中華大學

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中華大學

博士論文

電磁誘導透明材料組成之光子晶體頻帶結構及光子邏輯閘應用

Band Structure of EIT-Based Photonic Crystals and Photonic Logic Gate Design

系所別：工程科學博士學位學程學號姓名：D09824002 廖德超指導教授：高曜煌博士共同指導教授：楊宗哲博士沈建其博士

中華民國 103 年 6 月

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Band Structure of

EIT-Based Photonic Crystals and Photonic Logic Gate Design



Teh-Chau, Liau

2014/6/24

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Preface

This dissertation is submitted for the degree of Doctor of Philosophy at the Chung Hua University, Hsinchu, Taiwan, Republic of China. The related work was carried out between September 2009 and June 2014, under the supervision of Professor Tzong-Jer Yang, Professor Yao-Huang Kao, Professor Jian Qi Shen and Professor Jin-Jei Wu.

All other previous works are referred through the text according to the academic ethics.

Otherwise, the thesis is an original work and has not been put forward for any degree to other university neither academic institution.

Teh-Chau, Liau 2014/6/24

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摘要

原子的相位相干效應發展出許多有趣的物理現象，例如：哈里斯在 1997 年發表的電磁誘導透明(Electromagnetically Induced Transparency, EIT)^[1]以及相關的效應，包括同

一年由可漢與伯門發表的無反轉放大^[17]，1996 年由朱赫與史戈里發表的自然輻射相

消^[18]，2006 年千本諾發表的多光子布居捕獲^[19]，2006 年澤第可夫與 2007 年甘第門

的相干相位控制^[20],[21]，以及 2009 年克勞涅及沈的光子共振的左手材料^[23]。電磁誘導

透明是一種量子的光學現象，倘若一束共振雷射光穿過一種材料時，它會被吸收；但是，有兩束共振雷射光穿過同一種材料時，兩束光都不會被吸收。這時，原屬灰色的材料變成真空一般的透明。這麼有趣的光學反應引領了許多的應用面，例如嶄新的光子晶體與量子光學裝置。由於它展現了那麼奇妙的光學特性與效應，於是 EIT 引發了廣泛的注意並吸引許多研究人員投入到光學，原子物理，及凝聚態物理的世界，也讓許多的物理學家成就了許多重要的理論與實驗結果。舉例來說：一些與 EIT 現象相關

的物理效應包括超慢光脈波傳播，超光速光傳播，以及氣態原子的光能儲存^[24]~[27]，

主要是為了求取更有利也更夠力的量子光學與光子晶體的更大發展。

本文提議一種新的 EIT 的應用方式，那就是一 EIT 為基材的人造週期介質：挺特別的是這 EIT 材料(一種原子氣體材料或是一種半導體量子點材料)週期式嵌入一種主介電材料(例如：砷化鉀)。它就是眾所周知的光子晶體，它是介電材料的週期性結構，由於它的光傳播的潛力，在物理界，材料科學界與其他相關領域(例如：資訊科學)都引

發廣泛的注意^[28]~[30]。我們從 EIT 特性及介電材料的人造交疊式週期排列的結構裡發

現了新的光傳播模式以及新的操縱方法。這種效應反映著 EIT 與光子晶體的組合效果。在這一個新介電材料與 EIT 材料的週期性的交疊編排結構中，展現了可調的反射率與透射率(由外加的控制光場引發)以及對探測光場異常靈敏反應。舉例來說，這種

以 EIT 為基材的週期材料，假設探測光場的頻率為_p，只要有10^⁸的改變(即

p

108

 

  )就會戲劇性的極大的反射率與透射率的改變。因此，它可以被用來設計高

靈敏度的光開關，光子邏輯閘，以及可調的光子電晶體。在既有的文獻記載中，雖然

有一些有關可調的 EIT 基材的光子晶體特性^[31]~[34]，然而，鮮少文獻注意到這一種週

期結構對頻率變化甚為靈敏的顯著的光學特性。

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光子邏輯閘曾經用新的相干材料，例如奈米材料之間的近塲光藕合^[35],[36]以及多能級

原子的雙光場控制[37],[38],[41]，曾經在近幾年來被倡議出來。值得強調的是本文提議的

機制，是可以實踐這種光子與量子裝置光邏輯閘的另一溪徑。在最近期間內，亞伯當馬立可夫研究團隊提出單一人造原子的 EIT 的實驗報告，電磁波是可以被完全透射或

後向散射的^[42]。本文將指出這種光學特性的完全可控性是可以用我們提議的 EIT 為

基材的層狀週期材料所實現，使反射率為零抑或利用外加控制光場強度施加到這一 EIT 系統。我們也深信這樣的週期材料將開啟新的應用領域，例如光子晶體微型電路 (或積體型光路)。

由於控制光場強度可以調控氣態原子的光學響應，當介電材料(如砷化鉀)與電磁誘導透明材料組成層狀的晶胞結構，結果發現它的色散行為對於頻率非常敏感，可作為光邏輯閘或陡峭型光濾波器的原理依據，另一方面，它對拉比頻率變化也形成緩變卻又具備線性光放大器的基本特性。我們認為，這種結構可作為快速反應之全光型控制裝置。

