Line shapes in polarization spectroscopy for the rubidium D1 line in an external magnetic field

Optics Communications ] (]]]]) ]]]–]]]

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Line shapes in polarization spectroscopy for the rubidium D1 line in an external magnetic field Q1

Taek Jeong, Jun Yeon Won, Heung-Ryoul Noh n Department of Physics, Chonnam National University, Gwangju 500-757, Korea

a r t i c l e i n f o

abstract

Article history: Received 16 August 2012 Received in revised form 27 November 2012 Accepted 28 November 2012

We present a theoretical and experimental study of line shapes in polarization spectroscopy for the D1 transition line of 85Rb atoms in the presence (or absence) of a constant magnetic field. After the effect of the pump beam is calculated using rate equations (while the probe beam is neglected) the induced circular anisotropy is calculated by averaging the susceptibility over a Maxwell–Boltzmann velocity distribution and various atomic transit times. We find that the calculated results either in the absence or presence of an external magnetic field are in good agreement with experimental results. & 2012 Elsevier B.V. All rights reserved.

Keywords: Polarization spectroscopy Zeeman effect Rubidium

1. Introduction In polarization spectroscopy (PS), a circular anisotropy generated by a circularly polarized pump beam is detected by measuring the polarization rotation of the electric field of the probe beam [1]. Since in the counter-propagating geometry of two laser beams, only atoms belonging to specific velocity classes feel both the pump and the probe beams simultaneously, sub-Doppler resolution can be obtained in a Doppler-broadened line. Although saturated absorption spectroscopy (SAS) is widely used for active laser frequency stabilization [2], PS has also drawn much attention for this purpose owing to its modulation-free characteristics. After its first demonstration [1], there have been many experimental and theoretical reports on PS [3–11]. We should note that, as well as PS, there are several other modulation-free methods for active frequency stabilization, such as dichroic atomic vapor laser lock (DAVLL) [12], DAVLL in a transversal magnetic field [13], subDoppler DAVLL in a longitudinal magnetic field [10,14–17] or in a transversal magnetic field [18], and magnetically assisted rotation spectroscopy (MARS) [19]. As is well known, simple, theoretical method to predict line shapes in PS is Nakayama’s model [20,21]. Harris et al. also presented a numerical method for calculating line shapes in PS [5]. Accurate numerical and analytical studies have been produced by the authors of this paper [9,22]. Recently, Krezemien et al. reported MARS [19], which is in fact PS in a constant external magnetic field. They used Nakayama’s model to predict

n

Corresponding author. Tel.: þ82 62 5303366; fax: þ82 62 5303369. E-mail address: [email protected] (H.-R. Noh).

line shapes. In this paper we present a simple theory of calculating line shapes in PS in the presence of an external magnetic field. In this method the probe beam is neglected when the population of each relevant magnetic sublevel is calculated. The effect of the probe beam is considered when final optical coherences are calculated. Therefore, this method is quite general and is applicable to most sub-Doppler spectroscopies such as SAS and subDoppler DAVLL. It should be noted that PS [23] and SAS [24–28] in a magnetic field have been studied previously. Also, sub-Doppler spectroscopies in a magnetic field using a nanometric thin cell have also been reported [29–31]. The main aim of our study is not accurate measurements of Zeeman or Paschen–Back effects, but instead to provide a simple method of calculating line shapes in PS when there are moderate values for the external magnetic field. This paper is structured as follows: in Section 2, we present the theory for calculating the PS spectra. The experimental methods and calculated results are presented in Sections 3 and 4, respectively. The final section briefly summaries the results.

