Your propagation constant mismatch Delta-beta has units m-1 or km-1.  Keep it consistent.

The final Delta-lambda should be inside the brackets.  Depending which signal wavelength you measure dispersion D at, you may not need the slope term.  What is the dispersion at the signal wavelengths - you only specify slope?

Your FWM efficiency (eta) is dimensionless, but alpha is the Naperian loss coefficient, so  you need to scale dB/km by loge(10)/10 or 0.2303, so alpha becomes 0.048 km-1.  Note that this formula assumes L>>Leff.  For shorter spans, you may need to account for oscillations in power along the length.

It is too many decades since I had to convert between esu susceptibility and SI, so I suggest you dig into some EM or optics texts to sort this out.  I believe d is a dimensionless degeneracy factor, and d=6 corresponds to non-degenerate FWM with three discrete co-polarised input signals and an output product signal.

For Kerr non-linearity in single mode fibres, most folk find it easier to use the non-linear refractive index n2, rather than the non-linear susceptibility tensor.  For fibres which do not preserve polarisation state, including most telecommunications fibres, 2.35 10-20 m2/W is a reasonable value (G.P. Agrawal, "Nonlinear Fiber Optics", 2nd ed. 1995, Appendix B).

The fibre non-linear coefficient (gamma = 2 pi n2/ (wavelength x Aeff)) is the non-linear phase shift in radians per unit length per unit power.  Typical values are 0.0012 W-1m-1 for standard fibre, and closer to 0.0017-0.002 W-1m-1 for typical dispersion-shifted fibre or 55 um2 NZDSF.  The magnitude is sensitive to fibre composition and increases with germania content.

This gives a rather simpler expression for the FWM power at the end of your 80 km link:

PFWM = eta gamma2 Leff2 P3 exp(-alpha L)

Multiply by 4 for non-degenerate FWM in which all signals are in the same polarisation state and remain aligned over the first 20 km of fibre. 

FWM efficiency is lower if the 2 or 3 input signals are launched into different polarisation states.

thank sir,your explanation is totally helpful. I did the calculation and could u have a look at it?

Are there any standards or suggestion  for FWM in WDM system? For example, the power of FWM should be lower than X dBm. Or the SNR should be around Y value.

What are your input signal wavelengths?  You are working very close to the dispersion zero, and the phase mismatch could be exactly zero, depending how the fibre zero dispersion wavelength aligns with your signals.

You have dropped a factor 3 from your Delta-beta calculation, and another factor of 103 converting to km-1.

FWM efficiency (eta) is close to unity within 0.4 nm of the fibre dispersion zero, so you can tolerate some loss of precision in the phase mismatch.

I won't comment on the non-linear susceptibility calculation, other than to say that the result does not look familiar.  It is much easier to work with the non-linear refractive index in m2/W.  I believe the formula you are using is only valid when all signals are linearly polarised in the same polarisation orientation.  Polarisation states are not maintained over lengths greater than a few metres in conventional single mode fibre.

Your final calculation seems to be a summation over all possible FWM products between 3 input signals.  This is not valid.  You can't simply multiply by the number of products.  In general the phase mismatch and/or the degeneracy will be different for each product, so you need to calculate FWM efficiency for each product.   In your example the degeneracy is more important than the phase mismatch.

Are there any constraints on the polarisation states of the signals?  FWM is sensitive to the relative polarisations of the contributing signal, and close to zero for some orientations.

Signal to noise ratio is generally calculated on a per-channel basis.  In a dense WDM system, FWM products which fall outside the pass band of the demultiplexer filter are rejected.  Of those transmitted to the detector, only FWM products whose frequency offset is within the electrical bandwidth of the receiver will contribute significant beat noise.

Do you want a typical or worst case estimate of FWM?

There are no universal standards for acceptable levels of FWM.  Is this for a communications link?  What other noise sources are there? Is it single span or multi-span.  What ageing margins are required?  Is Forward Error Correction available to increase the tolerance to noise and non-linear crosstalk?  How much waveform distortion is likely?  An open eye will tolerate higher crosstalk than an eye  partially closed by distortion.  What modulation format is used.  Is there additional crosstalk from cross-phase modulation?