關鍵字:量子相干(Quantum coherence),頻率敏感之光線傳播操作(Frequency-sensitive light propagation manipulation), 可調諧光行為(Tunable optical behavior),電磁誘導透明 (Electromagnetically induced transparency), 左手材料(left-handed material), 左手材料- 電磁誘導透明材料層疊結構(LHM-EIT composite structure), 克萊茵穿透現象(Klein tunneling), 光子型電晶體(Photonic transistors), 橫電/橫磁模式(TE/TM mode) , 電磁誘導透明材料為基礎之週期型層疊材料(EIT-based periodic layered medium), 三能級原子系統(Three-level atomic system), 電磁誘導透明光子晶體(EIT photonic crystal), 類 EIT 電路(Circuit analog of EIT), 功率頻域分析(Power spectrum analysis), 電容藕合 (Capacitor coupling), 古典 EIT(Classical EIT).

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ABSTRACT

The effects of atomic phase coherence have exhibited a number of physically interesting phenomena such as electromagnetically induced transparency (EIT)^[1] and the effects that are relevant to EIT, including light amplification without inversion^[17], spontaneous emission cancellation^[18], multi-photon population trapping^[19], coherent phase control^[20],[21] as well as photonic resonant left-handed media^[23]. EIT is such a quantum optical phenomenon that if one resonant laser beam propagates in a medium (e.g., an atomic vapor or a semiconductor- quantum-dot material), the beam will get absorbed; but if two resonant laser beams instead propagate inside the same medium, neither would be absorbed. Thus the opaque medium becomes a transparent one. Such an interesting optical behaviour would lead to many applications, e.g., designs of new photonic and quantum optical devices. Since it can exhibit many intriguing optical properties and effects, EIT has attracted extensive attentions of a large number of researchers in a variety of areas of optics, atomic physics and condensed state physics^[1], and this enables physicists to achieve new novel theoretical and experimental results. For example, some unusual physical effects associated with EIT include the ultraslow light pulse propagation, the superluminal light propagation, and the light storage in atomic vapors[24],[25],[26],[27], some of which are expected to be beneficial (and powerful) for developing new technologies in quantum optics and photonics.

The optical response of an atomic vapor can be controlled by using tunable quantum interference induced by external control field. A periodic layered medium whose unit cells consist of dielectric (e.g., GaAs) and EIT (electromagnetically induced transparency) atomic vapor is suggested. It is demonstrated that such an EIT-based periodic layered medium shows more flexible optical response (sensitive to frequency) than a conventional photonic crystal does. The controllable band structure that depends on the external control field can be applicable to designs of new devices such as photonic switches and photonic logic gates, where one laser field can be controlled by the other one, and would have potential applications in the field of integrated optical circuits and other related areas, e.g., the all- optical technique.

This dissertation proposes a new application of EIT, i.e., EIT-based artificial periodic dielectric: specifically, the EIT medium (an atomic vapor or a semiconductor-quantum-dot material) is embedded in a periodic host dielectric (e.g., GaAs). As is well known, the photonic crystals, which are periodic arrangements of dielectrics, have captured wide attention in physics, materials science and other relevant fields (e.g., information science)

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due to its capacity of controlling light propagations[28],[29],[30]. Here, we find some new effects relevant to light propagation manipulation via EIT responses in an artificial periodic dielectric. Such effects result from the combination of EIT and photonic crystals. In this new application of EIT for manipulating light wave propagations, the periodic dielectric can exhibit a tunable reflectance and transmittance (induced by an external control field) and can show extraordinary sensitivity to the frequency of the applied probe field. For example, a change of one part in 10⁸ in the probe frequency _pwould lead to a dramatic change in the reflectance and transmittance of the EIT-based periodic layered medium, and therefore, it can be used for designing sensitive optical switches, photonic logic gates as well as tunable photonic transistors. In the literature,although there have been some investigations that are relevant to the tunable photonic crystals based on EIT media[31],[32],[33],[34], yet less attention has been paid to the frequency-sensitive optical behavior that would be the most remarkable property of such a kind of periodic layered media.

The photonic logic gates designed based on new coherent materials, such as near-field optically coupled nanometric materials^[35],[36] and double-control multilevel atomic media[37],[38],[41], have been suggested during the past few years. It should be emphasized that the mechanism presented in this paper can be considered an alternative way to realize such a kind of photonic and quantum optical devices. Very recently, Abdumalikov et al. reported an experimental observation of EIT on a single artificial atom, and found that the propagating electromagnetic waves are allowed to be fully transmitted or backscattered^[42]. We will demonstratein this dissertation that a full controllability of optical property of artificial media could also be achieved in the EIT-based layered structure, of which the reflectance can be either zero or large depending sensitively on the intensity of the external control field applied in the EIT system. We believe that this would open a good perspective for its application in some new fields such as photonic microcircuits (or integrated optical circuits).

