**1. Introduction**

132 Photonic Crystals – Introduction, Applications and Theory

Mekis, A. & Joannopoulos, J. D. Tapered couplers for efficient interfacing between dielectric

Pierantoni, L.; Massaro, A. & Rozzi, T. Efficient Modeling of a 3D-Photonic Crystal for

Pottier, P.; Ntakis, I.; De La Rue, R. M. Photonic crystal continuous taper for low-loss direct

Sanchis, P.; Garcia, J.; Marti, J.; Bogaerts, W.; Dumon, P.; Taillaert, D.; Baets, R.; Wiaux, V.;

Sanchis, P.; Martì, J.; Blasco, J.; Martinez, A.; & Garcia, A. Mode Matching technique for

Scully, M.O. & Zubairy, M.S. Quantum Optics. *Cambridge University press*, Cambridge, 1997. Talneau, A.; Mulot, M.; Anand, S.; Olivier, S.; Agio, M.; Kafesaki, M. & C.M. Soukoulis.

Wang, J., Ibanez, A., Chatrathi, M.P., & Escarpa, *A., Anal. Chem*. Vol. 73, (2001), pp. 5323. Xing, A.; Devanço, M.; Blumenthal, D. J. & Hu, E. L. Transmission measurements of tapered-

functionality. *Opt. Comm*., Vol. 223,(2003), pp. 339.

Scherer, J.R.et al. *Electrophoresis* Vol. 20, (1999), No. 7, pp. 1508.

*Photon. Nanostruct*., (2004), pp. 1.

Yablonovitch, E., *Phys. Rev. Lett*. Vol. 58, (1987), pp. 2059.

(2005),no. 10, pp. 2092.

*IEEE Photon. Techn. Lett.*, Vol. 16,(2004), No. 10, pp. 2272.

crystal circuit. *OSA Opt. Express*, (2002), Vol. 10, No. 24, pp. 1391.

861.

pp.319.

and photonic crystal waveguide. *IEEE J. Light. Technol*., Vol. 19, No.,(2001), pp.

Integrated Optical Devices. *IEEE Photonics Technology Letters*,Vol.18, No.2, (2006),

coupling into 2D photonic crystal channel waveguides and further device

Wouters, J. & Beckx, S.. Experimental demonstration of high coupling efficiency between wide ridge waveguides and single-mode photonic crystal waveguides.

highly efficient coupling between dielectric waveguides and planar photonic

Modal behavior of single-line photonic crystal guiding structures on InP substrate.

single line defect photonic crystal waveguide. *IEEE Photon. Techn. Lett*.,Vol. 17,

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) (Harris, 1997) and the effects that are relevant to EIT, including light amplification without inversion (Cohen & Berman, 1997), spontaneous emission cancellation (Zhu & Scully, 1996), multi-photon population trapping (Champenois et al., 2006), coherent phase control (Zheltikov, 2006; Gandman et al., 2007) as well as photonic resonant lefthanded media (Krowne & Shen, 2009). 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 (Harris, 1997), 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 (Schmidt & Imamoğlu, 1996; Wang et al., 2000; Arve, 2004; Shen et al., 2004), some of which are expected to be beneficial (and powerful) for developing new technologies in quantum optics and photonics.

In this chapter, 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)

EIT-Based Photonic Crystals and Photonic Logic Gate Design 135

advantage of the effect of such an optical switching control. In Sec. 9 we close the chapter

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.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). The off-diagonal

condition of weak probe field (Scully & Zubairy, 1997), and the atomic system can be characterized by an SU(2) time-dependent model when the control field intensity varies adiabatically. The present atomic system interacting with two light fields ( <sup>c</sup> and <sup>p</sup> ) is

> <sup>2</sup> \* 21 21 31 3 31

> > <sup>2</sup> <sup>2</sup>

<sup>i</sup> <sup>i</sup>

<sup>i</sup> <sup>i</sup>

pc c

It can be verified that the atomic microscopic electric polarizability of the 1 - 3 transition

i i

dephasing rate, respectively. The Rabi frequency c of the control field is defined by c c <sup>32</sup>*E* / with *E*c the slowly-varying amplitude (envelope) of the control field. 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

31 

1 , 1

 <sup>a</sup> <sup>r</sup> <sup>a</sup>

where *Na* denotes the atomic concentration (atomic number per unit volume) of the EIT

*N N* 

3

0 3 2 \* | | <sup>2</sup> . <sup>1</sup>

22 4

c p

<sup>0</sup> 2 2 .

<sup>2</sup> 2 2

p c

p p c cc

stand for the spontaneous emission decay rate and the collisional

 , c c 32 

<sup>i</sup> <sup>i</sup> <sup>i</sup>

31 can form a closed set of equations under the

 

 

p

 with p and (1)

(2)

(3)

<sup>c</sup> the

with some concluding remarks.

density matrix elements

*t*

p 

governed by

is of the form

Here, 3 and 2

probe frequency (

atomic vapor.

**2. Optical properties of an EIT medium** 

21 and

13

two frequency detunings are defined as p p

<sup>31</sup> <sup>p</sup> ) is given by

due to its capacity of controlling light propagations (Yablonovitch, 1987; Joannopoulos et al., 1995; Joannopoulos et al., 1997). 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 <sup>8</sup> 10 in the probe frequency **<sup>p</sup>** would 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 (Forsberg & She, 2006; He et al., 2006; Zhuang et al., 2007; Petrosyan, 2007), 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 (Sangu et al., 2004; Kawazoe et al., 2003) and double-control multilevel atomic media (Shen, 2007; Shen & Zhang, 2007; Gharibi et al., 2009), have been suggested during the past few years. It should be emphasized that the mechanism presented in this chapter 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 (Abdumalikov et al., 2010). We will demonstrate in the present chapter 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).

This chapter is organized as follows. In Sec. 2 we shall discuss the characteristic optical property of an EIT medium (e.g., an atomic vapor), and in Sec. 3 we 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 in Sec. 4 and Sec. 5, respectively, where 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 <sup>c</sup> / 3 of the control field and the normalized probe frequency detuning <sup>p</sup> /<sup>3</sup> will be addressed. The frequency-sensitive tunable band structure of TM wave in the EIT-based periodic structure containing a left-handed medium is discussed in Sec. 6, 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). In Sec. 7 and Sec. 8, a potential application, 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 Sec. 9 we close the chapter with some concluding remarks.

## **2. Optical properties of an EIT medium**

134 Photonic Crystals – Introduction, Applications and Theory

due to its capacity of controlling light propagations (Yablonovitch, 1987; Joannopoulos et al., 1995; Joannopoulos et al., 1997). 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

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 (Forsberg & She, 2006; He et al., 2006; Zhuang et al., 2007; Petrosyan, 2007), yet less attention has been paid to the frequency-sensitive optical behavior that would be the most remarkable property of such a

We should point out that photonic logic gates designed based on new coherent materials, such as near-field optically coupled nanometric materials (Sangu et al., 2004; Kawazoe et al., 2003) and double-control multilevel atomic media (Shen, 2007; Shen & Zhang, 2007; Gharibi et al., 2009), have been suggested during the past few years. It should be emphasized that the mechanism presented in this chapter 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 (Abdumalikov et al., 2010). We will demonstrate in the present chapter 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

This chapter is organized as follows. In Sec. 2 we shall discuss the characteristic optical property of an EIT medium (e.g., an atomic vapor), and in Sec. 3 we 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 in Sec. 4 and Sec. 5, respectively, where 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 <sup>c</sup> / 3 of the control field and the normalized probe frequency detuning <sup>p</sup> /<sup>3</sup> will be addressed. The frequency-sensitive tunable band structure of TM wave in the EIT-based periodic structure containing a left-handed medium is discussed in Sec. 6, 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). In Sec. 7 and Sec. 8, a potential application, i.e., photonic transistors and logic gates (tunable photonic logic gates) are suggested by taking full

**<sup>p</sup>** would lead to a dramatic change in the

of one part in <sup>8</sup> 10 in the probe frequency

kind of periodic layered media.

circuits).

