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Theoretical Investigation of Induced-Dichroism-Excited Atomic Line Filter

Yun-Dong Zhang*, Xu-Tao Sun, Zhu-Song He, Ping Yuan

Institute of Optoelectronics, Harbin Institute of Technology, State Key Laboratory of Tunable Technology, Harbin150001, China

Abstract: A theoretical model for the Laser induced (LIDOF) is presented. The filter has a higher transmission and a narrower line width than excited-state Faraday anomalous dispersion optical-filter (ESFADOF). The theoretical treatment is valid for different atoms LIDOF systems.

Keywords: LIDOF, polarizability, transmission

1. Introduction Narrowband filters with a wide field of view and high transmission play a key role in free space communication, and communication under water and so on. Comparing with the Faraday anomalous dispersion optical filter (FADOF)[1-4], the Laser induced dispersion optical filter (LIDOF) does not need an external , and has higher transmission with narrower linewidth at excited state operation. Their merits are better than that of excited-state- FADOF, and have a broaden application in the future. Laser-induced dispersion optical filter (LIDOF) was first described and experimentally studied by R.I.Billmers et al in 1995[5]. However, at present, it has not a perfectly theoretical model of the LIDOF. In this paper, the semi-classical theory was utilized to theoretically analyze LIDOF, and the model was given. By solving the density matrix equations, we gained the system induced susceptibility, and eventually the transmission spectra were gained. The theoretical model is in universality, adapting to different atoms LIFOF systems. 2. Theoretical consideration As working media of LIDOF are gaseous, it needs an source. Usual way of dealing with M= +1/2 -1/2

8S1/2

RCP probe 532nm

4P1/2 LCP pump 770nm

4S1/2 Fig.1 Simplified relevant energy-level diagram of K atom. interaction between three-level system and two light fields is used by density matrix method of semi-classical

National Natural Science Fund of China (60272075), Electronic address: [email protected]

Information Optics and Photonics Technology, edited by Guoguang Mu, 391 Francis T. S. Yu, Suganda Jutamulia, Proceedings of SPIE Vol. 5642 (SPIE, Bellingham, WA, 2005) · 0277-786X/05/$15 · doi: 10.1117/12.575862 theory. Hänsch.ThˈRichard L.Panock and Richard G.Brewer discussed this method respectively [6-8].

For the transition of 4S1/2ĺ4P1/2ĺ8S1/2 for atom system, potassium atoms only interact with right circularly polarized component of linearly polarized probe pulse according to the selection rules for the transition, provided the left circular polarization light is utilized for optical pumping source. Simplified potassium atoms level is shown in Fig 1. G G Consider electromagnetic waves of pump light E1 and probe light E2 : K 1 E ( z , t ) {E (1) ( z , t )eˆ exp[  i(k z  : t )]  c.c} (1) 1 2   1 1 K 1 E (z,t) {[E (2) (z,t)eˆ  E (2) (z,t)eˆ ]exp[i(k z  : t)] c.c} (2) 2 2     1 1 Three-level system described in the form of the general density matrix is given by

§ w w · i v ( H H ) i (3) =¨  ¸U mn ¦ ns U sm  U ns sm  =J nm U nm  =/G n,1G m,1 © wt wz ¹ s According to rotary wave approximation, non-diagonal elements are defined as ~ ~ U 21 U 21 exp[i(k1 z  :1t)],U 32 U 32 exp[i(k2 z  : 2t)], ~ (4) U 31 U 31 exp{i[(k1  k2 )z  (:1  : 2 )t]} By introducing the following quantities for electric dipole matrix elements and Rabi 1 P e 1 x  iy 2 ,D E (1) P 2=, 1 2 1  1 (5) 1 P e 2 x  iy 3 ,D E (2) P 2= 2 2 2  2 Solving Eqs(3) at steady state, we obtain ~ ~ J1U11 /iD1( U12  U21 ), ~ ~ ~ ~ J 2U22 iD1( U12  U21 )iD2( U23  U32 ), ~ ~ J 3U33 iD2( U23  U32 ), ~ ~ (6) ('1 k1v iJ 21 )U21 D2U31 D1( U22  U11 ), ~ ~ ('2 k2v iJ 32 )U32 D1U31 D1( U33  U22 ), ~ ~ ~ ('3 k3v iJ 31 )U31 D2U21 D1U32 where

