Three-level Rate Equations Research Articles

Sensitivity enhancement in erbium-doped fiber laser intra-cavity SPR sensor

To enhance sensitivity of fiber-based surface plasmon resonance (SPR) sensor for refractive index detection, we propose a concept of fiber laser intra-cavity SPR sensor that is constructed by placing a hollow fiber-based SPR sensor in an erbium-doped fiber laser cavity. The hollow fiber-based SPR sensor is firstly designed and optimized to achieve maximum amplitude sensitivity of 65.576 RIU−1 in a detection range of 1.6–1.61 at the operation wavelength of 1530 nm, and then placed in the erbium-doped fiber laser cavity to form the fiber laser intra-cavity SPR sensing system. The relationships between the detection sensitivity, pump power, and SPR sensor length of the laser intra-cavity SPR sensor are investigated in detail by using three-level rate equations, and the numerical results demonstrate that the sensitivity is enhanced 82 times, achieves 5375.627 RIU−1 when the fiber-based SPR sensor is placed in the laser cavity, and can be further enhanced by tuning the pump power to close to threshold value. The work described here demonstrates a possible solution to enhance the sensitivity by combining the fiber-based SPR sensor and the fiber laser.

Dynamic modulation and impedance characteristics of a terahertz quantum cascade laser

The modulation and impedance properties of a terahertz (THz) quantum cascade laser (QCL) are investigated theoretically based on a three-level rate equation model. The effect of different device parameters, namely, facet reflectivity, injection efficiency, spontaneous emission factor, and operating current, on the modulation and impedance characteristics of the QCL is analyzed in detail. The device shows a maximum modulation bandwidth (f3dB) of 21 GHz and an intrinsic impedance of 3.8 mΩ when operating under the designed conditions. The modulation bandwidth and impedance increase with the increase in facet reflectivity and injection efficiency. However, the modulation bandwidth increases but the impedance decreases with the increase in operating current. The spontaneous emission factor has no effect on both the modulation bandwidth and impedance. The theoretical model will aid in the design of THz QCLs requiring a large modulation bandwidth and the external circuit design to match the standard 50 Ω source for reducing reflections and improving the coupling efficiency.

2.7 μm mid-infrared Er-doped thin-film LiNbO3-on-insulator photonic wire laser

The 2.7 μm continuous-wave laser was demonstrated theoretically on the basis of a 1480-nm-pumped Er-doped thin-film LiNbO3-on-insulator photonic wire (Er:LNOI PW) and a three-level rate equation model of Er3+. Under 1480 nm pumping, the excited state absorption (4I13/2 → 4I9/2) and cooperative upconversion process (4I13/2 → 4I15/2: 4I13/2 → 4I9/2) populate the 4I11/2 level meanwhile bypass significantly the 4I13/2 level, enabling a population inversion between these two levels. The cavity length and output coupler's reflectivity at laser wavelength were optimized for maximizing laser output. Under the optimum configuration, dependences of laser power on both pump power and Er3+ concentration were investigated. Simulation results show that lower threshold pump power, higher laser output and slope efficiency are obtained for a higher Er3+ concentration. The laser output reveals a linear relationship to the Er3+ concentration and the relationship is strong at higher pump level. Simulation results show that the threshold pump power, laser output and slope efficiency are on orders of several tens of mW, sub-mW and 3.1%, respectively, for an Er3+ concentration of 1.5 mol%. In addition, there is no need to account for photorefractive effect on the laser performance as it is far smaller than the refractive index contrast of the LNOI PW, ∼0.7. Present work opens up the possibility of implementing an on-chip 2.7 μm mid-infrared laser based on an Er:LNOI.

