Question

1. ) An Erbium Doped Fibre Amplifier (EDFA) consists of glass rod of length L = 7 m and effective cross-sectional area A = 12.5 um? (so that the optical volume is AxL). The concentration of Er3+ ions in the glass is N = 109 cm? We can represent Er3+ as a 3-level system as shown in the diagram below. The pump wavelength is 1480 nm and the emission wavelength is 1550 nm. Let Ni and N2 be the concentrations of Erat energy levels E, and E2, respectively, then we have N = Ni + N2 (since the population at Ez is assumed to quickly decay to E2). E3 - Pump beam at 1480 nm E2 Non-radiative decay N2 Stimulated emission at 1550 nm Ni Energy diagram of Er3+ ions in glass rod (a) Given that the absorption and emission cross-sections of Er3 at 1550 nm wavelength are ab = 3x 10-21 cm' and Gem = 4x10-21 cm, determine the population N2 required to reach transparency (gnet = 0). (b) Suppose that initially all Er* atoms are at energy Ey. Determine the total pump energy required to achieve transparency (i.e., energy required to achieve the threshold value of N2 in (a)). (c) The Er3+ population at level E2 decays to E1 due to spontaneous emission as N2(t) = N20e-t/tsp, where tsp = 1/A21 is the decay time (or lifetime) of population N2 and A21 = 100 s! (Einstein's coefficient for spontaneous emission). Considering that the pump energy in (b) must be applied to the EDFA over a duration equal to Tsp in order to maintain the threshold population N2, determine the optical pump power needed to achieve transparency in the EDFA. (d) Determine the optical pump power needed to achieve complete population inversion (i.e., N2 = N and Ni = 0). What percentage of the pump power is lost as heat? (e) What is the maximum possible gain coefficient (gnet) that can be achieved in the EDFA? What is the maximum optical amplification in dB provided by the EDFA?

Answer 1

Length 2= 7m cross-section area um? A = 12.5 Concentration of E3+ = 10 cm3 =N. pumping wavelength = 1480 nm emission wavelength = 1550 nm Here N= N, TN2 (9) absorption cross section oa== 3x10 cm? emission cross-section de = 4 x 102 cm? at wavelength 1550 nm we know the gain 9 = N e - Noa and gto given in the condition In the begining the value of Ni= N but we can write NAN₂ = N (Griven) N, = N-N2 so ga Nave - Noa (gzo) or o= N₂ ve- (N-N2) oa o= N ve + Nava - Nora

N₂ = a N etoa 19 No - 3x10-21 10 3x10^2 + 4x1021 3 x 1027 x 10" 78107 N = 4.286x1018 cin3 (b) It is assumed that in the begining the all atoms are in State E, he. N = N The energy required to transfer L atom from State E, to Er is given by a he pumping wavelength = 6.67X10²34 x 3x108 1480 X1099 = 0.01352 X10-17 So the total energy reanined = 1.352 x 10 19 Joule to raise Ma no. of atoms from E, to Ez = 4.286x1018 x 1.352 81019 = 0.5794 Joule mu bu eo

-Hesp ( N (4) = N(0) é resp e ct] Csp = 1 = 1 (A2i = 100 seet] A/ESP N210) é esp/Esp Tsp = 0.01 sec 6 Nale) = N₂(o) é res But the time duration is given as eta Esp so N₂ (8) = N₂10) e. Na (4) = N₂0/e » N₂ (0) = N2 (4) xe So the no. of decay in esp time = N2(0) - NZ (0) : = N2 (0) - Nz (D) Nz (0) - N2(0)%e = N2(0) (1 - 1) = Nzco). el 1-ė) = N2 (2) (e-1) To het No be the no. of atoms jump from E to Ez states then N2(t) = No - N₂ (4) (end)

M(I) = No No = e N (4) e x 4.286 x 1018 1.1650 x 1019 cm3 - The in raising No atoms energy = required No x ha = 1.1650 8109 x 6.67810 34 x² 3x108 1480 x 109 z 1.57511149 Joule cm3 1157511ugx 10 Joule / m3

M(I) = No No = e N (4) e x 4.286 x 1018 1.1650 x 1019 cm3 - The in raising No atoms energy = required No x ha = 1.1650 8109 x 6.67810 34 x² 3x108 1480 x 109 z 1.57511149 Joule cm3 1157511ugx 10 Joule / m3

Tae e enorgy required by whole laser materical - 1.5751149x108 AL = 1.57511ugx 12.58106 x 78106 = 137.8225 Joule Time duration of pulse (tsiz o.ol see so power = energy time = 13782,225 Joule /sec

Tage 10. d) Ý N = N then No = N₂ e = No e energy required - Nex he & LA 18x 109 xe x 6.67810 34 x 3x108 X 7 X 12.5 x 106 1480 x16 g - 3215791 Joule lower = 3215791 Watt