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.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). The off-diagonal density matrix elements 21 and 31 can form a closed set of equations under the condition of weak probe field (Scully & Zubairy, 1997), and the atomic system can be characterized by an SU(2) time-dependent model when the control field intensity varies adiabatically. The present atomic system interacting with two light fields ( <sup>c</sup> and <sup>p</sup> ) is governed by

$$
\frac{\partial}{\partial t} \begin{pmatrix} \rho\_{21} \\ \rho\_{31} \end{pmatrix} = \begin{pmatrix} -\left(\frac{\mathcal{V}\_{2}}{2} + \mathrm{i}\left(\Delta\_{\mathrm{p}} - \Delta\_{\mathrm{c}}\right)\right) & \frac{\mathrm{i}}{2}\Omega\_{\mathrm{c}}^{\*} \\\\ \frac{\mathrm{i}}{2}\Omega\_{\mathrm{c}} & -\left(\frac{\Gamma\_{3}}{2} + \mathrm{i}\Delta\_{\mathrm{p}}\right) \end{pmatrix} \begin{pmatrix} \rho\_{21} \\ \rho\_{31} \end{pmatrix} + \begin{pmatrix} 0 \\ \mathrm{i}\Delta\_{\mathrm{p}} \end{pmatrix}.\tag{1}
$$

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

$$\beta = \frac{\mathbf{i} \left| \wp\_{13} \right|^2}{\varepsilon\_0 \hbar} \frac{\frac{\mathcal{I}\_2}{2} + \mathbf{i} \left( \Delta\_\mathrm{p} - \Delta\_\mathrm{c} \right)}{\left( \frac{\Gamma\_3}{2} + \mathrm{i} \Delta\_\mathrm{p} \right) \left[ \frac{\mathcal{I}\_2}{2} + \mathbf{i} \left( \Delta\_\mathrm{p} - \Delta\_\mathrm{c} \right) \right] + \frac{1}{4} \Delta\_\mathrm{c}^\dagger \Omega\_\mathrm{c}}.\tag{2}$$

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 c <sup>32</sup>*E* / with *E*c the slowly-varying amplitude (envelope) of the control field. The two frequency detunings are defined as p p 31 , c c 32 with p and <sup>c</sup> 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 <sup>31</sup> <sup>p</sup> ) is given by

$$\varepsilon\_{\rm r} = 1 + \frac{N\_{\rm a} \beta}{1 - \frac{N\_{\rm a} \beta}{3}} \,\prime\,\tag{3}$$

where *Na* denotes the atomic concentration (atomic number per unit volume) of the EIT atomic vapor.

EIT-Based Photonic Crystals and Photonic Logic Gate Design 137

(a) (b) Fig. 3. The dispersion of the relative electric permittivity of the EIT atomic medium versus the frequency detuning **<sup>p</sup>** of the probe field and the Rabi frequency **<sup>c</sup>** of the control field.

The 1D periodic (D|E) cells shown in Fig. 4 are composed of two kinds of media: a dielectric (e.g., GaAs dielectric with the relative refractive index <sup>1</sup> *n* 3.54 ) and a typical Lambdaconfiguration three-level EIT medium whose electric permittivity is determined by Eqs. (2) and (3). Here, the characters "D" and "E" in "(D|E)" denote the dielectric (GaAs) and the EIT, respectively. Assume the two materials are both homogeneous along y-direction (i.e. / 0 *y* ) and the probe signal wave travels in the (...D|E|D|E…) structure always along x-direction. The reflection coefficient (Yeh, 2005) on the left side interface ( 0 *x* ) of such an EIT-based periodic medium, which is in fact a 1D *N* -layer (D|E) layered structure

Fig. 4. The 1D *N* -layer structure of (D|E) cells embedded in GaAs homogeneous dielectric. The dielectrics *D*1 and *D*2 stand for the GaAs and EIT atomic media, respectively. A (D|E) cell consists of GaAs dielectric (D) and EIT medium (E). The lattice constants of the (D|E)

In order to make the chapter 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 (Yeh, 2005) for a more complete and detailed formalism). According to

bounded by the GaAs dielectric material, will be addressed.

m .

**3. The electromagnetism of periodic layered medium** 

cells are chosen as *a b* 0.1

Fig. 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 because of the destructive quantum interference between the 1 - 3 and 2 - 3 transitions when the control laser beam is present.

The tunable dispersive behavior of the bulk EIT atomic vapor is shown in Figs. 2 and 3. The typical atomic and optical parameters chosen for Figs. 2 and 3 are as follows: the atomic number density <sup>20</sup> *N* 5.0 10 **<sup>a</sup>** -3 m , the electrical dipole moment <sup>29</sup> 1.0 10 **<sup>31</sup>** C m , the frequency detuning of the control field <sup>7</sup> 1.0 10 **<sup>c</sup>** <sup>s</sup> <sup>1</sup> , the spontaneous emission decay rate <sup>7</sup> 2.0 10 **Γ3** <sup>s</sup> 1 and the dephasing rate <sup>5</sup> 1.0 10 **<sup>2</sup> <sup>γ</sup>** <sup>s</sup> <sup>1</sup> . Fig. 3 shows the threedimensional behavior of the real part (a) and the imaginary part (b) of the relative electric permittivity of the EIT atomic vapor (bulk). As the dispersive curve of the refractive index of the EIT bulk is a function of <sup>p</sup> and <sup>c</sup> , in the section that follows, we shall consider a band structure (versus both <sup>p</sup> and <sup>c</sup> ) of the EIT-based periodic medium (see Fig. 4 for its schematic diagram).

Fig. 2. The relative electric permittivity of the three-level EIT atomic vapor as a function of the probe frequency detuning **p** and the Rabi frequency **c** of the control field. In (a) the Rabi frequency of the control field is <sup>7</sup> 2.0 10 **<sup>c</sup>** <sup>s</sup> <sup>1</sup> . In (b) the probe frequency detuning is <sup>6</sup> 3.4 10 **<sup>p</sup>** s <sup>1</sup> .

Fig. 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 tunable dispersive behavior of the bulk EIT atomic vapor is shown in Figs. 2 and 3. The typical atomic and optical parameters chosen for Figs. 2 and 3 are as follows: the atomic

dimensional behavior of the real part (a) and the imaginary part (b) of the relative electric permittivity of the EIT atomic vapor (bulk). As the dispersive curve of the refractive index of the EIT bulk is a function of <sup>p</sup> and <sup>c</sup> , in the section that follows, we shall consider a band structure (versus both <sup>p</sup> and <sup>c</sup> ) of the EIT-based periodic medium (see Fig. 4 for its

(a) (b) Fig. 2. The relative electric permittivity of the three-level EIT atomic vapor as a function of the probe frequency detuning **p** and the Rabi frequency **c** of the control field. In (a) the


**<sup>31</sup>** C m , the

<sup>1</sup> . Fig. 3 shows the three-

<sup>1</sup> , the spontaneous emission decay

<sup>1</sup> . In (b) the probe frequency detuning

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 because of the destructive quantum interference between the 1 - 3 and 2 - 3 transitions when the control laser

1 and the dephasing rate <sup>5</sup> 1.0 10 **<sup>2</sup> <sup>γ</sup>** <sup>s</sup>

beam is present.

rate <sup>7</sup> 2.0 10 **Γ3** <sup>s</sup>

schematic diagram).

is <sup>6</sup> 3.4 10 **<sup>p</sup>** s

number density <sup>20</sup> *N* 5.0 10 **<sup>a</sup>**

frequency detuning of the control field <sup>7</sup> 1.0 10 **<sup>c</sup>** <sup>s</sup>

Rabi frequency of the control field is <sup>7</sup> 2.0 10 **<sup>c</sup>** <sup>s</sup>

<sup>1</sup> .