Quadratic Zeeman spectral diffusion of thulium ion population in an yttrium gallium garnet crystal

The creation of well understood structures using spectral hole burning is an important task in the use of technologies based on rare earth ion doped crystals. We apply a series of different techniques to model and improve the frequency dependent population change in the atomic level structure of Thulium Yttrium Gallium Garnet (Tm:YGG). In particular we demonstrate that at zero applied magnetic field, numerical solutions to frequency dependent three-level rate equations show good agreement with spectral hole burning results. This allows predicting spectral structures given a specific hole burning sequence, the underpinning spectroscopic material properties, and the relevant laser parameters. This enables us to largely eliminate power dependent hole broadening through the use of adiabatic hole-burning pulses. Though this system of rate equations shows good agreement at zero field, the addition of a magnetic field results in unexpected spectral diffusion proportional to the induced Tm ion magnetic dipole moment and average magnetic field strength, which, through the quadratic Zeeman effect, dominates the optical spectrum over long time scales. Our results allow optimization of the preparation process for spectral structures in a large variety of rare earth ion doped materials for quantum memories and other applications.

Ultrafast intramolecular charge transfer dynamics and nonlinear optical properties of phenothiazine-based push–pull zinc porphyrin

Comprehensive investigations of the ultrafast photophysical properties and charge injection dynamics (adsorbed on mesoporous TiO2) of phenothiazine-functionalized push–pull Zinc porphyrin (LG6) utilizing a benzothiadiazole (BTD) as an acceptor unit is presented herewith. This includes density functional theory (DFT) calculations, fluorescence measurements, time-dependent fluorescence measurements, and femtosecond transient absorption (fs-TA) spectroscopy measurements. The charge injection dynamics of LG6 were compared with another porphyrin, LG5, with a similar molecular structure but thiophene as the acceptor unit instead of BTD. The multicomponent electron injection analysis from different excited electronic energy levels reveals ultrafast electron injection times of 434 fs and 336 fs for LG6 and LG5, respectively, revealing the impact of acceptor units on charge injection times. Further, femtosecond nonlinear optical (NLO) properties were characterized using the femtosecond Z-scan measurements with 70 fs, 800 nm pulses. The analysis of Z-scan open aperture data has been performed by solving three-level rate equations incorporating excited-state lifetimes calculated from the target/global analysis of fs-TA data set. The estimated NLO parameters and ultrafast NLO response times of LG6 indicated the strong potential for optical limiting and optical signal processing applications. The second hyperpolarizability was estimated to be γ=2.5×10-31esu, which was found to be in close agreement with the theoretically calculated value of 0.4×10-31e.s.u.

Comparative investigation into key optoelectronic characteristics of semipolar InGaN blue laser diodes: A strategy to mitigate quantum-confine stark effect

In recent years, semipolar InGaN-based blue laser diodes (LDs) have raised indescribable appeal within solid-state lighting industry as promising alternatives for currently available polar c-plane LDs due to reduction in internal polarization field and related quantum-confined stark effect. However, any comparative study in terms of energy band gap, optical emission profile, power output (P) and forward bias response (I-V) etc. between semipolar-oriented InGaN blue LDs is inherently unattainable. This work aims to demonstrate the impact of internal polarization field on aforementioned key optoelectronic parameters of In0.17Ga0.83N/GaN double quantum well (DQW) blue LD along semipolar (101-2), (112-2), (101-1), (202-1), (303-1-), (202-1-) and (101-1-) crystal orientations by figuring out six-band k.p Hamiltonian at the brillouin zone center aided from Euler’s tensor transformation method. Three-level rate equations based LD analogous circuit model and state-space matrix are employed here to obtain P-I-V and frequency characteristics. Radiative as well as non-radiative recombination mechanisms for LD systems are portrayed by addressing modified ABC model. Our numerical investigation illustrates that best hole effective mass, optical gain, lasing power, threshold current and forward voltage can be attained from (112-2)-oriented InGaN blue LD due to insignificant presence of spontaneous plus piezoelectric (PZ) polarization charge. In addition with these, the inspection of bode plot and relevant optical power gain (dB) confirms the stability and superiority of this semipolar blue LD system for achieving high-speed visible-light communications at relatively low current densities.

