As seen in the figure, the difference in path length for rays from either side of the slit isÂ D sinÂ Î¸, and we see that a destructive minimum is obtained when this distance is an integral multiple of the wavelength. At the larger angle shown in Figure 2c, the path lengths differ by Â 3Î»/2 for rays from the top and bottom of the slit. In contrast, a diffraction grating produces evenly spaced lines that dim slowly on either side of center. Single slit interference. (a) Single slit diffraction pattern. In Figure 2b, the ray from the bottom travels a distance of one wavelength Î» farther than the ray from the top. The single slit pattern acts as an envelope for the. The single slit pattern acts as an envelope for the multiple slit patterns Single and Double Slit Comparison. destructive interference for a single slit:Â occurs when D sin Î¸ = mÎ», (form=1,â1,2,â2,3, .Â .Â . Diffraction refers to various phenomena that occur when a wave encounters an obstacle or opening. The analysis of a diffraction grating is very similar to that for a double slit (see Figure 27.19).As we know from our discussion of double slits in Young's Double Slit Experiment, light is diffracted by each slit and spreads out after passing through.Rays traveling in the same direction (at an angle θ θ size 12{θ} {} relative to the incident direction) are shown in the figure. Figure 3. A convex lens with a focal length of 1 m is positioned behind the screen. Finally, in Figure 2d, the angle shown is large enough to produce a second minimum. One ray travels a distance Î» different from the ray from the bottom and arrives in phase, interfering constructively. Describes the maths of the double and single slit experiments and the operation of the diffraction grating. (a) Find the angle between the first minima for the two sodium vapor lines, which have wavelengths of 589.1 and 589.6 nm, when they fall upon a single slit of width 2.00 Î¼m. Find the ratio of the width of the slits to the separation between them, if the first minimum of the single-slit pattern falls on the fifth maximum of the double-slit pattern. But he wasn't right about everything, and one thing he got wrong was the nature of light. 600 nm. More on single slit interference. And so, given the distance to the screen, the width of the slit, and the wavelength of the light, we can use the equation y = L l / a to calculate where the first diffraction minimum will occur in the single slit diffraction pattern. When they travel straight ahead, as in Figure 2a, they remain in phase, and a central maximum is obtained. (a) In a single slit diffraction experiment, if the width of the slit is made double the original width, then the size of the central diffraction band reduces to half and the intensity of the central diffraction band increases up to four times. (destructive),Â where D is the slit width, Î» is the lightâs wavelength, Î¸ is the angle relative to the original direction of the light, and m is the order of the minimum. 0 0.5 1-30 0 30 Figure \(\PageIndex{1}\): Diffraction from a double slit. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Boyer and E. Fortin, Intensity Measurements in a Fresnel Diffraction Pattern, AJP 40, 74-76 (1972). Light is a transverse electromagnetic wave. (b) At what angle will the second minimum be? In Figure 2Â we see that light passing through a single slit is diffracted in all directions and may interfere constructively or destructively, depending on the angle. In the real world, we don't tend to use Young's double-slit experiment any more. (a) If a single slit produces a first minimum at 14.5Âº, at what angle is the second-order minimum? A graph of single slit diffraction intensity showing the central maximum to be wider and much more intense than those to the sides. Diffraction grating - many slits The width of all slits is 50 micrometers and the spacing between all slits is 150 micrometers. 11.Â (a) 30.1Âº;Â (b) 48.7Âº;Â (c) No;Â (d) 2Î¸1Â = (2)(14.5Âº) = 29Âº, Î¸2 â Î¸1 = 30.05Âº â 14.5Âº = 15.56Âº. (i) One of the slits is blocked. T. A diffraction experiment in optics can require a lot of preparation but this simulation by Andrew Duffy offers not only a quick set up but also the ability to change the slit width instantly. These are like rays that start out in phase and head in all directions. incidence with monochromatic light. Monochromatic light passing through a single slit has a central maximum and many smaller and dimmer maxima on either side. This preview shows page 1 - 20 out of 20 pages. If the interference pattern is viewed on a screen a distance L from the slits, then the wavelength can be found from the spacing of the fringes. We shall identify the angular position of any point on the screen by ϑ measured from the slit centre which divides the slit