Analogy of transistor function with modulating photonic band gap in electromagnetically induced grating

Optical transistor is a device used to amplify and switch optical signals. Many researchers focus on replacing current computer components with optical equivalents, resulting in an optical digital computer system processing binary data. Electronic transistor is the fundamental building block of modern electronic devices. To replace electronic components with optical ones, an equivalent optical transistor is required. Here we compare the behavior of an optical transistor with the reflection from a photonic band gap structure in an electromagnetically induced transparency medium. A control signal is used to modulate the photonic band gap structure. Power variation of the control signal is used to provide an analogy between the reflection behavior caused by modulating the photonic band gap structure and the shifting of Q-point (Operation point) as well as amplification function of optical transistor. By means of the control signal, the switching function of optical transistor has also been realized. Such experimental schemes could have potential applications in making optical diode and optical transistor used in quantum information processing.

In analogy to their electronic counterparts, optical transistors and switches are required as fundamental building blocks for classical as well as quantum optical information processing 1,2 . In particular, an optical transistor and switch operated by a single photon stored in an atomic ensemble inside a cavity has recently been demonstrated 3 . In this article we will theoretically and experimentally demonstrate the analogy of an optical transistor function with the enhancement and suppression of multi-wave mixing process through the modulation of a photonic band gap structure. Four-wave mixing is a well-known nonlinear optical effect which can be enhanced in an electromagnetically-induced transparency (EIT) medium [4][5][6] . Four wave mixing 7-10 and six wave mixing 11 have been individually studied in multi-level atomic systems. By choosing appropriate atomic level schemes and driving fields, one can generate controllable nonlinearities with very interesting applications in designing novel nonlinear optical devices. This motivates the studies of enhanced higher-order nonlinear wave-mixing processes. Enhanced six-wave mixing via induced atomic coherence was experimentally observed in a four level inverted Y-type atomic system 12 . Such six-wave mixing signal can be made to even coexist, compete, and spatially interfere with the four-wave mixing signal in the same system 13 by the assistance of EIT. The EIT-based nonlinear schemes can be driven both by traveling wave beams and standing wave beams. The large nonlinearity was obtained in an atomic system driven by two counter propagating coupling fields of the same frequency which form a standing wave 14,15 . Standing wave interacts with atomic coherent medium to result in an electromagnetically induced grating 16,17 . The electromagnetically induced grating possesses medium results into electromagnetically induced grating. Furthermore electromagnetically induced grating will lead to a photonic band gap structure as shown in Fig. 1(b). The probe field E 1 propagates in the direction of ′ E 3 through the 85 Rb vapors with approximately 0.3° angle between them. The control field E 2 propagates in the opposite direction of ′ E 3 through 85 Rb with approximately 0. 3° angle between them. E r (The reflected band gap signal from the photonic band gap structure) and E 1 is another pair of the counter-propagating beams, but with a small angle between them as shown in Fig. 1(c). Due to the small angle between E 1 and ′ E 3 , the geometry not only satisfies the phase-matching condition but also provides a convenient spatial separation of the applied laser and generated signal beams. Thus we can easily detect the generated beams which are highly directional 22 . The probe transmission signal and generated band gap signal are detected by photodiode detectors PD1 and PD2, respectively.
A block-diagram of the analogy of the behavior of modulating the photonic band gap structure with an optical transistor amplification function is shown in Fig. 1(d), where FWM BGS generated by E 1 , E 3 and ′ E 3 in the medium is analogous to the input signal a in in the optical amplification experiment; enhanced FWM BGS is analogous to the output signal a out and G is analogous to the gain factor of transistor. The mathematical model of this analogy is given as = ⁎ a G a out i n . E 2 is a control signal, power of which can be used to modulate the photonic band gap structure. This behavior is analogous to change the Q-point of electrical transistor.
