In this work, a detailed theoretical analysis of 1529 nm ES-FADOF (excited state Faraday anomalous dispersion optical filter) based on rubidium atoms pumped by 780 nm laser is introduced, where Zeeman splitting, Doppler broadening, and relaxation processes are considered. Experimental results are carefully compared with the derivation. The results prove that the optimal pumping frequency is affected by the working magnetic field. The population distribution among all hyperfine Zeeman sublevels under the optimal pumping frequency has also been obtained, which shows that 85Rb atoms are the main contribution to the population. The peak transmittance above 90% is obtained, which is in accordance with the experiment. The calculation also shows that the asymmetric spectra observed in the experiment are caused by the unbalanced population distribution among Zeeman sublevels. This theoretical model can be used for all kinds of calculations for FADOF.
© 2016 Optical Society of America
Faraday anomalous dispersion optical filter (FADOF) is an atomic optical filter with high transmittance, narrow bandwidth and excellent out-of-band rejection . It acts as an important device in free-space optical communication , lidar [3,4], laser frequency locking [5,6], single photon generation system , as well as for quantum key distribution and ghost imaging in recent years [8,9].
The working wavelength of FADOF is restricted by the numbered transition lines of atoms. In order to achieve more working wavelengths to broaden applications of the filter, ES-FADOF was presented in 1995 , and various working wavelengths have been realized since then [11–17]. One of the main challenges for ES-FADOF is optical pumping. A highly efficient pumping method is critical to the high transmittance. The working magnetic field of ES-FADOF and the hyperfine splitting can affect the pumping efficiency . Previous theoretical analyses on ES-FADOF calculate the population without considering the influence of factors such as the magnetic field, the hyperfine splitting and the depolarization [11,14]. If those factors are not included in the calculation, the optimal pumping frequency which is important for the pumping efficiency cannot be provided.
In this work, the 1529 nm ES-FADOF based on the transition 5P3/2 to 4D5/2 of rubidium atoms is analyzed. We take use of the density matrix method to calculate the population of 5P3/2 pumped by a 780 nm laser. The calculation has taken into account factors such as Zeeman splitting, Doppler broadening, and the relaxation. We calculate the population in the working magnetic field and compare it to the experimentally measured absorption signal, and the distribution of atoms among all hyperfine Zeeman sublevels at the highest population point is also obtained. Based on the pumping calculation, the transmittance spectra at different experimental conditions are calculated and they match the experiment well. The characters of the transmittance spectra are also analyzed.
2. Analysis of the pumping process
The energy levels of 1529 nm ES-FADOF are shown in Fig. 1. Rubidium atoms are pumped from the ground state 5S1/2 to the excited state 5P3/2. The analysis is based on the density matrix equation involving all hyperfine Zeeman sublevels. We choose the direct sum space of 5S1/2 and 5P3/2 to be the Hilbert space of the system. The basis vectors are the eigenstates of the Hamiltonian 20,21].
The eigenstates and corresponding energy values are obtained by diagonalizing the matrix of Hamiltonian Ĥ0 in the uncoupled basis vectors which are the direct product state of the nuclear and electronic wave functionEq. (4) to the space of the eigenstates of the Hamiltonian Ĥ0. In the direct sum space of the ground state and the excited state, the total unitary transformation operator Û is the direct sum of Ûg and Ûe
The density matrix ρ̂ of the system can be got from the eigenstates of Ĥ0. Considering the interaction between the light and atoms, the evolution equation for the system can be written asEq. (7) is  Eq. (8) is that we first write out the matrix form of −er̂ in the electric space, and then change it into the space of the uncoupled basis vectors Eq. (4) by the direct product with I. At last we transform it into the space of the eigenstates of the Hamiltonian Ĥ0 through the unitary transformation. The matrix in the electric space can be written as Eq. (9) is the dimensionless dipole moment operator, and it has three components. If we choose the spherical basis vectors 18]. E⃗ in Eq. (6) is the electric field, and it is determined by the intensity of the pumping laser Spump = c|E⃗|2/2π. In our experiment, the pumping laser is linear polarized and the polarization direction is chosen along the unit vector x⃗.
In practice, various relaxation process such as the transit relaxation, wall collisions and the spontaneous emission should be included [22,23]. These relaxation terms are phenomenologically added in the density matrix equation. The evolution equations of the system can be written asEq. (8) Eq. (16), is the unit matrix of the ground state. Ng is the number of sublevels of the ground state. Γc is the total relaxation rate which includes the transit relaxation rate and the wall collision rate  18].
Considering Doppler broadening, the result should integrate over the velocity distribution function, and we take the numerical method. The discrete velocity distribution is adopted to approximate the continuous distribution . The Maxwell distribution function are divided into a lot of small regions, and the velocity in each small region is represented by its center velocity vs. The calculated results are contributed by the atoms in each small region
In our experiment, the magnetic field equals 650 G. The intensity of the pumping laser equals 800 mW/cm2, which is determined by 26 mW laser power and 2 mm spot diameter. The temperature equals 110 °C. The length of the cell is 6 cm and the diameter of which is 2 cm. The natural rubidium mainly contains two isotopes 85Rb and 87Rb. The abundance for 85Rb is 72.17% and for 87Rb is 27.83%. The contribution of the two isotopes is included in the pumping calculation.
