Coherent control of Optical limiting in atomic systems

Generation and control of the reverse saturable absorption (RSA) and optical limiting (OL) are investigated in a four-level Y-type quantum system. It is demonstrated that the applied laser fields induce the RSA and it can be coherently controlled by either intensity or frequency of the applied laser fields. The effect of the static magnetic field on the induced RSA is studied and we obtain that it has a constructive role in determining the intensity range in which the OL is established in the system. In addition, we find that the transmission of the suggested optical limiter can be decreased either by increasing the length of the medium or by getting the atomic system denser. Finally, the Z-scan technique is presented to confirm our theoretical results. The proposed scheme can be used in designing the coherent optical limiters with controllable threshold and intensity range of the OL.

studied and it is illustrated that the RSA and the OL regions are extended by increasing the magnitude of the static magnetic field. In addition, we show that the transmission of the suggested optical limiter decreases either by increasing the length of the medium or by getting the atomic medium denser. Finally, the Z-scan technique is presented to confirm our theoretical results.

Model and equations
The proposed realistic atomic system is a four-level Y-type quantum system, which can be established in 5S 1∕2 , 5P 1∕2 and 5D 3∕2 lines of 87 Rb atoms as shown in Fig. 1  ). ε p and E i (i = s, c) are the amplitude of the probe and coupling fields, respectively. ε ± stands for the polarization unit vectors of the coupling fields. A static magnetic field is also employed to remove the degeneracy of the higher excited states 3 and 4 by 2ℏΔ B = 2m F g s μ B B where μ B is Bohr magneton and g s is Landé factor. This system can make it possible that the intermediate state 2 to be depleted by applying two strong coupling fields to the higher excited states in the presence of a static magnetic field. Thus the necessary condition is provided for the RSA.
The Hamiltonian of the considered system in the dipole and rotating wave approximations can be written as where Δ p = ω p − ω 21 , Δ s = ω s − ω 32 and Δ c = ω c − ω 42 are the detunings. ω p , ω s and ω c are the frequencies of the probe and coupling fields, respectively. Also, ω 21 , ω 32 and ω 42 are the central frequencies of the corresponding transitions. The density matrix equations of motion can be written as follows where Γ 1 = γ 31 + γ 32 and Γ 2 = γ 41 + γ 42 . The parameter γ i1 (γ i2 )(i = 3, 4) denotes the spontaneous decay rate from the excited state i to the lower states 1 ( 2 ). The polarization vector in the atomic medium is given by Here, χ p is the susceptibility representing the response of the medium to the probe field. Let us now solve the wave equation for the probe field, which can be written as  www.nature.com/scientificreports www.nature.com/scientificreports/ χ p can be related to the probe transition coherence ρ 21 defined as where n is density of atoms and ρ 21 is calculated from equation (2). Therefore, the equation 6 reduces to is the resonant absorption. By introducing the normalized susceptibility S p = ρ 21 γ/Ω p , the output probe field amplitude takes the form Finally, the normalized transmission of the probe field is given by The normalized susceptibility S p is clearly a complex quantity that its imaginary part stands for the absorption of the probe field. The intensity region in which the imaginary part of S p increases with the increase of the input intensity denotes the RSA region. Equation 10 displays the transmission behavior of the light, which is going to be based on the study of the OL properties of the quantum system. The Z-scan technique is widely used to study the nonlinear refractive index 27 as well as the OL properties of various materials 21 . In experiment, a Z-scan setup includes a laser field with a transverse Gaussian profile focused by using a lens. The sample is then moved along the propagation direction of the focused Gaussian field. It is clear that the sample experiences maximum intensity at the focal point (z = 0), which gradually decreases in either direction from the focus. The Z-scan technique shows the transmission based on the scanning of the sample position relative to the focal plane of the lens. The incident probe field is a Gaussian laser field with the Rabi frequency Ω p where Ω p0 is the probe Rabi frequency at the focal point (beam waist), is the beam radius at z (the distance of the sample from the focal point) and www.nature.com/scientificreports www.nature.com/scientificreports/ is the diffraction length of the beam. It should be noted that the Z-scan measurements in our work are carried out for the probe field at 800 nm wavelength corresponding to the transition 5S 1∕2 ↔ 5P 1∕2 .
