Jounal of Physics: Confeence Seies PAPER OPEN ACCESS Modeling and Calculation of Optical Amplification in One Dimensional Case of Lase Medium Using Finite Diffeence Time Domain Method To cite this aticle: Okky Faja Ti Mayana and Rahmat Hidayat 06 J. Phys.: Conf. Se. 79 000 Related content - Tunable Infaed Luminescence and Optical Amplification in PbS Doped Glasses Huang Wei, Chi Ying-Zhi, Wang Xi et al. - Optical Amplification in Oganic Dyedoped Polymeic Channel Waveguide unde CW Optical Pumping Kenichi Yamashita, Kengo Hase, Hisao Yanagi et al. - Enhanced ulta-boadband optical amplification in Yb-Bi codoped magnesium gemanate glasses Jian Ruan, E Wu, Heping Zeng et al. View the aticle online fo updates and enhancements. This content was downloaded fom IP addess 8.5..8 on 0/05/08 at :6
Modeling and Calculation of Optical Amplification in One Dimensional Case of Lase Medium Using Finite Diffeence Time Domain Method Okky Faja Ti Mayana, Rahmat Hidayat Physics of Magnetism and Photonics Reseach Division, Faculty of Mathematics and Natual Sciences, Institut Teknologi Bandung, Jl. Ganesha 0, 0, West Java, Indonesia E-mail: ahmat@fi.itb.ac.id Abstact. Finite Diffeence Time Domain (FDTD) method has been much employed fo studying light popagation in vaious stuctues, fom simple one-dimensional stuctues up to thee-dimensional complex stuctues. One of challenging poblems is to implement this method fo the case of light popagation in amplifying medium o stuctues, such as optical amplifie and lases. The implementation is hindeed by the fact that the dielectic constant becomes a complex numbe when optical gain paamete is involved in the calculation. In geneal, complex dielectic constant is elated to complex susceptibility, in which the imaginay pat is elated to optical gain. Hee, we then modify the fomulation fo updating electic field in the calculation algoithm. Using this appoach, we then finally can calculate light amplification in lase active medium of Nd + ion doped glass. The calculation esult shows an ageement with the esult fom the calculation using diffeential equation fo intensity. Although this method is moe time consuming, the method seem pomising fo optical complex mico- and nano-stuctues, such quantum dot lases, mico-ing lases, etc.. Intoduction Finite Diffeence Time Domain (FDTD) is a numeical method fo calculating the popagation of electomagnetic wave. This method has been much employed fo vaious studies because of its easiness to be applied fo vaious complicated stuctues []. In most cases, this method is applied fo tanspaent dielectic mateials with optically lossless chaacteistics. Howeve, in paticula cases, the medium is optically absobing medium o optically amplifying medium, such as in lases and optical amplifies. In such cases, the medium pemittivity is no longe eal numbe, but it is a complex numbe. Diect implementation of complex pemittivity in the FDTD calculations leads to numeical eos. Theefoe, many appoaches to solve this poblem have been poposed [, ]. In a lase medium, the medium is iadiated with a light pumping to poduce a population invesion, in which moe valence electons ae in the excited state athe than in the gound state. These electons will then poceed in a stimulated emission emitting coheent photon. This event can be tiggeed by the pesence of the signal light enteing fom the outside medium. The pocess poducing lase emission is called as photo-pumped lase. The medium fo this pupose is commonly ae-eath based lase medium, such as Nd + in cystals o glasses. Such lase mediums ae often used fo diode pumped solid state lase systems, optical amplifies etc []. Content fom this wok may be used unde the tems of the Ceative Commons Attibution.0 licence. Any futhe distibution of this wok must maintain attibution to the autho(s) and the title of the wok, jounal citation and DOI. Published unde licence by Ltd