The present structure provides an EIT-based photonic logic gate, which is constituted by EIT-based stack layers of periodic array of photonic crystal (PCs) layers and EIT (Electromagnetic Induced Transparent) material layers. The input probe signals are incident on the first photonic crystal layer, passing through one or more than one PCs-EIT interfaces and transmitted out from the last EIT material layer. Control filed as the enable signals are incident on each EIT layer to activate the optical logic gate. By varying the frequency detuning of probe field and Rabi frequency of control field, its band gap structure can be adjusted. Henceforth, the tunable optical EIT-based photonic logic gates can be achieved as

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vii user required.

Keywords:Quantum coherence,Frequency-sensitive light propagation manipulation, Tunable optical behavior,Electromagnetically induced transparency, left-handed material, LHM-EIT composite structure, Klein tunneling, Photonic transistors, TE/TM mode, EIT- based periodic layered medium, Three-level atomic system,EIT photonic crystal, Circuit analog of EIT, Power spectrum analysis, Capacitor coupling, Classical EIT.

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DEDICATION

This thesis is dedicated to:

-My gone parents, who raised me in childhood and morally have supported me at youth.

- My wife, Chieu-Ying, Lieu who has supported and encouraged me in each step for my living and healthy caring after I were 30 years old.

- My good friends, Ping-Yu, Wang and Kuen-Su, Hwang, Mohammad Jamal,Mohannad Bosta, who have encouraged me and shared my feelings.

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ACKNOWLEDFEMENTS

I would like to appreciate my advisor Professor Tzong-Jer Yang and Professor Yao-Huang Kao for their Baton-style funding support, guidance, encouragement,untiring efforts, assistance and protection during my Ph.D. study.

I would also like to appreciate Professor Jian Qi Shen, Centre for Optical and Electromagnetic Research, ZJUfor his detailed guiding, revising, and correcting in each statement, figure, programming in each paper about EIT.

Thank Professor Yuli Lin, Dean of the College, CHU for his resources and stand-alone green energy and intelligence laboratory rationing, that I can have a very quiet and cool thinking place to read papers, books and study every problem.

Thank Professor Bor-Jang Tsai, Department of Mechanical Engineering, CHU for his long term spiritual encouragement, explanation to green building and phase-change material issue, trend of current and future affairs that I can have a world-wide view.

The work is supported by National Science Council under Grant Nos. NSC 99-2811-M- 216-001,NSC 99-2112-M-216-002, NSC 100-2112-M-216-002.

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ABBREVIATIONS

| n Quantum state of atom

3/4 Spontaneous emission decay rate of state | 3/| 4

m Decay rate of state | m

ph, m

 Dephasing rate of state | m

p Rabi frequency of probe field

c, c_

  Rabi frequency of control fields Probe frequency detuning

c, c_

  Control frequency detunings

RT Real part of total impedance of the circuit analogue X T Imaginary part of total impedance of the circuit

analogue

 Angular frequency of a radiation field (s^¹)

 Wavelength of a radiation field (m)

 

Ps  Efficient power under frequency  (Watts) cos Power factor of the circuit analogue

N ^a Atomic density (m^³) Re x

 

Real part of the quanty x _{Im x}

 

Imaginary part of the quanty x

 Susceptibility of the EIT atom vapor

r Relative permittivity of the EIT atom vapor

_p

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1 Introduction

1.1 Background and Motivation

Over the past two decades, the effects of atomic phase coherence have exhibited a number of physically interesting phenomena such as electromagnetically induced transparency (EIT)^[1] and the effects that are relevant to EIT, including light amplification without inversion^[17], spontaneous emission cancellation^[18], multi-photon population trapping^[19], coherent phase control^[20],[21] as well as photonic resonant left-handed media^[23]. EIT is such a quantum optical phenomenon that if one resonant laser beam propagates in a medium (e.g., an atomic vapor or a semiconductor-quantum-dot material), the beam will get absorbed; but if two resonant laser beams instead propagate inside the same medium, neither would be absorbed. Thus the opaque medium becomes a transparent one. Such an interesting optical behavior would lead to many applications, e.g., designs of new photonic and quantum optical devices. Since it can exhibit many intriguing optical properties and effects, EIT has attracted extensive attentions of a large number of researchers in a variety of areas of optics, atomic physics and condensed state physics^[1], and this enables physicists to achieve new novel theoretical and experimental results. For example, some unusual physical effects associated with EIT include the ultraslow light pulse propagation, the superluminal light propagation, and the light storage in atomic vapors[24],[25],[26],[27], some of which are expected to be beneficial (and powerful) for developing new technologies in quantum optics and photonics.

