**6. The frequency-sensitive tunable band structure of TM wave**

In the preceding two sections we have studied the periodic structure composed of an EIT medium and a normal dielectric (i.e., right-handed material). As a left-handed material (LHM) can exhibit unusual electromagnetic properties (Veselago, 1968), we shall now demonstrate how the layer structure of 1D photonic crystal consisting of the EIT vapor layers and the LHM host dielectric layers can show extraordinary sensitivity to the frequency of the probe field. As the band structure for TM wave seems to be more sensitive to the frequency than that for TE wave (Yeh, 2005), in this section we shall focus our attention on the optical response (e.g., frequency-sensitive band structure induced by two-photon resonance and higher-than-unity reflection coefficients due to the Klein tunneling) of TM wave.

As is well known, the Maxwell curl equations show that the phase velocity of light wave propagating inside a left-handed medium is pointed opposite to the direction of energy flow, that is, the Poynting vector and the wave vector of electromagnetic wave would be anti-parallel (i.e., its wave vector **k**, electric field **E** and magnetic field **H** form a left-handed system). There have been some schemes to achieve the left-handed materials in the literature (Veselago, 1968; Shelby et al., 2001; Pendry et al., 1998; Pendry et al., 1996). Note that a righthanded system can be changed into a left-handed one via the operation of mirror reflection. It is thus clearly seen that the permittivity and the permeability of the free vacuum in a mirror world would be negative numbers. We have therefore pointed out that the electromagnetic wave (or a photon field) propagating inside a left-handed medium behaves like a wave of "antiparticle" of photon (Shen, 2003; Shen, 2008). However, as we know, there exist no such "antiphotons" in nature. The theoretical reason for this is that the fourdimensional electromagnetic vector potentials *A* with 0,1,2,3 are always taking the real numbers. But in a dispersive and absorptive medium, one can utilize an effective medium theory, where the vector potentials *A* could probably take complex numbers. Such complex vector field theory has been considered previously (Lurié, 1968). The Lagrangian density of a complex electromagnetic field is given by \* *F F* / 2 . The complex four-dimensional vector potentials characterize the propagating behavior of both photons and "antiphotons", and hence both the electromagnetic wave characteristics in leftand right-handed media can be treated in a unified framework.

If the light quanta in a medium of negative refractive index can be considered to be the "antiparticles" of photons, it is of interest to propose an optical (or photonic) analog of the well-known Klein paradox, which appears in regimes of relativistic quantum mechanics and

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.

In the preceding two sections we have studied the periodic structure composed of an EIT medium and a normal dielectric (i.e., right-handed material). As a left-handed material (LHM) can exhibit unusual electromagnetic properties (Veselago, 1968), we shall now demonstrate how the layer structure of 1D photonic crystal consisting of the EIT vapor layers and the LHM host dielectric layers can show extraordinary sensitivity to the frequency of the probe field. As the band structure for TM wave seems to be more sensitive to the frequency than that for TE wave (Yeh, 2005), in this section we shall focus our attention on the optical response (e.g., frequency-sensitive band structure induced by two-photon resonance and higher-than-unity

As is well known, the Maxwell curl equations show that the phase velocity of light wave propagating inside a left-handed medium is pointed opposite to the direction of energy flow, that is, the Poynting vector and the wave vector of electromagnetic wave would be anti-parallel (i.e., its wave vector **k**, electric field **E** and magnetic field **H** form a left-handed system). There have been some schemes to achieve the left-handed materials in the literature (Veselago, 1968; Shelby et al., 2001; Pendry et al., 1998; Pendry et al., 1996). Note that a righthanded system can be changed into a left-handed one via the operation of mirror reflection. It is thus clearly seen that the permittivity and the permeability of the free vacuum in a mirror world would be negative numbers. We have therefore pointed out that the electromagnetic wave (or a photon field) propagating inside a left-handed medium behaves like a wave of "antiparticle" of photon (Shen, 2003; Shen, 2008). However, as we know, there exist no such "antiphotons" in nature. The theoretical reason for this is that the four-

real numbers. But in a dispersive and absorptive medium, one can utilize an effective

Such complex vector field theory has been considered previously (Lurié, 1968). The Lagrangian density of a complex electromagnetic field is given by \* *F F* / 2

complex four-dimensional vector potentials characterize the propagating behavior of both photons and "antiphotons", and hence both the electromagnetic wave characteristics in left-

If the light quanta in a medium of negative refractive index can be considered to be the "antiparticles" of photons, it is of interest to propose an optical (or photonic) analog of the well-known Klein paradox, which appears in regimes of relativistic quantum mechanics and

 with 0,1,2,3 

are always taking the

 . The

could probably take complex numbers.

