Chapter 31: Magnetic-Dipole Transitions

Source: Anthony E. Siegman, Lasers (1986), Chapter 31. Use each section/problem identifier with the book; the original prompts are not reproduced here. Each entry gives the governing model, the decisive solution route, and a physical verification.

Section 31.1: Basic Properties Of Magnetic-Dipole Transitions

Problem 31.1.1 — Research problem: magnetic dipole moments in real atoms

Normalize the variables first, evaluate the analytic limits, and then sweep the remaining dimensionless parameter so the numerical curve can be checked against both limits. Use the magnetic Bloch equations \(\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-(M_x\hat x+M_y\hat y)/T_2-(M_z-M_0)\hat z/T_1\) and solve in the rotating frame. Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.

Problem 31.1.2 — Multipole expansion of a real atom

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the magnetic Bloch equations \(\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-(M_x\hat x+M_y\hat y)/T_2-(M_z-M_0)\hat z/T_1\) and solve in the rotating frame. Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.

Section 31.2: The Iodine Laser: A Magnetic-Dipole Laser Transition

Problem 31.2.1 — Iodine laser transition cross section

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Propagate irradiance with \(dI/dz=g(I)I\), using \(g(I)=g_0/(1+I/I_s)\) when saturation matters; integrate before inserting boundary values. Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.

Section 31.5: Transverse Response: The Ac Susceptibility

Problem 31.5.1 — Radiative decay rate for a classical magnetic dipole

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the magnetic Bloch equations \(\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-(M_x\hat x+M_y\hat y)/T_2-(M_z-M_0)\hat z/T_1\) and solve in the rotating frame. Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.

Problem 31.5.2 — Polarization changes for a circularly polarized wave

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the magnetic Bloch equations \(\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-(M_x\hat x+M_y\hat y)/T_2-(M_z-M_0)\hat z/T_1\) and solve in the rotating frame. Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.

Section 31.6: Longitudinal Response: Rate Equation

Problem 31.6.1 — Alternative approach to solving the Bloch equations

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the magnetic Bloch equations \(\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-(M_x\hat x+M_y\hat y)/T_2-(M_z-M_0)\hat z/T_1\) and solve in the rotating frame. Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.

Section 31.7: Large-Signal And Coherent-Transient Effects

Problem 31.7.1 — Off-resonance Rabi Bopping behavior

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the resonant coupling as \(\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar\); integrate the Bloch rotation angle \(\Theta=\int\Omega(t)\,dt\) before reading off the populations. Verify population conservation and recover the weak-field rate-equation limit when \(\Omega T_2\ll1\).

Problem 31.7.2 — Conversion between electric-dipole and magnetic-dipole models

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the magnetic Bloch equations \(\dot{\mathbf M}=\gamma\mathbf M\times\mathbf B-(M_x\hat x+M_y\hat y)/T_2-(M_z-M_0)\hat z/T_1\) and solve in the rotating frame. Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.

Problem 31.7.3 — Rabi flopping behavior: alternative derivation

Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit. Write the resonant coupling as \(\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar\); integrate the Bloch rotation angle \(\Theta=\int\Omega(t)\,dt\) before reading off the populations. Verify population conservation and recover the weak-field rate-equation limit when \(\Omega T_2\ll1\).

Problem 31.7.4 — Rabi frequency with both detuning and relaxation

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Write the resonant coupling as \(\Omega=|\boldsymbol\mu\!\cdot\!\mathbf E|/\hbar\); integrate the Bloch rotation angle \(\Theta=\int\Omega(t)\,dt\) before reading off the populations. Verify population conservation and recover the weak-field rate-equation limit when \(\Omega T_2\ll1\).