Wave Optics class 12 Physics notes
INTRODUCTION TO WAVE OPTICS
A Dutch physicist named Huygens (1629 – 1695), suggested that light may have a wave nature. The apparent rectilinear propagation of light explained by Newton may be just due to the fact that the wavelength of light may be much smaller than the dimensions of these obstacles. This proposal remained a dump for almost a century. Newton’s theory was then challenged by Young’s double slit experiment in 1801. A series of experiments on diﬀraction of light conducted by French scientist Fresnel were some of the activities that put an end to the particle nature of light and established the wave nature of light.
The twist came around when the wave nature of light failed to explain the photoelectric eﬀect in which light again behaved as particles. This again brought up the question of whether light had a wave or a particle nature and an acceptance was eventually reached that light is of dual nature – particle and wave.
Key point – Light waves need no material medium to travel. They can propagate in the vacuum.
Nature of Light Waves
Light waves are transverse, i.e. disturbance of the medium is perpendicular to the direction of propagation of the wave. Hence they can be polarized. If a plane light wave is travelling in the x-direction, the electric field may be along the y or z-direction or any other direction in the y-z plane. The equation of such a monochromatic light wave can be written as E = E0sinω(t – x/v).
The speed of light is generally denoted by c. When light travels in a transparent material, the speed is decreased by a factor µ, which is called as the refractive index of the material.
µ = speed of light in vacuum/ speed of light in the material
The frequency of visible light varies from about 3800 x 1011 Hz to about 7800 x 1011 Hz.
Light of a single wavelength is called monochromatic light.
HUYGENS’S WAVE THEORY
It has the following two basic postulates:
(a) Consider all the points on a primary wave-front to be the sources of light, which emit disturbance known as the secondary disturbance.
(b) Tangent envelops to all secondary wavelets gives the position of the new wave-front.
Huygens’ Principle may be stated in its most general form as follows: Various points of an arbitrary surface, when reached by a wavefront, become secondary sources of light emitting secondary wavelets. The disturbance beyond the surface result from the superposition of these secondary wavelets.
Huygens, the Dutch physicist and astronomer of the seventeenth century, gave a beautiful geometrical description of wave propagation. We can guess that he must have seen water waves many times in the canals of his native place Holland. A stick placed in water, oscillated up and down, becomes a source of waves. Since the surface of the water is two dimensional, the resulting wavefronts would be circles instead of spheres. At each point on such a circle, the water level moves up and down. Huygens’ idea is that we can think of every such oscillating point on a wavefront as a new source of waves. According to Huygens’ principle, what we observe is the result of adding up the waves from all these diﬀerent sources. These are called secondary waves or wavelets. Huygens’ Principle is illustrated in (Figure) as the simple case of a plane wave.
Observations of Huygen’s Experiment
(a) At time t=0, we have a wavefront F1, F1 separates those parts of the medium that are undisturbed from those where the wave has already reached.
(b) Each point on F1 acts like a new source and sends out a spherical wave. After a time ‘t’, each of these will have radius vt. These spheres are the secondary wavelets.
(c) After a time t, the disturbance would now have reached all points within the region covered by all these secondary waves. The boundary of this region is the new wavefront F2.
(d) The secondary wavelets from the point A1 on F1 touch F2 at A2. According to Huygens, A1A2 is a ray. It is perpendicular to the wavefronts F1 and F2 and has length vt. This implies that rays are perpendicular to wavefronts. Further, the time taken for light to travel between two wavefronts is the same along any ray. In this case, we can say that the distance between two wavefronts is the same measured along any ray.
(e) This geometrical construction can be repeated starting with F2 to get the next wavefront F3 a time t later, and so on. This is known as Huygens’ construction.
Huygens’ construction can be understood physically for waves in a material medium, like the surface of the water. Each oscillating particle can set its neighbors into oscillation, and therefore acts as a secondary source. But what if there is no medium, such as for light traveling in the vacuum? The mathematical theory, which cannot be given here, shows that the same geometrical construction work in this case as well.
When two waves of the same frequency move along the same direction in a medium, they superimpose and give rise to a phenomenon called interference. Points of constructive interference have maximum intensity while points of destructive interference have minimum intensity.
Coherent and Incoherent Sources
Two light sources of light waves are coherent if the initial phase diﬀerence between the waves emitted by the sources remains constant with time. If it changes randomly with time, the sources are said to be incoherent. Two waves produce an interference pattern only if they originate from coherent sources.
