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Journal of Optoelectronical Nanostructures

Numerical Calculation of Resonant Frequencies and Modes of a Three-Atom Photonic Molecule and a Photonic Crystal in an External Cavity

Masoud Rezvani Jalal, Fahime Fuladvand, Mahshid Habibi, Saba Fathollahi

Published Jun 1, 2016• 0 citations
OpenAlex

Abstract

In the present paper, resonant frequencies and modes of a three-atom photonic molecule and a photonic crystal placed within a cavity are numerically calculated. First, governing formulation in transverse electric field mode (TE) is obtained using Maxwell equations. Then, an algorithm based on a finite difference scheme and matrix algebra is presented. The algorithm is then implemented in a computer code developed by the authors. In the first run, resonant modes and frequencies of a linear three-atom molecule in the cavity are calculated. Calculations show that by increasing molecule length,the frequencies first come closer to each other but further increase reverses the process.Such a behavior is related to external cavity walls. In the case of photonic crystal in cavity, calculations show that the field distributions in the ground and first excited states are similar to the case where the cavity is homogeneously filled. On the other hand, calculations show that increasing the optical constant of the crystal decreases the frequencies. Such phenomenon is justified by the fact that light speed is slow in medium of high optical constants.

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Journal of Modern Optics•Nov 1, 1999•9 citations
In the present paper, resonant frequencies and modes of a three-atom photonic molecule and a photonic crystal placed within a cavity are numerically calculated. First, governing formulation in transverse electric field mode (TE) is obtained using Maxwell equations. Then, an algorithm based on a finite difference scheme and matrix algebra is presented. The algorithm is then implemented in a computer code developed by the authors. In the first run, resonant modes and frequencies of a linear three-atom molecule in the cavity are calculated. Calculations show that by increasing molecule length,the frequencies first come closer to each other but further increase reverses the process.Such a behavior is related to external cavity walls. In the case of photonic crystal in cavity, calculations show that the field distributions in the ground and first excited states are similar to the case where the cavity is homogeneously filled. On the other hand, calculations show that increasing the optical constant of the crystal decreases the frequencies. Such phenomenon is justified by the fact that light speed is slow in medium of high optical constants.
Introduction The propagation of light in periodically ordered photonic crystals bears a strong analogy to the wave propagation of conduction electrons in atomic crystals. The development of Bloch modes and dispersion are determined by the interference of waves that are diffracted by lattice planes [19, 52]. The periodicity commensurate with the wavelength (a ≈ λ/2) gives rise to Bragg diffraction that is associated with frequency windows for which waves are forbidden to propagate in a particular direction. Such stop gaps have long been known to arise for electromagnetic waves, notably for X-rays in atomic crystals [214]. At optical frequencies, the stop gaps that occur in one-dimensional periodic structures, known as Bragg stacks, are widely used in optical laboratories as broadband highly reflecting dielectric mirrors [556]. The distinguishing feature of three-dimensional (3D) photonic crystals is that a common stop gap can be achieved for all directions and for all polarizations simultaneously: the widely pursued 3D photonic band gap . Historically a 3D band gap has never been considered for the propagation of X-rays in periodic media, since a gap requires a high refractive index contrast of order unity, whereas the refractive index varies by less then 10 −4 in the X-ray range. At frequencies inside the 3D photonic band gap, the density of optical states (DOS) completely vanishes. As the density of states can also be interpreted as the density of vacuum fluctuations [334], a 3D band gap thus serves as an effective shield to these fluctuations. The total absence of optical modes in a photonic band gap has implications beyond classical optics that break the analogy between the behavior of light in photonic crystals and the behavior of electrons in atomic crystals. 3D photonic band gap crystals play an important role in cavity quantum electrodynamics (cQED) [64, 549], where they offer at least five prospects for new physics. Firstly, probably the most eagerly pursued phenomenon is the complete inhibition of spontaneous emission: an excited quantum emitter – such as an atom, molecule, or quantum dot – embedded in a crystal with its transition frequency tuned to within the 3D band gap remains forever excited since it cannot decay to the ground state by emitting a photon.
We investigate quantum interference and classical interference effects when a three-level system interacts with both a cavity field mode and an external driving field mode, within the confines of a photonic crystal material. In freespace, we found that under certain circumstances the cavity field evolves to be equal in magnitude to, but 180° out-of- phase with the external pump field when the pump field frequency equals the cavity frequency. The better the cavity, the quicker this build-up occurs. When the cavity field reaches this out-of-phase condition, the resonance fluorescence from the atom in the cavity goes to zero. This is a purely classical interference effect between the two out-of-phase fields, with the resonance fluorescence going to zero at the same time as the two excited state populations go to zero. This is quite different from the quantum interference that occurs under the right circumstances, when the state populations are coherently driven into a linear combination that is decoupled from any applied field - and population is trapped in the excited states, thus allowing for a population inversion and an amplification of incoming optical signals. In this paper, we investigate the additional effects due to the presence of the altered photon density of states in a photonic crystal.
