Advances in Optical Amplifiers Part 10 pptx

Advances in Optical Amplifiers 256 We also put into evidence the existence of a strong interdependence between active medium parameters having important role in the designing of the erbium laser systems with given functional parameters. We demonstrated the stable, nonchaotic operation of the analyzed laser systems and the modulation performance of them using the communications theory methods. For the erbium doped fiber laser we explained the complex dynamics of this type of device by simulating the time dependence of the output power correlated with the corresponding changes in the populations of the implied levels. Another author's numerical simulations refers to nonlinear effects in optical fibers systems (Sanchez et al., 1995; Ninulescu & Sterian, 2005; Ninulescu et al., 2006;). Self - pulsing and chaotic dynamics are studied numerically in the rate equations approximation, based on the ion - pair formation phenomena (Sanchez et al., 1993; Sterian & Ninulescu, 2005; Press et al., 1992). The developed numerical models concerning the characterization and operation of the EDFA systems and also of the laser systems, both of the "crystal type" or "fiber type" realized in Er 3+ doped media and the obtained results are consistent with the existing data in the literature. That was possible due to the valences of the computer experiment method which permits a complex study taking into account parameters intercorrelations by simulating experimental conditions, as have been shown. The used fourth order Runge - Kutta method for the numerical simulation demonstrates the importance of the "computer experiments" in the designing, improving and optimization of these coherent optical systems for information processing and transmission (Stefanescu et al., 2000, 2002, 2005; Sterian, 2002; Sterian, 2007). Some new feature of the computer modeled systems and the existence of new situations have been put into evidence, for designers utility in different applications (Petrescu, 2007; Sterian, 2008). Our results are important also for the optimization of the functioning conditions of this kind of devices. 2. Fiber amplifier 2.1 Transport equations for signal and pumping Let us consider an optical fiber uniformly doped, the concentration of the erbium ions being 0 N . The pumping is done with a laser radiation having p λ wavelength and the pumping power p P , the absorption cross - section being a p σ . The population densities of the atoms on each of the three levels involved in laser process are: ( ) ( ) 12 ,, ,NtzNtz respectively ( ) 3 ,Ntz which verify the equations: ( ) 3 ,0Ntz ≅ (1) ( ) 1 ,Ntz+ ( ) 20 ,Ntz N= . (2) The necessary condition for radiation amplification in this kind of systems is as in the laser case the population inversion. In the next presentation we refer to the energy levels diagram presented in figure 1 where: a s σ is the absorption cross-section for the signal; e s σ is the stimulated emission cross-section Coherent Radiation Generation and Amplification in Erbium Doped Systems 257 corresponding to the signal; a p σ is the absorption cross-section for the pumping radiation and τ is the relaxation time by spontaneous emission. Fig. 1. The diagram of the energy levels involved in radiation amplification For this system of energy levels on can write three rate equations: one for the population of the E 2 level and two transport equations for the fluxes of the signal and pumping. These rate equations are respectively (Agrawal, 1995): () ( ) ( ) ()() ()() 1 12 2 2 ,, ,, ,, ,; a ae pp ss ss ps s NtzItz N tz I tz N tz I tz N Ntz th h h σ σσ νντν ⋅ ⋅⋅ ∂ =+−− ∂ (3) () () () () 1 1 ,,,,; a pppp I tz I tz N tz I tz ct z σ ∂∂ ⋅=−−⋅⋅ ∂∂ (4) () () ()() ()() 21 1 , , ,, ,,; ea ssssss I tz I tz N tz I tz N tz I tz ct z σσ ∂∂ ⋅=−+⋅⋅−⋅⋅ ∂∂ (5) where: ( ) , a pp p p Itz W h σ ν ⋅ = is the absorption rate for the pumping; ( ) , a ss a s s Itz W h σ ν ⋅ = - is