Modeling & Simulation of Erbium Doped Fiber Amplifier (EDFA) for Amplification of Optical Chaos

Abstract

In this paper we demonstrate the amplification of chaos by means of EDFA. Chaos is generated by using the Optical Fiber Ring Resonator. By optimizing the different parameters of EDFA we obtained the accurate amplified chaos at the output of EDFA. Different parameters of EDFA like pump power, signal power, amplifier length and erbium ions concentration affect the performance of EDFA. Therefore the right choice of the value of these parameters gives unchanged and appropriate amplified chaos at the output of EDFA.

1 Introduction:

The successful demonstration of chaotic communication in electronic circuits [1], paved the way to the exploration of optical chaos as a candidate for secure optical communication. Optical chaos gained much attention during the last decade and various practical schemes based on gas lasers, semiconductor lasers and erbium doped fiber lasers have been demonstrated for secure communication [2-5]. Message retrieval ranging from few Kb/s to few Gb/s with acceptable signal to noise ratio at single wavelength have been achieved [6].

The chaos is either being generated through Fiber ring lasers or semiconductor lasers and maximum transmission distance is limited by the signal strength. The transmission link can be extended to any distance by appropriate choice of the amplifiers which can be placed at the transmitter, receiver and along the transmission fiber.  Erbium doped fiber amplifiers have gained the attention of the researchers because of their simple structure, high gain, polarization independence and low noise.

2 EDFA Theory

2.1 Energy Levels

An Erbium Doped Fiber Amplifier consists of a nominally 10- to -30m length of optical fiber that has been highly doped (e.g., 1000 parts per million weight) with rare earth element Erbium. Erbium added to the glass in the form of an ion (Er3+).The host fiber material can be standard silica, a fluoride-based glass, or a telluride glass [8].

This ‘doped fiber’ is used as an amplifying medium. It gives an amplifying output around 1550nm. When the erbium ion is introduced into the host medium the energy levels are modified by local electric fields through Stark-splitting. These levels are in thermal equilibrium due to nonradiative transitions between these levels. The amplifier is assumed to have homogenous broadening but if the local electric field is different at various sites along or across the fiber due to impurities, or other glass structural disorders, then inhomogeneous broadening occurs resulting in different electronic transition at respective sites. The incorporation of network modifier such as Aluminum (Al) to enhance the solubility of Er⁺³ ions in the glass structure changes each energy level’s Stark-splitting and increase the in homogeneity of the medium. The energy transition typically associated with Er⁺³ in silicate glass are the ⁴I_11/2, ⁴I1_3/2 and ⁴I_15/2 states, as illustrated in the figure 1[7].

                                             Fig(1) Energy Diagram of EDFA

2.2 Amplification Mechanism in EDFA

Generally EDFA is pumped with 980nm or 1480nm.When incident photon energy is absorbed in doped fiber, electrons in the medium are made to move to higher energy levels[7].

The whole operation of EDFA when you pump with light at 980nm is described below. The photon at 980 nm interacts with electrons in the ground state. The electrons absorb the energy and jump to upper energy level or I_13/2 band. This band is unstable and electron decay into the band below (the I_11/2 band).This is called nonradiative transition.Electrons in the metastable state are able to stay for a long time. If an electron decay back to ground level in the absence of signal photon, this decaying phenomena is known as spontaneous emission which is a source of noise in EDFA.When a photon (signal) interacts with the excited electron in metastable state, the electron drops to the ground state which leads to stimulated emission .The most important point is that erbium ions give up their energy in from of additional photons which are exactly in same phase and directions of signal to be amplified. This phenomenon of stimulated emission is depicted in figure 2[9].

Fig(2) Atomic stimulated emission 

And when excited electrons drop into ground state without triggering the signal photon, undergoes a spontaneous emission. Thus a photon emitted in random phase and direction.A very small proportion of emitted photons will move along the length of fiber, and these photons are in indistinguishable from signal and being amplified [10].For meaningful amplification there must be more erbium ions in excited state than in the ground state. This condition is known as population inversion [9].

2.3 Mathematical Model of EDFA

To study the behavior of EDFA, a rate-equation is used to relate the various processes such as absorption, spontaneous and stimulated emissions and pumping etc [11].