2. Theory A schematics of the PS setup is shown in Fig. 1(a). We assume a constant external magnetic field along the direction of the laser beams. As shown in Fig. 1(a), the polarization of the linearly polarized probe beam is rotated by the anisotropy induced by a circularly polarized pump beam. The rotation angle is given by ðkL=4Þðw r w rþ Þ, where kð ¼ 2p=lÞ is the wave vector, lð ¼ 795 nmÞ is the wavelength, L is the length of the cell, w r7 is the averaged real part of the susceptibility for the s 7 component of the probe beam. Using this, we measure the circular birefringence

0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.11.069

Please cite this article as: T. Jeong, et al., Line shapes in polarization spectroscopy for the rubidium D1 line in an external magnetic field, Optics Communications (2012), http://dx.doi.org/10.1016/j.optcom.2012.11.069i

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2

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T. Jeong et al. / Optics Communications ] (]]]]) ]]]–]]]

factor of the excited (ground) state. In Eq. (2), Fe ¼2 and 3. In addition to Eqs. (1) and (2), we add the following differential equations:

Iy

PBS

Ix Pump + (σ)

Fe = 3

5P1/2

Δe

Fe = 2

Probe (x+y)/21/2

Cell

m U_ F g ¼ Pm Fg ,

5S1/2

z

y

x

Fg = 2

Fig. 1. (a) Schematics of the PS setup. (b) Energy level diagram of the D1 transition line of 85Rb (De ¼ D32 ).

ð3Þ

U and V are the integration functions of P and Q, respectively. The reason for introducing U and V is explained later. In reality, we solve Eqs. (1)–(3), and obtain U and V for a finite transit time as functions of dpu . The variation of the probe beam can be calculated from the following susceptibility:

wq ¼ N at Fg = 3

m V_ F e ¼ Q m Fe :

3 3 3 X 3l G X e ,m þ q C F e ,m þ q sF3,m 4p2 cq Opr F ¼ 2 m ¼ 3 3,m

ð4Þ

e

where snm ( ¼ /n9s9mS) is the optical coherence. cq are the coefficients of the electric field for the probe beam, where qð ¼ 71,0) denotes the polarization of the probe beam. Opr ð ¼ Gðs0 =2Þ1=2 Þ is the Rabi frequency of the probe beam and Nat is the atomic density. Using the rate equation approximation, the optical coherence is simply given in terms of the populations m

generated in the vapor cell. It is noteworthy that the absorption coefficient, kw i7 or kw i0 , is measured when the probe beam is s 7 or p polarized, respectively, where w i7 ð0Þ is the averaged imaginary part of the susceptibility. We present a general method for calculation of the line shapes for the transitions from the upper ground state (Fg ¼ 3) of the D1-line of 85Rb atoms as shown in Fig. 1(b). It is straightforward to apply this method to the transitions from the lower ground state (Fg ¼2). We first assume that there exists a pump beam alone whose polarization is s 7 or p where the probe beam is neglected. In other cases such as elliptical polarization, it is not possible to formulate simple rate equations for the populations. Of course, the polarization of the pump beam in PS is s þ , and the results for other polarizations can be used for calculating SAS. After the populations are adjusted from the interaction with a pump beam, a weak probe beam detects these variations. Here, we assume that the probe beam is weak enough not to affect the populations perturbed by the pump beam. The rate equations in the presence of the pump beam alone are given by [9] m P_ 3 ¼ 

0

m þ q0 e ,m þ q ðGs0 =2ÞRF3,m ðP m Þ 3 Q F e

3 X

oFme ,m þ q Þ2 =G2 0

3 F e ¼ 2 1 þ4ðdpu þ DF e D

þ

3 X

m þ1 X

e e ,me GRF3,m Qm Fe ,

ð1Þ

F e ¼ 2 me ¼ m1

m Q_ F e ¼

o

mB B _

ðg F e me g F g mg Þ,



X

m þ1 X

m GRFF eg ,m ,mg Q F e ,

me g e cme mg Opr C FF eg ,m ,mg ðP F g Q F e Þ

2ðdpr þ D3F e DoFmeg,me Þ þ iG

,

ð5Þ

where dpr ð ¼ dkv) is the effective detuning of the probe beam frequency. Thus, Eq. (4) becomes

wq ¼ N at

mþq e ,m þ q 3 3 3 X RF3,m ðP m Þ 3l X 3 Q F e : F e ,m þ q 4p2 F ¼ 2 m ¼ 3 iþ 2ðdpr þ D3F Dom Þ=G e