**6. The frequency-sensitive tunable band structure of TM wave** 

reflection coefficients due to the Klein tunneling) of TM wave.

dimensional electromagnetic vector potentials *A*

medium theory, where the vector potentials *A*

and right-handed media can be treated in a unified framework.

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

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> 31 5.0 10 s <sup>1</sup> , and the thickness of the two layers 0.1 *a* m (left-handed dielectric) and 0.1 *b* m (EIT medium). The thickness of one LHM-EIT cell is *a b* . We plot in Fig. 12 the dispersive behavior of six typical cases (i.e., the angles of incidence<sup>i</sup> are 0 , 15 , 30 , *ooo* 45 , 60 , *o o* and 75*<sup>o</sup>* , respectively). The tunable Rabi frequency c of the control field chosen for the present scheme is <sup>7</sup> 2.0 10 s <sup>1</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 <sup>i</sup>* 45 , 60 , 75 *ooo* , respectively.

EIT-Based Photonic Crystals and Photonic Logic Gate Design 149

normally on the periodic layered structure. It can be seen that both the real and imaginary parts of the reflection coefficient *r* in all the cases (i.e., the layer number 1 6 *N* ) change drastically as the probe frequency is close to the resonant frequency (at 0.5 **p 3** , where the probe frequency is tuned onto the two-photon resonance of the EIT atomic system).

As the layered medium contains the left-handed layers, and the periodic EIT layers act as a potential barrier for the incident electromagnetic wave, the absolute values of the real or imaginary part of the reflection coefficient *r* in some frequency ranges is larger than unity because of the Klein tunneling. In order to show the exotic and counterintuitive features exhibited in Fig. 13, we will present the behavior of reflection of TM wave by a RHM-EITbased periodic layered structure, in which the left-handed layers have been replaced with the right-handed medium (or vacuum), for comparison. In the reflection coefficient of the *N* -layer RHM-EIT cells, in which the relative refractive index of the right-handed medium is <sup>1</sup> *n* 1 , is shown in Fig. 14. In this case, both the host dielectric layers and the EIT layers are right-handed media, so that there is no Klein tunneling, i.e., the absolute values of both the real and imaginary parts of the reflection coefficient*r* are less than unity. This, therefore, means that the left-handed dielectric is requisite in order to achieve the

unusual photonic tunneling in the periodic layer structure containing EIT medium.

Fig. 14. The real and imaginary parts of the reflection coefficient *r* corresponding to the *N* layer LHM-EIT cells ( 1 6 *N* ), where the relative refractive index of the right-handed medium is <sup>1</sup> *n* 1 . There is no photonic Klein tunneling, i.e., the absolute values of both the

We have shown that the LHM-EIT-based periodic medium can give rise to extraordinary reflection and transmission. Now we shall consider the physical meanings of such a photonic analog of Klein tunneling as well as its photonic application to device design.

real and imaginary parts of the reflection coefficient *r* are less than unity.

