\r\n\tManagement of these disorders requires good clinical evaluation, diagnostic tests, appropriate therapy and huge healthcare cost. Sometimes multiple specialties (gastroenterologists,
\r\n\tgastrointestinal motility specialists, otolaryngologists, surgeons, speech therapists, medical oncologists and radiation oncologists) are involved in the management of dysphonia and dysphagia. In the recent years, there have been many updates in the management of these disorders. This book will discuss systematically the different etiologies and management of dysphonia, maxillofacial, oropharyngeal and esophageal dysphagia. This book will be a good
\r\n\tguide to the practicing physicians for the management of voice and swallowing disorders.
Unmanned aerial vehicles (UAVs) are aircrafts that do not require a pilot on board to be controlled. In the beginning, they were solely used for military purposes. One of the first applications of these vehicles was aerial photography. In the 1883, an Englishman named Douglas Archibald provided one of the world’s first reconnaissance UAVs. However, it was not until the World War I that UAVs became recognized systems. Since then, they are being widely used in military missions such as surveillance of enemy activity, airfield base security, airfield damage assessment, elimination of unexploded bombs, etc.
\nIn the last decades, advances in technology and costs reduction permitted to adopt this technology in civil applications such as aerial photography, video and mapping, pollution and land monitoring, powerline inspection, fire detection, agriculture, and among other applications [1].
\nQuadrotors are a kind of mini-UAV’s with vertical take-off and landing, controlled completed through four rotors mounted on each end of the crossed axes, which provide lift forces for the quadrotor move, this vehicle is very popular in the research community due to their special features like strong coupling subsystems, unknown physical parameters, and nonparametric uncertainties in inputs and external disturbances. Therefore, a suitable control system for stabilizing the closed loop control system is required; to do this, various control techniques, linear and nonlinear, have been implemented such as control PD [2, 3], control PID [4, 5], control of position and orientation by vision [6], sliding mode control [1, 7], fuzzy logic [8, 9], and adaptive control in [10].
\nThe dynamic behavior of quadrotor has been published in varying complexity [11, 12]; particularly, the model we used is based on [13], where an extended mathematical description by the full consideration of nonlinear coupling between the axes is presented. We assume elastic deformations sufficient stiffness and realistic flight maneuvers were omitted, mass distributions of the quadrotor are symmetrical in the x-y plane, drag factor and thrust factor of the quadrotor are constant, and air density around the quadrotor is constant.
\nIn this work, we develop an adaptive control strategy to stabilize the attitude dynamics of a quadrotor UAV. The adaptive control permit deals with modeling errors and disturbance uncertainty, variations of the mass, inertia, actuators faults, nonlinear aerodynamics, etc.
\nThis chapter is organized as follows: Section 2 presents the mathematical model of the quadrotor obtained using Newton-Euler equations. Section 3 provides brief introduction about the adaptive control theory and the methodology used is based to obtain the adaptive law equations. In Section 4, simulations and analysis are presented to verify the performance of MRAC schema, and finally in Section 5, conclusions of this work are presented.
\nThe rotors of quadrotor are modeled by
\nFor i = (1, 2…, 4), where ui\n represents the PWM input, the gain K > 0 and ωM\n are the rotor bandwidth, vi\n is the actuator dynamics. We assume K and ωM\n are the same for all rotors [5].
\nThe configurations of quadrotor are described in \nFigure 1\n and consist of a body-fixed frame denoted by Fb = {Fbx\n, Fby\n , Fbz\n}, the center mass that coincident with the body-frame origin denoted by Cm\n, the earth-fixed inertial frame is denoted by Fe = {Fex, Fey , Fez}, the position in earth-fixed inertial frame is given by a vector rI\n = (x, y, z)\nT\n, four rotors denoted each one as Mi\n with i = (1, 2, …, 4).
\nQuadrotor configuration.
To obtain the position of CM in the NED (North-East-Down) coordinate system with respect to the inertial frame Fe\n, we use the Euler angles (roll ϕ, pitch θ, yaw ψ) combined in a vector Ω = (ϕ, θ, ψ)T, where roll angle is generated by differential thrust betweet M\n2 and M\n4, pitch angle is generated by differential thrust between M\n1 and M\n3, and yaw angle is generated by diferential torque between clock wise and anticlock wise rotors, i.e., (M\n1 − M\n2 + M\n3 − M\n4). This kind of device is considered an underactuated system with six degree of freedom (DOF). Rotation matrix R is used to map from Fb to Fe, applying three consecutive rotations denoted by
\nwhere S and C refer to sin and cos function, respectively.
