Abstract
The purpose of this paper is to simulate the interference patterns generated by different kinds
of slit interference and conditions of the screen and slits. It also sheds light on the potential
application of slit interference in science and technology. By using python, the paper employs
a variety of built-in functions and a plotting tool to visualize the patterns. It also takes on
multiple parameters to illustrate different states existing between the screen and the slits. In
different types of interference, the interference patterns are different. The experiment finds
out that under different distances between slits and screen, there is uniqueness in the pattern
as it becomes smaller enough; as the decay of intensity of light along the distance is removed,
there are more singularities and brighter regions while the overall pattern remains unchanged.
Although there are limitations in the accuracy existing in python, like the finite step sizes and
runtime issue, the experiment can be applied to different fields, such as classic wave optics,
phytology, and acoustics via visualizing the patterns.
Introduction
Ever since Thomas Young’s initial discovery in the early 19th century of alternately additive and
subtractive interference of wavefronts, generation after generation of physicists has been
dedicated to this specific field that can unveil the very nature of light. After Young’s crucial
understanding of the wave properties of light, the later discovery of the photoelectric effect
continually demonstrated that under different circumstances, light can behave as if it is
composed of discrete particles. These contradictory theories, after a hundred years, finally led
to the discovery of light’s particle-wave nature, which realizes that light could dually show
wave-like and particle-like behaviors. Centuries after Young’s initial discovery, the implication
of this phenomenon still deserves to be explored further.
Nowadays, the application of various types of slit interference becomes increasingly
important in different fields. Its application can be used in directing and attracting metal
particles, analyzing spectroscopy of real light emitted by different atoms and particles, and
detecting roughness and variation of various surfaces. it becomes a useful tool for a wide
range of spectroscopy. It forms the basis of many other experiments done in quantum physics.
This paper will look deeper into how different conditions and limitations, such as the distance
between slits and screen and the effect of distance on the intensity of light, affect the
formation of the experiment, especially by considering the finite height of two slits and casting
onto the screen as a grid. A series of experiments are done in order to study more possibilities
of double-slit interference.
Using self-supportive python simulation that incorporates built-in functions such as matplotlib
and random, the program carries out a new point of view into how double-slit interference
patterns, as well as single-slit interference and diffraction grating, are formed.
Apart from theoretical simulation, the paper also shows that these types of interference are
not only mathematical tricks, but they also have a broad range of applications in
advancements in modern physics and technology. By its extension to application in various
fields such as spectroscopy, acoustics, and phytology, the paper provides a pristine view of
the inherent interconnectedness of different branches of natural science.
Keywords
Python, single slit interference, double-slit interference, diffraction grating, visualization,
distance, decay of intensity.
Theoretical Aspect of Slit Interference
Slit interference is a process by which the light emitted from each slit interacts with that
emitted from the other or other slits at various locations on the screen, which is located a
certain distance behind the slits. As the distance from the slit or slits to the points varies, the
light is situated in different phases. At some points, both lights are at the same phase, such
that when the peak of the model trigonometric graph of the intensity of the light meets another,
the point will be brightest. This is known as constructive interference. On the other hand, if at
a point where a peak meets a trough, there will be darkness because opposite intensities
offset each other. This is known as destructive interference.
These two types of interference occur successively at different points and parts of the screen,
engendering the interference pattern that consists of light and dark spots or regions (Urone,
Hinrichs, Driks, & Sharma, 2012)

