# Huygens Diffraction

Diffraction, the spreading of waves around obstacles.Diffraction takes place with sound; with electromagnetic radiation, such as light, X-rays, and gamma rays; and with very small moving particles such as atoms, neutrons, and electrons, which show wavelike properties.One consequence of diffraction is that sharp shadows are not produced. The phenomenon is the result of interference (i.e., when. Huygens Principle and Diffraction When light passes through an aperture, every point on the light wave within the aperture can be viewed as a source creating a circular wave that propagates outward from the aperture. The aperture thus creates a new wave source that propagates in the form of a circular wavefront. Huygens’s principle applied to a plane wave front striking an opening. The edges of the wave front bend after passing through the opening, a process called diffraction. The amount of bending is more extreme for a small opening, consistent with the fact that wave characteristics are most noticeable for interactions with objects about the same. Incident wavefront, decomposed into Huygens point sources transmitted wavefront, decomposed into Huygens point sources Each Huygens point source is a divergent spherical wave; therefore, after propagating by distance zin free space, it can be expressed as The entire propagated wavefront is the superposition of the propagated Huygens point sources at the wavefront transmitted by the transparency, so it may be expressed as This result is known as. Diffraction, shadows, beams, Huygens' construction. Diffraction of light and sound. Water waves in a ripple tank. Shadows and beams with water waves of short wavelength.

## Huygens principle of Wave Theory Huygens Principle helps us to find the position of a given wavefront at any instant if we know its present position. A wavefront is defined as the continuous locus of all particles of a medium vibrating in the same phase. For example, when a stone is dropped into a Pound of water, circular ripples are produced on the surface of the water. These ripples spread outward. Similarly, a point source of light emits a spherical wave with the source at the center, and these waves spread outward in all directions.

if the medium is isotropic, then the disturbance after time t will lie on a sphere of radius vt, where v is the velocity of the wave in the medium. Since the disturbance from the source moves in all directions, therefore, the particles of the medium at a distance vt from the center of the source will vibrate in the same phase. Any surface passing through these points or particles is called a wavefront. The wavefront is spherical as shown in figure 1.

At a very large distance from the source, any small part of the spherical wavefront appears plane, and part of the spherical wavefront is called plane wavefront as shown in figure 2.

According to Huygens principle of secondary wavelets :
All the points on the primary wavefront can be regarded as a point of new disturbance or source. Each point gives rise to secondary wavelets which spread out in all directions with the velocity of the wave.

The surface obtained by joining the tangents drawn on each secondary wavelet will be the new position of the wavefront after a time t.

This can be understood with the following example. Consider a point source S which gives the spherical wavefront AB at the instant t = 0. According to Huygens principle, every point on this wavefront is considered as the center of secondary wavelets. Since the medium is homogeneous and isotropic, the medium has the same property at all points and the speed of propagation of the wave is the same in all the directions, so the light wave travels an equal distance in equal time in all directions.
After time t, each secondary wavelet will travel a distance equal to vt. By taking every point as the center, draw a sphere of radius equal to vt. Now draw the tangents on every sphere and join them by freehand. Thus, we get the curve CD which is again a sphere with S as its Centre. Hence the new position of the wavefront AB after time t is CD.

## Drawbacks of Huygens principle

Huygens principle suffers from two serious drawbacks:
Why is the wavefront EF which shows backward propagation of light not considered?
Why is only the effective portion of the wavelets are touched by the enveloping surface and a large portion untouched by the surface is taken as ineffective?
The explanation for the first drawback was given on the basis of Stokes’s law. According to this, the amplitude and hence the intensity due to the secondary wavelets depend upon the obliquity ( 1+ cosθ ), where θ is the angle between the line drawn from the origin of the secondary wavelets to the point where the intensity is to be determined, and the normal to the wavefront drawn outwards.

Thus, along the forward direction or along the direction of the normal, θ=0, and hence the intensity has some specific value. Along a direction opposite to the normal, along the backward direction, θ=180 and the obliquity factor ( 1+cosθ ) = ( 1+cos 180 ) = 1 – 1 = 0. Hence the resultant intensity in the backward direction is zero and, therefore, back wavefront EF does not exist.

The second drawback can be explained on the basis of the principle of superposition. Due to the superposition of secondary wavelets, their major portions cancel Each Other effect by interference. Hence only the effective portions cancel Each Other effect by interference. Therefore, the only effective portions left are those where these secondary wavelets do not superpose each other.

## Diffraction

The phenomenon by virtue of which the light bends into the region of Geometrical shadow and there is a periodic variation in the intensity of light outside the Geometrical shadow is called diffraction of light.