Fig. 3. The dispersion of the relative electric permittivity of the EIT atomic medium versus the frequency detuning **<sup>p</sup>** of the probe field and the Rabi frequency **<sup>c</sup>** of the control field.

The 1D periodic (D|E) cells shown in Fig. 4 are composed of two kinds of media: a dielectric (e.g., GaAs dielectric with the relative refractive index <sup>1</sup> *n* 3.54 ) and a typical Lambdaconfiguration three-level EIT medium whose electric permittivity is determined by Eqs. (2) and (3). Here, the characters "D" and "E" in "(D|E)" denote the dielectric (GaAs) and the EIT, respectively. Assume the two materials are both homogeneous along y-direction (i.e. / 0 *y* ) and the probe signal wave travels in the (...D|E|D|E…) structure always along x-direction. The reflection coefficient (Yeh, 2005) on the left side interface ( 0 *x* ) 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.

Fig. 4. The 1D *N* -layer structure of (D|E) cells embedded in GaAs homogeneous dielectric. The dielectrics *D*1 and *D*2 stand for the GaAs and EIT atomic media, respectively. A (D|E) cell consists of GaAs dielectric (D) and EIT medium (E). The lattice constants of the (D|E) cells are chosen as *a b* 0.1m .

### **3. The electromagnetism of periodic layered medium**

In order to make the chapter 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 (Yeh, 2005) for a more complete and detailed formalism). According to

EIT-Based Photonic Crystals and Photonic Logic Gate Design 139

of the column vectors between the left side interface (at 0 *x* ) and in the *N*th unit cell is

*a AB a b CD b* 

sin *<sup>N</sup>*

0 12 1 0 1 12

*a AU U a BU b b CU a DU U b*

*A B AU U BU C D CU DU U*

*N N N*

12 1

*N K*

*K*

*N N N NN NN N N N*

> 1 1 2 , *<sup>N</sup>*

*N N*

*CU*

*AU U* 

where *bN* 0 has been substituted (since the present periodic layered medium is composed of *N* unit cells and is bounded by the medium of the refractive index *n*<sup>1</sup> , the reflected amplitude of the electric field in the last unit cell vanishes). It should be emphasized that the

deriving the atomic microscopic electric polarizability (2) of EIT. Such a convention is often used by physicists. In the convention of engineers, however, the time dependence

*<sup>t</sup>* e (Yeh, 2005; Caloz & Itoh, 2006). As we shall employ the formalism in the reference of Yeh (Yeh, 2005) for treating the wave propagation in the periodic layered medium, we need to convert the convention of physicists to that of engineers. This can be easily accomplished

In the sections that follow, we shall concentrate our attention on the influence of the external control field on the probe wave propagation inside the EIT-based periodic layered medium. It should be noted that we only consider a passive multilayered structure in this chapter. Although there are control and probe laser beams exciting the two electric-dipole allowed transitions, it is a passive atomic system because of the large spontaneous emission decay from the excited states to the ground state. If, however, there is an extra strong pumping laser beams driving the atomic system (Wu, 2004), we should address its optical response relevant to gain factor. But here such a strong pumping interaction is not taken into account.

In order to show how sensitive (to the probe frequency) the band structure of the EIT photonic crystal is, let us first see the dispersive relation of the 1D infinite periodic (D|E)

 

*N NN*

*NN N*

sin 1

1 12

. With the help of the relations

(13)

*<sup>t</sup>* e , has been adopted for the time harmonic wave in

p is far from the resonant frequency of the atomic

, ,

(10)

(12)

. (11)

0 0

*U*

the coefficient of reflection of an *N* -layer periodic medium is given by (Yeh, 2005)

*N*

*r*

 

*N*

given by (Yeh, 2005)

Here, the explicit expression for*UN* is

factor of phasor time dependence, <sup>i</sup>

cells, in which the probe frequency

by the imaginary variable substitution, i.e., i j .

**4. The frequency-sensitive tunable band structure** 

with

is +j

the theory of electromagnetism in photonic crystals, the electric field in the *m*th unit cell can be expressed by (Yeh, 2005)

$$E(\mathbf{x}) = \begin{cases} a\_m \mathbf{e}^{-\|\mathbf{k}\_{1x}(\mathbf{x} - m\boldsymbol{\Lambda})\|} + b\_m \mathbf{e}^{\|\mathbf{k}\_{1x}(\mathbf{x} - m\boldsymbol{\Lambda})\|}, & m\boldsymbol{\Lambda} - a < \mathbf{x} < m\boldsymbol{\Lambda} \\ c\_m \mathbf{e}^{-\|\mathbf{k}\_{2x}(\mathbf{x} - m\boldsymbol{\Lambda} + a)} + d\_m \mathbf{e}^{\|\mathbf{k}\_{2x}(\mathbf{x} - m\boldsymbol{\Lambda} + a)}. & (m - 1)\boldsymbol{\Lambda} < \mathbf{x} < m\boldsymbol{\Lambda} - a \end{cases} \tag{4}$$

Here, the wave vectors *knc* 1 1 *<sup>x</sup>* / , *knc* 2 2 *<sup>x</sup>* / . By using the matrix formalism for treating the wave propagation in layered media, one can arrive at the equation

$$
\begin{pmatrix} a\_{m-1} \\ b\_{m-1} \end{pmatrix} = \frac{1}{2} \begin{pmatrix} \mathbf{e}^{j\mathbf{k}\_{2x}b} \left( 1 + \frac{k\_{2x}}{k\_{1x}} \right) & \mathbf{e}^{-j\mathbf{k}\_{2x}b} \left( 1 - \frac{k\_{2x}}{k\_{1x}} \right) \\\\ \mathbf{e}^{j\mathbf{k}\_{2x}b} \left( 1 - \frac{k\_{2x}}{k\_{1x}} \right) & \mathbf{e}^{-j\mathbf{k}\_{2x}b} \left( 1 + \frac{k\_{2x}}{k\_{1x}} \right) \end{pmatrix} \begin{pmatrix} c\_m \\ d\_m \end{pmatrix} \tag{5}
$$

of electric field amplitudes as well as the eigenvalue equation

$$
\begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} a\_m \\ b\_m \end{pmatrix} = \mathbf{e}^{\mathsf{i}K\mathsf{A}} \begin{pmatrix} a\_m \\ b\_m \end{pmatrix} \tag{6}
$$

