IB DP Physics: Topic 9: Wave phenomena: 9.4 Resolution: Study Notes

WAVE OPTICS

WAVEFRONT

• Spherical wavefront : A spherical wavefront is produced by a point source of light. This is because the locus of all such points which are equidistant from the point source will be a sphere. Spherical wavefronts are further divided into two headings: (i) converging spherical and (ii) diverging spherical wavefront.

• Cylindrical wavefront : When the source of light is linear in shape such as a slit, the cylindrical wavefront is produced. This is because all the points equidistant from a line source lie on the surface of a cylinder.

• Plane wavefront : A small part of a spherical or cylindrical wavefront due to a distant source will appear plane and hence it is called plane wave-front. The wavefront of parallel rays is a plane wavefront.

HUYGENS WAVE THEORY

• Each point source of light is a centre of disturbance from which waves spread in all directions.
• Each point on primary wavelets acts as a new source of disturbance and produces secondary wavelets which travel in space with the speed of light.
• The forward envelope of the secondary wavelets at any instant gives the new wavefront.

           

• In a homogeneous medium the wavefront is always perpendicular to the direction of wave propagation.

DRAWBACKS OF HUYGENS WAVE THEORY

• This theory cannot explain photoelectric effect, Compton, and Raman effect.
• Hypothetical medium in vacuum is not true imagination.
• The theory predicted the presence of back wave, which proved to be failure.

REFLECTION AND REFRACTION OF PLANE WAVES USING HUYGENS PRINCIPLE

REFLECTION ON THE BASIS OF WAVE THEORY

∴  … (i)

angle of incidence, i =

and angle of reflection, r =

AA′ is common,  and

∴ Δs are congruent ∴ … (iii)

REFRACTION ON THE BASIS OF WAVE THEORY

µ =   … (iv)

AB is a plane wave front incident on XY at = . 1, 2, 3 are the corresponding incident rays normal to AB.

Let, angle of refraction.

In  ΔAA′B,   

In Δ AA′B′,  

∴    [using (iv)]

Hence       … (vi)

1. In 1873, Maxwell showed that light is an electromagnetic wave i.e. it propagates as transverse non-mechanical wave at speed c in free space given by

2. There are some phenomenon of light like photoelectric effect, Compton effect, Raman effect etc. which can be explained only on the basis of particle nature of light.
3. Light shows the dual nature i.e. particle as well as wave nature of light. But the wave nature and particle nature both cannot be possible simultaneously.
4. Interference and diffraction are the two phenomena that can be explained only on the basis of wave nature of light.

INTERFERENCE OF LIGHT WAVES AND YOUNG’S DOUBLE SLIT EXPERIMENT

 …(1)

 …(2)

 ….(3)

where (from fig)  =

Similarly,

so,   ….(4)

(A) CONDITIONS FOR MAXIMUM & MINIMUM INTENSITY

or, path difference

     

or

or  ….(5)

or  

or   …(6)

(B) POSITION OF FRINGE

or       …(7)

If, then we obtain dark fringe at point P on the screen and corresponds to destructive interference. So from equation (4), the position of, nth dark fringe is

or  …(8)

(C) SPACING OR FRINGE WIDTH

&

 …(9)

 …(10)

(D) CONDITIONS FOR SUSTAINED INTERFERENCE

• The two sources should be coherent i.e. they should have a constant phase difference between them.
• The two sources should give light of same frequency (or wavelength).
• If the interfering waves are polarized, then they must be in same state of polarization.

(E) CONDITIONS FOR GOOD OBSERVATION OF FRINGE

• The distance between two sources i.e. d should be small.
• The distance of screen D from the sources should be quite large.
• The two interfering wavefronts must intersect at a very small angle.

(F) CONDITIONS FOR GOOD CONTRAST OF FRINGE

• Sources must be monochromatic i.e. they emit waves of single wavelength.
• The amplitude of two interfering waves should be equal or nearly equal.
• Both sources must be narrow.
• As Intensity I is directly proportional to the square of amplitude, hence Intensity of resultant wave at P,  

, then

, if I1 = I2 = I0, then Imin = 0

• 
• Angular fringe-width
• The width of all interference fringes are same. Since fringe width β  is proportional to λ, hence fringes with red light are wider than those for blue light.
• If the interference experiment is performed in a medium of refractive index μ instead of air, the wavelength of light will change from λ to .

i.e.

• If a transparent sheet of refractive index μ and thickness t is introduced in one of the paths of interfering waves, then due to its presence optical path will become μt instead of t. Due to this a given fringe from present position shifts to a new position. So the lateral shift of the fringe,

• In Young’s double slit experiment (coherent sources in phase): Central fringe is a bright fringe. It is on the perpendicular bisector of coherent sources. Central fringe position is at a place where two waves having equal phase superpose.
• Young’s experiment with the white light will give white central fringe flanked on either side by coloured bands.

