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Spectropolarimetric measurement of populations in degenerated atomic systems with metastable lower level during optical pumping in intense laser fields
1 September 2002
Optics Communications 210 (2002) 67–73 www.elsevier.com/locate/optcom
Spectropolarimetric measurement of populations in degenerated atomic systems with metastable lower level during optical pumping in intense laser fields A.G. Dvoryanchikov, I.A. Kartashov, A.V. Shishaev * Institute of Semiconductor Physics, SB RAS, 630090 Novosibirsk, Russia Received 29 November 2001; received in revised form 18 March 2002; accepted 3 June 2002
Abstract The prospects for application of a nonlinear laser polarization spectroscopy for measuring the steady-state population characteristics in the degenerated atomic systems with metastable lower state during optical pumping in intense laser fields are considered. It is shown that a combination of the polarization spectroscopy of a probe field and the Faraday effect allows to measure immediately the population differences for ‘‘enriched’’ and ‘‘depleted’’ Zeeman’s sublevels during the process of optical pumping as well as for optical coupled sub-levels of lower and upper states of the transition. The experimental results for the 1s5 ! 2p2 ðJ ¼ 2 ! J ¼ 1Þ and 1s5 ! 2p4 ðJ ¼ 2 ! J ¼ 2Þ Ne20 transitions and their comparison are given. 2002 Elsevier Science B.V. All rights reserved. PACS: 32.60.+i; 32.80.Bx; 33.55.Fi; 42.62.Fi Keywords: Polarization spectroscopy; Optical pumping; Faraday effect; Metastable atoms
1. Introduction Degenerated atomic systems with the long-lived lower state have been a topic of active interest in resent years. The relevant information on such an atomic ensemble can be gained from the spectroscopic features arising at interaction of radiation with degenerated atomic systems and, conditioned largely by a noticeable role of effect of optical
*
Corresponding author. E-mail address: [email protected] (A.V. Shishaev).
pumping [1] in the considered processes. Depending on specific tasks and experiment conditions these features can be revealed, for example, in the form of narrow resonances of different physical origin in absorption spectra [2–4]. A major parameter describing these resonances and processes responsible for their origin is population of degenerated sub-levels of a lower transition state found during interaction. The polarization spectroscopy of a probe field appears by the most suitable method here. We have found that at a certain combination of the polarization spectroscopy of a probe field with Faraday effect it becomes possible to measure a
0030-4018/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 6 8 5 - 1
68
A.G. Dvoryanchikov et al. / Optics Communications 210 (2002) 67–73
population difference not only of ‘‘enriched’’ and ‘‘depleted’’ sub-levels of the lower state of transition during optical pumping, but also one’s of sub-levels of the lower and upper states optically connected by a strong field of pumping. Outcomes of these researches were used as the basis of the present article.
2. Technique for carrying out experimental researches As a whole, experimental setups for observation of polarization spectra are similar [5–7] except for some methodological differences. They propose to record rotation angle of polarization plane of a probe field as a function of laser radiation frequency by means of an analyzer. The analyzer is placed at the probe radiation output and is either rotated by a small angle relative to the initial polarization or crossed with it. In approximation, that environment anisotropy induced by a strong field of pumping is small, the radiation intensity of the probe field (I0 ) passing through the analyzer was analyzed repeatedly. Depending on the analyzer rotation angle h 1 and allowing for the terms of the same order it can be written as [5–7]: I=I0 f1 þ iðd d Þ þ f f gh2 þ ðf þ f Þf1 þ iðd d Þgh þ f f ;
ð1Þ
where I0 is the initial intensity of the probe radiation, f ¼ lxðvþ v Þ=4;
d ¼ lx½1 þ ðvþ þ v Þ=4=c:
Here ‘ is the length of an absorbing layer of gas, x is the laser radiation frequency. The complex susceptibilities v for intrinsic circular polarizations of environment are determined by an expression using refractive indexes (n þ 1) and absorption factors a c v 2ðn 1Þ þ i a : ð2Þ x It should be noted that the expression (1) was obtained with the assumption of ideality of the polaroid-analyzer and for the case of a plane light wave although it is inconsistent with the real ex-
periment conditions. In order to take account of the finite value of the polarizer transmission and of laser radiation divergence let us introduce jh0 j – the effective angle of permanent deviation of the analyzer from the crossed position by substituting 2 h ! jh0 j þ h in Eq. (1) [8]. Thus the value jh0 j will be consistent with the finite transmission of the polarizer in the crossed position. For real polarizer 2 jh0 j is about 10 7 –10 5 . One of the most effective ways of registration of alternating-sign polarization spectra is the method of the polarization modulation of a probe field radiation [8,9]. The useful signal is selected by phase detection at the modulation frequency of XM . So, takes h ¼ hM cos ðXM tÞ (here hM – the modulation angle amplitude), then from Eq. (1) and in terms of redefinition h, it is quite easy to determine the functional dependence of the spectra being recorded: l I=I0 2hM jh0 j 1 ðaþ þ a Þ 2 1 lx 1þ ð nþ n Þ : ð3Þ 2jh0 j c The physical sense of the obtained expression is