Treating the laser medium quantum mechanically and the laser electric field classically, a mathematical model was developed to study a laser field inside a single mode optical cavity by numerical and analytical techniques. The simulations for threshold population, population inversion and average population with electric field frequency for 500 kHz resonance frequency of the atom are presented. The gain coefficient and the pump parameter with the frequency of the cavity electric field for 1000 kHz resonance frequency of the atom were simulated. The threshold population is a minimum when the frequency of electric field is equal to the resonance frequency of the atom. The population inversion varies sinusoidal with time and the frequency of the electric field and is a maximum at the threshold frequency of the atom. The saturated laser intensity at steady state increases linearly with the pump parameter and for larger values of the pump parameter, a smaller time is needed to reach the saturation of the laser field.
We consider a one-dimensional photonic crystal (1DPC) composed of double-layered dielectrics. Electric permittivity and magnetic permeability of this crystal depends on the incident electromagnetic wave frequency. We suppose that three level atoms have been added to the second layer of each dielectric and this photonic crystal (PC) has been doped. These atoms can be added to the layer with different rates. In this paper, we have calculated and compared the band structure of the mentioned PC considering the effect of added atoms to the second layer with different rates through the Fresnel coefficients method. We find out that according to the effective medium theory, the electric permittivity of the second layer changes. Also the band structure of PC for both TE and TM polarizations changes, too. The width of bandgaps related to “zero averaged refractive index” and “Bragg” increases. Moreover, new gap branches appear in new frequencies at both TE and TM polarizations. In specific state, two branches of “zero permittivity” gap appear in the PC band structure related to TM polarization. With increasing the amount of the filling rate of total volume with three level atoms, we observe a lot of changes in the PC band structure.
This paper deals with atomic systems coupled to a structured reservoir of quantum EM field modes, with particular relevance to atoms interacting with the field in photonic band gap materials. The case of high Q cavities has been treated elsewhere using Fano diagonalization based on a quasimode approach, showing that the cavity quasimodes are responsible for pseudomodes introduced to treat non-Markovian behaviour. The paper considers a simple model of a photonic band gap case, where the spatially dependent permittivity consists of a constant term plus a small spatially periodic term that leads to a narrow band gap in the spectrum of mode frequencies. Most treatments of photonic band gap materials are based on the true modes, obtained numerically by solving the Helmholtz equation for the actual spatially periodic permittivity. Here the field modes are first treated in terms of a simpler quasimode approach, in which the quasimodes are plane waves associated with the constant permittivity term. Couplings between the quasimodes occur owing to the small periodic term in the permittivity, with selection rules for the coupled modes being related to the reciprocal lattice vectors. This produces a field Hamiltonian in quasimode form. A matrix diagonalization method may be applied to relate true mode annihilation operators to those for quasimodes. The atomic transitions are coupled to all the quasimodes, and the true mode atom-EM field coupling constants (one-photon Rabi frequencies) are related to those for the quasimodes and also expressions are obtained for the true mode density. The results for the one-photon Rabi frequencies differ from those assumed in other work. Expressions for atomic decay rates are obtained using the Fermi Golden rule, although these are valid only well away from the band gaps.
Specific forms of the travelling and trapped vector mode functions for a three-dimensional Fabry-Perot cavity are developed from the general results of the preceding paper, with parameters describing the output cavity mirror chosen for a typical high Q cavity case. Cavity and external quasi-mode functions associated with the quasi-mode theory of macroscopic canonical quantization are then obtained via an idealized choice of output mirror parameters. The coupling constants describing photon exchange processes between the single cavity quasi-mode associated with each Fabry-Perot resonance and various external quasi-modes are calculated, and their slow dependence on the external quasi-mode frequency shows that the conditions for irreversible Markovian damping of the cavity quasi-mode are satisfied. For radiative atoms placed in the cavity the coupling constants for energy exchange processes with sideways travelling external quasi-modes also vary slowely, so that Markovian spontaneous emission damping occurs for the radiative atoms. However, their coupling with the isolated cavity quasi-modes is associated with reversible photon exchanges as represented via one photon Rabi frequencies. The standard model in cavity quantum electrodynamics, in which the basic processes are described by a cavity damping rate, a radiative atom spontaneous decay rate and an atom-cavity mode coupling constant has now been justified in terms of the quasi-mode theory of macroscopic canonical quantization.