In this thesis, we shall consider a new application of EIT, i.e., EIT-based artificial periodic dielectric: specifically, the EIT medium (an atomic vapor or a semiconductor- quantum-dot material) is embedded in a periodic host dielectric (e.g., GaAs). As is well known, the photonic crystals, which are periodic arrangements of dielectrics, have captured wide attention in physics, materials science and other relevant fields (e.g., information science) due to its capacity of controlling light propagations[28],[29],[30]. Here, we shall propose some new effects relevant to light propagation manipulation via EIT responses in an artificial periodic dielectric. Such effects result from the combination of EIT and photonic crystals. In this new application of EIT for manipulating light wave propagations, the periodic dielectric can exhibit a tunable reflectance and transmittance (induced by an external control field) and can show extraordinary sensitivity to the frequency of the applied probe field. For example, a change of one part in 10⁸ in the probe frequency _pwould lead to a dramatic change in the

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reflectance and transmittance of the EIT-based periodic layered medium, and therefore, it can be used for designing sensitive optical switches, photonic logic gates as well as tunable photonic transistors. In the literature,although there have been some investigations that are relevant to the tunable photonic crystals based on EIT media[31],[32],[33],[34], yet less attention has been paid to the frequency-sensitive optical behavior that would be the most remarkable property of such a kind of periodic layered media.

We should point out that photonic logic gates designed based on new coherent materials, such as near-field optically coupled nanometric materials^[36],[36] and double-control multilevel atomic media[37].[38],[41], have been suggested during the past few years. It should be emphasized that the mechanism presented in this thesis can be considered an alternative way to realize such a kind of photonic and quantum optical devices. Very recently, Abdumalikov et al. reported an experimental observation of EIT on a single artificial atom, and found that the propagating electromagnetic waves are allowed to be fully transmitted or backscattered^[42]. We will demonstrate in the present thesis that such a full controllability of optical property of artificial media could also be achieved in the EIT-based layered structure, of which the reflectance can be either zero or large depending sensitively on the intensity of the external control field applied in the EIT system. We believe that this would open a good perspective for its application in some new fields such as photonic microcircuits (or integrated optical circuits).

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1.2 Structure of the Thesis

This thesis is organized as follows.

In Chapter 2 we shall discuss the characteristic optical property of an EIT medium (e.g., an atomic vapor) and review a formulation for treating the electromagnetic wave propagation in a periodic layered medium. The frequency-sensitive tunable band structure as well as the behavior of frequency-sensitive reflectance and transmittance of such an EIT-based periodic layered medium are presented. The spectrum of the reflectance as well as the transmittance of the EIT-based periodic structure (when the TE wave of the probe beam is normally incident on the layered medium) versus the normalized Rabi frequency  _c/ ₃of the control field and the normalized probe frequency detuning _p / ₃ will be addressed in Chapter 3. The frequency-sensitive tunable band structure of TM wave in the EIT-based periodic structure containing a left-handed medium is discussed in Chapter 4, where the reflection coefficient exceeding unity would occur in some frequency ranges. This will lead to a negative transmittance (so-called photonic analog of Klein tunneling in an LHM-EIT- based periodic layered medium). Some potential applications, i.e., photonic transistors and logic gates (tunable photonic logic gates) are suggested by taking full advantage of the effect of such an optical switching control.

In Chapter 5, we show some EIT-like circuit which is accomplished by automatic test hardware of GPIB as interface with different software, i.e. Matlab, Multisim (Electronic Workbench), and ADS(Advanced Design System).

In Chapter 6 we close the thesis with some conclusions.

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2 Optical properties of EIT Medium

2.1 Three-Level Atomic System

Here we shall address the intriguing optical behavior of an EIT atomic vapor. Consider a Lambda-configuration three-level atomic system with two lower levels |1, | 2 and one upper level | 3 (see Fig. 2.1 for its schematic diagram). This atomic system interacts with the electric fields of the two applied light waves (probe and control fields), which drive the

|1 | 3 and | 2 | 3 transitions, respectively. Note that the parity of level | 3 needs to be opposite to levels |1 and | 2, since the level pairs |1 | 3 and | 2 | 3 can be coupled to the electric fields of the probe and control waves, respectively. Such a three-level system can be found in metallic alkali atoms (e.g., Na, K, and Rb).

Fig. 2.1 The schematic diagram of a three-level EIT atomic system. The parity of upper level | 3 is opposite to that of lower levels |1 and | 2. The control and probe laser beams drive the | 2 | 3 and |1 | 3 transitions, respectively.

Once the control laser beam _c is switched off, the vapor will be a resonantly absorptive medium for the probe light. However, the vapor would be transparent to the probe light when the control laser beamis present because of the destructive quantum interference between the |1 | 3 and | 2 | 3 transitions.