obvious. The first multiplier in the braces describes the transmitted radiation intensity with regard to the medium dichroism. The second one defines the radiation transmission by the analyzer with regard to the polarization plane turn owing to the medium birefringence. As a result, the product of Lorentzian and dispersion contours defines the spectral line shape. Let us analyze the main properties of the spectral dependence (3), on the basis of a qualitative consideration of a radiation interaction with the simplest three-level K-like scheme of transitions on the assumption of the identity of transition widths for the right- and leftcircular polarization of radiation. The form of the spectral dependence (3) is given directly by the frequency properties of two factors taken in braces. The former is always distinguished from zero and has a constant sign regardless of laser frequency position x relative to the central frequency of an absorption line x0 . Unlike the former, the latter presents a function, which is able to take zero value:
A.G. Dvoryanchikov et al. / Optics Communications 210 (2002) 67–73
1þ
1 lx ðnþ n Þ ¼ 0: 2jh0 j c
ð4Þ
Zero value can be obtained in two cases, which makes it possible to determine directly (for the above K-like scheme) Zeeman sub-level populations N1;2 of the lower state of transition. Consider these cases. A. Optical pumping effect in polarization spectroscopy of transitions from the ground atom state in case of a small saturation parameter of pumping field has been considered in [6,7]. Problems concerned with probe field spectroscopy of degenerated atomic systems with a metastable lower level during optical pumping in a laser field of an arbitrary intensity have been considered in [10,11]. As is shown in these investigations, an absorption spectrum of probe field would be a sharp structure, similar to Bennet’s structure, against the backgrounds of Doppler-broadened line of unsaturated absorption. In accordance with the above a frequency dependence of value (n 1) is a superposition of two dispersion contours. Under small magnetic fields (D Cg , where D-Zeeman splitting) the contour corresponding to unsaturated absorption is removed from recorded signal owing to the value of (nþ n ) in Eq. (3). An absorbed power of probe field (Ppr ), calculated in [10], has a rather complicated form and is defined by specific scheme of investigated transitions (over total angular momentum J of transition levels). However, the frequency dependence of Ppr is noticeably simplified for K-like scheme of transition and can be written for branch ratio of a ¼ Aps =Cu < 0; 8 as: " 2 # pffiffiffi p X PprðÞ ¼ 2h x G2pr exp k v k v ( )
C ðCS þ CÞ2 0 0 1 N N 1;2 u ; 2 ðCS þ CÞ þ X2 ð5Þ where C – the natural half-width of the transition, C2S C2 ½1 þ ð1 aÞ j – the saturated halfwidth of the transition, Cu – the relaxation rate of 0 the upper level, N1;2 and Nu0 – populations of levels inpthe ffiffiffi absence of orienting field. Gpr ¼ lps Epr = 2 6 h – the interaction parameter for probe field,
69
C – coefficients depending on a and j (saturation parameter for pumping field), lps – the dipole moment of the transition, Aps – the first Einstein’s coefficient for the transition, X ¼ x0 x. Assuming N1;2 and Nu as steady level populations of K-like scheme during optical pumping one 0 can write C ðN1;2 Nu0 Þ ¼ N1;2 Nu . Considering relation of an absorption factor and a refractive index (Kramers–Kronig’s relationship) and using the above equation a nonlinear part of the refractive index Dn can be written in the form which is similar to the standard expression for a linear refractive index [12]: 2
Dn ¼
jlps j x0 x ðN1;2 Nu Þ; 6h ðx0 xÞ2 þ C2g
ð6Þ
where Cg ¼ C þ CS – the homogeneous half-width of the transition. By substituting (6) into (4), obtain DN ¼ N1 N2 ¼ 12jh0 j
2
ðx0 xÞ þ C2g : 2 ðx0 xÞ lxjlps j ch
ð7Þ
Thus measurement of frequency detuning of laser radiation (dx) relative to the absorption line center when expression (3) takes zero value allows to carry out a direct measurement of population differences of the Zeeman sub-levels of the lower transition state during optical pumping. B. Insertion of the absorbing medium into an longitudinal magnetic field results in the wellknown Faraday effect and the spectral dependence, conditioned by (nþ n ) difference, experiences a sign alternation within the absorption line (Macaluso–Corbino effect). Since the magnetic field is an axial vector value, the sign of the second term in Eq. (4) is defined by the reciprocal direction of the magnetic field and of the wave vector of the probe field. Therefore under appropriate choice of the value and direction of the magnetic field it becomes possible to ensure the fulfillment of condition (4): ( lxjlps j2 x1 x 1þ ðN1 Nu Þ 12jh0 jch ðx1 xÞ2 þ C2g ) x2 x ðN2 Nu Þ ¼ 0: ð8Þ ðx2 xÞ2 þ C2g
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A.G. Dvoryanchikov et al. / Optics Communications 210 (2002) 67–73
Here, x1;2 ¼ x0 D are frequencies of r – components of the transition, D is Zeeman’s splitting of the transition frequency. Recall that DN ¼ N1 N2 and, as indicated above, can be measured. Therefore, determination of the magnitude D and measurement of x, when Eq. (8) is held true allow to obtain the values of N1;2 Nu for the degenerated atomic system during optical pumping in intense laser fields. It should be noted once again that all the quantities appearing in Eqs. (7) and (8) can be either computed using the known data or measured directly. In particular, in con2 ditions of our experiments the value jh0 j 4 6 10 .