ð6Þ

e

After being averaged over the Maxwell–Boltzmann velocity distribution and the transit times, Eq. (6) can be written as Z tav Z 1 2 1 dv pffiffiffiffi eðv=uÞ wq ðv,tÞ wq ¼ dt t av 0 pu 1 Z 1 3 3 3 X 3l Nat X ðv=uÞ2 dve ¼  2 pffiffiffiffi 4p pu F e ¼ 2 m ¼ 3 1 Z tav F e ,m þ q R3,m 1 mþq   dtðP m ðtÞÞ, ð7Þ F ðtÞQ F e F e ,m þ q 3 iþ 2ðdpr þ DF Dom Þ=G t av 0 e

where t av is the average transit time for crossing the laser beam and u is the most probable speed of the atom [32]. Changing the integration variable and using U and V instead of P and Q, Eq. (7) becomes "   # Z 1 3 3 3 X X 3l N at dpu d 2 w q ¼  2 pffiffiffiffi  dd exp  ku 4p pkutav F ¼ 2 m ¼ 3 1 pu e



mq0 e ,m ðGs0 =2ÞRF3,mq Q m 0 ðP 3 Fe Þ 3 F e ,m 2 1þ 4ðdpu þ DF e D mq0 Þ =G2

DoFmeg,me ¼

e sFF eg ,m ,mg ¼

mþq e ,m þ q RF3,m  ðU m ðdpu ÞÞ 3 ðdpu ÞV F e

iþ 2ð2ddpu þ D3F e DoFme ,m þ q Þ=G

:

ð8Þ

Finally the rotation angle is given by

F g ¼ 2,3 mg ¼ m1

ð2Þ

where P and Q denote the populations of the ground and the excited states, respectively, G is the decay rate of the excited state, dpu ð ¼ d þkv) is the effective detuning of the pump beam frequency, and d is the laser detuning as defined in Fig. 1(b). s0 ð ¼ I=Is Þ is the on-resonance saturation parameter where I is the pump beam intensity and Is ð ¼ 16 W=m2 Þ is the saturation intensity. q0 ð ¼ 71,0Þ denotes the polarization of the pump beam. Dnm is the frequency spacing between the states 9F e ¼ mS and 9F e ¼ nS F e ,me 2 F e ,me e (n Z m). RFF eg ,m ,mg ¼ ðC F g ,mg Þ , where C F g ,mg is the normalized transition

coefficient [32]. mB is the Bohr magneton, B is the magnitude of the magnetic field applied along the axis, and g F e (g F g ) is the g

y¼

kL r ðw w rþ Þ: 4 

ð9Þ

3. Experimental methods We briefly describe the experimental methods. The pump and the probe beams were produced by a commercial external cavity diode laser (ECDL; Toptica, DL100). Those beams were expanded to Gaussian beams with an e1=2 diameter of 11 mm and passed through irises with diameters of approximately 6.0 mm each. The intensity of the pump (probe) beam was approximately 0.2Is (0:02Is ). The pump beam passes a quarter-wave plate and becomes circularly polarized, whereas the electric field of the linearly polarized probe beam is at approximately 451 with

Please cite this article as: T. Jeong, et al., Line shapes in polarization spectroscopy for the rubidium D1 line in an external magnetic field, Optics Communications (2012), http://dx.doi.org/10.1016/j.optcom.2012.11.069i

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T. Jeong et al. / Optics Communications ] (]]]]) ]]]–]]]

