Refraction:
Velocity of light in different media is different. As a Ray of light goes from medium 1 (velocity of light v₁) to medium 2 (velocity of light v₂), its velocity for changes on crossing the interface XY. The phenomenon is called refraction.
Refraction is the phenomenon by virtue of which a ray of light going from one medium to the other undergoes a change in its velocity.
If the ray is incident, obliquely as shown in Fig.1,
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| Fig. 1. Refraction at a plane surface |
the change in velocity of light results in a change in its path also while there is no change in path if the ray is incident normally. The ray which approaches the interface is called incident ray (SO). Ray which goes into the second medium is called refracted ray. Angle between the incident ray and the normal to the interface at the point of incidence is called angle of incidence (i). Angle between the refracted ray and the normal to the interface is called the angle of refraction (r).
If light travels from an optically rarer medium to a denser medium, it bends towards the normal, i.e., r is less than i. If it travels from denser to rarer medium, it bends away from the normal, i.e., r is greater than i.
Important Notes
During refraction the path of light may or may not change.
- For beam of light incident normally, there is no change of path.
- For a beam going from rarer to denser medium, the beam bends towards the normal.
- For a beam going from denser to rarer medium, the beam bends away from the normal.
Law of Refraction:
Phenomenon of refraction is governed by the following two laws, called laws of refraction:
(i) The sine of the angle of incidence bears a constant ratio with the sine of the angle of refraction.
i.e., sin i/sin r = constant
The law is often termed as Snell's law.
(ii) The incident ray, the refracted ray and the normal to the interface at the point of incidence all lie in one plane and that plane is perpendicular to the interface separating the two media.
Refractive Index:
Refractive index of a medium is a characteristic of medium which determines its behaviour to propagation of light. It is sometimes, taken to be a measure of the optical density of the medium. A medium having a greater value of refractive index is said to be a optically denser then that having a lower value. Refractive index of vacuum if the smallest value and is equal to one. All other medii have refractive indices greater than one when measured with respect to vacuum. In such a case in refractive index of the medium is termed as absolute refractive index. It can be defined as a number of ways.
(i) Definition in terms of angles of incidence and refraction
Let XY be the interface separating the two media 1 and 2 from each other [in Fig.1]. Let i and r be the angle of incidence and refraction respectively.
According to Snell's law,
sin i/sin r = constant = ¹μ₂ ... (1)
where ¹μ₂ is called the refractive index of second medium with respect to first. It is always referred to as the refractive index of that medium to which the light goes with respect to the medium from which it comes.
Refractive index of a medium with respect to another is defined as the ratio between sine of the angle of incidence to the sine of angle of refraction.
(ii) Definition in terms of velocity of light
Refractive index of medium 2 with respect to 1 is also defined as the ratio between velocity of light in medium 1 to the velocity of light in medium 2.
If v₁ and v₂ are the velocities of light in first and second medium,
¹μ₂ = v₁/v₂ .... (2)
If the first medium is air or vacuum, the refractive index is written as ⁰μ₂ or simply as μ and is known as the absolute refractive index.
μ = c/v .... (3)
where 'c' is the velocity of light in vacuum and 'v' is the velocity of light in that medium.
(iii) Definition in terms of wavelength of light
As light goes from one medium to another, there is no change in its frequency. Since v = nλ, a change if medium shall result in the change in wavelength of light.
v₁ = nλ₁, v₂ = nλ₂
Substituting for v₁ and v₂ in equation (2),
¹μ₂ = nλ₁/nλ₂ = λ₁/λ₂ ... (4)
Refractive index of second medium with respect to first is defined as the ratio between wave-length of light in medium 1 to the wavelength of light in medium 2.
(iv) Definition in terms of absolute refractive indices of the medium
Dividing the numerator and denominator of equation (2) by c, we get
¹μ₂ = v₁/c / v₂/c [μ₁ = c/v₁ ∴ 1/μ₁ = v₁/c]
= 1/μ₁ / 1/μ₂ [μ₂ = c/v₂ ∴ 1/μ₂ = v₂/c]
or ¹μ₂ = μ₂/μ₁ .... (5)
Equation (5) provides another definition of refractive index.
Refractive index of second medium with respect to first is defined as the ratio between absolute refractive index of second medium to the absolute refractive index of first medium.
Relative Magnitudes of i and r
Relative magnitudes of i and r, during refraction, depend upon the nature of two media across an interface. Let XY be an interface separating the media 1 and 2 from each other such that
Case (i) Light travelling from rarer to denser medium [Fig.2(i)].