Terahertz Plasmonic Quantum Cascade Lasers: a Comprehensive Physics Study

Generation of terahertz (THz) radiation has been a hot research topic in recent years. Plasmonic quantum cascade lasers (QCLs) are among the most compact and efficient sources to generate THz radiation. In this paper, we comprehensively study plasmonic QCLs designed based on the antenna-feedback structure to generate efficient radiation about the center frequency of 3 THz. By changing the geometric structure of the plasmonic cavity and using two-dimensional simulation, a minimum loss less than 5.9 cm−1 is achieved at the lasing frequency. It is also possible to control the orientation of the output beam either vertically or tilted by changing the geometry of the antenna design via chirped or non-chirped grating scheme. Moreover, the output characteristics of the QCL are simulated based on the three-level rate equations through which the dynamics of the laser, as well as the P-I curve, are investigated. Also, the gain spectra for two laser designs (with chirped and non-chirped gratings) are simulated and compared to each other. The results of this paper may provide deep insight into designing efficient laser sources in the THz region.

Numerical investigation into optical and electronic performance of crystal orientation-dependent InGaAs/InP near-infrared laser

The energy band dispersion profile and their dependence on crystal orientation for III-V zinc-blende compound quantum wells (QW) have earned significant attention in recent years due to reduced band mixing effects emerging from increased energy separation between valence subbands. So QW like InGaAs grown in non-conventional orientation seems to be quite optimistic to serve as active region in near-infrared optoelectronic applications due to small piezoelectric polarization, improved confinement of high energy states and increased favorable spectral range than traditional (100)-oriented growth. Here, crystal orientation-dependent electronic and optical performance of InGaAs-InP laser subjected to 1.60% compressive strain and emitting around 2 µm is studied numerically after working out an eight-band k.p Hamiltonian with the help of finite difference method and Tensor rotation technique. An equivalent circuit model using three-level laser rate equations is employed here to reveal optical output power and steady state frequency response. It is noticed that the wave functions of hole are more confined in (111) orientation than (100) orientation. Also, there is a noteworthy reliance of the energy band gap, optical gain spectra and output lasing power profile on change in crystal orientation. The estimated gains are found to be 4250, 3900, 3555, 3210 and 2950 cm−1 in (111), (100), (131), (110), and (113) orientations. The topmost lasing power and lowest threshold current are noted to be 56.4mW and 4.5 mA, respectively, in the (111) crystal orientation. Further, the highest optical emission point is observed to be moved towards longer wavelength for the alteration in crystal orientation from (131) to (111).

Photoluminescence of InAs/GaAs quantum dots under direct two-photon excitation

Self-assembled quantum dots grown by molecular beam epitaxy have been a hotbed for various fundamental research and device applications over the past decades. Among them, InAs/GaAs quantum dots have shown great potential for applications in quantum information, quantum computing, infrared photodetection, etc. Though intensively studied, some of the optical nonlinear properties of InAs/GaAs quantum dots, specifically the associated two-photon absorption of the wetting and barrier layers, have not been investigated yet. Here we report a study of the photoluminescence of these dots by using direct two-photon excitation. The quadratic power law dependence of the photoluminescence intensity, together with the ground-state resonant peak of quantum dots appearing in the photoluminescence excitation spectrum, unambiguously confirms the occurrence of the direct two-photon absorption in the dots. A three-level rate equation model is proposed to describe the photogenerated carrier dynamics in the quantum dot-wetting layer-GaAs system. Moreover, higher-order power law dependence of photoluminescence intensity is observed on both the GaAs substrate and the wetting layer by two-photon excitation, which is accounted for by a model involving the third-harmonic generation at the sample interface. Our results open a door for understanding the optical nonlinear effects associated with this fundamentally and technologically important platform.

Resonant two-photon spectroscopy of the 2s3d 1D2 level of neutral Be9

We report an absolute frequency measurement of the $2s3d{\phantom{\rule{0.16em}{0ex}}}^{1}{D}_{2}$ state in neutral $^{9}\mathrm{Be}$ using two-photon spectroscopy with a resonant intermediate state. The absolute center-of-gravity energy is determined to be 64 428.40 321(55) ${\mathrm{cm}}^{\ensuremath{-}1}$, a factor of 180 more precise than the previous experimental measurement. We also confirm our previous result for the energy of the intermediate $2s2p{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}$ level. Precision is limited by unresolved hyperfine structure and the complications of performing resonant two-photon spectroscopy on an atomic beam. A three-level rate-equation analysis is presented to explore, and minimize, systematic uncertainties arising from small deviations of the angle between the atomic and optical beams off-perpendicular.

