**W A V E / P H Y S I C A L O P T I C S**

**Def **– *The branch of the study of light,where the properties of light due to it’s wave nature like diffraction, interference and polarization are dealt with*.

**Huygens Wave Theory ( Huygens–Fresnel principle )**

This theory postulates that *light propagates in the form of waves *. Accordingly , *each point of an advancing wave front is in fact the center of a fresh disturbance and the source of a new train of waves; also, the advancing wave as a whole may be regarded as the sum of all the secondary* *waves arising from points in the medium already traversed*. The *locus of all the particles of the medium vibrating in the same phase at a given instant is called a wavefront .*

Considering a spreading wave , all points on the spreading wave circle are oscillating in phase , since they are at the same distance from the source. Hence a wavefront is a surface of constant phase. The speed with which the wavefront takes to travel outward from the source is called the **phase speed**. The energy of the wave travels in a direction perpendicular to the wavefront.

The light waves given by a point source forms a spherical wavefront in three-dimensional space.The energy is considered to travel outward along straight lines ( called rays ) from the point source called the radii of the spherical wavefront.Note – the spacing between a pair of wavefronts along any ray is always a constant. Points to note –

*Every point on a wavefront can independently produce secondary wavefronts.**Rays are always perpendicular to wavefronts.**All points of a wavefront has the same phase of vibration and same frequency.**The velocity of a wave is equivalent to that of it’s wavefronts in a particular medium.**The time needed by light to travel frome one wavefront to the next one is the same along any ray.*

**Principle of Linear Superposition** –

In Physics ,the *superposition principle*, also known as *superposition property*, states that, for **all linear systems**, the *net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually. This means that,* if input *A* produces response *X* and input *B* produces response *Y* then input (*A* + *B*) produces response (*X* + *Y*). Mathematically, for a linear system , *F*, defined by *F*(*x*) = *y*, where *x* is some sort of stimulus (input) and *y* is some sort of response (output), the superposition (i.e., sum) of stimuli yields a superposition of the respective responses:

- .

The viability of the superposition principle is based on the fact that ,* by definition, a linear system must be additive *. Superposition may sometimes imply linearity, depending on whether homogeneity is included or implied in the definition of superposition.

In wave like propagation –

*Waves motion denote variations in some parameter through space and time*—eg- height in a water wave, pressure in a sound wave, or the electromagnetic feild in a light wave. This parameter is the amplitude of the wave, and the wave itself is a function specifying the amplitude at each point. Now , in any system with waves, the waveform at a given time is a function of the sources (i.e., external forces, if any, that create or affect the wave) and initial conditions of the system. In cases like the *classic wave equation* , the equation describing the wave is linear. When this is true, the superposition principle can be applied. This *means that the net amplitude caused by two or more waves traversing the same space, is the sum of the amplitudes which would have been produced by the individual waves separately*. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side . Since light is considered to be an electromagnetic wave , it obeys this principle. When two or more light waves pass through a point , their electrical feilds combine to give a resultant electrical feild.

Naturally , we are referring to **Interference **. Now , electromagnetic wave theory states that the square of the electric feild strength is proportional to the intensity of light , which in turn is related to it’s brightness. Hence , interference alters the brightness of light. Therefore , the principle of linear superposition can account for phenomena like double – slit and thin film interference.

**Is there a condition in which the principal of linear superposition can’t be applied in Optics** ? Yes , there is . **Nonlinear optics** (NLO) deals with the behaviour of light in *nonlinear media*, i.e, *media in which the dielectric polarization P responds nonlinearly to the electric feild E of the light*. This nonlinearity is usually observed only at very high light intensities (values of the electric field comparable to interatomic electric fields, typically 10

^{8}V/m) like those provided by pulsed lasers. Thus , in nonlinear optics, the principle of superposition is not applicable .

**In Sound waves** –

Most of the phenomena associated with sound waves like *beats , diffraction* and *transverse / longitudinal standing waves* can be accounted by both the principle of *linear superposition and by the constructive / destructive interference of sound waves* .

**INTERFERENCE ** –

**Constructive Interference** – when two identical waves , i.e , having the *same wavelength and amplitude* arrive at a particular point in phase i.e* their crests and troughs align* , obeying the principle of linear superposition , they reinforce each other resulting in constructive interference. The resultant wave will have an *amplitude that is twice the amplitude *of either of the incoming waves.

**Visual Eg** . Drop a stone in a pond whose water remains still . Now, waves / ripples start to spread out. While the first wave is still rippling across the water, drop another stone close to the place where the first one was dropped. This reults in two surface waves, crests and troughs colliding and interfering. In some places, *they will interfere constructively, producing a wave—or rather, a portion of a wave—that is greater in amplitude than either of the original waves *.

Interference of waves from 2 point sources. Crests are blue, troughs red/yellow.

Interference involves two sinusoidal waves . Let them be –

y_{1} = a_{1} cos ( ωt + θ_{1} ) and y_{2} = a_{2} cos ( ωt + θ_{2}) . Now, the resultant displacement based on the principle of superposition would be –

y ( t ) = a cos ( ωt + θ ) where , a cos θ = a_{1}cos θ_{1} + a_{2} cos θ_{2} and a sin θ = a_{1} sin θ_{1} + a_{2} sin θ_{2} .

Note that the resultant wave is also sinusoidal with the same frequency but different amplitude. On squaring and adding a cos θ and a sin θ ,

we get – a = [ a_{1}² + a_{2}² + 2 a_{1} a_{2} cos ( θ_{1} – θ_{2} ) ] ½

and tan θ = [ a_{1} sin θ_{1} + a_{2} sin θ_{2}] / [ a_{1} cos θ_{1} + a_{2} cos θ_{2} ] . Now , if θ_{1} = 0 and θ_{2} = θ , we get –

** a = [ a _{1}² + a_{2}² + 2 a_{1}a_{2} cos θ ] ½** , => from this equation it can be seen that when θ = 0 ,2π , 4π …. then,

**a = a**. Hence , when the waves are in phase , the resultant amplitude will be the sum of the incoming two amplitudes i.e Constructive interference.

_{1}+ a_{2}[ Also note that tan θ = a_{2} sin θ / a_{1} + a_{2} cos θ ]

( to continue ……… )