the absorption rate for the signal; ( ) , e ss e s s Itz W h σ ν ⋅ = - is the stimulated emission rate; 1 τ - is the spontaneous emission rate; a p σ ( ) 1 ,Ntz⋅ - is the rate of pumping diminishing by absorption; ( ) 2 , e s Ntz σ ⋅ - rising rate of the signal by stimulated emission and ( ) 1 , a s Ntz σ ⋅ - is the rate of signal diminishing by absorption. (It admit that ae ss p WWW==). In the same time the initial condition are: ( ) ( ) ,0 pp It It= (6) Transfer of excitatio n Advances in Optical Amplifiers 258 ( ) ( ) ,0 ss It It= . (7) If the next conditions are fulfilled: () 2 ,0Ntz t ∂ = ∂ , (8) () () ,0 , 0 pp It Itz tt ∂∂ = = ∂∂ , (9) () () ,0 , 0, ss It Itz tt ∂∂ = = ∂∂ (10) one obtain the steady state equations: ( ) ( ) ()() ()() 1 12 2 ,, ,, ,, 0 a ae pp ss ss ps s NtzItz NtzItz NtzItz N hh h σ σσ νντν ⋅ ⋅⋅ + −− =, (11) () () () 1 ,,, a pp p Itz NtzItz z σ ∂ =− ⋅ ⋅ ∂ , (12) () ()() ()() 21 ,,,,, ea ss ss s I tz N tz I tz N tz I tz z σσ ∂ =⋅ ⋅ −⋅ ⋅ ∂ . (13) By eliminating of the populations ( ) 1 ,Ntz and ( ) 2 ,Ntz, it results the equivalent system of nonlinear coupled equations: 0 d d 1 a a pp ss pps a pp a ae pp ss ss p ss I I Ihh IN z I II hh h σ σ νν σ σ σσ ν ντν ⋅ ⋅ + =− ⋅ ⋅ ⋅ ⋅ ⋅ +++ , (14) 0 d 1 d 1 a a pp ss ea ps a sss ss aa ae spp ss ss ps s I I hh I IN z I II hh h σ σ νν σσ σ σσ σσ νντν ⎡ ⎤ ⋅ ⋅ ⎢ ⎥ + ⎢ ⎥ + = ⋅⋅ ⋅ − ⎢ ⎥ ⋅ ⋅⋅ ⎢ ⎥ +++ ⎢ ⎥ ⎣ ⎦ . (15) In the upper equations, there are involved the parameters: 34 6,626 10 Jsh − =⋅ - the Planck constant; 8 2,99 10 m/sc =⋅ - the light velocity in vacuum; 2 10 s τ − = - the relaxation time for spontaneous emission; 16 2 210 m a p σ − =⋅ -the absorption cross-section for pumping; 16 2 510 m a s σ − =⋅ -the absorption cross-section for signal; 15 2 710 m e s σ − =⋅ -the stimulated emission cross-section for signal; 9 980 10 m p λ − =⋅ - the pumping radiation wavelength; 9 1550 10 m s λ − =⋅ - the signal radiation wavelength; L - the amplifier length; 3 10 mz − Δ= - the quantization step in the long of the amplifier. We consider also the parameters: Coherent Radiation Generation and Amplification in Erbium Doped Systems 259 () 1 1 18 18 ; ; 4,947 10 ; 7,824 10 ps hc hc αβ α β λλ − − ⎛⎞ ⎛⎞ ⎜⎟ == =⋅=⋅ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ . 2.2 Numerical simulation Numerical modeling of the upper rate equations was realized using the MATHLAB programming medium. The base element of the program was the function ode 45, which realize the integration of the right side expressions of the nonlinear coupled equations using Runge - Kutta type methods, for calculation time reducing. The program was applied for many values of the amplifier length for each of them resulting different sets of results, for the photon fluxes, both for the signal and pumping as well as for the gain coefficients and signal to noise ratio. From the obtained results by numerical integration of the transport equations, it results that the intensity of the output signal rise with the amplifier length but the pumping diminish in the some time. The calculated gain coefficients of the amplifier have a similar variation as was expected. We observe also the rising of the signal to noise ratio, resulting an improving of the amplifier performances (Sterian, 2006). The obtained value of the gain coefficient for the signal, of the 40 dB is similar to published values (Agrawal, 1995) So that, the results can be very useful for designers, for example, to calculate the optimum length of the amplifier for maximum efficiency. 3. Laser system in erbium doped active media 3.1 The interaction phenomena and parameters We analyze the laser systems with Er 3+ doped active media by particularizing the models and the method of computer simulation for the case of the Er 3+ continuous wave laser which operate on the 3μm wavelength. This laser system is interesting both from theoretical and practical point of view because the radiation with 3μm wavelength is well absorbed in water. For this type of laser system don't yet completely are known the interaction mechanisms, in spite of many published works. Quantitative evaluations by numerical simulations are performed, refering to the representative experimental laser with Er 3+ :LiYF 4 , but we analyse also the codoping possibilities of the another host materials: Y 3 Al 5 O 12 (YAG), YAIO 3 , Y 3 Sc 2 Al 2 O 12 (YSGG) and BaY 2 F 8 . The energy level diagram for the Er 3+ :LiYF 4 system and the characteristics processes which interest us in that medium are presented in figure 2. The energy levels of the Er 3+ ion include: the ground state in a spectroscopic notation 4 15/2 I , the first six excited levels 4444 13/2 11/2 9/2 9/2 ,,,IIIF, the thermally coupled levels 42 9/2 11/2 SH+ and the level 4 7/2 F . The possible mechanisms for operation in continuous wave on 3μm of this type of amplifying media are (Pollnau et al., 1996): a. the depletion of the lower laser level by absorption in excited state (ESA) 42 13/2 11/2 IH→ for pumping wavelength of 795 nm; Advances in Optical Amplifiers 260 Fig. 2. The energy diagram of the Er 3+ ion and the characteristic transitions b. the distribution of levels excitation 4 3/2 S and 2 11/2 H between laser levels due to cross relaxation processes ( ) ( ) 42 4 44 9/2 11/2 15/2 9/2 19/2 ,,SHI II+→ and multiphoton relaxation 44 9/2 11/2 II→ ; c. the depletion of the lower laser level and enrichment of the upper laser level due to up- conversion processes ( ) ( ) 44 44 13/2 13/2 15/2 9/2 ,,HI II→ and multiphoton relaxation 44 9/2 11/2 II→ ; d. the relatively high lifetime for the upper laser level in combination with low branching ratio of the upper laser level to lower laser level. These mechanisms, separately considered can't explain satisfactory the complex behavior of the erbium doped system, as has been shown (Pollnau et al., 1996; Maciuc et al., 2001). That is way it is necessary to put into evidence the most important parameters of the system and to clarify the influence of these non-independent parameters on the amplification conditions as well as the determining the optional conditions of operation. The levels 4 11/2 H and 4 3/2 S being thermally coupled, will be treated as combined a level, having a Boltzmann type distribution of the populations. For numerical simulation the parameters of the Er 3+ :LiYF 4 were considered because that medium presents a high efficiency for 3μm continuous wave operation, if the pumping wavelength is 970 nm λ = on the upper laser level 4 9/2 I , or on the level 4 11/2 I in the case of the pumping wavelength 970nm λ = . The Active Medium Parameters. Corresponding to the energy levels diagram presented in figure 2, the lifetimes of the implied levels, for low excitations and dopant concentrations have the values: 1 10ms τ = ; 2 4,8ms τ = ; 3 6,6 s τ μ = ; 4 100 s τ μ = ; 5 400 s τ μ = and 6 20 s τ μ = . Just the variations of these intrinsic lifetimes due to ion-ion interactions or ESA will be considered in the rate equations. Coherent Radiation Generation and Amplification in Erbium Doped Systems 261 The radiative transitions on the levels 4 3/2 S and 2 11/2 H are calculated taking into account the Boltzmann contributions of these levels for 300K: 0,935 respectively 0,065 for each transition. The nonradiative transitions are described through the transition rates ,iNR A of the level i, calculated with formula: 1 1 , 0 i iNR i i j j AA τ =− − = =− ∑ , (16) where i j A are the radiative transition rates from level i to level j. In the same time, the branching rations i j β of the level i through the another lower levels are given by: i j β = ( ) 1 ,i j iNR i AA τ − + , for 1ij − = (17) respectively: i j β = 1 i j i A τ − , for 1ij − > . (18) The values of the branching ratios have been calculated (Desurvire, 1995; Pollnau et al., 1996). The considered ion-ion interaction processes are: ( ) ( ) ()() ()() 44 44 13/2 13/2 15/2 9/2 44 44 11/2 11/2 15/2 7/2 42 4 44 3/2 11/2 15/2 9/2 13/2 ,, ,, ,, ,, II II II IF SH I II ↔ ↔ ↔ (19) being characterized by the next values of the transition rates: 12331 1 2331 11 11 22 22 310 ms ; 1,810 ms ;WW WW − −− − −− ==⋅ ==⋅ 12331 50 50 210 ms ,WW − −− ==⋅ where the 50 W parameter take into account the indiscernible character of the corresponding relaxation processes. The Resonator Parameters. The resonator parameters used in the realized computer experiments are consistent with operational laser systems, as: the crystal length: l = 2 mm; the dopant concentration: 21 3 0 210 cmN − =⋅ ; the pumping wavelength: 795nm p λ = ; we consider for ground state absorption (GSA) 44 15/2 9/2 II→ the cross section 21 2 03 510 cm σ − =⋅ and for excited state absorption (ESA), 442 13/2 3/2 11/2 ISH→+ , the cross section 20 2 15 110 cm σ − =⋅ . (The ESA contribution of the level 4 11/2 I was neglected for that wavelength.) Another considered parameter values are presented in literature, being currently used by researchers. Advances in Optical Amplifiers 262 In the literature (Pollnau et al., 1996; Maciuc et al., 2001). we found also the values of the energy levels populations reported to the dopant concentration and the relative transition rates, for different wavelength used for pumping: 795nm λ = and 970nm λ = . 3.2 Computational model The presented model, include eight differential equations which describes the population densities of each Er 3+ ion energy levels presented in figure 2 and the photon laser densities inside the laser cavity. We take i N for 1,2, ,6i = to be the population density of the i level and 0 N the population density of the ground state, the photonic density being φ . That model consisting of eight equation system is suitable for crystal laser description (Pollnau et al., 1994). For the fiber laser, the model must be completed with a new field equation to describe the laser emission on 1,7 μm λ = between the fifth and the third excited levels. The rate equations corresponding to energy diagram with seventh levels, for Er 3+ systems are presented below: () 5 12 6 66622206 0 d ; d ii i N RN N W N NN t τ − = =−+− ∑ (20) () 4 11 53 5 5 565 55 5666 50 50 31 0 d ; d ii SE i N RN R N N N W NN NN R t τβτ −− → = =−−+ − −− ∑ (21) 36 6 11 4 444444 05 4 d d ii j ii i ij i N RN RN N N t τβτ −− == = =− −+ ∑∑ ∑ ; (22) () () 36 1 3 33333 04 6 1253 350503111103 4 d d ; ii j ij ii i SE i N RN RN N t NWNNNN WN NN R τ βτ − == −→ = =− −+ ++−+−+ ∑∑ ∑ (23) () 16 1 2 22222 03 6 12 222206 3 d d 2 ; ii j ij ii i SE i N RN RN N t NWNNNR τ βτ − == − = =− −+ +−−− ∑∑ ∑ (24) () () 6 1 1 01 0 1 1 1 1 2 6 12 150503111103 2 d d 2 ; j j ii i SE i N RN RN N t N W NN NN W N NN R τ βτ − = − = =− −+ ++−−−+ ∑ ∑ (25) Coherent Radiation Generation and Amplification in Erbium Doped Systems 263 () ()() 66 1 0 00 0 50 50 31 01 22 11 1 0 3 22 2 0 6 d d . jiii ji N RN N W NN NN t W N NN W N NN βτ − == = −+ − −+ +−+− ∑∑ (26) () () {} 1 21 21 2 2 d ln 1 1 2 d2 l SE r o p t o p t P lc NR T L l tl P l φ φ γβτ κ − ⎛⎞ ⎡⎤ =+−−−−+ ⎜⎟ ⎣⎦ ⎝⎠ ; (27) () () {} 53 53 153 53 53 5 5 d ln 1 1 2 . d2 l SE r o p t o p t P lc NR T L l tlP l φφ γβτ κ → → −→ ⎛⎞ ⎡⎤ =+−−−−+ ⎜⎟ ⎣⎦ ⎝⎠ (28) A similar models are given in the references ( Pollnau et al., 1996; Maciuc et al., 2001,a & b). In the field equations (27) and (28), the parameters ,, ,, . / roptl LTL l P P κ are considered the same for the two type of laser studied. In the equations system (20) ÷ (28) the parameters are: R is the pumping rate from lower levels to the higher ones; τ is the life-times for each corresponding level; W is associated with the transition rates of the ion-ion up-conversion and the corresponding inverse processes; - i j β are the branching ratios of the level i through the other possible levels j; SE R is the stimulated emission rate; l and o p t l are the crystal length and the resonator length; 21 γ is an additional factor for the spontaneous radiative transition fraction between the levels: 4 11/2 I and 4 13/2 I ; / l PP is the of spontaneous emission power emitted in laser mode; ,,, r TL c κ are the transmission of the output coupling mirror, the scattering losses and the diffraction - reabsorption losses respectively, c being the light speed in vacuum. The pumping rates depend on the corresponding cross-section and of the other parameters (Maciuc et al., 2001). The parameters for the lasing in an Er:LiYF 4 crystal system are considered the same and for the fiber laser. 