Wavelength Conversion Four Wave Mixing in Silicon Waveguide
Wavelength Conversion by Degenerate Four Wave Mixing in Silicon Waveguide Abstract – Four-wave mixing (FWM) is one of the interesting nonlinearities in optical systems.It is mainly used for wavelength conversion.To investigate the factors that affect the wavelength conversion efficiency, the evolution of Four-wave mixing (FWM) in silicon waveguide is modeled using matlab.
The method of modeling is described. The effects of input pump power and waveguide length on the conversion efficiency are investigated.
Results show that when propagating along a 0. 048m silicon waveguide, both the input pump power and stroke power decreases, while anti-stroke power increases first and then decreases along the waveguide. It is also shown that for a 0. 048 silicon waveguide, output anti-stroke power is the maximum when the input pump power is 3W. Also, when the input pump power is kept constant, there is a most effective waveguide length for wavelength conversion. Keywords -FWM; model; conversion efficiency; input pump power; waveguide length 1 Introduction
Four-wave mixing (FWM) is an inter modulation phenomenon in optical systems, whereby interaction between three waves (two pump waves and a signal wave) produce a fourth wave (idler wave) . This phenomenon can be used for all optical wavelength conversion (AOWC) and entangled photon generation [2, 3]. As extremely small core of si wires produce the nonlinear optical effect even under low optical power, Silicon is used as waveguide in our project for practical wavelength conversion by FWM process with longer waveguide lengths and smaller propagation loss.
Factors that affect optical wavelength conversion are being studied to enhance the conversion efficiency. It has therefore become important to study FWM in silicon waveguide theoretically to increase the conversion efficiency for further experiment. In our project, FWM matlab to study the factors that affect the conversion efficiency. This paper discusses the factors that affect FWM’s conversion efficiency in silicon waveguide. Theoretical treatment is presented in section 2, where FWM in silicon waveguide is described. The method to model FWM in silicon waveguide using matlab is described in section 3.
Results are shown in section 4. Results show that both the input pump power and the waveguide length play an important part in the FWM’s conversion efficiency. 2 THEORY The FWM process involves the interaction of four waves (two Pump waves, one signal and one idler wave) as they propagates along a medium. In our project, silicon waveguide is used as the medium. The schematic diagram of FWM in silicon waveguide is shown in figure 1. Here, E represents the electric field of the respective waves and normalized such that power P=|E|^2. Subscripts ‘p’, ‘s’ and ‘a’ represent pump, signal and idler respectively.
The superscript ‘f’ represents forward propagating waves. [pic] Figure 1 Schematic diagram of FWM in silicon waveguide . 3 METHODOLOGY The evolution of the three waves along the silicon waveguide can be modeled by the following differential equations . [pic][pic][pic][pic] where Aeff is the waveguide effective core area, ? is the wavelength, ? is the linear propagation loss and ? is the TPA coefficient, ? is the FCA cross section and ? eff is the effective carrier lifetime. h and c follow their usual physical meaning of Plank’s constant and free-space speed of light respectively. k denotes the linear phase mismatch and can be expressed as[pic]. ? is the nonlinear parameter assumed to be the same for three wavelengths and defined as [pic] where n2 is the nonlinear refractive index. To simulate the evolution of the three waves along the silicon waveguide, the above four differential equation are solved simultaneously using Runge-Kutta-Fehlberg (RKF) method . |Parameters |Input-Output simulation values | |? |100/4. 34 m-1 | |Aeff |0. 17? 10^(-12) m2 | |? 0. 7? 10^(-11) m/W | |? p |1310? 10^(-9) m | |? eff |1? 10^(-9) s | |c |2. 998? 10^(8) | |h |6. 626? 10^(-34) Js | |? k |0 m/s | |? p |1. 0297? 10-21m2 | |? |2. 43 ? 10^(-11) m/W | 4 RESULTs and discussion . 1 Modelling of FWM in silicon waveguide Given Pp=1W, Ps=0. 001W, Pa=0W and L=0. 048m, Pump power, stroke power and anti-stroke power are drawn with respect to the position in the waveguide. [pic][pic][pic]The figures above show that when propagating in the waveguide, both the pump power and stroke power decrease while the anti-stoke power increases. This is as expected, as the interaction of the pump wave and stroke wave produce the anti-stroke wave. The increase of the anti-stroke power comes from the decrease of the pump and stroke power.
