# תורת הקרינה, 2023א

#### About course - General

#### Topic outline

### Topics

Week Topic Additional reading 1 Introduction to electrodynamics: Maxwell equations, vector and scalar potentials, presentation of the electromagnetic field as a superposition of the plane waves, electromagnetic field energy. Landau & Lifshitz, § 522 Canonical variables, electromagnetic field as a superposition of harmonic oscillators, quantization of the field, photons. Thermal (black) radiation, Plank equation, Einstein theory of spontaneous and stimulated emission. Landau & Lifshitz, § 52

Demtroder, 2.2-2.3

Corney, 9.23 Spontaneous emission, expression for the spontaneous emission probability. Oscillator strength. Introduction to line broadening. Natural broadening. Corney, 4.1-4.3, 4.6, 4.8, 8, 8.1. 4 Pressure (collisional) broadening for the impact approximations. Pressure shift. Corney, 8.2 (excluding 8.2.4 b and c)

Demtroder, 3.3 (pressure shift)5 Doppler broadening. Voigt function, transit-time broadening. Corney, 8.3-8.5

Demtroder, 3.4 (transit-time broadening)6 Radiation transfer. Beer-Lambert law and expression for the absorption coefficient. Optically thick and thin layers. Self-reversal of the line. Corney, 10.1-10.3 7 Rate equations. Three and four level systems. Conditions for population inversion in both systems. Siegman, 6.1 8 Gain saturation for the cases of Doppler and Lorentz broadening. Hole burning. Absorption spectroscopy in gases. Yariv, 5.6-5.7 Corney, 13.3 -13.49 Laser amplification. Amplifier as a filter: narrowing of the line in the amplifier. Amplification of the strong signal: homogeneous and inhomogeneous broadening. Siegman, 7.7 10 Laser oscillation. Self-excitation condition. Single and multimode oscillation for homogeneous and inhomogeneous saturation, respectively. Optical resonator modes, width of the mode, Q-factor. Yariv, 6.0-6.1

Siegman, 12.111 Power extraction in the case of homogeneous broadening. Small output coupling and constant intraresonator intensity approximation. Optimal mirror transmission and optimal power. Siegman, 12.3

Yariv, 6.4-6.512 Rigrod model for the laser power in the case of the arbitrary output coupling. Siegman, 12.4

### Textbooks

There is no one main textbook in the course. Some of the books, which are relevant to the course are listed bellow. The relevant chapters could be read from the table above.

- Corney, Atomic and laser spectroscopy
- Siegman, Lasers
- Yariv, Introduction to optical electronics
- Demtroder, Laser spectroscopy
- Landau & Lifshitz, The Classical Theory of Fields

### Lecture, Tutorial and office hours

#### Office hours

Name Day Hours Building/Room E-mail Prof. Boris Barmashenko By appointment - 54/323 barmash@bgu.ac.il #### Lecture/Tutorial

Group What? Name Day Hours Building/Room 1 Lecture Prof. Boris Barmashenko Thursday 9-12 32/209 ### 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})^{1/2}exp[-4ln2(ω-ω_{0})^{2}/Δω_{D}^{2}].8. Show that in the line center (ω = ω

_{0}) the value of the Voigt function is smaller than the value of the Lorenz line shapeand that in the

_{ }wings of the line (ω - ω_{0}>>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 k

_{0}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 ω

_{0}) is g_{0}, 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 ω_{0}) is g_{0}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 α

_{0}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 α_{0}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 α

_{0}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.### Lecture notes

Lecture 1:

https://moodle.bgu.ac.il/moodle/pluginfile.php/3582000/course/section/4925131/rad%201%202022.pdf

Lecture 2:

https://moodle.bgu.ac.il/moodle/pluginfile.php/3582000/course/section/4925131/rad%202%202022.pdf

Lecture 3:

https://moodle.bgu.ac.il/moodle/pluginfile.php/3582000/course/section/4925131/rad%203%202022.pdf

Lecture 4:

https://moodle.bgu.ac.il/moodle/pluginfile.php/3582000/course/section/4925131/rad%204%202022.pdf

Lecture 5:

Lecture 6:

https://moodle.bgu.ac.il/moodle/pluginfile.php/3582000/course/section/4925131/rad%206%202022.pdf

Lecture 7:

Lecture 8:

https://moodle.bgu.ac.il/moodle/pluginfile.php/3582000/course/section/4925131/rad%208%202022.pdf

Lecture 9:

https://moodle.bgu.ac.il/moodle/pluginfile.php/3582000/course/section/4925131/rad%209%202022.pdf

Lecture 10:

https://moodle.bgu.ac.il/moodle/pluginfile.php/3582000/course/section/4925131/rad%2010%202022.pdf

Lecture 11:

https://moodle.bgu.ac.il/moodle/pluginfile.php/3582000/course/section/4925131/rad%2011%202022.pdf

Lecture 12:

https://moodle.bgu.ac.il/moodle/pluginfile.php/3582000/course/section/4925131/rad%2012%202022.pdf:

Lecture 13:

https://moodle.bgu.ac.il/moodle/pluginfile.php/3582000/course/section/4925131/rad%2013%202022.pdf

### Class exercises

### Home work solutions

Home work 1:

Home work 2:

Home work 3:

Home work 4:

### Course policy

##### Lectures

The lectures will be held in hybrid mode

##### Grading

**Homework (10 points)**:

**Exam (90%):**The exam will include:- One question on theoretical derivations
- 3 open questions on all the subjects of the course
- You can prepare the equation sheet of 2 pages (דף אחד, 2 עמודים)
- Total mark of the exam is 105

##### General

### Past exams