The cross-section is crucial for quantitative characterization and analysis of the absorption process. A model on the absorption cross-section of the simultaneous two-photon excitation in Rb-vapor four-wave mixing process is established by using the coupled-wave equation. Taken into account of the hyperfine structures for 85Rb and 87Rb, the third-order susceptibility and hyperfine line strength are calculated respectively. Then, the influences of hyperfine transition on cross section are investigated and simulation results agree well with the experiment results. The calculated results suggest that high pumping power intensity is essential in Rb two-photon excitation, while narrow linewidth is the limiting factor of high absorption efficiency by comparing normalized absorption profile between pumping beam and two-photon excitation process. Additionally, two approaches to improving absorption efficiency, linewidth narrowness of the pumping beam and absorption linewidth broadening, are proposed.
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Simultaneous two-photon absorption (2PA) has a broad range of applications in fluorescence microscopy, 3D optical memory, optical power limiting and photodynamic therapy since the first verified by Kaiser and Garret in 1961 [1–3]. And many theoretical and experimental research studies have been carried out in this field [4–6]. The two-photon absorption process of Rb has been paid more attention excitation in recent alkali vapor studies [7–10]. One of its advantages is the realization of cascading and coaxial output of blue-violet photons and mid-infrared photons, which can be used in many fields of military and scientific research, such as multi-wavelength detection, and atomic physics. Vernier achieved conversion efficiency of 2.6% in Rb by two-photon absorption experimentally in 2010 . 420nm light with power of 9.1mW was obtained by two-photon excitation and the power efficiency was 1.5% in 2014 . The energy efficiency over than 1% in Rb by single-wavelength conversion was realized in 2016 . The research on new atomic transition frequency and spectroscopy of Rb which is realized by two-photon transition from the 5S1/2 to 4D5/2, was carried out in 2017 . In order to quantitatively describe and analyze the absorption process, exact estimation of the 2PA spectra and the absorption cross-section values are of critical importance. Previously, Florescu used a modified precise one-electron model to calculate 2PA cross section of Rb 5S1/2-5D5/2 as 0.57 × 10−26 m4/W , and furthermore, Collins also measured the two-photon cross section for 5S1/2(F = 2)-5D5/2(F = 4) in 85Rb by a differential absorption technique experimentally . The spectral characteristics of 2PA of alkali vapor, such as Rb 5S1/2-5D, Cs 6S-8S and 6S-6D have been measured [16,17]. However, more unambiguous and systematic theory models are still needed to obtain the spectra and cross-section values of simultaneous two-photon cross section of Rb 5S1/2-5D.
In this paper, a theoretical study of the simultaneous two-photon excitation in the Rb four-wave mixing process is carried out. Based on the coupled-wave equation and definition of 2PA cross section, the intelligible expression of the 2PA in Rb four-wave mixing process is depicted and derived. The third-order susceptibility of Rb four-wave mixing process is calculated with the contribution of the intermediate 5P3/2, 5P1/2, 6P1/2 and 6P3/2 energy states. In addition, the nuclear spin and two isotopes of Rb are innovatively taken into account in four-wave mixing process with total angular momentum. The influences of hyperfine structures on 2PA cross section are analyzed, and the main reason of relatively low absorption efficiency is found out by comparing line shape spectrum of Rb absorption and profile of pumping light.
2. Description of the model
The Rb energy levels and four-wave mixing process with single-wavelength laser excitation are shown in Fig. 1. Due to small energy level spacing between 5D3/2 state and 5D5/2 state, and considering spectral linewidth of pumping laser and atom energy states, the Rb atoms can be excited to 5D states (5D3/2 and 5D5/2 state) by 778.1nm two-photon absorption with the help of intermediate virtual level showed as dotted line in Fig. 1. For Rb four-wave mixing process, the atoms in 5D3/2 state radiate two kinds of photons with wavelengths of 5.24μm and 5.04μm to the states of 6P3/2 and 6P1/2 respectively, and the atoms in 5D5/2 state radiate photons with a wavelength of 5.23μm to 6P3/2 state. And then the blue-violet light of 420nm or 422nm from state of 6P3/2 or 6P1/2 back to ground state is radiated. The wave vectors of two pumping photons, the mid-infrared photon and the blue-violet photon are satisfied with phase-matching in the four-wave mixing process. The partial hyperfine structures are given in the right half of Fig. 1. The values in middle of two lines are energy differ of adjacent hyperfine structures. It should be noted that the two-photon absorption in our text refers to the excitation in the four-wave mixing process of Rb, which is different from ordinary two-photon absorption process.
