Conclusion

125

126

effect are developed (in Chapter 5). The voltage source on the left side of the two-port network is analog to the probe field on the regarded EIT material. The capacitors (C C ) ₁, ₂ inside the RLC-resonant circuit is corresponding to two Rabi frequencies (the intensity control fieldsemployed between | 2 | 3 and | 2 | 3 respectively). The total capacitive susceptance is relevant to electrical dipole which is calculated through the square of off-diagonal density matrix element (i.e.,₃₁) and achieved the relative electric permittivity of the EIT vapor at probe frequency through equation (2-12). Therefore, the Autler-Townes doublet and triplet in EIT transparency windows can be inspected by any kind of response of the EIT-Like circuit through total current passing through, total impedance, output voltage, output current,..etc. All the responding parameters have the Autler-Townes doublet and triplet allocation defined by the resonant frequency points (valley/peak of responding curve).

The present scheme can be generalized to the cases of four-level EIT systems, where two control fields and one probe field drive the atomic level transitions[37],[38],[41],[63]. Obviously, the optical response in such a four-level EIT-based photonic crystal would be more sensitive to the probe frequency than in a three-level EIT photonic crystal presented in this paper. Apart from this intriguing property, there are also interesting applications based on the four-level EIT photonic crystal, e.g., some examples of photonic devices (e.g., multi-input logic gates), in which the control fields and the transmitted probe field act as the multi-input and output signals, respectively, can be designed. We expect that all these new optical properties relevant to quantum coherence, including their applications to photonic devices, could be realized experimentally in the near future.

127

Appendix A The steps to find t

21

and s

21

Fig. A-1 The capacitive susceptance relevant to bulk-EITcan analogue to the dipole due to the transition of electron from the ground state to excited state inside the atomic system. The transmission voltage reached the maximum point while the susceptance reached the minimum allocation (resonant frequency, i.e., the power dip allocation for EIT material)

step 1: Findt ：Use the voltage division law ₂₁

    

^{p p}



^{p p 1 p}

o

21 2 2

in p p p p p p p p

with angle tan

t R R R X

t V

V t R jX R jX R X R



 ^

  

          ,

Step 2: Substitute _p ^p _p ^p

o o

R , X

r x

Z Z

  into ₂₁ ^p

p p

t R

R jX

  , we get

p 21

p p

t r

r jx

 

step 3: Find s ：Refer to reference [104],[105] ₂₁

(a) ，

(b) ，

(c) ₂₁ ^{2 2 2}

1 1 1

50 50

b V I

s a V I

  

 ，from Fig. A-1，

Let _p ^p _p ^p

o o

R , X

r x

Z Z

  ，then

  

^*

 ^{ }  

1 1 50 1 / 2 o1 o1 1 50 1 / 10 2 , when o1 50

a  V  I Z Z  V  I Z 

  

^*

 ^{ }  

2 2 50 2 / 2 o2 o2 2 50 2 / 10 2 , when o2 50

b  V  I Z Z  V  I Z 

128

   

^{p p}

 

^{p p}

1 p p 1

p p p p

1 || ,

1 1

1 1 || 1 1 ||

r jx m V V

V V V I

m m

r jx r jx

    

           

   

^{p p}

^{ }

^p

2 p p 2 1

p p p

1 || , ,

1 1

1 1 ||

r r r

V V V I I

m r

r jx

      

           

   

p p

p p

p p

2 2 2

21

p p

1 1 1 p p

2 2

1 1

1 1

1 1

r V

V r

r r

m m

b V I

s a V I mV V m r jx

m m

     

       

    

    

     

    

 

      

p

p p p

p p p p p p

p

p p

2 1 2 2

1 1 1 2 1

1 1

r

r r r

r jx r r jx r

r jx r

 

 

  

 

  

       

  

step 4: Check when resonance undergoes (x_p 0):

 

p p p

21

p p p p

1 i.e., 0 dB 0

r r r

t  r jx  r j  r 

  

 

^p

   

^p

  

^p



21

p p p p p p

2 2 2

1 2 1 1 2 0 1 1 2

r r r

s  r jx r  r j r  r

        

IfR_p 51.7 then _p ^p

o

51.7 1.034 50

r R

 Z   ，



^p

 ^{ }

21

p

2 2 1.034 2.068

0.674 i.e. -3.7261 dB 1 2 1.034 3.068

1 2 s r

r

    

   and

 

p 21

p p

1.034

1 i.e. 0 dB

1.034 0

t r

r jx j

  

