopes and microscopes. It is also used in the study of wave behavior and in the design of diffraction gratings for use in optical devices.

What is the mathematical equation for calculating the location of the bright fringes in a single-slit diffraction pattern?

The mathematical equation for calculating the location of the bright fringes in a single-slit diffraction pattern is given by d*sin(θ) = m*λ, where d is the width of the slit, θ is the angle of diffraction, m is the order of the fringe, and λ is the wavelength of the wave.

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