Two photons of same frequency vP and pumping light intensity I(vP) are simultaneously absorbed by an Rb atom in its ground state leading to the excitation of 5D state. In coupled-wave equation of simultaneous two-photon absorption theory, pumping beam-intensity I(vP, z) change along the propagation direction (z axis) in Rb vapor cell can be expressed [18,19]
Where kp is pumping laser wavenumber, and the speed of light in vacuum c is 3 × 108m/s. The permeability of vacuum μ0 is 4π × 10−7 H/m. Im[χ(3)] is the imaginary part of third-order susceptibility of Rb in four-wave mixing process, which relates to the 2PA.
The nonlinear absorption coefficient due to simultaneous two-photon absorption is defined by β , and corresponding 2PA cross sections σTA (in units of m4/W) of the Rb atoms can be determined by using the similar solute molecules relationship 
Here N is atomic density. The np is the linear index of refraction. The σTA is proportional to frequency and imaginary part of third-order susceptibility.
To figure out the cross sections, we assume that incident pumping laser field is linearly polarized for simplification, and third-order susceptibility in four-wave mixing process is given by [21,22]
Where vUV and vIR are frequency of blue-violet light and mid-infrared light generated by four-wave mixing process, respectively, the vi, vj and vk are respectively transition frequency of 5S to 5P, 5S to 5D and 5S to 6P. The e is the electric charge and ε0 is permittivity of vacuum. is the electric-dipole matrix element between levels i and j, which is listed in Table 1. The Γgi reflects the linewidth of 5D energy level at frequency vp due to various broadening mechanisms. is the occupation probability of atomic state |g˃ showed in Fig. 1.
To obtain the imaginary part of third-order susceptibility, the fine structures of 5D states need to be taken into consideration in Eq. (4) as
Here the coefficient 6 and 4 are degeneracy of 5D5/2 and 5D3/2, respectively. Σ is summation for the intermediate 5P1/2, 5P3/2, 6P1/2 and 6P3/2 states located between state |g˃ and state |j ˃. Furthermore, for simultaneous two-photon absorption process, the influences of hyperfine structures (represent by the total angular momentum F) of states resulted from the coupling between electronic angular momentum J and the nuclear spin I (I = 3/2 for 87Rb and I = 5/2 for 85Rb) cannot be neglected. The hyperfine structures, shown in 85Rb energy level diagram of Fig. 1, are described by the magnetic dipole A and electric quadrupole constant B. Considering the influence of hyperfine structures, we innovatively borrow the thought from .Thus Eq. (5) is expanded as
The hyperfine line strengths SFF″ specifies relative intensities of the F→F″ (5S-5D) transitions of the two Rb isotopes, which represents the relative probability of transitions among hyperfine states. The SFF″ is expressed as 
The hyperfine line strengths and the frequency offset from center frequency are listed in Table 2. It contains all possibilities of two-photon hyperfine transition from ground state 5S1/2 to 5D in two Rb isotopes. The center frequency corresponds to the transition frequency between 5S1/2 and 5D in Fig. 1 when only fine structures are considered. And a summary of other critical parameters used in the calculation are listed in Table 3. The former two parameters are calculated according to the data in , and the another two parameters are valued when the vUV is set up as intermediate value of v6P1/2 and v6P3/2.
3. Result and discussions
The linewidth Γgi of 5D states is related to many factors such as temperature of vapor, buffer gas. The influences of buffer gas are neglected for lacking of relevant experimental data. The parameter Γgi exists different values when Rb vapor is excited by single-wavelength pumping beam in two different ways, which are only co-propagating laser beam, and both co- and counter-propagating laser beams equally existence, respectively. In the first case, the Γgi is determined by the Doppler broadening of energy level, which is about 1.2GHz calculated by formula of Gauss linewidth  when the temperature of Rb vapor is supposed to 443K. In the other case, in order to study the influence of narrow linewidth Γgi on the cross section σTA, we suppose that the co- and counter-propagating laser beams are separated by minimum angle (several milliradians) which is to ensure the phase-matching. And we still think that the first-order Doppler effect is eliminated considering of the minimum angle of co- and counter-propagating laser beams, so the main contribution to the linewidth Γgi is excited-state lifetime and linewidth caused by collisions within Rb atoms is ignored . The linewidth Γgi of the excited 5D5/2 state and 5D3/2 state are approximately 1.5MHz calculated by the excited-state lifetime with 230 ± 23 ns .