  

step 5: Check when load impedance is shorted (r_p 0), botht and ₂₁ s are zero. ₂₁

 

21 0, 21 0 i.e. dB

t  s   

step 6: Check when load impedance is opened (r_p  ),

 

p p

21

p p p

p

p

1 1 i.e., 0 dB

1 0

r r

t r jx x j

r j r

   

    

   

129

 

^p

 

^{1 p}

21 2

p p p p p

2 2 2

= angle tan

2 2

1 2 1 4

r x

s r jx r jx x

   

         

step 7: Check when power is half transfer to the port 2 (load port), r_p x_p，then

 

p o

21

p p

1 1

angle 90

1 1 2

t r

r jx j

   

  

 

^p

   

^p

 

21

p p p p p p

2 2

1 2 1 1 2 1

r r

s  r jx r  r jr r

     

 



^{p p}



p 1

2 3 4

p p p p p

2 1

angle tan 1 4 5 2 1 2

r r

r

r r r r r

    

 

        

130

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140

Table of Figures

Figure 2.1 The schematic diagram of a three-level EIT atomic system……… 4 Figure 2.2 The relative electric permittivity of the three-level EIT atomic vaporas a

function of the probe frequencydetuning _p and the Rabi frequency

_c of the control field………... 8 Figure 2.3 The typical behavior of the refractive index of the EIT vapor versus the

normalized probe frequency detuning ………. 9 Figure 2.4 The typical behavior of the refractive index of the EIT vapor versus the

normalized Rabi frequency of the control field………... 9 Figure 2.5 The typical behavior of the refractive index of the EIT vapor versus the

normalized frequency detuning of the control field………. 10 Figure 2.6 The schematic diagram of a double-control four-level system…………... 11

Figure 3.1 The 1D -layer structure of (D|E) cells embedded in GaAs

homogeneous dielectric………... 14 Figure 3.2 The bandgap structure of the 1D infinite periodic (D|E) cells when the

probe frequency of TE waves is far from the resonance……… 21 Figure 3.3 The Bloch wave number K of the 1D infinite periodic (D|E) cells when

the EIT atomic transition is on resonance………. 23 Figure 3.4 The real part (a) and the imaginary part (b) of the normalized Bloch wave

number K (in the units of 2 /) of the 1D infinite periodic (D|E) cells

versus _pand _c………... 23

Figure 3.5 The reflection coefficient of the 1D N-layer periodic (D|E) cells as_p changes. (N 1, 2, 3, 4, 5, 6)………... 25 Figure 3.6 The reflectance and transmittance on the left interface of the 1D N-layer

periodic (D|E)cells as_pchanges. (N1, 2, 3, 4, 5, 6)……… 26 Figure 3.7 The fine details of two typical cases for showing sensitivity of the

reflectance to thefrequency of the probe field in a narrow bandwidth…. 27 Figure 3.8 The real and imaginary parts of the reflection coefficient r versus the

normalized probe frequencydetuning_{ p/ 3} in the frequency range of two-photon resonance caused by the destructive quantum interference between the |1 | 3 and | 2 | 3 transitions………. 30 Figure 3.9 The real and imaginary parts of the reflection coefficient

r

versus the

normalizedRabi frequency

/ 3

 c of the control field………. 32 Figure 3.10 The three-dimensional behavior of the reflectance of the EIT-based

layered medium versus the normalized control Rabi frequency  _c/ ₃ and the normalized probe frequency detuning _p/ ₃………. 33 Figure 3.11 The reflectance and transmittance versus the normalized Rabi frequency

c/ 3

  of the control field………. 34

Figure 3.12 Bloch wave number versus the probe frequency detuning _pand Rabi frequency of the control field………... 36

/ 3

 _p

/ 3

 _c

c/ 3

 

N

K

_c

141

Figure 3.13 The real and imaginary parts of the Bloch wave number versus the normalized frequency detuning of the control field……….. 38 Figure 3.14 The real and imaginary parts of the reflection coefficient versus the

normalized frequency detuning of the control field in the frequency range of two-photon resonance caused by the destructive quantum interference between the - and - transitions………. 39 Figure 3.15 The reflectance of transmittance of the EIT-based layered medium versus

the normalized frequency detuning of the control field…………. 41 Figure 3.16 The transmittance of the EIT-based layered medium versus the

normalized probe frequency detuning and the normalized

control frequency detuning ……….. 42

Figure 3.17 The reflectance and transmittance versus the normalized probe frequency detuning . The layer number of the EIT-based

periodic mediumN1, 5, 20, 100………. 44 Figure 3.18 The reflectance and transmittance versus the normalized Rabi

frequency of the control field……… 46

Figure 3.19 The three-dimensional behavior of the transmittance of the EIT-based layered medium versus the normalized control Rabi frequency

and the normalized probe frequency detuning ……….. 47 Figure 3.20 The schematic diagram of a NOT gate designed based on the EIT periodic

layered structure……….