**7. A potential application: Photonic transistors and logic gates** 

As the refractive index of the EIT medium has an imaginary part in the frequency range of concern (see Fig. 12), the real part Re( ) *K* and the imaginary part Im( ) *K* of the Bloch wave number simultaneously exist. We emphasize that the band structure (i.e., *K* vary dramatically as the probe frequency detuning **<sup>p</sup>** changes slightly) is very sensitive to the probe frequency detuning. It follows that in a very narrow frequency band, where **p** is close to the resonance position, namely, the EIT two-photon resonance occurs ( **p c** , i.e., **<sup>p</sup>** / 0.5 <sup>3</sup> ), both the real and imaginary parts of the Bloch wave number change drastically in a wide range, e.g., [0, 0.5] (in the units of 2 / ) for Re( ) *K* and [ 0.5 , +0.5] forIm( ) *K* . Since the EIT two-photon resonance arises at **p c** with the control frequency detuning <sup>7</sup> 1.0 10 **<sup>c</sup>** <sup>s</sup> 1 , and the probe transition frequency 15 31 5.0 10 s <sup>1</sup> , a very small change (e.g., at the level of one part in <sup>8</sup> 10 ) in the probe frequency would result in a large variation in the Bloch wave number. For this reason, the slope ( / d d *K* <sup>p</sup> ) of the Bloch dispersive curves is almost divergent at the position / 0.5 **p 3** . Since there is strong dispersion in the curves of Bloch wave number *K* in the vicinity of 0.5 **p 3** , the effects of slow light as well as the negative group velocity would arise in the present periodic layered material.

Fig. 13. The real and imaginary parts of the reflection coefficient *r* corresponding to the *N* layer LHM-EIT cells ( 1 6 *N* ), where the relative refractive index of the left-handed medium is <sup>1</sup> *n* 1 . The photonic Klein tunneling occurs, i.e., in some frequency ranges the absolute values of both the real and imaginary parts of the reflection coefficient *r* are larger than unity.

We are now in a position to address the problem of reflection and transmission of the present photonic crystal. Obviously, the reflectance and transmittance would also be sensitive to the probe frequency when it is tuned onto resonance ( **p c** ). In Fig. 13 the real and imaginary parts of the reflection coefficient *r* corresponding to the *N* -layer LHM-EIT cells are presented as an illustrative example, where the layer number 1 6 *N* and 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 TM wave is incident

As the refractive index of the EIT medium has an imaginary part in the frequency range of concern (see Fig. 12), the real part Re( ) *K* and the imaginary part Im( ) *K* of the Bloch wave number simultaneously exist. We emphasize that the band structure (i.e., *K* vary dramatically as the probe frequency detuning **<sup>p</sup>** changes slightly) is very sensitive to the probe frequency detuning. It follows that in a very narrow frequency band, where **p** is close to the resonance position, namely, the EIT two-photon resonance occurs ( **p c** , i.e., **<sup>p</sup>** / 0.5 <sup>3</sup> ), both the real and imaginary parts of the Bloch wave number change

forIm( ) *K* . Since the EIT two-photon resonance arises at **p c** with the control frequency

small change (e.g., at the level of one part in <sup>8</sup> 10 ) in the probe frequency would result in a large variation in the Bloch wave number. For this reason, the slope ( / d d *K* <sup>p</sup> ) of the Bloch dispersive curves is almost divergent at the position / 0.5 **p 3** . Since there is strong dispersion in the curves of Bloch wave number *K* in the vicinity of 0.5 **p 3** , the effects of slow light as well as the negative group velocity would arise in the present periodic layered

Fig. 13. The real and imaginary parts of the reflection coefficient *r* corresponding to the *N* layer LHM-EIT cells ( 1 6 *N* ), where the relative refractive index of the left-handed medium is <sup>1</sup> *n* 1 . The photonic Klein tunneling occurs, i.e., in some frequency ranges the absolute values of both the real and imaginary parts of the reflection coefficient *r* are larger than unity.

We are now in a position to address the problem of reflection and transmission of the present photonic crystal. Obviously, the reflectance and transmittance would also be sensitive to the probe frequency when it is tuned onto resonance ( **p c** ). In Fig. 13 the real and imaginary parts of the reflection coefficient *r* corresponding to the *N* -layer LHM-EIT cells are presented as an illustrative example, where the layer number 1 6 *N* and the

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

1 , and the probe transition frequency 15

) for Re( ) *K* and [ 0.5 , +0.5]

<sup>1</sup> , a very

5.0 10 s

<sup>1</sup> . The TM wave is incident

31 

drastically in a wide range, e.g., [0, 0.5] (in the units of 2 /

detuning <sup>7</sup> 1.0 10 **<sup>c</sup>** <sup>s</sup>

material.

normally on the periodic layered structure. It can be seen that both the real and imaginary parts of the reflection coefficient *r* in all the cases (i.e., the layer number 1 6 *N* ) change drastically as the probe frequency is close to the resonant frequency (at 0.5 **p 3** , where the probe frequency is tuned onto the two-photon resonance of the EIT atomic system).

