Chapter 7. Laser and Black Body
PL1. Rayleigh-Jeans Law and Planck’s Law.(see p.271)
Go to FileFig.(L1RAJEANS) and write du/d = 8kT2/c3 as dLB/d. Use the ration 4/c
with respect to LB as derived on page 274.
a. Go to FileFig.(L2BBLS) and plot Rayleigh-Jeans Law and Plancks’s Law on the same
graph and indicate the frequency region of reasonable agreement of the two formulas.
b. Rewrite the Rayleigh-Jeans Law in wavelength dependence and go to FileFig.
(3BBFS) and plot Rayleigh-Jeans Law and Plancks’s Law on the same graph and indicate
the wavelength range of reasonable agreement of the two formulas.
PL2. Graph of Black Body Radiation depending on Wavelength and Frequency.(see
p.275-277)
a. Wavelength
Go to FileFig.(L2BBLS) and calculate the Radiance and Radiant energy for the
wavelength interval from 5·10-6 m to 2 ·10-5 m and area A = 1 ·10-6m and Solid angle
SA = 3.14/4 (SterRad).
b. Frequency
Go to FileFig.(L3BBFS) and calculate the Radiance and Radiant energy for the
frequency interval corresponding to the wavelength interval from 5 ·10-6 m to 2· 10-5 m
and use the same values for the area A and SA.
The results must be the same!
PL3. Lorentzian Line Shape with angular Resonance frequency o and Lifetime .
(see p.284)
Go to file L6BANDS for calculation of the Lorentzian line shape.
a. For the Lorentzian line shape, change the lifetime  to more realistic values in the
range of 10-3 to 10-8 and find a corresponding frequency range.
b. Change the lifetime and find the band width. Calculate Q as o/ and compare with
the data from a graph.
PL4. Calculation of (N2-N1)/No.(see p.292)
a. Calculate {(N2-N1)/N1}( N1/No) and arrive at
(N2-N1)/No ={(W13 – A21)/( A21 + W12)}(N1/No).
a. Calculate
(No/N1) = 1 +( N2/N1) +}( N3/N1) and arrive at
(No/N1) = 1 + {(W13 + W12}/( A21 + W12) + a term neglected,
because S32 is very large.
a. Insert the result of b. into the result of a. and get
(N2 – N1)/N0 = {(W13 - A21)/( W13 + A21 + 2W12)
PL5. Paraxial Wave Equation.(see p.294)
Start from the wave equation 2u/x2 +2u/y2 +2u/z2 + k2u = 0
where k = 2/ and introduce u = (x,y,z)exp(-ikz).
Neglecting 2/z2 arrive at
2 /x2 +2 /y2 -2ik/z = 0
Show that  = exp{-i(P(z) + k(x2 + y2)/2q(z))}
is a solution with
q(z)/z = 1 and
P(z)/z = -i/q(z)
PL6. Calculation of w2(z) and R(z) for Confocal Cavity.(see p.296)
a. Modify FileFig.(L10WRS) for the calculation of three curves of the radius of curvature
of the wave front R(z) = z + zR2/z for the chosen values of zR = 80, 100, 120 cm. Read off
the graph the value of R(z) corresponding to the values of zR = 80, 100, 120 cm.
Compare to the radius of curvature of the mirror Rm =2 zR.
b. Modify FileFig.(L10WRS) to calculate w2(z) = w02 { 1 + (z/zR)2 } with w0 = 0.1 for the
three values of zR = 80, 100, 120. Reqad off the graph the waist at the mirrors and find
that it is twice as large as in the middle at z = 0.
PL7. Modes for Confocal Cavity and rectangular Mirrors.(see p.300)
Consider FileFig.(L11MOCONFCS).
Surface plots of Hermitian polynomials with exponential factor for indices 0, 1, and 2.
Scaling factors X and Y.
a. Compare with Fig.7.20, and the number of zero intensity lines with the mode numbers.
b .Extend the modes using H3(x) and H3(y) and complete all graphs until indices 0, 1, 2,
and 3.
c. Convert surface to contour plots.
PL8. Modes for Confocal Cavity and circular Mirrors.(see p.305)
Consider FileFig.(L12MOCY1to4S) and (L12MOCY5to9S)
Contour plots of generalized Laguerre polynomials for indices l and p equal 0, 1, and 2.
Scaling factor is X.
a. Compare the number of zero intensity lines with the mode numbers and with Fig.7.21.
b. Convert surface to contour plots and try to identify the modes.