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

$$\begin{split} A &= \mathbf{e}^{\left|k\_{1x}a\right|} \left[ \cos k\_{2x}b + \frac{1}{2} \mathbf{j} \left( \frac{k\_{2x}}{k\_{1x}} + \frac{k\_{1x}}{k\_{2x}} \right) \sin k\_{2x}b \right] \\ B &= \mathbf{e}^{-\left|k\_{1x}a\right|} \left[ \frac{1}{2} \mathbf{j} \left( \frac{k\_{2x}}{k\_{1x}} - \frac{k\_{1x}}{k\_{2x}} \right) \sin k\_{2x}b \right] \\ C &= \mathbf{e}^{\left|k\_{1x}a\right|} \left[ -\frac{1}{2} \mathbf{j} \left( \frac{k\_{2x}}{k\_{1x}} - \frac{k\_{1x}}{k\_{2x}} \right) \sin k\_{2x}b \right] \\ D &= \mathbf{e}^{\left|k\_{1x}a\right|} \left[ \cos k\_{2x}b - \frac{1}{2} \mathbf{j} \left( \frac{k\_{2x}}{k\_{1x}} + \frac{k\_{1x}}{k\_{2x}} \right) \sin k\_{2x}b \right]. \end{split} \tag{7b}$$

Note that the eigenvalue equation yields

$$\det \begin{pmatrix} A - \mathbf{e}^{\mathbb{J}K\boldsymbol{\Lambda}} & B \\ \mathbf{C} & D - \mathbf{e}^{\mathbb{J}K\boldsymbol{\Lambda}} \end{pmatrix} = \mathbf{0} \, \cdot \tag{8}$$

This can be rewritten as a well-known form

$$\cos K\Lambda = \cos k\_{1x}a\cos k\_{2x}b - \frac{1}{2}\left(\frac{n\_2}{n\_1} + \frac{n\_1}{n\_2}\right)\sin k\_{1x}a\sin k\_{2x}b. \tag{9}$$

From this relation, one can obtain the Bloch wave number *K* . Now we are in a position to derive the coefficient of reflection, which is defined as *r ba <sup>N</sup>* 0 0 / . It follows that the relation of the column vectors between the left side interface (at 0 *x* ) and in the *N*th unit cell is given by (Yeh, 2005)

$$
\begin{pmatrix} a\_0 \\ b\_0 \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}^N \begin{pmatrix} a\_N \\ b\_N \end{pmatrix} \tag{10}
$$

with

(4)

(5)

(7a)

(7b)

(9)

138 Photonic Crystals – Introduction, Applications and Theory

the theory of electromagnetism in photonic crystals, the electric field in the *m*th unit cell can

*a b m axm*

*c d m xm a*

2 2

*k k*

1 1

*k k*

*m m K m m*

2 1 2 2 1 2

*x x*

cos sin , <sup>2</sup>

 

*k k*

2

2

2 1 2 2 1 2

*x x*

*K*

2 1 1 2 1 2 1 2

2 *x x x x*

*n n*

j

e

cos sin .

 

*k k*

*x x*

/ . By using the matrix formalism for

<sup>i</sup> e (6)

. (8)

1 1

*k b x x k b m m x x m m k b x x k b*

1 1

1 1 2 2

j j j j e e e e .

 

*m m*

*m m*

1 2

of electric field amplitudes as well as the eigenvalue equation

layered structure. The matrix elements are given by (Yeh, 2005)

1

j

*x*

1

=e j

j

1

j

*x*

e j

1


Note that the eigenvalue equation yields

This can be rewritten as a well-known form

*x*

*x*

 

() () () () , ( ) ( 1) *x x x x k xm k xm*

*k xm a k xm a*

treating the wave propagation in layered media, one can arrive at the equation

/ , *knc* 2 2 *<sup>x</sup>*

2 2

*x x*

*a c k k b d k k*

1 1 1 1 2 2

j -j

e e

j -j

e e

2 2

*x x*

*AB a a CDb b* 

for the column vector characterizing the electromagnetic field strengths in the periodic

1

*k k A kb k b*

2 1

*k a x x <sup>x</sup> x x*

*k k <sup>B</sup> k b k k*

*k k <sup>C</sup> k b k k*

e j

e j

<sup>1</sup> sin , <sup>2</sup>

<sup>1</sup> sin , <sup>2</sup>

*k a x x x x*

 

*k a x x x x*

1 2

2 1

*k a x x <sup>x</sup> x x*

 

1 2

1

*k k D kb k b*

2

det 0 *K*

1 cos cos cos sin sin .

From this relation, one can obtain the Bloch wave number *K* . Now we are in a position to derive the coefficient of reflection, which is defined as *r ba <sup>N</sup>* 0 0 / . It follows that the relation

*n n K ka kb ka kb*

*A B C D*

j

e

be expressed by (Yeh, 2005)

*E x*

Here, the wave vectors *knc* 1 1 *<sup>x</sup>*

$$
\begin{pmatrix} A & B \\ C & D \end{pmatrix}^N = \begin{pmatrix} A\mathcal{U}\_{N-1} - \mathcal{U}\_{N-2} & B\mathcal{U}\_{N-1} \\ C\mathcal{U}\_{N-1} & D\mathcal{U}\_{N-1} - \mathcal{U}\_{N-2} \end{pmatrix}. \tag{11}
$$

Here, the explicit expression for*UN* is sin 1 sin *<sup>N</sup> N K U K* . With the help of the relations

$$\begin{aligned} a\_0 &= \left( AUI\_{N-1} - U\_{N-2} \right) a\_N + BUI\_{N-1} b\_{N'} \\ b\_0 &= \mathbf{C} \mathbf{U}\_{N-1} a\_N + \left( D \mathbf{U}\_{N-1} - \mathbf{U}\_{N-2} \right) b\_{N'} \end{aligned} \tag{12}$$

the coefficient of reflection of an *N* -layer periodic medium is given by (Yeh, 2005)

$$r\_N = \frac{C \mathcal{U}\_{N-1}}{A \mathcal{U}\_{N-1} - \mathcal{U}\_{N-2}} \, \prime \tag{13}$$

where *bN* 0 has been substituted (since the present periodic layered medium is composed of *N* unit cells and is bounded by the medium of the refractive index *n*<sup>1</sup> , the reflected amplitude of the electric field in the last unit cell vanishes). It should be emphasized that the factor of phasor time dependence, <sup>i</sup>*<sup>t</sup>* e , has been adopted for the time harmonic wave in deriving the atomic microscopic electric polarizability (2) of EIT. Such a convention is often used by physicists. In the convention of engineers, however, the time dependence is +j*<sup>t</sup>* e (Yeh, 2005; Caloz & Itoh, 2006). As we shall employ the formalism in the reference of Yeh (Yeh, 2005) for treating the wave propagation in the periodic layered medium, we need to convert the convention of physicists to that of engineers. This can be easily accomplished by the imaginary variable substitution, i.e., i j .

In the sections that follow, we shall concentrate our attention on the influence of the external control field on the probe wave propagation inside the EIT-based periodic layered medium. It should be noted that we only consider a passive multilayered structure in this chapter. Although there are control and probe laser beams exciting the two electric-dipole allowed transitions, it is a passive atomic system because of the large spontaneous emission decay from the excited states to the ground state. If, however, there is an extra strong pumping laser beams driving the atomic system (Wu, 2004), we should address its optical response relevant to gain factor. But here such a strong pumping interaction is not taken into account.

