Realization Of Negative Refractive Index With Little Loss Via Coherently Induced Chirality

Inspired by the pioneer work of Pendry 10 years ago , materials of negative refraction have become an active area of research due to their counter intuitive electromagnetic properties and their potential applications. In such peculiar materials, for example, the phase velocity of an electromagnetic wave and its energy flowing are in the opposite directions, and a perfect lens could be developed with its imaging resolution not restricted by the wavelength. Negative refraction materials, first introduced as left-handed media by Veselago in the 1960s, usually possess both negative electric permittivityεand negative magnetic permeabilityμin a certain frequency band. Recently it was shown that, however, negative refraction of electromagnetic waves could be achieved in the so-called chiral materials without requiring both negativeεand negativeμ. In a chiral medium, the refractive index n also depends on two chirality coefficientsξEHandξHEbecause the polarization P is coupled with the magnetic field H of an electromagnetic wave and the magnetization M is coupled with its electric field E .So far, several interesting approaches have been proposed to realize negative refraction, among which the method using artificial composite metamaterials and the method using specific photonic crystals are the most popular ones. Negative refraction in the microwave regime has been experimentally demonstrated with a periodic structure composed of split-ring resonators and conducting wires, and the refractive index may be tuned between negative and positive values by modulating certain characteristics of the involved metamaterials. To realize negative refraction in the optical regime, one can resort to another method depending on quantum coherence established in dense atoms with notable local fields by applying laser fields on appropriate energy levels. This method may be regarded as a modified version of the well-known electromagnetically induced transparency (EIT) technique, where the dilute atoms are made highly transparent and dispersive to a weak probe in a narrow frequency window by a moderate driving field. And, the negative refraction is sensitive to atomic density, decay rates, probe detuning, driving phase and so on. So, if the driving phase is changed, the refractive index will change from negative to positive values. This thesis mainly discussed the phase sensitive negative refraction based on local-field enhanced interaction and dynamically induced chirality, then discussed some parameters'answer to the negative refraction. Finally, we analyzed the feasibility of the new method..Recently, Thommen and Mandel discussed a novel scheme to induce left handedness and negative refraction in an atomic four level scheme .We here show that although the main conclusion of the possibility to create negative refraction in the driven four-level scheme remains valid, the results obtained are quantitatively not correct. Fig. 1 4-level scheme,Ω13andΩ34denote rabifrequencies of weak external drive fields,Let's consider the four-level scheme, shown in Fig. 1, in order to calculate the complex index of refraction. Thommen and Mandel solved the density matrix equations in the weak-excitation limit, i.e., setting . However, whenωa =ωc, probeΩ34produced electric dipole transition between simultaneity magnetic dipole transition between .The coherent cross coupling between electric and magnetic dipole transitions leads to chirality, where the magnetic component H of the probe field couples to the electric polarization P and, correspondingly, the electric component E to the magnetization M , namely: HereξEHandξHEare the chirality coefficients. In a chiral medium the expression for the index of refraction reads:Rather than applying Eq. (2), Thommen and Mandel used the relation B = k×Eto calculate the permeabilityμand from that the refractive index n . Although this captures the most important contributions, it neglects the modification of the electric polarization by the magnetic component of the probe field. So the conclusion is exactness.Under this illumination, we introduce a new method to realize negative refraction.Firstly, we investigate the new atom system. As seen in FIG.2, the electric E and magnetic B components of the probe field couple state 1 to state 3 by an electric dipole E transition, and to state 2 by a magnetic dipole M transition, respectively. We neglect the electric quadrupole coupling on the 1 ? 2transition since it contributes insignificantly to induced chirality .There is also a strong resonant coherent field coupling states 2 and 3 with Rabi frequencyΩc. The parities in this system are 1 even, 2 even, 3 odd or vice versa. Besides:An analogous parity distribution will apply to the more realistic system of Fig. 2(b), to enable electromagnetically induced chiral negative refraction in realistic media,。we modify the simplistic scheme of Fig. 2(a) to satisfy three criteria: (1)Ωcmust be an ac field, so that its phase can be adjusted to induce chirality; (2) there must be high-contrast EIT for the probe field; and (3) the energy level structure must be appropriate for media of interest (atoms, molecules, excitons, etc.). This scheme employs strong coherent Raman coupling by two coherent fields with complex Rabi frequenciesΩ1andΩ2and carrier frequenciesω1 andω2, which creates a dark superposition of states 1 and 4 . This dark state takes over the role of the ground state 1 in the three-level scheme of Fig. 2(a).From Fig. 3 we can see the above atom system can be characterize in fact two coupled EIT systems mediated by the coherence termρ14: one is in the Ladder configuration, described by the first two equationsρ21,ρ31, while the other is in the Lambda configuration, described by the last two equationsρ24,ρ34. In the Ladder system, the probe magnetic transition can be finished through either the direct coupling of B por the cross coupling ofρ14?E p?Ωc. Similarly, the probe electric transition in the Lambda system also has two paths: the direct coupling one bridged by E pand the cross coupling one bridged byρ14?B p?Ωc. These four transition paths are expected to interact and interfere to give surprising and interesting results on the probe response in some situations. For black state, there isSecondly, in Fig. 4 we show the real and imaginary parts of the refractive index (including local-field effects) as functions of the density. This peculiar behavior is a result of the combined electric and magnetic local-field corrections. We find substantial negative refraction and minimal absorption for this density, which is largely smaller than the density needed without taking chirality into account. These results should be contrasted to previous theoretical proposals and experiments on negative refraction in the optical regime. Therefore, the negative refraction shown in Fig. 4 is clearly a consequence of chirality. Then we found the complex refractive index governing the probe refraction and absorption depends critically on the atomic density, the steady population distribution, the coherence dephasings, and the frequency detunings, and is also sensitive to the phase of the driving field because the photonic transition paths form a close loop. Thus, we can periodically tune the refractive index at a fixed frequency from negative to positive values and vice versa just by modulating the driving phase.Moreover, we investigate how to achieve and control the negative refraction in a dense atomic ensemble with local-field effects by providing detailed treatments on the complex refractive index. It is shown that the negative refraction accompanied by a transparency window can be attained as the probe and driving fields induce notable chirality in the dense atoms with a close-loop transition path. The spectral curves describing the probe refraction and absorption change dramatically as one varies the steady population at the two lowest levels, which tells us that the two chirality coefficients are critical for the negative refraction, the electric susceptibility answers for the transparency window, while the magnetic susceptibility is negligibly small.We first discuss the importance of dynamically induced chirality for achieving negative refraction by changing the steady population at levelsρ11andρ44. From Fig. 5 we can see that the negative refraction are obtained around a transparency window with appropriately chosen parameters. In particular, the maximal magnitude of negative refraction corresponds to the case ofρ11 =ρ44=ρ14=0.5where the chirality coefficientsξEHandξHEhave the largest absolute values. As we gradually decreaseρ11, the transparency window in the Im(n) curve becomes narrower and the negative-valued part of the Re(n) curve tends to the zero line. In this case, the electric susceptibilityχe is enhanced while the magnetic susceptibilityχmand the two chirality coefficientsξEHandξHEare weakened. In the ultimate situation ofρ11 =0, the transparency window are narrowest and no negative refraction exists around it as a result ofχm =ξEH=ξHE=0. Conversely, if we increaseρ11starting from the balanced case ofρ11 =0.5, the negative-valued part of the Re(n) curve once again tends to the zero line, but the transparency window becomes wider instead of narrower. In this case, the enhanced susceptibility is the magnetic oneχm but not the electric oneχe. Finally, we find that Im(n)≡0 and Re(n)≡1. Whenχmgets the largest absolute value andχe =ξEH=ξHE=0 in the ultimate situation ofρ11 =1.0. This could be better understood if we decompose the close-loop atomic system into two coupled three-level EIT systems sharing the same driving electric field but utilizing the probe electric and magnetic transitions, respectively. Then we may conclude that: (a) the direct coupling coefficientχmin the Ladder EIT system is always negligible for the probe response; (b) the direct coupling coefficientχein the Lambda EIT system answers for the probe transparency; (c) the cross coupling coefficientsξEHandξHEare critical for achieving the negative refraction around a transparency window. In addition, the magnetic dephasing rate on the two-photon transition in one EIT system should be kept very small to have notable transparency and negative refraction. If this dephasing rate is not small enough, one can increase the driving Rabi frequency to suppress the absorption and use a denser atomic ensemble to enhance the negative refraction. The probe response is also found to be sensitive to the driving phase in a periodic mode, which is one important characteristic of any close-loop atomic systems.