The cross section of two-photon absorption σTA can be obtained by substituting Eq. (5) into Eq. (3) with different values of Γgi. The results are shown in Fig. 2. The σTA nearly become vertical line in Fig. 2(a) for the very narrow linewidth of 5D level about 1.5MHz, and partial enlarged drawing is located in the middle. The σTA of 5S1/2-5D5/2 transition is about 26 times larger than that of 5S1/2-5D3/2, which is collectively determined by two parameters μ and v5P-vP (transition frequency difference between the 5P state and intermediate virtual state) listed in Table 1 and Table 3, respectively. Therefore, it is efficient to elect the 5D5/2 state as the upper level. As shown in Figs. 2(a) and 2(b), there has different σTA of 5S1/2-5D5/2 transition when two distinct pumping schemes are taken into account. According to Eq. (5), the maximum cross section is inversely proportional to linewidth Γgi, and therefore the σTA of 5S1/2-5D5/2 in Fig. 2(a) is larger orders of magnitude than that in Fig. 2(b). It can be seen that the maximum σTA about 2.5 × 10−26 m4/W in Fig. 2(a) is too small, and the result suggests that there must be high pumping power intensity comparing with single photon-atom interaction transition, which will be restricted in applications.
The influences of hyperfine structures on two-photon excitation are analyzed. As shown in Fig. 2, the σTA of 5S1/2-5D5/2 transition is far larger than that of 5S1/2-5D3/2, so we neglect the influence of 5S1/2-5D3/2 transition to the σTA. The linewidth Γgi of 5D5/2 level is about 1.5MHz when co- and counter-propagating laser beams equivalently coexist. The linewidth of 5D5/2 level is relatively smaller compared with the energy difference of hyperfine structures, so the influences of these hyperfine structures are obvious. It is shown in Figs. 3(a)-3(d) that the 2PA cross sections of 5S1/2-5D5/2 transition of Rb isotopes versus the frequency offset from center transition frequency. From Fig. 3, two isotopes have distinct hyperfine structures for each state and transitions of two-photon absorption obey the rule |ΔF| = 0, ± 2. The superior of σTA for 85Rb and 87Rb is transition of F = 3-F = 5 and F = 2-F = 4, which are respectively correspond to transition between the higher F of ground 5S1/2 state and terminal 5D5/2 state. We deduce that the reason of larger σTA between higher hyperfine transitions might result from more atoms accumulating in higher F of ground 5S1/2 state according to the Eq. (7). And the σTA gradually reduces from the higher hyperfine structure F to lower F of 5D5/2 state for 85Rb and 87Rb. However, frequency offset are significant differences between 85Rb and 87Rb.
In order to validate the results, we compare our simulation results with some experimental values reported so far. Zapka et al. have experimentally measured the value of σTA of 5S1/2(F = 2)→5D5/2(F = 4) transition in 87Rb and the value is 4 × 10−28 m4/W . Our calculated result of σTA for 87Rb in Fig. 3(d) is about 5.9 × 10−27 m4/W which remains within the error range caused by different linewidth Γgi in distinct experimental circumstances. The maximum σTA of 85Rb 5S1/2-5D5/2 is hyperfine transition of F = 3-F = 5 with 1.41 × 10−26 m4/W in Fig. 3(b), which is close to the 2PA cross section of Rb 5S1/2-5D5/2 as 0.57 × 10−26 m4/W in  and the experimentally measured value of (1.2 ± 0.5) × 10−26 m4/W in . And in addition, Nez et al. have experimentally measured the absolute frequencies and relative strength of hyperfine structure of the 5S1/2-5D5/2 two-photon transition in rubidium . By comparison, our calculated results agree well with the experimental outcomes reported in  both relative height and frequency interval of each hyperfine energy level transition.
The 2PA cross section σTA of 5S1/2-5D5/2 hyperfine transition versus the frequency offset from center transition frequency in Fig. 4 when the linewidth Γgi is 1.2GHz. The lineshape of total absorption cross section is a whole for wide linewidth of every hyperfine structure. It can be seen that the σTA has four main peaks in Fig. 4. The two peaks closed to center transition frequency are caused by hyperfine transition in 85Rb, and the other two peaks are caused by transition in 87Rb. The total absorption cross section covers up more details about the intensity and number of hyperfine transition. Moreover, the value of σTA in Fig. 4 is much smaller comparing with Fig. 3, which results from cross section is inversely proportional to linewidth Γgi of 5D state.
