Wave Optics
Prepared by
Sandhya Velayudhan
PGT Physics, JNV Kozhikode
HUYGEN’S PRINCIPLE
Wavelet and wavefront:
A wavelet is the point of disturbance due to propagation of light.
A wavefront is the locus of all points in space having the same phase of vibration.
A line perpendicular to a wavefront is called a ‘ray’.
Spherical Wavefront from a point source
Cylindrical Wavefront from a linear source
Plane Wavefront
Pink Dots – Wavelets
Blue Envelope– Wavefront Red Line – Ray
•
Refraction of a plane wave
PP’ - the surface separating medium 1 and medium 2
v1- the speed of light in medium 1
v2 – speed of light in medium 2,
AB - incident plane wavefront
EC – refracted wavefront
τ− the time taken to travel BC
BC = v1τ
AE = v2τ
Sini/ Sinr = v1/v2
Sini/ Sinr = n2/n1
Snell’s law
n1 – refractive index of medium 1
n2- refractive index of medium 2
Refraction of a plane wave
Sini/ Sinr = v1/v2
Sini/ Sinr = n2/n1
Snell’s law
Reflection of a plane wave
MN – reflecting surface
AB – plane incident wavfront
EC- reflected wavfront
i- angle of incidence
r- angle of reflection
v- the speed of light
τ – time taken
AE = BC = vτ
Since the triangles EAC and BAC are congruent,
i = r , the law of reflection
�Refraction of a plane wave by (a) a thin prism, (b) a convex lens. (c) Reflection of a�plane wave by a concave mirror.�
(a)
(b)
(c)
Interference
The phenomenon of one wave interfering with another and the resulting redistribution of energy in the space around the two sources of disturbance is called interference of waves.
Coherent Sources
The sources emitting light waves of same frequency or wavelength having either a zero or constant phase difference are said to be coherent
Interference of Waves:
Destructive Interference E = E1 - E2
Crest Trough Bright Band Dark Band
1st Wave (E1) 2nd Wave (E2) Resultant Wave Reference Line
E2
E2
The phenomenon of one wave interfering with another and the resulting redistribution of energy in the space around the two sources of disturbance is called interference of waves.
E1 + E2
E1
Constructive Interference E = E1 + E2
E1
E1 - E2
Bright Band
Dark Band
Dark Band
Bright Band
Bright Band
S2
S1
Condition for Constructive and destructive Interference
For two coherent sources vibrating
in phase, constructive interference occurs when
Path difference = n λ
(n = 0, 1, 2, 3,...)
For two coherent sources vibrating
in phase, destructive interference occurs when
Path difference =( n +1/2 ) λ (n = 0, 1, 2, 3,...)
λ − wavelength
Intensity in interference
Consider two coherent light waves with intensity I0 each and phase difference φ between them. The resultant intensity at the point of interference will be
I = 4I0 Cos2 (φ/2)
Condition for constructive interference : φ = 0, ±2π,±4π…….
Condition for destructive interference : φ = ±π,±3π,…….
For incoherent waves the average intensity will be
< I > = 4I0 < Cos2 (φ/2) >
= 2I0
Double slit experiment�
Path difference
The waves from S1 and S2 reach the point P with
Path difference ∆ = S2P – S1P
S2P2 – S1P2 = [D2 + {x + (d/2)}2] - [D2 + {x - (d/2)}2]
(S2P – S1P) (S2P + S1P) = 2 x d
∆ = x d / D
Positions of Bright Fringes:
For a bright fringe at P,
∆ = xd / D = nλ
where n = 0, 1, 2, 3, …
For n = 0, For n = 1, For n = 2, For n = n,
x = n D λ / d x0 = 0
x1 = D λ / d
x2 = 2 D λ / d …… xn = n D λ / d
x0’ = D λ / 2d x1’ = 3D λ / 2d
x2’ = 5D λ / 2d …..
For n = 0, For n = 1, For n = 2, For n = n,
xn’ = (2n+1)D λ / 2d
Positions of Dark Fringes:
For a dark fringe at P,
∆ = xd / D = (2n+1)λ/2
where n = 0, 1, 2, 3, …
x = (2n+1) D λ / 2d
Expression for Dark Fringe Width:
βD = xn – xn-1
= n D λ / d – (n – 1) D λ / d
= D λ / d
Expression for Bright Fringe Width:
βB = xn’ – xn-1’
= (2n+1) D λ / 2d – {2(n-1)+1} D λ / 2d
= D λ / d
The expressions for fringe width show that the fringes are equally spaced on the screen.
Distribution of Intensity:
Intensity
0
x
x
Suppose the two interfering waves have same amplitude say ‘a’, then
Imax α (a+a)2 i.e. Imax α 4a2
All the bright fringes have this same intensity.
Imin = 0
All the dark fringes have zero intensity.
Conditions for sustained interference:
A computer generated pattern
Some Examples of Interference
One of the best examples of interference is demonstrated by the light reflected from a film of oil floating on water. Another example is the thin film of a soap bubble, which reflects a spectrum of beautiful colors when illuminated by natural or artificial light sources.
Diffraction
The phenomenon of bending of light around the sharp edges of an obstacle or aperture is called diffraction.
Condition for diffraction of light
The wavelength of incident light should be comparable to the size of the obstacle or aperture
The single slit experiment
Theory:
The path difference between the wavelets is NQ.
If ‘θ’ is the angle of diffraction and ‘á’ is the slit width, then NQ = a sin θ
To establish the condition for secondary minima, the slit is divided into 2, 4, 6, … equal parts such that corresponding wavelets from successive regions interfere with path difference of λ/2.
