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Optical Electronics Course: Modulation Instability and Wavelength Conversion In Optical Fibers By: Hassan Pakarzadeh Winter-Spring 1 Department of Physics, Shiraz University of Technology, Shiraz, Iran.
OUTLINE 1-Introduction, Applications and Motivation -Theory of Modulation Instability 3- Results and Discussions
1- Introduction Purpose: Access to new wavelength regions Variety of Applications: Agriculture Medical and Surgery Police Industry and
Modulation Instability Modulation Instability (MI) is a general feature of wave propagation in dispersive nonlinear media and manifests itself in several branches of physics such as: Nonlinear Optics Fluid Dynamics Plasma Physics
Scalar Modulation Instability (SMI) Scalar modulation instability, which leads to the breakup of an intense CW beam is the simplest form of MIs which can occur in optical fibers. It can be considered scalar in the sense that it is the only MI which can be described by the scalar single polarization nonlinear Schrodinger equation (NLSE).
Applications FOPAs ( Fiber-Optic Parametric Amplifiers ) FOPOs ( Fiber-Optic Parametric Oscillators ) Advantages compared with bulk OPAs /OPAs Wide Tunability Efficient Conversion Low Pump Power
H. Pakarzadeh, H. Shahmirzaee, and A. Zakery, Scalar Modulation Instability in Highly Dispersive Photonic Crystal Fibers, 3rd Iranian Student Conference of Payame Noor University (PNU) on Physics, Khouzestan Payame Noor University, Ahwaz, Iran, (8). H. Pakarzadeh, H. Shahmirzaee, and A. Zakery, The Role of Phase Mismatching in Optimized Design of Parametric Photonic Crystal Fiber Amplifiers, 1st Iranian Conference on Photonics Engineering, Isfahan University, Isfahan, Iran, (9). H. Pakarzadeh, and A. Zakery, Dynamics of Scalar Modulation Instability in Presence of Raman Scattering in Photonic Crystal Fibers, Proceedings of 16th Conference of Iranian Optics and Photonics Society, Yazd University, Yazd, Iran (1). H. Pakarzadeh, and A. Zakery, Investigation of Scalar Modulation Instability in the Presence of Raman Scattering in Photonic Crystal Fibers, International Journal of Optics and Photonics (IJOP), Vol., No. 1, pp. 31-38 (1). تبديل طول موجي با استفاده از پديدهي تركيب چهار موج در تارهاي نوري براي دستيابي به نواحي طول موجي مناسب در پزشكي پوست-زهرا خدير-پايان نامه كارشناسي ارشد تاريخ دفاع بهمن 91
Dispersion GVD parameter The traditional theoretical approach to the SMI analysis involves only terms up to in the Taylor series expansion.
Dispersion Regimes Normal: > SMI can be observed only if higher dispersion terms ( ) are included. Anomalous: < SMI can be observed up to.
Photonic Crystal Fibers Lattice Pitch: Λ (PCFs) Air Hole Diameter: d Air Hole Shape Refractive Index of the Glass n core Type of Lattice The large waveguide contribution to the dispersion properties of PCFs is non-negligible
- Theory Nonlinear Schrodinger Equation describing the propagation of light in an optical fiber : i A z A T 3 3 i + + γ A A = 3 6 A T A T A is the slowly varying pulse envelope of the electric field T = γ = t z ( υ g n ca ω eff ) is the local time parameter The nonlinear coefficient A eff is the effective area of the core
Steady state solution: ), exp( ) ( z P i P z A γ = where P is the incident power of the CW light at z =. the stability of this CW solution to small perturbations can be examined by adding a small perturbation to the steady state solution such that ), )]exp(, ( [ z P i T z a P A γ + = where a (z, T) is the perturbation and a (z, T) << P. ) ( 6 T z a i 3 3 3 = + + + a a P T a T a i a γ
Assuming a general solution for the small perturbation of the form: a( z, T ) = a exp[ i( kz Ωt)] + a exp[ i( kz Ωt)] 1 where the coefficients a 1 and a are real, k is the wave number and is the frequency of the perturbation. b11 b1 a1 =, b 1 b a b b b 1 11 = k + Ω + Ω + γ P 1 = b1 = γ P 1 = k + Ω + Ω + γ P..
A nontrivial solution exists only when the determinant of the matrix is zero: ] ) 8 1 1 Im[( 1 1 ) ( Im ) ( Ω + Ω + Ω + Ω ± = = Ω P k g γ ) 6 9 (3 γ P + ± Ω = The peak frequency shift of the SMI sidebands
Phase-Matching Analysis of Scalar Modulation Instability =, +Δ =Δ Δ NL k L k k ). ( ) ( ) ( p a s L k ω ω ω + = Δ.! 1 3! 1 1 ) ( 3 3 1 Ω + Ω + Ω Ω + + = ω p a. ) (! 1 ) ( 3! 1 ) ( 1 ) ( ) ( 3 3 1 Ω + Ω + Ω + Ω + = ω p s
Δ k L = 1 Ω Ω + 1. Δk NL = Δk a, NL + Δk s, NL Δk p, NL = = γ P γ P +. γ P γ P So, only the even-order terms of contribute to SMI gain.