It shall be observed that the ray, on entering the second medium bends towards the normal. So, in this case r is less than i. Angle of deviation = (i - r)
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| Fig. 2. Bending of light due to refraction |
Case (ii) Light travelling from denser to rarer medium [Fig.2(ii)].
Principle of reversibility
It states that if a ray of light, after suffering a number of reflections and/or refractions has its path reversed at any stage, it will retrace itself back to the source along the same path.
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| Fig. 3. Principle of reversibility. |
Consider a ray SO, travelling in medium 2 striking the interface XY [Fig. 3]. The ray changes its path after refraction. If the path of the ray is reversed by placing a plane mirror normal to its path, the ray goes from medium 2 to medium 1 and reaches the source S.
Considering journey of ray from medium 1 to 2
¹μ₂ = sin i/sin r ... (6)
As the ray goes from medium 2 to medium 1, the angle of incidence and the refraction get their roles interchanged. Therefore, while going from medium 2 to medium 1.
²μ₁ = sin r/sin i ... (7)
Multiplying equations (6) and (7), we get
¹μ₂ × ²μ₁ = sin i/sin r × sin r/sin i = 1,
¹μ₂ = 1/²μ₁
For example,
airμglass = aμg = 1.5
∴ glassμair = gμa = 1.5
Important notes
Velocity of light 'c' is maximum in vaccum. Since refractive index μ = c/v, there is no existence of a medium having μ < l.
Optical Fibre
According to the property of rectilinear propagation of light, light travels in a straight line. Making use of the phenomenon of total internal reflection, it can be made to go in a curved path. Consider, a curved rod AB [Fig. 4(i)] made up of a transparent material like glass or optical grade plastic. A ray of light entering the pipe through face A undergoes successive total internal reflections at points P₁, P₂, .... and emerges out of face B. AB is called a light pipe in the sense that it has allowed light to flow through it from end A to B just like a hollow pipe allowing flow of water through it. We can use this pipe in the case where the order of transmission of different parts of an image is not important, but we require only a beam of light.
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| Fig. 4. Optical fibre. |
An optical fibre consists of a glass or a plastic core surrounding by a cladding made up of a similar material but with a lower refractive index [Fig. 4(ii)]. Light propagates through an optical fibre in two modes.
(i) Monomode propagation. In this case light has only one propagation path along the length of the core. As light propagates, cladding causes total internal reflection.
(ii) Multimode propagation. There are many propagation paths while the reflection takes place from the edges of the core. The core in this case, is of greater diameter. Due to pulse dispersion in this case, the data is transmitted at a lesser rate than that in monomode propagation.
There are three types of optical fibres which are in common use.
(i) Stepped index multimode optical fibre. The core and cladding, both are of uniform refractive index. Thus, there is an abrupt change in refractive index at the interface. The diameter of the core is about 50-60 μm while that of cladding is 125 μm. Because of high pulse dispersion bandwidth is limited between 10 MHz km to 50 MHz km.
(ii) Stepped index monomode optical fibre. The core and cladding both are again of uniform refractive index. Core diameter is small, 1-10 μm and cladding diameter is 125 μm. Due to smaller pulse dispersion in it, bandwidth of several GHz km is attainable.
(iii) Graded index multimode optical fibre. The core consists of a material having non uniform refractive index being maximum at the centre and minimum at the interface. The distribution of refractive index is parabolic. The core diameter is 50-60 μm while cladding diameter is 125 μm. In this case, velocity of propagation is higher than that in 'stepped index' multimode. Therefore, there is lesser pulse dispersion with a consequent increase in bandwidth up to 1 GHz km.
Following comparison of transmission through optical fibre and through electrical cable shall illustrate the superiority of optical fibre.
Key Formulae
1. Law of refraction :
Snell's law, sin i/sin r =constant
2. Refractive index :
¹μ₂ sin i/sin r = v₁/v₂ = μ₂/μ₁
Absolute refractive index,
μ = real depth/apparent depth
3. Principle of reversibility :
¹μ₂ × ²μ₁ = 1
4. Lateral shift through a glass slab :
d = t sec r sin (i - r)
5. Critical angle :
sin C = 1/μ
6. Refraction at a single surface :
(i) Light going from rarer to denser medium
(General formula)
μ₂/v - μ₁/u = μ₂ - μ₁/R
(Air to Glass)
μ/v - 1/u = μ - 1/R
(ii) Light going from denser to rarer medium
(General formula)
μ₂/u - μ₁/v = μ₂ - μ₁/R
(Glass to Air)
μ/u - 1/v = μ - 1/R
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