'1211 ZZ :, ' 2322 : '312312 '', kkk 

'1 , '2 are detunings and '33,k are two-photon coefficients. The diagonal elements of density matrix are real

392 Proc. of SPIE Vol. 5642 * number, and non-diagonal elements have such the relation U ij U ji . Linking with Equation (6), there are nine

Equations and nine unknown parameters. We can theoretically solve these equations, but it is very difficult. Owing to LIDOF is used to detect weak optical signal, we can consider probe light field as a perturbation, via

~ (1) (1) (1) approximately considering D2 0 , consequently can obtain U21 , U22 , U11 in the case of first approximation.

~ (1) N D 1 (' 1  k1v  iJ 21 ) U 21  2 2 (' 1  k1v)  *1 2 (1) 2D 1 J 21 N U 11  N  2 2 J 1[( ' 1  k1v)  *1 ] (7) 2 (1) 2D 1 J 21 N U 22  2 2 J 2 [( ' 1  k1v)  *1 ] (1) U 33 0

2 2 J 12 (J 1  J 2 ) / where *1 J 12  2D1 , N  (8) J 1J 2 J 1

~(2) According to Eqs. (6), (7) and (8), U32 is given by

2 ~ (2) D 1 D 2 U 32  N 2 2 (' 1  k1v)  *1 (9) 2(J 12 J 2 )(' 3  k 3v  iJ 31 )  (' 1  k1v  iJ 21 ) u 2 (' 2  k 2 v  iJ 32 )(' 3  k 3v  iJ 31 )  D 1

2 In Eqs.(9), we can see two physical phenomena: pump light power broaden represented by *1 and line shift

2 effect represented by D1 in denominator. If we neglect these effects, Eqs.(9) can be simplified as

2 ~ (2) 2/D1 D 2J 12 1 U32 { J J [('  k v) 2  J 2 ]('  k v  iJ ) 1 2 1 1 12 2 2 32 (10) J 1  ( 2 ) } 2J 12 ('1  k1v  iJ 21 )(' 2  k2v  iJ 32 )(' 3  k3v  iJ 31 ) Then we can obtain polarizability K  520232 z,t N PU e  c.c (11) For probe light, polarizability can also be solved by K K 5 z,t HF02 E z,t (12)

Proc. of SPIE Vol. 5642 393 Comparing Eqs. (11) with Eqs. (12), polarizability F  is gained 2N P 02 exp i k z t FU ()2 32¬¼ªº: 2 2 H 0 E (13) 2N02P ()2 U32 H 0 E For gaseous medium, we should take account of velocity distribution of atoms. Assuming medium at thermal balance, velocity of the atom is one dimension Maxwellian distribution, the number distribution of atoms is given by

N v 2 N(v) 0 exp( ) (14) S u u 2

where u is the maximum probability velocity given by u 2k BT M , M ,T , k B are the particle mass,

, and Boltzmann constant, respectively. In Eqs.(13), we can substitute N(v) for N 0 and integrate

over the velocity of the atom. Then we can use plasma dispersion function to deal with this integral, but it is relatively complicated. When Doppler-broadened width is much bigger than that of corresponding atom velocity distribution, the exponential term approximately closes to zero. So it can be taken out of sign of integration. Then