3. Experimental investigation results of 1s5 2p2 ðJ = 2 fi J = 1) and 1s5 2p4 ðJ = 2 fi J = 2ÞNe20 transitions A setup scheme is given in Fig. 1. A tunable cw single-frequency dye-laser on R6G was used in the work. The output power of the laser radiation could reach over 100 mW at k ¼ 590 nm with the linewidth 2 MHz. The control of laser frequency tuning was accomplished by a semiconfocal Fabry–Perot interferometer with a free spectral range of 862 MHz.
A strong saturating field with a circular polarization and a weak probe field with a linear one were obtained by dividing laser radiation with a beam splitter and were directed opposite to each other at an angle of 10 2 radian. A quarter-wave plate realized the circle polarization of the strong field. Having passed through a polarizer (1), a polarized modulator (3), an absorbing cell (4) and an analyzer (5) the probe field radiation was registered by a photodetector using a phase-sensitive amplifier on the polarization modulation frequency ðXM ¼ 425 HzÞ. The signal calibration against a rotation angle of the polarization plane was carried out with the help of a calibrating silica plate placed into a longitudinal magnetic field. The output signal was recorded by a plotter, with the voltage of laser frequency tuning being supplied to the X-input. The absorbing cell was a discharge tube 25 mm in diameter with a hollow cathode and anode placed axially and an absorption region length of L 10 cm. The cell faceplates were tilted at a little angle. The cell was inserted into a solenoid able to induce a longitudinal magnetic field with an intensity in the range from 0 to 500 gauss. The investigations have been made on two transitions Ne20 : 1s5 –2p2 ðJ ¼ 2 ! J ¼ 1Þ, k ¼ and 1s5 –2p ðJ ¼ 2 ! J ¼ 2Þ; k ¼ 5881; 89 A 4
Fig. 1. Scheme of an experimental setup (1. polarizer, 2. calibrating silica plate with solenoid, 3. a polarized modulator, 4. absorbing cell inserted into solenoid, 5. analyzer, 6. aperture, 7. photodetector, 8. quarte-wave plate).
A.G. Dvoryanchikov et al. / Optics Communications 210 (2002) 67–73
71
Fig. 2. (a) Transition diagram 1s5 ! 2p2 . (b) Transition diagram 1s5 ! 2p4 .
. The transition diagrams taking to ac5944; 83 A count of the Zeeman’s level structure are shown in Figs. 2(a) and (b), respectively. The bold arrows correspond to transitions induced by the strong field, the thin ones to the probe field and the wavy ones to the spontaneous transitions. The asterisks depict the sub-levels enriched owing to the optical pumping. A spectra were recorded for isotopic pure Ne20 at the pressure of 0,1 torr and discharge current of 30 ma. The specific spectrum records versus the magnetic field magnitude are given in Fig. 3 (transition 1s5 –2p2 ) and in Fig. 4 (transition 1s5 –2p4 ). Note at once that a zero signal value was defined in the absence of the discharge current in a cell and a zero frequency value corresponded to the maximum of the saturated absorption resonance for each transition in the absence of the magnetic field.
Fig. 3. Polarization spectra versus magnetic field magnitude for 1s5 ! 2p2 transition.
Fig. 4. Polarization spectra versus magnetic field magnitude for 1s5 ! 2p4 transition.
The common property of the observed spectral dependence is its alternating-sign character and transformation into a spectral line similar to Maccaluso–Korbino contour with increasing axial magnetic field. Depending on the mutual direction of the magnetic field and probe laser one this transformation occurs either without sign alternation of the central maximum or with its alternation as expected while analyzing Eqs. (3) and (4). The research results of Faraday effect for degenerated atomic systems during optical pumping in intense laser fields are a theme for a special article. The present work goals were achieved by the analysis of shape of spectral lines for the cases of a zero magnetic field and magnetic field wherein the central maximum amplitude takes a zero value.
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A.G. Dvoryanchikov et al. / Optics Communications 210 (2002) 67–73
3.1. 1s5 –2p2 transition In the absence of a magnetic field (Fig. 3, H ¼ 0) the spectral dependence takes a zero value while detuning the laser radiation frequency relative to the line center dx 1:8 109 s 1 ðdm 296 MHzÞ. Based upon the first Einstein’s coefficient value for the transition Aps ¼ 1:16 107 s 1 and branch ratio a ¼ Aps =Cu 0; 214 [13], the orienting radiation intensity (I ¼ 0,1 W=cm2 ) and the decay rate of the lower metastable level 1s5 ðC1;2 105 s 1 Þ [10], one can determine the values of the dipole moment 2 square jlj , the saturation parameter (j) and homogeneous half-width of the transition (Cg ), entering Eqs. (7) and (8): 3 hc3 ð2J1 þ 1ÞAps 4x3 ¼ 2:2 10 35 erg cm3 ;
amplitude while dx 1:4 109 s 1 ðdm 230 MHzÞ. The main features of the transition are: 2 Aps ¼ 1:38 107 s 1 , a 0:24, jlj ¼ 4:62 10 35 erg cm3 , j 100, CS ¼ 2:67 108 s 1 . As a result from (7) one takes DN ¼ N1 N2 ¼ 0:75 109 cm 3 . At the line center the resonance amplitude takes a zero value when H ¼ 2 G (Fig. 4). Zeeman’s shift of the line is 4 MHz/G, and therefore D ¼ 2:5 107 s 1 and we derive from Eq. (8): N1 Nu 0:35 109 cm 3 ; N2 Nu 1:1 109 cm 3 :