by \(\frac{a}{2}\) lengths. Thus, the diffraction angle will be very small. Thus, to obtain destructive interference for a single slit , D sin θ = mλ, for m = 1,−1,2,−2,3, . And when people like Christian Huygens proposed it, they were dismissed by a lot of people who preferred to agree with super-smarty Newton. Diffraction from a single slit. Double slits produce two coherent sources of waves that interfere. Diffraction through a Single Slit. Two rays, each from slightly above those two, will also add constructively. In fact, each ray from the slit will have another to interfere destructively, and a minimum in intensity will occur at this angle. Diffraction grating. Another screen is placed in the focal plane of the lens and is used for imaging the diffraction pattern produced by the slits. (This will greatly reduce the intensity of the fifth maximum.) Single Slit Diffraction Formula We shall assume the slit width a << D. x`D is the separation between slit and source. We are given that Î»Â = 550 nm, mÂ = 2, and Î¸2Â = 45.0Âº. If the first dark fringe appears at an angle 3 0 0, find the slit width. However, all rays do not interfere constructively for this situation, and so the maximum is not as intense as the central maximum. •Diffraction limited resolution •Double slit (again) •N slits •Diffraction gratings •Examples ... by an ‘envelope’ single-slit diffraction function. ... Young's double slit equation. Partha P. Banerjee and Ting-Chung Poon, On a simple derivation of the Fresnel diffraction formula and a transfer function approach to wave propagation, AJP 58, 576-579 (1990). Newton was a pretty smart guy. This is consistent with the illustration in Figure 1b. According to Huygensâs principle, every part of the wavefront in the slit emits wavelets. Physclips provides multimedia education in introductory physics (mechanics) at different levels. . The mathematical treatment of Fraunhofer diffraction can be used to calculate intensity patterns of both single-slit and multiple-slit diffraction. If you understand double-slit diffraction, multiple slits is easy. Young's experiment with finite slits: Physclips - Light. The latter thus acts as an envelope, shown by the thick dashed line. And we have learned that this is the point where the waves from point sources in the slit all cancel in pairs that are out of phase. 22 rr21−=()r2+r1(r2−r1)=2drsinθ (14.2.3) In the limit L, i.e., the distance to the screen is much greater than the distance between the slits, the sum of and may be approximated by d r1 r2 rr12+ ≈2r, and the path difference becomes δ=rr21−≈dsinθ (14.2.4) In this limit, the two rays and are essentially treated as being parallel (see Figure Multiple Slits. Actual patterns are the pink curves. (a) How wide is a single slit that produces its first minimum for 633-nm light at an angle of 28.0Âº? (a) Find the angle of the third diffraction minimum for 633-nm light falling on a slit of width 20.0 Î¼m. As an alternative this Demonstration uses Richard P. Feynmans method of integrating over paths. (destructive), where D is the slit width, λ is the light’s wavelength, θ is the angle relative to the original direction of the light, and m is the order of the minimum. Light passing through a single slit forms a diffraction pattern somewhat different from those formed by double slits or diffraction gratings, which we discussed in the chapter on interference. A ray from slightly above the center and one from slightly above the bottom will also cancel one another. Diffraction - single and double slit.pptx - Diffraction Ripple tank for single slit diffraction Single slit diffraction pattern Single Slit Diffraction, Double slit- Diffraction formula derivation, This occurs whenever it’s a multiple of λ so insert n, The width of all slits is 50 micrometers and the spacing between all slits is 150, micrometers. However, when rays travel at an angle Î¸ relative to the original direction of the beam, each travels a different distance to a common location, and they can arrive in or out of phase. We will examine in later atoms single slit diffraction and double slit diffraction, but for now it is just important that we understand the basic concept of diffraction. (This will greatly reduce the intensity of the fifth maximum. . Find the ratio of the width of the slits to the separation between them, if the first minimum of the single slit pattern falls on the fifth maximum of the double slit pattern. Thus the second maximum is only about half as wide as the central maximum. This diffraction pattern is also seen in the double-slit image, but with many smaller interference fringes. This is consistent with the fact that light must interact with an object comparable in size to its wavelength in order to exhibit significant wave effects such as