#### **4. The frequency-sensitive tunable band structure**

In order to show how sensitive (to the probe frequency) the band structure of the EIT photonic crystal is, let us first see the dispersive relation of the 1D infinite periodic (D|E) cells, in which the probe frequency p is far from the resonant frequency of the atomic

EIT-Based Photonic Crystals and Photonic Logic Gate Design 141

photon resonant frequency ( **p c** ). As there is almost divergent dispersion close to 0.5 **p 3** , the effects of slow light and the negative group velocity in such an EIT-based periodic layered material deserve consideration. This would lead to promising applications in designing devices for slowing down light speed. Besides, the EIT-based band structure is tunable in response to the intensity (characterized by the Rabi frequency **<sup>c</sup>** ) of the external control field, since the refractive index of the EIT medium can be controlled by the control field. In Fig. 6 (b) the real part of the Bloch wave number *K* decreases as **<sup>c</sup>** increases from 0 to <sup>3</sup> 4 , and then increases when **<sup>c</sup>** / 4 <sup>3</sup> ; the absolute value of the imaginary part of the Bloch wave number increases first in the range **<sup>c</sup>** / [0,1] 3 and then decreases when **<sup>c</sup>** / 1 <sup>3</sup> . This, therefore, means that one can use one optical field to controllably manipulate the wave propagation of the other optical field via such an effect of sensitive

(a) (b)

(c) (d)

details exhibited in the EIT-based band structure in the probe frequency detuning ranges (in

As we have shown the characteristics of both sensitivity and tunability of the EIT-based band structure in Fig. 5(b) and Fig. 6, we shall present its three-dimensional behavior as both the probe frequency detuning and the Rabi frequency of control field vary. The sensitivity and the tunability versus the probe frequency detuning **p** and the Rabi

<sup>8</sup> 2.5 10 ]. In (b), (c) and (d) are the fine

Fig. 5. The bandgap structure of the 1D infinite periodic (D|E) cells when the probe frequency of TE waves is far from the resonance. In (a) is the band structure in the probe

units of **<sup>3</sup>** ), i.e., / **p 3** <sup>7</sup> [ 1.7 10 , <sup>7</sup> 2.0 10 ], 7 7 [ 11.0 10 , 1.0 10 ] and

frequency **c** of control field, respectively, are shown in Fig. 7.

frequency detuning range <sup>8</sup> / [ 2.5 10 , **p 3**

8 8 [ 2.5 10 ,0.0 10 ] , respectively.

switching control exhibited in the EIT-based periodic layered structure.

1 - 3 transition. We shall plot the band structure by using Eq. (9), which is an equation of dispersion of a 1D infinite periodic structure. Since the permittivity of EIT also depends upon the Rabi frequency of the control field, **c** is a tunable parameter involved in the equation of dispersion. Then we will also present the three-dimensional behavior of the Bloch wave number versus both **c** and p with the help of Eqs. (2), (3) and (9). Here, we choose the typical atomic transition frequency <sup>15</sup> 31 5.0 10 s <sup>1</sup> , and the thickness of the two layers 0.1 *a* m (GaAs dielectric) and 0.1 *b* m (EIT medium).

As the probe frequency detuning of TE waves in Fig. 5 is quite large ( <sup>p</sup> *c* / with *a b* ), the strong dispersion of EIT cannot be exhibited, and the present (D|E) layered structure behaves like a conventional 1D photonic crystal. However, when the probe frequency detuning <sup>p</sup> approaches zero (or negligibly small compared with *c* / having the order of magnitude <sup>15</sup> 10 s <sup>1</sup> , e.g., <sup>p</sup> is tuned onto resonance, i.e., p c that equals <sup>7</sup> 1.0 10 s <sup>1</sup> ), it would exhibit a band with a fine structure (and hence remarkable frequency-sensitive reflectance and transmittance). The band structure in the probe frequency detuning range 8 8 / [ 2.5 10 , 2.5 10 ] **p 3** is plotted in Fig. 5 (a). The typical atomic and optical parameters such as the atomic number density *N***<sup>a</sup>** , the electrical dipole moment **<sup>31</sup>** , the control frequency detuning **<sup>c</sup>** , the spontaneous emission decay rate **Γ3** and the dephasing rate **<sup>2</sup> γ** are chosen exactly the same as in Figs. 2 and 3 (these typical parameters are also used throughout the chapter). The Rabi frequency of the control field is <sup>7</sup> 2.0 10 **<sup>c</sup>** <sup>s</sup> <sup>1</sup> . Since, seen from Fig. 5 (a), there are some fine structures of the band in the frequency range ( 8 8 / [ 2.5 10 , 2.5 10 ] **p 3** ) that need to be addressed, we present the intricate structures in Fig. 5(b)-(d) and demonstrate them in more details. In Fig. 5(b), for example, as the probe frequency detuning tends to the resonant frequency p c (i.e., **p 3** / approaches almost zero compared with / *c* ), both the real and imaginary parts of the Bloch wave number *K* would arise, because the strong dispersion of the EIT medium, of which the relative refractive index is a complex number, plays a key role for creating such a band structure.

In Fig. 5(b) we have shown the fine structure of the band induced by the EIT resonance. However, the detailed fine texture cannot be signified by the coarse curves in Fig. 5(b), since the band structure is plotted within a large range of probe frequency detuning. We shall in what follows treat further the fine structure of the band of EIT-based photonic crystal when the 1 - 3 transition of the EIT atomic levels is on resonance. It follows from Fig. 6(a) that after aligning dielectric GaAs side by side with EIT medium there are three extreme values of imaginary part *K* in the Bloch wave number (or *K kn*<sup>0</sup> r,(D|E) ). Coincidently, there are also three extreme values of real part *K* in the Bloch wave number (or *K kn*<sup>0</sup> r,(D|E) ). However, neither of them reaches the band edge 0 *K* or 0.5 *K* (in the units of 2 / ). Note that *K* and *K* simultaneously exist, since the refractive index of the EIT medium has an imaginary part. It should be emphasized that the band structure (i.e., *K* and *K* vary as the probe frequency detuning **<sup>p</sup>** changes slightly) is very sensitive to the probe frequency detuning. From Fig. 6 (a) one can see that the real part of the Bloch wave number changes drastically from 0 to 0.5 (in the units of 2 / ) and the imaginary part changes from 0.5 to 0 (in the units of 2 / ) within a very narrow probe frequency band (namely, a very small change, e.g., at the level of one part in <sup>8</sup> 10 in the probe frequency, gives rise to a large variation in the Bloch wave number). In particular, the slope ( / d d *K* <sup>p</sup> ) is almost divergent at the position / 0.5 **p 3** . The reason for this is because 0.5 **p 3** is exactly the two-

1 - 3 transition. We shall plot the band structure by using Eq. (9), which is an equation of dispersion of a 1D infinite periodic structure. Since the permittivity of EIT also depends upon the Rabi frequency of the control field, **c** is a tunable parameter involved in the equation of dispersion. Then we will also present the three-dimensional behavior of the Bloch wave number versus both **c** and p with the help of Eqs. (2), (3) and (9). Here, we

As the probe frequency detuning of TE waves in Fig. 5 is quite large ( <sup>p</sup>

frequency detuning <sup>p</sup> approaches zero (or negligibly small compared with

 *a b* ), the strong dispersion of EIT cannot be exhibited, and the present (D|E) layered structure behaves like a conventional 1D photonic crystal. However, when the probe

frequency-sensitive reflectance and transmittance). The band structure in the probe frequency detuning range 8 8 / [ 2.5 10 , 2.5 10 ] **p 3** is plotted in Fig. 5 (a). The typical atomic and optical parameters such as the atomic number density *N***<sup>a</sup>** , the electrical dipole moment **<sup>31</sup>** , the control frequency detuning **<sup>c</sup>** , the spontaneous emission decay rate **Γ3** and the dephasing rate **<sup>2</sup> γ** are chosen exactly the same as in Figs. 2 and 3 (these typical parameters are also used throughout the chapter). The Rabi frequency of the control field

the frequency range ( 8 8 / [ 2.5 10 , 2.5 10 ] **p 3** ) that need to be addressed, we present the intricate structures in Fig. 5(b)-(d) and demonstrate them in more details. In Fig. 5(b), for example, as the probe frequency detuning tends to the resonant frequency p c (i.e.,

of the Bloch wave number *K* would arise, because the strong dispersion of the EIT medium, of which the relative refractive index is a complex number, plays a key role for creating such