jlps j2 ¼
4. Discussion
ð9Þ p ffiffi ffi 2 j ¼ 2jGj =CC1;2 100, G ¼ lE=2 3 h, 2C ¼ C1;2 þ Cu , CS ¼ 2:72 108 s 1 . Substituting the obtained values in (7) one gets DN ¼ N1 N2 ¼ 1:9 109 cm 3 . While switching on the longitudinal magnetic field the resonance amplitude at the frequency xH near the line center takes a zero value when H ¼ 1:38 G (Fig. 3). Direct measurements allow to determine Zeeman’s shift of the line 1s5 –2p2 to be of 5.2 MHz/G. Thus D ¼ 2:27 107 s 1 . Since the populations N1;2 are not equal, the frequency xH 6¼ x0 . However this distinction doesn’t exceed D=2 and is within the limits of measurement error which is confined by the width of laser radiation line. From expression (8), such an uncertainty in measuring xH is responsible for a precision loss in definition of the populations, which is within 0.1 of the measured value. For xH ¼ x0; one gets:
The distinctive feature of the transitions under consideration is a lower metastable state. It defines a nature of optical orientation of an atomic system showing an evident dissimilarity from the case of absorption from a ground state. A small part of atoms is in the metastable state and this part is a negligible quantity in comparison with the total balance of particles. This opens possibility to apply a classic kinetic model for analyzing the process, the model being typical for small subsystem interacting with a thermostat. Furthermore, after excitation the atoms do not necessarily go back to the initial metastable state, since several relaxation channels can be essential. And so their relative probabilities (the branch ratio a) take on special significance. The optical orientation problem of metastable atoms has been researched in [10]. Let us make use of some conclusions of this work for the qualitative consideration of our experimental results. Referring to [10], the efficiency of optical pumping is characterized by the branch ratio value and slumps at small values of 1 a 0:1–0:15. Such a conclusion is conditioned by atom ‘‘wandering’’ throughout magnetic sub-levels nM of the lower state as a result of the spontaneous transitions of mM ! nM and mM ! nM 1 and its leaving through the other channels. In our case the value 1 a 0:76–0:79, and only the closest magnetic sub-levels of the ground state linked by a
N1 Nu 0:7 109 cm 3 ; N2 Nu 2:6 109 cm 3 : 3.2. 1s5 –2p4 transition Let us carry out an analysis of the experimental data shown in Fig. 4 similarly to the above one. With no magnetic field a spectral line has zero
A.G. Dvoryanchikov et al. / Optics Communications 210 (2002) 67–73
spontaneous transition with enriched sub-level make the principal contribution to the population of a magnetic sub-level enriched during optical pumping. In terms of the rates of spontaneous transitions between magnetic sub-levels of the upper and lower states of the transition the model of a K-like scheme of transition chosen for estimation of a recorded signal is quite acceptable. The principle distinction for the transitions of J ¼ 2 ! J ¼ 1 and J ¼ 2 ! J ¼ 2 manifests itself in the fact that during optical pumping in the circular polarization field, in the first case, two extreme magnetic sublevels of the lower metastable state are enriched and in the second case only one is (Figs. 2(a) and (b)). And so the measurement results of population characteristics for 1s5 ! 2p2 transition must be practically twice as large as the similar ones for 1s5 ! 2p4 . It was the result that has been obtained in our investigations. A population of 1s5 Ne20 metastable level was measured repeatedly [14,15]. Under conditions of our experiment (p 0:1 Torr, I ¼ 30 mA) the population is to have a value of N 1010 cm 3 . Assuming an excitation of magnetic sub-levels isotropic for J ¼ 2 the average population of Zeeman sub-levels undisturbed by the pumping field is Nm 2 109 cm 3 . Based on this magnitude, moderately small measured values of DN can be interpreted in two ways. On the one hand, a selective occupancy efficiency of individual Zeeman sub-levels of metastable state 1s5 during optical pumping for 1s5 ! 2p2 and 1s5 ! 2p4 transitions is moderate owing to a small branching ratio. On the other hand, our results can be considered as evidence in favor of processes compensating a nonequilibrium population of depleted magnetic sub-levels, for example, a collisional mixing in the bloc of closely spaced levels 1si [11]. Summary We have demonstrated that the modulation saturation PS in a combination with a Faraday
73
effect enables making measurement of steady-state populations of degenerated atomic systems having a metastable lower state during optical pumping in intense laser fields. The measurements allow to receive an important spectroscopic information about the influence of different factors (branching ratio, interatomic collisions etc.) on occupancy efficiency of individual magnetic sub-levels of the lower state of the investigated transition.
Acknowledgements Authors acknowledge prof. Rautian S.G. and prof. Shalagin A.M. for discussion of results. This work was supported by programs Universities of Russia and Physics of Quantum and Wave Processes.