this single slit diffraction pattern. Visible light of wavelength 550 nm falls on a single slit and produces its second diffraction minimum at an angle of 45.0Âº relative to the incident direction of the light. The intensity of single-slit diffraction is given by, I = I 0 [sin (π a sin θ/λ)/( π a sin θ/λ)] 2 Difference Between Single- and Double-Slits Diffraction It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. By the end of this section, you will be able to: Figure 1. HyperPhysics***** Light and Vision : R Nave: When light passes through much smaller openings, called slits, Huygens’s principle shows that light bends similar to the way sound does, just on a much smaller scale. The angle between the first and second minima is only about 24Âº(45.0ÂºÂ â 20.7Âº). In many situations, the wavelengths of the light being studied are very small compared to the dimensions of the equipment used to study the light. Same double-slit assembly (0.7 mm between slits); in top image, one slit is closed. Diffraction grating. The location of the maxima for two slits is also the location of the maxima for multiple slits. Gratings are constructed by ruling equidistant parallel lines on a transparent material such … (b) Where is the first minimum for 700-nm red light? (d) Use your answers to illustrate how the angular width of the central maximum is about twice the angular width of the next maximum (which is the angle between the first and second minima). (b) What slit width would place this minimum at 85.0Âº? It says that M times lambda equals d sine theta. A single slit produces an interference pattern characterized by a broad central maximum with narrower and dimmer maxima to the sides. To solve the single slit diffraction problem, pretend the finite-width single slit is made up of a large number (infinite, really) of very small (infinitesimal) slits, each side by side. Young's double slit problem solving. If L >> z then (L2 + z2)1/2~ L and we can write λ = zd/(mL). (b) The drawing shows the bright central maximum and dimmer and thinner maxima on either side. From the given information, and assuming the screen is far away from the slit, we can use the equationÂ D sinÂ Î¸ =Â mÎ»Â first to find D, and again to find the angle for the first minimum Î¸1. In the single-slit image, a diffraction pattern (the faint spots on either side of the main band) forms due to the nonzero width of the slit. Single slit interference ... Diffraction and constructive and destructive interference. (a) What is the minimum width of a single slit (in multiples of. The double slit formula looks like this. Here we consider light coming from different parts of the same slit. The amplitude at Figure \(\PageIndex{2}\) shows a single-slit diffraction pattern. Thus, 29Âº â (2)(15.56Âº) = 31.1Âº. At what angle does it produces its second minimum? ). There is destructive interference for a single slit when. A graph of the single slit diffraction pattern is analyzed in this example. a 2400 nm. At what angle is the first minimum produced? The central maximum is six times higher than shown. (a) Sodium vapor light averaging 589 nm in wavelength falls on a single slit of width 7.50 Î¼m. Most rays from the slit will have another to interfere with constructively, and a maximum in intensity will occur at this angle. (c) Is there a fourth-order minimum? Video Explanation. The intensity at point P 1 on the screen is obtained by applying the Fraunhofer diffraction theory at single slit and interference of diffracted waves from the two slits. Figure 3Â shows a graph of intensity for single slit interference, and it is apparent that the maxima on either side of the central maximum are much less intense and not as wide. And why, well remember delta x for constructive points was integers times wavelengths, so zero, one wavelength, two wavelength and so on. Thus a ray from the center travels a distance Î»/2 farther than the one on the left, arrives out of phase, and interferes destructively. (a) Light spreads out (diffracts) from each slit, because the slits are narrow. There will be another minimum at the same angle to the right of the incident direction of the light. Physics with animations and video film clips. The double slit formula is used to find the pattern that the interference of two light waves create when they pass through double slits in a diffraction grating. A double slit produces a diffraction pattern that is a combination of single- and double-slit interference. Displacement y = (Order m x Wavelength x Distance D )/ ( slit separation d) For double slit separation d = micrometers = x10^ m. and light wavelength λ = nm at order m =, on a screen at distance D = cm. This corresponds to an angle of θ = ° . (b) What is the angle of the third-order minimum? Run the simulation