Considering the time-dependent Schrödinger equation, using the perturbation chain (i.e., Liouville pathways with perturbation theory 23 and satisfying phase-matching condition) and rotating wave approximation, we can obtain a series of density matrix equations by using the way with combining the coupling method and the perturbation theory as follows 10   3   10 10  3  3  30   2  2  20  3 3 30 Where the superscript (0), (1), (2) or (3)   (which is reasonable since the probe field is always weak, compared with other fields), we finally obtain the first order and the third order density matrix elements ρ ( )   , we can also obtain the fifth order density matrix element ρ ( ) 10 5 as follows ] 11 5 2 0 30 20 10 3 The condition of generating the photonic band gap structure is that the medium is of periodic refractive index. In order to get the periodic refractive index, the susceptibility should be periodic according the relation of refractive index with susceptibility, i.e., χ = + ( ) n 1 Re . To get periodic susceptibility we should generate the periodic energy levels structure. Hence, by introducing periodic standing wave field, we can obtain the periodic energy levels as shown in Fig. 2. In Fig. 2 values are also periodic. Thus we can obtain the periodic energy levels as shown in Fig. 2(b). When the probe reaches two-photon resonance Δ 1 − Δ 3 = 0, the absorption will be suppressed, i.e. the probe transmission signal becomes strong. At the same time, the band gap signal will be suppressed correspondingly. Thus, we define Δ 1 − Δ 3 = 0 as the suppression condition. When E 2 is turn on, due to the second level dressing effect of E 2 . The two dressed states . In our system, the normalized total susceptibility is 1  3  3  2  3  2  5  3  4  3  4  3  2  3 2 , which determines the refractive index of the system according to χ = + ( ) n 1 Re . For the system generating the band gap signal, the real part of susceptibility is periodic.
In order to estimate the reflection efficiency of band gap signal, we start with the wave equation in the following form 24 , Where P is polarization of the medium given by 3 is the strong coupling fields and Δ = + E E E r 1 is the weak signal fields. Substituting P and E into Eq. (12) and using the slowly varying amplitude approximation , after is attenuation of the field due to the absorption of the medium and is the gain due to the nonlinear susceptibilities in which four-wave and six-wave mixing are considered. χ′ ( ) 1 , χ′ ( ) 3 and χ′ ( ) 5 are the zero order coefficients from Fourier expan- is the phase mismatch magnitude, in which θ is the angle between probe E 1 and ′ E 3 . Equations (13) and (14) describe the mutual generation process of ( ) represents the generating efficiency. The generating efficiency will be high when the phase matching condition is satisfied (Δ = k 0 x ). In order to estimate the efficiency of the reflected band gap signal and probe transmission signal, we solve Eqs. (13) and (14) as follows. We differentiate Eq. (13) with respect to x and simplify it by using Eq. (14) to get the following second order differential equation (13) and (15), we get the following equation The general solution of Eq. (16) is . We assume that length of the 85 Rb sample is d x and apply initial conditions ( ) = E E 0 . We define the reflection efficiency of band gap signal from the photonic band gap structure with respect to the incident probe field as While the probe transmission efficiency across the medium with respect to the incident probe field is defined as First, we observe the variations in the intensities of band gap signal and probe transmission signal by scanning Δ 2 at different discrete values of Δ 1 as shown in Fig. 3. Note that variations in the intensities discussed here are displayed by the efficiencies of probe transmission signal and reflected band gap signal in the following figures according to the above theory. In Fig. 3(a), each sub curve's peak is the dressed probe transmission signal induced by the second-level dressing effect of E 2 , which meets the condition Δ + Δ = 0  Fig. 3(b) the baselines show the intensity of FWM BGS, which is the reflection from photonic band gap structure. The dip in each sub curve shows that FWM BGS is suppressed while the peak within each sub curve represents that