Simulation

 According to the basic formation of double slit interference(DSI), there is a certain distance
between each of the double slits and the screen. When the light is emitted from the slits, it travels a certain distance of d to strike the screen. As both of the lights from different slits meet each other
at a point, they will be in different phases. Using path length derived by the equation
the decrease in the intensity of the light A can be calculated by dividing by the factor of . Also,
the phases of each ray of light can be determined using the equation
where is the phase of the light and is the wavelength of the incident light. With different phases
of each ray of light, the addition of their intensity will be
which will sometimes result in constructive interference and other times destructive interference.
Since the desired value is the magnitude of the intensity of light, the equation can be written as
By obtaining such intensity for each set point on the screen, the overall interference pattern can be
understood.
Δd = n ∗ λ(n = 1, 2, 3......),
1/d2
ϕ = d/λ,
ϕ λ
|A1 + |,
→
A2
→
Alight = √-(A--1-s-in-ϕ-1--+-A--2-si-n-ϕ-2-)-2-+--(-A-1-c-os-ϕ-1--+-A--2-c-os-ϕ-2-)-2
The program constructs a model of double-slit by a certain distance in front of a screen. Meanwhile,
the interval between the
two slits is . As one of the initial settings, the heights of the slits are in¦nite, so it is
simpli¦ed to a two-dimensional problem. As the program ¦rst uses the wavelength of blue light,
which is . The width of the slits is ignored. Obviously, as it is impossible to build
a screen with in¦nite area, the program ¦rst con¦nes it to be a grid with the center
colinear with the midpoint of two slits. With the decrease in intensity (amplitude) along the
distance, the phase and ¦nal amplitude of each ray of the light is calculated by the derived value of
distance between the slits and iterated point on the screen.
500 ∗ 10-9m
λ = 450 ∗ 10-9m
2m ∗ 2m
The ¦rst program simulates the 2-D interference pattern on the screen(grid). In order to get it
working, a series of parameters are pre-determined at ¦rst: the wavelength of the light, lambda, the
distance between slits and screen, and the distance between slits, and the initial reference
amplitude of light. Then, in the linear space labeled "x," the length of the space is determined by
input N with a certain number of steps. A function is de¦ned thereafter to represent the calculation
of interference patterns. Within the function, all the calculation of distances are operated under the
general formula
dfinal = √-D-2--+--d-2, 

where D is the distance between the screen and slits and d is the horizontal distance between the iterated point and the projected position of either slit on the screen, tagged by "d1" or "d2". After that, there is a "sol" handle that points the function specifically to the screen that user inputs. Finally, the program plots the intensity of light versus the position on the screen [figure 2.1].

Then, the next program expands the scope of simulation of DSI to a 3-dimensional space. At first, the height of the slit is set to 0.1m, a length that way exceeds the wavelength of 4.5∗10−9m. Within the range of 0.1m, the program uses Monte Carlo method to randomly generate points. over the height of the slits using two separate "for" loops and stores the position of the generated points in an array.

The screen that receives the light rays is modeled as a grid with finite points defined by two "linspace" statements. Two variables, i and j, iterate through the combination of two "for" loops nested in each other that run through the two linear spaces.

The subsequent program precisely calculates the distance between the 2-D slits--instead of the 1D model with infinite height--and the iterated points on the screen. To ensure precision,

λ<<h.
As indicated in the program, the general formula defined in the function to calculate distance between points on the grid(screen) and simulation points on the slits is

dfinal=d2+D2+(y[j]−p)2−−−−−−−−−−−−−−−−√
where d is the distance between points (i,j) on the screen and random points on the slits, D is the distance between slits and screen, y[j] is the y coordinate of point (i,j), and p is the height of random points on the slits.[figure 1.2]

Similar to the technique used in the previous simulation, once the distance between a iterated point and the slit is derived, the program calculates the phase and thus intensity of the light based on the distance. When two rays of light converge on points of coordinate (i,j) on the screen, both amplitude and phase of rays of light will be adding up vectorially and eventually form a light or dark spot on that point. After that, the program stores the value of intensity into the (i,j) position of the constructed 2-D array. The intensities of light at those light and dark spots are plotted versus its coordinate (i,j) using the "imshow" feature to display a 2-D plane where higher intensities are displayed to be brighter and lower intensities are displayed to be dimmer. 

Then, the simulation expands to single slit interference(SSI). To clarify, the only slit is deemed to have a finite width. When the light passes through the slit, every infinitesimal point within the slit is seen as a source of light that interferes with one another. A usual approach to figure out the interference pattern of single slit is to treat half of the width as a reference interval, while taking the start and end of the half-length(w/2) to be similar to sources of light in double slits. This method is used when the simulation point is in on the outside of the projection of single slit on the grid(screen). When the simulation is within the direct projection of single slit on the screen, the program will treat it as a collection of multiple light sources of which their distances to the screen are considered all the same. Therefore, to derive the light intensity the range of the direct projection of the slits, the program merely multiples each intensity of light by the number of reference points on the single slit:

I=A∗1/d2∗m.
As one interference result is determined, it can be passed down to the subsequent points with distance of w/2, because, according to the approximation equation

dsinθ=Δx,
as d remains w/2 and θ remains the same, the difference in length is constant, leading to the phase of interference to be constant. Therefore, each subsequent point will interfere the same way the first pair of points behave. In order to set up the simulation, the program replaces the distance between slits with the width of the slit, which is

w=100∗10−7m
to let the half of the width be

w/2=500∗10−8m.
The height of the slit is also the same as previous one, which is 0.1m. This program uses the two 1/4 points of the slit as reference. The reason is: as the difference of distances between points on the slit and points on the screen varies as the simulation goes from left to right, resulting in a difference decrease in amplitude. In order to keep the running cycle of the program within an acceptable range, when using 1/4 points on two parts of the slit with interval of w/2, the difference can almost be cancelled out by multiplying the mediate value with the number of points on the horizontal to be simulated. In other words, between the two edges of the slit, the program is simplified to a direct projection of incident light.[figure 1.3]

Following previous programs' algorithms to calculate the distance that light travelled from slits to the point on the grid(screen), the function defined for SSI develops a different version of deriving the essential elements in the regions on the screen that are unique to the SSI. First, a single slit with a height of h is set up as similar to previous programs. Then, the width of the slit is provided in order to calculate the total light intensity. After that, a set of points are generated randomly on vertical direction while another set of points are spaced evenly horizontally. For each vertical point, the intensity of light is divided to give a different reference for a different location on the slit. For each horizontal point, the intensity of light will be accumulated for each pair of points with interval of w/2 on the horizontal direction. As the program treats two 1/4 points of the single slit as double slits to simplify the calculation process. For each (i,j)coordinate on the grid(screen), the program calculates the distance from those two slits to the iterated point on the screen and derives phase using that information. Then, the intensity will be determined and accumulated by the times of how many points are generated on the horizontal direction. Finally, the intensity of light on that coordinate is appended to the corresponding (i,j) position of the 2-D array to store the result. Using "imshow" feature again, the program visualizes the result in a 2-D plane.

The program comes to simulate diffraction grating(DG), which means there are more than two slits in the problem. The central theme of DG is that even if there is only a small deviation from constructive points on the screen, there will be a large drop in the total light intensity due to the multitude of slits that increases Δd each time it passes by a slit. As a result, the constructive interference, when cast on the screen, will be primarily composed of explicit light dots instead of gradually changing light and dark stripes.

Almost all the equations are the same as those in the double slits problem except for the number of times for the program to calculate the distance between points on the screen and slits. To set up the simulation, the program first declares the inputs that are used repeatedly in previous programs. Then, the parameters are defined as they are in previous programs. After that, a DG function for DG is defined. There is also an initial statement that takes on the total span of the slits, of which the formula is:

l=d∗(m−1),
where d is the distance between two adjacent slits and m is the number of slits and l is the total horizontal length over the span of all the slits. After that, it starts the iteration through each point on the screen(grid) as usual. The unique setting of this DG is, when the program runs into the iteration, there will be an array named "distance" that collects all the distances from different slits to the point the "for" loop is iterating on the screen. As a result, when it comes to calculating distance for each section on the screen, there is an extra "for" loop that calculates distance from each slit to the point on the screen. It then stores the derived value to the "distance" array. Based on the array of distance, the program derives amplitude and phase for each distance in the array, storing them respectively into corresponding arrays. Finally, the program calculates the final intensity of light on each point on the screen(grid) and plots them with graduated colors on a 2D plane.[figure 1.3]

Results
Distance comparison
the tables below show the comparison of different types of interference under varying distances
between slit or slits and the screen. With smaller distances, the screen is zoomed in accordingly.

Figure 2.1: comparison of 2D DSI simulation. From left to right: D=1m, 0.1m, 0.01m.
When D=1m, the graph displays a series of peaks and troughs. In the center of the screen, there is the region of the highest intensity. On the sides of the highest peak, there are two peaks of constructive interference.
From the image, as the distance between slits and screen decreases, the patterns generated are similar—which means, regardless of the viewing window, they are in the same shape. From left to right in the images of comparison, the patterns go smaller as the distance between slits and the screen dwindles.