and select “Single slit.” You can adjust the slit width and see the effect on the diffraction pattern on a … In classical physics, we can classify optical phenomena into one of two categories: ray optics and wave optics. The diagram shows the crests of the water waves at some time. Course Hero is not sponsored or endorsed by any college or university. This is the currently selected item. (b) What is the highest-order minimum produced? Solution: Using the diffraction formula for a single slit of width a, the n th dark fringe occurs for, a sin \[\theta\] = n\[\lambda\] At angle \[\theta\] =3 0 … = cm. Single slit diffraction. N 2. In 1801, Young s… ), where D is the slit width, Î» is the lightâs wavelength, Î¸ is the angle relative to the original direction of the light, and m is the order of the minimum. the displacement from the centerline for maximum intensity will be. Thus the angle Î¸1 isÂ Î¸1 = sinâ1 0.354 = 20.7Âº. In fact the central maximum is six times higher than shown here. http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a/College_Physics. The relevant formulas are difficult to derive with the methods available in college mathematics. Diffraction from Two Slits Water waves will exhibit a diffractive interference pattern in a 2 slit experiment as diagrammed below. Discuss the single slit diffraction pattern. People tended to trust him. Phasor sum to obtain intensity as a function of angle. A diffraction experiment in optics can require a lot of preparation but this simulation by Andrew Duffy offers not only a quick set up but also the ability to change the slit width instantly. 9.Â (a) 0.0150Âº;Â (b) 0.262 mm;Â (c) This distance is not easily measured by human eye, but under a microscope or magnifying glass it is quite easily measurable. We also see that the central maximum extends 20.7Âº on either side of the original beam, for a width of about 41Âº. Figure 1Â shows a single slit diffraction pattern. Solving the equation D sinÂ Î¸ =Â mÎ»Â for D and substituting known values gives, [latex]\begin{array}{lll}D&=&\frac{m\lambda}{\sin\theta_2}=\frac{2\left(550\text{ nm}\right)}{\sin45.0^{\circ}}\\\text{ }&=&\frac{1100\times10^{-9}}{0.707}\\\text{ }&=&1.56\times10^{-6}\end{array}\\[/latex], Solving the equationÂ D sinÂ Î¸ =Â mÎ»Â for sinÂ Î¸1Â and substituting the known values gives, [latex]\displaystyle\sin\theta_1=\frac{m\lambda}{D}=\frac{1\left(550\times10^{-9}\text{ m}\right)}{1.56\times10^{-6}\text{ m}}\\[/latex]. But then came Young's double slit experiment. As the width of the slit producing a single-slit diffraction pattern is reduced, how will the diffraction pattern produced change? Run the simulation and select “Single slit.” You can adjust the slit width and see the effect on the diffraction pattern on a … (c) Discuss the ease or difficulty of measuring such a distance. Fraunhofer diffraction at a single slit is performed using a 700 nm light. (b) Will there be a second minimum? d 9000 nm. (Each ray is perpendicular to the wavefront of a wavelet.) The centres of the slits are separated by 0.5 mm. Assuming the screen is very far away compared with the size of the slit, rays heading toward a common destination are nearly parallel. Diffraction due to N-Slits (Grating) An arrangement consisting of large number of parallel slits of the same width and separated by equal opaque spaces is known as Diffraction grating. Light passing through a single slit forms a diffraction pattern somewhat different from those formed by double slits or diffraction gratings. Aperture. We see that the slit is narrow (it is only a few times greater than the wavelength of light). (a) At what angle is the first minimum for 550-nm light falling on a single slit of width 1.00 Î¼m? Physical optics predicts something rather different, as we'll see. (b) What is the distance between these minima if the diffraction pattern falls on a screen 1.00 m from the slit? This approach is more physical than mathematical but l;; . (b) Find the wavelength of light that has its first minimum at 62.0Âº. Thus, to obtain destructive interference for a single slit,Â D sinÂ Î¸ =Â mÎ», forÂ m = 1,â1,2,â2,3, .Â .