Until now, we have assumed that slits are point sources of light. In this topic, we abandon that assumption and see how the finite width of slits is the basis for understanding Fraunhofer diffraction. We can deduce some important features of this phenomenon by examining waves coming from various portions of the slit, as shown in Figure 38.5. According to Huygens’s principle, each portion of the slit acts as a source of light waves. Hence, light from one portion of the slit can interfere with light from another portion, and the resultant light intensity on a viewing screen depends on the direction θ. To analyze the diffraction pattern, it is convenient to divide the slit into two halves, as shown in Figure 38.5. Keeping in mind that all the waves are in phase as they leave the slit, consider rays 1 and 3. As these two rays travel toward a viewing screen far to the right of the figure, ray 1 travels farther than ray 3 by an amount equal to the path difference (a/2) sin θ, where a is the width of the slit. Similarly, the path difference between rays 2 and 4 is also (a/2) sin θ. If this path difference is exactly half a wavelength (corresponding to a phase difference of 180°), then the two waves cancel each other and destructive interference results. This is true for any two rays that originate at points separated by half the slit width because the phase difference between two such points is 180°. Therefore, waves from the upper half of the slit interfere destructively with waves from the lower half when or when If we divide the slit into four equal parts and use similar reasoning, we find that the viewing screen is also dark when Likewise, we can divide the slit into six equal parts and show that darkness occurs on the screen when Therefore, the general condition for destructive interference is This equation gives the values of θ for which the diffraction pattern has zero light intensity—that is, when a dark fringe is formed. However, it tells us nothing about the variation in light intensity along the screen. The general features of the intensity distribution are shown in Figure 38.6. A broad central bright fringe is observed; this fringe is flanked by much weaker bright fringes alternating with dark fringes. The various dark fringes occur at the values of θ that satisfy Equation 38.1. Each bright-fringe peak lies approximately halfway between its bordering darkfringe minima. Note that the central bright maximum is twice as wide as the secondary maxima.

Intensity of single-slit diffraction patterns

We can use phasors to determine the light intensity distribution for a single-slit diffraction pattern. Imagine a slit divided into a large number of small zones, each of width Δy as shown in Figure 38.7. Each zone acts as a source of coherent radiation, and each contributes an incremental electric field of magnitude ΔE at some point P on the screen. We obtain the total electric field magnitude E at point P by summing the contributions from all the zones. The light intensity at point P is proportional to the square of the magnitude of the electric field.
The incremental electric field magnitudes between adjacent zones are out of phase with one another by an amount Δβ, where the phase difference Δβ is related to the path difference Δy sin θ between adjacent zones by the expression
To find the magnitude of the total electric field on the screen at any angle θ, we sum the incremental magnitudes ΔE due to each zone. For small values of θ, we can assume that all the ΔE values are the same. It is convenient to use phasor diagrams for various angles, as shown in Figure 38.8. When θ = 0, all phasors are aligned as shown in Figure 38.8a because all the waves from the various zones are in phase. In this case, the total electric field at the center of the screen is E_{0} = N ΔE, where N is the number of zones. The resultant magnitude E_{R} at some small angle θ is shown in Figure 38.8b, where each phasor differs in phase from an adjacent one by an amount Δβ. In this case, E_{R} is the vector sum of the incremental magnitudes and hence is given by the length of the chord. Therefore, E_{R} < E_{0}. The total phase difference β between waves from the top and bottom portions of the slit is
Where a = N Δy is the width of the slit.
Figure 38.8 Phasor diagrams for obtaining the various maxima and minima of a single-slit diffraction pattern.

As θ increases, the chain of phasors eventually forms the closed path shown in Figure 38.8c. At this point, the vector sum is zero, and so E_{R} = 0, corresponding to the first minimum on the screen. Noting that β = N Δβ = 2π in this situation, we see from Equation 38.3 that
That is, the first minimum in the diffraction pattern occurs where sin θ = λ/a; this is in agreement with Equation 38.1.

At greater values of θ, the spiral chain of phasors tightens. For example, Figure 38.8d represents the situation corresponding to the second maximum, which occurs when β = 360° + 180° = 540° (3π rad). The second minimum (two complete circles, not shown) corresponds to β = 720° (4π rad), which satisfies the condition sin θ = 2λ/a.

We can obtain the total electric field magnitude E_{R} and light intensity I at any point P on the screen in Figure 38.7 by considering the limiting case in which Δy becomes infinitesimal (dy) and N approaches ∞. In this limit, the phasor chains in Figure 38.8 become the red curve of Figure 38.9. The arc length of the curve is E_{0} because it is the sum of the magnitudes of the phasors (which is the total electric field magnitude at the center of the screen). From this figure, we see that at some angle θ, the resultant electric field magnitude E_{R} on the screen is equal to the chord length. From the triangle containing the angle β/2, we see that
where R is the radius of curvature. But the arc length E_{0} is equal to the product Rβ, where β is measured in radians. Combining this information with the previous expression gives
Because the resultant light intensity I at point P on the screen is proportional to the square of the magnitude E_{R} , we find that
where I_{max} is the intensity at θ = 0 (the central maximum). Substituting the expression for β (Eq. 38.3) into Equation 38.4, we have
From this result, we see that minima occur when
in agreement with Equation 38.1.

Figure 38.10a represents a plot of Equation 38.5, and Figure 38.10b is a photograph of a single-slit Fraunhofer diffraction pattern. Note that most of the light intensity is concentrated in the central bright fringe.

Intensity of two-slit diffraction patterns

When more than one slit is present, we must consider not only diffraction due to the individual slits but also the interference of the waves coming from different slits. You may have noticed the curved dashed line in Figure 37.13, which indicates a decrease in intensity of the interference maxima as θ increases. This decrease is due to diffraction. To determine the effects of both interference and diffraction, we simply combine Equation 37.12 and Equation 38.5:
Although this formula looks complicated, it merely represents the diffraction pattern (the factor in brackets) acting as an “envelope” for a two-slit interference pattern (the cosine-squared factor), as shown in Figure 38.11.

Equation 37.2 indicates the conditions for interference maxima as d sin θ = mλ, where d is the distance between the two slits. Equation 38.1 specifies that the first diffraction minimum occurs when a sin θ = λ, where a is the slit width. Dividing Equation 37.2 by Equation 38.1 (with m = 1) allows us to determine which interference maximum coincides with the first diffraction minimum:
In Figure 38.11, d/a = μm/3.0 μm = 6. Thus, the sixth interference maximum (if we count the central maximum as m = 0) is aligned with the first diffraction minimum and cannot be seen.
Figure 38.8 The combined effects of diffraction and interference. This is the pattern produced when 650-nm light waves pass through two 3.0-m slits that are 18 m apart. Notice how the diffraction pattern acts as an “envelope” and controls the intensity of the regularly spaced interference maxima.

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