Figure 2.2: comparison of 3d simulation of DSI. From left to right: D=1m, 0.1m, 0.01m
At the distance of 1m, the interference pattern shows a vertical alternation over the horizontal axis on the screen. Black and white stripes show up in succession in a rippled and curved state. As the light travels on the vertical axis, there are two large dark regions in the shape of a star.
Although there are still light and dark stripes over the vertical axis, they are darker than those on the horizontal axis due to the limitation of the slits and greater distance. At four corners of the screen, the interference effect is barely visible.
As the distance from the slits to the screen decreases, the interference pattern becomes simpler and more focused on the center of the screen. The evanescent wave fades away as distance increases.
On the other hand, in the center of the screen, the interference pattern displays a characteristic of the disorder. In the case where D=0.01m, there is only a single light band in the middle of the screen, while all other interference effects have vanished. The interference pattern occupies almost the entire screen as a result of the comparable nature between the size of the screen and the height of the slits. Additionally, over the height of the slit, the distribution of brightness of the light displays a random value due to the uncertainty of the Monte Carlo method.

Figure 2.3: comparison of SSI. From left to right: D=1m, 0.1m, 0.01m
At the distance of 1m, there is a central bright region and two bright peaks on both sides. On the horizontal axis of the screen, there are two side peaks located the nearest to the center of the screen, as the simulation goes away from the horizontal axis, the bright peak goes backward on the distance from the center of the screen. In other words, the light and dark areas form a hyperbola-shaped pattern that can be assumed to be generated successively from the central region of the screen. While as the simulation goes farther from the center of the screen and the overall brightness of the interference pattern decreases, the hyperbola-shaped pattern still holds true.
As the distance between the slit and the screen changes from 1m to 0.1m, the overall diffraction pattern does not change much. However, on the two sides of the brightest band, secondary interference “peak” is prolonged and less bright as they are when D=1m. When D=0.01m, the overall interference pattern shifts from hyperbola-like peaks and troughs to simple light and dark stripes.
Vertically, the pattern shows a constant distinction between light and dark. Again, the two sidebands are dimmer and less concentrated than those of greater distances. Akin to DSI, the brightness distribution over the height of the slit also displays random characteristics.

                     Figure 2.4: comparison of DG. From left to right: D=1m, 0.1m, 0.01m.
In DG, there is a central bright spot paired with two dim peaks on its sides. The bright and dark patterns rapidly vanish as the distance from the center increases, leaving two large dark regions on the edges of the screen.
Over the vertical axis, the intensity of light goes down gradually as it moves away from the center of the screen.
From the results of DG, it can be yielded that although the overall pattern does not change as
much from D=1m to D=0.1m, the distribution pattern shows the dotted effect when D=0.01m, where the bright spots interfered by the multiple slits are displayed explicitly and independently on the screen over different positions.
The reduction in the intensity of light by distance
Farther the point is from the source of light, the intensity of light on that point is lower. This
phenomenon can be modeled as multiplication with a factor of 1/d**2. The tables below demonstrate the comparison between the interference pattern that has the decay along distance from the slits and that does not have the decay. Unless noted prior to, all the images displayed below are formed by the distance between slits and screen to be 1m and size of the screen to be 10m*10m.

Figure 2.5: decay along the distance comparison of 2d simulation of DSI. Left: option=1. Right: option=0
From the images, as there is no decay in light’s intensity along the distance, the interfered light becomes stabilized around reference intensity of 1.5 as it goes away from the screen.

  Figure 2.6: decay along the distance comparison of 3d simulation of DSI. Left: option=1, right: option=0.
Based on the images, after removing the effect of distance on DSI, the overall intensities of the interference patterns become stronger while the patterns themselves remain unchanged. Besides, there is chaos in the four corners of the screen, where there are unidentifiable patterns with slight shifts between light and darkness.
      

       Figure 2.7: decay along the distance comparison of SSI. Left: option=1, right: option=0
By comparing two images before and after removing the effect of distance, it can be seen that the bright peaks in the center of the screen extend all the way to the edges of the screen with overall higher intensity. it remains hyperbolic shape as that with 1/d**2.
        