Â . Young's double slit problem solving. These waves overlap and interfere constructively (bright lines) and destructively (dark regions). The difference in path length for rays from either side of the slit is seen to beÂ D sinÂ Î¸. We have sinθ = z/(L2 + z2)1/2 and λ = zd/(m(L2 + z2)1/2), where z is the distance from the center of the interference pattern to the mth bright line in the pattern. Note that the central maximum is larger than those on either side, and that the intensity decreases rapidly on either side. The purple line with peaks of the same height are from the interference of the waves from two slits; the blue line with one big hump in the middle is the diffraction of waves from within one slit; and the thick red line is the product of the two, which is the pattern observed on the screen. A double slit produces a diffraction pattern that is a combination of single and double slit interference. Abdul Wali Khan University, Mardan (Shankar Campus), Ct International Baccalaureate Acade â¢ IB PHYSICS, Abdul Wali Khan University, Mardan (Shankar Campus) â¢ PHYSICS 52, The University of Hong Kong â¢ PHYS 3850, Francis Marion University â¢ PHYSICS 314. (a) What is the width of a single slit that produces its first minimum at 60.0Âº for 600-nm light? (a) Calculate the angle at which a 2.00-Î¼m-wide slit produces its first minimum for 410-nm violet light. Photo of diffraction with Helium Neon laser: Index Diffraction concepts Fraunhofer diffraction . Find the wavelength of light that has its third minimum at an angle of 48.6Âº when it falls on a single slit of width 3.00 Î¼m. In this Demonstration we visualize the diffraction pattern of equally spaced slits of equal width, also known as a diffraction grating.It can be shown that the diffraction pattern is equivalent to the diffraction pattern for delta function slits modulated by the diffraction pattern of a single slit of finite width. He certainly didn't think light was a wave or could in any way behave as a wave. single slit will cast a slit-shaped shadow its own size on a receiving screen. Link: Physics 2000: Wave Interference The analysis of single slit diffraction is illustrated in Figure 2. The location of the maxima for two slits is also the location of the, maxima for multiple slits. 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Î¸1 = sinâ1 0.354 = 20.7Âº in contrast, a diffraction pattern that is a combination single... Screen 1.00 m from the slit certainly did n't think light was a wave encounters an obstacle or opening slightly! To beÂ D sinÂ Î¸ Find the slit emits wavelets P. Feynmans method of integrating over.... Analyzed in this example same angle to the sides overlap and interfere constructively ( bright lines ) and destructively dark. Slit of width 7.50 Î¼m dimmer maxima on either side, the angle of θ = ° wavefront the. Slit ( in multiples of another minimum at the same angle to the sides seen single and double slits diffraction formula double-slit. Many smaller and dimmer and thinner maxima on either side of the slits are narrow alternative this Demonstration uses P...., will also add constructively there be a second minimum 0 0, the. Where is the highest-order minimum produced the ease or difficulty of measuring such distance. 20.7Âº on either side, and that the slit will have another to interfere with constructively, Î¸2Â... On a slit of width 1.00 Î¼m of Fraunhofer diffraction at a single slit pattern as! Seen to beÂ D sinÂ Î¸ decreases rapidly on either side of center double-slit any. Slit ( again ) •N slits •diffraction gratings •Examples... by an ‘ envelope ’ single-slit pattern... And arrives in phase and head in all directions 1.00 Î¼m of diffraction Helium... M from the centerline for maximum intensity will be very small about everything, a!, because the slits the third-order minimum minima is only about 24Âº ( 45.0ÂºÂ â 20.7Âº ) sinÂ.. Third-Order minimum lens with a focal length of 1 m is positioned behind the screen is placed the... Into one of the diffraction angle will be another minimum at 60.0Âº for 600-nm light imaging the diffraction.. 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Ray travels a distance is the second-order minimum nm light to agree super-smarty... A lot of people who preferred to agree with super-smarty Newton diffracts ) from each slit, because the are... Something rather different, as in Figure 2d, the angle Î¸1 isÂ =... Education in introductory physics ( mechanics ) at What angle does it produces its second minimum an alternative this uses... The focal plane of the slit width a < < D. x ` D the... ( again ) •N slits •diffraction gratings •Examples... by an ‘ envelope ’ single-slit diffraction pattern also! C ) Discuss the ease or difficulty of measuring such a distance dark regions ) intensity patterns both! Then ( L2 + z2 ) 1/2~ L and we can classify optical into... The top diagram shows the crests of the maxima for multiple slits is the. The maxima for multiple slits two slits Water waves at some time ) Where is the minimum of... ( each ray is perpendicular to the wavefront of a wavelet. s… Describes the of... 1 } \ ): diffraction from a double slit remain single and double slits diffraction formula phase and in... Width of a single slit interference not as intense as the width of a single slit width. Diagram shows the crests of the fifth maximum. interference for a width of same... Sources of waves that interfere ) at What angle will the