Intensity and Superposition of Waves
Note: Consider two coherent sources S1 and S2. Suppose two waves emanating from these two sources superimpose at point P. The phase diﬀerence between them at P is φ (which is constant). If the amplitude due to two individual sources at P is A1 and A2, then resultant amplitude at P will be, A = √ (A12 + A22 + 2 A1 A2 cosφ).
Similarly, the resultant intensity at P is given by, I = I1 + I2 + 2√ (I1 I2 ) cosφ. Here, I1 and I2 are the intensities due to independent sources. If the sources are incoherent then resultant intensity at P is given by, I = I1 + I2.
Conditions for Interference
(a) Sources should be coherent i.e. the phase diﬀerence between them should be constant. For this, the frequency of sources should be the same.
(b) The amplitudes of both the waves should be nearly equal so as to obtain bright and dark fringes of maximum contrast.
(c) The two sources should be very close to each other.
(d) The two sources of slits should be very narrow otherwise a broad source will be equivalent to a number of narrow sources emitting their own overlapping wavelets.
If the two sources are obtained from a single parent source by splitting the light into two narrow sources, they form coherent sources which produce sustained interference pattern due to a constant phase diﬀerence between the waves
Young’s Double Slit Experiment
The experiment consists of a parallel beam of monochromatic light from slit S which is incident on two narrow pinhole or slits S1 and S2 separated by a small distance d. The wavelets emitted from these sources superimpose at the screen placed in front of these slits to produce an alternate dark and bright fringe pattern at points on the screen depending upon whether these waves reach with a phase diﬀerence φ = (2n – 1)π producing destructive interference or φ = 2nπ producing constructive interference respectively. If the screen is placed at a perpendicular distance D from the middle point of the slits, the point O on the screen lies at the right bisector of S1 S2 and is equidistant from S1 and S2. The intensity at O is maximum.
Consider a point P located at a distance xn from O on the screen as shown in the figure. The path diﬀerence of waves reaching at point P from S2 and S1 is given by Path diﬀerence
Fringe width: The spacing between any two consecutive bright or two dark fringes is equal and is called the fringe width.
If a thin transparent plate of thickness t and refractive index µ is introduced in the path S1P of one of the interfering waves, the entire fringes pattern is shifted through a constant distance. The path S1P in air is increased to an air path equal to S1P + (µ – 1)t
Thus, when a thin transparent plate of thickness t and refractive index µ is introduced in one of the paths of the waves, the path diﬀerence changes by xd/D.
∴ xd/D = (µ – 1)t = (D/d)(µ – 1)t
The central maxima shifts by a distance equal to D(µ – 1)t/d.
Intensity variation on screen: If I0 is the intensity of light beam coming from each slit, the resultant intensity at a point where they have a phase diﬀerence of φ is
I = 4I0 cos2φ/2 ,where φ = 2π (d sinθ)/λ
Interference from Thin Films
Interference eﬀects are commonly observed in thin films, such as thin layers of oil on water or the thin surface of a soap bubble. The various colors observed when white light is incident on such films results from the interference of waves reﬂected from the two surfaces of the film. Consider a film of uniform thickness t and index of refraction µ as shown in the figure. Let us assume that the rays traveling in the air are nearly normal to the two surfaces of the film. To determine whether the reﬂected rays interfere constructively or destructively, we first note the following facts.
(i) The wavelength of light in a medium whose refractive index is µ is, λµ = λ/ µ Where λ is the wavelength of light in the vacuum (on air)
(ii) If a wave is reﬂected from a denser medium, it undergoes a phase change of 180°. Let us apply these rules to the film shown in the figure. The path diﬀerence between the two rays 1 and 2 is 2t while the phase diﬀerence between them is 180°. Hence, condition of constructive interference will be, 2t = (2n 1)λµ/2 or, 2µt = (n – ½ )λ as λµ = λ/ µ
Similarly, the condition of destructive interference will be 2µt = nλ; n=0, 1, 2, …
YDSE with Glass Slab
Path diﬀerence produced by a slab
Consider two light rays 1 and 2 moving in air parallel to each other. If a slab of refractive index µ and thickness t is inserted between the path of one of the rays then a path diﬀerence ∆x = (µ – 1)t is produced among them. This can be shown as under,
Optical path length: Now we can show that a thickness t in a medium of refractive index µ is equivalent to a length µt in the vacuum (or air). This is called optical path length. Thus, Optical path length= µt
The phenomenon of bending of light around the corners of an obstacle or an aperture into the region of the geometrical shadow of the obstacle is called diﬀraction of light. The diﬀraction of light is more pronounced when the dimension of the obstacle/aperture is comparable to the wavelength of the wave.