\nThe dynamics of a rigid body under external forces and moments in the f\nb\n can be formulated as follows [14]
\nwhere i = (1, 2, …, 4), ∑Fi\n ∈ \n
To obtain the thrust force contribution of each rotor applied to Fb\n, we use the following formula
\nwith fxi\n force from rotor i applied in Fb\n respect x axes, fyi\n force from rotor i applied in Fb\n respect y axes and fzi\n force from rotor i applied in Fb\n respect z axes, ωi\n as rotor’s angle speed, k as proportional gain related with air density, the geometry of the rotor blade and its pitch angles. As fxi\n and fyi\n is zero, since they form an angle of 90 degrees respect to the rotor thrust, rewriting Eq. (6), we have
\nBy transforming Fi\n to F\ne\n and making use of the principle of linear momentum, the following equations can be introduced for a quadrotor of mass m under gravity \n
The rotatory moment of the body is described by Eq. (4) can be rewrite as
\nwhere MB\n is the vector of external torques and is composed of the thrust differences and drag moments of the individual rotors and under considerations of the rotatory directions and can be calculated by
\nwhere d is called drag factor and it is related with the air resistance, L is a parameter which represents the length of the lever between center of mass and the four rotors. MG\n is the vector of gyroscopic torques generated due to the propellers rotational movements and can be calculated by the follow formula
\nwhere JR\n represents the inertia of rotating rotors. The follow equations permit us to determine the rate of change of the Euler angles in the inertial frame Fe\n\n
\nThe relationship that maps rotor angular velocities to forces and moments on the vehicle is
\nwhere the variable u\n1 = F\n1 + F\n2 + F\n3 + F\n4. The variables u\n2 and u\n3 correspond the forces from rotors necessary to generate the pitch and roll moments, and u\n4 represents the yaw moment.
\nThe translation motion is obtained combining Eqs. (3), (7), (8), and (13), as a result, we have the follow equations
\nOn the other hand, if we assume small perturbations in hover flight, \n
To simplify the earlier equations, we have linearized the rotatory system around hover state assuming small change in Euler angles, u\n1 ≈ mg in x and y directions, cos(α) = 1, sin(α) = α and neglected the gyroscopic torques; therefore,
\nThis work only considers the angular momentum corresponding to the orientation of the vehicle, so that the displacement in (x, y, z) axes will not be used for mathematical analysis.
\nTherefore, the transfer function for dynamic correspondence of orientation is expressed as
\nand adding rotor’s model, we have
\nwhere K\n\np1 = Lkϕ\n/Jx\n, K\n\np2 = Lkθ\n/Jy\n, K\n\np3 = Lkψ\n/Jz\n. These parametres were computed for a X4-flyer and have the following values: K\n\np1 = K\n\np2 = 33.23, K\n\np3 = 16.95, aϕ\n = aθ\n = aψ\n = 4.1 and τ = 0.017. This mathematical model will be used to synthesize the control law to stabilize the closed-loop system [15].
\nThe overall control schema is showed in \nFigure 2\n, where position and attitude control are presented. This schema consist in two loops, first one is used to perform the quadrotor tracking of desired trajectory rd\n, while the second one is used to achieve the desired Euler (ϕd\n, θd\n, ψd\n). Due that we are only interested in attitude control, the above schema is redrawn as is showed in \nFigure 3\n.
\nOverall control system.
Attitude control system of a quadrotor.
The adaptive control is an advance control technique which provides a systematic approach for automatic adjustment of controllers in real time, in order to achieve or to maintain a desired level of control system performance, when the parameters of the plant dynamic model are unknown and/or change in time [16]. Two different approaches can be distinguished: indirect and direct approaches. In the first approach, the plant parameters are estimate online and used to calculate the controller parameters. In the second, the plant model is parameterized in terms of the controller parameters that are estimated directly without intermediate calculations involving plant parameter estimates [17].
\nThe model reference adaptive control or MRAC is a direct adaptive strategy which consists of some adjustable controller parameters and an adjusting mechanism to adjust them. The goal of the MRAC approach is adjusting the controller parameters so that the output of the plant tracks the output of the reference model having the same reference input. The MRAC schema is combine two loops: the inner or primary loop where controller and plant are feedback as in normal loop and outer loop or also called adjustment loop where some adaptive mechanisms and a model reference are used to obtain the some performance [18]. In \nFigure 4\n, an overall MRAC schema is presented.