The narrow linewidth of two-photon absorption process of 5D level in Rb have a great help to spectral resolution and measurement. However, it is also the disadvantage of high absorption efficiency. As shown before, two pumping schemes not only affect the absorption linewidth, but have an impact on the σTA. To analyze the influence of linewidth on absorption process when hyperfine structures are taken into account, normalized profile function of pumping beam and two-photon absorption linewidth according to former results are shown in Fig. 5. It is shown that Figs. 5(a) and 5(b) are normalized profile function of linewidth Γgi with 1.5MHz and 1.2GHz, respectively. The profile function of pumping beam is considered as Gaussian lineshape and linewidth is set up as 0.001nm or 0.007nm. It can be seen from Fig. 5 that the two absorption lineshapes are quite different that depend on pumping schemes. The ratio of overlap of pumping lineshape and absorption lineshape increase as linewidth of pumping beam reduces, and absorption profile is beyond pumping spectrum range when linewidth of pumping beam is 0.001nm. From Fig. 5(b), hyperfine transition is indistinguishable because of larger linewidth, and the normalized profile is much smaller than that of Fig. 5(a). Moreover, red shift about 1.1GHz of frequency for pumping light could increase the overlap of pumping lineshape and absorption lineshape to obtain large absorption rate.
As we all know, wide absorption linewidth is guarantee of high absorption efficiency for the laser gain medium. If two-photon excitation of Rb vapor is used as laser medium, the problem of linewidth matching between medium absorption and pumping beam need to be solved. We can know that there are two ways to improve the problem from Fig. 5. One is linewidth narrowing of pumping beam by volume grating. However, it has the minimum limitation from Fig. 5. And it also needs to balance the relationship of narrow linewidth and large pumping power intensity. The other way is appropriate buffer gas filled in vapor to wide absorption linewidth. But more experiment data about influences of buffer gas to two-photon absorption is needed. Combining two ways might be an effective method to solve the problem.
In this study, we report an absorption cross section model in Rb simultaneous two-photon excitation process. The absorption cross section σTA is calculated when the hyperfine structures of two isotopes are taken into account. The calculated results show that it is efficient to elect the 5D5/2 state as the upper level for the σTA of 5S1/2-5D5/2 transition being far larger than that of 5S1/2-5D3/2, and it is essential to pumping Rb vapor using extremely strong pumping power intensity for simultaneous two-photon excitation process. In addition, hyperfine structures have an important influence on the σTA value and profile for simultaneous two-photon absorption. The model and simulation results of σTA could provide support for study and design in simultaneous two-photon absorption process. Two ways, linewidth narrowness of the pumping beam and absorption linewidth broadening by appropriate buffer gas filled, are proposed to improve absorption efficiency. The influences of types and amount of buffer gas need to be more researched in the future study.
The National Defense Science and Technology Innovation Fund of the Chinese Academy of Sciences (Grant No.CXJJ-16M228); the Major Science and Technology Biding Project of Jilin Province (No.20160203016GX); the Young and Middle-Aged Science and Technology Innovation Leader and Team Project of Jilin Province (No.2017051901JH); and the Youth Innovation Promotion Association of CAS (No. 2017259).
References and links
1. B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. S. Lee, and D. Mccordmaughon, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature 31, 52 (1999).