In Fig. 5(b) we have shown the fine structure of the band induced by the EIT resonance. However, the detailed fine texture cannot be signified by the coarse curves in Fig. 5(b), since the band structure is plotted within a large range of probe frequency detuning. We shall in what follows treat further the fine structure of the band of EIT-based photonic crystal when the 1 - 3 transition of the EIT atomic levels is on resonance. It follows from Fig. 6(a) that after aligning dielectric GaAs side by side with EIT medium there are three extreme values of imaginary part *K* in the Bloch wave number (or *K kn*<sup>0</sup> r,(D|E) ). Coincidently, there are also three extreme values of real part *K* in the Bloch wave number (or *K kn*<sup>0</sup> r,(D|E) ). However, neither of them reaches the band edge 0 *K* or 0.5 *K* (in the units of 2 /

Note that *K* and *K* simultaneously exist, since the refractive index of the EIT medium has an imaginary part. It should be emphasized that the band structure (i.e., *K* and *K* vary as the probe frequency detuning **<sup>p</sup>** changes slightly) is very sensitive to the probe frequency detuning. From Fig. 6 (a) one can see that the real part of the Bloch wave number changes

small change, e.g., at the level of one part in <sup>8</sup> 10 in the probe frequency, gives rise to a large variation in the Bloch wave number). In particular, the slope ( / d d *K* <sup>p</sup> ) is almost divergent at the position / 0.5 **p 3** . The reason for this is because 0.5 **p 3** is exactly the two-

31 

<sup>1</sup> ), it would exhibit a band with a fine structure (and hence remarkable

<sup>1</sup> . Since, seen from Fig. 5 (a), there are some fine structures of the band in

5.0 10 s

m (EIT medium).

<sup>1</sup> , e.g., <sup>p</sup> is tuned onto resonance, i.e., p c that equals

<sup>1</sup> , and the thickness of the

*c* ), both the real and imaginary parts

) and the imaginary part changes from 0.5

) within a very narrow probe frequency band (namely, a very

*c* / with

*c* / having

).

choose the typical atomic transition frequency <sup>15</sup>

**p 3** / approaches almost zero compared with /

drastically from 0 to 0.5 (in the units of 2 /

to 0 (in the units of 2 /

m (GaAs dielectric) and 0.1 *b*

two layers 0.1 *a*

<sup>7</sup> 1.0 10 s

is <sup>7</sup> 2.0 10 **<sup>c</sup>** <sup>s</sup>

a band structure.

the order of magnitude <sup>15</sup> 10 s

photon resonant frequency ( **p c** ). As there is almost divergent dispersion close to 0.5 **p 3** , the effects of slow light and the negative group velocity in such an EIT-based periodic layered material deserve consideration. This would lead to promising applications in designing devices for slowing down light speed. Besides, the EIT-based band structure is tunable in response to the intensity (characterized by the Rabi frequency **<sup>c</sup>** ) of the external control field, since the refractive index of the EIT medium can be controlled by the control field. In Fig. 6 (b) the real part of the Bloch wave number *K* decreases as **<sup>c</sup>** increases from 0 to <sup>3</sup> 4 , and then increases when **<sup>c</sup>** / 4 <sup>3</sup> ; the absolute value of the imaginary part of the Bloch wave number increases first in the range **<sup>c</sup>** / [0,1] 3 and then decreases when **<sup>c</sup>** / 1 <sup>3</sup> . This, therefore, means that one can use one optical field to controllably manipulate the wave propagation of the other optical field via such an effect of sensitive switching control exhibited in the EIT-based periodic layered structure.

Fig. 5. The bandgap structure of the 1D infinite periodic (D|E) cells when the probe frequency of TE waves is far from the resonance. In (a) is the band structure in the probe frequency detuning range <sup>8</sup> / [ 2.5 10 , **p 3** <sup>8</sup> 2.5 10 ]. In (b), (c) and (d) are the fine details exhibited in the EIT-based band structure in the probe frequency detuning ranges (in units of **<sup>3</sup>** ), i.e., / **p 3** <sup>7</sup> [ 1.7 10 , <sup>7</sup> 2.0 10 ], 7 7 [ 11.0 10 , 1.0 10 ] and 8 8 [ 2.5 10 ,0.0 10 ] , respectively.

As we have shown the characteristics of both sensitivity and tunability of the EIT-based band structure in Fig. 5(b) and Fig. 6, we shall present its three-dimensional behavior as both the probe frequency detuning and the Rabi frequency of control field vary. The sensitivity and the tunability versus the probe frequency detuning **p** and the Rabi frequency **c** of control field, respectively, are shown in Fig. 7.

EIT-Based Photonic Crystals and Photonic Logic Gate Design 143

drastically in the frequency detuning range of concern. We plot in Fig. 8 the dispersive behavior of *r* in the range of <sup>p</sup> /<sup>3</sup> [0.3, 0.7], i.e., the probe frequency detuning changes

from about 0.25 to 0.95 and from about 0.25 to 0.40, respectively. As is expected, such a dramatic change in the coefficient of reflection results from the two-photon resonance (because of the destructive quantum interference between the 1 - 3 and 2 - 3 transitions). In general, the more layers there are in the dielectric-EIT cell structure, the more drastic change there would be in the reflection coefficient on the left-side interface of this EIT-based periodic layered medium. Thus, the total number of valleys and peaks in the curve of the reflection coefficient *r* in a narrow band close to 0.5 **p 3** becomes more and more as the total layer number *N* increases. However, such valleys and peaks in the reflection coefficient are no longer conspicuous for the cases of large *N* , since the amplitudes of fluctuation become smaller when the layer number *N* is adequately large. If, for example, the layer number 100 *N* , the small fluctuations tend to efface themselves

Fig. 8. The real and imaginary parts of the reflection coefficient *r* versus the normalized probe frequency detuning <sup>p</sup> /3 in the frequency range of two-photon resonance caused by the destructive quantum interference between the 1 - 3 and 2 - 3 transitions (close to 0.5 **p 3** ). The layer number of the EIT-based periodic medium *N* 1, 5, 20, 100 . The

We have demonstrated the probe *frequency-sensitive* behavior of the EIT-based periodic layered material. It can exhibit another effect (field-controlled *tunable* optical response), where the control field can be used to manipulate the photonic band structure, and therefore the reflection coefficient would vary as we tune the control Rabi frequency <sup>c</sup> . It follows from Fig. 9 that the tunable reflection coefficient of the EIT-based periodic layered medium

<sup>1</sup> .

Rabi frequency of the control field is chosen as <sup>7</sup> 2.0 10 **<sup>c</sup>** <sup>s</sup>

<sup>1</sup> ). It follows from Fig. 8 that the real and imaginary parts of *r* change

**<sup>p</sup>** (the typical value of the probe

at the level of one part in <sup>8</sup> 10 in the probe frequency

frequency <sup>15</sup> 10 **<sup>p</sup>** s

(see Fig. 8).

Fig. 6. The Bloch wave number *K* of the 1D infinite periodic (D|E) cells when the EIT atomic transition is on resonance. The curves in (a) indicate the real and imaginary parts of the normalized Bloch wave number *K* sensitive to the probe frequency detuning **<sup>p</sup>** , where the Rabi frequency of the control field is chosen as <sup>7</sup> 2.0 10 **<sup>c</sup>** <sup>s</sup> <sup>1</sup> . The curves in (b) show the tunable Bloch wave number *K* at the frequency detuning <sup>7</sup> 2.0 10 **<sup>p</sup>** s 1 when the Rabi frequency **c** of the control field changes.