Various EDFA models are available. Two-level atomic model is especially accurate for 1480 nm pump wavelength. For the two-level system the rate of change in populations in two –levels can be written as[9].

s_s^{(a )} -   s_s^{(f )}  ) f_s  -(   s_p^(e)-   s_p^(a) ) f_p                                                                                (2.1)

= s_s^(e) –N₁ s_s^(a))f_s –(N1  s_p^(a) –N₂s_p^(e))f_p                                                                                                   (2.2)

Where  s_s^{(a )}_, s_s^{(e )}  _, s_p^{(a )}and  s_p^{(e )}  represent the signal and pump absorption and emission cross section. f_s =I _s/hv_s and f_p=I _p/hv_p are signal and pump flux intensities (in number of photons per unit time per unit area).  is the  transition probability from level 2 to level 1 which can also be given as . Here t is the life time of level 2.Since total population is given as N=  and differentiating we get

                                                                                                                           (2.3)

This equation shows that only one of the equation from (1) and (2) is an independent equation. And we can calculate  in terms of signal and pump intensities.  =  .By substituting these relation in equation (1) and eliminating , the population density ,as a function of position ‘z’ along the fiber is, given by,

=[ts_s^{(a )} I _s(z)/hv_s +ts_p^{(a )} I _p(z)/hv_p /t(s_s^{(a )}₊ s_s^{(e )}₎ I _s(z)/hv_s +t(s_p^{(a )}₊ +s_p^{(e )} ) I _p(z)/hv_p+1]N            (2.4) 

Using the relation I (z)=P(z) /pR²=P(z) /A, we can express the effective average intensity throughout the erbium doped region, where A=pR² is an effective area of erbium ion distribution. Therefore we can rewrite the equation (4) in terms of corresponding signal and pump powers, P_s (z), P_p (z).

=[ts_s^{(a )} P_s/Ahv_s +ts_p^{(a )} P_p A/hv_{p /}t(s_s^{(a )}₊ s_s^{(e )}  )  P_s /Ahv_s +t(s_p^{(a )}₊ +s_p^{(e )} ) ⁾  P_p / Ahv_p+1]N (2.5)

The signal and pump components experience gain due to emission and attenuation. Due to absorption as they propagate along the fiber, the possible intrinsic background loss in the fiber in included by means of parameters α_p  and α_s for pump and signal components.

=( s_p^{(e )}- s_p^(a)) P_p- α_pP_p                                                                                                                                                (2.6)

=( s_s^(e) - s_s^{(a )}) - α_s P_s                                                                                                                                                   (2.7)

The gain coefficient g₀ can be calculated while using the above model.

g₀₌G(s_s^{(e )}+ s_s^{(a )})N₂-Gs_s^{(a )}N                                                                                                         (2.8)

 3 Chaotic optical Communication

What exactly is chaos? .The systems that the theory describes are apparently disordered, but chaos theory is really about finding the underlying order in apparently random data. Recently, chaos has been found in various fields in optics.

3.1 Mathematical Model for Chaos Generation using Optical Fiber Ring Resonator

In an optical ring resonator which is one of key optical devices, chaotic feature has been found first by Ikeda et al [12]. From a view point of optical interference, an input field and circulated fields with a nonlinear phase shift superpose in the resonator, which yields a complicated dynamics in an interference output. The OFRR model consists of an optical fiber ring and a fiber coupler as shown in Fig. 3.

Fig (3)  Optical fiber ring resonator

An output optical field in OFRR can be given by the superposition among the input light transmitted through the coupler and multiple circulated light components in the resonator. Then, the output electric field of OFRR is written as [12]

E_out=iκ (1-ρ)E_in(t)+Σ_n(i( ^n-1 (1-κ )(1- ρ)^(n+1)/2 *E_in (t-nτ)*exp(-n αL/2)*exp[-i(nΦ +Σⁿ_l=1 ∆   Φ(t-τ))]   (3.1)

where j is the coupling coefficient of the fiber coupler in the resonator, ρ is the loss of the fiber coupler, α is the fiber loss in the ring resonator, L is the length of the fiber ring resonator, Φ  is the linear phase shift in the resonator, and ∆Φ  is the nonlinear phase shift in the resonator and is given by,   ∆ Φ=kLn_{‌‌ 2} ‌‌| E_t  (t)|² ,k the wave number of the input light in vacuum, n₂ the nonlinear refractive index, E_t(t) the circulated light in the resonator. Here, E_in, the input optical field, and is given by E₀exp {iθ (t)}, E₀ is amplitude of input field and θ (t) represent the random phase fluctuation which give rise to coherence reduction in input light.The final output power of ring resonator can be expressed as