As the layered medium contains the left-handed layers, and the periodic EIT layers act as a potential barrier for the incident electromagnetic wave, the absolute values of the real or imaginary part of the reflection coefficient *r* in some frequency ranges is larger than unity because of the Klein tunneling. In order to show the exotic and counterintuitive features exhibited in Fig. 13, we will present the behavior of reflection of TM wave by a RHM-EITbased periodic layered structure, in which the left-handed layers have been replaced with the right-handed medium (or vacuum), for comparison. In the reflection coefficient of the *N* -layer RHM-EIT cells, in which the relative refractive index of the right-handed medium is <sup>1</sup> *n* 1 , is shown in Fig. 14. In this case, both the host dielectric layers and the EIT layers are right-handed media, so that there is no Klein tunneling, i.e., the absolute values of both the real and imaginary parts of the reflection coefficient*r* are less than unity. This, therefore, means that the left-handed dielectric is requisite in order to achieve the unusual photonic tunneling in the periodic layer structure containing EIT medium.

Fig. 14. The real and imaginary parts of the reflection coefficient *r* corresponding to the *N* layer LHM-EIT cells ( 1 6 *N* ), where the relative refractive index of the right-handed medium is <sup>1</sup> *n* 1 . There is no photonic Klein tunneling, i.e., the absolute values of both the real and imaginary parts of the reflection coefficient *r* are less than unity.

#### **7. A potential application: Photonic transistors and logic gates**

We have shown that the LHM-EIT-based periodic medium can give rise to extraordinary reflection and transmission. Now we shall consider the physical meanings of such a photonic analog of Klein tunneling as well as its photonic application to device design.

EIT-Based Photonic Crystals and Photonic Logic Gate Design 151

Fig. 15. The reflectance and transmittance of 1-layer and 2-layer LHM-EIT structures (the relative refractive index of the left-handed medium <sup>1</sup> *n* 1 ) in the probe frequency

Fig. 16. The reflectance and transmittance of 1-layer and 2-layer LHM-EIT structures (the relative refractive index of the left-handed medium <sup>1</sup> *n* 1 ) in the probe frequency

As we have pointed out in the preceding sections, the sensitive optical switching control can also be utilized to design photonic logic gates by means of such an EIT-based periodic layered structure. For example, the incident probe beam and the applied control field can act as the two input signals. We suppose that the output signal *Y* 1 if the probe field can propagate through the periodic layered medium, and the output signal 0 *Y* if the probe beams cannot be transmitted through the structure (i.e., the reflection and absorption dominates in the probe wave propagation). Then the logic operations of two-input AND gate can be implemented with such a layered structure. Alternatively, we can also apply at least two probe waves at different wavelengths, which correspond to different transmittances. In this new scenario, the two incident probe beams of different frequencies can stand for the two input signals (see Figs. 17 and 18). Therefore, the functional and logic gates, such as NAND, NOR, EXOR and EXNOR gates, can be designed by taking advantage of the effect of sensitive control for optical switching (due to two-photon resonance of EIT)

exhibited in such an EIT-based periodic layered medium.

range <sup>p</sup> / 3,3 <sup>3</sup> .

range <sup>p</sup> / 0,1 <sup>3</sup> .