F can be written as

2PSD22N / ªº' 2 1 F 20 1uexp «»1 (15)  ukkikk=J JJJ2 ''  12¬¼«» ku1 21 12 132 221 System refractive indices and absorption coefficients are given by

nr 1 2S Re(F r ) (16) Z D 4S Im(F ) (17) r c r LIDOF and FADOF are both filtering by making polarization plane of probe light which is in incidence into vapor cell rotated. The similar transmission spectra equation, is given by

1 L : Tr {exp(D L)  exp(D L)  2exp[(D  D ) ]cos[ 2 (n  n )L]} (18) 4     2 C  

Using Eqs.(15),(16) and(17), we can obtain transmission spectra. 3. Calculated Results In order to indicate operation mechanism of LIDOF, we calculated the filter transmission, the rotation angle and the absorption coefficient at strong pump light power density, which are given in Fig.2, respectively.

394 Proc. of SPIE Vol. 5642 57% 0.6 T=473k, Tra L=0.075cm 0.4 pump intensity=800mW/cm2 0.2

Transmission 0.0 4 -0.2 -0.1 0.0 0.1 0.2 2 0 -2 -4

Rotation angel( rad) -0.2 -0.1 0.0 0.1 0.2 14 12 10 8 6 4 2 0 Absorption coefficient Absorption -0.2 -0.1 0.0 0.1 0.2 difference( GHz)

Fig.2 The calculated line-wing transmission spectra, rotation angel and absorption coefficient of K LIDOF at 532nm For strong pump light power density, a great deal population of first excited-state results in increasing of the number of the atoms which participate in filtering and enhancing of atomic absorption at the center frequency, therefore transmission is very low at the center frequency. Contrarily, because the frequency of incident light is off from the center frequency of the atom, atomic absorption is small at Tra in Fig2. The transmission peak appears because the corresponding rotation angle 1.3 is close to ʌ/2 and the absorption is very weak. 4. Conclusions In this paper, the model and transmission mechanism of the K 532nm LIDOF system which match the double frequency output of the Nd:YAG laser is analyzed. It indicates that LIDOF is an optical filter with high transmission and ultra-narrow linewidth when it operates at excited-state. Investigation of the LIDOF is in significance for actual laser application. 5. Acknowledgements This work was financially supported by National Natural Science Foundation of China, grant no. 60272075, and done in National Key Laboratory of Tunable Laser Technology. References 1 Zhang Y D, Jia X L, Ma Z G and Wang Q. Optical Filtering Characteristic of Potassium Faraday Optical Filter IEEE Journal of Quantuim Electronics 2001,37(3):372-37 2 Zhang Y D, Jia X L, Ma Z G and Wang Q. Potassium Faraday Optical Filter in Line-center Operation Optics Communications 2001,194:147-150 3 Zhang Y D, Jia X L, Bi Y, Ma Z G and Wang Q. Filter performance of a Cesium Faraday Optical Filter at 852nm Chin.Phys.Lett 2002, 19(6):807-809 4 Wang Qi, Jia Xiaoling, Zhang Yundong, Ma Zuguang, The study of Tunable Ultra-narrow Band Optical Filter of K Atoms at 532nm, Acta Physica, 2003ˈ52(5):1151-1156, in Chinese 5 S.K.Gayen, R.I.Billmers, V.M.Contarino, M.F.Squicciarini, W.J.Scharpf, Guangning Yang, P.R.Herczfeld, D.M.Allocca.

Proc. of SPIE Vol. 5642 395 Induced—dichroism—excited atomic line filter .Opt.lett, 1995,20(12):1427-1429 6 Th.Hänsch et al.Theory of a Three-Level Gas Laser Amplifier [J].Z. Physik 1970,236,213 7 Richard L.Panock and R.J.Temkin Interaction of Two Laser Fields with a Three-level Molecular System[J]IEEE .Journal of Quantum Electronics 1977,QE13(6),425 8 Richard G.Brewer and E.L.Hahn Coherent two –photon processes:Transient and steady-state cases[J]physical review A 1975,11(5),1641

396 Proc. of SPIE Vol. 5642