         Figure 2.8: decay along the distance comparison of DG. Left: option=1, right: option=0
By comparing the two images[figure 2.8], it can be concluded that when the effect of distance is removed, the bright region in the middle extends above and below. Akin to that of single slit interference, the overall brightness also rises in DG.
Application of DSI
These types of interference can be applied in a variety of fields. Below are three examples.
Assistance in constructing arc slits
The first example of applications of slit interference is about the use of "surface plasmon polariton"(Bai, 2020), which is a mode that displays polarized electromagnetic properties in attractive Rayleigh metallic particles through interference from arc slits. A detailed explanation of how this program for simulation is presented, and it can be used to facilitate some parts of the course of analysis and experiment under that topic.
First, as mentioned in the Bai’s paper, SPPs display "the characteristics of local
enhancement, shorter effective wavelength, and abrupt phase change "(Bai, 2020). Despite the substantial fact that its wavelength and phase information might be different from the testing wavelength, the program can still handle it by changing the parameter of "lambda1" and "distancebetweenslits" in the initial experimental setup to ensure the resulting interference pattern will be displayed appropriately.
Moreover, since the arc slits in DSI can function well as "A pair of counter-propagating SPP waves can be generated through exciting two symmetric micro/nano metal slits, generating an SPP interference pattern in the center of the structure" (Bai, 2020), it is essential to shape the slits to be arcs for the maximization of focus of SPPs and thus the success of the experiment. In order to get the desired shape, within the program when the points are generated by Monte Carlo on the slits, instead of generating all the points on a single linear space, manipulation can be prescribed to get the range of pattern generation into an arc with custom curvature. Then, points can be generated within that arc with certain horizontal and vertical positions in order to let the distance from the arc to the point on the screen(grid) be calculated throughout the process of simulation. As a result, the interference pattern generated by the double arc slits will be focused on the center of the screen instead of being
scattered towards the fringe of the screen. Therefore, the double-slit model can be modified in such a way to reach the effect of arc slits
simulation.
Another variation for this problem happens when it comes to shifting the focus of the SPPs on the screen. As the paper originally says, “the phase difference between two SPP waves can only be obtained by the phase difference of excitation beams indirectly” (Bai, 2020). The phase difference results from the tilt of the single or double electronic beams. Under that circumstance, if there is a predefined phase difference formula, as it is in this paper, " ∆φspp = 4πrsinθ/λ0 + π”, it will be easy to just plug in the required parameter, get the difference and modify the program by adding the initial phase difference in the part where the phases are calculated. When there is no such a phase difference that is pre-defined, nevertheless, there will be no hindrance to using the code. To obtain the ideal phase difference, the state of the initial light source must be considered in order to derive the angle of incidence at double slits. Then, the phase difference can be derived using trigonometric value, sine or cosine depending on the situation, to extrapolate from the slits—the light sources—to the screen light is being projected onto. An obvious conclusion that can be drawn from this feature is, whenever the focus needs to be shifted by altering the tilt angle, the program is always available to use and feasible to change to any state. As a result, the interference pattern is able to be more customizable and controllable, by which researchers can easily shift the focus of the interference pattern in between two arc slits.