\nMRAC control schema.
This section presents the design of an adaptive controller MRAC using the Lyapunov‘s stability theory. This allows us to ensure the tracking trajectory of an X4 to our reference model and makes the system insensitive to parameter variation and external disturbances, leading the state error to zero. Based on this, the process model can be represented by state space as follows:
\nwhere Ap\n and Bp\n represent the matrix and the vector of unknown constant parameters of the system, u is the output signal of the controller, and x is the state vector.
\nThe reference model is defined as follows:
\nThen, the control law is selected as:
\nwhere Lr\n and L are the matrix containing the parameters of the controller, which can be freely selected, and uc\n is the reference signal.
\nSubstituting (Eq. (22)) in (Eq. (18)), the closed loop system is expressed as follows:
\nNow we introduce the error equation as follows
\nDifferentiating the error with respect to time, we obtain:
\nand adding Amx and subtracting to the left side of the equation
\nThe error goes to zero if Am\n is stable and
\nIf we assume that the closed-loop system can be described by (Eq. (14)), where the matrices A and B depend on the parameter θ, and it is some combination of Lr\n and L, then we can define the following Lyapunov function for the parameter adaptation law:
\nwhere \n
where Q is a positive definite matrix such that meet the follow equation:
\nTherefore, if we choose a stable Am\n matrix, we will get always a P and Q positive definite matrix.
\nAccordingly, the derivative to time of function V is
\nWhere the function V is a Lyapunov function negative semi-definite ensures the output error between the real system and the reference model will tend to be zero, and the system is asymptotically stable.
\nThus, we obtain the following parameter adaptation laws
\nThen, the resulting control diagram is shown in \nFigure 5\n.
\nMRAC control of quadrotor: block diagram.
This section presents several simulations test made to prove the performance of MRAC controller to stabilize a mini-UAV quadrotor. As mentioned before, only orientation dynamic (angle position, angular velocity and acceleration) are considered. Analyzing Eq. (18), it is easy to see that roll, pitch, and yaw dynamics are very similar; for this reason, only roll moment is used as example in simulation.
\nThe test begins considering controller parameters are unknown and by using an online adaptive mechanism to determine the values that permit the convergence of plant response to reference model response. It is important to note that the MRAC approach seeks to keep the tracking error (x − xm\n) equal to zero by an adjustment of the controller parameters and do not to seek to identify the real parameters of the plant.
\n\n\nFigures 6\n–\n8\n shows a comparative between the states response of the plant and the state response of the model reference, where can be observer than all states of the plant converge asymptotical to the reference model states. To verify this, the \nFigures 9\n–\n11\n are presented; these figures show the tracking error of states goes to zero. Additionally, in \nFigure 12\n, the reference input is compared to the plant and model reference output.
\nComparison between x\n1 and x\n\nm1.
Comparison between x\n2 and x\n\nm2.
Comparison between x\n3 and x\n\nm3.
Error between x\n1 to x\n\nm1.
Error between x\n2 to x\n\nm2.
Error between x\n3 to x\n\nm3.
Roll response.
Tuning of controller parameters.
Simulink diagram.
\n\nFigure 13\n shows the variations of the controller parameters during the adjustment process. This mechanism started many times is necessary to assure the perfect tracking of the plant to the desire response. Finally, in \nFigure 14\n, the simulation diagram is presented.
\nThis work presents an adaptive control technique to stabilize the attitude of a quadrotor UAV using MRAC schema, which requires no information of the plant model. The asymptotic stability was demonstrated using the well-known Lyapunov’s theory, obtaining in this way the adaptation law of the controller parameters. Simulations results demonstrate that the adaptive control approach proposed have a good performance to perform the asymptotic tracking of model reference output. It is important to note that the adaptive mechanism is started only when it is needed.
\nControlling the flow of light is extremely essential for quantum information processing in integrated optical circuits. Nonreciprocal propagation of light at the single-photon level is in great demand for applications in quantum networks [1, 2], quantum computing [3], quantum entanglement [4], and quantum measurement [5]. For this purpose, nonreciprocal photonic elements, such as optical isolators and circulators, processing and routing of photonic signals at ultralow light level, or single-photon level in integrated optical circuits has been attracting a lot of interest.