4. Calculated and experimental results The typical results for the upper ground state (Fg ¼3) and lower ground state (Fg ¼2) for 85Rb atoms in the absence of the external magnetic field are shown in Fig. 2(a) and (b), respectively. In each figure, both the experimental and calculated results are plotted. As has been seen many times before, we observe two resonance signals and one crossover signal. Although there exists a slight discrepancy in the linewidths between the experimental and calculated spectra, which may result from neglect of laser linewidths or collisional broadening in the calculation, we observe a good agreement between calculated and experimental results. The linewidths of the laser beam and the collisional broadening can be included in the calculation by replacing G in the denominators in Eqs. (1) and (2) by G þ g þ2gc where g is the laser linewidth and gc refers to additional damping of the optical coherence due to collisions [33,34]. However, since our main aim is to obtain an easy method to calculate the line shape in PS, the additional broadening mechanisms are not taken into account in the calculation. The vertical offsets between experimental and calculated results may result from slight difference in the intensities between the s þ and s components of the probe beam. The assumption where the probe beam is neglected is only valid when it is very weak compared to the pump beam intensity. Otherwise we must solve accurate density matrix equations to predict accurate line shapes for PS spectra. It should be also noted that the pump beam intensity is smaller or comparable to the saturation intensity (Is). Fig. 3(a) and (b) shows the experimental and calculated results for the upper ground state of 85Rb in the presence of an external magnetic field, respectively. The saturation parameter for the pump beam is s0 ¼0.2. In each plot of Fig. 3, we can see the PS spectra for B ¼0, 1.0 G, 5.0 G, 10.0 G, 20.0 G, 30.0 G, 40.0 G, and 50.0 G from the top of each panel. In Fig. 3, we can see good agreement between experimental and calculated results. When the magnetic field is greater than  20 G, the signal shows various peaks corresponding to shifted Zeeman energy levels.

15 10 5 0 -5 -10

85

Rb Fg = 3 - Fe = 2,3 B = 0G

(i) Experiment (ii) Calculation

(i) (ii) Fg = 3 - Fe = 3 Crossover Fg = 3 - Fe = 2

-500 -400 -300 -200 -100 Detuning (MHz)

0

100

We find that our simple model is able to predict PS spectra in the presence of a magnetic field quite well. The results for the lower ground state of 85Rb are shown in Fig. 4. The scheme in Fig. 3 is same as that in Fig. 4. We can see good agreement between experimental and calculated results. The PS spectra in the presence of a magnetic field exhibit very complicated line shapes. We can, however, identify where the signals originate. As an example, consider the transition F g ¼ 3-F e ¼ 3 at B ¼50 G, which corresponds to the rightmost signal of the lowest traces in Fig. 3(a) and (b). The results are shown in Fig. 5. As it is somewhat difficult to identify resonances in dispersive PS spectra, we calculated the SAS spectra for the s þ polarized pump beam. In Fig. 5(a), the upper [lower] trace shows the results when the probe beam is s þ [s ] polarized. In Fig. 5(a), the thick curves denote the total SAS spectra, while the thin curves denote a decomposition of the spectra for various transitions. In the case of the s þ s þ polarization configuration (upper trace), we can find six resonances for mg ¼ 3, . . . ,2. In contrast, we can find six resonances for mg ¼ 2, . . . ,3 in the case of s þ s polarization configuration (lower trace). For the upper trace, the resonance conditions for the pump and probe beams are given by

dkv

mB B _

d þ kv

½g e ðmg þ 1Þg g mg  ¼ 0,

mB B _

½g e ðmg þ 1Þg g mg  ¼ 0,

ð10Þ

respectively, where g e ð ¼ g 3 ¼ 1=9Þ is the g-factor of the state 9F e ¼ 3S and g g ð ¼ g 3 ¼ 1=3Þ is the g-factor of the state 9F g ¼ 3S. From Eq. (12), we can find six resonances

d ¼ ½g e ðmg þ 1Þg g mg 

mB B _

12mg mB B , ¼ _ 9

ð11Þ

for mg ¼ 3, . . . ,2, and the corresponding velocity groups are kv ¼ 0. We can clearly see that six resonances in Eq. (11) exist in the upper trace of Fig. 5(a). For the lower trace in Fig. 5(a), the resonance conditions for the pump and probe beams are given by

