Continuous light entering a Michelson interferometer has a spectrum described by

The emerging light has average intensity where

Show that the Fringe visibility is given by:

Sketch the output intensity of the interferometer as a function of delay and give an expression for the locations of the minima.

a) Light emerging from a hot gas has a spectrum described by After passing through a Michelson interferometer, the emerging light has average intensity where

Derive an expression for the average power arriving at the detector as a function of delay.

Use

b) Find the Fringe visibility as a function of .

a) What is the spectral content (i.e.) of a square laser pulse given by

Make a sketch of showing the location of the first zeros. `

b) What is the temporal shape (i.e. ) of a light pulse with frequency content

Make a sketch of showing the location of the first zeros.

c) If is known (not same as above) and light passes through material of thickness l, and refractive index , how would you find the form of the pulse after passing through material. Please set up integral.

Nb. Fourier transform of a) A region of air above the desert on a hot day has an index of refraction that varies with heightaccording to . Show that is a solution of the Eikonal equation , where is a unit vector pointing in the direction .

b) Compute for y = h, y = h/2 and y = h/4 for both + and - cases. Represent these vectors graphically and place them sequentially point-to-tail for both + and - cases to depict how the light bends as it travels.
By applying Stokes Theorem to the Eikonal equation, , show that the ray direction lies along the path of minimal optical length (Fermat's principle).

Nb. Stokes Theorem: and by definition.

Use Fermat's principle to derive a simple law of
a) reflection. b) refraction

Sample question

a) Given that and are the ABCD matrices for propagation through a distance d and a thin lens (focal length = f), derive an expression for the ABCD matric of a simple telescope consisting of two lenses separated by a distance and hence propose a simple expression for the telescope's angular magnification

b) A complex imaging system is formed by separating two identical simple lenses by their focal length, f. Find the locations of the principle planes and effective focal length, given the relationships and illustrate these on a simple diagram.

The Fresnel-Kirchoff formula for the electric field after diffraction by an aperture is written:

By integrating in polar co-ordinates or otherwise, show that the on-axis intensity after a circular aperture of diameter, is given by;
.
Show, by use of Babinet's principle or otherwise, that the on-axis intensity after a circular block is constant

Sample question

a)Use Fermat's principle to derive Snell's law of refraction.

b)Given that and are the ABCD matrices for propagation and reflection from a curved surface respectively, show that the ABCD matrix that describes a ray which propagates through a distance , reflects from a curved mirror and then travels through a distance is given by

c)Use this ABCD matrix to derive the formula for image formation in a mirror and the magnification of the image, , where u, v and f are object, image and focal distances respectively.

The Fresnel-Kirchoff formula for the electric field after diffraction by an aperture is written

By integrating in polar co-ordinates or otherwise, show that the on-axis intensity after a circular aperture of diameter, is given by;

Show, by use of Babinet's principle or otherwise, that the on-axis intensity after a circular block is constant

Sample question

a)Use Fermat's principle to derive Snell's law of refraction.

b)Given that and are the ABCD matrices for propagation and reflection from a curved surface respectively, show that the ABCD matrix that describes a ray which propagates through a distance , reflects from a curved mirror and then travels through a distance is given by

c)Use this ABCD matrix to derive the formula for image formation in a mirror and the magnification of the image, , where u, v and f are object, image and focal distances respectively.

The Fresnel-Kirchoff formula for the electric field after diffraction by an aperture is written:

By integrating in polar co-ordinates or otherwise, show that the on-axis intensity after a circular aperture of diameter, is given by;

Show, by use of Babinet's principle or otherwise, that the on-axis intensity after a circular block is constant

Sample question

a)Use Fermat's principle to derive Snell's law of refraction.