## Abstract

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 ^{85}Rb 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

## 1. Introduction

Faraday anomalous dispersion optical filter (FADOF) is an atomic optical filter with high transmittance, narrow bandwidth and excellent out-of-band rejection [1]. It acts as an important device in free-space optical communication [2], lidar [3,4], laser frequency locking [5,6], single photon generation system [7], 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 [10], 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 [17]. 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 5P_{3/2} to 4D_{5/2} of rubidium atoms is analyzed. We take use of the density matrix method to calculate the population of 5P_{3/2} 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 5S_{1/2} to the excited state 5P_{3/2}. The analysis is based on the density matrix equation involving all hyperfine Zeeman sublevels. We choose the direct sum space of 5S_{1/2} and 5P_{3/2} to be the Hilbert space of the system. The basis vectors are the eigenstates of the Hamiltonian [18]

The Hamiltonian

*μ*

_{B}= 1.4 MHz/G, and the Lande g factor g

_{J}is 2 for 5S

_{1/2}and 4/3 for 5P

_{3/2}. The second term can be ignored because the contribution from the nuclear magnetic moment is negligibly small.

The eigenstates and corresponding energy values are obtained by diagonalizing the matrix of Hamiltonian *Ĥ*_{0} in the uncoupled basis vectors which are the direct product state of the nuclear and electronic wave function

*Ĥ*

_{0}, we can construct two unitary transformation operators Û

_{g}and Û

_{e}for the ground state and the excited state respectively. They will transform operators and wave functions from the space of the uncoupled basis vectors Eq. (4) to the space of the eigenstates of the Hamiltonian

*Ĥ*

_{0}. In the direct sum space of the ground state and the excited state, the total unitary transformation operator Û is the direct sum of Û

_{g}and Û

_{e}

The density matrix *ρ̂* of the system can be got from the eigenstates of *Ĥ*_{0}. Considering the interaction between the light and atoms, the evolution equation for the system can be written as

*D̂*is the electric dipole operator, and the matrix form for it in the direct sum space of the ground state and the excited state is

*D̂′*in Eq. (7) is [18]

*S*and

*J*represent the angular momentum quantum number of the ground state and excited state respectively.

**I**represents the unit matrix of the nuclear space. The symbol ⊗ represents the direct product.

*r̂*is the position operator and e is the elementary charge. The meaning of Eq. (8) is that we first write out the matrix form of −e

*r̂*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

*Ĥ*

_{0}through the unitary transformation. The matrix in the electric space can be written as

*α*is the amplitude

*ω*

^{{eg}}is the transition angular frequency of the D2 line of Rubidium atoms. Θ̂ in Eq. (9) is the dimensionless dipole moment operator, and it has three components. If we choose the spherical basis vectors

*= Θ̂ ·*

_{m}*ξ*, and the matrix element of which is

_{m}*E⃗*in Eq. (6) is the electric field, and it is determined by the intensity of the pumping laser

*S*=

_{pump}*c*|

*E⃗*|

^{2}/2

*π*. 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 as

*ρ̂*

^{{ee}}is the density matrix of the excited state,

*ρ̂*

^{{gg}}is the density matrix of the ground state, and

*ρ̂*

^{{ge}}and

*ρ̂*

^{{eg}}are the coherence between the ground state and the excited state. The terms like

*V̂ρ̂*

^{{ge}}represent the matrices multiplication, and the terms like

*ξ*

^{{ee}}·

*ρ̂*

^{{ee}}represent the product of the matrix elements that have the same indexes of the row and the column.

*V̂*is obtained through the matrix form of the electric dipole operator Eq. (8) The term ${\widehat{\rho}}_{s}^{\left\{gg\right\}}$ represents the population increased by the spontaneous emission

*N*is the number of sublevels of the ground state. Γ

^{g}*is the total relaxation rate which includes the transit relaxation rate and the wall collision rate [22] where*

_{c}*l*is the laser spot diameter and L is the length of the cell.

*v̄*is the mean thermal velocity in which

*T*is the temperature,

*M*is the molar mass of the atom, R = 8.314J/(mol · K) is the universal gas constant. We have assumed that atoms have the same mean velocity in each direction. The matrix elements of

*ξ*

^{{ee}},

*ξ*

^{{ge}},

*ξ*

^{{eg}}and

*ξ*

^{{gg}}are

*E*, ${E}_{\overline{{u}^{\prime}}}$ are the energies of the excited state sublevels, and

_{ū}*E*,

_{u}*E*are the energies of the ground state sublevels, which we have obtained by diagonalizing the Hamiltonian

_{u′}*Ĥ*

_{0}in the uncoupled basis vectors. The term

*h̄*(Δ

*ω*−

*k⃗*·

*v⃗*) represents the influence of the velocity

*v⃗*of atoms and the detuning Δ

*ω*of the pumping laser.

*k⃗*is the wave vector of the laser(axis

*z*).

*γ*

_{1}is the spontaneous emission rate which is also called the longitudinal relaxation rate.