dkv

mB B _

d þ kv

½g e ðmg þ 1Þg g mg  ¼ 0,

mB B _

½g e ðmg 1Þg g mg  ¼ 0,

ð12Þ

respectively. Thus, the resonances are given by

d ¼ ðg e g g Þmg

mB B _

¼

2mg mB B , 9 _

ð13Þ

for mg ¼ 2, . . . ,3, and the velocity groups are given by

PS Signals (Arb.Units)

respect to the horizontal axis, in order to make the background signal zero. The crossing angle between the counter propagating pump and probe beams is less than 10 mrad inside the cell. The external magnetic field was generated using a Helmholtz coil surrounding the cell, whose dimensions were 230 mm in length and 55 mm in diameter, and number of turns was 227. The magnetic field at the place of the cell was uniform within 0.5%. The cell and the Helmholtz coil were enclosed by a three-layer m-metal sheet to eliminate the terrestrial magnetic field. The cell was kept at room temperature. The two outputs of the signal were detected using a balanced photo-detector.

PS Signals (Arb.Units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

3

2 1 0 -1

85Rb

Fg = 2 - Fe = 2,3

(i) Experiment (ii) Calculation

B = 0G (i) (ii) Fg = 2 - Fe = 2

-2

Crossover Fg = 2 - Fe= 3

-500 -400 -300 -200 -100 Detuning (MHz)

Fig. 2. Typical results for (a) the upper ground state and (b) the lower ground state of

85

0

100

Rb atoms in the absence of an external magnetic field.

Please cite this article as: T. Jeong, et al., Line shapes in polarization spectroscopy for the rubidium D1 line in an external magnetic field, Optics Communications (2012), http://dx.doi.org/10.1016/j.optcom.2012.11.069i

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4

0.02

Experiment 85Rb

0.4

Fg = 3 - Fe = 2,3

Calculation 85Rb

0.2

B (G)

Fg = 3 - Fe = 2,3 B (G)

0.00

0

0.0

0

-0.01

1

-0.2

1

-0.02

5

-0.4

5

-0.6

10

-0.8

20

-1.0

30

-1.2

40

-1.4

50

-0.03

10

-0.04

20

PS Signals (Arb.Units)

PS Signals (Arb.Units)

0.01

30

-0.05

40

-0.06

50

-0.07

-1.6

-0.08 -400

-200 0 Detuning (MHz)

-400

200

Fig. 3. (a) Experimental and (b) calculated results for the upper ground state of each panel.

0.2

PS Signals (Arb. Units)

-0.02

10

-0.03

20 30

-0.05

Calculation

85

Rb Fg = 2 - Fe = 2,3

B (G) 0

0.0

5

-0.04

-0.2

1

-0.4

5

-0.6

10

-0.8

20

-1.0

40

-1.2

50

-1.4

30 40 50

-1.6 -400

-200 0 Detuning (MHz)

200

Fig. 4. (a) Experimental and (b) calculated results for the lower ground state of each panel.

kv ¼ g e

mB B _

¼

200

Rb where B¼ 0, 1.0 G, 5.0 G, 10.0 G, 20.0 G, 30.0 G, 40.0 G and 50.0 G from the top of

1

-0.01

-200 0 Detuning (MHz)

85

0.01 Experiment 85Rb F = 2 - F = 2,3 g e B (G) 0 0.00 PS Signals (Arb. Units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

T. Jeong et al. / Optics Communications ] (]]]]) ]]]–]]]

-400

-200 0 Detuning (MHz)

200

85

1 mB B : 9 _

The identification of the resonances in PS spectra is presented in Fig. 5(b) and (c). Fig. 5(b) shows the s 7 components of the PS spectra and their decomposition into each transition. As depicted in Fig. 5(a), we can clearly see six resonances for the s 7 components. The comparison with experimental results is shown in Fig. 5(c). We can see good agreement between these two results.

5. Conclusions In this paper we presented a simple method for numerical calculation of line shape in polarization spectroscopy for the D1 transition line of 85Rb atoms. The method of calculation is general, and thus it can be easily applied to SAS and sub-Doppler DAVLL.

