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Performance analysis of double-amplitude population gratings by non-overlapping unipolar pulses
Optics Communications 462 (2020) 125182
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Performance analysis of double-amplitude population gratings by non-overlapping unipolar pulses Huadi Zhang, Shuanggen Zhang ∗, Shengdong Li, Xiurong Ma Tianjin University of Technology, School of Electrical and Electronic Engineering, Engineering Research Center of Optoelectronic Devices & Communication Technology, Ministry of Education, Tianjin Key Laboratory of Film Electronic and Communication Device, No. 391 Binshui West Street, Xiqing District, Tianjin 300384, China
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Keywords: Population gratings Double-amplitude gratings Unipolar pulses Pulse propagation Time delay
ABSTRACT We investigate the performance of the induced double-amplitude population grating (DAPG) based on spatially periodic modulation by the optical non-overlapping interaction of unipolar pulses. Considering the two typical allowed pulse configurations, the time delay requirements and the relationship between the grating amplitude, phase shift and the time delay were analyzed in detail. The results reveal that the grating amplitudes and phase can be dynamically controlled by the suitable time delay between the excited pulses. The population gratings presented here maybe offer more potential applications in nonlinear optics.
1. Introduction
the resonant propagation of several periodic pulses in an asymmetric medium having a periodic sub-wavelength structure [19]. Most
Periodic spatial population gratings are typically generated in resonant mediums via interference of two or more spatially overlapping light waves [1], but in this situation, the spatial period is limited by the light wavelength and pulses have to be long in duration. These gratings found lots of applications in the spectroscopy of different species and in nonlinear optics etc. [2,3]. Multi-beam optical pulses that do not overlap in time domain can interact with non-uniform broadening medium [4–6], and one pulse may induce polarization oscillations that remain after the pulse passage. Such induced polarization gratings can interact with the second-temporally delayed pulse, thus changing the overall inversion and forming transient population distribution gratings similar to coherent fringe structures in the medium. So far, ultrashort optical pulses have been widely used in the field of ultrafast optics [7–9], and usually the generated pulses are bipolar which have zero electric field area, i.e., integral of the electric-field strength with respect to time at a given point in space. In recent years, the generation of unipolar pulses has aroused people’s interest [10– 13], and such pulses contain a single half-period electric field so that the integral of the electric field intensity over the duration of the pulse is not equal to zero [12,14–16]. Unipolar pulses has a number of advantages not only due to their short duration, but also it allows the effective transfer of kinetic energy to the particles and effective control over the motion of the charges [10,11]. This circumstance will allow one to produce and control gratings, which is considerably faster than in the case of bipolar pulses [17,18]. Generally, such pulses can exist as soliton solutions to nonlinear optical equations [10,11], and it is also possible to generate extremely short unipolar pulses during
recently, some methods have been proposed for generating unipolar pulses in Raman-active media [20–22] or by using nonlinear field coupling technique [23,24]. It has been predicted that the indirect interactions of the light pulses with polarization waves by the other pulse can create, erase and manipulate the population density gratings [25,26]. Gratings formation and control by the unipolar pulses was studied both theoretically and numerically in [17,18,27]. The above work mainly discussed the possibility of the uniform gratings creation and erasure in a resonant medium by the pulse trains. In Ref. [28], the overlapping interaction of the bipolar pulse and the unipolar pulse in a resonant medium has been studied. However, characteristic analysis of DAPGs by the non-overlapping unipolar pulses has not yet been clarified so far. In this paper, we presented and evaluated an scheme to generate DAPGs based on spatially periodic modulation by the optical nonoverlapping interaction of unipolar pulses passing through a resonant medium, taking the Tm3+ : YAG crystal [29,30] for an example. Two typical allowed pulse configurations are considered via the numerical treatment of the Bloch equations for the density matrix elements, and the time delay requirements for the respective generation of DAPG were confirmed. Additionally, we detailed analyses on relationship between the amplitude, phase shift of DAPG and the time delay between the excited pulses.
∗ Corresponding author. E-mail address: [email protected] (S. Zhang).
https://doi.org/10.1016/j.optcom.2019.125182 Received 11 November 2019; Received in revised form 5 December 2019; Accepted 23 December 2019 Available online 21 January 2020 0030-4018/© 2019 Published by Elsevier B.V.
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Optics Communications 462 (2020) 125182
2. Theoretical model and qualitative analysis Our work will focus on the case where the unipolar pulses are coherent with the resonant medium and the applied pulses are identical in shape and amplitude. Before proceeding to numerical calculations, we consider the notion of the pulse area that is an important quantity ∞ to describe the pulse dynamics, 𝜃 = 𝜇ℏ ∫−∞ 𝜀(𝑡)𝑑𝑡, where 𝜀(𝑡) is the pulse envelope. For long pulses, the pulse with the 𝜋∕2 area fully saturates the medium. However, the action of ultrashort pulses can strongly differ from those for long pulses, and a real-valued envelope is not well defined, thus, the determination of the pulse area should be performed taking into account the phase change of light wave oscillations. Assume that the unipolar pulses have a brief Gaussian profile as following: ( ( )2 ) 𝑁 𝑁 ∑ ∑ 𝑡 − 𝜏𝑖 𝐸(𝑡) = 𝐸𝑖 (𝑡) = 𝐸0 exp − (1) 𝜏𝑝 2 𝑖=1 𝑖=1 where, 𝐸0 is the amplitude and N is the number of pulses. Time delay 𝜏𝑖 between the 1st pulse and the 𝑖th pulse can be adjusted so that the pulses did not overlap in the medium, and 𝜏𝑝 is the pulse duration.