*γ*

_{2}is the relaxation rate of the coherence between the excited state and the ground state and it is also called the transverse relaxation rate. If the collision between atoms is not included, we have the relationship ${\gamma}_{1}=2{\gamma}_{2}={\mathrm{\Gamma}}_{s}^{\left\{ge\right\}}$ [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 [22]. 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 *v _{s}*. The calculated results are contributed by the atoms in each small region

*f*(

*v*) is the probability of atoms with the velocity

_{s}*v*. Δ

_{s}*v*is the width of the small region. ${\widehat{\rho}}_{\overline{uu}}^{{\left\{ee\right\}}^{\prime}}({v}_{s},\mathrm{\Delta}\omega )$ represented the calculated density matrix element when the velocity is set to

*v*and the detuning is set to Δ

_{s}*ω*.

In our experiment, the magnetic field equals 650 G. The intensity of the pumping laser equals 800 mW/cm^{2}, 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 ^{85}Rb and ^{87}Rb. The abundance for ^{85}Rb is 72.17% and for ^{87}Rb is 27.83%. The contribution of the two isotopes is included in the pumping calculation.

The calculated population of 5P_{3/2} 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 ^{85}Rb atoms, and shifts about 2.5 GHz from the strongest absorption point of ^{87}Rb atoms.

The shape of the absorption spectrum is produced by the splitting of the hyperfine transition lines of 5S_{1/2} to 5P_{3/2} 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 5S_{1/2}. In the magnetic field, the transitions of the higher sublevels of 5S_{1/2} (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 5P_{3/2} is required. We calculate the population distribution of ^{85}Rb and ^{87}Rb 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 ^{85}Rb atoms when the pumping laser is tuned to the highest pumping efficiency point. If we add these distribution, we can get the proportion of ^{85}Rb and ^{87}Rb atoms pumped to 5P_{3/2}. Adding the distribution in Fig. 2(b.1) shows that 3.05% of ^{85}Rb atoms are pumped to 5P_{3/2}, and adding the distribution in Fig. 2(b.2) shows that 0.046% of ^{87}Rb atoms are pumped to 5P_{3/2}. Because the abundance of ^{85}Rb in natural rubidium vapor cell is 72.17%, the proportion above can reveal that pure ^{85}Rb 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 ^{85}Rb atoms at this condition. The pure ^{85}Rb vapor cell is adopted in the analysis of the transmittance spectra.

## 3. Analysis of the transmittance

The expression of the transmittance is [24]

*L*is the length of the cell,

*ν*

_{0}is the transition center frequency,

*c*is the speed of the light and

*χ*

_{±}are susceptibility tensors of right and left circularly polarized components

*q*represents ±1 for left and right circularly polarized components respectively, |

*γ*,

*M*〉 represents the Zeeman sublevel,

*M*

_{0}is the mass of the atom,

*k*is the Boltzmann constant,

*T*is the temperature, and

*W*represents the plasma dispersion function.

*N*represents the population of each Zeeman sublevel of 5P

_{γ,M}_{3/2}, which we can get through the optical pumping calculation (the population distribution multiply the density of atoms which is determined by the temperature).

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]

in which*N*is Avogadro constant,

_{A}*R*is the universal gas constant,

*T*is the temperature, and

*N*(

*T*) is the density of rubidium atoms in the vapor cell. The corresponding atomic densities that are 3.56 × 10

^{18}/

*m*

^{3}, 7.16 × 10

^{18}/

*m*

^{3}, 1.38 × 10

^{19}/

*m*

^{3}and 2.55 × 10

^{19}/

*m*

^{3}for these four temperatures respectively. We put the results of the calculation and the experiment in the same coordinate frame. The calculated transmittance spectra for

^{85}Rb are the dashed blue lines in 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 [1]. 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 5P_{3/2}. 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 ^{85}Rb 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 ^{85}Rb vapor do have an advantage over nature rubidium vapor in which the abundance of ^{85}Rb is 72.17%.

## 4. Conclusion

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 ^{85}Rb 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.

## Acknowledgments

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.

## References and links

**1. **P. Yeh, “Dispersive magnetooptic filters,” Appl. Opt. **21**(11), 2069–2075 (1982). [CrossRef] [PubMed]

**2. **J. Tang, Q. Wang, Y. Li, L. Zhang, J. Gan, M. Duan, J. Kong, and L. Zheng, “Experimental study of a model digital space optical communication system with new quantum devices,” Appl. Opt. **34**(15), 2619–2622 (1995). [CrossRef]

**3. **A. Popescu and T. Walther, “On an ESFADOF edge-filter for a range resolved Brillouin-lidar: the high vapor density and high pump intensity regime,” Appl. Phys. B **98**(4), 667–675 (2010). [CrossRef]

**4. **W. Huang, X. chu, B. P. Williams, S. D. Harrell, J. Wiig, and C. Y. She, ‘Na double-edge magneto-optic filter for Na lidar profiling of wind and temperature in the lower atmosphere,” Opt. Lett. **34**(2), 199–201 (2009). [CrossRef] [PubMed]