FWM BGS is enhanced. It is worth mentioning that, in the case of scanning the dressing frequency detuning Δ 2 , the suppression and enhancement of FWM BGS are caused by the same dressing term / G d gives us very interesting information about band gap signal, which is  Figure 3(b) shows that the probe detuning can be used to find the optimal conditions for the reflected band gap signal. This information can be used to give the analogy of the enhancement and suppression of the reflected band gap signal with optical amplification and switching. Maximum enhancement of band gap signal occurs at particular detuning (Δ 1 = 280 MHz), while the maximum suppression occurs at Δ 1 = 235 MHz, as shown in Fig. 3(b). 235 MHz and 280 MHz are the optimal values of Δ 1 which are used to demonstrate the optical switch and amplification functions. Next, we observe the variations in the intensities of probe transmission signal and band gap signal versus dressing frequency detuning Δ 2 by blocking different beams at two optimal values of Δ 1 as shown in Fig. 4. Based on the values of Δ 1 , we classify the signals in two different groups. First, Fig. 4(a1-b1),(a2-b2) shows probe transmission signal and band gap signal with no blocking and blocking E 2 when Δ 1 is set as 235 MHz. In Fig. 4(a1), when all the beams are turned on, there is a peak in the probe transmission signal because of the dressing term / G d  1 , which is of the same intensity with the baseline of sub curve in Fig. 4(a1). To demonstrate the switching function of a transistor, the value of Δ 1 should be set as about 235 MHz. At the same time, we define the intensity of the band gap signal lower than reference level (RL) as OFF-State while the intensity higher than reference level as ON-State of the switch. Two extreme values 0 mW and 21 mW of the power of control signal (P 2 ) are used to turn on or off the switch. The two extreme values of P 2 are analogous to the digital values of gate voltage or base current in the case of MOSFET and BJT respectively. In Fig. 4(b2), when P 2 is set to 0 mW, the band gap signal is analogous to the input signal of a transistor. This band gap signal is FWM BGS generated by E 1 , E 3 and ′ E 3 without E 2 according to ρ ( ) 10 3 of Eq. (7) in the medium. When P 2 is set to 0 mW, the input signal directly passes to the output (the detector PD2 (Fig. 1(c))). Since the output signal intensity is higher than reference level, the switch is ON-State. When P 2 is set as 21 mW, the baseline in Fig. 4(b1) has the same intensity with the band gap signal in Fig. 4(b2). The dip shows that FWM BGS is suppressed  7). Physically, due to the modulation effect of E 2 on the photonic band gap structure at Δ 2 = − Δ 1 , the reflected FWM BGS will get weak. At the moment, the output signal intensity (which is given by the lowest point of curve in Fig. 4(b1)) at the detector PD2 ( Fig. 1(c)) is lower than the reference level and the switch is OFF-State. Next, from Fig. 4(a3-b3) to Fig. 4(a7-b7), we observe the variations of probe transmission signal and reflected band gap signal by blocking E 3 , blocking ′ E 3 , blocking E 1 , blocking E 2 and no blocking with Δ 1 = 280 Mhz from left to right, respectively. The results in Fig. 4(a6,a7) are similar with the ones in Fig. 4(a1,a2). In Fig. 4(a3,a4), intensities of the peaks in the probe transmission signal increases by blocking E 3 and ′ E 3 , respectively, compared with Fig. 4(a7) (no blocking). This is because of the decreasing suppression effect of G s . Intensity of the probe transmission signal will become zero by blocking the incident probe beam (E 1 ) because of G 1 is zero in Eq. (6) as shown in Fig. 4(a5). The reflected band gap signal can be enhanced with increasing P 2 when Δ 1 is set as about 280 MHz as shown Fig. 4(b7), whose efficiency is measured by R in Eq. (17). This behavior is analogous to the amplification function of transistor. To demonstrate the analogy of the enhancement of the band gap signal with amplification function of a transistor, we need set Δ 1 as about 280 MHz. Compared with Fig. 4(b7) (no blocking), the enhancement peaks in the band gap signal disappear by blocking ′ E 3 , E 3 , E 1 or E 2 , respectively, as shown in Fig. 4(b3-b6). As shown in Fig. 4(b3,b4), when any one of ′ E 3 or E 3 is blocked, the photonic band gap structure is deformed and therefore the reflected band gap signal disappears. In Fig. 4(b5), when the incident probe beam is blocked, there is still no reflection because of the absence of incident signal source E 1 according to ρ ( ) 10 3 in Eq. (7) although the photonic band gap structure is there. When E 2 is turned off, Fig. 4(b6) shows the FWM BGS generated by E 1 , E 3 and E 3 according to ρ ( ) 10 3 of Eq. (7). The FWM BGS is analogous to the input signal in the optical amplification experiment. When E 2 is turned on, the FWM BGS can be obviously enhanced as shown in Fig. 4(b7). Physically, at the large detuning Δ 1 = 280 Mhz, due to the modulation effect of E 2 on the photonic band gap structure at Δ = −Δ − G 2 1 2 , the reflected FWM BGS will become strong. The highest point of peak in Fig. 4(b7) gives the amplified intensity of the band gap signal, which is the output in the optical amplification experiment. Furthermore, the baseline in Fig. 4(b7) has the same intensity with the band gap signal in Fig. 4(b6), which can also be viewed as the input signal in the optical amplification experiment. It is clear from the above discussion, to operate the system as a switch, we need set Δ 1 as about 235 MHz; while to operate it as an amplifier, we need set Δ 1 as about 280 MHz and make the power of incident probe smaller at the same time.
Next, we further demonstrate the analogy of modulating the band gap signal with the optical transistor amplification function with the power dependences of probe transmission signal and band gap signal versus Δ 2 . Variations in the two types of signals are shown from right to left with increasing power of E 2 (P 2 ) as shown in Fig. 5(a1-b1). In Fig. 5(a1), the peak in each baseline shows the enhancement of the probe transmission signal induced by the dressing effect of E 2 . The efficiency of the probe transmission signal is given by T in Eq. (18). The dressing term / G d 2 2 20 of ρ ( ) 10 1 has an enhancement effect on the probe transmission signal. Changing P 2 from small to large values, the peak becomes higher with the increasing dressing effect of / G d 2 2 20 in ρ ( ) 10 1 of Eq. (6). In Fig. 5(b1), the baseline of each sub curve shows intensity of FWM BGS generated by E 1 , E 3 and ′ E 3 which is analogous to the input signal in the optical amplification experiment according to the discussion about Fig. 4(b6,b7). Dips at Δ 2 = − Δ 1 show the suppression of reflected FWM BGS because of the dressing effect of E 2 according to the dressing term / G d 2 2 20 in ρ ( )  3 . Therefore, the band gap signal with the higher intensity corresponds to the stronger probe transmission signal as shown in Fig. 5. Thus it is important to measure the probe transmission signal because of its strong relation with the band gap signal. Here we give an analogy of the amplification function of the transistor with modulating the ban gap signal. This analogy of the amplification function is demonstrated by two alternative ways which are modulating photonic band gap structure and changing the power of the incident probe field (E 1 ). Here we modulate the photonic band gap structure by changing the power of control signal (P 2 ) to get the amplified output signal, while powers of the incident probe and coupling fields are held constant, i.e. the FWM BGS generated by the probe and coupling fields is constant, which is viewed as the input here. As mentioned earlier band gap signal is the reflection of a controllable photonic band gap structure. Therefore the intensity of the reflection can be changed by controlling the photonic band gap structure or by changing power of the incident probe field (E 1 ). The photonic band gap structure can be controlled by changing P 2 . As shown in Fig. 2(d), P 2 further modulates the periodic energy levels to change the generated photonic band gap structure. Like the electrical transistors, changing P 2 is analogous to changing biasing of electrical transistor. When the biasing of a transistor is changed, its operation point (Q-point) is shifted. As shown in Fig. 5(c), the gradient of gain factor (the ratio of the amplified band gap signal intensity (the highest point in the peak) to the original FWM BGS intensity (the baseline)) versus P 2 curve is gradually decreasing as the amplification of band gap signal tends toward saturation. Similarly the relative amount of amplification (Δ r) also decreases with P 2 increasing for a fixed input signal as the bang gap signal intensity reaches the saturation region as shown in Fig. 5(b1). This behavior is similar to the variation of amplifier output by changing Q-point or biasing of a transistor.  