References [1] W. Happer, Rev. Mod. Phys. 44 (1972) 169. [2] E. Arimondo, in: E. Wolf (Ed.), Progress in Optics, vol. 35, 1996, p. 257. [3] W. Gawlik, G. Series, in: H. Walter, K. Rothe (Eds.), Forward Scattering and Polarization Spectroscopy, Laser Spectroscopy IV, Springer Series in Optical Sciences, vol. 21, Springer, Berlin, 1979, p. 210. [4] I.A. Kartashov, A.V. Shishaev, JETP Lett. 58 (1993) 501. [5] C. Wieman, T. Hansch, Phys. Rev. Lett. 36 (1976) 1170. [6] W. Gawlik, Acta Phys. Polonica A 66 (1984) 401. [7] S. Nakayama, J. Phys. Soc. Japan 50 (1981) 609–614. [8] E. Gillham, R. King, J. Sci. Instrum. 38 (1961) 21. [9] L.S. Vasilenko, L.N. Guskov, A.V. Shishaev, Sov. J. Quantum Electron. 5 (1978) 1746. [10] S.G. Rautian, A.V. Shishaev, JETP 108 (1995) 807. [11] I.A. Kartashov, S.G. Rautian, A.V. Shishaev, Physics of Vibrations, Allerton Press, New York 6(2), 1998, p. 143. [12] R. Pantell, H. Puthoff, Fundamentals of Quantum Electronics, Wiley, New York, 1969. [13] L.A. Vainstein, V.R. Mironenko, S.G. Rautian, E.G. Saprykin, Opt. Spectrosc. 87 (1999) 372. [14] I.M. Beterov, V.P. Chebotaev, Opt. Spectrosc. 23 (1967) 854. [15] R. Nakata, M. Suemitsu, K. Fukuda, J. Appl. Phys. 15 (1979) 1199.
Optics Communications 210 (2002) 67–73 www.elsevier.com/locate/optcom
Spectropolarimetric measurement of populations in degenerated atomic systems with metastable lower level during optical pumping in intense laser fields A.G. Dvoryanchikov, I.A. Kartashov, A.V. Shishaev * Institute of Semiconductor Physics, SB RAS, 630090 Novosibirsk, Russia Received 29 November 2001; received in revised form 18 March 2002; accepted 3 June 2002
Abstract The prospects for application of a nonlinear laser polarization spectroscopy for measuring the steady-state population characteristics in the degenerated atomic systems with metastable lower state during optical pumping in intense laser fields are considered. It is shown that a combination of the polarization spectroscopy of a probe field and the Faraday effect allows to measure immediately the population differences for ‘‘enriched’’ and ‘‘depleted’’ Zeeman’s sublevels during the process of optical pumping as well as for optical coupled sub-levels of lower and upper states of the transition. The experimental results for the 1s5 ! 2p2 ðJ ¼ 2 ! J ¼ 1Þ and 1s5 ! 2p4 ðJ ¼ 2 ! J ¼ 2Þ Ne20 transitions and their comparison are given. 2002 Elsevier Science B.V. All rights reserved. PACS: 32.60.+i; 32.80.Bx; 33.55.Fi; 42.62.Fi Keywords: Polarization spectroscopy; Optical pumping; Faraday effect; Metastable atoms
1. Introduction Degenerated atomic systems with the long-lived lower state have been a topic of active interest in resent years. The relevant information on such an atomic ensemble can be gained from the spectroscopic features arising at interaction of radiation with degenerated atomic systems and, conditioned largely by a noticeable role of effect of optical
*
Corresponding author. E-mail address: [email protected] (A.V. Shishaev).
pumping [1] in the considered processes. Depending on specific tasks and experiment conditions these features can be revealed, for example, in the form of narrow resonances of different physical origin in absorption spectra [2–4]. A major parameter describing these resonances and processes responsible for their origin is population of degenerated sub-levels of a lower transition state found during interaction. The polarization spectroscopy of a probe field appears by the most suitable method here. We have found that at a certain combination of the polarization spectroscopy of a probe field with Faraday effect it becomes possible to measure a
0030-4018/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 6 8 5 - 1
68
A.G. Dvoryanchikov et al. / Optics Communications 210 (2002) 67–73
population difference not only of ‘‘enriched’’ and ‘‘depleted’’ sub-levels of the lower state of transition during optical pumping, but also one’s of sub-levels of the lower and upper states optically connected by a strong field of pumping. Outcomes of these researches were used as the basis of the present article.