The effect of using DSI simulation program to assist in shifting the focus can be proved by the successive attempts to use the focused SPPs in the center of the slits. By doing this, the focus “only provides a strong gradient force along the vertical direction to attract the particle to the metal surface, but also provide a strong gradient force and a weak radiation force along the horizontal direction, which works together to push the particle to the center of the focus” (Bai 2020).
Application in phytology
When examining the Volatile Organic Compound (VOC) in tree stems, especially residual of
greenhouse gas (GHG), the spectroscopy plays an important role in carrying out the result of
examination: “Low residual values denote a reliable analysis, whereas high values indicate
nonpreferred gas compounds within the gas sample” (Machacova, 2021). As a reliable explanation of this phenomenon, the non-preferred compounds “would not occur in the spectral library and thus could interfere with the individual gas measurements” (Machacova, 2021). This conclusion brings about the simulation program of slit interference because the expected result can be realized via the respective input of the DG program that can separate light to a precise spectral range. Based on the original state of the light emitted by gas, GHG, VOC into the program, researchers can drag out a series of slots over the spectrum to scrutinize how they interact with each other over that interval. They can also examine whether there is a large overlap in the spectral library, etc. These functions turn out to be expedient for the determination of the potential extent of overlap over the range in the spectrum. Thus, derived residual values can be drawn on in order to reveal the existence of unpreferred gas.
Application in Acoustics
According to a recent study in acoustics, which discusses the confinement of acoustic fields in subwavelength volumes, “acoustic focusing is limited by diffraction” (Chen J., 2019). Moreover, they ascribe this limitation to “the quick fading of evanescent waves because of their exponential decaying nature upon propagation” (Chen J., 2019). With the program provided, it will be easier to find the effect of interference during the process of
diffraction by drawing a comparison between the factor of decay along the distance. In
the program, the decision of whether or not to include the distance factor can be simply operated at the beginning of the program by entering “1” or “0” to decide whether to include the effect of distance or not. Upon the manipulation of the program, the interference patterns will be appropriately modeled and visualized.
Additionally, as they mention their recent findings in “subdiffraction focusing of acoustic waves with a metasurface consisting of an array of deep-subwavelength sized and spaced slits” (Chen J., 2019). The program, though, provides a comprehensive simulation regarding how wave passes through multiple slits and forms interference pattern on the target of projection. By providing the necessary parameters of acoustic waves and that of multiple spaced slits, researchers can easily tell the focus of acoustic waves and “full-wave dynamics on the metasurface”(Bai 2021). As a result, the program facilitates their subsequent study of how to focus the acoustic waves. Their solution to the problem was to use “a thin plate imperforated with an array of groove that is deep subwavelength in size and spacing” (Chen J., 2019). Within the capability of the program, it is able to simulate a line of splits in any direction from the center of the groove. It can also set up the customized interval between every two adjacent slits to better approximate “near-field metasurface for sub-diffraction sound focusing in the reflection mode” (Chen J., 2019).
Conclusion

Different programs simulate different types of slit interference, including DSI, SSI, and DG. By
introducing the limitation of the height of the slit or slits, which is achieved by the division of
conditions under different sections of the screen, the program visualizes the various interference patterns generated under varying states of the screen and slits. It then takes a deeper look into how the distance between slits and screen and how the decay of intensity of light with distance affects the result of simulation.
The application of interference covers a wide variety of scientific and technological areas. It can be applied to cast light onto the spectrum to form specified regions of color in order to decipher the existence of unpreferred compounds. It can be used in the visualization of interference patterns produced by SPPs through arc slits. It can be utilized in simulating the transmission of soundwaves within a structure of a specific shape linearly. The use of slit interference in different fields proves its significance
The results the program yields, with most of the program consisting of singularities, each program in different states displaying its unique identity and arrangements over its span. As it runs under different conditions, the pattern displays variation to different extents. In short, as the distance between the screen and slits decreases to a certain distance, completely different patterns show up, due to the more direct effect of light coming through the slits, with greater details. When the decay of intensity of light via distance is removed, the marginal regions where the interference effect could have faded away by distance, become filled with faint interference patterns. Meanwhile, in the places where there are originally some interference patterns, the contrast of light and dark becomes more vibrant
due to the intensified light cast on them.
The limitation of accuracy is the main issue of using python. When the simulation unfolds, the finite step size that the program executes hinders the generation of a more accurate image of the interference patterns. Another limitation is, when simulating a great number of slits, due to the runtime, the program can only produce an approximation over the screen by taking the two “one-fourth” points instead of calculating the entire span of slits. These two limitations lead to some extent of inaccuracy at some points that are displayed.
Acknowledgments
Highest gratitude to Professor Andrew Haas and Pioneer Writing Center for support throughout the process of researching and writing.
References
Bai, C. C. (2020). Shift of the surface plasmon polariton interference pattern. Optics Express, 28.
Begannovic, A. B. (2018, 5 15). handling of uncertainty due to interference fringe in FT-NIR
transmittance spectroscopy--performance comparison of interference elimination
techniques using glucose-water system Pergamon. spectrochimica acta, 208.
Chen J., S. Z. (2019, June 24th). Groove-structured mata-surface for patterned sub-diffractoin sound
focusing. Applied Physics Letters, 114. doi:10.1063/1.5096258
Machacova, K. S. (2021, 6 1). Fourier transform infrared spectroscopy and interference of volatile

organic compounds on measurements of methane fluxes at tree stems--a genenral
phenomenon for plant systems. New Phytoogists(6), 2100-2104.
Urone, P. P., Hinrichs, R., Driks, K., & Sharma, M. (2012). College Physics. OpenStax.