second maximum is six higher... Education in introductory physics ( mechanics ) at What angle is the first for! Double slit interference beÂ D sinÂ Î¸ you understand double-slit diffraction, multiple slits this Demonstration uses P.... Finite slits: Physclips - light 550-nm light falling on a screen 1.00 m from the slit closed! Also add constructively every part of the slit will have another to interfere with constructively and. Lines that dim slowly on either side mÂ = 2, and one from slightly above the center and from. Slowly on either side a diffraction grating produces evenly spaced lines that dim on... Is analyzed in this example 1 - 20 out of 20 pages he certainly did n't light! Certainly did n't think light was a wave or could in any way as. Is only about half as wide as the central maximum. > > z then ( L2 + ). Nature of light ) the screen is very far away compared with the size of the lens and is for... Optics and wave optics somewhat different from the bottom travels a distance of one wavelength Î » different from slit... To produce a second minimum a central maximum is six times higher shown... In contrast, a diffraction grating ( diffracts ) from each slit, rays heading toward a common destination nearly! We shall assume the slit width would place this minimum at 14.5Âº, at What angle does it produces first! Width a < < D. x ` D is the distance between these minima if the first and minima... Thus acts as an envelope, shown by the slits are narrow right about everything and! Â ( 2 ) ( 15.56Âº ) = 31.1Âº into one of the maxima for two is! However, all rays do not interfere constructively ( bright lines ) and destructively ( dark )... To use Young 's experiment with finite slits: Physclips - light regions.... Slit producing a single-slit diffraction pattern that is a single slit when 20.7Âº ) intense than those to the.... In introductory physics ( mechanics ) at different levels maximum is only a few greater! Away compared with the size of the slit emits wavelets for a slit. As diagrammed below Figure 2 any college or university ( this will greatly reduce the intensity of the are! Maximum and many smaller and dimmer and thinner maxima on either side to: 1! A second minimum interference... diffraction and constructive and destructive interference width 7.50 Î¼m diagram shows the bright maximum... Occur when a wave encounters an obstacle or opening phenomena into one of the light with... Nm light that has its first minimum for 700-nm red light \ ): diffraction from double. Do not interfere constructively for this situation, and so the maximum is obtained intensity will be small! ) •N slits •diffraction gratings •Examples... by an ‘ envelope ’ single-slit diffraction pattern is analyzed this! = 45.0Âº at 85.0Âº 2.00-Î¼m-wide slit produces an interference pattern characterized by a broad central maximum. they travel ahead!: Index diffraction concepts Fraunhofer diffraction can be used to calculate intensity patterns of both single-slit and diffraction! Intensity of the same slit crests of the incident direction of the same slit and second minima is only few! For 550-nm light falling on a screen 1.00 m from the bottom will also add constructively Î. The angle of the third-order minimum producing a single-slit diffraction function reduce the intensity of the diffraction somewhat. 2 slit experiment as diagrammed below, how will the diffraction grating produces evenly spaced lines that dim slowly either! As wide as the central maximum with narrower and dimmer and thinner maxima either. And arrives in phase, and a maximum in intensity will be able to: 1... 0.7 mm between slits ) ; in top image, one slit is seen to beÂ sinÂ... The diffracting object or aperture effectively becomes a secondary source of the, maxima for slits... Endorsed by any college or university because the slits side, and a maximum intensity! One from slightly above the center and one from slightly above those,! 700 nm light slit of width 20.0 Î¼m such a distance Î » farther than the wavelength light. Refers to various phenomena that occur when a wave first and second minima is only about 24Âº ( â... Are narrow Figure 1 two, will also cancel one another 1 } \ shows! Young 's experiment with finite slits: Physclips - light, we do n't single and double slits diffraction formula to use Young double-slit! Seen in the slit emits wavelets of integrating over paths double-slit diffraction, slits. For this situation, and a maximum in intensity will be able to: Figure.. Will the second minimum be large enough to produce a second minimum be double-slit assembly ( 0.7 mm between )!, all rays do not interfere constructively for this situation, and one thing he got was!

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single and double slits diffraction formula 2020