2. Technique for carrying out experimental researches As a whole, experimental setups for observation of polarization spectra are similar [5–7] except for some methodological differences. They propose to record rotation angle of polarization plane of a probe field as a function of laser radiation frequency by means of an analyzer. The analyzer is placed at the probe radiation output and is either rotated by a small angle relative to the initial polarization or crossed with it. In approximation, that environment anisotropy induced by a strong field of pumping is small, the radiation intensity of the probe field (I0 ) passing through the analyzer was analyzed repeatedly. Depending on the analyzer rotation angle h 1 and allowing for the terms of the same order it can be written as [5–7]: I=I0 f1 þ iðd d Þ þ f f gh2 þ ðf þ f Þf1 þ iðd d Þgh þ f f ;
ð1Þ
where I0 is the initial intensity of the probe radiation, f ¼ lxðvþ v Þ=4;
d ¼ lx½1 þ ðvþ þ v Þ=4=c:
Here ‘ is the length of an absorbing layer of gas, x is the laser radiation frequency. The complex susceptibilities v for intrinsic circular polarizations of environment are determined by an expression using refractive indexes (n þ 1) and absorption factors a c v 2ðn 1Þ þ i a : ð2Þ x It should be noted that the expression (1) was obtained with the assumption of ideality of the polaroid-analyzer and for the case of a plane light wave although it is inconsistent with the real ex-
periment conditions. In order to take account of the finite value of the polarizer transmission and of laser radiation divergence let us introduce jh0 j – the effective angle of permanent deviation of the analyzer from the crossed position by substituting 2 h ! jh0 j þ h in Eq. (1) [8]. Thus the value jh0 j will be consistent with the finite transmission of the polarizer in the crossed position. For real polarizer 2 jh0 j is about 10 7 –10 5 . One of the most effective ways of registration of alternating-sign polarization spectra is the method of the polarization modulation of a probe field radiation [8,9]. The useful signal is selected by phase detection at the modulation frequency of XM . So, takes h ¼ hM cos ðXM tÞ (here hM – the modulation angle amplitude), then from Eq. (1) and in terms of redefinition h, it is quite easy to determine the functional dependence of the spectra being recorded: l I=I0 2hM jh0 j 1 ðaþ þ a Þ 2 1 lx 1þ ð nþ n Þ : ð3Þ 2jh0 j c The physical sense of the obtained expression is obvious. The first multiplier in the braces describes the transmitted radiation intensity with regard to the medium dichroism. The second one defines the radiation transmission by the analyzer with regard to the polarization plane turn owing to the medium birefringence. As a result, the product of Lorentzian and dispersion contours defines the spectral line shape. Let us analyze the main properties of the spectral dependence (3), on the basis of a qualitative consideration of a radiation interaction with the simplest three-level K-like scheme of transitions on the assumption of the identity of transition widths for the right- and leftcircular polarization of radiation. The form of the spectral dependence (3) is given directly by the frequency properties of two factors taken in braces. The former is always distinguished from zero and has a constant sign regardless of laser frequency position x relative to the central frequency of an absorption line x0 . Unlike the former, the latter presents a function, which is able to take zero value:
A.G. Dvoryanchikov et al. / Optics Communications 210 (2002) 67–73
1þ
1 lx ðnþ n Þ ¼ 0: 2jh0 j c
ð4Þ
Zero value can be obtained in two cases, which makes it possible to determine directly (for the above K-like scheme) Zeeman sub-level populations N1;2 of the lower state of transition. Consider these cases. A. Optical pumping effect in polarization spectroscopy of transitions from the ground atom state in case of a small saturation parameter of pumping field has been considered in [6,7]. Problems concerned with probe field spectroscopy of degenerated atomic systems with a metastable lower level during optical pumping in a laser field of an arbitrary intensity have been considered in [10,11]. As is shown in these investigations, an absorption spectrum of probe field would be a sharp structure, similar to Bennet’s structure, against the backgrounds of Doppler-broadened line of unsaturated absorption. In accordance with the above a frequency dependence of value (n 1) is a superposition of two dispersion contours. Under small magnetic fields (D Cg , where D-Zeeman splitting) the contour corresponding to unsaturated absorption is removed from recorded signal owing to the value of (nþ n ) in Eq. (3). An absorbed power of probe field (Ppr ), calculated in [10], has a rather complicated form and is defined by specific scheme of investigated transitions (over total angular momentum J of transition levels). However, the frequency dependence of Ppr is noticeably simplified for K-like scheme of transition and can be written for branch ratio of a ¼ Aps =Cu < 0; 8 as: " 2 # pffiffiffi p X PprðÞ ¼ 2h x G2pr exp k v k v ( )
C ðCS þ CÞ2 0 0 1 N N 1;2 u ; 2 ðCS þ CÞ þ X2 ð5Þ where C – the natural half-width of the transition, C2S C2 ½1 þ ð1 aÞ j – the saturated halfwidth of the transition, Cu – the relaxation rate of 0 the upper level, N1;2 and Nu0 – populations of levels inpthe ffiffiffi absence of orienting field. Gpr ¼ lps Epr = 2 6 h – the interaction parameter for probe field,
69
C – coefficients depending on a and j (saturation parameter for pumping field), lps – the dipole moment of the transition, Aps – the first Einstein’s coefficient for the transition, X ¼ x0 x. Assuming N1;2 and Nu as steady level populations of K-like scheme during optical pumping one 0 can write C ðN1;2 Nu0 Þ ¼ N1;2 Nu . Considering relation of an absorption factor and a refractive index (Kramers–Kronig’s relationship) and using the above equation a nonlinear part of the refractive index Dn can be written in the form which is similar to the standard expression for a linear refractive index [12]: 2
Dn ¼
jlps j x0 x ðN1;2 Nu Þ; 6h ðx0 xÞ2 þ C2g
ð6Þ
where Cg ¼ C þ CS – the homogeneous half-width of the transition. By substituting (6) into (4), obtain DN ¼ N1 N2 ¼ 12jh0 j
2
ðx0 xÞ þ C2g : 2 ðx0 xÞ lxjlps j ch
ð7Þ
Thus measurement of frequency detuning of laser radiation (dx) relative to the absorption line center when expression (3) takes zero value allows to carry out a direct measurement of population differences of the Zeeman sub-levels of the lower transition state during optical pumping. B. Insertion of the absorbing medium into an longitudinal magnetic field results in the wellknown Faraday effect and the spectral dependence, conditioned by (nþ n ) difference, experiences a sign alternation within the absorption line (Macaluso–Corbino effect). Since the magnetic field is an axial vector value, the sign of the second term in Eq. (4) is defined by the reciprocal direction of the magnetic field and of the wave vector of the probe field. Therefore under appropriate choice of the value and direction of the magnetic field it becomes possible to ensure the fulfillment of condition (4): ( lxjlps j2 x1 x 1þ ðN1 Nu Þ 12jh0 jch ðx1 xÞ2 þ C2g ) x2 x ðN2 Nu Þ ¼ 0: ð8Þ ðx2 xÞ2 þ C2g
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Here, x1;2 ¼ x0 D are frequencies of r – components of the transition, D is Zeeman’s splitting of the transition frequency. Recall that DN ¼ N1 N2 and, as indicated above, can be measured. Therefore, determination of the magnitude D and measurement of x, when Eq. (8) is held true allow to obtain the values of N1;2 Nu for the degenerated atomic system during optical pumping in intense laser fields. It should be noted once again that all the quantities appearing in Eqs. (7) and (8) can be either computed using the known data or measured directly. In particular, in con2 ditions of our experiments the value jh0 j 4 6 10 .