COHERENCE

TEMPORAL COHERENCE

• The duration of this transition is about 10–9 to 10–10 sec. Thus the emitted wave remains sinusoidal for this much time. This time is known as coherence time ().
• Definite phase relationship is maintained for a length called coherence length.

For neon ,   and L = 0.03 m.

For cadmium , and L = 0.3 m

For Laser  sec and L = 3 km.

• The spectral lines width is related to coherence length L and coherence time τc.

or

SPATIAL COHERENCE

Points P and P′ will be spatially coherent if the distance between P and P′ is much less than the coherence length i.e.

METHODS OF OBTAINING COHERENT SOURCES

• By division of wavefront
• By division of amplitude.

INCOHERENCE OF TWO CONVENTIONAL LIGHT SOURCES

LLOYD’S MIRROR

• If screen is moved so that, point O touches the edge of glass plate, the geometrical path difference for two wave trains is zero. The phase change of π radian on reflection at denser medium causes a dark fringe to be formed.
  • The fringe width remains unchanged on introduction of transparent film.
  • If the film is placed in front of upper slit S1, the fringe pattern will shift upwards. On the other hand if the film is placed in front of lower slit S2, the fringe pattern shifts downwards.
• This interference pattern is frequently seen in a ripple tank when one uses a wave train to demonstrate the law of reflection.
• In this case, fringe width

INTERFERENCE IN THIN FILMS

• reflected light
• transmitted light

INTERFERENCE DUE TO REFLECTED LIGHT

• If , where n = 0,1,2, …………..then constructive interference takes place and the film appears bright.
• If , where n = 0, 1, 2, 3,………… then destructive interference takes place and the film appears dark.

INTERFERENCE DUE TO TRANSMITTED LIGHT

The optical path difference between the reflected ray (DQ) and the transmitted ray (NR) is given by

1. If , where n = 0, 1, 2, 3, ………….then constructive interference takes place and the film appears bright.

2. If , n = 0, 1, 2, ……. then destructive interference takes place and the film appears dark.

NEWTON’S RINGS

NEWTON’S RINGS BY REFLECTED LIGHT

for bright rings,……

for dark rings, ……

for bright rings,

for dark rings,
where R = radius of curvature of lens.

• The centre is dark and alternately dark and bright rings are produced.
• While counting the order of the dark rings 1, 2, 3, etc. the central ring is not counted. Therefore,

for 1st dark ring, n = 1 and

for 2nd dark ring, n = 2 and

NEWTON’S RINGS BY TRANSMITTED LIGHT

for bright rings,

for dark rings,

Radius of bright ring,

• The centre is bright and alternately bright and dark rings are obtained.
• The ring pattern due to reflected light is just opposite to that of transmitted light.  

• If Dn and Dn + m be the diameters of n th and (n + m)th dark rings then the wavelength of light used is given by

• If  Dn = diameter of nth dark ring when air is present between the glass plate and the lens

DIFFRACTION

FRAUNHOFER DIFFRACTION BY SINGLE SLIT

In general, b sin θ = nλ for minima and, for maxima.

• The points of the maximum intensity lie nearly midway between the successive minima. The amplitude E0‘ of the electric field at a general point P is

where and

The intensity at a general point P is given as

• The graph for the variation of intensity as a function of sinθ is as follows :

• The width of the central maxima is and angular width of central maxima is.

FRAUNHOFER DIFFRACTION BY A CIRCULAR APERTURE

• The 1st dark ring is formed by the light diffracted from the circular aperture at an angle θ with the axis where

where λ = wavelength of light used, b = diameter of circular aperture

• If the screen is at a distance D (D >> b) from the circular aperture, the radius of the 1st dark ring is,
• If the light transmitted by the hole is converged by a converging lens at the screen placed at the focal plane of the lens, the radius of the 1st dark ring is  

DIFFERENCE BETWEEN INTERFERENCE AND DIFFRACTION OF LIGHT

POLARISATION

The plane of vibration is that which contains the vibrations of electric vector and plane of polarisation is perpendicular to the plane of vibration

POLARIZATION BY REFLECTION (BREWSTER’S LAW)

LAW OF MALUS

• Plane polarized : oscillating electric field is in a single plane.
• Circularly polarized : tip of oscillating electric field describes a circle.
• Elliptically polarized : tip of oscillating electric field describes an ellipse.

RESOLVING POWER OF AN OPTICAL INSTRUMENT

The limit of resolution of a microscope where a is the aperture of the microscope.

DOPPLER’S EFFECT FOR LIGHT WAVES

When the source moves away from the stationary observer or vice-versa, the apparent frequency