The present atomic system interacting with two light fields (_c and _p, the Rabi frequency of control field and probe field respectively) is governed by



^*



11 21 22 31 33 p 31 p 13

i 2 d

dt           (2-1)

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   

21 ph *

12 p c 12 p 32 c 13

i i

2 2

d dt

 

  ^ ^     ^      

  (2-2)

 

* * *

31 32

13 p 13 p 33 p 11 c 12

i i

2 2

d dt

 

  ^ ^   ^  ^        ^ (2-3)

   

21 ph *

21 p c 21 c 31 p 23

+i i

2 2

d dt

 

  ^ ^    ^      

  (2-4)



^*



22 32 33 21 22 c 32 c 23

i 2 d

dt           (2-5)

 

* * *

31 32 21

23 c 23 c 33 p 21 c 22

i i

2 2

d dt

  

  ^ ^ ^   ^  ^        ^ (2-6)

 

31 32

31 p 31 p 33 p 11 c 21

+i i

2 2

d dt

 

  ^ ^  ^  ^        ^ (2-7)

 

31 32 21

32 c 23 c 33 p 12 c 22

+i i

2 2

d dt

  

  ^ ^ ^  ^  ^        ^ (2-8)

  

^*

 

^*



33 31 32 33 p 13 p 31 c 23 c 32

i i

2 2

d

dt                  (2-9)

where, _ij are the decay rate between energy levels (relevant to the spontaneous emission)

phis the collisional dephasing rate, and



^, ^{1, 2,3}



ij i j

  are the density matrix elements with conditions ij 

 

ji ^*^,i j and



11 22 33



0

t   

   

 (i.e., ₁₁₂₂₃₃1)

Let ph 0, 11

 

0 1, 22

 

0 33

 

0 _{i j}, 0, for i j



,i j1, 2,3



and ₃ ₃₁

32, ₂ ₂₁_ph ₂₁, the off-diagonal density matrix elements ₂₁ and ₃₁ can be formed as a closed set of matrix equation under the condition of weak probe field^[18], and the atomic system can be characterized by a time-dependent model when the control field intensity varies adiabatically.

 

^*

2

21 21

31 3 31 p 11

c

2 2 0

.

2 2 2

t



 

  

         

   

   

        

            

p c c

p

i i i i

i

(2-10)

It can be verified that the atomic microscopic electric polarizability of the |1 | 3 transition is of the form

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6

 

2 2 13

3 2 *

0

| | 2 .

1

2 2 4



  

   

 

           

   

 

p c

p p c c c

i i

(2-11)

Here, ₃ and ₂ stand for the spontaneous emission decay rate and the collisional dephasing rate, respectively. The Rabi frequency _c of the control field is defined by

c 32Ec/

  with E the slowly-varying amplitude (envelope) of the control field. In _c the same system, the Rabi frequency _p of the probe field is defined by  _p ₃₁E_p / with E the slowly-varying amplitude (envelope) of the probe field and _p ₃₁ the absolute value of electrical dipole moment of electron when it transit from |1 to | 3. The two frequency detunings are defined as  _p ₃₁_p,  _c ₃₂_c with _p and _c the mode frequencies of the probe field and the control field, respectively. By using the Clausius- Mossotti relation (governing the local field effect due to the dipole-dipole interaction between neighboring atoms), the relative electric permittivity of the EIT vapor at probe frequency ( _p ₃₁ _p) is given by

1 ,

1 3

N N

^r ^{ } _ ^a_a (2-12)

where, N denotes the atomic concentration (atom number per unit volume) of the EIT _a atomic vapor.

The tunable dispersive behavior of the bulk EIT atomic vapor is shown in Figs. 2.2. The typical atomic and optical parameters chosen for Figs. 2.2(a), (b), (c) are as follows: the atomic number density N_a 5.0 10 m ²⁰ ^³ , the electrical dipole moment

29 31 1.0 10^ C m

    , the frequency detuning of the control field _c 1.0 10 s ⁷ ^¹, the spontaneous emission decay rate  ₃ 2.0 10 s ⁷ ^¹ and the dephasing rate

7 1

γ21.0 10 s ^ . Fig. 2.2(d), (e) shows the three-dimensional behavior of the real part (d) and the imaginary part (e) of the relative electric permittivity of the EIT atomic vapor (3D).

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7 (a)

(b)

(21)

8 (c)

(d) (e)

Fig. 2.2 The relative electric permittivity of the three-level EIT atomic vaporas a function of the probe frequencydetuning _p and the Rabi frequency _c of the control field, In (a) the Rabi frequency of the control field is  _c 2.0 10 s ⁷ ^¹. In (b) the probe frequency detuning is  _p 3.4 10 s ⁶ ^¹. In (c)  _p 10 s⁷ ^¹ and  _c 2.0 10 s ⁷ ^¹. In (d) and (e)the 3D real part and imaginary part of the relative electric permittivity of the EIT atomic medium versus the probe frequency _p detuning and the control Rabi frequency _c under the condition  _c 2.0 10 s ⁷ ^¹.

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9

Fig. 2.3 The typical behavior of the refractive index of the EIT vapor versus the normalized probe frequency detuning _p / ₃. The two-photon resonance occurs when   _p _c, i.e.,  _p / ₃ approaches 0.5. The Rabi frequency of the control field is chosen as  _c 2 10 s⁷ ^¹ and the frequency detuning

7 1

c 0.5 3 1 10 s^

      .

Fig. 2.4 The typical behavior of the refractive index of the EIT vapor versus the normalized Rabi frequency  _c/ ₃ of the control field. The frequency detunes of the probe and control fields are  _p 3.4 10 s ⁶ ^¹and  _c 0.5  ₃ 10 s⁷ ^¹ respectively.