3. Experimental investigation results of 1s5 2p2 ðJ = 2 fi J = 1) and 1s5 2p4 ðJ = 2 fi J = 2ÞNe20 transitions A setup scheme is given in Fig. 1. A tunable cw single-frequency dye-laser on R6G was used in the work. The output power of the laser radiation could reach over 100 mW at k ¼ 590 nm with the linewidth 2 MHz. The control of laser frequency tuning was accomplished by a semiconfocal Fabry–Perot interferometer with a free spectral range of 862 MHz.
A strong saturating field with a circular polarization and a weak probe field with a linear one were obtained by dividing laser radiation with a beam splitter and were directed opposite to each other at an angle of 10 2 radian. A quarter-wave plate realized the circle polarization of the strong field. Having passed through a polarizer (1), a polarized modulator (3), an absorbing cell (4) and an analyzer (5) the probe field radiation was registered by a photodetector using a phase-sensitive amplifier on the polarization modulation frequency ðXM ¼ 425 HzÞ. The signal calibration against a rotation angle of the polarization plane was carried out with the help of a calibrating silica plate placed into a longitudinal magnetic field. The output signal was recorded by a plotter, with the voltage of laser frequency tuning being supplied to the X-input. The absorbing cell was a discharge tube 25 mm in diameter with a hollow cathode and anode placed axially and an absorption region length of L 10 cm. The cell faceplates were tilted at a little angle. The cell was inserted into a solenoid able to induce a longitudinal magnetic field with an intensity in the range from 0 to 500 gauss. The investigations have been made on two transitions Ne20 : 1s5 –2p2 ðJ ¼ 2 ! J ¼ 1Þ, k ¼ and 1s5 –2p ðJ ¼ 2 ! J ¼ 2Þ; k ¼ 5881; 89 A 4
Fig. 1. Scheme of an experimental setup (1. polarizer, 2. calibrating silica plate with solenoid, 3. a polarized modulator, 4. absorbing cell inserted into solenoid, 5. analyzer, 6. aperture, 7. photodetector, 8. quarte-wave plate).
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Fig. 2. (a) Transition diagram 1s5 ! 2p2 . (b) Transition diagram 1s5 ! 2p4 .
. The transition diagrams taking to ac5944; 83 A count of the Zeeman’s level structure are shown in Figs. 2(a) and (b), respectively. The bold arrows correspond to transitions induced by the strong field, the thin ones to the probe field and the wavy ones to the spontaneous transitions. The asterisks depict the sub-levels enriched owing to the optical pumping. A spectra were recorded for isotopic pure Ne20 at the pressure of 0,1 torr and discharge current of 30 ma. The specific spectrum records versus the magnetic field magnitude are given in Fig. 3 (transition 1s5 –2p2 ) and in Fig. 4 (transition 1s5 –2p4 ). Note at once that a zero signal value was defined in the absence of the discharge current in a cell and a zero frequency value corresponded to the maximum of the saturated absorption resonance for each transition in the absence of the magnetic field.
Fig. 3. Polarization spectra versus magnetic field magnitude for 1s5 ! 2p2 transition.
Fig. 4. Polarization spectra versus magnetic field magnitude for 1s5 ! 2p4 transition.
The common property of the observed spectral dependence is its alternating-sign character and transformation into a spectral line similar to Maccaluso–Korbino contour with increasing axial magnetic field. Depending on the mutual direction of the magnetic field and probe laser one this transformation occurs either without sign alternation of the central maximum or with its alternation as expected while analyzing Eqs. (3) and (4). The research results of Faraday effect for degenerated atomic systems during optical pumping in intense laser fields are a theme for a special article. The present work goals were achieved by the analysis of shape of spectral lines for the cases of a zero magnetic field and magnetic field wherein the central maximum amplitude takes a zero value.