Fig. 1. Typical pulse configurations to generate transient population gratings. The applied pulse 1, pulse 2, pulse 3 and pulse 4 are represented in orange, blue, red, and purple, respectively. The resonant medium is drawn as a dotted and light purple rectangle. . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The resonant medium is described as a two-level system, and the wave function[of two-level] structure can be expressed by density op𝜌 (𝑡) 𝜌12 (𝑡) erator: 𝜌(𝑡) = 11 , 𝜌11 and 𝜌22 indicate the probability that 𝜌21 (𝑡) 𝜌22 (𝑡)
3. Results and discussion
the atom is in the ground state and the excited state, respectively. 𝜕𝑡 𝜌(𝑡) = − ℏ𝑖 [𝐻(𝑡)𝜌(𝑡) − 𝜌(𝑡)𝐻(𝑡)], and H(t) is the Hamiltonian of the [ ] ℏ𝜔2 −𝜇21 𝐸(𝑡) system under the action of the light field, 𝐻(𝑡) = , ∗ −𝜇12 𝐸 (𝑡) ℏ𝜔1
3.1. DAPG generated by three pulses Firstly, two applied pulses propagate toward each other and pass through the resonant medium but do not meet in it. The first pulse will be a source of the polarization wave, and the polarization wave interfere with pulse 2. It should be noted that the phase velocity of light is significantly greater than the phase velocity of polarized waves. The optical field can be written in the form: ( ) 2 ∑ 𝑡2 𝐸0 exp − 𝑖 𝐸(𝑡) = (4) 𝜏𝑝 2 𝑖=1
where 𝜔1 , 𝜔2 are the frequencies of upper and lower energy levels, and 𝜇12 , 𝜇21 are electric dipole moments. Thus, the coherent interaction of atomic system with the unipolar pulses can be evaluated by the following density matrix equations: 𝜌12 (𝑡) 𝑖 − 𝜇𝐸(𝑡)𝑤(𝑡) + 𝑖𝜔0 𝜌12 (𝑡) 𝑇2 ℏ [ ] 𝑤(𝑡) − 𝑤0 4 𝜕𝑡 𝑤(𝑡) = − + 𝜇𝐸(𝑡)Im 𝜌12 (𝑡) 𝑇1 ℏ
𝜕𝑡 𝜌12 (𝑡) = −
(2) (3)
where 𝑡1 = 𝑡, 𝑡2 = 𝑡1 − 𝜏 = 𝑡 − 𝜏. 𝜏 is the time delay between the two applied pulses. The effective area of the two pulses strongly depends on the phase of the second pulse, i.e., the time delay 𝜏. Other parameters are as following: 𝜔0 = 2.69 × 1015 Hz, 𝜏p = 7.37 × 10−16 s, 𝜇 = 5 × 10−18 𝐶𝐺𝑆𝐸, 𝑇 = 2𝜋∕𝜔0 , and 𝑇0 is set to be 1.5T. The total integration time is set to be 10T 0 . Since the two unipolar pulses shoot into the medium successively from two opposite directions, the arrival time delay of the pulse in the medium decreases linearly in the transmission direction, which causes the periodic change of the particle population distribution along transmission direction, and thus forms the population grating. Subsequently, the third pulse is identical to the first pulse and moves behind it with a time delay 𝜏13 . The positions of three pulses with electric field (5) on the time axis are shown in Fig. 1(a). The expression of the electric field of three pulses has the form:
where 𝜌12 is the non-diagonal element, 𝑤 = 𝜌22 − 𝜌11 is the population difference between the ground and excited states of the two-level system, 𝑤0 is the population difference in the absence of an electric field (for the absorbing medium 𝑤0 = 1), 𝜔0 is the resonant transition frequency of the medium, and 𝜇 is the transition dipole moment of the atoms. 𝑇1 and 𝑇2 are the relaxation times, ℏ is the Planck constant, and 𝐸(𝑡) is the electric field. Eqs. (2) and (3) describe the evolution of the off-diagonal element and the difference between the diagonal elements of the density matrix, which can be numerically solved by the fourth-order Runge–Kutta method [31]. For simplicity, we consider the case of a thin medium and restrict ourselves to a one-dimensional geometry with light propagating in 𝑧direction. The identical unipolar pulses act on the resonant medium in different directions shown in Fig. 1. In Fig. 1(a), pulse 3 is in the same direction as pulse 1, and time interval is 𝜏13 , and pulse 2 enters into the medium in the opposite direction. Fig. 1(b) shows that pulse 3 propagates along the medium in the direction of pulse 2 with time delay 𝜏23 . As shown in Fig. 1(c), the unipolar pulses [(1) and (4)] and [(2) and (3)] propagating in the opposite directions through the medium. Configurations (a) and (c) fulfill Eqs. (5) and (6), and configuration (b) is out of the conditions for generating DAPG. Unlike the coherent interaction of the bipolar pulses with a resonant medium to create and erase the population density gratings, the unipolar pulses can create a new type of grating under the certain configurations and time delay setting.