3. W. K. Kaiser and C. G. B. Garrett, “2-Photon Excitation in Caf2 - Eu2+,” Phys. Rev. Lett. 7(6), 229–231 (1961). [CrossRef]
4. M. Göppert-Mayer, “Über Elementarakte mit zwei Quantensprüngen,” Ann. Phys.-Berlin 401(3), 273–294 (1931). [CrossRef]
5. G. Grynberg and B. Cagnac, “Doppler-free multiphotonic spectroscopy,” Rep. Prog. Phys. 40(7), 791–841 (1977). [CrossRef]
7. A. M. Akulshin, A. A. Orel, and R. J. Mclean, “Collimated blue light enhancement in velocity-selective pumped Rb vapour,” J. Phys. At. Mol. Opt. Phys. 45(1), 015401 (2012). [CrossRef]
8. A. M. Akulshin, N. Rahaman, S. A. Suslov, and R. J. Mclean, “Amplified spontaneous emission at 5.23 um in two-photon excited Rb vapour,” J. Opt. Soc. Am. B 34(12), 2478 (2017). [CrossRef]
9. M. P. Moreno, G. T. Nogueira, D. Felinto, and S. S. Vianna, “Two-photon transitions driven by a combination of diode and femtosecond lasers,” Opt. Lett. 37(20), 4344–4346 (2012). [CrossRef] [PubMed]
12. J. F. Sell, M. A. Gearba, B. D. DePaola, and R. J. Knize, “Collimated blue and infrared beams generated by two-photon excitation in Rb vapor,” Opt. Lett. 39(3), 528–531 (2014). [CrossRef] [PubMed]
13. R. C. Cao, B. G. Baodong Gai, J. Y. Jie Yang, T. L. Tong Liu, J. L. Jinbo Liu, S. H. Shu Hu, J. G. Jingwei Guo, Y. T. Yannan Tan, S. H. Shan He, W. L. Wanfa Liu, H. C. Hongxing Cai, and X. Z. and Xihe Zhang, “Efficient generation of collimated frequency upconversion blue light in rubidium vapor,” Chin. Opt. Lett. 13(12), 121903 (2015). [CrossRef]
14. R. Roy, P. C. Condylis, Y. J. Johnathan, and B. Hessmo, “Atomic frequency reference at 1033 nm for ytterbium (Yb)-doped fiber lasers and applications exploiting a rubidium (Rb) 5S1/2 to 4D5/2 one-colour two-photon transition,” Opt. Express 25(7), 7960–7969 (2017). [CrossRef] [PubMed]
16. C. V. Sulham, G. A. Pitz, and G. P. Perram, “Blue and infrared stimulated emission from alkali vapors pumped through two-photon absorption,” Appl. Phys. B. 101(1-2), 57–63 (2010). [CrossRef]
17. Y. H. Chen, T. W. Liu, C. M. Wu, C. C. Lee, C. K. Lee, and W. Y. Cheng, “High-resolution 133Cs 6S-6D, 6S-8S two-photon spectroscopy using an intracavity scheme,” Opt. Lett. 36(1), 76–78 (2011). [CrossRef] [PubMed]
18. S. Shi, G. Chen, W. Zhao, and J. Liu, Nonlinear Optica, 2nd ed. (Xidian University, 2012), Chap. 5.
19. L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerence, semiconductors, and other materials,” Prog. Quantum Electron. 17(4), 299–338 (1993). [CrossRef]
20. G. S. He, L. Yuan, N. Cheng, J. D. Bhawalkar, P. N. Prasad, L. L. Brott, S. J. Clarson, and B. A. Reinhardt, “Nonlinear optical properties of a new chromophore,” J. Opt. Soc. Am. B 14(5), 1079–1087 (1997). [CrossRef]
21. J. Heinrich and W. Behmenburg, “Two-photon resonant four-wave mixing processes in atomic barium,” Appl. Phys. (Berl.) 23(3), 333–339 (1980). [CrossRef]
22. A. Smith and J. Ward, “Doubly Resonant Four-Photon Interactions in Cesium Vapor,” IEEE J. Quantum Electron. 17(4), 525–529 (1981). [CrossRef]
23. G. A. Pitz and G. P. Perram, “Pressure broadening of the D1 and D2 lines in diode pumped alkali lasers,” Proc. SPIE 7005, 700526 (2008). [CrossRef]
24. O. Axner, J. Gustafsson, N. Omenetto, and J. D. Winefordner, “Line strengths, -factors and absorption cross-sections for fine structure lines in multiplets and hyperfine structure components in lines in atomic spectrometry—a user’s guide,” Spectroc. Acta Pt. B-Atom. Spectr. 59, 1–39 (2004).
25. T. William, Laser Fundamentals, 2nd ed. (Cambridge University, 2004), Chap. 4.
26. K. H. Weber and K. Niemax, “Self-broadening and shift of Doppler-free two-photon lines of Rb,” Opt. Commun. 31(1), 52–56 (1979). [CrossRef]
27. C. E. Theodosiou, “Lifetimes of alkali-metal-atom Rydberg states,” Phys. Rev. A 30(6), 2881–2909 (1984). [CrossRef]
28. W. Zapka, M. D. Levenson, F. M. Schellenberg, A. C. Tam, and G. C. Bjorklund, “Continuous-wave Doppler-free two-photon frequency-modulation spectroscopy in Rb vapor,” Opt. Lett. 8(1), 27–29 (1983). [CrossRef] [PubMed]
29. F. Nez, F. Biraben, R. Felder, and Y. Millerioux, “Optical frequency determination of the hyperfine components of the 5S1/2-5D3/2 two-photon transitions in rubidium,” Opt. Commun. 102(5-6), 432–438 (1993). [CrossRef]