Three-level rate equations in cold, disordered Rydberg gases

We have investigated formation of structures of Rydberg atoms excited from a disordered gas of ultracold atoms, using rate equations (REs) for two-photon Rydberg excitation in a single atom without eliminating the intermediate state. We have explored the validity range of these REs and defined a simple measure to determine, whether our model is applicable for a given set of laser parameters. We have applied these REs in Monte Carlo simulations of ultracold gases, for different laser beam profiles, and compared these simulations to experimental observations and find a general agreement.

Dynamical Characteristics of External Cavity Quantum Cascade Lasers

In this paper, we calculate the dynamical characteristics such as delay time, rise time, and time to steady state establishment, of a mid infrared quantum cascade laser coupled to external cavity. The approach is based on the three-level rate equations including the dependence of the loss on external cavity parameters. We find in particular that the threshold current of external cavity is strongly influenced by the external reflectivity and external cavity length. In addition, the equations that allow for the determination of the dynamical characteristics are derived within the premises of our model in the general case. The effects of the external cavity parameters on dynamical characteristics are also explored.

Gain Switching of Monolithic 1.3 μm InAs/GaAs Quantum Dot Lasers on Silicon

We report the first demonstration of gain-switched optical pulses generated by continuous-wave 1.3 μm InAs/GaAs quantum dot (QD) broad-area lasers directly grown on silicon. The shortest observed pulses have typical durations between 175 and 200 ps with peak output powers of up to 66 mW. By varying the drive current pulsewidth and amplitude systematically, we find that the peak optical power is maximized through sufficiently long high-amplitude drive pulses, whereas shorter drive pulses with high amplitudes yield the narrowest achievable pulses. A three-level rate equation travelling-wave model is used for the simulation of our results in order to gain a first insight into the underlying physics and the laser parameters responsible for the observed behavior. The simulations indicate that a limited gain from the InAs QDs and a very high gain compression factor are the main factors contributing to the increased pulsewidth. As the optical spectra of the tested broad-area QD laser give a clear evidence of multitransverse-mode operation, the laser's dynamic response could be additionally limited by transversal variations of the gain, carrier density, and photon density over the 50 μm wide laser waveguide.

The effect of external cavity reflectivity on the dynamic behaviors of quantum cascade laser

The effect of external reflectivity on the dynamic behaviors of quantum cascade laser coupled to external cavity is theoretically investigated. Our model is based on three-level rate equations including the dependence of the loss on external cavity parameters. We find in particular that the effective reflectivity, photon lifetime and threshold current are strongly influenced by the external cavity length and external reflectivity. The optimum external cavity length (Lext) is about 4 cm. We also show that the external cavity reflectivity influences significantly the dynamics of the electron number in the upper and lower laser levels, population inversion, and photon numbers in the Fabry Perot and in the external cavities. In addition, the external cavity reflectivity dependences of turn-on delay time (tth) is also studied and discussed.

Rate equation modeling of the frequency noise and the intrinsic spectral linewidth in quantum cascade lasers.

This work theoretically investigates the frequency noise (FN) characteristics of quantum cascade lasers (QCLs) through a three-level rate equation model, which takes into account both the carrier noise and the spontaneous emission noise through the Langevin approach. It is found that the power spectral density of the FN exhibits a broad peak due to the carrier noise induced carrier variation in the upper laser level, which is enhanced by the stimulated emission process. The peak amplitude is strongly dependent on the gain stage number and the linewidth broadening factor. In addition, an analytical formula of the intrinsic spectral linewidth of QCLs is derived based on the FN analysis. It is demonstrated that the laser linewidth can be narrowed by reducing the gain coefficient and/or accelerating the carrier scattering rates of the upper and the lower laser levels.