48 Figure 3.21 The schematic diagram of a two-input NOR gate designed based on the

EIT periodic layered structures………

48 Figure 3.22 The schematic diagram of an EIT-based periodic layered medium. (TE

wave of the probe beam is normally incident on the left side interface (x0) of the layered medium)……… 52 Figure 3.23 The reflectance and transmittance of the 4 -layer and 6 -layer periodic

(D|E)cells in a narrow probe frequency band………

53 Figure 3.24 The dispersive and tunable optical response in the transmittance of the

4 -layerand 6 -layer periodic (D|E) cells with    c



2 3, 20 3



and  p



0.2 3, 0.7 3



……… 55 Figure 3.25 General behavior of tunable optical response of the -layer periodic

(D|E) cells with ……….. 56

Figure 4.1(a) The structure of finite layer periodic (D|E) cells andan obliquely incident visible ray………. 60 Figure 4.1(b) The structure of finite layer periodic (LHM|EIT) cells are embeded

inside the same LHM material and an obliquely incident visible ray

(probe field) imping on the most left side interface……… 61

c/ 3

 

r / 3

 _c

1 3 2 3

/ 3

 _c

p/ 3

 

c/ 3

 

R T

/ 3

 _p

R T

/ 3

 _c

/ 3

 _c / 3

 _p

N

1, 5, 20, 100 N

142

Figure 4.1(c) The structure of finite layer periodic (LHM|EIT) cells are embeded

inside the vacuum and an obliquely incident visible ray (probe field)

imping on the most left side interface……… 61 Figure 4.2 Bloch wave number versus normalized_pand normalized_c………….. 63 Figure 4.3 The bandgap structure of the 1D infinite periodic (LHM|EIT) cells when

the probe frequency of TM waves is far from the resonance………

64 Figure 4.4The band structure of the 1D infinite periodic LHM-EIT cells when the

angles of incidence of the TM wave of the probe beam are,

o o o o o o

in 0 , 15 , 30 , 45 , 60 , 75

  respectively………... 67

Figure 4.5 The real and imaginary parts of the reflection coefficient

r

corresponding to theN-layer LHM-EIT cells (N 1 ~ 6), where the relative refractive index of the left-handed medium isn₁ 1……….. 68 Figure 4.6 The real and imaginary parts of the reflection coefficient

r

corresponding

to theN-layer Dielectric-EIT cells (N 1 ~ 6), where the relative

refractiveindex of the left-handed medium isn₁ 1……….. 70 Figure 4.7 Reflection coefficients of one (D|E) cell with     _p 1 ₃ 2 10 s^{7 1}..

72 Figure 4.8 Reflection coefficients of finite layer (D|E) cells



N 1, 5, 20, 100



with

8 1

p 5 3 10 s^

     ……… 73

Figure 4.9 Reflectance and transmittance of one (D|E) cell with     _p 1 ₃

7 1

2 10 s ^ ……… 74

Figure 4.10 Reflectance and transmittance of finite layer (D|E) cells



N 1, 5, 20, 100



with    _p 5 ₃ 10 s^{8 1}………... 75 Figure 4.11 Reflection coefficients of (Left-Handed Material|EIT) cells with

7 1

p 1 3 2 10 s^

      ……… 75 Figure 4.12 Reflection coefficients of finite layer (Left-Handed Material |EIT) cells



N 1, 5, 20, 100



with    _p 5 ₃ 10 s^{8 1}……….. 77 Figure 4.13 Reflectance and transmittance of (Left-Handed Material|EIT) cells with

7 1

p 1 3 2 10 s^

      ……….. 78 Figure 4.14 Reflectance and transmittance of finite layer (Left-Handed Material |EIT)

cells



N 1, 5, 20, 100



with    _{p 5 3 10 s}^{8 1}……… 79 Figure 4.15 The reflectance and transmittance of 1-layer and 2-layer LHM-EIT

structures (the relative refractive index of the left-handed medium

1 1

n   ) in the probe frequency rangep/ 3

 

0, 1 ………. 80 Figure 4.16 The reflectance and transmittance of 1-layer and 2-layer LHM-EIT

structures (the relative refractive index of the left-handed medium

1 1

n   ) in the probe frequency range p/ 3

 

0, 1 ………... 82 Figure 4.17 The schematic diagram of a photonic transistor designed based on the

LHM-EIT layered structure……….. 82 Figure 4.18(a) Scheme for all optics logic gates design……….