Rb where B ¼0, 1.0 G, 5.0 G, 10.0 G, 20.0 G, 30.0 G, 40.0 G and 50.0 G from the top of

In normal experimental conditions a very weak probe beam is used, this makes it possible to neglect the probe beam when the populations of the sublevels are calculated. We were able to predict PS line shape very accurately up to a magnetic field of 50 G. As the typical magnetic field in experiment for PS or subDoppler DAVLL is less than this value, we can use this method for calculating the signals in those spectroscopies. We are currently preparing sub-Doppler DAVLL for the D1 and D2 transition lines of 85 Rb atoms and will carry out the comparison of experimental and calculated results. In order to prove the generality of the method of calculation, we will also perform experiments of PS for Cs and K. The purpose of this paper is not to calculate the exact line shape of the spectra, but to calculate the approximate line shape in an efficient and amenable manner. It is true that there exist cases where the approximations used in this paper no longer hold. For example, when the laser is elliptically polarized, we must calculate complicated density matrix equations. Moreover,

Please cite this article as: T. Jeong, et al., Line shapes in polarization spectroscopy for the rubidium D1 line in an external magnetic field, Optics Communications (2012), http://dx.doi.org/10.1016/j.optcom.2012.11.069i

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T. Jeong et al. / Optics Communications ] (]]]]) ]]]–]]]

SAS Signals (Arb. Units)

85

Rb Fg = 3 - Fe = 3

0.2 σ + - σ+ 0.0

-0.2

σ+ - σ-

-0.4 -100

-500

0 Detuning (MHz)

0.05

PS Signals (Arb. Units)