𝐸(𝑡) =
3 ∑ 𝑖=1
𝐸0 exp(−
𝑡𝑖 2 𝜏𝑝 2
)
(5)
where 𝑡1 = 𝑡, 𝑡2 = 𝑡 − 𝜏, 𝑡3 = 𝑡 − 𝜏13 . Fig. 2 shows the population periodically varies according to the time delay between two counter-propagating pulses (pulse 1 and pulse 2). In Fig. 2(a), the DAPG begins to appear at 𝜏13 = 2T0 , and with the increase of 𝜏13 , the DAPG gradually replaces the single amplitude grating. However, 𝜏13 cannot increase indefinitely, and when it continues to increase to 29𝑇0 , the DAPG will disappear and becomes a single-amplitude grating again, as shown in Fig. 2(d). It should be noted that the irregular oscillation emerges in small time delay range in Fig. 2(a) and (b). It can be seen that if the time interval between the 2
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Optics Communications 462 (2020) 125182
Fig. 4. Dependence of the peak-to-peak value of grating amplitudes on time delay 𝜏13 between pulse 1 and pulse 3.
Fig. 2. Dependence of the inversion population on the time delay. The horizontal coordinate is scaled to time constant 𝑇0 . (a) 𝜏13 = 2𝑇0 , (b) 𝜏13 = 5𝑇0 , (c) 𝜏13 = 11𝑇0 , and (d) 𝜏13 = 29𝑇0 .
Fig. 5. Population inversion in dependence on the distance 𝜏 between pulses 1 and 2 and 𝜏13 . All the time durations are normalized to 𝑇0 .
Fig. 3. Comparison of the inversion population as a function of time delay 𝜏 when 𝜏13 changed from 15.1 𝑇0 to 15.9 𝑇0 .
more intuitively shows the amplitudes and phase changes of the DAPG as the time delay 𝜏13 is varied. Under the influence of these two aspects, the grating can be regarded as moving horizontally and accompanied by the periodic change of amplitudes.
first pulse and the third pulse is small, the third pulse and the second pulse will overlap at a certain position in the medium, which causes the irregular oscillations, and it can be eliminated easily by selection of an appropriate time delay between pulses 1 and 3. By comparing and analyzing Fig. 2(a)–(c), it can be found that there is no significant change in the DAPG when 𝜏13 takes the integer multiple of 𝑇0 . In Fig. 3, we demonstrated that the effect on the grating is significant when 𝜏13 is non-integer multiples of 𝑇0 under the condition that DAPG can appear. Taking 15T0 ∼16T0 as an example, Fig. 3 shows the inversion population as a function of the time delay 𝜏 for different values of 𝜏13 , and it also shows that the DAPG can be effectively modulated by time delay. Dependence of the amplitudes of DAPG on time delay 𝜏13 in each 𝑇0 period is plotted in Fig. 4, and the two amplitudes will undergo non-monotonic changes due to the increase of 𝜏13 . Two amplitudes of the DAPG are called A1 and A2 respectively, and they are exchanged and repeat the other change process in two adjacent 𝑇0 cycles, which varies periodically with the increase of the time delay. For above-mentioned amplitudes variation, we also showed that the dependence of the inversion population on the time delay 𝜏 between pulses 1 and 2 and 𝜏13 , visible in Fig. 5. Compared with Fig. 3, Fig. 5
3.2. DAPG generated by four pulses We now discussed the other allowed pulse configuration to generate the DAPG by the four applied pulses. Different to the mentioned threepulse geometry, the third applied pulse is identical to pulse 2 and moves behind it with a time interval 𝜏23 = 2.5𝑇0 , which is a constant. Finally, the fourth pulse passes through the medium in the opposite direction after the third pulse is presented in Fig. 1(c). The expression for the electric field of four pulses can be written as the following form: 𝐸(𝑡) =
4 ∑ 𝑖=1
𝐸0 exp(−
𝑡𝑖 2 𝜏𝑝 2
)
(6)
where 𝑡1 = 𝑡, 𝑡2 = 𝑡 − 𝜏, 𝑡3 = 𝑡2 − 𝜏23 = 𝑡 − 𝜏 − 𝜏23 , 𝑡4 = 𝜏14 − 𝑡. 𝜏14 is the time delay between pulse 1 and pulse 4. The positions of four pulses with electric field (6) on the time axis are shown in Fig. 1(c). The effect of the time delay 𝜏14 on inversion population created by four unipolar pulses is shown in Fig. 6. In Fig. 6(a), the DAPG begins to appear when 𝜏14 = 5T0 , and the DAPG gradually replaces the single-amplitude grating as 𝜏14 increases. As described in Section 3.1, 3
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Optics Communications 462 (2020) 125182
Fig. 8. Time dependence of the two amplitudes of the DAPG on 𝜏14 .
Fig. 6. Inversion population as a function of 𝜏 for different values of 𝜏14 . (a) 𝜏14 = 5𝑇0 , (b) 𝜏14 = 9𝑇0 , (c) 𝜏14 = 13𝑇0 , and (d) 𝜏14 = 29𝑇0 .
Fig. 9. Dependence of population inversion on the distance 𝜏 between pulses 1 and 2 and 𝜏14 , where 𝜏14 is a non-integer multiple of 𝑇0 . Fig. 7. Comparison of the inversion population as a function of 𝜏 when 𝜏14 varying from 15.1𝑇0 to 15.9𝑇0 . The horizontal coordinate is scaled to time constant 𝑇0 .
by pulse time delay 𝜏14 , and the two amplitudes can be converted to each other, as mentioned above. In other words, with the increase of 𝜏14 , the grating can be seen to move horizontally along the transmission
when 𝜏14 continues to increase to 29𝑇0 , the DAPG will disappear and becomes a single-amplitude grating again, as shown in Fig. 6(d). There are also some irregular oscillation emerges as described above in the transition area of the generated single-amplitude grating and DAPG in Fig. 6(a)–(b). The reason for the irregular oscillation is that the fourth pulse overlaps with the second pulse and the third pulse, and it can be eliminated easily by selection of an appropriate time delay between pulses 1 and 4. Similarly, Fig. 6(a)–(c) shows that when 𝜏14 takes an integer multiple of 𝑇0 , it has little effect on the generated DAPG, nevertheless, the grating amplitudes are greatly influenced by the time delay 𝜏14 when it takes a non-integer multiple of 𝑇0 . Fig. 7 shows the inversion population as a function of 𝜏 when 𝜏14 varying from 15.1𝑇0 to 15.9𝑇0 , and with the increase of the time delay 𝜏14 , the grating amplitude changes periodically. Dependence of time delay 𝜏14 on amplitudes of DAPG is shown in Fig. 8, and the two amplitudes will undergo non-monotonic changes as 𝜏14 is varied. Although the variation trend of amplitudes with time delay in a 𝑇0 period is different from that described in Section 3.1, the amplitudes A1 and A2 would also exchange and repeat the change process of the other in the next adjacent phase. Fig. 9 illustrates the dependence of the population inversion 𝑤 after the passage of the pulses as a function of 𝜏14 and of the delay 𝜏, and the inversion depends periodically on 𝜏. Here, the grating amplitudes are greatly influenced
direction of pulse 1, and the amplitude will change periodically. As long as the pulse time delay is selected reasonably, we can effectively control the two amplitudes of the grating. In terms of the generation conditions of the DAPGs, two allowed configurations mentioned above have the similar time delay range, however, the grating characteristics by the two pulse configurations are different. The grating amplitude fluctuation caused by the four pulses is smaller than that of the three pulses, as shown in Figs. 4 and 8, which indicates the DAPGs produced by the four pulses are relatively stable. Fig. 10(a) and (b) shows the dependence of the phase shift 𝜙 of the DAPG on 𝜏13 and 𝜏14 , respectively. In Fig. 10(a), the relationship between phase shift and time delay is linear and monotonic, and it can be seen that as 𝜏14 increases, so does the change of the phase shift 𝜙 in Fig. 10(b). The phase shift here is relative phase shift, that is, the shift of the grating phase relative to the first grating within a 𝑇0 period, as shown in Fig. 3. It is because of such monotonic change in phase shift that explains why the grating is almost unaffected when the time delays 𝜏13 and 𝜏14 take an integer multiple of 𝑇0 . 4
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Optics Communications 462 (2020) 125182
Fig. 10. (a). Phase shift of DAPGs as a function of time delay 𝜏13 . (b). Dependence of phase shift of DAPGs on time delay 𝜏14 .
4. Conclusion
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In summary, we proposed an efficient scheme to generate DAPGs by means of spatially periodic modulation during the optical nonoverlapping interaction of unipolar pulses, and evaluated the grating performance. Two allowed pulse configurations are considered to obtain the respective time delay requirements by numerically solving the density matrix equations. Comparative analysis of the grating amplitudes and phase shift shows that their behavior depends sensitively on time delay, and the grating amplitudes exhibit a temporally periodic evolution, while the phase shift increases linearly with time delay. It is noted that the grating amplitudes are almost immune to time delay which takes the integer multiples of 𝑇0 , however, there emerges significant fluctuations between two adjacent moments with integer multiples of 𝑇0 , and the grating amplitudes change alternately in the two adjacent stages. Ultrafast modification of the spatial frequency of the gratings presented here can be used for the fabrication of ultrafast laser beam deflectors, which makes it possible to implement spatial-angle scanning of laser beams [32,33]. The physical mechanism governing such kind of gratings with different amplitudes originates from the atomic response to the coherent optical fields and need to be further quantitative studies in the future work. Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 11004152) and Natural Science Foundation of Tianjin, China (No. 19JCYBJC16100). References [1] H.J. Eichler, A. Hermerschmidt, Light-induced dynamic gratings and photorefraction, in: Photorefractive Materials and their Applications, Vol. 1, Springer, 2006, pp. 7–42. [2] G.D. Scholes, J. Kim, C.Y. Wong, Exciton spin relaxation in quantum dots measured using ultrafast transient polarization grating spectroscopy, Phys. Rev. B 73 (2006) 195325. [3] L. Van Dao, M. Lowe, P. Hannaford, H. Makino, T. Takai, T. Yao, Femtosecond three-pulse photon echo and population grating studies of the optical properties of CdTe/ZnSe quantum dots, Appl. Phys. Lett. 81 (2002) 1806–1808. [4] I. Abella, N. Kurnit, S. Hartmann, Photon echoes, Phys. Rev. 141 (1966) 391. [5] V. Da Silva, Y. Silberberg, Accumulated photon echo in er-doped fibers, Braz. J. Phys. 26 (1996) 471–481. [6] E.I. Shtyrkov, Optical echo holography, Opt. Spectrosc. 114 (2013) 96–103. [7] J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena, Elsevier, 2006. [8] F. Krausz, M. Ivanov, Attosecond physics, Rev. Modern Phys. 81 (2009) 163–234. 5
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[28] R.M. Arkhipov, M.V. Arkhipov, A.V. Pakhomov, D.O. Zhiguleva, N.N. Rosanov, Collisions of single-cycle and subcycle attosecond light pulses in a nonlinear resonant medium, Opt. Spectrosc. 124 (2018) 541–548. [29] M. Colice, F. Schlottau, K.H. Wagner, Broadband radio-frequency spectrum analysis in spectral-hole-burning media, Appl. Opt. 45 (2006) 6393–6408. [30] N. Ohlsson, M. Nilsson, S. Kröll, R.K. Mohan, Long-time-storage mechanism for Tm: YAG in a magnetic field, Opt. Lett. 28 (2003) 450–452.
6
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Performance analysis of double-amplitude population gratings by non-overlapping unipolar pulses Huadi Zhang, Shuanggen Zhang ∗, Shengdong Li, Xiurong Ma Tianjin University of Technology, School of Electrical and Electronic Engineering, Engineering Research Center of Optoelectronic Devices & Communication Technology, Ministry of Education, Tianjin Key Laboratory of Film Electronic and Communication Device, No. 391 Binshui West Street, Xiqing District, Tianjin 300384, China
ARTICLE
INFO
Keywords: Population gratings Double-amplitude gratings Unipolar pulses Pulse propagation Time delay
ABSTRACT We investigate the performance of the induced double-amplitude population grating (DAPG) based on spatially periodic modulation by the optical non-overlapping interaction of unipolar pulses. Considering the two typical allowed pulse configurations, the time delay requirements and the relationship between the grating amplitude, phase shift and the time delay were analyzed in detail. The results reveal that the grating amplitudes and phase can be dynamically controlled by the suitable time delay between the excited pulses. The population gratings presented here maybe offer more potential applications in nonlinear optics.
1. Introduction
the resonant propagation of several periodic pulses in an asymmetric medium having a periodic sub-wavelength structure [19]. Most
Periodic spatial population gratings are typically generated in resonant mediums via interference of two or more spatially overlapping light waves [1], but in this situation, the spatial period is limited by the light wavelength and pulses have to be long in duration. These gratings found lots of applications in the spectroscopy of different species and in nonlinear optics etc. [2,3]. Multi-beam optical pulses that do not overlap in time domain can interact with non-uniform broadening medium [4–6], and one pulse may induce polarization oscillations that remain after the pulse passage. Such induced polarization gratings can interact with the second-temporally delayed pulse, thus changing the overall inversion and forming transient population distribution gratings similar to coherent fringe structures in the medium. So far, ultrashort optical pulses have been widely used in the field of ultrafast optics [7–9], and usually the generated pulses are bipolar which have zero electric field area, i.e., integral of the electric-field strength with respect to time at a given point in space. In recent years, the generation of unipolar pulses has aroused people’s interest [10– 13], and such pulses contain a single half-period electric field so that the integral of the electric field intensity over the duration of the pulse is not equal to zero [12,14–16]. Unipolar pulses has a number of advantages not only due to their short duration, but also it allows the effective transfer of kinetic energy to the particles and effective control over the motion of the charges [10,11]. This circumstance will allow one to produce and control gratings, which is considerably faster than in the case of bipolar pulses [17,18]. Generally, such pulses can exist as soliton solutions to nonlinear optical equations [10,11], and it is also possible to generate extremely short unipolar pulses during
recently, some methods have been proposed for generating unipolar pulses in Raman-active media [20–22] or by using nonlinear field coupling technique [23,24]. It has been predicted that the indirect interactions of the light pulses with polarization waves by the other pulse can create, erase and manipulate the population density gratings [25,26]. Gratings formation and control by the unipolar pulses was studied both theoretically and numerically in [17,18,27]. The above work mainly discussed the possibility of the uniform gratings creation and erasure in a resonant medium by the pulse trains. In Ref. [28], the overlapping interaction of the bipolar pulse and the unipolar pulse in a resonant medium has been studied. However, characteristic analysis of DAPGs by the non-overlapping unipolar pulses has not yet been clarified so far. In this paper, we presented and evaluated an scheme to generate DAPGs based on spatially periodic modulation by the optical nonoverlapping interaction of unipolar pulses passing through a resonant medium, taking the Tm3+ : YAG crystal [29,30] for an example. Two typical allowed pulse configurations are considered via the numerical treatment of the Bloch equations for the density matrix elements, and the time delay requirements for the respective generation of DAPG were confirmed. Additionally, we detailed analyses on relationship between the amplitude, phase shift of DAPG and the time delay between the excited pulses.
∗ Corresponding author. E-mail address: [email protected] (S. Zhang).
https://doi.org/10.1016/j.optcom.2019.125182 Received 11 November 2019; Received in revised form 5 December 2019; Accepted 23 December 2019 Available online 21 January 2020 0030-4018/© 2019 Published by Elsevier B.V.
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Optics Communications 462 (2020) 125182
2. Theoretical model and qualitative analysis Our work will focus on the case where the unipolar pulses are coherent with the resonant medium and the applied pulses are identical in shape and amplitude. Before proceeding to numerical calculations, we consider the notion of the pulse area that is an important quantity ∞ to describe the pulse dynamics, 𝜃 = 𝜇ℏ ∫−∞ 𝜀(𝑡)𝑑𝑡, where 𝜀(𝑡) is the pulse envelope. For long pulses, the pulse with the 𝜋∕2 area fully saturates the medium. However, the action of ultrashort pulses can strongly differ from those for long pulses, and a real-valued envelope is not well defined, thus, the determination of the pulse area should be performed taking into account the phase change of light wave oscillations. Assume that the unipolar pulses have a brief Gaussian profile as following: ( ( )2 ) 𝑁 𝑁 ∑ ∑ 𝑡 − 𝜏𝑖 𝐸(𝑡) = 𝐸𝑖 (𝑡) = 𝐸0 exp − (1) 𝜏𝑝 2 𝑖=1 𝑖=1 where, 𝐸0 is the amplitude and N is the number of pulses. Time delay 𝜏𝑖 between the 1st pulse and the 𝑖th pulse can be adjusted so that the pulses did not overlap in the medium, and 𝜏𝑝 is the pulse duration.
Fig. 1. Typical pulse configurations to generate transient population gratings. The applied pulse 1, pulse 2, pulse 3 and pulse 4 are represented in orange, blue, red, and purple, respectively. The resonant medium is drawn as a dotted and light purple rectangle. . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The resonant medium is described as a two-level system, and the wave function[of two-level] structure can be expressed by density op𝜌 (𝑡) 𝜌12 (𝑡) erator: 𝜌(𝑡) = 11 , 𝜌11 and 𝜌22 indicate the probability that 𝜌21 (𝑡) 𝜌22 (𝑡)
3. Results and discussion
the atom is in the ground state and the excited state, respectively. 𝜕𝑡 𝜌(𝑡) = − ℏ𝑖 [𝐻(𝑡)𝜌(𝑡) − 𝜌(𝑡)𝐻(𝑡)], and H(t) is the Hamiltonian of the [ ] ℏ𝜔2 −𝜇21 𝐸(𝑡) system under the action of the light field, 𝐻(𝑡) = , ∗ −𝜇12 𝐸 (𝑡) ℏ𝜔1
3.1. DAPG generated by three pulses Firstly, two applied pulses propagate toward each other and pass through the resonant medium but do not meet in it. The first pulse will be a source of the polarization wave, and the polarization wave interfere with pulse 2. It should be noted that the phase velocity of light is significantly greater than the phase velocity of polarized waves. The optical field can be written in the form: ( ) 2 ∑ 𝑡2 𝐸0 exp − 𝑖 𝐸(𝑡) = (4) 𝜏𝑝 2 𝑖=1
where 𝜔1 , 𝜔2 are the frequencies of upper and lower energy levels, and 𝜇12 , 𝜇21 are electric dipole moments. Thus, the coherent interaction of atomic system with the unipolar pulses can be evaluated by the following density matrix equations: 𝜌12 (𝑡) 𝑖 − 𝜇𝐸(𝑡)𝑤(𝑡) + 𝑖𝜔0 𝜌12 (𝑡) 𝑇2 ℏ [ ] 𝑤(𝑡) − 𝑤0 4 𝜕𝑡 𝑤(𝑡) = − + 𝜇𝐸(𝑡)Im 𝜌12 (𝑡) 𝑇1 ℏ
𝜕𝑡 𝜌12 (𝑡) = −
(2) (3)
where 𝑡1 = 𝑡, 𝑡2 = 𝑡1 − 𝜏 = 𝑡 − 𝜏. 𝜏 is the time delay between the two applied pulses. The effective area of the two pulses strongly depends on the phase of the second pulse, i.e., the time delay 𝜏. Other parameters are as following: 𝜔0 = 2.69 × 1015 Hz, 𝜏p = 7.37 × 10−16 s, 𝜇 = 5 × 10−18 𝐶𝐺𝑆𝐸, 𝑇 = 2𝜋∕𝜔0 , and 𝑇0 is set to be 1.5T. The total integration time is set to be 10T 0 . Since the two unipolar pulses shoot into the medium successively from two opposite directions, the arrival time delay of the pulse in the medium decreases linearly in the transmission direction, which causes the periodic change of the particle population distribution along transmission direction, and thus forms the population grating. Subsequently, the third pulse is identical to the first pulse and moves behind it with a time delay 𝜏13 . The positions of three pulses with electric field (5) on the time axis are shown in Fig. 1(a). The expression of the electric field of three pulses has the form:
where 𝜌12 is the non-diagonal element, 𝑤 = 𝜌22 − 𝜌11 is the population difference between the ground and excited states of the two-level system, 𝑤0 is the population difference in the absence of an electric field (for the absorbing medium 𝑤0 = 1), 𝜔0 is the resonant transition frequency of the medium, and 𝜇 is the transition dipole moment of the atoms. 𝑇1 and 𝑇2 are the relaxation times, ℏ is the Planck constant, and 𝐸(𝑡) is the electric field. Eqs. (2) and (3) describe the evolution of the off-diagonal element and the difference between the diagonal elements of the density matrix, which can be numerically solved by the fourth-order Runge–Kutta method [31]. For simplicity, we consider the case of a thin medium and restrict ourselves to a one-dimensional geometry with light propagating in 𝑧direction. The identical unipolar pulses act on the resonant medium in different directions shown in Fig. 1. In Fig. 1(a), pulse 3 is in the same direction as pulse 1, and time interval is 𝜏13 , and pulse 2 enters into the medium in the opposite direction. Fig. 1(b) shows that pulse 3 propagates along the medium in the direction of pulse 2 with time delay 𝜏23 . As shown in Fig. 1(c), the unipolar pulses [(1) and (4)] and [(2) and (3)] propagating in the opposite directions through the medium. Configurations (a) and (c) fulfill Eqs. (5) and (6), and configuration (b) is out of the conditions for generating DAPG. Unlike the coherent interaction of the bipolar pulses with a resonant medium to create and erase the population density gratings, the unipolar pulses can create a new type of grating under the certain configurations and time delay setting.
𝐸(𝑡) =
3 ∑ 𝑖=1
𝐸0 exp(−
𝑡𝑖 2 𝜏𝑝 2
)
(5)
where 𝑡1 = 𝑡, 𝑡2 = 𝑡 − 𝜏, 𝑡3 = 𝑡 − 𝜏13 . Fig. 2 shows the population periodically varies according to the time delay between two counter-propagating pulses (pulse 1 and pulse 2). In Fig. 2(a), the DAPG begins to appear at 𝜏13 = 2T0 , and with the increase of 𝜏13 , the DAPG gradually replaces the single amplitude grating. However, 𝜏13 cannot increase indefinitely, and when it continues to increase to 29𝑇0 , the DAPG will disappear and becomes a single-amplitude grating again, as shown in Fig. 2(d). It should be noted that the irregular oscillation emerges in small time delay range in Fig. 2(a) and (b). It can be seen that if the time interval between the 2
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Optics Communications 462 (2020) 125182
Fig. 4. Dependence of the peak-to-peak value of grating amplitudes on time delay 𝜏13 between pulse 1 and pulse 3.
Fig. 2. Dependence of the inversion population on the time delay. The horizontal coordinate is scaled to time constant 𝑇0 . (a) 𝜏13 = 2𝑇0 , (b) 𝜏13 = 5𝑇0 , (c) 𝜏13 = 11𝑇0 , and (d) 𝜏13 = 29𝑇0 .
Fig. 5. Population inversion in dependence on the distance 𝜏 between pulses 1 and 2 and 𝜏13 . All the time durations are normalized to 𝑇0 .
Fig. 3. Comparison of the inversion population as a function of time delay 𝜏 when 𝜏13 changed from 15.1 𝑇0 to 15.9 𝑇0 .
more intuitively shows the amplitudes and phase changes of the DAPG as the time delay 𝜏13 is varied. Under the influence of these two aspects, the grating can be regarded as moving horizontally and accompanied by the periodic change of amplitudes.
first pulse and the third pulse is small, the third pulse and the second pulse will overlap at a certain position in the medium, which causes the irregular oscillations, and it can be eliminated easily by selection of an appropriate time delay between pulses 1 and 3. By comparing and analyzing Fig. 2(a)–(c), it can be found that there is no significant change in the DAPG when 𝜏13 takes the integer multiple of 𝑇0 . In Fig. 3, we demonstrated that the effect on the grating is significant when 𝜏13 is non-integer multiples of 𝑇0 under the condition that DAPG can appear. Taking 15T0 ∼16T0 as an example, Fig. 3 shows the inversion population as a function of the time delay 𝜏 for different values of 𝜏13 , and it also shows that the DAPG can be effectively modulated by time delay. Dependence of the amplitudes of DAPG on time delay 𝜏13 in each 𝑇0 period is plotted in Fig. 4, and the two amplitudes will undergo non-monotonic changes due to the increase of 𝜏13 . Two amplitudes of the DAPG are called A1 and A2 respectively, and they are exchanged and repeat the other change process in two adjacent 𝑇0 cycles, which varies periodically with the increase of the time delay. For above-mentioned amplitudes variation, we also showed that the dependence of the inversion population on the time delay 𝜏 between pulses 1 and 2 and 𝜏13 , visible in Fig. 5. Compared with Fig. 3, Fig. 5
3.2. DAPG generated by four pulses We now discussed the other allowed pulse configuration to generate the DAPG by the four applied pulses. Different to the mentioned threepulse geometry, the third applied pulse is identical to pulse 2 and moves behind it with a time interval 𝜏23 = 2.5𝑇0 , which is a constant. Finally, the fourth pulse passes through the medium in the opposite direction after the third pulse is presented in Fig. 1(c). The expression for the electric field of four pulses can be written as the following form: 𝐸(𝑡) =
4 ∑ 𝑖=1
𝐸0 exp(−
𝑡𝑖 2 𝜏𝑝 2
)
(6)
where 𝑡1 = 𝑡, 𝑡2 = 𝑡 − 𝜏, 𝑡3 = 𝑡2 − 𝜏23 = 𝑡 − 𝜏 − 𝜏23 , 𝑡4 = 𝜏14 − 𝑡. 𝜏14 is the time delay between pulse 1 and pulse 4. The positions of four pulses with electric field (6) on the time axis are shown in Fig. 1(c). The effect of the time delay 𝜏14 on inversion population created by four unipolar pulses is shown in Fig. 6. In Fig. 6(a), the DAPG begins to appear when 𝜏14 = 5T0 , and the DAPG gradually replaces the single-amplitude grating as 𝜏14 increases. As described in Section 3.1, 3
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Optics Communications 462 (2020) 125182
Fig. 8. Time dependence of the two amplitudes of the DAPG on 𝜏14 .
Fig. 6. Inversion population as a function of 𝜏 for different values of 𝜏14 . (a) 𝜏14 = 5𝑇0 , (b) 𝜏14 = 9𝑇0 , (c) 𝜏14 = 13𝑇0 , and (d) 𝜏14 = 29𝑇0 .
Fig. 9. Dependence of population inversion on the distance 𝜏 between pulses 1 and 2 and 𝜏14 , where 𝜏14 is a non-integer multiple of 𝑇0 . Fig. 7. Comparison of the inversion population as a function of 𝜏 when 𝜏14 varying from 15.1𝑇0 to 15.9𝑇0 . The horizontal coordinate is scaled to time constant 𝑇0 .
by pulse time delay 𝜏14 , and the two amplitudes can be converted to each other, as mentioned above. In other words, with the increase of 𝜏14 , the grating can be seen to move horizontally along the transmission
when 𝜏14 continues to increase to 29𝑇0 , the DAPG will disappear and becomes a single-amplitude grating again, as shown in Fig. 6(d). There are also some irregular oscillation emerges as described above in the transition area of the generated single-amplitude grating and DAPG in Fig. 6(a)–(b). The reason for the irregular oscillation is that the fourth pulse overlaps with the second pulse and the third pulse, and it can be eliminated easily by selection of an appropriate time delay between pulses 1 and 4. Similarly, Fig. 6(a)–(c) shows that when 𝜏14 takes an integer multiple of 𝑇0 , it has little effect on the generated DAPG, nevertheless, the grating amplitudes are greatly influenced by the time delay 𝜏14 when it takes a non-integer multiple of 𝑇0 . Fig. 7 shows the inversion population as a function of 𝜏 when 𝜏14 varying from 15.1𝑇0 to 15.9𝑇0 , and with the increase of the time delay 𝜏14 , the grating amplitude changes periodically. Dependence of time delay 𝜏14 on amplitudes of DAPG is shown in Fig. 8, and the two amplitudes will undergo non-monotonic changes as 𝜏14 is varied. Although the variation trend of amplitudes with time delay in a 𝑇0 period is different from that described in Section 3.1, the amplitudes A1 and A2 would also exchange and repeat the change process of the other in the next adjacent phase. Fig. 9 illustrates the dependence of the population inversion 𝑤 after the passage of the pulses as a function of 𝜏14 and of the delay 𝜏, and the inversion depends periodically on 𝜏. Here, the grating amplitudes are greatly influenced
direction of pulse 1, and the amplitude will change periodically. As long as the pulse time delay is selected reasonably, we can effectively control the two amplitudes of the grating. In terms of the generation conditions of the DAPGs, two allowed configurations mentioned above have the similar time delay range, however, the grating characteristics by the two pulse configurations are different. The grating amplitude fluctuation caused by the four pulses is smaller than that of the three pulses, as shown in Figs. 4 and 8, which indicates the DAPGs produced by the four pulses are relatively stable. Fig. 10(a) and (b) shows the dependence of the phase shift 𝜙 of the DAPG on 𝜏13 and 𝜏14 , respectively. In Fig. 10(a), the relationship between phase shift and time delay is linear and monotonic, and it can be seen that as 𝜏14 increases, so does the change of the phase shift 𝜙 in Fig. 10(b). The phase shift here is relative phase shift, that is, the shift of the grating phase relative to the first grating within a 𝑇0 period, as shown in Fig. 3. It is because of such monotonic change in phase shift that explains why the grating is almost unaffected when the time delays 𝜏13 and 𝜏14 take an integer multiple of 𝑇0 . 4
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Fig. 10. (a). Phase shift of DAPGs as a function of time delay 𝜏13 . (b). Dependence of phase shift of DAPGs on time delay 𝜏14 .
4. Conclusion
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In summary, we proposed an efficient scheme to generate DAPGs by means of spatially periodic modulation during the optical nonoverlapping interaction of unipolar pulses, and evaluated the grating performance. Two allowed pulse configurations are considered to obtain the respective time delay requirements by numerically solving the density matrix equations. Comparative analysis of the grating amplitudes and phase shift shows that their behavior depends sensitively on time delay, and the grating amplitudes exhibit a temporally periodic evolution, while the phase shift increases linearly with time delay. It is noted that the grating amplitudes are almost immune to time delay which takes the integer multiples of 𝑇0 , however, there emerges significant fluctuations between two adjacent moments with integer multiples of 𝑇0 , and the grating amplitudes change alternately in the two adjacent stages. Ultrafast modification of the spatial frequency of the gratings presented here can be used for the fabrication of ultrafast laser beam deflectors, which makes it possible to implement spatial-angle scanning of laser beams [32,33]. The physical mechanism governing such kind of gratings with different amplitudes originates from the atomic response to the coherent optical fields and need to be further quantitative studies in the future work. Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 11004152) and Natural Science Foundation of Tianjin, China (No. 19JCYBJC16100). References [1] H.J. Eichler, A. Hermerschmidt, Light-induced dynamic gratings and photorefraction, in: Photorefractive Materials and their Applications, Vol. 1, Springer, 2006, pp. 7–42. [2] G.D. Scholes, J. Kim, C.Y. Wong, Exciton spin relaxation in quantum dots measured using ultrafast transient polarization grating spectroscopy, Phys. Rev. B 73 (2006) 195325. [3] L. Van Dao, M. Lowe, P. Hannaford, H. Makino, T. Takai, T. Yao, Femtosecond three-pulse photon echo and population grating studies of the optical properties of CdTe/ZnSe quantum dots, Appl. Phys. Lett. 81 (2002) 1806–1808. [4] I. Abella, N. Kurnit, S. Hartmann, Photon echoes, Phys. Rev. 141 (1966) 391. [5] V. Da Silva, Y. Silberberg, Accumulated photon echo in er-doped fibers, Braz. J. Phys. 26 (1996) 471–481. [6] E.I. Shtyrkov, Optical echo holography, Opt. Spectrosc. 114 (2013) 96–103. [7] J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena, Elsevier, 2006. [8] F. Krausz, M. Ivanov, Attosecond physics, Rev. Modern Phys. 81 (2009) 163–234. 5
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