3.3 Crystal laser simulation Laser Efficiency for Different Pumping Wavelength. In the simulation were used for pumping the radiations having λ = 795, 970 and 1570 nm, which are in resonance with the energy levels in diagram of Er 3+ ion presented in figure 2. The pumping radiation for 795nm λ = connect the ground state level 4 15 I with the third excited level 4 9/2 I and also the second level with the fifth one ( ) 44 2 13/2 3/2 11/2 ,IS I+ , processes. In the case of pumping radiation having 970 nm λ = the ground state absorption (GSA) corresponds to transition 44 15 11/2 II→ and excited state absorption (ESA) to transition 44 11/2 7/2 IF→ . Similarly the pumping for 1530nm λ = determine a single transition GSA that is 44 15/2 13/2 II→ . The dependence of the output power versus input power for different pumping wavelength (795 nm, 970 nm and 1530 nm) were plotted resulting the functioning thresholds and the slope efficiencies for each situation. For the crystal laser Er 3+ doped, the optimum efficiency results for the direct pumping on the upper laser level. Advances in Optical Amplifiers 264 The output power variation with the level lifetimes. The output power variation on the lifetimes for the upper levels having 456 ,, τ ττ was studied for an input pump power 5W p P = and 795nm p λ = . We found that radiative and nonradiative transitions from the fifth and the sixth levels, improve the population difference for the laser line and determine the raising of the output power of them, the variation of the fourth level lifetime, being without influence for the output power. The influence of the Er 3+ ion doped host material on the output power. A three dimensional study was done to investigate the influence in the laser output power due to parameters variations for the host material, using 795nm p λ = . The relative spontaneous transition rates were considered the same for all simulations. To determine the host material change influence on the laser output power the next variation scale of the lifetimes have been considered: () ( ) 12 115ms, 0,49,6ms, ττ =÷ = ÷ ( ) 3 0, 22 22 s, τ μ =÷ ( ) 4 3 300 s τ μ =÷ , ( ) 5 12 1200 s τ μ =÷ and ( ) 6 0,6 60 s τ μ =÷ . Similarly, the variations of the transition rates corresponding to up-conversion processes for different host materials are considered to span the intervals given bellow: ( ) 21 3 1 11 0,1 300 10 cm msW − − =÷⋅ , ( ) 21 3 1 22 1,8 180 10 cm msW − − =÷⋅ , ( ) 21 3 1 50 0,02 200 10 cm msW − − =÷⋅ . For the other parameters used in the numerical simulation the published data was the main source of reference. Fig. 3. Output laser power dependence on parameters 2 τ and 11 W for two values of 1 τ [...]... Amplification in Erbium Doped Systems Fig 9 Threshold pumping parameter vs ion pair percentage Fig 10 Laser intensity in the steady state versus the pumping strength Fig 11 (a) Calculated stability diagram for EDFL τ L = 10 −8 s ; (b) The influence of the photon lifetime on the margins of the stability domains 271 272 Advances in Optical Amplifiers Fig 12 Long term temporal evolution of laser intensity... 