. Optical amplification Fo the -level system, which is commonly used fo ae-eath based lase medium, the schematic diagam of the electonic levels involved in the pocess is shown in Figue. The ates of electon population change at those electonic levels ae dependent on the ates of upwad tansition (excitation) and downwad tansition (elaxation), including stimulated tansition. The optical tansition stimulated optical tansition W ij is dependent on the local light intensity (I), which is given Iij by Wij, whee ij indicates the tansition fom the i-th level to j-th level in the above diagam. It h ij is supposed that the non-adiative tansition (γ) is much faste in compaison to that of the adiative tansitions. In such case, theefoe, only the ates of electon population changes at the st level and d level ae needed to be consideed. The ates unde steady state condition ae given by dn WN W N dt dn WN W N dt NN Ntotal () The spontaneous emission ate, hence A, can be consideed much smalle unde lasing condition. W W, A W, A (a) Figue. (a) The geneal -levels system and (b) the -levels model fo Nd + ions. In ode to find the distibution of the amplified signal intensity o lasing intensity inside the medium, the simplest way is by solving the fist ode of diffeential equations fo both the excitation (pumping) beam and the amplified (signal) beam. When the light souce is a lase, fo a simplification, we may conside that both the pumping beam and the amplified signal beam ae popagating coaxially. In such case, we then have a set of diffeential equations of one dimensional case di p absn I p pi p dz dis em N I s si s dz () whee I p and I s ae the pumping beam and signal intensity, espectively. We may see that to solve this set of equation, we must also solve the set of equation (). It can be solved easily by numeical computation. The optical amplification o gain can then be found fom (b)
0 Is( L) gtot db/ cm log Lcm ( ) I s (0) (). Finite Diffeence Time Domain method Finite Diffeence Time Domain (FDTD) is a method to calculate electomagnetic popagation by solving numeically the diffeential fom of Maxwell s equation. This method fistly intoduced by Yee, who popose an algoithm of numeical calculation fo those diffeential foms of Maxwell s equation. If we see the Ampee and Faaday-Lenz laws in the Maxwell s equation, namely E H 0 t H E t () We may notice that those equations connect the electic field and magnetic field in diffeent domain. On the left side is in the spatial domain, while on the ight side is the time domain. Theefoe, by foming a discetization in both spatial and time domain, as illustated in Figue, the above equations can be witten as n/ n/ t n n Ex ( k) Ex ( k) Hy( k/) Hy( k/) 0 z n n t n/ n/ Hy ( k /) Hy( k/) Ez ( k) Ez ( k ) 0 z. (5) E x time: (n /)t k- k- k k+ H y time: (n)t k-/ k-/ k+/ k+/ E x time: (n+/)t k- k- k k+ Figue. Spatial and time discetization to be implemented in the FDTD method. We may notice hee that those equations contains mateial paametes only in the electic pemittivity () and magnetic pemeability (). Those equations ae commonly be implemented fo dielectic constants with eal numbe, that is fo simple tanspaent dielectic mateials.. A simple scheme fo employing FDTD in an amplifying medium When a medium exhibit optically absobing o amplifying chaacteistics, the electic pemittivity becomes a complex numbe, which can witten as i (6) The eal pat is elated to the efactive index of the mateial, while the imaginay pat is elated to the optical loss (absoption) o optical amplification gain (stimulated emission). The positive value is fo absoption, while the negative value fo amplification. This pemittivity is elated to the complex susceptibility by the following elationship
