# Simple Diffraction: Deriving Equations with Phasors

• Manchot
In summary, the conversation discusses a derivation of an equation for single-slit Fraunhoffer diffraction using phasors. One person notices a factor of a in the denominator that they believe to be a mistake, which leads to a discussion about integrating with respect to x instead of x-prime.

#### Manchot

Hello, I've recently been studying simple diffraction for an upcoming proficiency test, and in the lecture notes for the class, a simple equation regarding single-slit Fraunhoffer diffraction was derived. The derivation was a little weird, so as an exercise, I just decided to go ahead and use phasors to derive the formula myself. Anyway, when I was finished, I nearly had the exact same result as the notes, except for one thing: a factor of a, the width of the slit. Because I couldn't find my mistake anywhere, I decided to look up the result on Wikipedia, which I suspected would do it the same way that I did. Anyway, in the step where the factor of a in the denominator was introduced, there seems to be an integration error. Am I just missing something here?

$$= C \int_{-\frac{a}{2}}^{\frac{a}{2}}e^\frac{ikxx^\prime}{z} \,dx^\prime =C \frac{\left(e^\frac{ikax}{2z} - e^\frac{-ikax}{2z}\right)}{\frac{2ikax}{2z}}$$

where do you think there is a mistake ?

marlon

Looks to me like they snuck in that factor of a in the denominator where it doesn't belong.

you are right Doc Al

i did not even see that...i must be getting delirious again

marlon

In my notes, the LHS should be;

$$C \int_{-\frac{a}{2}}^{\frac{a}{2}}e^{ikxsin\theta} dx$$

Which is just a Fourier integral ($u = xsin\theta$ is the conjugate variable), whereby the correct result is the sinc function given.

It would appear that you should be integrating with respect to x rather than x-prime.

Claude.

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## 1. How is diffraction related to phasors?

Diffraction is the bending of waves around obstacles or through narrow openings. Phasors are mathematical tools used to represent the amplitude and phase of a wave. In the case of diffraction, phasors can be used to analyze and calculate the resulting diffraction pattern.

## 2. What is the difference between simple and complex diffraction?

Simple diffraction refers to the diffraction pattern produced by a single slit or a single point source. It follows a simple mathematical relationship and can be described using the equations derived from phasors. Complex diffraction, on the other hand, involves multiple slits or sources and requires more advanced mathematical techniques to analyze.

## 3. What are the key equations used in deriving the diffraction pattern with phasors?

The key equations used in deriving the diffraction pattern with phasors are the Huygens-Fresnel principle, which describes how each point on a wavefront acts as a source of secondary waves, and the principle of superposition, which states that the total wave disturbance at any point is the sum of the individual waves. These equations are used to derive the diffraction pattern for a single slit, double slit, and multiple slits.

## 4. Can phasors be used to analyze any type of diffraction?

No, phasors are only applicable to simple diffraction, where the diffracting object is much smaller than the wavelength of the incident wave. For more complex diffraction patterns, other mathematical techniques such as Fourier analysis must be used.

## 5. How does the width of the slit affect the resulting diffraction pattern?

The width of the slit has a direct impact on the diffraction pattern. A wider slit will produce a narrower central peak and wider secondary peaks, while a narrower slit will produce a wider central peak and narrower secondary peaks. This relationship is described by the equation for the width of the central peak, where a wider slit will result in a smaller value for the width.