FWM Simulations in O-and Dora van Veen & Vincent Houtsma, Bell Las dora.van_veen@nokia-ell-las.com, vincent.houtsma@nokia-ell-las.com,
Introduction FWM is a nonlinear process in which three waves o requencies i, j, and k : (k # i, j ) interact through the third-order electric susceptiility o the optical ier to generate a ourth wave o requency ijk = i + j - k For example 3 co-propagating waves generate 9 new waves (and 4 generate 4 new waves, etc.) The mixing products act as crosstalk (in and) and deplete power rom the 3 main optical carriers rom [] Chiata et al. # l s 3 4 6 8 # FWM products 9 4 9 4
3 Taylor expansion around o propagation constant (): When is chosen at zero-dispersion wavelength the phase-mismatching D is: FWM eiciency h: Model o FWM around zero-dispersion wavelength 4 3 3 D D c D c C C C l l l l k j k i j i C D c D 4 l l F k j i D D D exp sin 4exp L L L h See also [] Inoue et al. or urther details
Satisying phase-matching conditions around zero-dispersion wavelength Partially degenerate: one o wavelength channels on zero-dispersion wavelength (ZDW) with = i = j making D= Completely non-degenerate case: Zero-dispersion wavelength in etween two wavelengths: From model: Zero-dispersion wavelength exactly in etween two wavelength with D C ()= and ( i - )=-( j - ) making D= independent o grid spacing More practical case: Zero-dispersion wavelength in etween two wavelengths, positive and negative D C in nd order terms o expansion cancel each other (yellow-highlight on previous slide) also satisying D= 4
Simulations Two types o cases are simulated : Channels symmetric around the zero dispersion wavelength All channels on one side o the zero dispersion wavelength Simulation parameters zero dispersion wavelength =3 nm +9 dbm optical transmission power km o total ier length Wavelength grid spacing is varied 5
Optical Power [-] Eiciency [-] P wm/p und [db] Simulation Case : Wavelengths exactly symmetric around zero dispersion wavelength FWM Eiciency Power ratio in db -.5 - -3.5-4 -5 -.5 4 6 8 Grid Spacing [GHz] -6 4 6 8 Wavelength Spectrum Grid Spacing [GHz].5.5 6 3 35 3 35 3 35 Wavelength [nm]
Eiciency [-] Eiciency [-] Eiciency [-] Simulation case : Almost symmetric around zero dispersion wavelength Bandwidth o phase matching condition depends on grid spacing Eiciency Bandwidth Eiciency Bandwidth Eiciency Bandwidth.8.8.8.6.6.6.4.4.4... 7 - -5 5 Frequency oset [GHz] - -5 5 Frequency oset [GHz] - -5 5 Frequency oset [GHz] 5 GHz spacing GHz spacing GHz spacing Eiciency andwidth versus requency oset (rom symmetrical around zero dispersion wavelength)
Simulation case : Almost symmetric around zero dispersion wavelength I channels are exactly symmetric around zero dispersion, phase matching condition is always achieved independent o grid spacing Bandwidth o phase matching condition will however depend on channel spacing This condition is valid or the case where the contriution o dispersion slope to the phase matching is much larger than the dispersion contriution 8
Optical Power [-] Eiciency [-] P wm/p und [db] Simulation case 3 : Wavelengths all on one side o the zero dispersion wavelength FWM Eiciency Power ratio in db -.8 -.6-3.4-4. -5 4 6 8 Grid Spacing [GHz] -6 4 6 8 Wavelength Spectrum Grid Spacing [GHz].5.5 9 35 3 35 3 35 33 335 34 Wavelength [nm]
Simulation case 3 : wavelengths all on one side o the zero dispersion wavelength From this case it can e oserved that having a little it o dispersion will help a lot to avoid phase matching condition Four wave mixing eiciency in this case will drop rapidly with increasing grid spacing Increasing the dispersion around the zero dispersion wavelength will only help in reducing the phase matching andwidth
Eiciency [-] Eiciency [-] Eiciency [-] Simulation case 4: wavelengths symmetric around zero dispersion wavelength with ±Dc() Bandwidth o phase matching condition or 5 GHz grid spacing FWM Eiciency Eiciency Bandwidth Eiciency Bandwidth.5.8.8.6.6.5.4.4 -.5 4 6 8 Grid Spacing [GHz] Dc=Dc() & ±Dc(). - -5 5 Frequency oset [GHz] Dc=Dc() - -5 5 Frequency oset [GHz] ±Dc() Right igure : positive and negative dispersion added to wavelengths (±Dc()) which are symmetric around zero dispersion wavelength.
Unequal spacing I the requency separation o any two channels o a WDM system is dierent rom that o any other pair o channels, no FWM waves will e generated at any o the channel requencies, therey suppressing FWM crosstalk Even though the FWM waves do not interact with the channels, they are still generated at the expense o the transmitted power, so that ultimately the maximum launched power is limited y the channel depletion caused y the generation o the FWM waves See also [3] Forghieri et al.
Design rules or wavelength plan NG-EPON driven y FWM Higher dispersion reduces FWM eiciency except when the zero-dispersion wavelength is in etween wavelength channels, it is est to avoid zero-dispersion to avoid phase-matching Wider grid reduces FWM eiciency when sign o dispersion is same or all wavelengths Unequal grid reduces FWM crosstalk (still power depletion) Upstream wavelengths elow 3 nm (to avoid zero-dispersion) to enale DML at ONU (negative dispersion) First wavelength l ar away rom nd -4 th reduces FWM crosstalk (unequal grid) and also reduces cost o optical ilters or irst 5 Gps wavelength Unequal grid or l-l4 (or example skip channel on 4-channel WDM) First downstream wavelength l, could possily e in the zero dispersion wavelength window (etween 3-34 nm) i it is ar away rom the other 3 wavelengths (unequal spacing and low andwidth FWM eiciency) US l US l-l4 DS l DS l-l4 3 7 8 9 3 zero-dispersion 34 33 3-34 nm
Conclusions and Factors reducing FWM eiciency Zero dispersion wavelength assumed same along whole ier, multiple stretches o ier with dierent ZDW will reduce FWM eiciency Same polarization assumed or all waves. Dierent polarizations will reduce average (not peak) FWM eiciency Unmodulated carriers simulated, modulated carriers will have reduced FWM eiciency. Phase matching andwidth is narrow or grid spacing > GHz reducing likelihood o maximum FWM eiciency, however it is not ininite small. It is est to avoid zero dispersion wavelength region i possile, unequal channel spacing will e eneicial as well 4
Reerences [] N. Shiata, R. P. Braun, and R. G. Waarts, Phase-mismatch dependence o eiciency o wave generation through our-wave mixing in a singlemode optical ier, IEEE J. Quantum Electron., vol. 7, pp. 5-, July 987 [] K. Inoue, Four-wave mixing in an optical ier in the zero-dispersion wavelength region, J. Lightwave Technol., vol., pp. 553-56, Nov. 99 [3] F. Forghieri, R. W. Tkach and A. R. Chraplyvy, "WDM systems with unequally spaced channels," in Journal o Lightwave Technology, vol. 3, no. 5, pp. 889-897, May, 995 5
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