Or for nth secondary minimum, the slit can be divided into 2n equal parts.
For θ1, a sin θ1 = λ For θ2, a sin θ2 = 2λ For θn, a sin θn = nλ
Since θn is very small, a θn = nλ
θn = nλ / a (n = 1, 2, 3, ……)
To establish the condition for secondary maxima, the slit is divided into 3, 5, 7, … equal parts such that corresponding wavelets from alternate regions interfere with path difference of λ.
Or for nth secondary minimum, the slit can be divided into (2n + 1) equal parts.
For θ1’, a sin θ1’ = 3λ/2 For θ2’, a sin θ2’ = 5λ/2
For θn’, a sin θn’ = (2n + 1)λ/2
Since θn’ is very small, a θn’ = (2n + 1)λ / 2
θn’ = (2n + 1)λ / 2a
(n = 1, 2, 3, ……)
Diffraction pattern
Difference between Interference and Diffraction:�
Interference | Diffraction | ||
1. | Interference is due to the superposition of two different wave trains coming from coherent sources. | 1. | Diffraction is due to the superposition of secondary wavelets from the different parts of the same wavefront. |
2. | Fringe width is generally constant. | 2. | Fringes are of varying width. |
3. | All the maxima have the same intensity. | 3. | Central maximum has maximum intensity . The intensity falls as we go to successive maxima away from the centre on either side of central maximum |
Diffraction
The pattern
Width of central maximum
The first minima on either side is at θ = λ/a on either side of the central maximum
So the angular width of the central bright fringe is
2 θ = 2 λ/a
The linear width of the central bright fringe is β = 2 λD/a
a- the slit width,λ- the wavelength
θ- the angular position of the first minimum from the central maximum
Fresnel’s Distance:
Fresnel’s distance is that distance from the slit at which the spreading of light due to diffraction becomes equal to the size of the slit.
y1 = D λ / d
At Fresnel’s distance, y1 = d and D = DF
So, DF λ / d = d or DF = d2 / λ
If the distance D between the slit and the screen is less than Fresnel’s distance DF, then the diffraction effects may be regarded as absent.
So, ray optics may be regarded as a limiting case of wave optics.
Polarisation of Light
Polarisation is the phenomenon in which light or other radiation are restricted in direction of vibration
Polarized light has electric fields oscillating in one direction and Unpolarized light has electric fields oscillating in all directions.
Polarisation of Transverse Mechanical Waves:
Transverse disturbance (up and down)
Narrow Slit
Transverse disturbance (up and down)
Narrow Slit
Narrow Slit
90°
Polarisation of Light Waves:
•
S
Natural Light
Representation of Natural Light
In natural light, millions of transverse vibrations occur in all the directions perpendicular to the direction of propagation of wave. But for convenience, we can assume the rectangular components of the vibrations with one component lying on the plane of the diagram and the other perpendicular to the plane of the diagram.
• • • • • • • • • •
Wave
- Parallel to the plane
•
Polariser Tourmaline Crystal
Analyser Tourmaline Crystal
• • • • • •
Unpolarised light
Plane Polarised light
Plane Polarised light
Light waves are electromagnetic waves with electric and magnetic fields oscillating at right angles to each other and also to the direction of propagation of wave. Therefore, the light waves can be polarised.
Optic Axis
90°
• • • • • •
Unpolarised light
Plane Polarised light
No light
• • • • • •
Unpolarised light
Polariser
Plane Polarised light
Analyser
90°
Plane of Vibration Plane of Polarisation
When unpolarised light is incident on the polariser, the vibrations parallel to the crystallographic axis are transmitted and those perpendicular to the axis are absorbed. Therefore the transmitted light is plane (linearly) polarised.
The plane which contains the crystallographic axis and vibrations transmitted from the polariser is called plane of vibration.
The plane which is perpendicular to the plane of vibration is called plane of polarisation.
Malus’ Law:
When a beam of plane polarised light is incident on an analyser, the intensity I of light transmitted from the analyser varies directly as the square of the cosine of the angle θ between the planes of transmission of analyser and polariser.
I α cos2 θ
A
If E0 be the amplitude of the electric vector transmitted by the polariser, then only the component E0 cos θ will be transmitted by the analyser.
Intensity of transmitted light from the analyser is
I = k (E0 cos θ)2
I = I0 cos2 θ
Case I : When θ = 0° or 180°, I = I0
Case II : When θ = 90°, I = 0
Case III: When unpolarised light is incident on the analyser the intensity of the transmitted light is one-half of the intensity of incident light. (Since average value of cos2θ is ½)
E0 cos θ
P
θ
E0sin θ
E0
Polarisation by Reflection and Brewster’s Law:
r
μ
90°
a
b
θB
r
b
n21 = Sin θB / Sin r
= Sin θB / Sin (90 - θB )
n21 = tan θB
When unpolarised light is incident on the boundary between two transparent media, the reflected light is polarised with its electric vector perpendicular to the plane of incidence when the refracted and reflected rays make a right angle with each other. The angle of incidence in this case is called Brewster’s angle and is denoted by θB
Brewster’s Law
Polaroids:
H – Polaroid is prepared by taking a sheet of polyvinyl alcohol (long chain polymer molecules) and subjecting to a large strain. The molecules are oriented parallel to the strain and the material becomes doubly refracting. When strained with iodine, the material behaves like a dichroic crystal.
K – Polaroid is prepared by heating a stretched polyvinyl alcohol film in the presence of HCl (an active dehydrating catalyst). When the film becomes slightly darkened, it behaves like a strong dichroic crystal.
Uses of Polaroids: