电磁诱导的单光子和多光子带隙结构及相位光栅
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摘要
本论文以电磁感应光透明为背景,研究了驻波诱导的单光子和多光子带隙结构以及相位光栅,共分为四部分。
一、基于电磁感应透明和光子禁带的全光路由
在这部分我们研究了被一个行波场和一个完美驻波场共同耦合的四能级Tripod系统的光学响应。首先计算了两个耦合场失谐相等(△。=△g)情况下的布洛赫波矢以及反射率和透射率谱,发现若行波场拉比频率为零,即四能级系统退化为三能级Λ系统,这时由于驻波节点为零产生的残余吸收,带隙结构很差：若行波场拉比频率不等于零,带隙结构接近完美(反射率超过95%)。
其次我们考虑了两个耦合场失谐不相等(△c≠△g)的情况。通过计算系统的反射谱和透射谱,发现系统存在一个高反射带的同时还存在一个高透射带。这是因为当△c≠△g时,两个耦合场引起的EIT窗口不再重合,只有探测场失谐分别等于△c和△g时,探测场才会与行波场和驻波场产生有效的相互作用,从而在这两个位置分别产生一个EIT窗口(高透射带)和一个光子禁带(高反射带)。透射带和反射带的位置主要由△c和△g决定,因此两者的位置可以发生交换。
利用此光学可控的高反射带和透射带,我们设计了一种可以传输和分布弱光信号的全光路由。路由结构如图1,A、B、C是通讯网络中的三个结点,结点B代表我们所研究的被行波场和驻波场共同耦合的Tripod型系统。进入结点B的两个信号(信号1和2)在空间和时间上重合。若信号1的频率落入透射带,信号2的频率落入反射带,那么信号1将被传输(透射)至指向结点C的透射信道(t-channel),信号2将被传输至(反射)指向结点A的反射信道(r-channel)。若我们调节两个耦合场的参数,使两个带的位置交换,那么原本进入t-channel的信号1将被传输至r-channel,而原本进入r-channel的信号2将被传输至t-channel。
二、基于自发辐射相干的双光子带隙
在这部分我们在存在自发辐射相干(SGC)的系统中建立了驻波诱导的双光子带隙结构。所采用的能级系统是包含两个距离很近上能态的双A系统。两个上能态与两个基态分别被一个弱探测场和强驻波场驱动,并且两个上能态向两个基态的白发辐射存在相干(SGC系数分别用p1和p2表示)。通过计算发现,无论是对称结构还是反对称结构,当p1=p2=1时,即系统存在最大的SGC效应时,在共振点两侧各形成一个接近完美的光子带隙。为研究SGC对带隙的影响,我们分别计算了不同SGC参数p1和p2下的带隙结构。对于对称和反对称这两种结构,随着p1和p2的下降两个带隙反射率都降低,但是不同的结构,反射率受p1和p2降低的影响不同。对于反对称结构,p1的影响更主要,p1最大时就可以保证带隙具有高反射率；对于对称结构,p1和p2对带隙的影响相同,只有两者都最大时,才能获得高反射率的带隙。通过分析不同SGC强度下的吸收谱,我们讨论了具有高反射率的双光子带隙形成的原因：驻波场引起的非均匀量子干涉效应把原本的一个透明区域劈裂成两个EIT窗口,并且使每个窗口内介质的折射率在空间上呈周期变化,同时SGC效应进一步抑制了驻波节点处的吸收,提供了实现光子带隙所需的高透明背景。
最后我们研究了耦合场拉比频率和两个上能级的间隔对带隙的影响。随着拉比频率的增加,带隙逐渐变宽,但反射率随之降低。随着能级间隔逐渐增大,两个带隙的反射率先逐渐增大之后开始下降。
三、三光子带隙结构及扩展
目前关于电磁诱导的光子带隙的研究主要停留在单个或两个带隙的产生及应用上。在这部分里我们扩展了单光子及双光子带隙的研究,探讨了多光子带隙的建立。首先我们考虑了一个被驻波场和行波场共同驱动的三Λ系统,该系统同时存在探测跃迁的SGC和驱动跃迁的SGC。利用传输矩阵方法分析介质的光学响应发现,当探测跃迁和驱动跃迁的SGC都最大时,在三个不同区域出现了三个光子带隙,每个带隙的反射率均达到92%。然而,无论是当探测跃迁还是共振跃迁的SGC效应逐渐减弱时,EIT窗口处的吸收逐渐增强,以至于三个带隙的反射率都逐渐降低,带隙逐渐被破坏。考虑到在真实的原子中找不到存在SGC效应的系统,我们在一个在缀饰态图像下与三重Λ系统等价、被两个行波场和一个驻波场共同驱动的五能级链式M系统中同样获得了光学可调的三光子带隙,并重点研究了耦合场强度对带隙结构的影响以确定获得最佳带隙所需的条件。通过分析以上的研究结果,我们相信在多重Λ及链式Λ系统中能够建立多带隙结构。
四、电磁诱导相位光栅
在这部分,我们首先介绍了利用大失谐的双拉曼共振消除双光子吸收获得巨克尔非线性的方法。与用EIT产生巨克尔非线性的方法相比,这种方法不需要强的耦合激光,可实现几十个光子水平的非线性,但缺点是存在线性吸收,不过线性吸收可以通过增大拉曼跃迁同激发态的失谐来抑制。在此基础上,我们把该双拉曼结构中的控制场换成驻波场,设计出一种高衍射效率的相位光栅。我们利用傅里叶变换的方法计算了平面电磁波正入射的Frannhofer衍射,发现在一定参数下一级衍射效率超过30%。通过计算同样参数下的透射率及相移发现,透射的信号由于受到控制光的调制而获得π相移,同时透射率与控制光强度无关,即不依赖空间变量,因此该光栅接近理想的正弦光栅。之后我们详细分析了失谐、退相位速率及拉曼间隔等参数对衍射效率的影响,为实验上的验证提供了支持。与用基于EIT的三阶非线性建立的相位光栅相比,本章研究的相位光栅衍射效率更高,不需要大的光学深度,并且能实现弱控制光衍射。
In this thesis, based on electromagnetically induced transparency, we study the double and multiple photonic band gaps and phase grating induced by the standing-wave field, which consists of four parts.
I. All-optical routing using dynamically induced transparency windows and photonic band gaps
In this part, we investigate the optical response of a Tripod system coupled by a travelling-wave (TW) and a perfect standing-wave (SW) field. First, we calculate the Bloch vector and corresponding reflection and transmission spectra in the case of the same detunings of two coupling fields (Δc=Ag). We find that if the Rabi frequency of TW equals to zero, i.e., this four level Tripod-type system degenerates into a three levelΔ-type one, the photonic bandgap (PBG) cannot be well formed due to the absorption at nodes of perfect SW. However, if the Rabi frequency of TW is not zero, a nearly perfect photonic bandgap appears with reflectivity exceeding 95%.
Second, we consider the case that the SW grating and the TW coupling may be assigned to have different detunings (Δc≠Δg). From the reflection and transmission spectra, we find a high-transmission region with comparable width appear besides a high-reflection band gap. This is because the TW coupling and the SW grating effectively interact with the weak probe at different probe detuningsΔp via two-photon transitions whenΔc is different with Ag, and the two effectively interaction atΔc andΔg correspond to an electromagnetically induced transparency window (high-transmission region) and a photonic bandgap (high-reflection region). Moreover, the positions of transmission region and reflection region are mainly determined byΔc andΔg, respectively. Thus, the high-reflection and the high-transmission region may exchange their positions as we modulate the TW and SW detunings as well as the misaligned angle of the forward and backward components of SW.
We devise an efficient scheme for the all-optical routing of weak light signals using the high-transmission region and the high-reflection band gap with controllable positions and widths. The schematic is showed in Fig.1, where A, B and C are the nodes in information network. In particular, node B is the Tripod-type system dressed by a TW coupling and a perfect SW grating. The two signals labeled by 1 and 2 overlap in space and time. If the frequency of signal 1 falls in the high-transmission region while that of signal 2 falls in the high-reflection region, signal 1 will be routed (transmitted) into the t channel toward node C while signal 2 will be routed (reflected) into the r channel toward node A. If we make the two regions exchange their positions by modulating the parameters of two couplings, the routing process is totally inversed to allow signal 1 (signal 2) going into the r channel (the t channel).
II. Dynamically induced double photonic bandgaps in the presence of spontaneously generated coherence
In this part, we develop a double-PBG structure induced by a SW field in the presence of spontaneously generated coherence (SGC). The system we investigate is a double-A system consisting of two closely lying upper doublets coupled to a low level by a strong SW field and another low populated level by a weak probe field. There are coherences generated by spontaneous emission from doublets to two ground states, the intensities of which are represented by p1 and p2, respectively. By calculating the PBG structure and the reflection spectra, we find that no matter in symmetric or dissymmetric system, two symmetric PBGs with high reflectivities are generated at left and right wings of the probe resonance in the presence of p1=p2=1 and both PBGs become poorly developed and hardly defined with reduced probe reflectivities when either p1 or p2 is decreased to weaken the SGC effect. However, to obtain the good double-PBG structure, in symmetric system both p1 and p2 should be maximal, while in dissymmetric system only p1 need to be maximal. Through the reflection spectra in different SGC intensities, we explain the formation of two well developed PBGs. The underlying physics is that the SW coupling splits the transparent background into two space-dependent EIT windows, and SGC can suppress the absorption at SW nodes further which provided a high transparent background. At last we discuss the impacts of Rabi frequency of SW field and the interval of the doublets on the two PBGs. When Rabi frequency of SW gradually increases, the width of two PBGs get wider and wider accompanying with the reduction of their reflectivities. As the interval of the doublets increase gradually, the reflection rates of the two bandgaps, firstly, increase gradually, and then begin to decline.
III. Triple photonic band gap structure and expansion.
So far all investigations on dynamically induced PBGs are restricted in the controlled generation and potential application of one or two band-gaps. In this part, we extend our research on single and double PBGs and demonstrate a feasible scheme for generation of multiple PBGs. We first study a triple-A system interacting with a weak probe field and a strong driving field as well as exhibiting the SGC effects on both probe transitions and driving transitions. By analyzing the optical response of this system with transfer-matrix method, we can see that in the presence of maximal SGC on both probe and driving transitions, a triple-PBG structure manifests with probe reflectivities up to 92% in three different spectral regions. However, when the intensity of SGC on either probe or driving transition decreases, the probe absorption arising from nodes of SW within three EIT windows gets larger so that the reflectivities inside all the three bandgaps become smaller, which means the three PBGs are destroyed. Considering the difficulty of finding an atom or molecule existing SGC effect, we obtain the triple-PBG structure in a five-level chain-A system without SGC which in the dressed state representation of two additional TW coupling fields is equivalent to a five-level triple-A system with SGC. Then we study the impact of Rabi frequency of coupling fields on three bandgaps in order to determine the conditions for perfect bandgaps. Analyzing the results from the preceding investigations of double- and triple-PBG structures, we believe that more PBGs may be simultaneously generated in multiple-A system and chain-A system.
IV. Electromagnetically induced phase grating
We first introduce a method from a paper for achieving giant Kerr nonlinearities between two weak laser beams, which is based on Raman resonance with far detuning from the excited state to eliminate the two-photon absorption. The key advantage of our scheme over EIT is that the coupling laser beam can be arbitrary small even at the several photons level. The key disadvantage is that the single-photon linear absorption of beams is not eliminated. However the single-photon linear absorption can be suppressed by enlarging the detuning between the Raman transition and the excited state. On these bases, we introduce a SW control field instead of TW control field in this Raman transition and then propose an electromagnetically induced phase grating (EIG). For the case in which the incident probe laser is a plane wave, we show the Fraunhofer diffraction by the Fourier transform and find that the first-order diffraction efficiency of the grating exceeds 30%. From the analysis expression of the transmission and phase shift, we find a large phase modulation of the transmission function is observed reaching a peak phase shift value ofπ, and the transmissivity is not relevant to the intensity of control field, i.e., transmissivity is space-independent. Thus this phase grating is very close to an ideal sinusoidal phase grating. In order to find the experimentally achievable parameters, we show how the parameters such as detuning, the dephasing rate of Raman transition and interval of separation of the two Raman resonances affect the diffraction efficiency. Compared with the phase grating based on the Kerr nonlinearity of an atomic medium under electromagnetically induced transparency, the diffraction efficiency of this grating proposed here is high with a lower absorption length of sample and the intensity of the control field may be extremely weak.
一、基于电磁感应透明和光子禁带的全光路由
在这部分我们研究了被一个行波场和一个完美驻波场共同耦合的四能级Tripod系统的光学响应。首先计算了两个耦合场失谐相等(△。=△g)情况下的布洛赫波矢以及反射率和透射率谱,发现若行波场拉比频率为零,即四能级系统退化为三能级Λ系统,这时由于驻波节点为零产生的残余吸收,带隙结构很差：若行波场拉比频率不等于零,带隙结构接近完美(反射率超过95%)。
其次我们考虑了两个耦合场失谐不相等(△c≠△g)的情况。通过计算系统的反射谱和透射谱,发现系统存在一个高反射带的同时还存在一个高透射带。这是因为当△c≠△g时,两个耦合场引起的EIT窗口不再重合,只有探测场失谐分别等于△c和△g时,探测场才会与行波场和驻波场产生有效的相互作用,从而在这两个位置分别产生一个EIT窗口(高透射带)和一个光子禁带(高反射带)。透射带和反射带的位置主要由△c和△g决定,因此两者的位置可以发生交换。
利用此光学可控的高反射带和透射带,我们设计了一种可以传输和分布弱光信号的全光路由。路由结构如图1,A、B、C是通讯网络中的三个结点,结点B代表我们所研究的被行波场和驻波场共同耦合的Tripod型系统。进入结点B的两个信号(信号1和2)在空间和时间上重合。若信号1的频率落入透射带,信号2的频率落入反射带,那么信号1将被传输(透射)至指向结点C的透射信道(t-channel),信号2将被传输至(反射)指向结点A的反射信道(r-channel)。若我们调节两个耦合场的参数,使两个带的位置交换,那么原本进入t-channel的信号1将被传输至r-channel,而原本进入r-channel的信号2将被传输至t-channel。
二、基于自发辐射相干的双光子带隙
在这部分我们在存在自发辐射相干(SGC)的系统中建立了驻波诱导的双光子带隙结构。所采用的能级系统是包含两个距离很近上能态的双A系统。两个上能态与两个基态分别被一个弱探测场和强驻波场驱动,并且两个上能态向两个基态的白发辐射存在相干(SGC系数分别用p1和p2表示)。通过计算发现,无论是对称结构还是反对称结构,当p1=p2=1时,即系统存在最大的SGC效应时,在共振点两侧各形成一个接近完美的光子带隙。为研究SGC对带隙的影响,我们分别计算了不同SGC参数p1和p2下的带隙结构。对于对称和反对称这两种结构,随着p1和p2的下降两个带隙反射率都降低,但是不同的结构,反射率受p1和p2降低的影响不同。对于反对称结构,p1的影响更主要,p1最大时就可以保证带隙具有高反射率；对于对称结构,p1和p2对带隙的影响相同,只有两者都最大时,才能获得高反射率的带隙。通过分析不同SGC强度下的吸收谱,我们讨论了具有高反射率的双光子带隙形成的原因：驻波场引起的非均匀量子干涉效应把原本的一个透明区域劈裂成两个EIT窗口,并且使每个窗口内介质的折射率在空间上呈周期变化,同时SGC效应进一步抑制了驻波节点处的吸收,提供了实现光子带隙所需的高透明背景。
最后我们研究了耦合场拉比频率和两个上能级的间隔对带隙的影响。随着拉比频率的增加,带隙逐渐变宽,但反射率随之降低。随着能级间隔逐渐增大,两个带隙的反射率先逐渐增大之后开始下降。
三、三光子带隙结构及扩展
目前关于电磁诱导的光子带隙的研究主要停留在单个或两个带隙的产生及应用上。在这部分里我们扩展了单光子及双光子带隙的研究,探讨了多光子带隙的建立。首先我们考虑了一个被驻波场和行波场共同驱动的三Λ系统,该系统同时存在探测跃迁的SGC和驱动跃迁的SGC。利用传输矩阵方法分析介质的光学响应发现,当探测跃迁和驱动跃迁的SGC都最大时,在三个不同区域出现了三个光子带隙,每个带隙的反射率均达到92%。然而,无论是当探测跃迁还是共振跃迁的SGC效应逐渐减弱时,EIT窗口处的吸收逐渐增强,以至于三个带隙的反射率都逐渐降低,带隙逐渐被破坏。考虑到在真实的原子中找不到存在SGC效应的系统,我们在一个在缀饰态图像下与三重Λ系统等价、被两个行波场和一个驻波场共同驱动的五能级链式M系统中同样获得了光学可调的三光子带隙,并重点研究了耦合场强度对带隙结构的影响以确定获得最佳带隙所需的条件。通过分析以上的研究结果,我们相信在多重Λ及链式Λ系统中能够建立多带隙结构。
四、电磁诱导相位光栅
在这部分,我们首先介绍了利用大失谐的双拉曼共振消除双光子吸收获得巨克尔非线性的方法。与用EIT产生巨克尔非线性的方法相比,这种方法不需要强的耦合激光,可实现几十个光子水平的非线性,但缺点是存在线性吸收,不过线性吸收可以通过增大拉曼跃迁同激发态的失谐来抑制。在此基础上,我们把该双拉曼结构中的控制场换成驻波场,设计出一种高衍射效率的相位光栅。我们利用傅里叶变换的方法计算了平面电磁波正入射的Frannhofer衍射,发现在一定参数下一级衍射效率超过30%。通过计算同样参数下的透射率及相移发现,透射的信号由于受到控制光的调制而获得π相移,同时透射率与控制光强度无关,即不依赖空间变量,因此该光栅接近理想的正弦光栅。之后我们详细分析了失谐、退相位速率及拉曼间隔等参数对衍射效率的影响,为实验上的验证提供了支持。与用基于EIT的三阶非线性建立的相位光栅相比,本章研究的相位光栅衍射效率更高,不需要大的光学深度,并且能实现弱控制光衍射。
In this thesis, based on electromagnetically induced transparency, we study the double and multiple photonic band gaps and phase grating induced by the standing-wave field, which consists of four parts.
I. All-optical routing using dynamically induced transparency windows and photonic band gaps
In this part, we investigate the optical response of a Tripod system coupled by a travelling-wave (TW) and a perfect standing-wave (SW) field. First, we calculate the Bloch vector and corresponding reflection and transmission spectra in the case of the same detunings of two coupling fields (Δc=Ag). We find that if the Rabi frequency of TW equals to zero, i.e., this four level Tripod-type system degenerates into a three levelΔ-type one, the photonic bandgap (PBG) cannot be well formed due to the absorption at nodes of perfect SW. However, if the Rabi frequency of TW is not zero, a nearly perfect photonic bandgap appears with reflectivity exceeding 95%.
Second, we consider the case that the SW grating and the TW coupling may be assigned to have different detunings (Δc≠Δg). From the reflection and transmission spectra, we find a high-transmission region with comparable width appear besides a high-reflection band gap. This is because the TW coupling and the SW grating effectively interact with the weak probe at different probe detuningsΔp via two-photon transitions whenΔc is different with Ag, and the two effectively interaction atΔc andΔg correspond to an electromagnetically induced transparency window (high-transmission region) and a photonic bandgap (high-reflection region). Moreover, the positions of transmission region and reflection region are mainly determined byΔc andΔg, respectively. Thus, the high-reflection and the high-transmission region may exchange their positions as we modulate the TW and SW detunings as well as the misaligned angle of the forward and backward components of SW.
We devise an efficient scheme for the all-optical routing of weak light signals using the high-transmission region and the high-reflection band gap with controllable positions and widths. The schematic is showed in Fig.1, where A, B and C are the nodes in information network. In particular, node B is the Tripod-type system dressed by a TW coupling and a perfect SW grating. The two signals labeled by 1 and 2 overlap in space and time. If the frequency of signal 1 falls in the high-transmission region while that of signal 2 falls in the high-reflection region, signal 1 will be routed (transmitted) into the t channel toward node C while signal 2 will be routed (reflected) into the r channel toward node A. If we make the two regions exchange their positions by modulating the parameters of two couplings, the routing process is totally inversed to allow signal 1 (signal 2) going into the r channel (the t channel).
II. Dynamically induced double photonic bandgaps in the presence of spontaneously generated coherence
In this part, we develop a double-PBG structure induced by a SW field in the presence of spontaneously generated coherence (SGC). The system we investigate is a double-A system consisting of two closely lying upper doublets coupled to a low level by a strong SW field and another low populated level by a weak probe field. There are coherences generated by spontaneous emission from doublets to two ground states, the intensities of which are represented by p1 and p2, respectively. By calculating the PBG structure and the reflection spectra, we find that no matter in symmetric or dissymmetric system, two symmetric PBGs with high reflectivities are generated at left and right wings of the probe resonance in the presence of p1=p2=1 and both PBGs become poorly developed and hardly defined with reduced probe reflectivities when either p1 or p2 is decreased to weaken the SGC effect. However, to obtain the good double-PBG structure, in symmetric system both p1 and p2 should be maximal, while in dissymmetric system only p1 need to be maximal. Through the reflection spectra in different SGC intensities, we explain the formation of two well developed PBGs. The underlying physics is that the SW coupling splits the transparent background into two space-dependent EIT windows, and SGC can suppress the absorption at SW nodes further which provided a high transparent background. At last we discuss the impacts of Rabi frequency of SW field and the interval of the doublets on the two PBGs. When Rabi frequency of SW gradually increases, the width of two PBGs get wider and wider accompanying with the reduction of their reflectivities. As the interval of the doublets increase gradually, the reflection rates of the two bandgaps, firstly, increase gradually, and then begin to decline.
III. Triple photonic band gap structure and expansion.
So far all investigations on dynamically induced PBGs are restricted in the controlled generation and potential application of one or two band-gaps. In this part, we extend our research on single and double PBGs and demonstrate a feasible scheme for generation of multiple PBGs. We first study a triple-A system interacting with a weak probe field and a strong driving field as well as exhibiting the SGC effects on both probe transitions and driving transitions. By analyzing the optical response of this system with transfer-matrix method, we can see that in the presence of maximal SGC on both probe and driving transitions, a triple-PBG structure manifests with probe reflectivities up to 92% in three different spectral regions. However, when the intensity of SGC on either probe or driving transition decreases, the probe absorption arising from nodes of SW within three EIT windows gets larger so that the reflectivities inside all the three bandgaps become smaller, which means the three PBGs are destroyed. Considering the difficulty of finding an atom or molecule existing SGC effect, we obtain the triple-PBG structure in a five-level chain-A system without SGC which in the dressed state representation of two additional TW coupling fields is equivalent to a five-level triple-A system with SGC. Then we study the impact of Rabi frequency of coupling fields on three bandgaps in order to determine the conditions for perfect bandgaps. Analyzing the results from the preceding investigations of double- and triple-PBG structures, we believe that more PBGs may be simultaneously generated in multiple-A system and chain-A system.
IV. Electromagnetically induced phase grating
We first introduce a method from a paper for achieving giant Kerr nonlinearities between two weak laser beams, which is based on Raman resonance with far detuning from the excited state to eliminate the two-photon absorption. The key advantage of our scheme over EIT is that the coupling laser beam can be arbitrary small even at the several photons level. The key disadvantage is that the single-photon linear absorption of beams is not eliminated. However the single-photon linear absorption can be suppressed by enlarging the detuning between the Raman transition and the excited state. On these bases, we introduce a SW control field instead of TW control field in this Raman transition and then propose an electromagnetically induced phase grating (EIG). For the case in which the incident probe laser is a plane wave, we show the Fraunhofer diffraction by the Fourier transform and find that the first-order diffraction efficiency of the grating exceeds 30%. From the analysis expression of the transmission and phase shift, we find a large phase modulation of the transmission function is observed reaching a peak phase shift value ofπ, and the transmissivity is not relevant to the intensity of control field, i.e., transmissivity is space-independent. Thus this phase grating is very close to an ideal sinusoidal phase grating. In order to find the experimentally achievable parameters, we show how the parameters such as detuning, the dephasing rate of Raman transition and interval of separation of the two Raman resonances affect the diffraction efficiency. Compared with the phase grating based on the Kerr nonlinearity of an atomic medium under electromagnetically induced transparency, the diffraction efficiency of this grating proposed here is high with a lower absorption length of sample and the intensity of the control field may be extremely weak.
引文
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