The calculated population of 5P3/2 of natural rubidium atoms as a function of the detuning of the pumping laser is in Fig. 2(a.1). We also experimentally measure the absorption ratio of the pumping laser in Fig. 2(a.2). The population is proportional to the absorption of the pumping laser. The highest points in Figs. 2(a.1) and 2(a.2) both represent the highest pumping efficiency and they have the same frequency. In comparison to the situation without the magnetic field, the frequency of the highest point shifts about 1.2 GHz from the strongest absorption point of 85Rb atoms, and shifts about 2.5 GHz from the strongest absorption point of 87Rb atoms.
The shape of the absorption spectrum is produced by the splitting of the hyperfine transition lines of 5S1/2 to 5P3/2 in the working magnetic field. In the usual D2 absorption spectrum of the rubidium atoms without Zeeman splitting, there are two absorption lines on both sides of the fine transition frequency corresponding to two hyperfine sublevels of 5S1/2. In the magnetic field, the transitions of the higher sublevels of 5S1/2 (level1 and level2 in Fig. 1) induced by the left circularly polarized light will shift toward the higher frequency, and transitions of the lower sublevels (level3 and level4 in Fig. 1) induced by the right circularly polarized light will shift toward the lower frequency. As the magnetic field increasing, the frequencies of the two transition bunches can get close to each other and superpose at the adjacency of the fine transition frequency at a appropriate magnetic field value. The superposition forms the highest absorption point. If the frequency of the pumping laser is tuned to the highest point, the highest pumping efficiency can be reached. The calculated population curve matches the experimentally measured absorption curve in our work.
In order to calculate the transmittance spectra of the 1529 nm ES-FADOF, the population distribution among all hyperfine Zeeman sublevels of 5P3/2 is required. We calculate the population distribution of 85Rb and 87Rb atoms at the highest pumping efficiency point respectively, which are Figs. 2(b.1) and 2(b.2).
The distribution bar graphs also indicate that the population is mainly from the 85Rb atoms when the pumping laser is tuned to the highest pumping efficiency point. If we add these distribution, we can get the proportion of 85Rb and 87Rb atoms pumped to 5P3/2. Adding the distribution in Fig. 2(b.1) shows that 3.05% of 85Rb atoms are pumped to 5P3/2, and adding the distribution in Fig. 2(b.2) shows that 0.046% of 87Rb atoms are pumped to 5P3/2. Because the abundance of 85Rb in natural rubidium vapor cell is 72.17%, the proportion above can reveal that pure 85Rb vapor cell is more efficient than the natural rubidium vapor cell when the pumping laser is tuned to the highest pumping efficiency point in our experimental condition, which also means that the transmittance spectra are mainly determined by 85Rb atoms at this condition. The pure 85Rb vapor cell is adopted in the analysis of the transmittance spectra.
3. Analysis of the transmittance
The expression of the transmittance is 
The experiment setup is Fig. 3(a). The temperatures are set to 90 °C, 100 °C, 110 °C, and 120 °C, and the relationship between the atomic density and the temperature can be obtained according to the previous work [20,21]Figs. 3(b.1)–3(b.4) and the measured transmittance spectra are the red solid lines. The power losses caused by the optical components is 39%, which is not considered in the results.
In Figs. 3(b.1) and 3(b.2), the density of the atoms is insufficient, thus the optical rotation efficiency is weak and cannot provide the high transmittance. In Fig. 3(b.3), the temperature is appropriate, so is the density of atoms, thus, the rotation of the polarization at the center approaches 90°, which cause most of the signal light pass the second polarizer. The peak transmittance is higher than 90%. The transmittance of the sidebands also improve observably as the density of atoms increases. In Fig. 3(b.4), the surplus atoms cause the rotation angle exceeds 90°, which reduces the transmittance at the center. However, they also make the rotation angles at the positions of the sidebands approach 90°.
One may notice the asymmetry of the transmittance spectra. If the population is uniformly distributed in Zeeman sublevels, the sidebands will symmetrically located at the two sides of the center peak . However, in our work, the sidebands are asymmetric. The calculated distribution in Fig. 2(b.1) and the transmittance spectra calculation which match the experiment indicate that the asymmetric property is caused by the unbalanced distribution of atoms in the working lower level 5P3/2. This property has not been pointed out in the former work of ES-FADOF, and is important for the application of the optical filter.
We also compare the peak transmittance of 85Rb vapor cell and the natural rubidium cell experimentally at different pumping laser power, which is Fig. 2(c). The result shows that in our experimental condition, pure 85Rb vapor do have an advantage over nature rubidium vapor in which the abundance of 85Rb is 72.17%.
We theoretically analyze the rubidium 1529 nm ES-FADOF pumped by a 780 nm laser. The density matrix method is adopted to calculate the pumping process, in which Zeeman splitting, Doppler broadening, and the relaxation process are considered. The results prove that the optimal pumping frequency is affected by the working magnetic field. Under the optimal frequency point, the population distribution among all hyperfine Zeeman sublevels has been obtained. The result also shows that mainly 85Rb atoms contribute to the population in the optimal pumping frequency. Based on the population distribution, the transmittance of the filter is analyzed. The calculated transmittance spectra match well with the experiment, and under certain circumstance, the peak transmittance above 90% is obtained, which is in accordance with the experiment. The calculation also shows that the asymmetric spectra observed in experiment are caused by the unbalanced population distribution among Zeeman sublevels.
This work is supported by the National Science Fund for Distinguished Young Scholars of China (61225003), the National Natural Science Foundation of China (61401036, 61531003, 61571018), the China Postdoctoral Science Foundation (2015M580008), and the National HiTech Research and Development (863) Program.
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