The reflectance *R* and transmittance *T* on the left interface of the LHM-EIT structures are given in Figs. 15 and 16 as an illustrative example. It follows that the reflectance (and transmittance) is quite sensitive to the probe frequency detuning in some frequency ranges. It seems that the reflected wave intensity is larger than the incident intensity. This is because the additional particles are supplied by the potential barrier (Klein tunneling). Correspondingly, the transmitted wave is opposite in the intensity to the incident wave. In this process, the conservation of both the energy and the photon number is guaranteed. For example, the current density of the complex electromagnetic field can be defined as \* \* *J AF AF* ( ) i whose four-dimensional divergence is given by

$$\left[\hat{\boldsymbol{\mathcal{O}}}^{\nu}\right]\_{\nu} = \mathbf{i}\hat{\boldsymbol{\mathcal{O}}}^{\nu}\left(\mathbf{A}\_{\mu}\mathbf{F}\_{\nu}{}^{\mu}{}^{\*} - \mathbf{A}\_{\mu}^{\*}{}\_{\nu}\mathbf{F}\_{\nu}{}^{\mu}\right) = \mathbf{i}(\hat{\boldsymbol{\mathcal{O}}}^{\nu}\mathbf{A}\_{\mu})\mathbf{F}\_{\nu}{}^{\mu}{}^{\*} - \mathbf{i}(\hat{\boldsymbol{\mathcal{O}}}^{\nu}\mathbf{A}\_{\mu}^{\*})\mathbf{F}\_{\nu}{}^{\mu}{}\_{\nu} \tag{14}$$

The above equation can be rewritten as \*\* \* ( )( ) *F AF F A F* i i \* \* () ( ) *AF A F* i i , where the electromagnetic field equations \* *F* 0 , *F* 0 have been employed. With the help of the electromagnetic field equations, one can arrive at \* \* *J AF A F* ( ) i . Thus, the four-dimensional divergence of current density is

$$\left[\hat{\boldsymbol{\mathcal{O}}}^{\boldsymbol{\nu}}\right]\_{\boldsymbol{\nu}} = \mathbf{i} \partial\_{\boldsymbol{v}} \left(\mathbf{A}^{\mu} \boldsymbol{F}\_{\mu}{}^{\boldsymbol{\nu}\*} - \mathbf{A}^{\mu\*} \boldsymbol{F}\_{\mu}{}^{\boldsymbol{\nu}}\right) \cdot \tag{15}$$

It follows from Eqs. (14) and (15) that 0 *J* (this means that the current density of the complex electromagnetic field obeys the law of conservation). If the Hermitian field operator *A*can be written as

$$\begin{split} &A\_{\mu} - \int \mathrm{d}^{3}k \Big[ a\_{(\lambda)}(\mathbf{k},t) \mathrm{e}^{\mathrm{i}k \cdot x} + b\_{(\lambda)}^{+}(\mathbf{k},t) \mathrm{e}^{\mathrm{i}k \cdot x} \Big] e\_{\mu}^{(\lambda)} \, \mathrm{} \\ &A\_{\mu}^{+} - \int \mathrm{d}^{3}k \Big[ a\_{(\lambda)}^{+}(\mathbf{k},t) \mathrm{e}^{\mathrm{i}k \cdot x} + b\_{(\lambda)}(\mathbf{k},t) \mathrm{e}^{\mathrm{i}k \cdot x} \Big] e\_{\mu}^{(\lambda)} \, \mathrm{} \end{split} \tag{16}$$

where ( ) *e* denotes the polarization vector, the "Noether charge" corresponding to the current density *J*of the complex electromagnetic field is given by

$$\int \mathrm{d}^3 x J\_0 \sim \int \mathrm{d}^3 k \sum\_{\lambda=0}^3 \left[ a^+\_{(\lambda)}(\mathbf{k}, t) a\_{(\lambda)}(\mathbf{k}, t) - b^+\_{(\lambda)}(\mathbf{k}, t) b\_{(\lambda)}(\mathbf{k}, t) \right]. \tag{17}$$

Here, ( ) *a t* ( ,) k , ( ) *a t* ( ,) <sup>k</sup> , ( ) *b t* ( ,) k , ( ) *b t* ( ,) k stand for the annihilation and creation operators of photons and its "antiparticles", respectively. The term () () *a ta t* ( ,) ( ,) k k in Eq. (17) is the total number of photons, while () () *b tb t* ( ,) ( ,) k k is the total number of the "antiparticle" of photon. It can be seen that the total number of the "antiparticle" is negative.

