# How Does Changing Wavelength Affect the Central Peak in Single-Slit Diffraction?

• dreaan
In summary: Fourier}(\Delta x)##The number of fringes is then given by$$N(\Delta x)=\frac{\pi}{2}\left (\frac{\Delta x k d}{2L}\right )$$As the wavelength is decreased, the width of the central peak decreases, but the number of interference fringes remains the same.
dreaan
In my physics book, one of the basic quick quizzes checkpoints asks what happens to the central peak in a diffraction envelope when you decrease the wavelength of light (from 650 nm to 450 nm, for reference).

My understanding is that the width of the peak would decrease, while the number of interference fringes would remain the same.
This is because the number of interference fringes depends only on the width of the slit and the distance of the screen based on m=d/a.
The width of the peak decreases, because using the equation asin(theta)=m(lambda), m and a are constant, while changing lambda would change theta, thereby changing the height of the central maximum. The first order minimum would be closer to the center with a smaller wavelength.

However, my book says the answer is that the width of the peak remains the same (along with the number of interference fringes).

This seems inherently wrong. Based on my research on the internet, as well as the interactive picture my book included with the question, I feel like the publishers made a mistake in the answer. Clearly, wavelength affects the diffraction pattern. Right? Please help. My brain hurts.

oops, this question is regarding the combined effects of two-slit and single-slit interference, not just single slit. That makes a pretty big difference, but I am still pretty confused. Any help would be appreciate.

Thanks, Bill. I don't really see two much in there that answers my question. Here is where I am having trouble grasping the supposed easy question in my book. Interference pattern depends on wavelength. Diffraction pattern depends on wavelength. How does the combined effects not depend on wavelength. This seems counter-intuitive.

too*

dreaan said:
Thanks, Bill. I don't really see two much in there that answers my question. Here is where I am having trouble grasping the supposed easy question in my book. Interference pattern depends on wavelength. Diffraction pattern depends on wavelength. How does the combined effects not depend on wavelength. This seems counter-intuitive.

You are stuck in the wave-particle paradigm by thinking in terms of wavelength. Forget it. My link gives the correct analysis.

That said, in that paradigm obviously the interference pattern will depend on wavelength so your objection leaves me scratching my head.

Thanks
Bill

I'm sorry, but that article does not make sense to me. This is supposedly a somewhat simple understanding in my introductory quantum physics class.

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dreaan said:
I'm sorry, but that article does not make sense to me. This is supposedly a somewhat simple understanding in my introductory quantum physics class.

As far as I can see that's purely a classical wave analysis.

What I am saying is this wave-particle duality stuff is simply a way-station to the correct theory which doesn't really have it. The link I gave is the correct explanation.

But as a wave analysis I can't see anything wrong or what your issue is.

Thanks
Bill

The chapter I am studying primarily has to do with wave analysis. My problem is that based on the information I have learned, with a combination of single-slit and multi-slit pattern, if the wavelength is decreased, the central maximum should also decrease in width. The interference fringes should also decrease in width, although there are the same number of them within the central diffraction envelop. My book asks this question. It says the answer is that everything remains the same. I just don't see how this could be the correct, based on the equations provided.

So if the width of the peak remains constant and the number of fringes remains constant, what WOULD change if the wavelength changes. Surely something has to. I am wondering if there is a mistake in the book.

dreaan said:
So if the width of the peak remains constant and the number of fringes remains constant, what WOULD change if the wavelength changes. Surely something has to. I am wondering if there is a mistake in the book.

I would need to go through the book.

But if the wavelength changes what doesn't change is getting only single particle hits - of course the pattern changes with particle wavelength.

But I want to emphasise this wave-particle duality is a crock. Its not the real explanation.

Thanks
Bill

okay, thank you, I appreciate your help. based on what I have researched online, their answer is incorrect, which is even more frustrating. I will go back and try to comprehend the article you sent me after my final exams.

The single-slit interference pattern in Fraunhofer observation is described as the Fourier transform of the opening, i.e., the amplitude is given by
$$A(x) \propto \int_{-d/2}^{d/2} \mathrm{d} y \exp(\mathrm{i} k x y/L)=\frac{L}{\mathrm{i} k x}[\exp[\mathrm{i} k x d/(2L)]-\exp[\mathrm{i} k x d/(2L)]]=\frac{2L}{kx} \sin \left (\frac{k x d}{2L} \right ).$$
The width is usually taken as the distance of the first minimum relative to the main maximum at ##x=0##, which is thus found by
##\frac{\Delta x k d}{2L}=\pi \; \Rightarrow \; \Delta x=\frac{2 \pi L}{k d}=\frac{\lambda L}{d}.##
This shows that the width is proportional to the wave length ##\lambda## and the distance from the slit to the screen and inversely proportional to the width of the slit (the approximations involved require ##\lambda \ll d \ll L##.

For more details, see
http://en.wikipedia.org/wiki/Diffraction#Single-slit_diffraction
and
http://en.wikipedia.org/wiki/Diffraction_formalism

dreaan,
what you are saying is correct for the width (horizontally). BUT the question is asking about the central peak which is intensity at theta=0 (width vertically!, there is no angle so it has nothing to do with wavelength)
Of course the central peak remains the same!

heehe, thank you. this question will forever haunt me as i missed a similar question on the final exam last semester, worth about 1/3 of the points, dropping me a whole letter grade and even more in invaluable self confidence, haaha. at least i know now. pain is the greatest teacher.

## 1. What is single-slit diffraction?

Single-slit diffraction is a phenomenon that occurs when a beam of light passes through a narrow slit, causing the light waves to spread out and interfere with each other. This results in a diffraction pattern on a screen or surface behind the slit.

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

Single-slit diffraction differs from other types of diffraction, such as double-slit diffraction, because it involves only one narrow slit instead of multiple slits. This results in a simpler and more easily observable diffraction pattern.

## 3. What factors affect the diffraction pattern in single-slit diffraction?

The diffraction pattern in single-slit diffraction is affected by several factors, including the size of the slit, the wavelength of the light, and the distance between the slit and the screen. Additionally, the intensity of the light and the angle at which the light passes through the slit can also impact the diffraction pattern.

## 4. How is single-slit diffraction used in scientific research?

Single-slit diffraction is commonly used in scientific research to study the properties of light and to measure the size of small objects. It is also used in fields such as optics, astronomy, and biology to analyze the diffraction patterns of different materials and structures.

## 5. How can I observe single-slit diffraction in everyday life?

Single-slit diffraction can be observed in everyday life through various means, such as looking at the diffraction pattern created by a laser pointer passing through a small opening or observing the diffraction of light passing through a grating or mesh. It can also be seen in natural phenomena, such as the colorful patterns created by light passing through a bird's feathers or a soap bubble.

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