83

143

Figure 4.18(b) Scheme for optical-transistor amplifier...

83

Figure 5.1 The circuit analog of the three- and four-level quantum coherent effects... 90 Figure 5.2 The ADS simulation results of Autler-Townes doublet (triplet) and EIT

transparency windows for the circuit analog of two-, three- and four-level quantum coherent effects (corresponding to the various

capacitances chosen for )………... 93

Figure 5.3 The configuration of experimental setup for measuring the voltage ….. 94 Figure 5.4 The lumped circuit for findingt and ₂₁ s ………. ₂₁ 97 Figure 5.5 EIT can be represented byRp jXp

 

 and calculate outt without 50 ₂₁

Ohm………. 98 Figure 5.6 EIT-like RLC series circuit with switch S , S_{1 2}………. 100 Figure 5.7 EIT-like RLC series circuit with switch S ON and ₁ S OFF………….. 102 ₂ Figure 5.8 EIT-like RLC series circuit with switches S , S both ON………... 105 _{1 2} Figure 5.9 The circuit analog of the three- and four-level quantum coherent effects... 108 Figure 5.10 The simplified the EIT-like RLC circuit………... 110 Figure 5.11The Matlab simulation results for power absorption in the circuit analog

of two-/three-/four-levelatomicsystem………..

113 Figure 5.12 The ADS stimulation results for the circuit analog of power absorption

in the circuitanalog of two-/three-/four-levelatomicsystem………….. 114 Figure 5.13 The Matlab simulation results for the absorbed power (versus the work-

ing frequency) when the coupling capacitors have different capacitan-

ces……….. 116 Figure 5.14 The absorbed power versus the working frequency (simulated by ADS)

when the coupling capacitors have different capacitances………... 117 Figure 5.15 The active and reactive powers in the circuit analog of the four-level

System……….. 118

Fig. 5.16 Another draw of schematic diagram for a double-control four-level system. 120 Figure 5.16 Two-ports circuit analog of the three- and four-level quantum coherent

effects……… 122 Figure 5.17 Power dips in two/three/four-level EIT-like RLC circuit………. 124

Cx

V12

144

Tables

Table 1 Parameters setting of the bulk EIT material for finding the permittivity… 10 Table 2 Typical physical parameters setting for peridical (D|E) layered structure.. 24 Table 3 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)………. 55

Table 4 Typical physical parameters setting for peridical (LHM|EIT) layered

Structure……… 63

Table 5 Specification of the electronic components applied in the EIT-Like circuit 114

145

Papers Published

1. Book chapter 7, " EIT-Based Photonic Crystal and Photonic Logic Gate Design" , pp. 133-156, ISBN 978-953-51-0431-5, InTech, March, 2012

2 . "Frequency-Sensitive Optical Response via Tunable Band Structure in an EIT- Based Layered Medium” International Conference onMaterials Science And Engineering Applications (ICMSEA 2011), Advanced MaterialsResearch Vols.160-162 (2011) p1432-1439

3. "The Sensitive Band Structure of an EIT Photonic Crystal and Its Application to Photonic Logic Gate Design ", pp 243-250, ISSN 1971-680X Vol. 4 N. 5, October 2010, International Review of PHYSICS (IREPHY).,

4. ”Photonic analog of Klein paradox in a periodic layered material with composite structure of left handed medium and electromagnetically induced transparency”, pp 43-52, ISSN 1971-680X Vol. 5 N. 2, Apirl 2011, International Review of PHYSICS (IREPHY).,

5. "EIT-based Coherent Control Effect Sensitive to Probe Frequency and Control Field Intensity in a Periodic LayeredMedium”, Progress In Electro-magnetics Research Symposium (PIERS,2011)

6. "Probe frequency- and field-intensity-sensitive coherent control effect in an EIT-based periodic layered medium", pp. 010201 Vol. 10, Issue 01, (2012), Chinesee Optics Letters

7. Extraordinary Ray Responseon LHM and EIT Bi-LayeredMedium”, No.117, D00-8, The15th National Conferenceon Vehicle Engineering

8. "Band Structure of 1D Infinite Periodic Dielectric and EITBi-Layered Medium”, Conference, OPT5-P-032, International Conference on Optics and Photonics in Taiwan (OPT’10)

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