To understand the diffraction more clearly, let us consider a monochromatic source (S) of light placed near the narrow slit AB as shown in the figure. According to geometrical optics, the region between P and Q on the screen should be uniformly illuminated and the portion above point P and below point Q should be completely dark. The portion above point P and below point Q is known as the geometrical shadow.

However, observations taken minutely show that the region between P and Q is not uniformly illuminated and there is also some intensity of light inside the geometrical Shadow if the aperture of slit AB is small as compared to the wavelength of light used. It is observed that if the aperture AB is made sufficiently narrow, the reason PQ is marked by equidistant dark and bright fringes where there exist a few and unequally spaced fringes above P and below Q (but near these points), in the regions of geometrical shadow.

Thus bending of lightwave when it passes through a narrow slit is known as the phenomena of diffraction and the intensity distribution on the screen is known as the diffraction pattern.

## The phenomena of diffraction are classified into two categories

1. Fresnel class of diffraction.
2. Fraunhofer class of diffraction.

Fresnel class of diffraction.

In the fresnel class of diffraction, the source and the screen are placed at a finite distance from the slit. This is also known as near field diffraction. In this case, incident wavefronts and diffracted wavefronts are taken as spherical wavefronts.

Fraunhofer class of diffraction.

In Fraunhofer diffraction, the source and the screen are placed at an infinite distance from the slit. This can be achieved by placing the screen on the focal plane of another convex lens as shown in fig. Since the source and the screen are infinitely distant apart, so this is also known as far-field diffraction. In Fraunhofer diffraction, incident and reflected wavefronts are planes.

## Huygens Fresnel diffraction theory

According to Huygens principle of wave theory, the future position of the wavefront may be derived from a past position by considering every point of the wavefront to be a source of secondary waves. The progress of the wavefront can often be described by ray optics. We now turn to situations where rays provide totally inadequate description.

Diffraction effects occur when part of the wavefront is removed by an obstacle in an otherwise infinite wavefront, or when all but a part of the wavefront is removed by an aperture or stop. The importance of the diffraction effect depends on the size of the obstacle or aperture compared with the wavelength.

The wavefront passing through the aperture is initially divided into equal elements small compared with the wavefront. Each such element is considered to be a source of secondary Huygens wave. To determine the amplitude at any point beyond the aperture these waves are summed. The phase of wavelets is behind that of the portion of the original wavefront emitting it by an amount determined by its distance and the wavelength λ.

This summation of Huygens wavelets taking account of their phase is referred to as Huygens Fresnel diffraction theory. It was Huygens who first saw the new wavefront as the envelope of the wavelets. Fresnel contributed the essential idea of the interference of the Huygens wavelets.

Suppose W is an incident wavefront and AB is the aperture. P is a point where we want to study the effect. C is any point in the aperture as shown in the figure.

let CP = L

The amplitude and phase at P may be considered to be the sum of Huygens wavelets from points such as C. The phase at P of the wavelet originating at C is behind that at B by 2πl/λ radians.

To calculate the amplitude at P all such contributions from each equal element have to be taken. Such a process first involves the calculation of the phase of each contribution, relative to some phase reference, often the face of the contribution from the center of the aperture. Then follows the summation of all such contributions either by an integral, a series, or possibly by the application of the trigonometry.

Learning Objectives

By the end of this section, you will be able to:

• Describe Huygens’s principle
• Use Huygens’s principle to explain the law of reflection
• Use Huygens’s principle to explain the law of refraction
• Use Huygens’s principle to explain diffraction

So far in this chapter, we have been discussing opticalphenomena using the ray model of light. However, some phenomenarequire analysis and explanations based on the wave characteristicsof light. This is particularly true when the wavelength is notnegligible compared to the dimensions of an optical device, such asa slit in the case of diffraction. Huygens’s principle isan indispensable tool for this analysis.  Figure (PageIndex{1}) shows how a transverse wave looks asviewed from above and from the side. A light wave can be imaginedto propagate like this, although we do not actually see it wigglingthrough space. From above, we view the wave fronts (or wave crests)as if we were looking down on ocean waves. The side view would be agraph of the electric or magnetic field. The view from above isperhaps more useful in developing concepts about wave optics.

The Dutch scientist Christiaan Huygens (1629–1695) developed auseful technique for determining in detail how and where wavespropagate. Starting from some known position, Huygens’s principlestates that every point on a wave front is a source of waveletsthat spread out in the forward direction at the same speed as thewave itself. The new wave front is tangent to all of thewavelets.