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10

Fig. 2.5. The typical behavior of the refractive index of the EIT vapor versus the normalized frequency detuning  _c/ ₃ of the control field. The frequency detune of the probe is   _p 1 10 s⁷ ^¹and the Rabi frequency of the control field is   _c 2 10 s⁷ ^¹.

Table 1 Parameters setting of the bulk EIT material for finding the permittivity

Parameter Value Unit Parameter Value Unit

3 _{2 10}_ ⁷ _s^¹ ₂ 1 10 ⁵ s^¹

p _{2 10}_ ⁷ _s^¹ _c



0, 5 3



_s^¹

N a _{5 10}_ ²⁰ m^³ 31 _{1 10}_ ^²⁹ C m

31 _{5 10}_ ¹⁵ _s^¹ _p _{ }_{5 10}¹⁵ _s^¹

32 _{ }_{5 10}¹⁵ _s^¹ _c _{ }_{5 10}¹⁵ _s^¹

c 32 c

  



  5 3, 5 3



_s^¹  _p ₃₁_p



  5 3, 5 3



_s^¹

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11

2.2 Four-Level Atomic System

The destructive and constructive quantum interferences would be exhibited in a four- level double-control atomic system, where the probe transition (driven by the probe field) can be manipulated by the quantum interferences between two control transitions (driven by the two control fields) of the four-level system^[27]. The atomic vapor is opaque (or transparent) to the probe field if the destructive (or constructive) quantum interference between the control transitions emerges. Consider a four-level atomic system with three lower levels |1,

| 2, | 2 and one upper level | 3 (see Fig. 2.6). Such an atomic system interacts with three optical fields, i.e., the two control laser beams and one probe laser beam, which couple the level pairs | 2 | 3 , | 2 | 3 and |1 | 3 respectively. The three frequency detunings _c , _c and _p are defined as follows:  _c ₃₂_c ,  _c_ ₃₂__c_ , and

p 31 p

   . According to the Schrödinger equation, one can obtain the equation of motion of the density matrix elements of the present atomic system

 

^*

2

21 p c 21 c 31

i i

2 2

  ^     ^    (2-13)

 

^*

2

2 1 p c 2 1 c 31

i i

2 2

 _  ^ ^    _ ^ _  _ (2-14)

 

3

31 p 21 p 11 c 21 c 2 1

i i

2 2

  ^^   ^         _ _ (2-15)

Fig. 2.6 The schematic diagram of a double-control four-level system. The two control laser beams, _c and _c , drive the | 2 | 3 and | 2 | 3 transitions,

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12

respectively. The probe transition |1 | 3 can be controllably manipulated via the destructive and constructive quantum interferences between the | 2 | 3 and | 2 | 3 transitions. If levels |1 , | 2, and | 2 form a three-level dark state, the atomic vapor is transparent to the probe field, while it is opaque to the probe field when levels | 2 and | 2 form a two-level dark state.

We assume that the intensity of the probe beam is sufficiently weak and therefore nearly all the atoms remain in the ground state, i.e., the atomic population in level |1 is unity.

The density matrix elements of levels | 2, | 2and | 3are given by

 

* 2

21 p c p c

1 i

4D 2

      ^      (2-16)

 

* 2

2 1 p c p c

1 i

4D 2

 _    _^     ^ (2-17)

   

2 2

31 p p c p c

i i i

4D 2 2

 

   ^     ^{ }   ^    _ ^ (2-18)

where, ³ i p ² i



p c



² i



p c



2 2 2

D   _



     

                 

   

* 2 * 2

c c p c c c p c

1 1

i i

4 2 4 2

  _

     

                (2-19)

The atomic electric polarizability of the double-control four-level atomic vapor is of the form



p c c



¹³ ²



p c



²



p c



0

, , i i i

2 2

D

 

 ^  ^ ^

    

                 (2-20)

The relative electric permittivity with local field correction is

   



^p ^c ^c



r p c c

p c c

, ,

, , 1

, ,

1 3

N N

 





  

    

  

 (2-21)

The most remarkable feature of the present scheme is that the optical properties (absorption, transparency and dispersion) of the atomic system can be controllably

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13

manipulated by the double-control multi-pathway interferences (multiple routes to

excitation). The four-level system will exhibit two-level resonant absorption when the two control levels (driven by the two control fields) form a dark state (and hence the destructive quantum interference occurs between the two control transitions). However, the present four-level system will exhibit electromagnetically induced transparency to the probe field when the three lower levels (including the probe level and two control levels) form a three- level dark state. Similar to the preceding three-level EIT photonic crystals, an alternative scenario to manipulate the probe transitions (and hence the optical properties) via the combination of double-control (destructive and constructive) quantum interferences and photonic crystal can be suggested. The optically sensitive responses due to the double- control quantum interferences can also be utilized to realize the logic-gate devices.

