Lakhasly

We discussed the fact that light of wavelength comparable to or larger than the width of a slit spreads out in all forward directions upon passing through the slit.Note that the central bright maximum is twice as wide as the secondary maxima.The pattern consists of a broad, intense central band (called the central maximum), flanked by a series of narrower, less intense additional bands (called side maxima or secondary maxima) and a series of intervening dark bands (or minima).We can deduce some important features of this phe- nomenon by examining waves coming from various portions of the slit, as shown in Figure 2.A diffraction pattern consisting of light and dark areas is observed, somewhat similar to the interference patterns discussed earlier.Figure 2.= +- ?

We discussed the fact that light of wavelength comparable to or larger than the width of a
slit spreads out in all forward directions upon passing through the slit. We call this phenomenon
diffraction. This behavior indicates that light, once it has passed through a narrow slit, spreads
beyond the narrow path defined by the slit into regions that would be in shadow if light traveled
in straight lines. Other waves, such as sound waves and water waves, also have this property of
spreading when passing through apertures or by sharp edges.
We might expect that the light passing through a small opening would simply result in a
broad region of light on a screen, due to the spreading of the light as it passes through the
opening. We find something more interesting, however. A diffraction pattern consisting of light
and dark areas is observed, somewhat similar to the interference patterns discussed earlier. For
example, when a narrow slit is placed between a distant light source (or a laser beam) and a
screen, the light produces a diffraction pattern. The pattern consists of a broad, intense central
band (called the central maximum), flanked by a series of narrower, less intense additional bands
(called side maxima or secondary maxima) and a series of intervening dark bands (or minima).
Diffraction Patterns from Narrow Slits
Let us consider a common situation, that of light passing through a narrow opening
modeled as a slit, and projected onto a screen. To simplify our analysis, we assume that the
observing screen is far from the slit, so that the rays reaching the screen are approximately
parallel. This can also be achieved experimentally by using a converging lens to focus the parallel21
rays on a nearby screen. In this model, the pattern on the screen is called a Fraunhofer
diffraction pattern. Figure (1a )shows light entering a single slit from the left and diffracting
as it propagates toward a screen. Figure (1b) is a photograph of a single-slit Fraunhofer.
Figure1 (a) Fraunhofer diffraction pattern of a single slit.. (b) Photograph of a single-slit Fraunhofer diffraction pattern.
If the screen is brought close to the slit (and no lens is used), the pattern is a Fresnel
diffraction pat- tern. The Fresnel pattern is more difficult to analyze, so we shall restrict our
discussion to Fraunhofer diffraction.
A bright fringe is observed along the axis at  = 0, with alternating dark and bright fringes on
each side of the central bright fringe.
Until now, we have assumed that slits are point sources of light. In this section, we abandon that
assumption and see how the finite width of slits is the basis for under- standing Fraunhofer
diffraction. We can deduce some important features of this phe- nomenon by examining waves
coming from various portions of the slit, as shown in Figure 2. 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 . Based on this analysis, we recognize that a
diffraction pattern is actually an interference pattern, in which the different sources of light are
different portions of the single slit.
To analyze the diffraction pattern, it is convenient to divide the slit into two halves, as shown in
Figure 2. Keeping in mind that all the waves are in phase as they leave the slit, consider rays 122
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, as is that between rays 3 and 5.
Figure 2. Paths of light rays that encounter a narrow slit of width a and diffract toward a screen in the direction described by angle .
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. If this is true
for two such rays, then it 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
𝒂 𝟐
𝐬𝐢𝐧 𝜽 = ±
𝝀 𝟐
Or when
𝐬𝐢𝐧 𝜽 = ±
𝝀 𝒂
If we divide the slit into four equal parts and use similar reasoning, we find that the viewing screen
is also dark when
𝐬𝐢𝐧 𝜽 = ±
𝟐𝝀
𝒂
Likewise, we can divide the slit into six equal parts and show that darkness occurs on the screen
when23
𝐬𝐢𝐧 𝜽 = ±
𝟑𝝀
𝒂
Therefore, the general condition for destructive interference is
𝐬𝐢𝐧 𝜽𝒅𝒂𝒓𝒌 = 𝓶
𝛌 𝐚
(𝟏) 𝓶 = ±𝟏, ±𝟐, ±𝟑, … … ..
This equation gives the values of dark 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 3. 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 dark that satisfy Equation.1. Each bright-fringe peak lies approximately halfway
between its bordering dark-fringe minima. Note that the central bright maximum is twice as wide
as the secondary maxima.