Theoretical study of laser-mode competition in quantum-dot semiconductor lasers using a self-consistent electro-opto-thermal model

We present a new model for quantum-dot (QD) semiconductor lasers which includes the self-consistent treatment of electrical, optical, and thermal interactions. The approach is developed from the energy balance equation to incorporate the carrier temperature and a three-level rate equation to describe the carrier and photon dynamics within the QD active region operating simultaneously on two longitudinal modes: via ground-state (GS) and first-excited-state (ES) transitions. Using the presented model, the steady-state, small-signal modulation response and transient device characteristics are calculated and analyzed. Similar to the situation of mode competition of laser modes, it is found that the light output power of the GS mode enters the roll-off regime, while the power of the ES mode increases. The model also predicts the negative temperature dependency of the power threshold for the ES mode, which can be understood from the negative temperature-dependent behavior of incomplete clamping of the ES population above the GS threshold. The simulated results are in good agreement with experimental data from the devices reported earlier.

Three-level rate-equations-based model of quantum cascade lasers with a single solution regime

The three-level rate-equations-based QCL models can possess multiple DC solution regimes for nonnegative values of injection currents. We show that using the proposed variable transformation, the three-level rate-equations based models, considering the gain-saturation terms, can possess single solution regime and hence, the circuit-level simulation results will always converge to the positive laser output powers. By using the proposed equivalent circuit-level model, the effects of injection current on QCL static and dynamic behaviors are investigated. It has been shown that the proposed circuit-level model accurately predicts the operating characteristics of the QCLs.

Second-Order Correlation Function for Asymmetric-to-Symmetric Transitions due to Spectrally Indistinguishable Biexciton Cascade Emission**Supported by the National Key Basic Research Program of China under Grant No 2013CB922304, and the National Natural Science Foundation of China under Grant Nos 11474275 and 11464034.

We report the observed photon bunching statistics of biexciton cascade emission at zero time delay in single quantum dots by second-order correlation function g(2)(τ) measurements under continuous wave excitation. It is found that the bunching phenomenon is independent of the biexciton binding energy when it varies from 0.59 meV to nearly zero. The photon bunching takes place when the exciton photon is not spectrally distinguishable from the biexciton photon, and either of them can trigger the ‘start’ in a Hanbury–Brown and Twiss setup. However, if the exciton energy is spectrally distinguishable from the biexciton, the photon statistics will become asymmetric and a cross-bunching lineshape can be obtained. The theoretical calculations based on a model of three-level rate-equation analysis are consistent with the result of g(2) (τ) correlation function measurements.

Multiparameter-dependent spontaneous emission in PbSe quantum dot-doped liquid-core multi-mode fiber

A theoretical model was established in this paper to analyze the properties of 3.50 and 4.39 nm PbSe quantum dot-doped liquid-core multi-mode fiber. This model was applicable to both single- and multi-mode fiber. The three-level system-based light-propagation equations and rate equations were used to calculate the guided spontaneous emission spectra. Considering the multi-mode in the fiber, the normalized intensity distribution of transversal model was improved and simplified. The detailed calculating results were thus obtained and explained using the above-mentioned model. The redshift of the peak position and the evolution of the emission power were observed and analyzed considering the influence of the fiber length, fiber diameter, doping concentration, and the pump power. The redshift increased with the increases of fiber length, fiber diameter, and doping concentration. The optimal fiber length, fiber diameter, and doping concentration were analyzed and confirmed, and the related spontaneous emission power was obtained. Besides, the normalized emission intensity increased with the increase of pump power in a nearly linear way. The calculating results fitted well to the experimental data.

Optical rogue waves in an all-solid-state laser with a saturable absorber: importance of the spatial effects

We study the features of the optical rogue waves (ORWs) observed in an all-solid-state (Cr:YAG+Nd:YVO4) passively-Q-switched laser, which is a system of wide practical interest. The extreme events appear as isolated pulses of extraordinary intensity during the chaotic regime of this laser. The standard theoretical description (three-level rate equations for a single mode of the field and a two-level system for the absorber) does predict the existence of many of the observed dynamical features, including chaos, but it fails to predict the existence of ORWs. Faced with the problem of improving the theoretical description, we find that ORWs are observed only when the Fresnel number of the laser cavity and the embedding dimension of the attractor reconstructed from the experimental time series are high, and the laser spot profile has a spatially complex structure. These results suggest that spatial effects are an essential ingredient in the formation of ORWs in this type of laser.