1Nanophotonics lab, Department of Physics, Indian Institute of Technology, Delhi, New Delhi- 1100 16, 2Laser Instrumentation Design Centre, Instruments Research & Development Establishment, Dehradun-248 008, India 1 Introduction Optical amplifiers are of potential use in wide variety of optoelectronic and optical communication applications, particularly for Wavelength Division Multiplexing (WDM) to increase the... the fibers in such conditions Above threshold there exists one steady-state intensity given by eq (5), the other two solutions being unphysical (negative) The numerical calculation of the intensity in a significant range of the pumping parameter (Fig .10) gives a laser intensity following a straight line dependence The influence of the ion pairs is again disadvantageous and manifests in reducing the slope... in Er3+ doped media and the obtained results are consistent with the existing data in 276 Advances in Optical Amplifiers the literature Our results put into evidence the existence of the new situations which are important for the optimization of the functioning conditions for this kind of devices That was due to the valences of the computer experiment method which make possible a complex study taking... possible a complex study taking into account parameters intercorrelations by simulating experimental conditions, as have been shown The erbium ion pairs in a laser fiber can explain the experimentally observed nonlinear dynamics of the system, apart from the intrinsic nonlinearity of a multiple-mode laser The existence of the erbium ion pairs introduces a supplementary nonlinearity with a saturable absorber... presented in Fig 11.b This proves that a cavity with low losses makes it possible to preserve the cw dynamics at larger doping levels Fig.12 clarifies the quantitative changes of the laser intensity inside the self-pulsing domain At a fixed value of the ion pair concentration, the increase in pumping gives rise to pulses of a higher repetition rate and close to the bifurcation point the intensity becomes sinusoidal... regarded as one of the most influencing parameter and many physical, linear and nonlinear optical properties of materials are strongly dependent on it Duffy, Dimitrov and Sakka correlated many independent linear optical entities to the oxide ion polarizability of single component oxides (Dimitrov & Sakka, 1996; Duffy, 1986) This polarizability approach, predominantly gives the insight into the strong relation... d2 )I 2 + Axyd− I 2 (2f) In the above, I 1,2 are the normalized intensities and d1,2 are the normalized population inversions of the two modes The supplementary parameter γ takes into account the anisotropy in pumping for the two modes System (2) is to be investigated for typical parameters: τ L = 200 ns, τ 2 = 10 ms, τ 22 = 2 µs, N 0 = 5 × 10 18 cm −3 , σ L = 1.6 × 10 10 cm 3 s −1 , y = 0.2 , β... pumping parameter for the laser action and two-mode states vs ion pair percentage 274 Advances in Optical Amplifiers Fig 14 Steady-state intensities of the two-mode laser versus the pumping parameter, x = 0.1 Fig 15 (a) Transitory regime to a two-mode stationary state x = 0.03 and r = 1.5 A thorough investigation of the laser dynamics in the domain where both modes are active seems to be a hard task Instead,... various atomic interactions between electrons and can be found by solving the time dependent Schrodinger equation The weak interaction of the 4f electrons with electrons from other ions (in any environment) allows a Hamiltonian to be written for an individual rare earth ion as (Wybourne, 1965) ˆ ˆ ˆ H = HFI + H CF (1) ˆ ˆ where HFI is defined to incorporate the isotropic parts of H (including the spherically . Advances in Optical Amplifiers 256 We also put into evidence the existence of a strong interdependence between active medium parameters having important role in the designing of the. effective cross section, 23 2 27 210 cm σ − =⋅ . Increasing the value of the second Advances in Optical Amplifiers 268 process(ESA) (Fig. 7) on obtain the same behavior of the output power. solutions being unphysical (negative). The numerical calculation of the intensity in a significant range of the pumping parameter (Fig .10) gives a laser intensity following a straight line dependence.