The photonic analog of the Klein tunneling presented here can be used to design the so-called frequency-sensitive photonic transistors (see Fig. 17(a) for a schematic diagram) that can switch the photonic signals: specifically, the incident probe beam, the reflected probe beam, and the transmitted probe beam can mimic the operation of the three terminals (i.e., base, collector and emitter, respectively) of a bipolar transistor (a semiconductor device for amplifying and switching electronic signals). A small intensity of probe beam at the base terminal can manipulate (or switch) a much larger intensity between the terminals of collector (reflected probe beam) and the emitter (transmitted probe beam), since the incident probe beam can control the reflected wave in proportion to the input signal (incident probe beam).

The reflectance *R* and transmittance *T* on the left interface of the LHM-EIT structures are given in Figs. 15 and 16 as an illustrative example. It follows that the reflectance (and transmittance) is quite sensitive to the probe frequency detuning in some frequency ranges. It seems that the reflected wave intensity is larger than the incident intensity. This is because the additional particles are supplied by the potential barrier (Klein tunneling). Correspondingly, the transmitted wave is opposite in the intensity to the incident wave. In this process, the conservation of both the energy and the photon number is guaranteed. For example, the current density of the complex electromagnetic field can be defined

\* \* \* \* *J AF AF A F A F* ( )( ) ( )

have been employed. With the help of the electromagnetic field equations, one can arrive

i . Thus, the four-dimensional divergence of current density is

\* \* ( ) *<sup>v</sup> J AF A F*

complex electromagnetic field obeys the law of conservation). If the Hermitian field operator

( ) ( )

de e

*A ka t b t e*

*A ka t b t e*

de e

0 () () () ()

"antiparticle" of photon. It can be seen that the total number of the "antiparticle" is negative. The photonic analog of the Klein tunneling presented here can be used to design the so-called frequency-sensitive photonic transistors (see Fig. 17(a) for a schematic diagram) that can switch the photonic signals: specifically, the incident probe beam, the reflected probe beam, and the transmitted probe beam can mimic the operation of the three terminals (i.e., base, collector and emitter, respectively) of a bipolar transistor (a semiconductor device for amplifying and switching electronic signals). A small intensity of probe beam at the base terminal can manipulate (or switch) a much larger intensity between the terminals of collector (reflected probe beam) and the emitter (transmitted probe beam), since the incident probe beam can control the reflected wave in proportion to the input signal

*xJ k a t a t b t b t* ( ,) ( ,) ( ,) ( ,) 

of the complex electromagnetic field is given by

operators of photons and its "antiparticles", respectively. The term () () *a ta t* ( ,) ( ,)

3 ( )

k k

k k

3 ( ) ( ) ( )

The above equation can be rewritten as \*\* \* ( )( ) *F AF F A F*

 

> 

 

( ,) ( ,) ,

i -i

 

> 

denotes the polarization vector, the "Noether charge" corresponding to the


*k x k x*

( ,) ( ,) ,

*k x k x*

 

 

d d <sup>k</sup> <sup>k</sup> <sup>k</sup> <sup>k</sup> . (17)

 

i ii . (14)

 

i i

   

 

, *F* 0

 

 

 

(16)

 

 

i . (15)

(this means that the current density of the

k stand for the annihilation and creation

k k is the total number of the

 

k k in Eq.

 

 

 

i whose four-dimensional divergence is given by

 

i i , where the electromagnetic field equations \* *F* 0

 

3 3

 <sup>k</sup> , ( ) *b t* ( ,) 

3

(17) is the total number of photons, while () () *b tb t* ( ,) ( ,)

0

 k , ( ) *b t* ( ,) 

as \* \* *J AF AF* ( ) 

\* \* () ( ) *AF A F*

at \* \* *J AF A F* ( )

 k , ( ) *a t* ( ,) 

can be written as

 

 

*A*

where ( ) *e* 

current density *J*

Here, ( ) *a t* ( ,) 

(incident probe beam).

