2: Maxwell-Bloch Equations
- Page ID
- 44641
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 2.1: Maxwell's Equations
- This page covers Maxwell's equations, outlining the fundamental relationships between electric and magnetic fields. It details the equations' curl and divergence relationships, as well as material responses for dielectrics and magnets. Key variables like the electric field (\(\vec{E}\)), magnetic field (\(\vec{H}\)), polarization (\(\vec{P}\)), and magnetization (\(\vec{M}\)) are discussed.
- 2.2: Linear Pulse Propagation in Isotropic Media
- This textbook page elaborates on wave equations in dielectric non-magnetic media, exploring the relationships between electric field \(\vec{E}\), polarization \(\vec{P}\), refractive index, and dispersion for plane waves. It explains the interaction with magnetic fields \(\vec{H}\) and introduces complex notation for physical fields like energy density and Poynting vectors.
- 2.3: Bloch Equations
- This page explores the behavior of two-level atoms, detailing their line spectra, eigenstates, and transitions using operators. It covers the influence of lattice symmetry on dipole moments, introduces Rabi-oscillations, and examines density operators characterizing quantum systems. The von Neumann equation is discussed in the context of population dynamics and dipole oscillation, alongside phase and energy relaxation rates impacted by environmental interactions.
- 2.4: Dielectric Susceptibility
- This page covers the interaction between a monofrequent electric field and an atomic dipole moment, explaining how both oscillate at the same frequency. It introduces the normalized Lorentzian lineshape, deriving formulas for saturated inversion, susceptibility, and dipole polarization.
- 2.5: Rate Equations
- This page examines the Maxwell-Bloch equations that describe how an electromagnetic field interacts with a collection of atoms. It derives equations for electric field polarization, dipole moments, and their temporal and spatial evolution. The discussion narrows down to a specific electromagnetic wave, resulting in a simplified rate equation that reflects the gain in an inverted medium, which plateaus as electromagnetic power density increases.
- 2.6: Pulse Propagation with Dispersion and Gain
- This page covers the time-domain analysis of mode locking in lasers and nonlinear wave propagation, discussing pulse dynamics in different media, the effects of frequency-dependent refractive index, group velocity dispersion, and medium loss on pulses. It also examines optical transition properties, using complex Lorentzian susceptibility to explain modifications to the refractive index and losses in noninverted media.
- 2.7: Kramers-Kroenig Relations
- This page covers the linear susceptibility and its role in a system's frequency response to an electric field, following Kramers-Kroenig Relations. It explains the use of delta functions in approximating the imaginary part for transparent media, leading to the Sellmeier Equation for the refractive index, with examples from fused quartz and sapphire.
- 2.8: Pulse Shapes and Time-Bandwidth Products
- This page includes a detailed table (Table 2.2) that outlines different pulse forms—Gaussian, hyperbolic secant, rectangular, Lorentzian, and double-exponential—along with their characteristics such as pulse shape, spectrum, temporal width (\(\Delta t\)), and time-bandwidth product (\(\Delta t \cdot \Delta f\)).