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
Brief solution
1. Method.
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.
2. Decisive step.
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.
3. Verification.
Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.
Show detailed steps
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
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
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.
3. Verification.
Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.
Show detailed steps
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
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
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.
3. Verification.
Verify that the small-signal limit is exponential, while extracted energy never exceeds the stored inversion energy.
Show detailed steps
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
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
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.
3. Verification.
Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.
Show detailed steps
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
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
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.
3. Verification.
Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.
Show detailed steps
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
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
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.
3. Verification.
Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.
Show detailed steps
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
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
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.
3. Verification.
Verify population conservation and recover the weak-field rate-equation limit when \(\Omega T_2\ll1\).
Show detailed steps
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
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
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.
3. Verification.
Verify the weak-drive susceptibility, conservation in the no-relaxation limit, and the correct resonant phase quadrature.
Show detailed steps
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
Brief solution
1. Method.
Begin with the stated physical law, keep the derivation symbolic, and introduce each approximation only where its limiting condition is explicit.
2. Decisive step.
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.
3. Verification.
Verify population conservation and recover the weak-field rate-equation limit when \(\Omega T_2\ll1\).
Show detailed steps
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
Brief solution
1. Method.
List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
2. Decisive step.
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.
3. Verification.
Verify population conservation and recover the weak-field rate-equation limit when \(\Omega T_2\ll1\).
Show detailed steps
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 \(\Omega T_2\ll1\).