תורת הקרינה, 2023א

Topics

Textbooks

Lecture, Tutorial and office hours

Theoretical questions for exam

1.      Derive Plank’s equation for the energy density of thermal radiation.

2.      Derive Wien displacement lаw: show that \( \lambda \)_m corresponding to the maximum intensity of radiation is given by λ_m≈0.2hc/kT.

3.      Using phenomenological approach of Einstein find relations between the coefficients of absorption (B_lu), spontaneous (A_lu) and stimulated (B_ul) emission.

4.      Using the model of damped harmonic oscillator find the natural line shape.

5.      Show that in the case of collisional broadening in the impact approximation the line profile is Lorentzian (Lorentz theory) (Corney)

6.      Derive an expression for the line profile in the case of Doppler broadening.

7.      Derive an expression for the line profile in the gas if the Doppler and Lorentz line-widths are of the same order of magnitude (Voigt function). (Doppler line profile is given by G(w)  = (4 ln2/πΔω_D)^1/2exp[-4ln2(ω-ω₀)²/Δω_D²].

8.   Show that in the line center (ω = ω₀) the value of the Voigt function is smaller than the value of the Lorenz line shape 

and that in the wings of the line (ω - ω₀>>Δω_D, Δω_L) the line profile is always Lorentzian.

9.   Derive an expression for absorption coefficient k in terms of the wavelength l, the spontaneous coefficient A, lineshape functiom g(ν) and the number densities of the particles in the lower and upper level, N_l and N_u, respectively.

10.  Derive an expression for the flux of radiation energy of frequency w emitted by the layer of depth l. Assume that the layer consists of the two-level molecules with the energy of transition hω.  The temperature is T. Consider only the limiting cases of the optically thick and thin layer.

11.  Estimate the linewidth of the radiation emitted by the optically thick body in the case of Lorentzian broadening. The product of the absorption coefficient in the center of line k₀ and the dimension of the body l is much greater than unity.

12.  Find dependencies of the population inversion on the pumping rate for 4- and 3- level systems using rate equations. For what system (3- or 4- level) is it easier to get population inversion and why?

13.  Derive the homogeneous saturation law: find the gain at frequency \( \omega \), when a strong saturating signal of intensity I is applied at the same frequency w in the case of homogeneous (Lorentzian) broadening. Assume that the small signal gain in the line center (frequency ω₀) is g₀, the life time of the upper level \( \tau \) and stimulated emission cross section \( \sigma \).

14.  Derive the inhomogeneous saturation law: find the gain at frequency \( \omega \) when a strong saturating signal of intensity I is applied at the same frequency \( \omega \) in the case of homogeneous (Doppler) broadening (Δω_D>>Δω_L). Assume that the small signal gain in the line center (frequency ω₀) is g₀ and the saturation parameter is I_s,res.

15.  Line narrowing in the amplifier. Find the linewidth after passing through the laser amplifier with the unsaturated small signal gain α₀ and length l; assume that before the amplifier the line has a uniform spectral intensity profile and that homogeneous (Lorentz) broadening takes place. Neglect saturation effects.

16.  Saturation in laser amplifier. Laser light with intensity I_in is passing through the amplifier with unsaturated small signal gain α₀ and length l. 

i) Find the output intensity I_out assuming homogeneous broadening with saturation parameter I_S. Find maximum available power which can be extracted from the amplifier; for which I_in this power can be extracted. Assume that the frequency of the amplified beam is equal to the central frequency of the laser transition.

ii) The same for in-homogeneous broadening in the gain medium.

17.  Laser oscillation. Find the output power of the laser with small signal gain α₀ and the gain length l. The transmission of the output mirror is t, the absorption/scattering losses of this mirror are equal to a. Find optimal transmission of the mirror, assume that intensity is uniform inside the resonator.

Class exercises

Homework exercises

• 

  Homework 1 Quiz
• 

  Submission box homework 1 Assignment
• 

  Homework 2 Quiz
• 

  Submission box homework 2 Assignment
• 

  Homework 3 Quiz
• 

  Submission box homework 3 Assignment
• 

  Homework 4 Quiz
• 

  Submission box homework 4 Assignment

Home work solutions

Course policy

Past exams

Exam solutions