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To understand the double slit interference pattern, we consider how two waves travel from the slits to the screen, as illustrated in [link] . Each slit is a different distance from a given point on the screen. Thus different numbers of wavelengths fit into each path. Waves start out from the slits in phase (crest to crest), but they may end up out of phase (crest to trough) at the screen if the paths differ in length by half a wavelength, interfering destructively as shown in [link] (a). If the paths differ by a whole wavelength, then the waves arrive in phase (crest to crest) at the screen, interfering constructively as shown in [link] (b). More generally, if the paths taken by the two waves differ by any half-integral number of wavelengths [ ( 1 / 2 ) λ size 12{ \( 1/2 \) λ} {} , ( 3 / 2 ) λ size 12{ \( 3/2 \) λ} {} , ( 5 / 2 ) λ size 12{ \( 5/2 \) λ} {} , etc.], then destructive interference occurs. Similarly, if the paths taken by the two waves differ by any integral number of wavelengths ( λ size 12{λ} {} , 2 λ size 12{2λ} {} , 3 λ size 12{3λ} {} , etc.), then constructive interference occurs.

Take-home experiment: using fingers as slits

Look at a light, such as a street lamp or incandescent bulb, through the narrow gap between two fingers held close together. What type of pattern do you see? How does it change when you allow the fingers to move a little farther apart? Is it more distinct for a monochromatic source, such as the yellow light from a sodium vapor lamp, than for an incandescent bulb?

Both parts of the figure show a schematic of a double slit experiment. Two waves, each of which is emitted from a different slit, propagate from the slits to the screen. In the first schematic, when the waves meet on the screen, one of the waves is at a maximum whereas the other is at a minimum. This schematic is labeled dark (destructive interference). In the second schematic, when the waves meet on the screen, both waves are at a minimum.. This schematic is labeled bright (constructive interference).
Waves follow different paths from the slits to a common point on a screen. (a) Destructive interference occurs here, because one path is a half wavelength longer than the other. The waves start in phase but arrive out of phase. (b) Constructive interference occurs here because one path is a whole wavelength longer than the other. The waves start out and arrive in phase.

[link] shows how to determine the path length difference for waves traveling from two slits to a common point on a screen. If the screen is a large distance away compared with the distance between the slits, then the angle θ size 12{θ} {} between the path and a line from the slits to the screen (see the figure) is nearly the same for each path. The difference between the paths is shown in the figure; simple trigonometry shows it to be d sin θ size 12{d`"sin"θ} {} , where d size 12{d} {} is the distance between the slits. To obtain constructive interference for a double slit    , the path length difference must be an integral multiple of the wavelength, or

d sin θ = , for m = 0, 1, 1, 2, 2, (constructive). size 12{d`"sin"θ=mλ,`m=0,`1,`-1,`2,`-2,` dotslow } {}

Similarly, to obtain destructive interference for a double slit    , the path length difference must be a half-integral multiple of the wavelength, or

d sin θ = m + 1 2 λ , for m = 0, 1, 1, 2, 2, (destructive), size 12{d`"sin"θ= left (m+ { {1} over {2} } right )λ,`m=0,`1,` - 1,`2,` - 2,` dotslow } {}

where λ size 12{λ} {} is the wavelength of the light, d size 12{d} {} is the distance between slits, and θ size 12{θ} {} is the angle from the original direction of the beam as discussed above. We call m size 12{m} {} the order    of the interference. For example, m = 4 size 12{m=4} {} is fourth-order interference.

The figure is a schematic of a double slit experiment, with the scale of the slits enlarged to show the detail. The two slits are on the left, and the screen is on the right. The slits are represented by a thick vertical line with two gaps cut through it a distance d apart. Two rays, one from each slit, angle up and to the right at an angle theta above the horizontal. At the screen, these rays are shown to converge at a common point. The ray from the upper slit is labeled l sub one, and the ray from the lower slit is labeled l sub two. At the slits, a right triangle is drawn, with the thick line between the slits forming the hypotenuse. The hypotenuse is labeled d, which is the distance between the slits. A short piece of the ray from the lower slit is labeled delta l and forms the short side of the right triangle. The long side of the right triangle is formed by a line segment that goes downward and to the right from the upper slit to the lower ray. This line segment is perpendicular to the lower ray, and the angle it makes with the hypotenuse is labeled theta. Beneath this triangle is the formula delta l equals d sine theta.
The paths from each slit to a common point on the screen differ by an amount d sin θ size 12{d`"sin"θ} {} , assuming the distance to the screen is much greater than the distance between slits (not to scale here).

The equations for double slit interference imply that a series of bright and dark lines are formed. For vertical slits, the light spreads out horizontally on either side of the incident beam into a pattern called interference fringes, illustrated in [link] . The intensity of the bright fringes falls off on either side, being brightest at the center. The closer the slits are, the more is the spreading of the bright fringes. We can see this by examining the equation

Practice Key Terms 5

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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