Phase Matching Analysis of SMI Δ k=δ k +Δ k = L NL Nonlinear Wave-vector Mismatch Δ k = ( ω ) + ( ω ) ( ω ) L s a p 1 1 1 ( ω a ) = p + Ω + Ω + Ω + Ω 3!! 3 1 3 1 1 3 1 ( ωs) = p + 1( Ω ) + ( Ω ) + 3( Ω ) + ( Ω). 3!! 1 Δ k L = Ω + Ω 1 Linear Wave-vector Mismatch Δ k =Δ k +Δk Δ k = γ P + γ P γ P = γ P NL a, NL s, NL p, NL 1
Normal vs. Anomalous Dispersion Regime Δ k=δ k +Δ k = L Peak Frequency of SMI Sidebands g = γ P max NL Anomalous Dispersion Regime: Normal Ωmax =± Dispersion Regime: Ω max =± γ P ( ) 3 + 9 6 γp
FWM Analysis of SMI The propagation of CW light inside an optical fiber in the absence of loss is given by: A A i + γ A A= z T Total Transverse Electric Field Propagating along the Fiber Local Time Parameter E ( ) ( ) z exp ikz E ( xyz,, ) = + E( ) ( ) ( ) 1 zexp ikz 1 + iω t f xy, exp( iωt) + E ()exp( z ikz iω t) Common Transverse Modal Profile 3
Coupled Mode Equations de dz de dz de dz 1 ( ) * = iγ E + E1 + E E + γe1eeexp( + iδ klz) ( ) * = iγ E 1 + E + E E1+ γeeexp( iδ klz) ( ) * = iγ E + E1 + E E + γe1eexp( iδ klz) E = A exp ( iϕ ) ; for j =,1,. j j j Conservation of the total power among the waves: Introducing normalized power for each wave: η ( z ) = [ A P ( z ) ] / 1, z ) A1, / 1 / Total Input Pump Power = + 1 + P A A A a ( = P η + a + a 1 1 = G. Cappellini and S. Trillo, Third order three-wave mixing in single-mode fibers: Exact solutions and spatial instability effects, J. Opt. Soc. Amer. B 8, pp. 8-838 (1991).
Four Coupled Equations dη = dz γ P η a1a sinφ da 1 = γ P η dz a sin φ da = γ P η a dz 1 sin φ dϕ a a =Δ kl + γ P [ η ( a + a )] + γ P [ η ( + ) a a ]cosϕ dz a a φ 1 1 1 1 ( 1 z z) = Δk L z + φ ( z) + φ ( z) φ ( ) Total phase difference of the process 5
3-Results and Discussions Calculations have been done for a highly dispersive PCF: =.5 ps / km =.9 1 5 ps / km = γ 188 1 W / km P = 1 W
SMI gain for both normal and anomalous dispersion regimes for P = 1W
Power dependency of the peak frequency shifts in both dispersion regimes
With more details:
H. Pakarzadeh, A. Keshavarz, and Z. Khadir, "Wavelength conversion based on four-wave mixing in photonic crystal fibers for achieving suitable wavelength region for skin applications,"proceedings of 19 th Iranian Conference on Optics and Photonics, ICOP 13, University of Sistan and Balouchestan, Zahedan (13).
H. Pakarzadeh, A. Keshavarz, and Z. Khadir, "Wavelength conversion based on four-wave mixing in photonic crystal fibers for achieving suitable wavelength region for skin applications,"proceedings of 19 th Iranian Conference on Optics and Photonics, ICOP 13, University of Sistan and Balouchestan, Zahedan (13).
طول بهينه (m) بازدهی بيشينه (درصد) توان پمپ (W).5 7. 15.6.7.8.9 7.3 7.5 7.6 7.7 15 11 95 85 1. 7.8 75 H. Pakarzadeh, A. Keshavarz, and Z. Khadir, "Wavelength conversion based on four-wave mixing in photonic crystal fibers for achieving suitable wavelength region for skin applications,"proceedings of 19 th Iranian Conference on Optics and Photonics, ICOP 13, University of Sistan and Balouchestan, Zahedan (13).
A low-amplitude, periodic perturbation can be amplified to form a train of high-intensity pulses through the modulation instability. The left panel shows how a small temporal modulation on a constant background (not visible on the scale used in the figure) evolves into a train of high power pulses at z= before decaying. In the frequency domain (right panel), this modulation is associated with multiple harmonic generation. Distance is plotted relative to a characteristic nonlinear length (LNL). Time and frequencies are normalized relative to the fundamental modulation period (Tmod) and frequency (fmod) respectively. Benjamin Wetzel, et al., New analysis of an old instability, Optoelectronics & Communications, SPIE Newsroom, 11.
D. R. Solli et al., Fluctuations and correlations in modulation instability, Nature Photonics, Vol. 6, pp. 63-68 (1).