1. Nonlinear Optics Lab. Hanyang Univ. Chapter 8. Semiclassical Radiation Theory 8.1 Introduction Semiclassical theory of light-matter interaction (Ch. 6-7) - Ignores the quantum-mechanical nature (ex : quantum fluctuations) of the EM field - Treats the matter quantum-mechanically through the Schrodinger equation => Semiclassical theory is not perfect to describe fully the light-matter interactions (ex : spontaneous emission), but successful to describe all of atomic radiation when the number of photons are much larger than unity. In this chapter, we will complete our development of the semiclassical theory ; Maxwell-Bloch equation.

2. Nonlinear Optics Lab. Hanyang Univ. 8.2 Optical Bloch Equation : An equivalent set of vector equations of the density-matrix equations ; (6.5.14) (6.5.17) (6.5.2) : without relaxation : with relaxation

3. Nonlinear Optics Lab. Hanyang Univ. For two-level atomic system, Define, (8.2.1) 1) In the case of no relaxation (6.5.2) => (8.2.2)

4. Nonlinear Optics Lab. Hanyang Univ. Consider a fictitious space with unit vectors, and define a “coherence vector” (Pseudo spin, Bloch vector), and a “torque vector” (Axis vector), (8.2.2) => (8.2.5)

5. Nonlinear Optics Lab. Hanyang Univ. ※ The effect of is only to rotates about the direction of It cannot lengthen or shorten ※ : Conservation of probability in the two-level atom. (in the absence of collision or relaxation) ※ : Degree of inversion : population is entirely in the upper level : population is entirely in the lower level

6. Nonlinear Optics Lab. Hanyang Univ. Ex) : Bloch vector rotates about axis (8.2.2c) => : Same form to (6.3.18) when

7. Nonlinear Optics Lab. Hanyang Univ. Ex) : “area” of the pulse where, [ pulse] If, external wave inverts the atomic population from the lower to the upper level.

8. Nonlinear Optics Lab. Hanyang Univ. If E 0 is time-dependent, ※ 1) The rotating wave approximation in going from (6.3.13) to (6.3.14) fails if  itself contributes rapid temporal variation. => We must assume that E 0 (t) is a slow varying time function. 2) If not just the amplitude but the phase of E 0 changes in time, we can no longer assume that an adjustment of the wave function phase will make  (t) real. => , E should be complex. [Remarks]

9. Nonlinear Optics Lab. Hanyang Univ. 2) In the case considering the relaxation processes (6.5.14), (6.5.17) => By definitions,

10. Nonlinear Optics Lab. Hanyang Univ. where, : longitudinal lifetime of spin : transverse lifetime of spin

11. Nonlinear Optics Lab. Hanyang Univ. 8.3 Maxwell-Bloch Equations Atomic state under the influence of an external perturbation (EM wave or light) can be described by Schrodinger equation. But, in order to describe the atomic state exactly, we need additional equation to express the behavior of the light by the Interaction with the atoms. = Maxwell equation ! Assumptions : : monochromatic, plane wave propagating along the z-direction

12. Nonlinear Optics Lab. Hanyang Univ. Maxwell wave equation : where, (8.2.18) => Maxwell-Bloch Equations

13. Nonlinear Optics Lab. Hanyang Univ. 8.4 Linear Absorption and Amplification In practice, there may be background atoms in laser active medium, and we should add the background atom effect to the Maxwell equation, (8.3.6). (8.3.1) : background atom effect Background-atoms are far from resonance and come to steady state extremely quickly, so we can use the adiabatic result (7.2.1) for  21. And, these atoms are at most only slightly excited, so that (7.2.1) where,

14. Nonlinear Optics Lab. Hanyang Univ. (8.3.6) => [Quasi-steady solution] where,

15. Nonlinear Optics Lab. Hanyang Univ. 1) If the resonant atoms are all in their ground state,  11 ~1,  22 ~0 2) If the resonant atoms are all in their excited state,  11 ~0,  22 ~1 * Threshold condition for amplification ;  22 -  11

16. Nonlinear Optics Lab. Hanyang Univ. 8.5 Semiclassical Laser Theory Electric field in a laser cavity ; where, Polarization of m-th mode ; Maxwell wave equation considering cavity loss ; Actually, different cavity modes are coupled through the z-dependence of  m   z,t , but in many lasers this coupling is not important. => Take the average value of    t) instead of individual  m   z,t).

17. Nonlinear Optics Lab. Hanyang Univ. For quasisteady-state lase operation, and (7.2.1) : adiabatic approximation

18. Nonlinear Optics Lab. Hanyang Univ. where, : Fundamental equations of semiclassical laser theory

19. Nonlinear Optics Lab. Hanyang Univ. In steady state, Cf) section 3.5 : a positive sign of g is now possible ! ※

20. Nonlinear Optics Lab. Hanyang Univ. [Einstein laser model] (8.5.13) => (8.5.12) => Add a pumping term, K, and Assume N 2 >>N 1 condition is maintained by the pumping : Define atom number :, photon number :  Results of (1.5.1), (1.5.2) Einstein laser model