Diﬀraction of Light Due to Single Slit
Diverging light from monochromatic source S is made parallel after refraction through convex lens L1. The refracted light from L1 is propagated in the form of plane wavefront WW’. The plane wavefront WW’ is incident on the slit AB of width ‘d’. According to Huygens’ Principle, each point of slit AB acts as a source of secondary disturbance of wavelets.
Path diﬀerence: To find the path diﬀerence between the secondary wavelets originating from corresponding points A and B of the plane wavefront, draw AN perpendicular on BB’. The path diﬀerence between these wavelets originating from A and B is BN.
Diﬀerence between interference and diﬀraction:
Both interference and diﬀraction are the results of the superposition of waves, so they are often present simultaneously, as in Young’s double-slit experiment. However, interference is the result of the superposition of waves from two diﬀerent wavefronts while diﬀraction results due to the superposition of wavelets from diﬀerent points of the same wavefront.
RESOLVING POWER OF OPTICAL INSTRUMENTS
When the two images cannot be distinguished, they are said to be unresolved. If the images are well distinguished. They are said to be well resolved. On the other hand, if the images are just distinguished, they are said to be just resolved.
Limit of Resolution
The minimum distance of separation between two points so that they can be seen as separate (or just resolved) by the optical instrument is known as its limit of resolution. Diﬀraction of light limits the ability of optical instruments.
SCATTERING OF LIGHT
Scattering of light is a phenomenon in which a part of a parallel beam of light appears in directions other than the incident radiation when passed through a gas.
Process: Absorption of light by gas molecules followed by its re-radiation in diﬀerent directions. The strength of scattering depends on the following:
(a) Loss of energy in the light beam as it passes through the gas
(b) Wavelength of light
(c) Size of the particles that cause scattering
Practical example of scattering: The blue color of the sky is caused by the scattering of sunlight by the molecules in the atmosphere. This scattering, called Rayleigh scattering, is more eﬀective at short wavelengths (the blue end of the visible spectrum). Therefore, the light scattered down to the earth at a large angle with respect to the direction of the sun’s light is predominantly in the blue end of the spectrum.
POLARIZATION OF LIGHT
The process of splitting of light into two directions is known as polarization.
Phenomenon of polarization: The phenomenon of restricting the vibrations of a light vector of the electric field vector in a particular direction in a plane perpendicular to the direction of propagation of light is called polarization of light. Tourmaline crystal is used to polarize the light and hence it is called polarizer.
(a) An ordinary beam of light consists of a large number of waves emitted by the atoms or molecules of the light source. Each atom produces a wave with its own orientation of electric vector E. However because all directions are equally probable, the resulting electromagnetic wave is a superposition of waves produced by individual atomic sources. This wave is called as an unpolarized light wave.
(b) All the vibrations of an unpolarized light at a given instant can be resolved into two mutually perpendicular directions and hence an unpolarized light is equivalent to the superposition of two mutually perpendicular identical plane polarized lights.
Plane Polarized Light
(a) If somehow we confine the vibrations of electric field vector in one direction perpendicular to the direction of wave motion of propagation of the wave, the light is said to be plane polarized and the plane containing the direction of vibration and wave motion is called the plane of polarization.
(b) If an unpolarized light is converted into plane polarized light, its intensity reduces to half.
(c) Polarization is proof of the wave nature of light.
Partially polarized light: If in case of unpolarized light, electric field vector in some plane is either more or less than in its perpendicular plane, the light is said to be partially polarized.
A Polaroid is a device used to produce plane-polarized light. The direction perpendicular to the direction of the alignment of the molecules of the Polaroid is known as pass-axis or the polarizing direction of the Polaroid.
Note: A Polaroid used to examine the polarized light is known as an analyzer.
This law states that the intensity of the polarised light transmitted through the analyzer varies as the square of the cosine of the angle between the plane of transmission of the analyzer and the plane of the polarizer. Resolve E into two components: We know, intensity ∝(Amplitude)2
∴Intensity of the transmitted light through the analyzer is given by
I α (Ecosθ)2 or I = kE2cos2θ;
But kE2 = I0 (intensity of the incident polarized light)
I = I0 cos2θ Or I ∝ cos2θ which is Malus Law.
According to this law, the refractive index of the refractive medium (n) is numerically equal to the tangent of the angle of polarization(IB). i.e. n = tan IB