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3.1. 1s5 –2p2 transition In the absence of a magnetic field (Fig. 3, H ¼ 0) the spectral dependence takes a zero value while detuning the laser radiation frequency relative to the line center dx 1:8 109 s 1 ðdm 296 MHzÞ. Based upon the first Einstein’s coefficient value for the transition Aps ¼ 1:16 107 s 1 and branch ratio a ¼ Aps =Cu 0; 214 [13], the orienting radiation intensity (I ¼ 0,1 W=cm2 ) and the decay rate of the lower metastable level 1s5 ðC1;2 105 s 1 Þ [10], one can determine the values of the dipole moment 2 square jlj , the saturation parameter (j) and homogeneous half-width of the transition (Cg ), entering Eqs. (7) and (8): 3 hc3 ð2J1 þ 1ÞAps 4x3 ¼ 2:2 10 35 erg cm3 ;
amplitude while dx 1:4 109 s 1 ðdm 230 MHzÞ. The main features of the transition are: 2 Aps ¼ 1:38 107 s 1 , a 0:24, jlj ¼ 4:62 10 35 erg cm3 , j 100, CS ¼ 2:67 108 s 1 . As a result from (7) one takes DN ¼ N1 N2 ¼ 0:75 109 cm 3 . At the line center the resonance amplitude takes a zero value when H ¼ 2 G (Fig. 4). Zeeman’s shift of the line is 4 MHz/G, and therefore D ¼ 2:5 107 s 1 and we derive from Eq. (8): N1 Nu 0:35 109 cm 3 ; N2 Nu 1:1 109 cm 3 :
jlps j2 ¼
4. Discussion
ð9Þ p ffiffi ffi 2 j ¼ 2jGj =CC1;2 100, G ¼ lE=2 3 h, 2C ¼ C1;2 þ Cu , CS ¼ 2:72 108 s 1 . Substituting the obtained values in (7) one gets DN ¼ N1 N2 ¼ 1:9 109 cm 3 . While switching on the longitudinal magnetic field the resonance amplitude at the frequency xH near the line center takes a zero value when H ¼ 1:38 G (Fig. 3). Direct measurements allow to determine Zeeman’s shift of the line 1s5 –2p2 to be of 5.2 MHz/G. Thus D ¼ 2:27 107 s 1 . Since the populations N1;2 are not equal, the frequency xH 6¼ x0 . However this distinction doesn’t exceed D=2 and is within the limits of measurement error which is confined by the width of laser radiation line. From expression (8), such an uncertainty in measuring xH is responsible for a precision loss in definition of the populations, which is within 0.1 of the measured value. For xH ¼ x0; one gets:
The distinctive feature of the transitions under consideration is a lower metastable state. It defines a nature of optical orientation of an atomic system showing an evident dissimilarity from the case of absorption from a ground state. A small part of atoms is in the metastable state and this part is a negligible quantity in comparison with the total balance of particles. This opens possibility to apply a classic kinetic model for analyzing the process, the model being typical for small subsystem interacting with a thermostat. Furthermore, after excitation the atoms do not necessarily go back to the initial metastable state, since several relaxation channels can be essential. And so their relative probabilities (the branch ratio a) take on special significance. The optical orientation problem of metastable atoms has been researched in [10]. Let us make use of some conclusions of this work for the qualitative consideration of our experimental results. Referring to [10], the efficiency of optical pumping is characterized by the branch ratio value and slumps at small values of 1 a 0:1–0:15. Such a conclusion is conditioned by atom ‘‘wandering’’ throughout magnetic sub-levels nM of the lower state as a result of the spontaneous transitions of mM ! nM and mM ! nM 1 and its leaving through the other channels. In our case the value 1 a 0:76–0:79, and only the closest magnetic sub-levels of the ground state linked by a
N1 Nu 0:7 109 cm 3 ; N2 Nu 2:6 109 cm 3 : 3.2. 1s5 –2p4 transition Let us carry out an analysis of the experimental data shown in Fig. 4 similarly to the above one. With no magnetic field a spectral line has zero
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spontaneous transition with enriched sub-level make the principal contribution to the population of a magnetic sub-level enriched during optical pumping. In terms of the rates of spontaneous transitions between magnetic sub-levels of the upper and lower states of the transition the model of a K-like scheme of transition chosen for estimation of a recorded signal is quite acceptable. The principle distinction for the transitions of J ¼ 2 ! J ¼ 1 and J ¼ 2 ! J ¼ 2 manifests itself in the fact that during optical pumping in the circular polarization field, in the first case, two extreme magnetic sublevels of the lower metastable state are enriched and in the second case only one is (Figs. 2(a) and (b)). And so the measurement results of population characteristics for 1s5 ! 2p2 transition must be practically twice as large as the similar ones for 1s5 ! 2p4 . It was the result that has been obtained in our investigations. A population of 1s5 Ne20 metastable level was measured repeatedly [14,15]. Under conditions of our experiment (p 0:1 Torr, I ¼ 30 mA) the population is to have a value of N 1010 cm 3 . Assuming an excitation of magnetic sub-levels isotropic for J ¼ 2 the average population of Zeeman sub-levels undisturbed by the pumping field is Nm 2 109 cm 3 . Based on this magnitude, moderately small measured values of DN can be interpreted in two ways. On the one hand, a selective occupancy efficiency of individual Zeeman sub-levels of metastable state 1s5 during optical pumping for 1s5 ! 2p2 and 1s5 ! 2p4 transitions is moderate owing to a small branching ratio. On the other hand, our results can be considered as evidence in favor of processes compensating a nonequilibrium population of depleted magnetic sub-levels, for example, a collisional mixing in the bloc of closely spaced levels 1si [11]. Summary We have demonstrated that the modulation saturation PS in a combination with a Faraday
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effect enables making measurement of steady-state populations of degenerated atomic systems having a metastable lower state during optical pumping in intense laser fields. The measurements allow to receive an important spectroscopic information about the influence of different factors (branching ratio, interatomic collisions etc.) on occupancy efficiency of individual magnetic sub-levels of the lower state of the investigated transition.
Acknowledgements Authors acknowledge prof. Rautian S.G. and prof. Shalagin A.M. for discussion of results. This work was supported by programs Universities of Russia and Physics of Quantum and Wave Processes.
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