\nThe conventional implementations of nonreciprocal optical devices are achieved by using the Faraday magneto-optical effect. However, such Faraday-effect-based devices suffer large optical losses and conflict with miniaturization and integration. To date, integrated nonreciprocal photonic elements have been demonstrated via magneto-optical effect [6, 7], optical nonlinearity [8, 9, 10], and opto-mechanical system [11, 12, 13]. Very recently, Dong et al. proposed and experimentally realized a scheme to achieve a true single-photon non-reciprocity in a cold atomic ensemble [14]. However, most of these devices cannot achieve high isolations, low losses, and compatibility with single-photon level at the same time.
\nNanophotonic devices control and confine the flow of light at a subwavelength scale. The strong confinement in these structures yields optical chirality, which is an inherent link between local polarization and the propagating direction of light [15]. If quantum emitters are embedded in these structures, chiral light-matter interaction is obtained, leading to propagation-direction-dependent emission, absorption, and scattering of photons. As a result, chiral light-matter interaction can be used to break time symmetry and achieve on-chip single-photon isolation. Some feasible schemes based on chiral quantum optics have been proposed to realize non-reciprocity at the single-photon level [16, 17], and optical isolators and circulators have been experimentally demonstrated in full quantum regime [18, 19].
\nBesides strong confinement of light, atoms can induce optical chirality. Here, optical chirality is chiral cross-Kerr (XKerr) nonlinearity induced in atoms. As a result of the chirality of atomic nonlinearity, the phases and transmission amplitudes of the forward- and backward-moving probe fields are sufficiently different after passing through atoms in two opposite directions. Thus, chiral XKerr nonlinear can achieve chip-compatible optical isolation with high isolation and low insert losses [20]. And very recently, XKerr-based optical isolators and circulators for high isolation, low loss, and an ultralow probe field at room temperature have been experimentally demonstrated [21].
\nThe strong light confinement in subwavelength structures, e.g., nanofibers, nanowaveguides, or whispering-gallery mode (WGM) microresonators, can lock the local polarization of the light to its propagation direction. In these structures, the light is strongly confined transversely, leading to a longitudinal component of the electric field (e-field), which is parallel with the propagation direction. The longitudinal and transverse components, denoted as \n
As a consequence, the local polarization of the light is elliptical, yielding a transverse spin angular momentum component, whose e-field rotates around an axis perpendicular to the propagation direction. The transverse spin components flip sign when the propagation of light reverses. This correlation of the polarization and the propagation direction is named the spin-momentum locking (SML) [15]. For the ideal case, \n
In order to characterize what degree the e-field is locked to the momentum or what percentage of circular polarization of the local light in the nanostructures, the circular polarization unit vectors are defined as
\nwhere \n
Obviously, the OC is limited to a region from \n
Then we focus on the interaction between the light possessing photonic SML and chiral quantum emitters with polarization-dependent dipole transitions. If a pair of counter-propagating spin-momentum-locked light interacts with quantum emitters, the interaction becomes chiral. In other words, the interaction strength for forward- and backward-propagating light modes is different. In this case, photon emission, absorption, and scattering become unidirectional. As a result, optical nonreciprocal flow of light can be achieved when a quantum emitter with degenerate transitions is populated in a specific spin state or one can shift the transition energy to make it couple (decouple) with one of the two of counter-propagating modes. On the basis of these effects, optical isolation can be realized at the single-photon level, which enables nonreciprocal single-photon devices, e.g., single-photon isolator and circulator. Next, we introduce the realization of single-photon isolators and circulators based on chiral light-matter interaction.
\nA single-photon isolator and circulator can be achieved by chirally coupling a quantum emitter to a passive, linear nanophotonic waveguide or a WGM microresonator which possesses optical chirality.
