Research Background
The Semiconductor Optical Amplifier known as SOA is an active device. The current pumping source provides the device with abundant electronic carriers, which easily bind with holes and release energy as photons. For stimulated emission, it requires these generated photons to carry the same information with input light. The more currents come into SOA, the more photons will be stimulated by the input light and thus SOA can amplify it. XGM effect is short for cross-gain modulation effect. It occurs when two optical signals input a SOA and they compete carriers with each other. As the amount of carriers is stable, the increasing magnitude of the input information-carried signal will cause the decreasing magnitude of the other detecting signal. The name “cross-gain modulation” vividly describes this phenomenon. Differential equations can describe this effect in a mathematical way.
“N” represents carrier concentration, “A” represents the magnitude of the electric field. “ω” represents a symbol in a set in which two symbols of input signals represent the information-carried light and the detecting light respectively. “r”, “f”, “h” are functions to describe the XGM effect that is known.
Motivations of Simulating The XGM Effect
To numerically simulate XGM effect, there has been some work proposing the result of the simulation for two beams of input optical light (information-carried light and the detecting light) transmitting towards the same direction. However, some defects remain unsolved. For example, transmitting towards the same direction for information-carried light and the detecting light is not practical as the corresponding output light cannot be separated especially when the input wavelengths are the same. Instead, transmitting towards opposite directions is the case. However, it is more difficult to simulate the opposite-direction case, which is exactly what I focus on in this work.
Backward Propagation Difference Equations
For backward propagation, the difference equations cannot directly substitute the differential equations shown above without making any other changes. The reason is that the next moment of carrier concentration will be affected by current light intensity coming from both the information-carried light and the detecting light. Therefore, we introduce another coordinate for the usage of transferring differential equations into difference equations, and find the relations between two coordinate (black and red). The prime notation is used to distinguish parameters in the red coordinate and in the black coordinate.
Because A'(z’+Δz’,t)≠A(z+Δz,t), the equations are different for backward transmission from those that describe input signals transmitting towards the same direction. According to the formulas above, the correct backward propagation difference equations should be:
Analyses of Iterative Process in Simulation
To start an iterative process, the first step is to determine initial conditions. In this case A1(1,t), A2(end,t), N(z,1) are known. A1(1,t) means an optical waveform of left input port; A2(end,t) means the other optical waveform of right input port; N(z,1) means at the first moment the carrier concentration is a constant we know. Then all we have to do is to finish (z,t) matrices of A1, A2 and N. Pictures below show initial conditions.
The second step is to determine the iterative process. According to the formulas above, the cycle structure is established from the time dimension (cycle by column in a matrix). For a certain moment “t”, we can calculate A1(z+Δz) from A1(z,t) and N(z,t) and A2(z-Δz) from A2(z,t) and N(z,t). Calculations for a column are thus finished. The diagrammatic sketch shows below.
For continuously calculating A(z,t+Δt) in the next column, N(z,t+Δt) should be calculated before which should be referred from N(z,t) and A(z,t). The diagrammatic sketch shows below.
Then cycling the above steps can we get the complete (z,t) matrices for A1, A2 and N. A1(end,t) represents the output signal which inputs from the left port; A2(1,t) represents the other output signal which inputs from the right port. The results show below.
To prove the results are correct for any other input waveforms, the following results respectively show under the input of the sine function and the square wave function (still keep the detecting signal as a constant).
Appendix
Conditions of simulation:
