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Analytical Solutions of Birefringence and Dichroism Spectroscopy for the Jg = 0 → Je = 1 Transition
Analytical Solutions of Birefringence and Dichroism Spectroscopy for the Jg = 0 → Je = 1 Transition
Journal of the Optical Society of Korea. 2014. Aug, 18(4): 365-369
Copyright © 2014, Journal of the Optical Society of Korea
  • Received : May 05, 2014
  • Accepted : June 06, 2014
  • Published : August 25, 2014
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About the Authors
Heung-Ryoul Noh
hrnoh@chonnam.ac.kr
Abstract
We present accurate analytical solutions of the lineshapes of birefringence (rotation) and dichroism (absorption) spectroscopy for a circular anisotropic medium composed of atoms of the transition Jg =0 → Je =1. The susceptibility of a weak probe beam was analytically calculated and was averaged over a Maxwell-Boltzmann velocity distribution. The lineshapes of the two spectroscopies were then presented in analytical forms at arbitrary values of the linewidths of the inhomogeneous (Doppler) broadening and the homogeneous (natural) broadening of the atoms.
Keywords
I. INTRODUCTION
A circular anisotropy is generated in an otherwise isotropic atomic medium by applying a circularly polarized laser beam. This generated circular anisotropy is characterized by the susceptibility whose real (imaginary) part represents the circular birefringence (dichroism). The birefringence can be measured by detecting the polarization rotation of a counterpropagating weak probe beam. This is called polarization spectroscopy (PS) [1] . In contrast, the circular dichroism can be measured by detecting the difference in the transmissions of the opposite circular components of the probe beam, which has recently been called polarizationenhanced absorption spectroscopy (POLEAS) [2] although the similar nature was well-known in the spectroscopy such as dichroic atomic vapor laser lock (DAVLL) [3] . PS and POLEAS can be measured in a single experimental setup by changing analyzing optics after the probe beam passes through the cell. As in usual saturated absorption spectroscopy (SAS) [4] , a sub-Doppler resolution can be obtained in PS and POLEAS because only atoms belonging to specific velocity classes contribute to the signal.
In particular, PS has been widely used in laser frequency stabilization due to its non-modulation characteristic. There were many experimental and theoretical reports on PS for various atomic species [5 - 13] . From the perspective of an analytical study of the lineshape of PS, Nakayama's model was widely used for simple prediction of the PS spectra [14] . We presented also analytical solutions of the PS spectra for Rb atoms under the approximation of weak intensity [15] . When the pump beam intensity is arbitrarily large, it is impossible to obtain analytical solutions for the complicated transition lines of the atoms such as Rb or Cs. However, it is possible for the simple transition of Jg =0 → Je =1 of the atoms such as 88 Sr or Yb. We note that experimental demonstration of PS using Sr was reported [13 , 16] . Recently, we reported analytical solutions of the PS spectra for the transition Jg =0 → Je =1 using reasonable approximation that the Doppler broadening linewidth is much larger than the natural linewidth of the atoms [17] . Extending the previous study we consider arbitrary magnitudes of the two linewidths and obtain analytical solutions of the PS and POLEAS spectra. Since exact analytical solutions of the PS and POLEAS spectra are obtained regardless of the ratio between the Doppler broadening linewidth and the natural linewidth of the atoms, it is possible to obtain analytical forms of the spectra for various atomic samples such as an atomic vapor and cold atoms. It may also be possible to extend to the case of inhomogeneously broadened solids [18] . We also present analytical solutions of the SAS spectra whose analytical form under the narrow natural linewidth approximation is presented in the textbook [19] .
II. CALCULATION OF THE SUSCEPTIBILITY
The energy level diagram and schematic of the spectroscopic scheme are shown in Figs. 1(a) and (b) , respectively. We consider the atomic transition Jg =0 → Je =1 as shown in Fig. 1(a). The pump beam is σ + polarized, whereas the probe beam is linearly polarized with an inclination angle of π /4 with respect to the x axis. The propagation directions of the two beams are opposite. The detected signal is the difference in the intensities along the x and the y axes, i.e. Δ I = Ix Iy . The schematic of POLEAS is shown in Fig. 1(b) , where a quarter-wave plate is placed after the cell. In contrast, the quarter-wave plate is removed in the setup of PS. The variation of the arbitrarily polarized probe beam after traversing a circular anisotropic atomic medium was reported [20] .
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(a) Energy level diagram for the transition Jg=0 → Je=1. (b) Simple schematic for POLEAS. A quarter-wave plate (λ /4) behind the polarizing beam splitter (PBS) cube is removed in the case of PS.
Since the details of calculating the susceptibilities are described in [17] , we review briefly the method of calculating the susceptibilities in this paper. In the atomic rest frame moving at velocity v , the susceptibilities of the σ ± components of the probe beam are given by
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where p is the population of the ground state, q ± are the populations of the excited states ( e ± ), and N at represents the atomic density in the vapor cell. λ is the resonant wavelength, k is the wave vector, δ is the laser frequency detuning, Γ is the decay rate of the excited state, and γ t is the decay rate of the optical coherence which is normally Γ /2 in the absence of the collisional broadening. When the populations p and q ± are calculated, the effect of the probe beam is neglected. The populations can be analytically calculated as follows:
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where A = ( δ + kv ) 2 + γt 2 , B = Ω 1 2 γt / Γ , and Ω 1 is the Rabi frequency of the pump beam [21] . Therefore, the susceptibilities in Eq. (1) are given by
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where η + =1 and η =1/2 .
Now Eq. (2) is averaged over the Maxwell-Boltzmann velocity distribution as
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where u is the most probable speed of the atom. When the integration is performed, we have used the following result:
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with s =sign[Im( z )]. Finally the averaged susceptibility is given by
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where
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and s 0 (=Ω 1 2 /( γt Γ)) is the on-resonance saturation parameter. The first term on the right-hand side in Eq. (3) represents the background signal.
Just after the atomic cell, the electric field of the probe beam is changed slightly from the incident one due to interactions with the atoms. When the intensity and the inclination angle of the incident probe beam are I 0 and θ 0 (=π /4), respectively, the values I and θ after traversing the atomic cell of the length l are given by
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respectively, with a ± =exp[-( kl /2) Im( χ ± )] [20] . The rotation angle is given by Δ θ = ( kl /4) Re(Δ χ ) where the difference in the susceptibilities, Δ χ (≡ χ χ + ), is given by
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with
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w (=2 δ / γt ) is the normalized detuning and μ (= ku / γt ) is the ratio between the linewidths of the inhomogeneous and homogeneous broadenings. μ is approximately 30 for 88 Sr atoms placed at room temperature. At a typical experimental condition, when μ ≫1, the real and imaginary parts of the susceptibilities are given by explicitly
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respectively.
III. PS AND POLEAS SPECTRA
In the absence of an optical element after the cell, the PS signal, the difference in the intensities along the x and the y axes, is given by
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where the following approximation is used because of small absorption:
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In the presence of a quarter-wave plate whose optic axis has an angle of π /4 with respect to the x axis, the POLEAS signal, the difference in the intensities along the x and the y axes, is given by
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In Eq. (5), the PS signal is proportional to Re(Δ χ ) , and the POLEAS signal is proportional to Im(Δ χ ) in Eq. (6) where Δ χ is in Eq. (4). When μ ≫1, the PS and POLEAS signals can be expressed in a more convenient way as
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respectively.
We present typical plots of the real and the imaginary parts of accurate (Eq. (4)) and approximate (Eq. (7)) values of Δ χ in Fig. 2(a). In Fig. 2(a) , accurate and approximate values of Δ χ are shown in solid and dotted curves, respectively, where s 0 = 50 and μ = 15 . There is a slight discrepancy between the two results for the real part of Δ χ . We found no significant discrepancy for the imaginary part of Δ χ at this condition. Fig. 2(b) shows the maximum amplitude and the slope of the real part of Δ χ (PS spectrum) as a function of s 0 for the two different values of μ = 30 and μ = 10 . Accurate and approximate results coincide well at a usual experimental condition of μ ≃30. However, there are discrepancies between the two results, in particular, for the amplitude. The weak dependence of the slope on μ results from the fact that the discrepancy occurs at larger detunings.
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(a) The real and the imaginary parts of Δχ at s0 = 50 and μ =15 . (b) The dependence of the amplitude and the slope of the real part of Δχ on s0.
Finally we mention briefly the SAS spectrum for two-level atoms. When only the σ + component of the probe beam is considered, the scheme is identical to the case of two-level atoms. Therefore, we can obtain the absorption coefficient for two-level atoms as follows:
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The first term on the right-hand side in Eq. (8) represents the background signal in the absence of the pump beam. Eq. (8) can be simplified when the condition, μ ≫1, is satisfied as follows:
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which is the well-known expression found in the textbooks [19] .
IV. CONCLUSION
We have presented an analytical study of the lineshapes in PS and POLEAS for the atomic transition Jg =0 → Je =1. PS and POLEAS can measure the birefringence and dichroism, respectively. Eqs. (4)-(6) are the main results of this paper. Unlike the cases for the atoms with complicated structure such as Rb and Cs, the only mechanism responsible for the signals is the saturation effect caused by the pump beam, which enables us to calculate the signal regardless of the magnitude of the pump beam intensity. The results provided in this paper can be used in studying PS and POLEAS using 88 Sr or Yb atoms.
Color versions of one or more of the figures in this paper are available online.
Acknowledgements
This study was financially supported by Chonnam National University, 2013.
References
Wieman C. , Hӓnsch T. W. T. W. 1976 Doppler-free laser polarization spectroscopy Phys. Rev. Lett. http://dx.doi.org/10.1103/PhysRevLett.36.1170 36 1170 - 1173
Kunz P. D. , Heavner T. P. , Jefferts S. R. 2013 Polarizationenhanced absorption spectroscopy for laser stabilization Appl. Opt. http://dx.doi.org/10.1364/AO.52.008048 52 8048 - 8053
Corwin K. L. , Lu Z. , Hand C. F. , Epstein R. J. , Wieman C. E. 1998 Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor Appl. Opt. http://dx.doi.org/10.1364/AO.37.003295 37 3295 - 3298
Demtröder W. 1998 Laser Spectroscopy Springer Berlin
Pearman C. P. , Adams C. S. , Cox S. G. , Griffin P. F. , Smith D. A. , Hughes I. G. 2002 Polarization spectroscopy of a closed atomic transition: applications to laser frequency locking J. Phys. B http://dx.doi.org/10.1088/0953-4075/35/24/315 35 5141 - 5151
Yoshikawa Y. , Umeki T. , Mukae T. , Torii Y. , Kuga T. 2003 Frequency stabilization of a laser diode with use of light-induced birefringence in an atomic vapor Appl. Opt. http://dx.doi.org/10.1364/AO.42.006645 42 6645 - 6649
Harris M. L. , Adams C. S. , Cornish S. L. , McLeod I. C. , Tarleton E. , Hughes I. G. 2006 Polarization spectroscopy in rubidium and cesium Phys. Rev. A http://dx.doi.org/10.1103/PhysRevA.73.062509 73 062509 -
Tiwari V. B. , Singh S. , Mishra S. R. , Rawat H. S. , Mehendale S. C. 2006 Laser frequency stabilization using Dopplerfree bi-polarization spectroscopy Opt. Commun. http://dx.doi.org/10.1016/j.optcom.2006.01.028 263 249 - 255
Do H. D. , Moon G. , Noh H. R. 2008 Polarization spectroscopy of rubidium atoms: Theory and experiment Phys. Rev. A http://dx.doi.org/10.1103/PhysRevA.77.032513 77 032513 -
Ohtsubo N. , Aoki T. , Torii Y. 2012 Buffer-gas-assisted polarization spectroscopy of6Li Opt. Lett. http://dx.doi.org/10.1364/OL.37.002865 37 2865 - 2867
Carr C. , Adams C. S. , Weatherill K. J. 2012 Polarization spectroscopy of an excited state transition Opt. Lett. http://dx.doi.org/10.1364/OL.37.000118 37 118 - 120
Wu T. , Peng X. , Gong W. , Zhan Y. , Lin Z. , Luo B. , Guo H. 2013 Observation and optimization of4He atomic polarization spectroscopy Opt. Lett. http://dx.doi.org/10.1364/OL.38.000986 38 986 - 988
Javaux C. , Hughes I. G. , Locheada G. , Millen J. , Jones M. P. A. 2010 Modulation-free pump-probe spectroscopy of strontium atoms Eur. Phys. J. D http://dx.doi.org/10.1140/epjd/e2010-00029-4 57 151 - 154
Nakayama S. 1985 Theoretical analysis of Rb and Cs D2lines in Doppler-free spectrosopic techniques with optical pumping Jpn. J. Appl. Phys. 24 1 - 7
Do H. D. , Heo M. S. , Moon G. , Noh H. R. , Jhe W. 2008 Analytic calculation of the lineshapes in polarization spectroscopy of rubidium Opt. Commun. http://dx.doi.org/10.1016/j.optcom.2008.04.022 281 4042 - 4047
Shimada Y. , Chida Y. , Ohtsubo N. , Aoki T. , Takeuchi M. , Kuga T. , Torii Y. 2013 A simplified 461-nm laser system using blue laser diodes and a hollow cathode lamp for laser cooling of Sr Rev. Sci. Instrum. http://dx.doi.org/10.1063/1.4808246 84 063101 -
Noh H. R. 2013 Analytical study of polarization spectroscopy for the Jg= 0 → Je= 1 transition J. Opt. Soc. Korea http://dx.doi.org/10.3807/JOSK.2013.17.3.279 17 279 - 282
Berman P. R. , Malinovsky V. S. 2011 Principles of Laser Spectroscopy and Quantum Optics Princeton University Press Princeton
Foot C. J. 2005 Atomic Physics Oxford University Press New York, USA
Seo M. J. , Won J. Y. , Noh H. R. 2011 Variation in the polarization state of arbitrarily polarized light via a circular anisotropic atomic medium J. Korean Phys. Soc. http://dx.doi.org/10.3938/jkps.59.253 59 253 - 256
Cohen-Tannoudji C. , Dupont-Roc J. , Grynberg G. 1992 Atom-Photon Interactions, Basic Processes and Applications Wiley New York, USA