\nThe type-I single-photon isolator is based on a line defect photonic crystal waveguide [16]. By carefully engineering the photonic crystal waveguide, it can have an in-plane circular polarization, and counter-propagating modes are counter circulating [22]. A quantum emitter is doped at the position where the waveguide possesses only the right-propagating \n
Schematics of the single-photon isolators. (a) Single-photon isolator based on a photonic crystal waveguide asymmetrically coupling with a quantum emitter. The waveguide possesses local circular polarization, and its rotating direction is dependent on the propagating direction [16]. The quantum emitter is doped in these specific sites. (b) Single-photon isolator based on a photonic nanofiber asymmetrically coupling with quantum emitters. The nanofiber is realized as the waist of a tapered silica fiber, whose evanescent fields exhibit propagation-direction-dependent circular polarization [18]. Quantum emitters are located in the vicinity of the nanofiber. (c) Energy-level diagram for a quantum emitter with unbalanced decay rates \n\n\nγ\n+\n\n≫\n\nγ\n−\n\n\n and different coupling strengths with \n\n\nσ\n±\n\n\n-polarized light \n\n\ng\n+\n\n≫\n\ng\n−\n\n\n. The states \n\n∣\n\ne\n\n±\n1\n\n\n〉\n\n and \n\n∣\ng\n〉\n\n correspond to excited states and a ground state, respectively, and the transition \n\n∣\ng\n〉\n↔\n∣\n\ne\n\n+\n1\n\n\n〉\n\n is driven by \n\n\nσ\n+\n\n\n-polarized light, while \n\n∣\ng\n〉\n↔\n∣\n\ne\n\n−\n1\n\n\n〉\n\n transition is driven by \n\n\nσ\n−\n\n\n-polarized light. Here, quantum emitters can be Cs atoms [23], Rb atoms [24], or quantum dots [25].
The steady-state transmission for the two atomic transitions coupling with waveguide is calculated by using the photon transport method [16, 26, 27]
\nwhere \n
Due to the coupling of the atom to the waveguide and open environment, the ratio of atomic dissipation rate is set to \n
(a) Steady-state transmission as a function of \n\nΔ\n\n for \n\nα\n=\n1\n\n. Solid blue curve is for transmission \n\n\nT\n+\n\n\n, and dashed red curve is for transmission \n\n\nT\n−\n\n\n. (b) Isolation contrast as a function of \n\nα\n\n. Solid blue curve indicates \n\nϒ\n\n for \n\n∣\nΔ\n∣\n/\n\nγ\n−\n\n=\n10\nα\n\n and dashed red curve for \n\n∣\nΔ\n∣\n/\n\nγ\n−\n\n=\n5\nα\n\n. The two curves are mostly overlapped. Figures are reproduced with permission from [16].
Low-loss silica nanophotonic waveguides with a strongly nonreciprocal transmission controlled by the internal state of spin-polarized atoms have been demonstrated [18]. In the experiment, an ensemble of individual cesium atoms is located in the vicinity of a subwavelength-diameter silica nanofiber (250-nm radius) trapped in a nanofiber-based two-color optical dipole trap [28]. As shown in Figure 1b, a quasilinearly polarized light incident to the nanofiber exhibits chiral character at the position of the atoms: when the evanescent field propagates in the forward direction, it is almost fully \n
The type-II single-photon isolator is based on a WGM microresonator. In this setup, when the linear polarized light enters the bus-waveguide from port \n
The transmission into the bus and drop waveguides are calculated by [16].
\nwhere \n
In the configuration of this device, if the drop waveguide is removed, i.e., \n
Schematic of the single-photon isolator or circulator. There is a quantum emitter doped in a WGM microresonator, which possesses a \n\n\nσ\n+\n\n\n-polarized CCW mode and a \n\n\nσ\n−\n\n\n-polarized CW mode. The microresonator couples to a lower bus waveguide with a rate \n\n\nκ\n\nex\n1\n\n\n\n and a upper drop waveguide with a rate \n\n\nκ\n\nex\n2\n\n\n\n. Reproduced with permission from [16].
Steady-state transmissions of the single-photon isolator using a WGM microresonator in the absence of the drop waveguide, i.e., \n\n\nκ\n\nex\n2\n\n\n=\n0\n\n, and under the critical coupling condition \n\n\nκ\n\nex\n1\n\n\n=\n\nκ\ni\n\n\n. Dashed and solid curves are for the transmission \n\n\nT\n−\n\n\n when \n\n\ng\n−\n\n=\n0\n\n and \n\n\ng\n−\n\n=\n\ng\n+\n\n/\n\n45\n\n\n, while solid blue and dotted green curves are for the transmission \n\n\nT\n+\n\n\n when \n\n\nγ\n+\n\n=\n0\n\n and \n\n\nγ\n+\n\n=\n3\n\nκ\ni\n\n\n, respectively. \n\n\ng\n+\n\n=\n10\n\nκ\ni\n\n\n. Reproduced with permission from [16].
