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Assignment-1 Gaussian beams1. Calculate the collimation range (confocal beam parameter) for three Gaussian

beams with wo1=3mm, wo2=4cm and wo3=30cm for the case of i) a CO2 laser =10.6 m and b) a frequency doubled Nd: YAG laser =532 nm. Present the results in a table. (1.5 marks) Ans. Collimation Range (2zo) =2 2 w0

Collimation Range (in m) wo (in mm) 3 4 300 CO2, =10.6 m 5.33 9.48 53347.80 Nd:YAG, =.532 m 106.29 188.97 1062944.88

2. Derive analytically the expression for the optimum wo of a monochromatic

Gaussian beam in order to achieve the minimum spot size w(z) at a distance z away from the source. Apply the result to calculate wo for =488 nm and z=1000 km. What is the value for the spot size w(z) at that distance? (2 marks) Ans.2 w( z ) = ( wo +

z 2 2 1/2 ) 2 2 wo

For optimum wo differential of w(z) with respect to wo should be 0.2 w0 wo = ( 2 z 2 2 =0 3 2 wo z 1/2 )

For =488 nm and z=1000 km, minimum wo = 0.394m.3. Calculate the size of the mode on the mirrors of a symmetric (R1=R2=1.5d)

Nd:YAG laser (=1064 nm ) cavity with length d=30 cm. What should be the minimum size of the mirrors in order to have a loss of less than 0.001% per round trip? (2.5 marks) Ans. For stable symmetric resonator minimum spot size should be at centre. Distance of mirror from centre is 0.5d.

4 2 wo Using, R( z ) = z (1 + 2 2 ) for R(z)=1.5d at z=0.5d, d=30cm and =1064 nm we z 2 z 1/2 ) =.268mm get wo = (

Radius of the mode, W(z)=.328mm

For .001% round trip loss the loss at each mirror should be 0.0005%. Radius of the aperture of mirror = .810mm4. A common method for the evaluation of the spot size of a Gaussian beam

consists of measuring the power transmission of the beam that is being partially obstructed by scanning a knife edge as shown in the following image. Show analytically how this measurement is possible. Assume that:2 P wo 2 wo 2 . Consider movement of the knife edge along the x-axis (4 I= e e wo 2x2 y2


wo z

Ans. The Intensity function of the Gaussian beam is same as the Gaussian distribution function in two variables which are independent of each other and hence can be separated into two independent Gaussian functions.2 wo2 2 wo2 I = P( e )( e ) 2 2 w0 w02 x2 2 y2

In knife-edge experiment intensity of the beam passing to the detector can be obtained by integrating the above equation for -