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14

3 Band Structure of Periodic Dielectric and EIT Bi-layeredStructure

3.1 Frequency-sensitive tunable band structure

The optical response of an atomic vapor can be coherently manipulated by tunable quantum interference occurring in atomic transition processes. A periodic layered medium whose unit cells consist of a dielectric and an EIT (electromagnetically induced transparency) atomic vapor is designed for light propagation manipulation. Such an EIT-based periodic layered medium exhibits a flexible frequency-sensitive optical response, where a very small change in probe frequency can lead to a drastic variation of reflectance and transmittance.

The 1D periodic (D|E) cells shown in Fig. 3.1 are composed of two kinds of media: a dielectric (e.g., GaAs dielectric with the relative refractive index n₁3.54) and a typical Lambda-configuration three-level EIT medium whose electric permittivity is determined by equations (2-11) and (2-12). Here, the characters “D” and “E” in “(D|E)” denote the dielectric (GaAs) and the EIT material, respectively.

Fig. 3.1 The 1D -layer structure of (D|E) cells embedded in GaAs homogeneous dielectric. The dielectrics and stand for the GaAs and EIT atomic media, respectively. A (D|E) cell consists of GaAs dielectric (D) and EIT medium (E).

As the destructive quantum interference relevant to two-photon resonance arises in EIT atoms interacting with both control and probe fields, the controllable optical processes that

N

D1 D₂

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15

depend sensitively on the external control field will take place in this EIT-based periodic layered medium. Such a frequency-sensitive and field-controlled optical behavior of reflection and transmission in the EIT photonic crystal can be applicable to designs of new devices such as photonic switches, photonic logic gates and photonic transistors, where one laser field can be controlled by the other one, and would have potential applications in the areas of integrated optical circuits and other related techniques (e.g., all-optical instrumen- tations).

Since it can exhibit many intriguing optical properties and effects, EIT has attracted extensive attention of a large number of researchers in a variety of areas of optics, atomic physics and condensed state physics, and this enables physicists to achieve new novel results.

For example, some unusual physical effects associated with EIT include the ultraslow light pulse propagation, the superluminal light propagation, and the light storage in atomic vapors, some of which are expected to be beneficial (and powerful) for developing new technologies in quantum optics and photonics.

In addition to EIT (and hence quantum coherence), in this section, we shall also consider another kind of artificial dielectric, EIT-based periodic material: specifically, the EIT medium (an atomic vapor or a semiconductor-quantum-dot material) is embedded in a periodic host dielectric (e.g., GaAs). As is well known, the photonic crystals, which are periodic arrangements of dielectrics, have captured wide attention in physics, materials science and other relevant fields (e.g., information science) due to its capacity of controlling light propagations. Here, we shall propose some new effects relevant to light propagation manipulation via EIT responses in artificial periodic dielectric. Such effects result from the combination of EIT and photonic crystals. This can be viewed as a new application of EIT for manipulating light wave propagations, which can exhibit a tunable reflectance and transmittance (induced by an external control field) and show extraordinary sensitivity to the frequency of the probe field. For example, a change of one part in10 in the probe frequency ⁸

pwould lead to a dramatic change in the reflectance and transmittance of the EIT-based periodic layered medium, and therefore, it can be used for designing sensitive optical switches, photonic logic gates as well as tunable photonic transistor. Although there have been some investigations of photonic crystals based on EIT media in the literature, yet less attention has been paid to the frequency-sensitive optical behavior that would be the most remarkable property of such a kind of periodic layered media.

In the literature, photonic logic gates designed based on new coherent materials, such as

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16

near-field optically coupled nanometric materials and double-control multilevel atomic media, have been suggested. It should be emphasized that the mechanism presented in this thesis can be considered an alternative way to realize such a kind of photonic and quantum optical devices. Very recently, Abdumalikov et al. reported an experimental observation of EIT on a single artificial atom, and found that the propagating electromagnetic waves are allowed to be fully transmitted or backscattered. We will demonstrate in the present paper that such a full controllability of optical property of artificial media can also be achieved in the EIT-based layered structure, of which the reflectance can be either zero or large depending sensitively on the intensity of the external control field, and we expect that this would open a good perspective for its application in some new fields such as photonic microcircuits (or integrated optical circuits). It should be mentioned that, in the literature, the tunable photonic band structure was suggested for its realization via optical lattice formed by the standing waves of the control fields. However, this is different from our scenario, where the EIT-based photonic crystal is composed of two kinds of media: an ordinary dielectric and an EIT medium. In the present works, we shall study the transmittance depending upon a variety of parameters, including control field intensity, probe frequency detuning and layer number of the periodic structure, and suggest photonic logic gate design based on the optically tunable response of such an EIT-based layered medium. We will concentrate our attention on its application in photonic device design, including particularly the influence of layer number of periodic structure on the dramatic modification of the optical property of the layered medium.