A WGM bottle microresonator coupling to the optical fiber and a single \n
The single-photon circulator consists of two waveguides and a WGM microresonator, and both of the bus and drop waveguides overcouple to the resonator, i.e., \n
(a) Steady-state transmissions of the single-photon circulator using a WGM microresonator and both of the bus and drop waveguides overcouple to the resonator, i.e., \n\n\nκ\n\nex\n1\n\n\n=\n\nκ\n\nex\n2\n\n\n=\n3\n\nκ\ni\n\n\n. Solid blue (red) curve is for the transmission \n\n\nT\n\n+\n,\nB\n\n\n\n (\n\n\nT\n\n−\n,\nB\n\n\n\n) in the bus waveguide, while the dashed blue (red) curve is for the transmission \n\n\nT\n\n+\n,\nD\n\n\n\n (\n\n\nT\n\n−\n,\nD\n\n\n\n) in the drop waveguide. The gray curves are the results from numerical simulations, which mostly overlap with other curves. \n\n\ng\n+\n\n=\n5\n\nκ\ni\n\n\n, \n\n\ng\n−\n\n=\n\ng\n+\n\n/\n\n45\n\n\n, \n\n\nγ\n+\n\n=\n0.3\n\nκ\ni\n\n\n, and \n\n\nγ\n−\n\n=\n\nγ\n+\n\n/\n45\n\n. (b) Propagation of single-photon pulses with \n\n∼\n4\n\nκ\ni\n\n\n in the resonator for \n\nΔ\n=\n0\n\n and \n\n\nv\ng\n\n=\n1\n×\n\n10\n8\n\nm\n/\ns\n\n. Blue (red) lines are for the input and transmitted excitations for the left-handed (right-handed) input. The arrows indicate the moving directions of photons. Reproduced with permission from [16].
If \n
Note that if the states are initially populated to \n
A quantum optical circulator operated by a single atom has been demonstrated [19]. In the experiment, a single \n
In the experiment, the transmissions \n
giving the probability of the correct circulator operation average over various inputs. The minimum fidelity is \n
Furthermore, the circulator performance can also be quantified by the isolations [19].
\nFor the optimum working point, it achieves \n
Note that when the atom is prepared in the opposite Zeeman ground state, \n
The type-III single-photon isolator is based on a microring resonator coupling to a QD and a nearby waveguide [17]. In the approach, the silicon microring resonator in which light is tightly transversely confined has an exceptionally strong evanescent e-field and a near-unity OC surrounding the whole outside and inside walls of the resonator. By initializing a quantum dot (QD) in a specific spin ground state or using the optical Stark control, a broadband single-photon isolation over several gigahertz is achieved.
\nThe QD-resonator system consists of a silicon waveguide, a silicon microring resonator, and a single negatively charged quantum dot (QD). Numerical simulations using the finite-difference time-domain (FDTD) method are performed to calculate the properties of the resonator. At the resonant wavelength \n
When the light enters the waveguide from port \n
(a, c) Intensity difference \n\nD\n\n and (b, d) optical chirality \n\nC\n\n for light with \n\nλ\n=\n1.556\n\nμm\n\n. Here, to clearly show the chiral e-fields in the vicinity of the resonator and wipe off the negligibly weak background, we use the definition for \n\nC\n=\n\n\n\n\n\nE\n\nr\n\n⋅\n\n\nσ\n̂\n\n−\n\n\n\n2\n\n−\n\n\n\nE\n\nr\n\n⋅\n\n\nσ\n̂\n\n+\n\n\n\n2\n\n\n\n/\n\n\n\n\n\nE\n\nr\n\n\n\n2\n\n+\nϱ\n\n\n\n, where we introduce a small bias, \n\nϱ\n=\nmax\n\n\n\n\nE\n\nr\n\n\n\n2\n\n\n×\n\n10\n\n−\n4\n\n\n\n, in the denominator. Light incident to port \n\n\nP\n1\n\n\n (a, b) and port \n\n\nP\n2\n\n\n (c, d). White lines are for the waveguide boundaries. Reproduced with permission from [17].
As seen in Figure 7, a negatively charged QD is doped near the outside wall of the resonator. It has two energy-degenerate transitions at \n
Schematic of the single-photon isolation based on a microring resonator. The silicon resonator couples to a nearby silicon waveguide with refractive index \n\nn\n=\n3.48\n\n and a single negatively charged QD. The resonator and the waveguide are \n\n0.44\n\nμm\n\n wide and \n\n0.22\n\nμm\n\n thick. The resonator has a \n\n4.22\n\nμm\n\n radius. The light incident to port \n\n\nP\n1\n\n\n(\n\n\nP\n2\n\n\n) drives the counterclockwise (CCW) [clockwise (CW)] WGM. The polarization of the evanescent field of the CCW mode is \n\n\nσ\n+\n\n\n- (\n\n\nσ\n−\n\n\n-) polarized near the whole outside (inside) wall, while that for the CW mode is \n\n\nσ\n−\n\n\n- (\n\n\nσ\n+\n\n\n-) polarized. After initialization for the QD, it is treated as a two-level system with \n\n\nσ\n+\n\n\n-polarized transition. Reproduced with permission from [17].
Initialization of a negatively charged QD including two methods: Coherent population trapping (a–c) and optical Stark effect (d). (a) Four-level configuration of an electron spin in a single negatively charged QD. (b) Four-level configuration with dipole-allowed transitions, enabled by a magnetic field along the X direction. (c) The Trion system which has been pumped with linearly polarized light at the magnetic field can be treated as a two-level system only with \n\n\nσ\n+\n\n\n-polarized light excitation at zero magnetic field. (d) A \n\n\nσ\n+\n\n\n-polarized classical light \n\n\nΩ\ns\n\n\n with a detuning \n\n\nΔ\ns\n\n\n from the \n\n\nσ\n+\n\n\n-polarized transition \n\n∣\n1\n/\n2\n〉\n↔\n∣\n3\n/\n2\n〉\n\n is applied to shift the transition energy by \n\n\nΔ\nOSE\n\n∝\n\nΩ\ns\n2\n\n/\n\nΔ\nS\n\n\n. The \n\n\nσ\n−\n\n\n-polarized CW mode decouples from the QD because it is detuned by \n\n\nΔ\n−\n\n\n from the relevant transition \n\n∣\n−\n1\n/\n2\n〉\n↔\n∣\n−\n3\n/\n2\n〉\n\n. Figures are reproduced with permission from [17] and are slightly modified.
By initializing the QD in a specific spin ground state or shifting the transition energy with the optical Stark effect (OSE), chiral QD-resonator interaction can be achieved. As shown in Figure 8b, by applying a magnetic field along the direction perpendicular to the growth direction of the QD, the spin-flip Raman transitions are enabled and can couple to linearly polarized e-fields. In this case, the spin ground state \n
After QD spin ground state preparation, the QD strongly couples to the CCW mode with large strength \n
The steady-state forward (backward) transition amplitude \n
where \n
(a) Steady-state transmissions for \n\n∣\nD\n∣\n=\n0.99\n\n. Blue (red) curves are for transmissions \n\n\nT\n+\n\n\n (\n\n\nT\n−\n\n\n), \n\nh\n=\n0\n\n (solid curves) for \n\n∣\nh\n∣\n=\n\nκ\ni\n\n\n (dashed curves), and \n\n∣\nh\n∣\n=\n3\n\nκ\ni\n\n\n (dotted curves). (b) Blue curve is for isolation contrast, and red dashed curve is for insert loss as a function of the optical chirality \n\nD\n\n for \n\nh\n=\n0\n\n (solid curves) and \n\n∣\nh\n∣\n=\n\nκ\ni\n\n\n (dashed curves). (c) Propagation of single-photon pulses incident to ports \n\n\nP\n1\n\n\n and \n\n\nP\n2\n\n\n simultaneously. Red thin (blue thick) curves are for the propagation of the right-moving (left-moving) single-photon pulses. Solid curves are for incident single-photon wave function, and dashed curves are for transmitted wave function. \n\n∣\nD\n∣\n=\n1\n\n for simplicity.
