fix the overleaf physics report

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\begin{frontmatter}
    \title{Diffraction }
    \author{YU Fu}
    \date{\today}

    % \address{[email protected]}

    \begin{abstract}
       In this experiment, the behavior of light when it is illuminated through a two-slit obstacle was observed. Dark and bright lines were formed representing destructive and constructive interference respectively. There was one brighter line in the middle of the pattern.
    \end{abstract}

\end{frontmatter}

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\section{Introduction}
The purpose of this experiment was to obtain the position of maxima in double-slit interference. Interference of light is the superposition of light waves (Castañeda, Matteucci, & Capelli, 2016). The equation dsin\theta=m\lambda is used to estimate the position of maxima on the screen.

\textbf {Background and theory:} The equation dsin\theta=m\lambda     is used to estimate the position of maxima on the screen. The light was projected to an obstacle which had two small slits. The slits had a d distance between them. After the projection of light through the slits, wavelets were formed as interfered behind the obstacle. Patterns of bright and dark stripes were formed on the screen placed behind the obstacle. These bright and dark stripes formed a pattern known as the fringe pattern. These bright stripes will be the maxima. \\
          

\section {Method}
Part 1\\
PhET; https://phet.colorado.edu/en/simulation/wave-\\interference\\
Part 1; Slits-Single Slits Design an experiment to verify that for small angles, the angular spread is inversely proportional to the ratio of (a/λ), and collect data.\\

Solution\\
Assume the width of the slit to be finite and consider how Fraunhofer diffraction arises,
Let a source of monochromatic light be incident on a slit of limited width a, as shown below\\
\includegraphics[]{1.png}\\
In diffraction of Fraunhofer type, all the rays passing through the slit are almost parallel. Also, each portion of the slit will act as a source of light waves, according to Huygens’s principle. If the slit is divided into two halves. At the first minimum, each ray from the upper half will be exactly180 out of phase with a corresponding beam from the lower half with a path difference \sigma = \lambda /2=. \\
Therefore, the condition for the first minimum is:(a/2) sin \theta (\lambda /2)\\
which can be written as Sin \theta (\lambda /a)\\
We can then apply the same reasoning to the wavefronts from the four equally spaced points a distance apart, the path difference would be/nasin/ 4a \sigma \theta =, and the condition for destructive interference is usually given as 2sinaλθ (Bezugly & Petchenko, 2018)
Therefore, generalizing the argument will show that harmful interference will occur when a sin θ=mλ, m=±1, ±2, ±3….and is represented graphically as below: λ is the wavelength\\
\includegraphics[]{2.png}

\\

    

 
\section{Discussion}
\\
Where is the space between one wave and another called wavelength. And d is the distance between the two holes where the wave passed, and  is the angle between centerline optimism. So using this formula, we can get the order of the constructive point in the first length or the second length. The change of getting an error is 0.4 as due to the material used, the speed of the wave, and also the size of the two streams can affect the size of the wavelength, which may cause an error.
In conclusion, the young’s double-slit formula can be used to determine how close two holes are in the molecular structure or any other application without the need to measure the distance between the trenches which sometimes can be impossible.

\section{Conclusion}

Explain how the aperture Geometry relates to the diffraction pattern. \\
\\Aperture Geometry typically refers to the diameter of the light-collecting region; aperture Geometry might be in a circular geometry. For instance, an optical instrument has an aperture as its diameter of the objective lens. The aperture Geometry is essential as far as diffraction pattern is concerned. For instance, multiple, multiple openings/apertures result in an intricate diffraction pattern of varying intensity. In general, the aperture Geometry influences the nature of the diffraction pattern; it also influences the power of the diffraction pattern et cetera (Andrews, 1947)\\
\\Predict how changing the wavelength or aperture size affects the diffraction patterns.\\
\\Diffraction happens when light bends in the same medium. The bending is due to the light waves “squeezing” through small openings. Since light waves are small (on the order of 400 to 700 nanometers), diffraction only occurs through small openings or over tiny grooves. Conversely, as the wavelength decreases, the angle of diffraction decreases. In short, the angle of diffraction is directly proportional to the size of the wavelength `openings.  The formula for diffraction shows a direct relationship between the edge of diffraction (theta) and wavelength. However, increasing the wavelength or aperture size will then increase the angle of diffraction hence affecting the diffraction pattern (Asakura, 1973).

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\section{References}
\bibliography{references}

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\bibitem{pressure scanner} 

Andrews, C. L. (1947). undefined. Physical Review, 71(11), 777-786.\\DOI:10.1103/physrev.71.777\\
Asakura, T. (1973). Partially coherent Fresnel diffraction by a slit aperture. III. Fresnel \\
diffraction patterns. Optics Communications, 8(1), 33-36. DOI:10.1016/0030-4018(73)90175-2\\
Bezugly, A. V., & Petchenko, A. M. (2018). Photon flow density distribution in the diffraction pattern of single- And two parallel slits. Telecommunications and Radio Engineering, 77(1), 77-82. DOI:10.1615/telecomradeng. v77.i1.80

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