In section2.1 and section 2.2 we have presented the characteristic optical property of an EIT medium (e.g., an atomic vapor), and in section 3.2 to section 3.7, we shall address the behavior of frequency-sensitive reflectance and transmittance of the EIT-based periodic layered medium. The spectrum of the reflectance as well as thetransmittance of such a periodic structure (when the TE wave of the probe beam is normally incident on the layered medium) versus the normalized control Rabi frequency  _c/ ₃ and the normalized probe frequency detuning _p/ ₃is addressed. In Section 3.8, two tunable logic gates (NOT and NOR gates) are designed by taking full advantage of the effect of such an optical switching control, which is sensitive to the probe frequency due to the two-photon resonance.

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17

3.2 N-Layers Cell Level Analysis and Cascaded Matrix Formalism

3.2.1 Matrix Formalism (TE mode)

Assume that the two materials are both homogeneous along y-direction (i.e., /  y 0) and the probe signal wave travels in the (...D|E|D|E…) structure always along x-direction.

The reflection coefficient^[66] on the left side interface (x0) of such an EIT-based periodic medium, which is in fact a 1D N -layer (D|E) layered structure bounded by the GaAs dielectric material, will be addressed. (Refer to Fig. 3.1)

In order to make the section self-contained, we shall in this section review the formalism for treating the light wave propagation in a periodic layered medium (Readers are referred to e.g. Yeh’s reference^[66] for a more complete and detailed formalism). According to the theory of electromagnetism in photonic crystals, the electric field in the mth unit cell can be expressed by^[66]

 

_^ ^_ ^ _ ^ _

 

1 1

2 2

,

, 1

x x

ik x m ik x m

m m

ik x m a ik x m a

m m

a e b e m a x m

E x

c e d e m x m a

     

     

      

 

      

 (3-7)

where, the wave vectors k₁_x n₁/ ,c k₂_x n₂/c.

By using the matrix formalism for treating the wave propagation in layered media, one can arrive at the equation

2 2

+ 2 2

1 1

1

1 + 2 2

1 1

1

2 1 1

x x

ik b x ik b x

x x

m m

m ik b x ik b x m

x x

k k

e e

k k

a c

b k k d

e e

k k



 

    

 

    

       

       

         

(3-8)

of electric field amplitudes as well as the eigenvalue equation

m iK m

m m

a a

A B

b e b

C D

   

    

 

      (3-9)

for the column vector characterizing the electromagnetic field strengths in the periodic layered structure. The matrix elements are given by (Yeh, 2005)

¹ 

 

² ¹

 

TE 2 2

1 2

cos sin

2

i k ax x x

x x

i k k

A e k b k b

k k

   

      

 

  (3-10)

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18

¹  ² ¹

 

TE 2

1 2

2 sin

i k ax x x

x

x x

i k k

B e k b

k k

    

     

 

  (3-11)

¹ 

 

+ 2 1

TE 2

1 2

2 sin

i k ax x x

x

x x

i k k

C e k b

k k

   

     

 

  (3-12)

¹ 

 

² ¹

 

TE 2 2

1 2

cos sin

2

i k ax x x

x x

i k k

D e k b k b

k k

    

      

 

  (3-13)

Note that the eigenvalue equation yields

det 0

iK

A e B

C D e



  

 

  

  (3-14)

This can be rewritten as a well-known for



TE

 

1

 

2



² ¹



1

 

2



1 2

cos cos cos 1 sin sin

2

x x

x x x x

x x

k k

K k a k b k a k b

k k

 

     

  (3-15)

From this relation, one can obtain the Bloch wave numberK.

       

1 2 1

TE 1 2 1 2

1 2

1 1

cos cos cos sin sin

2

x x

x x x x

x x

k k

K k a k b k a k b

k k

    

        (3-16)

3.2.2 Bloch Analysis

Now we are in a position to derive the coefficient of reflection, which is defined as

0/ 0

rN b a . It follows that the relation of the column vectors between the left side interface (atx0) and in the Nthunit cell is given by

0

N N

N

a A B a

b C D b

    

     

 

    (3-17)

with

1 2 1

1 1 2

N

N N N

AU U BU

A B

CU DU U

C D

  

  

 

  

  

    (3-18)

Here, the explicit expression forU is _N ^sin



¹



N sin

N K

U K

 

 

 

  . With the help of the

relations

中華大學

中華大學

博士論文

Band Structure of

EIT-Based Photonic Crystals and Photonic Logic Gate Design

Teh-Chau, Liau

2014/6/24

Preface

ABSTRACT

DEDICATION

ACKNOWLEDFEMENTS

ABBREVIATIONS

 

 

 

CONTENTS

1 Introduction

1.1 Background and Motivation

1.2 Structure of the Thesis

2 Optical properties of EIT Medium

2.1 Three-Level Atomic System





   

 

   





 

 

 

  

 







 





 

 

 





 

 

 













2.2 Four-Level Atomic System

 

 

 

 

 

   









   













   





3 Band Structure of Periodic Dielectric and EIT Bi-layeredStructure

3.1 Frequency-sensitive tunable band structure

3.2 N-Layers Cell Level Analysis and Cascaded Matrix Formalism

3.2.1 Matrix Formalism (TE mode)

 

 