67 68 69 This research was supported by Basic Science Research Pro70 gram through the National Research Foundation of Korea (NRF) 71 funded by the Ministry of Education, Science and Technology 72 (2011-0009886). 73 74 References 75 76 ¨ [1] C. Wieman, T.W. Hansch, Physical Review Letters 36 (1976) 1170. 77 ¨ [2] W. Demtroder, Laser Spectroscopy, Springer, Berlin, 1998. 78 [3] G.P.T. Lancaster, R.S. Conroy, M.A. Clifford, J. Arlt, K. Dholakia, Optics 79 Communications 170 (1999) 79. 80 [4] C.P. Pearman, C.S. Adams, S.G. Cox, P.F. Griffin, D.A. Smith, I.G. Hughes, Journal of Physics B 35 (2002) 5141. 81 [5] M.L. Harris, C.S. Adams, S.L. Cornish, I.C. McLeod, E. Tarleton, I.G. Hughes, 82 Physical Review A 73 (2006) 062509. 83 [6] Y. Yoshikawa, T. Umeki, T. Mukae, Y. Torii, T. Kuga, Applied Optics 42 (2003) 6645. 84 [7] A. Ratnapala, C.J. Vale, A.G. White, M.D. Harvey, N.R. Heckenberg, 85 H. Rubinsztein-Dunlop, Optics Letters 29 (2004) 2704. 86 [8] V.B. Tiwari, S. Singh, S.R. Mishra, H.S. Rawat, S.C. Mehendale, Optics Communications 263 (2006) 249. 87 [9] H.D. Do, G. Moon, H.R. Noh, Physical Review A 77 (2008) 032513. 88 [10] C. Javaux, I.G. Hughes, G. Locheada, J. Millen, M.P.A. Jones, European Physical 89 Journal D 57 (2010) 151. 90 [11] N. Ohtsubo, T. Aoki, Y. Torii, Optics Letters 37 (2012) 2865. [12] K.L. Corwin, Z.T. Lu, C.F. Hand, R.J. Epstein, C.E. Wieman, Applied Optics 37 91 (1998) 3295. 92 [13] T. Hasegawa, M. Deguchi, Journal of the Optical Society of America B 26 93 (2009) 1216. [14] G. Wasik, W. Gawlik, J. Zachorowski, W. Zawadzki, Applied Physics B 75 94 (2002) 613. 95 [15] T. Petelski, M. Fattori, G. Lamporesi, J. Stuhler, G.M. Tino, European Physical 96 Journal D 22 (2003) 279. [16] M.L. Harris, S.L. Cornish, A. Tripathi, I.G. Hughes, Journal of Physics B 41 97 (2008) 085401. 98 [17] M. Pichler, D.C. Hall, Optics Communications 285 (2012) 50. 99 [18] S. Okubo, K. Iwakuni, T. Hasegawa, Optics Communications, submitted for 100 publication. Q2 [19] L. Krzemien´, K. Brzozowski, A. Noga, M. Witkowski, J. Zachorowski, 101 M. Zawada, W. Gawlik, Optics Communications 284 (2011) 1247. 102 [20] S. Nakayama, Japanese Journal of Applied Physics Part 1 24 (1985) 1. 103 [21] S. Nakayama, Physica Scripta T 70 (1997) 64. [22] H.D. Do, M.S. Heo, G. Moon, H.R. Noh, W. Jhe, Optics Communications 281 104 (2008) 4042. 105 [23] S. Nakayama, G.W. Series, W. Gawlik, Optics Communications 34 (1980) 382. 106 [24] H.S. Lee, S.E. Park, J.D. Park, H. Cho, Journal of the Optical Society of America B 11 (1994) 558. 107 [25] J. Bowie, J. Boyce, R. Chiao, Journal of the Optical Society of America B 12 108 (1995) 1839. 109 [26] M.U. Momeen, G. Rangarajan, P.C. Deshmukh, Journal of Physics B 40 (2007) 110 3163. [27] G. S˘kolnik, N. Vujic˘ic´, T. Ban, Optics Communications 282 (2009) 1326. 111 [28] S. Pradhan, B.N. Jagatap, Journal of the Optical Society of America B 28 (2011) 112 398. 113 [29] G. Hakhumyan, D. Sarkisyan, A. Sargsyan, A. Atvars, M. Auzinsh, Optics and Spectroscopy 108 (2010) 685. 114 [30] G. Hakhumyan, C. Leroy, Y. Pashayan-Leroy, D. Sarkisyan, M. Auzinsh, Optics 115 Communications 284 (2011) 4007. 116 [31] A. Sargsyan, G. Hakhumyan, C. Leroy, Y. Pashayan-Leroy, A. Papoyan, D. Sarkisyan, Optics Letters 37 (2012) 1379. 117 [32] H.R. Noh, H.S. Moon, Physical Review A 80 (2009) 022509. 118 [33] J. Gea-Banacloche, Y.Q. Li, S.Z. Jin, M. Xiao, Physical Review A 51 (1995) 576. 119 [34] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom–Photon Interactions, 120 Basic Processes and Applications, Wiley, New York, 1992. 121 122 123 124 125 126 127 128 129 130 131 132 Acknowledgments

0.4

85

50

100

Rb Fg = 3 - Fe = 3

0.00 σ+ component -0.05

σ- component

-0.10

-0.15 -100

0.2 PS Signals (Arb. Units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

-500

85Rb

0 Detuning (MHz)

50

100

Fg = 3 - Fe = 3 Experiment

0.1

σ- component

0.0 Calculation -0.1 σ+ component -0.2 -100

-500

0 Detuning (MHz)

50

100

Fig. 5. (a) Upper (lower) trace: calculated SAS spectra for the s þ polarized pump beam and s þ (s ) polarized probe beam. (b) The s 7 components of the PS spectra and their decomposition into each transition. (c) Comparison of the calculated PS signal with experimental results.

when the intensity is not much weaker than the saturation intensity, it may be not possible to use the method outlined in this paper. The calculations for these conditions are currently under progress.

5

Please cite this article as: T. Jeong, et al., Line shapes in polarization spectroscopy for the rubidium D1 line in an external magnetic field, Optics Communications (2012), http://dx.doi.org/10.1016/j.optcom.2012.11.069i