This device can achieve optical isolation when oppositely propagating photons enter the system at the same time, avoiding the dynamic reciprocity problem [36]. Numerical simulations for the propagation of single-photon wave packets incident to ports \n
In a waveguide embedded with \n
(a) Schematic of the realization of optical isolator and circulator by using chiral cross-Kerr nonlinearity. To realize an optical isolator, we use only the upper waveguide (WG) embedded with a cloud of \n\nN\n\n-type atoms. The photon passing through the atoms suffers an amplitude transmission of \n\nξ\n\n and a phase shift \n\nϕ\n\n, which are dependent on its propagation direction. To achieve optical circulator, the lower waveguide is added to form a Mach-Zehnder interferometer with the upper one by using two beam splitters BS1 and BS2. (b) Energy-level diagram of \n\nN\n\n-type atoms. The switching (carrier frequency \n\n\nΩ\ns\n\n\n), coupling (\n\n\nΩ\nc\n\n\n), and probe (\n\n\nΩ\np\n\n\n) fields couple to transition \n\n∣\n1\n〉\n↔\n∣\n2\n〉\n\n, \n\n∣\n3\n〉\n↔\n∣\n2\n〉\n\n, and \n\n∣\n3\n〉\n↔\n∣\n4\n〉\n\n, with detunings \n\n\nΔ\ns\n\n\n, \n\n\nΔ\nc\n\n\n, and \n\n\nΔ\np\n\n\n, respectively. Reproduced with permission from [20].
Rb atoms are used to create the chiral XKerr nonlinearity. In the \n
For a centimeter-scale medium, e.g., \n
(a) The transmission of the isolator for forward-moving (blue curves) and backward-moving (red curves) probe fields as a function of the probe detuning \n\n\nΔ\np\n\n\n. Green curves are for the isolations. Solid (dashed) curves are for the length of medium \n\nL\n=\n2\n\n4\n\n\n cm. (b) Amplitude transmissions (red curves) and phase shifts (blue curves) for forward-moving (solid curves) and backward-moving (dashed curves) probe fields as a function of \n\n\nΔ\np\n\n\n. (c) Green curves are for fidelities, and blue dashed curves are for average insertion loss as a function of \n\n\nΔ\np\n\n\n. The vertical black dashed lines in the two figures show the optimal detuning \n\n\nΔ\np\nopt\n\n=\n7.77\n\nγ\n0\n\n\n. The length of medium is 3.33 mm. Other parameters \n\n\nN\na\n\n=\n5\n×\n\n10\n12\n\n\ncm\n\n−\n3\n\n\n\n, \n\n\nΓ\n3\n\n=\n0.1\n\nγ\n0\n\n\n, \n\n\nΩ\nc\n\n=\n20\n\nγ\n0\n\n\n, \n\n\nΩ\ns\n\n=\n4\n\nγ\n0\n\n\n, and \n\nδ\n=\n0\n\n are fixed. Reproduced with permission from [20].
By carefully choosing the density and length of the atomic vapor, and properly arranging the switching and coupling fields, a phase shift difference, \n
For a short medium (\n
The proposal can achieve the nonlinear optical isolation without dynamic reciprocity [36], because the XKerr nonlinearity itself is chiral and the isolation is based on linear equations [20]. According to this proposed method, the device that uses XKerr nonlinearity to achieve cavity-free optical isolator and circulator at ultralow light level has been demonstrated experimentally [21].
\nIn this chapter, we introduce the optical chirality of light confined around nanophotonic structures and the chiral optical XKerr nonlinearity induced in atoms. Based on optical chirality, we propose single-photon isolators and circulators with chiral light-emitter interaction. These concepts have been demonstrated experimentally. Then we showed approaches to achieve an optical isolator and a circulator by using the chiral XKerr nonlinearity. All of these approaches can realize chip-comparable optical isolations with low insertion loss and high isolation performance. The methods also work at ultralow light level and even single-photon level. These optical isolators and circulators may pave the way for photon routing and information processing in a nonreciprocal way in integrated optical circuits and quantum networks.
\nK. X. thanks the support of the National Key R&D Program of China (Grant No. 2017YFA0303703) and the National Natural Science Foundation of China (Grants Nos. 11874212 and 61435007). He also thanks his co-workers, in particular, Prof. Jason Twamley, Prof. Yong Zhang, Prof. Min Xiao, and Prof. Franco Nori for their essential contribution to the original works.
\nOC | optical chirality |
WGM | whispering-gallery mode |
QD | quantum dot |
OSE | optical Stark effect |
CCW | counterclockwise |
CW | clockwise |
FDTD | finite-difference time-domain |
XKerr | cross-Kerr |
MZI | Mach-Zehnder interferometer |
BS | beam splitter |
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