wave optics

some microscopic phenomena explained by wave optics

• interference

• diffraction

• polarisation

wavefront

• it is the locus of all particles of a medium, which are vibrating in the same phase

• a line perpendicular to a wavefront is called a ray

spherical wavefront

• when the source of light is a point source

• its a sphere with the centre as source

(draw the diagram)

cylindrical wavefront

• when the source of light is linear (straight line/ slit etc.)

• it is cylindrical in shape

• all points equidistant from the source lie on the cylinder

(draw the diagram)

plane wavefront

• when the point/ linear source of light is at a very large distance

• a small portion of the spherical/ cylindrical wavefront appears to be a plane

(draw the diagram)

huygen’s principle

• each point on the wavefront (primary wavefront) is the source of a secondary disturbance (secondary wavelets), and these wavelets spread out in all directions with the speed of the wave

  • the energy of the wave travels perpendicular to the wavefront

• a surface touching these secondary wavelets tangentially in the forward direction at any instant of time gives a new wavefront (secondary wavelength)

(draw the diagram)

why is there no backward secondary wavefront?

this is because the amplitude of the secondary wavefront is maximum in the forward direction and zero in the backward direction

derivation of laws of reflection using huygen’s principle

derive it

(first & second laws)

derivation of snell’s law using huygen’s principle

derive this

also prove that the frequency doesnt change after refraction

behaviour of prism towards plane wavefront

• the speed of light waves is less in glass

• hence, the lower portion of the incident wavefront (which travels through the greatest thickness of the prism) will get delayed

• this results in a tilt in the emerging wavefront

(draw diagram)

behaviour of lens towards plane wavefront

• the speed of light is less in lens

• hence, the central portion of the incident wavefront travels through the thickest part and is delayed the most (convex lens)

• this results in a depression at the centre of the refracted wavefront

  • thus, the wavefront becomes spherical and converges at the focus

(draw the diagram)

behaviour of spherical mirror towards plane wavefront

• the central part of the incident wavefront travels the largest distance before reflection (concave mirror)

• hence, the reflected wavefront gets delayed at the centre

• this results in the reflected wavefront becoming spherical, which converges at the focus

coherent sources

they are sources of light which emit light waves of same wavelength, frequency and are in same phase or have constant phase difference

incoherent sources

they are sources of light which do not emit light waves with constant phase difference

interference term

• when 2 independent sources of light emit monochromatic waves of intensities I1 & I2, and phase difference \phi meet at a point, then

• resultant I = I1 + I2 + 2\sqrt{I1I2} \cos\phi

interference term when \cos\phi remains constant with time

in this case,

• maximum value: when \cos\phi is 1

  • I = (\sqrt{I1} + \sqrt{I2} )²

• minimum value: when \cos\phi is -1

  • I = (\sqrt{I1} - \sqrt{I2} )²

• hence, it is clear that we need 2 sources with same frequency & constant phase difference

  • so, we need 2 coherent sources

inference term when \cos\phi various with time

• assuming it varies with both positive and negative value, then the avg. value of \cos\phi over a full cycle is 0

• hence, resultant I = I1 + I2

• the 2 sources in this case are incoherent sources

conditions for obtaining 2 coherent sources of light

• they should be obtained from a single source from some device, such as

  • the source and its virtual image (lloyd’s mirror)

  • 2 virtual images of the same source (fresnel’s biprism)

  • two real images of the same source (young’s double slit)

• the 2 sources should give monochromatic light

• the path difference between light waves from both sources should be small

fringe

bright/ dark band on the screen formed due to the constructive/ destructive interference of light from the two slits

young’s double split experiment

supposing S1 and S2 are 2 slits distances d apart, with a screen distance D away. a source of light S with wavelength \lambda is passed through

• constructive interference (bright fringes)

  • distance of nth bright fringe from midpoint O on the screen,

    • y = nD\lambda / d

• destructive interference (dark fringes)

  • distance of nth dark fringe from midpoint O on the screen,

  • y = (2n - 1)D\lambda / 2d

where n = nth bright/ dark fringe

(draw diagram)

fringe width

• it is the separation between any 2 consecutive bright & dark fringes

• \beta = D\lambda / d

intensity of fringes

for bright fringe

• \phi = 2n\pi

  • \cos\phi = 1

• hence, I = I1 + I2 + 2\sqrt{I1I2}

• I = 4Io (since in YDSE, I1 = I2 = Io)

for dark fringe

• \phi = (2n - 1)\pi

  • \cos\phi = -1

• hence, I = I1 + I2 - 2\sqrt{I1I2} = 0 (I1 = I2 = Io)

in general, the intensity at any point is

I = 4Iocos²(\phi /2)

what happens to the fringe width when its immersed in a medium with R.I \mu ?

• it decreases \mu times

• this is because the wavelength decreases

distribution of intensity

• all bright interference fringes have intensity 4Io

• all dark fringes have intensity 0

(draw diagram)

conditions for sustained interference

in order to get a well defined interference pattern, intensity of constructive should be maximum and destructive should be 0. the following conditions must be satisfied

• the 2 sources of light should be coherent

• the 2 interfering ways must have same plane of polarisation

• the 2 sources must be very close to each other and the pattern must be observed at a larger distance

• the sources must be monochromatic

  • otherwise, the fringes of diff. colors would overlap

• the 2 waves must have same amplitude for better contrast btw. bright and dark