Dr Gillani"s Blog: Physics Made Simple : Refraction through a prism:

Refraction through a prism:

Prism is a transparent body having three rectangular and two triangular surfaces.

The angle of the triangular surface opposite the base is known as Angle of Prism.

If an incident ray PQ strikes prism on surface AB,on entering the prism,the ray bends towards the normal at its point of incidence Q.,ie towards the base of the prism.

The refracted ray QR on coming out of the prism bends away from the normal RN at the point of emergence R ie the emergent ray RS bends towards the base of the prism.

The incident ray PQ makes an angle of incidence I and r is its corresponding angle of refraction of the prism.

If n is the refractive index of the prism,

N = sin i/ sin r

The original direction of incidenr ray is PQT but it is turned through angle TDS on passing through the prism.

The angle TDS is known as the Angle of deviation.

THE VALUE OF ANGLE OF DEVIATION DEPENDS UPON THE VALUE OF ANGLE OF INCIDENCE.

When the angle of incidence is continuously increased from a small value,the angle of deivation first decreases,reaches a minimum value and then starts increasing.

The minimum value of angle of deviation is known as the angle of minimum deviation and is denoted by Dm.

The refractive index “n” of the material of the prism with respect to air canbe determined by the following relation:

N = Sin ( A + Dm/2) / sin(A/2)

Where A is angle of the prism and Dm is the angle of minimum deviation.

The refraction of waves depends on their wavelengths.Since sunlight consists of different colours,the waves of different wavelengths,thus when it passes through a prism,then the waves of different wavelengths deviate on different paths due to this white light disperses in different colours,which is called as dispersion and the band of colours which is seen on the screen is called solar spectrum.

A ray trace through a prism with apex angle α. Regions 0, 1, and 2 haveindices of refraction $n_0$ , $n_1$ , and $n_2$ , and primed angles $\theta'$ indicate the ray angles after refraction.

Types of prisms

Dispersive prisms

Comparison of the spectra obtained from a diffraction grating by diffraction (1), and a prism by refraction (2). Longer wavelengths (red) are diffracted more, but refracted less than shorter wavelengths (violet).

Dispersive prisms are used to break up light into its constituent spectral colors because the refractive index depends on frequency; the white light entering the prism is a mixture of different frequencies, each of which gets bent slightly differently. Blue light is slowed down more than red light and will therefore be bent more than red light.

Triangular prism
Abbe prism
Pellin–Broca prism
Amici prism
Compound prism
Grism, a dispersive prism with a diffraction grating on its surface

Reflective prisms

Reflective prisms are used to reflect light, in order to flip, invert, rotate, deviate or displace the light beam. They are typically used to erect the image inbinoculars or single-lens reflex cameras – without the prisms the image would be upside down for the user. Many reflective prisms use total internal reflection to achieve high reflectivity.

The most common reflective prisms are:

Beam-splitting prisms

Some reflective prisms are used for splitting a beam into two or more beams:

Polarizing prisms

There are also polarizing prisms which can split a beam of light into components of varying polarization. These are typically made of a birefringent crystalline material.

Nicol prism
Wollaston prism
Nomarski prism – a variant of the Wollaston prism with advantages in microscopy
Rochon prism
Sénarmont prism
Glan–Foucault prism
Glan–Taylor prism
Glan–Thompson prism

Deflecting prisms

Wedge prisms are used to deflect a beam of light by a fixed angle. A pair of such prisms can be used for beam steering; by rotating the prisms the beam can be deflected into any desired angle within a conical "field of regard". The most commonly found implementation is a Risley prism pair. Two wedge prisms can also be used as an anamorphic pair to change the shape of a beam. This is used to make a round beam from the elliptical output of a laser diode.

Rhomboid prisms are used to laterally displace a beam of light without inverting the image.

Deck prisms were used on sailing ships to bring daylight below deck, since candles and kerosene lamps are a fire hazard on wooden ships.

Deviation angle and dispersion

A ray trace through a prism with apex angle α. Regions 0, 1, and 2 haveindices of refraction $n_0$ , $n_1$ , and $n_2$ , and primed angles $\theta'$ indicate the ray angles after refraction.

Ray angle deviation and dispersion through a prism can be determined by tracing a sample ray through the element and using Snell's law at each interface. For the prism shown at right, the indicated angles are given by

$\begin{align} \theta'_0 &= \, \text{arcsin} \Big( \frac{n_0}{n_1} \, \sin \theta_0 \Big) \\ \theta_1 &= \alpha - \theta'_0 \\ \theta'_1 &= \, \text{arcsin} \Big( \frac{n_1}{n_2} \, \sin \theta_1 \Big) \\ \theta_2 &= \theta'_1 - \alpha \end{align}$ .

For a prism in air $n_0=n_2 \simeq 1$ . Defining $n=n_1$ , the deviation angle $\delta$ is given by

$\delta = \theta_0 + \theta_2 = \theta_0 + \text{arcsin} \Big( n \, \sin \Big[\alpha - \text{arcsin} \Big( \frac{1}{n} \, \sin \theta_0 \Big) \Big] \Big) - \alpha$

If the angle of incidence $\theta_0$ and prism apex angle $\alpha$ are both small, $\sin \theta \approx \theta$ and $\text{arcsin} x \approx x$ if the angles are expressed in radians. This allows the nonlinear equation in the deviation angle $\delta$ to be approximated by

$\delta \approx \theta_0 - \alpha + \Big( n \, \Big[ \Big(\alpha - \frac{1}{n} \, \theta_0 \Big) \Big] \Big) = \theta_0 - \alpha + n \alpha - \theta_0 = (n - 1) \alpha \ .$

The deviation angle depends on wavelength through n, so for a thin prism the deviation angle varies with wavelength according to

$\delta (\lambda) \approx [ n (\lambda) - 1 ] \alpha$ .

Dr Gillani"s Blog

Monday 23 December 2013

Physics Made Simple : Refraction through a prism: