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Abstract
Both integer and fractional electromagnetically induced Talbot effects are experimentally investigated in a coherent rubidium 5S1/2 − 5P3/2 − 5D5/2 ladder-type system. By launching a probe laser into a periodically modulated lattice constructed by two crossed coupling fields with a small angle inside the rubidium vapor, a high-resolution diffraction pattern is obtained. The diffraction pattern is reproduced completely at detection positions of an integer multiple of twice the Talbot lengths. Meanwhile, the fractional Talbot effect, presented as complicated subimages at special positions, is also clearly observed. Furthermore, the theoretical simulations are conducted and agree well with the experimental results. These results pave the way for studying the control of light dynamics based on the periodically modulated medium.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The Talbot effect, a near-field diffraction effect, arises when a grating or periodic structure medium is illuminated by a normally-incident monochromatic plane wave [1]. The original periodic object image is reproduced exactly at multiples of a certain distance away from the object plane in the Fresnel approximation regime (integer Talbot effect). Furthermore, the fractional Talbot effect refers to a complicated subimages when detection positions are located between every two adjacent self-images [2]. Owing to their wide practicability and simplicity, interesting applications have been proposed in many different areas, such as optical testing and metrology [3], Talbot array illumination [4], optical imaging and computing [5]. More recently, the Talbot effect has been demonstrated in several different systems, such as atomic waves [6], waveguide arrays [7], x-ray phase imaging [8,9], Bose-Einstein condensates [10], exciton polaritons [11], metamaterials [12], surface plasmonic [13]. In addition, the Talbot effect has been extended to temporal Talbot effect [14], nonlinear Talbot effect [15,16], angular Talbot effect [17,18] and PT-symmetric Talbot effects [19].
The coherent atomic system has drawn enormous attentions because of its easy reconfiguration, flexible tunability, and especially the various coherence control techniques enabled by electromagnetically induced transparency (EIT) [20–22]. In EIT related medium, a standing-wave coupling field renders electromagnetically induced grating (EIG) [23–25], electromagnetically induced focusing [26], stationary light pulse [27–30], controllable photonic crystal [31] and parity-time (PT) symmetry [32–35]. All of these intriguing observations resulting from the formation of periodically modulated refractive index in an EIT-related atomic medium. Thus it is natural to use the coherently prepared multilevel atomic system as a fertile platform to experimentally realize Talbot effect, which is known as the electromagnetically induced Talbot effect (EITE) [36].
The EITE has distinctive merits since the combination of the optical Talbot effect with the coherent atomic media, which possess coherent nature and multi-parameter tunable characteristics. The EITE was experimentally demonstrated for the first time in a Λ-type three-level atomic system just recently [37], however, a fully reproduce of the Talbot image was not observed and the fractional Talbot effect was only simply depicted. With great potential applications, an elaborate experiment research is urgently required, which is more challenging for accuracy of experiment. The key point of these research works is to make a periodically modulated struture so that the probe laser can be confined at the discrete sites in the weakly guiding waveguides of the lattice, which has been extensively studied in the former researches [38,39].
In this work, a standing-wave coupling field is used to construct the periodically modulated lattice and a probe laser incident on this particular periodic medium is utilized to study the integer and fractional electromagnetically induced Talbot effects in a coherent rubidium 5S1/2 − 5P3/2 − 5D5/2 ladder-type system. This standing-wave coupling field consists of two crossed 776 nm beams with a small angle. Then a clear diffraction pattern is obtained after a 780 nm probe laser passing through the standing-wave field. As a result, a fully experimental observation of the EITE and a high-resolution fractional Talbot effect have been observed for the first time to our knowledge. A comprehensive theoretical simulation has been conducted as well. The measurement of the diffraction patterns in a wide distance range becomes possible, thus the observation of full integer Talbot effect is realized. Furthermore, the high precision control of the experimental parameters (laser frequency detuning, measurement position and vapor temperature) and the high-resolution detection system ensured the observation of the fractional Talbot effect. These results will further stimulate the development of the theoretical research and practical application based on the Talbot effect.
2. Experiment setup
We employed a ladder-type coherent atomic system that is similar with our previous report [39]. The Doppler-free configuration in a ladder-type system can be achieved only when the probe and coupling fields are counter-propagating and having similar frequencies [40]. The related energy levels and the experiment setup are shown in Figs. 1(a) and 1(c). The probe laser couples the 5S1/2 − 5P3/2 transition and excites the atoms into 5P3/2 state, and the coupling laser resonances with the 5P3/2 − 5D5/2 transition. This coherent process can be detected for atoms in the 5D5/2 state having a significant probability of decaying into the 5S1/2 ground state via 6P3/2 state by emission of 420 nm fluorescence [41].
Fig. 1 (a) The related energy levels of 85Rb ladder-type atomic system. (b) The schematic plot of the periodically modulated lattice by two crossed coupling fields. (c) Experiment setup, HWP: half-wave plate, PBS: polarization beam splitter, QWP: quarter-wave plate, G: glass, L: lens, M: high reflection mirror, BS: beam splitter, PD: photodiode detector, CCD: charge-coupled device, AP: anamorphic prism, AOM: acousto-optic modulator, BB: beam block, SAS: saturation absorption spectroscopy, EIT: electromagnetically induced transparency.
Two independent external cavity diode lasers (DL pro, Toptica) with Gaussian profile provide the required laser beams, the probe laser (Rabi frequency Ωp, detuning from the atomic resonant frequency Δp) operating around the D2 line of rubidium (780 nm), the coupling laser operating around the 5P3/2 − 5D5/2 transition (776 nm). The frequency locking for the probe laser is achieved by a saturation absorption spectroscopy method. The coupling laser (Rabi frequency Ωc, detuning from the atomic resonant frequency Δc) frequency is monitored by a wavemeter (WS-7, Highfiness) and shifted by a double-pass acousto-optics modulator system in order to keep the same propagation direction [42]. The shifted coupling laser splits into two beams. One is used to obtain the electromagnetically induced transparency (EIT) signal together with the probe laser for monitoring the coherent of the system. The other one is shaped into an elliptical profile by an anamorphic prism pair, which is then split into two beams after passing through a beam splitter. These two beams, with same intensity and profile, recombine together with a small angle θ ≈ 0.2° in the center of the rubidium vapor cell. These two crossed coupling beams will interference with each other and induce a standing-wave field in the x direction perpendicular to the probe laser propagation with a period of d = λc/(2sin θ)≈110 μm, where λc is the wavelength of the coupling laser. The related schematic is shown in Fig. 1(b). The vapor is 10 cm long and 2.5 cm in diameter, and the temperature is controlled by a self-feedback system with a resolution of 0.1 K. The temperature of the cell is set around 377.0 K, so the atomic density of saturated Rb vapor is about 1.0 × 1013 cm−3. The probe and coupling laser beams are linearly polarized and counter-propagated through the vapor. The focused probe laser beam in the center of the vapor cell is 77 μm measured by the beam quality analyzer (BP209-VIS, Thorlabs). A charge-coupled device (CCD) along the direction of probe laser propagation is employed for detecting the beam profile of probe laser. An electronically controlled stage is used to move the CCD along the direction of probe laser propagation.
3. Theoretical analysis
In the presence of the rotating-wave approximation and a rotating frame, the Hamiltonian of this system can be written as
where Ωp = μ21Ep/ħ (Δp = ωp −ω21), Ωc = μ32Ec/ħ (Δc = ωc −ω32) are the Rabi frequencies (the detunings) of the probe and coupling fields, respectively, with μij (i, j = 1, 2, 3) being the transition dipole momentum between the levels |i〉 and |j〉 and ωij = ωi − ωj being the resonant transition frequency, and H.c. is the Hermitian conjugate.The density matrix master equation of the Hamiltonian Eq. (1) is governed by the following Lindblad form [20]
where Γ21 and Γ32 denotes the spontaneous emission rate from |2〉 to |1〉 and |3〉 to |2〉 respectively. σij = |i〉 〈j| is the atomic projection operator. The susceptibility, which is associated with the optical properties of the atomic medium, specifically the dispersion and absorbtion of the probe laser, can be obtained througth . Considering the steady state of Eq. (2) and the standing-wave field induced by two symmetrical coupling fields, the spatially modulated susceptibility can be expressed as where d is the spatial period of the standing-wave field along x direction. It should be noted that the periodicity of the optical lattice can be modulated by varying the angle between the two wave vectors of two coupling fields.From Eq. (3), we have periodically modulated the refractive index experienced by the probe laser based on the experimental scheme. For an atomic medium of thickness L along the z direction and modulated in the x direction, the transmission function of the probe laser at z = L can be given by
where χ′ (χ″) is the real (imaginary) part of the susceptibility χ, and kp = 2π/λp, Ep (x, 0) is the input profile of plane-wave Ep. Because of the periodicity of χ, the representation of probe transmission can be rewritten in the form of Fourier series. According to the Fresnel-Kirchhoff diffraction theory and under the paraxial approximation, the diffraction amplitude Ep(X, z) of the probe laser at the observation plane with a distance of z from the output surface can be expressed as where x and X are the coordinates in the object and observation planes, respectively. Completing the integral in Eq. (5) with the Fourier series expansion of Ep (x, L), we can obtain the Talbot effect as described by with En being the Fourier coefficient. The Talbot length is defined by zT = d2/λp, and then From Eq. (7), we can see the typical features of the EITE. At a certain distance z = mzT, where m denotes a positive integer referred to as the self-imaging number, all the diffraction orders are in phase. Specifically, for even integers, the probe transmission appears exactly the same as the initial one, while for odd integers, it is shifted by half a period (d/2) with respect to the even-integer case. Moreover, the fractional Talbot images appears at all rational multiples of zT, i.e., z = (p/q)zT, where p, q are coprime integers. To see these more clearly, we show the numerical results in Fig. 2. Since the optically induced lattice established by the interference of two lasers is in the x direction, the probe laser propagates (z direction) perpendicularly the lattice. Thus, the theoretical simulation is accomplished in x − z plane to show the Talbot image. Figure 2(a) displays the Talbot effect carpet based on Eqs. (3) and (7). Moreover, in Fig. 2(b), the intensity distributions are shown at different Talbot distances corresponding to z = 0, z = zT/2, z = zT, z = 3zT/2 and z = 2zT. Figure 2(c) shows the intensity distributions of several special distances corresponding to z = 0, z = zT/5, z = zT/4, z = zT/3, z = 2zT/5 and z = zT/2.Fig. 2 (a) The diffraction pattern in the near field of a periodic grating in the z-x plane, shown as a “Talbot carpet”. The plotted paramrters are chosen as Ωc/2π = 20 MHz, Δc = Δp = 0. The calculated intensity distributions of the probe field at the output surface (b) z = 0, z = zT/2, z = zT, z = 3zT/2 and z = 2zT (c) z = 0, z = zT/5, z = zT/4, z = zT/3, z = 2zT/5 and z = zT/2.
4. Results and discussions
Figure 3 shows the output and intensity profiles of the probe laser without/with the standing-wave field from CCD. The probe laser remains its original Gaussian profile with the absence of the coupling field (Figs. 3(a) and 3(b)). However, the high-resolution discrete diffraction pattern of the probe laser is observed with the standing-wave field, which induces the periodic modulated lattice. As shown in Figs. 3(c) and 3(d), the clear diffraction pattern is observed up to third order. Here, the powers of the coupling laser and the probe laser are 16 mW and 3 mW, respectively. The temperature of the cell is precisely controlled at 377.6 K. The two-photon detuning is 20 MHz throughout the experiment to ensure a better phase modulation of the medium.
Fig. 3 The output and intensity profiles of the probe laser without (a)–(b)/with (c)–(d) the standing-wave field.
The detection position with a clear diffraction pattern shown in Fig. 3(c) is defined as z = 0. Figures 4(a)–4(i) illustrate the diffraction pattern at different positions by precisely moving the CCD along the direction of probe laser propagation, which utilize same experimental parameters with that of Fig. 3.
Fig. 4 The diffraction patterns at different detection positions. (a)–(i) show the images at representative positions of z =0, 8, 15, 17, 23, 30, 32, 38, 45 mm, respectively. The crossed lines in each figure identify same position for reference.
The diffraction patterns at different positions show clear movement regularity in accordance with the theoretical simulation as shown in Fig. 2. The image at z = 30 mm (≈2zT) reproduces the pattern at z = 0 mm, indicating the full integer Talbot effect is realized. To our best knowledge, it is the first time that the full integer Talbot effect in coherent atomic system is clarified. The result gives a Talbot length of about zT =z /2 = 15 mm. Taking account of the experimental parameters, the calculated Talbot length is zT = d2/λp≈ 15.3 mm that agrees well with experiment result. The images at z = 15 mm (≈zT) and z = 45 mm (≈3zT) almost remain the same but each order of the diffraction patterns shifts about a half period of d along x direction compared to that at z = 0. The image at z = 17 mm shows the similar profile with z = 15 mm but the position of each order of the diffraction patterns is lower than that of z = 15 mm. When the observation position exceeds 2zT, the diffraction pattern restarts the next cycle of movement, which can be clearly seen from the Fig. 4(g). At odd multiple of half the Talbot length, the self-image is halved in size, and appears with half the period (thus twice as many images are seen), which can be seen in Figs. 4(b), 4(e) and 4(h) with observation positions around zT/2, 3zT/2, 5zT/2, respectively. The experimental results agree well with the theoretical simulation in Figs. 2(a) and 2(b). A slight error from the measured positions and the theoretical ones is inevitable, which is coming from the uncertainty of the estimation of the small angle. The crossed lines in each figure identify same position for reference.
To further investigate the fractional Talbot effect of the diffraction pattern, the diffraction patterns between 0 and 8 mm (≈zT/2) with a precise distance change were measured, which are illustrated in Fig. 5. The high-resolution fractional Talbot effect is clearly demonstrated at the predicted distances above. It can be understood as the coherent superposition of many waves. At the observation position of 5 mm (≈ zT/3), a superposition of three waves is obtained, which can be clearly shown in Fig. 5(d). At the position of 4 mm (≈zT/4) as shown in Fig. 5(c), the period and size of the images is halved again compared with half the Talbot length, and so forth creating a fractal pattern of sub-images with ever-decreasing size, often referred to as a Talbot carpet [2]. For the observation positions of the 3 mm (≈zT/5) and 6 mm (≈2zT/5), more complicated subiamges are obtained, which are shown in Figs. 5(b) and 5(e). Note that the experimental results in Fig. 5 are in great agreement with the theoretical simulation in Fig. 2(c). The experimental intensity distributions at positions of 0, 3, 5, 6 mm agree with the theoretical ones. The minor mismatch is coming from the small distance differences between the actual experimental measurement positions and the ones theoretical simulation employed.
Fig. 5 The diffraction patterns at different detection positions to demonstrate the fractional Talbot effect. (a)–(f) show the images at positions of 0, 3, 4, 5, 6, 8 mm, respectively.
5. Conclusion
In conclusion, the EIT assisted Talbot effect is observed for a wide range of measurement distance of three times Talbot length in a ladder-type three-level atomic vapor. The high-resolution fractional Talbot effect is obtained at special detection positions. Compared with traditional grating, the periodic structure used here is the atomic medium with periodically modulated refractive index induced by the interference of two coupling laser fields. The image is detected by the profile of the probe laser after transmitting the periodically modulated lattice. The self-replicate of the image and multiple subimages are clearly observed in the experiment, which agree well with the theoretical simulation. We strongly believe that this work will promote the applications based on the electromagnetically induced Talbot effect. It should be noted that the extension of the current technique to the cold atomic system needs high density atomic sample, more elaborate experimental parameter adjustment and multi-beam coordination. Additionally, our system can be used as an ideal platform to investigate the control of light dynamics based on the periodically modulated medium, such as nonlinear beam dynamics and non-Hermitian related phenomena.
Appendix A: Supplementary explanation of the experimental results
This appendix gives some detailed information about the theoretical simulation and experimental results. Throughout this paper, the theoretical calculations are shown in 2D (x − z) form. As shown in Fig. 1(b), the optically induced lattice established by the interference of two lasers is in the x direction, the probe laser propagates (z direction) perpendicularly the lattice. Thus, the theoretical simulation is accomplished in x − z plane to show the Talbot image. In the experiment, the detection position is along the z direction of probe laser propagation, the obtained images are x − y Talbot image. Since the crossing areas can be approximated by quasi-one-dimensional, the comparison between the theory and experiment is conducted in x − z plane. The x − y distribution of the diffraction pattern in Fig. 3 is shown in Fig. 6. The Talbot effect mainly focus on the x distributions at different z positions.
The experimental results of the z = 4, 8 mm have a distance differences of theoretical ones of z = 1/4zT, 1/2zT (3.75 mm, 7.5 mm). The comparison between the experimental and theoretical transversal intensity distributions at different positions is shown in Fig. 7. The experimental intensity distributions at z = 0, 3, 5, 6 mm agree with the theoretical ones.
Funding
National Key R&D Program of China (2017YFA0304203); National Natural Science Foundation of China (61575116, 61875112, 61705122, 61728502, 91736209, 11434007); Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (IRT 17R70); Program for Sanjin Scholars of Shanxi Province; Applied Basic Research Project of Shanxi Province (201701D221004); 111 project (D18001) and Fund for Shanxi “1331 Project’ Key Subjects Construction.
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