Fig. 7. The real part (a) and the imaginary part (b) of the normalized Bloch wave number *K* (in the units of 2 / ) of the1D infinite periodic (D|E) cells versus **<sup>p</sup>** and **<sup>c</sup>** . Both the real and imaginary parts of the Bloch wave number *K* are sensitive to the small change in the probe frequency (the slope / d d *K* p of the dispersive curve is much more larger than that in a conventional photonic crystal), and both the real and imaginary parts of the Bloch wave number at any fixed probe frequencies can be controllable by the control Rabi frequency **<sup>c</sup>** .

### **5. Probe-frequency-sensitive and field-intensity-sensitive coherent control effects in an EIT-based periodic layered medium**

We shall now show that the reflection coefficient would be sensitive to the probe frequency when it is tuned onto two-photon resonance ( **p c** ). The typical atomic and optical parameters for the numerical results are chosen exactly the same as those used in the preceding sections. In Fig. 8, the real and imaginary parts of the reflection coefficient *r* corresponding to *N* -layer (D|E) cells are presented as an illustrative example, where the layer number 1, 5, 20, *N* 100 . It can be seen that the reflection coefficient changes 142 Photonic Crystals – Introduction, Applications and Theory

(a) (b)

(a) (b) Fig. 7. The real part (a) and the imaginary part (b) of the normalized Bloch wave number *K*

real and imaginary parts of the Bloch wave number *K* are sensitive to the small change in the probe frequency (the slope / d d *K* p of the dispersive curve is much more larger than that in a conventional photonic crystal), and both the real and imaginary parts of the Bloch wave number at any fixed probe frequencies can be controllable by the control Rabi

**5. Probe-frequency-sensitive and field-intensity-sensitive coherent control** 

We shall now show that the reflection coefficient would be sensitive to the probe frequency when it is tuned onto two-photon resonance ( **p c** ). The typical atomic and optical parameters for the numerical results are chosen exactly the same as those used in the preceding sections. In Fig. 8, the real and imaginary parts of the reflection coefficient *r* corresponding to *N* -layer (D|E) cells are presented as an illustrative example, where the layer number 1, 5, 20, *N* 100 . It can be seen that the reflection coefficient changes

**effects in an EIT-based periodic layered medium** 

) of the1D infinite periodic (D|E) cells versus **<sup>p</sup>** and **<sup>c</sup>** . Both the

<sup>1</sup> . The curves in (b) show

1 when the

Fig. 6. The Bloch wave number *K* of the 1D infinite periodic (D|E) cells when the EIT atomic transition is on resonance. The curves in (a) indicate the real and imaginary parts of the normalized Bloch wave number *K* sensitive to the probe frequency detuning **<sup>p</sup>** , where

the Rabi frequency of the control field is chosen as <sup>7</sup> 2.0 10 **<sup>c</sup>** <sup>s</sup>

Rabi frequency **c** of the control field changes.

(in the units of 2 /

frequency **<sup>c</sup>** .

the tunable Bloch wave number *K* at the frequency detuning <sup>7</sup> 2.0 10 **<sup>p</sup>** s

drastically in the frequency detuning range of concern. We plot in Fig. 8 the dispersive behavior of *r* in the range of <sup>p</sup> /<sup>3</sup> [0.3, 0.7], i.e., the probe frequency detuning changes at the level of one part in <sup>8</sup> 10 in the probe frequency **<sup>p</sup>** (the typical value of the probe frequency <sup>15</sup> 10 **<sup>p</sup>** s <sup>1</sup> ). It follows from Fig. 8 that the real and imaginary parts of *r* change from about 0.25 to 0.95 and from about 0.25 to 0.40, respectively. As is expected, such a dramatic change in the coefficient of reflection results from the two-photon resonance (because of the destructive quantum interference between the 1 - 3 and 2 - 3 transitions). In general, the more layers there are in the dielectric-EIT cell structure, the more drastic change there would be in the reflection coefficient on the left-side interface of this EIT-based periodic layered medium. Thus, the total number of valleys and peaks in the curve of the reflection coefficient *r* in a narrow band close to 0.5 **p 3** becomes more and more as the total layer number *N* increases. However, such valleys and peaks in the reflection coefficient are no longer conspicuous for the cases of large *N* , since the amplitudes of fluctuation become smaller when the layer number *N* is adequately large. If, for example, the layer number 100 *N* , the small fluctuations tend to efface themselves (see Fig. 8).

Fig. 8. The real and imaginary parts of the reflection coefficient *r* versus the normalized probe frequency detuning <sup>p</sup> /3 in the frequency range of two-photon resonance caused by the destructive quantum interference between the 1 - 3 and 2 - 3 transitions (close to 0.5 **p 3** ). The layer number of the EIT-based periodic medium *N* 1, 5, 20, 100 . The Rabi frequency of the control field is chosen as <sup>7</sup> 2.0 10 **<sup>c</sup>** <sup>s</sup> <sup>1</sup> .

We have demonstrated the probe *frequency-sensitive* behavior of the EIT-based periodic layered material. It can exhibit another effect (field-controlled *tunable* optical response), where the control field can be used to manipulate the photonic band structure, and therefore the reflection coefficient would vary as we tune the control Rabi frequency <sup>c</sup> . It follows from Fig. 9 that the tunable reflection coefficient of the EIT-based periodic layered medium

EIT-Based Photonic Crystals and Photonic Logic Gate Design 145

Fig. 10. The three-dimensional behavior of the reflectance of the EIT-based layered medium versus the normalized control Rabi frequency <sup>c</sup> /3 and the normalized probe frequency detuning <sup>p</sup> / <sup>3</sup> . All the atomic and optical parameters such as**<sup>31</sup>** , **Γ3** , **<sup>2</sup> γ** , **<sup>c</sup>** , and *N***<sup>a</sup>**

Fig. 11. The reflectance and transmittance versus the normalized Rabi frequency <sup>c</sup> / <sup>3</sup> of the control field. The probe frequency detuning is chosen as <sup>s</sup> 8 1 <sup>10</sup> <sup>p</sup> . All the atomic and optical parameters such as**<sup>31</sup>** , **Γ3** , **<sup>2</sup> γ** , **<sup>c</sup>** , *N***a** are chosen exactly the same as those in Fig. 8.

are chosen exactly the same as those in Figs. 8 and 9.

is also sensitive to the Rabi frequency of the control field when the total layer number *N* increases. This means that the incident probe signal is either reflected or transmitted depending quite sensitively on the intensity of the external control field (characterized by \* c c ), and therefore it could be used for designing some sensitive photonic devices (e.g., optical switches, photonic logic gates as well as tunable photonic transistors). In addition, a full controllability of reflection and transmission of the present EIT-based layered structure can also be demonstrated in Fig. 9. It can be readily seen that both the real and imaginary parts of the reflection coefficient *r* are less than 0.1, and hence the reflectance ( \* *R rr* ) approaches zero (or almost zero) when the normalized control Rabi frequency <sup>c</sup> / 3 is taken to be certain values, such as <sup>c</sup> / 6.0 3 (for 5 *N* ), <sup>c</sup> / 8.5 3 (for 20 *N* ) and <sup>c</sup> / 10, 20 3 (for 100 *N* ). Thus, a field-intensity-sensitive switchable mirror can be fabricated with the EIT-based layered structure having a large total layer number *N* (e.g., 100 *N* ). The three-dimensional behavior of the reflectance of the EIT-based periodic layered medium as both the Rabi frequency c and the probe frequency detuning <sup>p</sup> change is indicated in Fig. 10. Besides, we also consider the reflectance and transmittance of 1-, 5-, 20-, 100-layer periodic structures at other probe frequency detuning, e.g., <sup>s</sup> 8 1 <sup>10</sup> <sup>p</sup> in Fig. 11 as an illustrative example of tunable field-intensity-sensitive coherent control effect.