Finally, we change the power of E 1 (P 1 ) and observe the variations in probe transmission signal and band gap signal versus Δ 2 . Power dependences of the two types of signals are shown from top to bottom with decreasing power of E 1 (P 1 ) as shown in Fig. 6(a,b). The intensity of the probe transmission signal decreases with decreasing power of E 1 according to ρ ( ) 10 1 in Eq. (6). Peaks in Fig. 6(a) become smaller from top to bottom with the decreasing power of E 1 because of G 1 in Eq. (6). In Fig. 6(b), the baseline of the signal represents the intensity of the FWM BGS generated by E 1 , E 3 and ′ E 3 which is analogous to the input here according to the discussion about Fig. 4(b6,b7). Dips show the suppression of the FWM BGS because of the dressing term / G d 2 2 20 of ρ ( ) 10 3 in Eq. (7). The suppression is further modulated by changing P 1 . The dip becomes shallow from top to bottom by changing P 1 from large to small values. Peaks in the baseline show the enhancement of the FWM BGS. The highest point on the peak gives the intensity of the amplified band gap signal, which is the output of the amplifier. Compared to the previous case where we changed the power of E 2 , here the band gap signal intensity is changed by changing P 1 . Since E 1 is the generating field for FWM BGS which is viewed as the input in the optical amplification experiment, the intensities of input signals increase as increasing P 1 , as shown by x coordinate values of solid circles in Fig. 6(c) which are obtained by measuring the intensities of baselines of Fig. 6(b). As a result, the output intensities (the highest points on the peaks) change in proportion to the input intensities when the power of the control signal (E 2 ) is fixed as shown by y coordinate values of solid circles in Fig. 6(c) which are obtained by measuring the intensities of highest points of peaks in Fig. 6(b). The gradient of curve in Fig. 6(c) is constant, which is analogous to the behavior of electrical transistor operated at a fixed Q-point with a constant gain (G) due to the fixed power of the control signal (E 2 ) which decides the Q-point.

Discussion
In summary, the double-dressed probe transmission signal and band gap signal are compared for the first time to deeply comprehend the double-dressing effect on the photonic band gap structure. We experimentally and theoretically demonstrated that, probe transmission signal and band gap signal can be manipulated by multiple parameters like changing power and frequency detuning. We demonstrate the analogy between switching and amplification function of the transistor with modulating the reflected band gap signal. Such research could find its applications in optical diodes and transistors which are used in quantum information processing.

Methods
In our experiment, there are four laser beams generated by three external cavity diode lasers (ECDL) with line width of less than or equal to 1 MHz. The probe laser beam E 1 is from an ECDL with a horizontal polarization. The two coupling laser beams E 3 and ′ E 3 with a vertical polarization are split from another ECDL. The dressing laser beam E 2 with a vertical polarization is from the third ECDL. The intensity of probe beam E 1 is the only weak laser beam while other laser beams are strong. The powers of E 1 , E 3 and ′ E 3 are 2.1 mW, 13.2 mW and 8.4 mW, respectively. And P 2 are set as 21 mW in the experi- (c) Output signal intensity (enhanced FWM BGS whose intensity is the one of the highest point on the peak of each sub curve in (b)) versus input signal intensity(FWM BGS whose intensity is the one of the baseline in each sub curve in (b)).
Scientific RepoRts | 5:13880 | DOi: 10.1038/srep13880 ment of changing probe frequency detuning. The atomic vapor cell has the typical density of 2 × 10 11 cm −3 . We measure the probe transmission signal and band gap signal in the inverted Y-type four level atomic system which can be dressed by fields E 3