Figure (PageIndex{2}) shows how Huygens’s principle isapplied. A wave front is the long edge that moves, for example,with the crest or the trough. Each point on the wave front emits asemicircular wave that moves at the propagation speed (v). We candraw these wavelets at a time (t) later, so that they have moveda distance (s=vt). The new wave front is a plane tangent to thewavelets and is where we would expect the wave to be a time (t)later. Huygens’s principle works for all types of waves, includingwater waves, sound waves, and light waves. It is useful not only indescribing how light waves propagate but also in explaining thelaws of reflection and refraction. In addition, we will see thatHuygens’s principle tells us how and where light raysinterfere.

## Huygens Diffraction Theory

Figure (PageIndex{3}) shows how a mirror reflects an incomingwave at an angle equal to the incident angle, verifying the law ofreflection. As the wave front strikes the mirror, wavelets arefirst emitted from the left part of the mirror and then from theright. The wavelets closer to the left have had time to travelfarther, producing a wave front traveling in the directionshown.

## Refraction

The law of refraction can be explained by applying Huygens’sprinciple to a wave front passing from one medium to another(Figure (PageIndex{4})). Each wavelet in the figure was emittedwhen the wave front crossed the interface between the media. Sincethe speed of light is smaller in the second medium, the waves donot travel as far in a given time, and the new wave front changesdirection as shown. This explains why a ray changes direction tobecome closer to the perpendicular when light slows down. Snell’slaw can be derived from the geometry in Figure (PageIndex{5})(Example (PageIndex{1})).

Example (PageIndex{1}): Deriving the Law ofRefraction

By examining the geometry of the wave fronts, derive the law ofrefraction. Strategy

Consider Figure (PageIndex{5}), which expands upon Figure(PageIndex{4}). It shows the incident wave front just reachingthe surface at point A, while point B is stillwell within medium 1. In the time (Δt) it takes for a waveletfrom (B) to reach (B') on the surface at speed (v_1=c/n_1), awavelet from (A) travels into medium 2 a distance of(AA'=v_2Δt), where (v_2=c/n_2). Note that in this example,(v_2) is slower than (v_1) because (n_1<n_2).

Solution

The segment on the surface AB' is shared by both the triangleABB' inside medium 1 and the triangle AA′B′ inside medium 2. Notethat from the geometry, the angle ∠BAB' is equal to the angle ofincidence, (θ_1). Similarly, (∠AB'A') is (θ_2).

The length of AB' is given in two ways as

[AB'=dfrac{BB'}{sin θ_1}=dfrac{AA'}{sin θ_2}.nonumber]

Inverting the equation and substituting AA'=cΔt/n2from above and similarly (BB'=cΔt/n_1), we obtain

[dfrac{sin θ_1}{cDelta t/n_1}=dfrac{sin θ_2}{cDeltat/n_2}. nonumber]

Cancellation of (cΔt) allows us to simplify this equation intothe familiar form

[underbrace{n_1sin θ_1=n_2 sin θ_2}_{text{Snell's law}}.nonumber]

Significance

Although the law of refraction was established experimentally bySnell, its derivation here requires Huygens’s principle and theunderstanding that the speed of light is different in differentmedia.

Exercise (PageIndex{1})

In Example (PageIndex{1}), we had (n_1<n_2). If (n_2)were decreased such that (n_1>n_2) and the speed of light inmedium 2 is faster than in medium 1, what would happen to thelength of AA'? What would happen to the wave front A'B' and thedirection of the refracted ray? Mysqli_affected_rows.

AA′ becomes longer, A'B' tilts further away from the surface,and the refracted ray tilts away from the normal.

This appletby Walter Fendt shows an animation of reflection and refractionusing Huygens’s wavelets while you control the parameters. Be sureto click on “Next step” to display the wavelets. You can see thereflected and refracted wave fronts forming.

## Diffraction

What happens when a wave passes through an opening, such aslight shining through an open door into a dark room? For light, weobserve a sharp shadow of the doorway on the floor of the room, andno visible light bends around corners into other parts of the room.When sound passes through a door, we hear it everywhere in the roomand thus observe that sound spreads out when passing through suchan opening (Figure (PageIndex{6})). What is the differencebetween the behavior of sound waves and light waves in this case?The answer is that light has very short wavelengths and acts like aray. Sound has wavelengths on the order of the size of the door andbends around corners (for frequency of 1000 Hz,

[lambda=dfrac{c}{f}=dfrac{330,m/s}{1000,s^{−1}}=0.33,m,]

about three times smaller than the width of the doorway).

If we pass light through smaller openings such as slits, we canuse Huygens’s principle to see that light bends as sound does(Figure (PageIndex{7})). The bending of a wave around the edgesof an opening or an obstacle is called diffraction. Diffraction is a wave characteristic and occurs for all typesof waves. If diffraction is observed for some phenomenon, it isevidence that the phenomenon is a wave. Thus, the horizontaldiffraction of the laser beam after it passes through the slits inFigure (PageIndex{7}) is evidence that light is a wave.