Numerical solutions to 2D Maxwell–Bloch equations

Xiong, Jingyi; Colice, Max; Schlottau, Friso; Wagner, Kelvin; Fornberg, Bengt

doi:10.1007/s11082-008-9227-4

Cited by 10 publications

(10 citation statements)

References 7 publications

(6 reference statements)

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“…[249,250] The propagation along z can be performed with a con-ventional ODE scheme where, for example, the Adams-Moulton method (with the trapezoidal rule as a widely used special case) or Adams-Bashforth method, Equation (146), have been employed, in both cases combined with fourth-order Runge-Kutta for the Bloch equations. [249,250] In the more general case where time derivatives have to be considered in Equation (112), for example, to incorporate chromatic dispersion, these can be handled in Fourier domain, similarly as for the x and y derivatives discussed in the previous paragraph. One option is to process all terms in Fourier domain, [255] which however complicates the treatment of expressions which are nonlinear in the field, such as the self-phase modulation term in Equation (112).…”

Section: Generalizations and Alternative Methodsmentioning

confidence: 99%

“…In Section , 1D propagation has been assumed, neglecting the transverse coordinates in the SVAA propagation equation, Equation . In reality, the field dependence, and thus also the temporal evolution of the quantum systems, is varying along the x and y coordinates, which must be explicitly considered for an inclusion of diffraction and other effects . As long as no transverse boundary conditions or material dependencies have to be considered, that is,

Δ_{n}

and σ in Equation are constant, the most straightforward approach is to Fourier‐transform Equation with respect to x and y before the time propagation step is carried out .…”

Section: Numerical Schemesmentioning

confidence: 99%

“…In reality, the field dependence, and thus also the temporal evolution of the quantum systems, is varying along the x and y coordinates, which must be explicitly considered for an inclusion of diffraction and other effects . As long as no transverse boundary conditions or material dependencies have to be considered, that is,

Δ_{n}

and σ in Equation are constant, the most straightforward approach is to Fourier‐transform Equation with respect to x and y before the time propagation step is carried out . The resulting equation then depends on z , t and the spatial Fourier frequencies

k_{x}

and

k_{y}

, converting the derivative operator

\nabla_{T}^{2}

into a multiplication with

- (k_{x}^{2} + k_{y}^{2})

.…”

Section: Numerical Schemesmentioning

confidence: 99%

“…In the absence of other time derivatives, for example, due to chromatic dispersion, this effectively reduces the propagation equation to an ODE. The solution is then marched in z direction in dependence of τ (and

k_{x, y}

Section: Numerical Schemesmentioning

confidence: 99%

“…The solution is then marched in z direction in dependence of τ (and

k_{x, y}

if applicable), and the density matrix is updated after every propagation step . The propagation along z can be performed with a conventional ODE scheme where, for example, the Adams–Moulton method (with the trapezoidal rule as a widely used special case) or Adams–Bashforth method, Equation , have been employed, in both cases combined with fourth‐order Runge–Kutta for the Bloch equations . In the more general case where time derivatives have to be considered in Equation , for example, to incorporate chromatic dispersion, these can be handled in Fourier domain, similarly as for the x and y derivatives discussed in the previous paragraph.…”

Section: Numerical Schemesmentioning

confidence: 99%

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Optoelectronic Device Simulations Based on Macroscopic Maxwell–Bloch Equations

Jirauschek

Riesch

Tzenov

2019

Advcd Theory and Sims

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Due to their intuitiveness, flexibility, and relative numerical efficiency, the macroscopic Maxwell-Bloch (MB) equations are a widely used semiclassical and semi-phenomenological model to describe optical propagation and coherent light-matter interaction in media consisting of discrete-level quantum systems. This review focuses on the application of this model to advanced optoelectronic devices, such as quantum cascade and quantum dot lasers. The Bloch equations are here treated as a density matrix model for driven quantum systems with two or multiple discrete energy levels, where dissipation is included by Lindblad terms. Furthermore, the 1D MB equations for semiconductor waveguide structures and optical fibers are rigorously derived. Special analytical solutions and suitable numerical methods are presented. Due to the importance of the MB equations in computational electrodynamics, an emphasis is placed on the comparison of different numerical schemes, both with and without the rotating wave approximation. The implementation of additional effects which can become relevant in semiconductor structures, such as spatial hole burning, inhomogeneous broadening, and local-field corrections, is discussed. Finally, links to microscopic models and suitable extensions of the Lindblad formalism are briefly addressed. of a lower bandgap material than the adjacent layers, which restricts the free electron motion in that layer to the in-plane directions and gives rise to quantized energy states in growth direction. As a consequence of the further restriction of the energy spectrum and the even stronger carrier localization, additional improvement can be expected from 2D or 3D confinement, resulting in quantum wire/dash and quantum dot (QD) structures, respectively. Indeed, QD [1][2][3] and quantum dash [4] lasers and laser amplifiers have been shown to exhibit excellent characteristics. In Figure 1, the formation of quantized states in quantum wells, wires, and dots is schematically illustrated. The term quantum dash refers to an elongated nanostructure, that is, some kind of short quantum wire. By contrast, the term nanowire does not necessarily indicate strong quantum confinement. For example, in nanowire lasers, the nanowire geometry typically serves as a single-mode optical waveguide resonator, while the active region is based on a heterostructure or quantum well, as in a conventional laser diode. [5,6] Semiconductor optoelectronic devices usually rely on electron-hole recombination, that is, optical transitions between conduction and valence band states. The associated resonance wavelength is largely determined by the semiconductor bandgap, which establishes a lower bound on the transition energy. Thus, the coverage of a certain spectral region depends on the existence of suitable semiconductor materials, which for example restricts the availability of practical optoelectronic sources and detectors in the mid-infrared and terahertz regions. An alternative concept is based on intersubband devices, which employ so-called i...

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Section: Generalizations and Alternative Methodsmentioning

confidence: 99%

Δ_{n}

and σ in Equation are constant, the most straightforward approach is to Fourier‐transform Equation with respect to x and y before the time propagation step is carried out .…”

Section: Numerical Schemesmentioning

confidence: 99%