**5. **X. Zhang, Z. Tao, C. Zhu, Y. Hong, W. Zhuang, and J. Chen, “An all-optical locking of a semiconductor laser to the atomic resonance line with 1 MHz accuracy,” Opt. Express **21**(23), 28010–28018 (2013). [CrossRef]

**6. **P. Wanninger, E. C. Valdez, and T. M. Shay, “Diode-laser frequency stabilization based on the resonant Faraday effect,” IEEE Photonics Technol. Lett. **4**(1), 94–96 (1992). [CrossRef]

**7. **P. Siyushev, G. Stein, J. Wrachtrup, and I. Gerhardt, “Molecular photons interfaced with alkali atoms,” Nature **509**(7498), 66–70 (2014). [CrossRef] [PubMed]

**8. **X. Shan, X. Sun, J. Luo, Z. Tan, and M. Zhan, “Free-space quantum key distribution with Rb vapor filters,” Appl. Phys. Lett. **89**, 191121 (2006). [CrossRef]

**9. **X. Liu, X. Chen, X. Yao, W. Yu, G. Zhai, and L. Wu, “Lensless ghost imaging with sunlight,” Opt. Lett. **39**(8), 2314–2317 (2014). [CrossRef] [PubMed]

**10. **R. I. Billmers, S. K. Gayen, M. F. Squicciarini, V. M. Contarino, W. J. Scharpf, and D. M. Allocca, “Experimental demonstration of an excited-state Faraday filter operating at 532 nm,” Opt. Lett. **20**(1), 106–108 (1995). [CrossRef] [PubMed]

**11. **Y. Peng, “Transmission characteristics of an excited-state Faraday optical filter at 532 nm,” J. Phys. B **30**(22), 5123–5129 (1997). [CrossRef]

**12. **G. Yang, R. I. Billmers, P. R. Herczfeld, and V. M. Contarino, “Temporal characteristics of narrow-band optical filters and their application in lidar systems,” Opt. Lett. **22**(6), 414–416 (1997). [CrossRef] [PubMed]

**13. **A. Rudolf and T. Walther, “High-transmission excited-state Faraday anomalous dispersion optical filter edge filter based on a Halbach cylinder magnetic-field configuration,” Opt. Lett. **37**(21), 4477–4479 (2012). [CrossRef] [PubMed]

**14. **Y. Peng, W. Zhang, L. Zhang, and J. Tang, “Analyses of transmission characteristics of Rb, ^{85}Rb and ^{87}Rb Faraday optical filters at 532 nm,” Opt. Commun. **282**(2), 236–241 (2009). [CrossRef]

**15. **A. Cer, V. Parigi, M. Abad, F. Wolfgramm, A. Predojevic, and M. W. Mitchell, “Narrowband tunable filter based on velocity-selective optical pumping in an atomic vapor,” Opt. Lett. **34**(7), 1012–1014 (2009). [CrossRef]

**16. **Q. Sun, Y. Hong, W Zhang, Z. Li, and J. Chen, “Demonstration of an excited-state Faraday anomalous dispersion optical filter at 1529 nm by use of an electrodeless discharge rubidium vapor lamp,” Appl. Phys. Lett. **101**, 211102 (2012). [CrossRef]

**17. **L. Yin, B. Luo, A. Dang, and H. Guo, “An atomic optical filter working at 1.5 μm based on internal frequency stabilized laser pumping,” Opt. Express **22**(7), 7416–7421 (2014). [CrossRef] [PubMed]

**18. **W. Happer, Y. Jau, and T. Walker, *Optically Pumped Atoms* (WILEY-VCH, 2010). [CrossRef]

**19. **J. Wang, H. Liu, G. Yang, B. Yang, and J. Wang, “Determination of the hyperfine structure constants of the ^{87}Rb and ^{85}Rb 4D_{5/2} state and the isotope hyperfine anomaly,” Phys. Rev. A **90**, 052505 (2014) [CrossRef]

**20. **Daniel A. Steck, “Rubidium 85 D Line Data,” http://steck.us/alkalidata.

**21. **Daniel A. Steck, “Rubidium 87 D Line Data,” http://steck.us/alkalidata.

**22. **M. Auzinsh, A. Berzins, R. Ferber, F. Gahbauer, U. Kalnins, L. Kalvans, R. Rundans, and D. Sarkisyan, “Relaxation mechanisms affecting magneto-optical resonances in an extremely thin cell: experiment and theory for the cesium D1 line,” Phys. Rev. A **91**023410 (2015). [CrossRef]

**23. **M. Auzinsh, D. Budker, and S. M. Rochester, *Optically Polarized Atoms* (Oxford University, 2010).

**24. **E. T. Dressler, A. E. Laux, and R. I. Billmers, “Theory and experiment for the anomalous Faraday effect in potassium,” J. Opt. Soc. Am. B **13**(9), 1849–1858 (1996). [CrossRef]