It can be seen that, at the end of the waveguide, the pump power is only 0. 26W and the stoke power is only 0. 026W. Both of them decrease 74% of their original power. Both the pump power and stroke power decrease fast at the beginning, and then their decrease rate becomes slower when propagating further in the waveguide. This implies that the higher the pump power and the stroke power, the higher the propagation loss. As a result, the anti-stroke power increases fast at the beginning and then its increasing rate slows down. At the length of 0. 42m, the power of the anti-stroke reaches its maximum value which is about 3. 2*10^-5W. Then the anti-stroke power starts to decrease slowly. This may be because when the pump and stroke power is small, the gain of the anti-stroke power is less than its propagation loss. 4. 2 Effects of input pump power on conversion efficiency Given Ps=0. 001W, Pa=0W and L=0. 048m, Pp changes from 0 to 10W with step 0. 2W. The graph of the output stroke power and the output anti-stroke power are drawn with respect to the input pump power. [pic] Figure 2. 1 Output stroke power with different input pump power
This graph shows that the larger the input pump power, the smaller the output stroke power. This is as expected, as the larger the input pump power, the larger the propagation loss. The output stroke decreases slower when the input pump power is higher. [pic] Figure 2. 2 Output anti-stroke power with different input pump power This graph shows that when the input pump power is less than3W, the higher the input pump power, the higher the output anti-stroke power. This is as expected, as more input power can be converted to anti-stroke power when the input pump power is larger.
When the input pump power is larger than3W, the output anti-stoke power decreases with the input pump power. As the higher the input pump power, the higher the propagation loss. When the input pump power is larger than3W, the propagation loss dominates. 4. 3 Effects of waveguide length on conversion efficiency To investigate the relationship between the waveguide length and the conversion efficiency, input power are keep constant, Pp=1W, Ps=0. 001W, Pa=0W, L changes from 0. 001m to 0. 1m with step 0. 001m. Output stroke power and output anti-stroke power are drawn with respect to different waveguide length. pic] Figure 3. 1 Output stroke power with different waveguide length This graph shows that the longer the waveguide length, the smaller the output stroke power. This is as expected, as the longer the waveguide length, the larger the propagation loss. The decreasing rate of the output stroke power decreases with the waveguide length. [pic] Figure 3. 2 Output anti-stroke power with different waveguide length This graph shows that when the waveguide length is less than 0. 048m, the output anti-stroke power increases with the waveguide length.
This implies that the gain is larger than the propagation loss in the waveguide. When the waveguide length is larger than 0. 48m, the output anti-stoke power decreases with the waveguide length. At waveguide length larger than 0. 048m, the propagation loss is larger than the gain of the anti-stroke power. The output anti-stroke power has a maximum value of 4. 5*10^3 when the waveguide is 0. 048m. Thus, the most effective waveguide length is 0. 048m. 5 Conclusion The conclusion serves the important function of drawing together the various sections of the written report.
The conclusion is a summary, and the developments of the previous sections or chapters should be succinctly restated, important findings discussed and conclusions drawn from the whole study. In addition, you may list questions that have appeared in the course of the study that require additional research, beyond the limits of the project being reported. Where appropriate, recommendations for future work may be included. The conclusion should, however, leave the reader with an impression of completeness and of gain. Acknowledgment
The author would like to express her deepest gratitude to A/P Luan Feng and PhD student Huang Ying for their guidance, assistance and advices. The author also wishes to acknowledge the funding support for this project from Nanyang Technological University under the Undergraduate Research Experience on Campus (URECA) programme. References The template will number citations consecutively within brackets . The sentence punctuation follows the bracket . Refer simply to the reference number, as in —do not use “Ref. ” or reference ” except at the beginning of a sentence: “Reference  was the first …”
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