Diffraction: Intensity in the double slit situation

In summary, the conversation discusses the equations for intensity in the double slit experiment, both with and without considering the effects of diffraction. The doubt arises when taking the limit case of a/λ going to 0 and d=0, which results in a different expression for intensity than the single slit case. The potential explanations for this discrepancy are either a mistake in taking the limit case or a flaw in the expression for intensity.
  • #1
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Hello. I have been studying interference and diffraction and one doubt has appeared. When you consider the double slit experiment forgeting the effects of diffraction you get the following equation for intensity

[itex]I^{}=4I_{0}cos^{2}(\frac{πdsin(θ)}{λ})[/itex]

where d is the distance between the slits.
For the single slit diffraction we get

[itex]I^{}=I_{0}(\frac{sin(x)}{x})^{2}[/itex]
[itex]x^{}=(\frac{asinθπ}{λ})[/itex]

where a is the width of the slit.

Then for the double-slit case considering diffraction we get

[itex]I^{}=4I_{0}cos^{2}(\frac{πdsin(θ)}{λ})(\frac{sin(x )}{x})^{2}[/itex]

My doubt raises when i consider the two limit cases:
1.For a/λ going to 0 the expression becomes that of the interference-only case.
2.But when we consider d=0(the distance between the centers of the slits) the expression obtained is

[itex]I^{}=4I_{0}(\frac{sin(x)}{x})^{2}[/itex]

which is different from that of the single slit case although doing d=0 we are turning two slits of width a in one slit of width a.

My thoughts trying to solve this problem have considered that maybe I am taking the limit case wrong (although I haven't found where) or some expression for the intensity is wrong.

Thanks in advance for your attention, expecting a good answer...
 
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  • #2
which is different from that of the single slit case although doing d=0 we are turning two slits of width a in one slit of width a.
... if d is the center-to-center distance between slits of width a, then this is the same as a single slit width d+a with a barrier of width d-a in the middle.

Even in the case where we let (d--->a) that is just the limit that the barrier gets very small ... there may still be a barrier in the math; just like the limiting case for a-->0 is not the same as for a solid wall with no slits.
 
  • #3
By merging the positions of the two slits you haven't turned either of them off. You have two superimposed slits. They are giving out waves in phase. The resultant amplitude at any point is twice that due to either slit by itself, so the intensity is 4 times that due to either by itself.
 

Related to Diffraction: Intensity in the double slit situation

1. What is the double slit experiment and why is it significant in understanding diffraction?

The double slit experiment is a classic experiment in which a beam of light is passed through two narrow slits and then projected onto a screen. It is significant in understanding diffraction because it demonstrates the wave-like behavior of light and how it can diffract and interfere with itself. This experiment helped to support the wave theory of light and is also important in understanding the properties of diffraction, such as intensity.

2. How does the distance between the two slits affect the intensity of the diffraction pattern?

The distance between the two slits plays a significant role in the intensity of the diffraction pattern. As the distance increases, the intensity of the pattern decreases. This is because the farther apart the slits are, the wider the interference pattern becomes, causing the individual peaks to be less intense.

3. What is the relationship between the wavelength of light and the intensity of diffraction?

The wavelength of light is directly proportional to the intensity of diffraction. This means that as the wavelength increases, the intensity of the diffraction pattern also increases. This is because longer wavelengths have a greater ability to diffract around obstacles, resulting in a more intense diffraction pattern.

4. How does the size of the slits impact the intensity of the diffraction pattern?

The size of the slits also plays a role in the intensity of the diffraction pattern. As the size of the slits decreases, the intensity of the diffraction pattern increases. This is because smaller slits create a narrower interference pattern, resulting in more intense peaks.

5. Can the intensity of the diffraction pattern be manipulated or controlled?

Yes, the intensity of the diffraction pattern can be manipulated or controlled by adjusting the experimental setup. For example, changing the distance between the slits, the size of the slits, or the wavelength of light can all affect the intensity of the pattern. Additionally, using different types of light sources, such as lasers, can also impact the intensity of the diffraction pattern.

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