Fig. 9. The real and imaginary parts of the reflection coefficient *r* versus the normalized Rabi frequency <sup>c</sup> / <sup>3</sup> of the control field. The probe frequency detuning is <sup>7</sup> 2.0 10 <sup>p</sup> s <sup>1</sup> . All the atomic and optical parameters such as**<sup>31</sup>** , **Γ3** , **<sup>2</sup> <sup>γ</sup>** , **<sup>c</sup>** , *<sup>N</sup>***a** are chosen exactly the same as those in Fig. 8. In the case of 100 *N* , the reflection coefficient depends quite sensitively on the Rabi frequency of the control field.

is also sensitive to the Rabi frequency of the control field when the total layer number *N* increases. This means that the incident probe signal is either reflected or transmitted depending quite sensitively on the intensity of the external control field (characterized by \* c c ), and therefore it could be used for designing some sensitive photonic devices (e.g., optical switches, photonic logic gates as well as tunable photonic transistors). In addition, a full controllability of reflection and transmission of the present EIT-based layered structure can also be demonstrated in Fig. 9. It can be readily seen that both the real and imaginary parts of the reflection coefficient *r* are less than 0.1, and hence the reflectance ( \* *R rr* ) approaches zero (or almost zero) when the normalized control Rabi frequency <sup>c</sup> / 3 is taken to be certain values, such as <sup>c</sup> / 6.0 3 (for 5 *N* ), <sup>c</sup> / 8.5 3 (for 20 *N* ) and <sup>c</sup> / 10, 20 3 (for 100 *N* ). Thus, a field-intensity-sensitive switchable mirror can be fabricated with the EIT-based layered structure having a large total layer number *N* (e.g., 100 *N* ). The three-dimensional behavior of the reflectance of the EIT-based periodic layered medium as both the Rabi frequency c and the probe frequency detuning <sup>p</sup> change is indicated in Fig. 10. Besides, we also consider the reflectance and transmittance of 1-, 5-, 20-, 100-layer periodic structures at other probe frequency detuning, e.g., <sup>s</sup> 8 1 <sup>10</sup> <sup>p</sup> in Fig. 11 as an illustrative example of tunable field-intensity-sensitive

Fig. 9. The real and imaginary parts of the reflection coefficient *r* versus the normalized

chosen exactly the same as those in Fig. 8. In the case of 100 *N* , the reflection coefficient

<sup>1</sup> . All the atomic and optical parameters such as**<sup>31</sup>** , **Γ3** , **<sup>2</sup> <sup>γ</sup>** , **<sup>c</sup>** , *<sup>N</sup>***a** are

Rabi frequency <sup>c</sup> / <sup>3</sup> of the control field. The probe frequency detuning

depends quite sensitively on the Rabi frequency of the control field.

coherent control effect.

is <sup>7</sup> 2.0 10 <sup>p</sup> s

Fig. 10. The three-dimensional behavior of the reflectance of the EIT-based layered medium versus the normalized control Rabi frequency <sup>c</sup> /3 and the normalized probe frequency detuning <sup>p</sup> / <sup>3</sup> . All the atomic and optical parameters such as**<sup>31</sup>** , **Γ3** , **<sup>2</sup> γ** , **<sup>c</sup>** , and *N***<sup>a</sup>** are chosen exactly the same as those in Figs. 8 and 9.

Fig. 11. The reflectance and transmittance versus the normalized Rabi frequency <sup>c</sup> / <sup>3</sup> of the control field. The probe frequency detuning is chosen as <sup>s</sup> 8 1 <sup>10</sup> <sup>p</sup> . All the atomic and optical parameters such as**<sup>31</sup>** , **Γ3** , **<sup>2</sup> γ** , **<sup>c</sup>** , *N***a** are chosen exactly the same as those in Fig. 8.

EIT-Based Photonic Crystals and Photonic Logic Gate Design 147

quantum field theory (Calogeracos & Dombey, 1999). In the Klein paradox, the relativistic wave equation can lead to so-called "negative probabilities" induced by certain energy potentials (e.g., the strong repulsive potential barrier with height exceeding the rest energy of particle) (Calogeracos & Dombey, 1999). Such a paradox can be interpreted based on the mechanism of particle-antiparticle pair production, which gives rise to higher-than-unity reflectance and negative transmittance. The Klein tunneling has been expected to be observed in QED regime, where an incoming electron wave function propagates and penetrates through a sufficiently high potential barrier. Though such a counterintuitive effect of relativistic quantum tunneling can be explained by using the notion of creation of electron-positron pairs, which is a physical process at the potential discontinuity, even today it is still referred to as "Klein paradox" in order to indicate its anomalous tunneling characteristics. Since the electron is massive, it is in fact quite difficult to realize the exotic Klein tunneling experimentally. Here, we shall suggest an alternative way to realize this intriguing effect, i.e., the photonic analog of Klein tunneling in an LHM-EIT-based periodic layered medium, where the reflection coefficient exceeding unity will also occurs in some

The 1D periodic LHM-EIT cells are embedded in a left-handed homogeneous dielectric (an LHM-EIT cell consists of a left-handed dielectric and an EIT atomic medium). Fig. 12 indicates the band structure of the 1D infinite periodic LHM-EIT cells (sketched in Fig. 4) when the TM wave of the probe beam whose magnetic field vector is perpendicular to the xz plane (Yeh, 2005) is incident normally or obliquely on such a periodic layered medium. Here we also choose the typical atomic ( 1 - 3 ) transition frequency <sup>15</sup>

Fig. 12. The band structure of the 1D infinite periodic LHM-EIT cells when the angles of

incidence of the TM wave of the probe beam are 0 , 15 , 30 , *ooo*

medium). The thickness of one LHM-EIT cell is *a b* . We plot in Fig. 12 the dispersive

75*<sup>o</sup>* , respectively). The tunable Rabi frequency c of the control field chosen for the present

31 

<sup>i</sup> are 0 , 15 , 30 , *ooo* 45 , 60 , *o o* and

*<sup>i</sup>* 45 , 60 , 75 *ooo* , respectively.

m (left-handed dielectric) and 0.1 *b*

5.0 10 s

m (EIT

<sup>1</sup> ,

frequency ranges, and this will lead to a negative transmittance.

behavior of six typical cases (i.e., the angles of incidence

and the thickness of the two layers 0.1 *a*

<sup>1</sup> .

scheme is <sup>7</sup> 2.0 10 s

It should be noted that the probe frequency detuning <sup>p</sup> does not equal the frequency detuning <sup>c</sup> of the control field in Figs. 9-11, which are some typical cases for exhibiting general optical behavior of EIT-based photonic crystals. The quantum interference between atomic transitions (particularly when the condition of two-photon resonance, c p , is fulfilled) can give rise to a strong dispersion that is tunable by the external control field (characterized by the Rabi frequency <sup>c</sup> ). The structure of the EIT-based photonic crystal can thus be designed by taking advantage of such an effect of quantum coherence. We expect that the present probe-frequency-sensitive and field-intensity-sensitive coherent control effect with an EIT-based periodic layered structure can be used as a fundamental mechanism for designs and fabrications of new quantum optical and photonic devices.
