ast step in addition to the previous 2 questions: a lengthy derivation using phasors that the first diffraction peak should have an intensity of I1 ≈ 0.045 I0. Using the equation provided below in the pictures and the statement that β = 1.5π, you can find out what θ value to choose. You can also go for the midpoint between the first two destructive interference locations m = 1, 2 in the equation provided in the last question in which I will provide in the second image here. Repeat your three-path amplitude calculations for this value of θ, and observe what the relative intensity is for the first side maximum according to your calculation. Also explain How do the results improved when subdividing the aperture with 5, or 7, or more paths?

In this comprehensive exploration, we embark on a journey to understand the intricate physics behind single-slit diffraction. We begin by revisiting the foundational principles of wave optics, introducing key concepts such as Huygens' principle and wave interference. As we venture deeper, we will derive the mathematical expressions that govern the intensity of diffraction patterns and pinpoint the locations of maxima and minima, shedding light on the mysterious world of diffraction angles and intensities.

Determine the meaning and significance of the calculated values:

1. β (Beta):
   - Beta (β) is a crucial parameter in the context of diffraction. It represents the angle at which light diffracts or spreads as it passes through an aperture. Specifically, β is the angle between the central maximum (the maximum brightness at the center of the diffraction pattern) and the first diffraction minimum (the first point of destructive interference). In other words, β indicates how much the light deviates from its original path due to the diffraction caused by the aperture. In this case, β is approximately 0.0314 radians or about 1.8 degrees, which means the spread of light due to diffraction is relatively small.

2. θ (Theta) for First Minimum:
   - Theta (θ) is the angle at which the first diffraction minimum occurs in the diffraction pattern. It's the angle at which destructive interference takes place, resulting in a dark region in the pattern. In this specific case, θ is approximately 5.71 degrees. This means that when you observe the diffraction pattern, the first dark region (minimum) will appear at an angle of about 5.71 degrees from the center of the pattern. It's a fundamental characteristic of the diffraction pattern associated with the given aperture size and wavelength.

3. Relative Intensity for First Side Maximum:
   - The relative intensity for the first side maximum quantifies the brightness of the first bright region (maximum) to the sides of the central maximum in the diffraction pattern. It's a measure of how intense this region is compared to the central maximum. In this case, the relative intensity is approximately 0.0025. This value indicates that the first side maximum has much lower intensity compared to the central maximum. The central maximum is usually the brightest part of the diffraction pattern, and the side maxima are progressively dimmer.

 The β characterizes the angular spread of light due to diffraction, θ represents the angle of the first dark region in the diffraction pattern, and the relative intensity quantifies the brightness of the first bright region to the sides of the central maximum. These values are essential in understanding and analyzing the behavior of light when it passes through an aperture and forms a diffraction pattern.