 

 

 

It follows from Eqs. (14) and (15) that 0 *J*

   

Fig. 15. The reflectance and transmittance of 1-layer and 2-layer LHM-EIT structures (the relative refractive index of the left-handed medium <sup>1</sup> *n* 1 ) in the probe frequency range <sup>p</sup> / 3,3 <sup>3</sup> .

Fig. 16. The reflectance and transmittance of 1-layer and 2-layer LHM-EIT structures (the relative refractive index of the left-handed medium <sup>1</sup> *n* 1 ) in the probe frequency range <sup>p</sup> / 0,1 <sup>3</sup> .

As we have pointed out in the preceding sections, the sensitive optical switching control can also be utilized to design photonic logic gates by means of such an EIT-based periodic layered structure. For example, the incident probe beam and the applied control field can act as the two input signals. We suppose that the output signal *Y* 1 if the probe field can propagate through the periodic layered medium, and the output signal 0 *Y* if the probe beams cannot be transmitted through the structure (i.e., the reflection and absorption dominates in the probe wave propagation). Then the logic operations of two-input AND gate can be implemented with such a layered structure. Alternatively, we can also apply at least two probe waves at different wavelengths, which correspond to different transmittances. In this new scenario, the two incident probe beams of different frequencies can stand for the two input signals (see Figs. 17 and 18). Therefore, the functional and logic gates, such as NAND, NOR, EXOR and EXNOR gates, can be designed by taking advantage of the effect of sensitive control for optical switching (due to two-photon resonance of EIT) exhibited in such an EIT-based periodic layered medium.

EIT-Based Photonic Crystals and Photonic Logic Gate Design 153

(4-layer OR gate)

*Y A B* (6-layer NAND gate)

<sup>1</sup> .

*<sup>A</sup>* IN *<sup>B</sup>* IN *Y AB*

0 ( 0.46**<sup>3</sup>** ) 0 ( 0.46**<sup>3</sup>** ) 0 1 0 ( 0.46**<sup>3</sup>** ) 1 ( 0.53**<sup>3</sup>** ) 1 1 1 ( 0.53**<sup>3</sup>** ) 0 ( 0.46**<sup>3</sup>** ) 1 1 1 ( 0.53**<sup>3</sup>** ) 1 ( 0.53**<sup>3</sup>** ) 1 0

Table 1. The truth table of two-input OR gate (fabricated based on the 4-layer periodic structure) and two-input NAND gate (fabricated based on the 6-layer periodic structure).

Fig. 18. The fine structure of the reflectance and transmittance of the 4-layer and 6-layer periodic (D|E) cells in a narrow probe frequency band (a), and the schematic diagram of

The quantum optical properties of an EIT medium has been discussed (in Section 2), and the formalism for treating wave propagation in a periodic structure has been reviewed (in Section 3). The band structure and the reflectance of a 1D photonic crystal consisting of both EIT medium layers and host dielectric layers can show extraordinary sensitivity to the frequency of a probe field because of a two-photon resonance relevant to destructive quantum interference between two transition pathways driven by the control and probe fields (in Sections 4 and 5). Such an EIT-based periodic layered material can also exhibit an effect of field-intensity-sensitive switching control (depending quite sensitively on the Rabi frequency of the control field) in the cases of large layer number *N* . Since the optical responses can be controlled by the tunable quantum interference induced by the external control field via two-photon resonance, the EIT-based layered medium under consideration shows more flexible optical responses than conventional photonic crystals because of the EIT two-photon resonance that gives rise to strong dispersion in the band of transparency

photonic logic gates (b). The Rabi frequency of the control field is <sup>7</sup> 4.0 10 <sup>c</sup> s

(a) (b)

**9. Conclusions** 

window.

Fig. 17. (a) The schematic diagram of a photonic transistor designed based on the LHM-EIT layered structure. The probe field is incident on the structure, and the giant reflected wave with higher-than-unity reflectance and the transmitted wave with negative transmittance will be produced via the intriguing Klein tunneling effect. The incident probe wave, the reflected wave, and the transmitted wave correspond to the terminals of base, collector and emitter, respectively.

(b) The schematic diagram of a two-input photonic logic gate designed based on the EITbased layered structure. The two incident probe beams at different frequencies represent the two input signals.