i ( ) It should be noted that equation (5) will poduce a numeical eo if the electic pemittivity is a complex numbe. Theefoe, we need to modify equation (5) so that it can accommodate the complex numbe of pemittivity dielectic. If we see equation (5) and equation (6), we may also notice that the pemittivity is fequency dependence. In such chaacteistics, the time esponse will be also vaies depend on the fequency so that the implementation of FDTD is not staightfowad. In ode to simplify the calculation, we made some assumptions. The fist assumption is that the whole pocess is unde steady conditions. The second one is that the popagating wave is a plane wave with single wavelength. Those assumptions, howeve, ae consistent with a paticula lasing mode, that is, the cw opeating mode with a vey shap single wavelength emission. With those assumptions, the Ampee law is the can be witten as H 0( ) E t E E E E 0 0 0 0( i) t t t t (8) whee χ is the susceptibility, which is elated to complex pemittivity by the elationship: ( i). (9) Fo the -level model, the complex susceptibility can be witten as nc abs N abs( ) N em( ), fo upwad tansition (0.a) and nc em N em( ) N abs( ), fo downwad tansition (0.b) Substituting eq. (0) into eq. (8) esults in E nc E 0 i0 N H t t () whee N N N is fo net upwad tansition (), while N N N is fo net downwad tansition. Theefoe, the FDTD fomula fo the electic field can be witten as n / n t ( ) / n n t c n E ( ) ( /) ( /) / x k Ex k Hy k Hy k N E () x z 0 By pefoming calculation of the electic field and magnetic field fom the one edge of the medium to the othe one edge position, we can get the intensities at those edges and optical gain fom equation (). 5. Calculation esults and Discussions In ode to see the esults of the above appoximation methods, the calculations have been pefomed fo a case of Nd + ions doped in glass lase medium. Figue shows the optical amplification by the fist method fo vaious concentations of Nd + ions. The lifetime paamete used in these calculations is 00 s with the total length is cm. Othe paametes used in the calculations ae listed in Table.. At the ion concentation of 50 0 cm, the optical amplification inceases quickly and satuates at about 0. cm fom the oigin edge. As the concentation become smalle, the optical amplification aises slowly. Figue (b) shows the optical gain at vaious pumping intensities. At pumping intensities of 800 mw, the optical gain up to about. db/cm can be obtained. (7)
5 5 Gain (db/cm) Gain (db/cm) 0 0.0 0.5.0.5.0.5.0 Medium length (cm) 0 0.0 0.5.0.5.0.5.0 Medium length (cm) (a) Figue. (a) Optical gain at diffeent Nd + ion concentations and (b) at vaious light pumping intensities. (b) Table. The paametes used in the calculations. abs ( cm ) emi cm 0 0 ( ) Signal ( mw ) Lcm ( ) ( nm ) ( nm ) pump signal 5 0 0 0.00 0 808 06 Figue shows the calculation esults by FDTD method, indicated by the squae symbol ( ). The thin solid line shows the calculation esult of the fist method as compaison. Because the FDTD method equied much longe time calculation, the calculations in this wok was limited fo medium length up to 0. mm only. Howeve, as evident in Figue, the calculation esults show a high ageement between this FDTD method and the fist method. Gain (db/cm) Medium Length (cm) Figue. The optical calculation esults by FDTD method (squae symbol) and the conventional method by solving numeically eq.() (dash line). 5
6. Conclusions We have pefomed optical gain calculations by FDTD method with a modification in its electic field fomula to accommodate the complex pemittivity of the medium. The calculation esults wee compaed with the calculation esults fom the common method based on the diffeential equation fo intensities. The simulation esults shows about. db/cm can be obtained in lasing medium containing Nd + ions of about 50 0 cm, unde light pumping intensities of 800 mw. The ageement with the fist method shows the validity of appoximation applied fo simplifying the FDTD calculations. Although this method equies a much longe time, this method may offe much flexibility and easiness fo calculating complicated mico- and nano-stuctues, such quantum dot lases, mico-ing lases, etc. Acknowledgment The authos deeply acknowledge the Indonesia Asahi Glass Foundation (contact No. 76d/I.C0/PL/0) fo the suppot to this eseach wok. Refeences [] Sullivan DM 0 Electomagnetic simulation using the FDTD method (John Wiley & Sons) [] Yang J, Diemee M B J, Sengo G, Pollnau M, and Diessen A 00 IEEE Jounal of Quantum Electonics 6 p.0 [] Fafin A, Cadin J, Dufou C and Goubilleau 0 Optics Expess p.96 [] Sennaoglu A 00 Solid-State Lases and Applications (CRC Pess) 6