With calculations of equation 11 numerically for a pulsed Gaussian beam, normalized transmission as a function of position can be obtained as

Results and Discussion
Here, we are going to present our numerical results describing the absorption behavior of the system. We are interested in investigating the SA and RSA regions in the system to provide the appropriate conditions for inducing the OL. All the parameters are scaled by γ, which is 2π × 5.75 MHz for the transition 5S 1∕2 ↔ 5P 1∕2 of 87 Rb atoms. Figure 2 shows the absorption of the probe field versus the incident intensity of the probe field for different values of the Ω s . The dotted line is for Ω s = 0, the dot-dashed line for Ω s = 0.5γ, the dashed line for Ω s = γ and the solid line for Ω s = 2γ. The other used parameters are Ω p = 0.01γ, Ω c = 65γ, Δ c = 100γ, Δ p = 1.5γ, Δ s = 0 and Δ B = 0. It is seen that when the coupling field Ω s is switched off, the absorption of the probe field decreases by increasing the intensity of the input probe field and the SA is dominant in the absence of the Ω s . By switching on the Ω s , the RSA is induced and a peak is generated in the absorption of the probe field, leading to separate the RSA and SA regions. Moreover, the absorption peak enhances by increasing the Ω s , so it allows us to control the RSA phenomenon. Generally, in the RSA region (left side of the peak), the absorption increases by growing the intensity of the input laser field. An investigation on Fig. 2 shows that the RSA can switch to the SA for the intense input laser field. The extension of the RSA with respect to the SA region is another scenario that we can do by applying the static magnetic field. After inducing the RSA in the system, it is important that the system maintains the RSA behavior in wider range of the intensity of the input field. The constructive role of the static magnetic field in the RSA is presented in Fig. 3 for Δ B = 0 (dotted), Δ B = γ (dot-dashed), Δ B = 2γ (dashed) and Δ B = 3γ (solid) corresponding to the static magnetic field B = 0, B = 2G, B = 4.3G and B = 6.4G, respectively. The other taken parameters are Ω p = 0.01γ, Ω c = 65γ, Δ c = 100γ, Δ p = 1.5γ, Δ s = 0 and Ω s = 2γ. We result that the RSA region is extended by increasing the magnitude of the static magnetic field and the RSA is established in a larger range of the input probe field intensity, which promises the extension of the OL range.
In the following, we investigate the effect of the coupling field, Ω s , on the transmission of the probe field. In Fig. 4, the transmission of the probe field versus the intensity of the input probe field is shown for αl = 800γ, Ω s = 0 (dotted), Ω s = 0.5γ (dot-dashed), Ω s = γ (dashed) and Ω s = 2γ (solid). The other used parameters are those taken in Fig. 2. Figure 4 shows that in the absence of the coupling field, the transmission of the probe field grows with the increase of the input probe field intensity going the system toward transparency. Thus, the system cannot be used as an optical limiter. As proved in Fig. 2, the RSA was induced and intensified by increasing the Ω s . In the RSA domain, the transmission of the probe field keeps constant or even reduces wity increasing the coupling field intensity. Hence, it is demonstrated that the OL is coherently induced and controlled in the system. In addition, a bird's eye view of Fig. 4 reveals that the increase of the Ω s can lead to decrease the threshold of the OL induced in the system. www.nature.com/scientificreports www.nature.com/scientificreports/ One of the important features that distinguish an optical limiter from the rest is the intensity range in which the optical limiter operates. Here, we show that the applying the static magnetic field can extend the OL range. Figure 5 depicts the transmission of the probe field versus input intensity for different values of the static magnetic field. The taken parameters are the same used in Fig. 3. Figure 5 shows that the OL range can be controlled by the static magnetic field. It is seen that in the absence of the static magnetic field, the OL is even established in a small range of the input intensity. Thus, in our suggested atomic optical limiter, applying the static magnetic field makes the optical devices and sensors safe from damages in a larger range of input intensity.
Control of the intensity of the transmission is another advantage of the suggested optical limiter. In Fig. 6, the effect of the resonant absorption, αl, is studied on the transmission of the probe field plotted versus the intensity of the input probe field. Resonance absorption is directly related to the length of the medium and density of atoms. It is observed that by increasing αl, the transmission decreases with the same OL thresholds. Decrease of the transmission makes it possible that the presented optical limiter can be set to use in optical devices, which need the optical limiters with lower transmissions.
In order to gain a deeper insight, the slope of transmission of the probe field, D(T), is displayed in Fig. 7 as a function of the intensity of the input probe field and the static magnetic field. The used parameters are  Ω p = 0.01γ, Ω c = 65γ, Δ c = 100γ, Δ p = 1.5γ, Δ s = 0, Ω s = 2γ and αl = 800γ. It is worth to note that for the OL range, the slope of the transmission is zero and even negative. Otherwise, the slope of the transmission is positive for the SA region. Figure 7 delineates the behavior of the RSA and the corresponding OL induced in the system as well as the SA for all values of the static magnetic field and intensity of the input field. This figure helps us to determine the OL range needed for different optical devices by selecting the appropriate parameters. The OL line, shown in the Fig. 7, presents the zero slope of transmission at the end of the RSA region. The left side of the OL line specifies the OL range, while the right side determines the SA region.
Z-scan technique. In Fig. 8, the schematic of experimental setup is displayed including the open aperture Z-scan technique. A diode laser at 780 nm passes through the Electro-Optic Modulator (EOM) to generate a probe field at 795 mm and coupling field at 762 nm. The generated fields then pass through a high-quality polarizer (P1) to have linear polarization. The probe field is sent to organize the open aperture Z-scan part of the setup. The coupling field passes through the EOM2 to generate the two linearly polarized coupling fields. These two fields then pass through the quarter wave plate 1 (QWP 1 ) and 2 (QWP 2 ) to form the right-(E c ) and left-(E s ) circularly polarized coupling field, respectively, and are applied to the medium. In addition, a static magnetic field is applied to the medium parallel to the coupling fields. The sample is moved around the focal point of the  focused probe field by a fine micropositioner. Finally the transmission of the probe field in each step is detected by the PhotoDiode (PD).
Finally, we employ the Z-scan technique to confirm the validity of the obtained theoretical results. The dip in the Z-scan transmission curve corresponds to the OL, while the peak stands for the SA effect. In Fig. 9, Z-scan technique measurement is displayed to investigate the z-dependent transmission for different values of input intensity chosen from the RSA and the SA region in Figs. 3 and 5. In Fig. 3, it was observed that the RSA was induced in the system up to a certain intensity called OL threshold. After the OL threshold, the SA is dominant in the system. These results was followed by the Fig. 5 in which it was observed that in the RSA region (I in < 15γ), limited by the OL threshold, the OL appears. These results are demonstrated in the Z-scan technique presented in Fig. 9. The parameters that their results are examined in Fig. 9 are Ω p = 0.01γ, Ω c = 65γ, Δ c = 100γ, Δ p = 1.5γ, Δ s = 0, Ω s = 2γ, Δ B = 3γ and αl = 800γ. It is seen in Fig. 9 that as the sample approaches and then moves away from the focal point, which its intensity is chosen from the RSA region in Figs. 3 and 5 (I in < 15γ), the transmission curve takes the form of a dip. On the contrary, when the intensity of the focal point is chosen from the SA region in Figs. 3 and 5 (I in > 15γ), the z-scan curve looks like a peak. Thus, the theoretical Z-scan experiment results are in good agreement with the results mentioned in Figs. 2-6. conclusion In summary, coherent generation and control of the RSA and OL are reported in a four-level Y-type quantum system. It was shown that the RSA is coherently induced by applied laser fields. We showed that, consequently, the OL is coherently induced through the RSA region so that all characteristics of the induced OL such as the intensity range and the threshold intensity can be controlled by either intensity or frequency of the laser fields. In addition, we proved that the static magnetic field has a constructive role in extending the RSA region and the  EOM is an electrooptical-modulator; P is a high-quality polarizer; QWP is a quarter wave plate; and PD is a photodiode.