P_out=     ,                                                                                                                     (3.2)

Where , represents the time averaging process. In the final output power the random phase factor leads to the coherence term that is given by

(exp {i[(θ(t)- θ(t-nτ)]})=exp(-2pvnτ),                                                                                       ( 3.3)

Where θ (t) is assumed to be a variable obeying the Gaussian random process. The coherence state of the input light is assumed to be sufficiently high so that the output power changes in its state from stable to periodic, and finally to chaotic as the input power Pin increases in OFRR[12].

3.2 Synchronization of Chaos at Transmitter and Receiver

Optical communications with chaotic carriers requires that the broad band chaotic emission of two spatially separated lasers, are synchronized to each other. Synchronization means that the irregular time evolution of the emitter laser, e.g. in the optical power, can be perfectly reproduced by the receiver laser. Once the two lasers have been synchronized, messages can be encrypted into the chaotic carrier.

The receiver system allows this encrypted message to be extracted. Decoding is based upon the non-linear phenomenon of chaos synchronization between the emitter and the receiver. By comparing the input (carrier + message) and the output (carrier only) of the receiver, the message can be extracted. Due to the high sensitivity of the synchronization process and the complexity of the carrier, effective decoding requires well-matched transmitters and receivers.

4 Amplification of Chaos

Chaos generated by the ring resonator is amplified by EDFA at the transmitting end. There are different parameters such as fiber length, pumping power and erbium ions concentration etc, which affects the performance of EDFA. The objective is to optimize these parameters, so that we can obtain amplified chaos without any change in their shape.

5 Results and Discussion

1: The effect of pump power on the amplification

• If pump power is 30mw, signal power is 0.5mw, fiber length is 12m and erbium ions concentration is 40e23 cm^-3 .Here fig(4)depicts that  the pulses represented by blue solid line are input chaos and the pulses which are presented by black dash line are amplified pulses. Fig (5) depicts the temporal evaluation of amplified pluses along the length of fiber that is 12 m long. Fig (6) shows that all chaos having low amplitude as well as high amplitude are amplified equally and their shape remain constant during amplification.

                          Fig (4) Amplification of chaos                                         Fig (5) Amplification of chaos in 3d domain                             

      Fig (6) Comparison b/w input& amplified chaos                     

• Here Pump power is 60mw and all other parameters are same.fig (7) shows the effect of pumping power on the amplifier gain.

         Fig(7)Amplification of chaos                                                    Fig(8) Amplification of chaos  in 3d domain                             

      Fig (9) Comparison b/w input& amplified chaos                     

• For 90mw other parameters are same

Now the pump power is 90mw and all other parameters are same.fig (10) shows an increase of chaos amplification. We can notice that amplification is high as pump power is increases .By increasing the pump power we get more excited erbium ions in meta-stable state, which are stimulated by incident signal photon .And as a result signal will amplify.

Fig(10) Amplification of chaos                                         Fig(11) Amplification of chaos  in 3d domain                               

    Fig (12) Comparison b/w input& amplified chaos                      

2: The effect of fiber length on the amplification

• Fiber length is 20 m and pump power is 90mw all other parameters are kept constant.

As we increase the amplifier length there is an increase in amplification of chaos.

            Fig (13) Amplification of chaos                                               Fig (14) Amplification of chaos in 3d domain                               

Fig (15) Comparison b/w input& amplified chaos                       

• Fiber length is 30 m and pump power is 90mw all other parameters are kept constant.

But as we increase the amplifier length as should increase the pump power because pump power decreases along the length of fiber ,and signal power is used to excite the erbium ions. As a result signal attenuates rather than to amplify. Hence, there should be an appropriate pump power for longer length of amplifier, so that we can achieve more amplified chaos.

              Fig(16)Amplification of chaos                                      Fig (17) Amplification of chaos in 3d domain                               

       Fig (18) Comparison b/w input& amplified chaos