What are the tolerances on the fibre dispersion?  Cables are often manufactured in 10-20 km lengths, and will have dispersion zero wavelengths which vary by a few nanometres, even when manufactured from nominally similar fibres.  Most new build systems will avoid fibres with low dispersion near the transmission band.

Note that the FWM efficiency formula you are using is an approximation, and only valid if the dispersion does not vary along the cable.  Changes in dispersion between cable segments can lead to higher than average FWM, varying with the exact dispersion map and channel spacings.

For three 0.4 nm channels very close to the fibre dispersion zero, the FWM efficiency is very high, but will be less susceptible to some of these variabilities. 

This is not a configuration I would choose for a 3-channel WDM communications like.  What is the aim of this work?

woa, u bring a ton of knowledge. Actually, i am doing my master dissertation and i haven't learnt it before. I also don't have much support from my supervisor. But i want to study about the non-linear effect because i think it would be good experience for me to apply a PHD in optical communications.

I just take this example to understand the calculation of FWM. I read a lot of books and articles but they didn't show how to do it. they also didn't explain in detail like u. many thanks.

In my dissertation, i will intend give a constant distance ' L ' from A to B. then the vary quantity is the length ' l ' of fibre and the number of amplifiers is ' m '.

=> L = l * m.

I would ignore the connection loss.fibre dispersion will be constant.

for example, I would have a diagram like the attached file ( just assumption ), calculate the ASE noise for each case, then finally power penalty or OSNR or probability of error ( i haven't decided yet !!! ) for each case => conclusion.

I don't know if it is ok or not, my supervisor didn't focus on this field so he didn't guide me much. If u have any other suggestion. Please let me know.

thank sir.

A few thoughts and suggestions:

Dispersion will be constant, but you presumably get to choose the value.  Operating at the dispersion zero wavelength (D=0), some of the FWM products have essentially zero phase mismatch.  In a multi-span system, this means that FWM from each span adds coherently to the FWM field from preceding spans.  The FWM power will scale as the square of the number of spans, m2, rather than adding incoherently and scaling linearly with m as the ASE noise does.

If you stick with this choice, and you have more than 3 channels, the dispersion will change significantly across the transmission band.  The FWM penalty will be very strong near the dispersion zero, falling off for more distant channels.

Dense WDM systems generally choose to operate with the zero dispersion wavelength outside the transmission band, either nearby (near-zero dispersion shifted fibre NZDSF), or using non-dispersion shifted fibre (NDSF) with a dispersion zero closer to 1300 nm.  If phase mismatch Delta-beta is much larger than alpha, it becomes more reasonable to assume FWM contributions from different spans add incoherently.  A more detailed analysis would take account of manufacturing tolerances on fibre dispersion and variations in span length.

With moderate dispersion, FWM is dominated by near-neighbour interactions.  At lower dispersion or with narrower channel spacing, more distant channels can contribute at relatively high FWM efficiency.  It is interesting to consider how the "non-linear bandwidth" varies with channel spacing, fibre dispersion coefficient and system length - though possibly outside the scope of your thesis.

Have you chosen a modulation format and bit rate?  If you are comparing ASE and non-linear noise sources, the FWM will have a bandwidth comparable with the signal bandwidth (generally somewhat wider).  The receiver electrical bandwidth is typically some fraction of the modulation rate, and will determine the effective ASE bandwidth contributing to signal-ASE beat noise.

At 10 Gbaud, NZDSF might take you a few hundred km without dispersion compensation, but with NDSF you need to worry about in-line dispersion compensation much sooner.  Dispersion penalty scales as the product of net dispersion and bit rate squared, so this is much less of a problem at 2.5 GBd.

Will you consider other non-linear effects, such as cross phase modulation (XPM) or self-phase modulation (SPM)?  For amplitude modulated signals, the penalty arises from the interplay of non-linear phase shift (Kerr effect), and chromatic dispersion which turns phase modulation into amplitude noise and distortion.  If this makes life too interesting, choose a combination of channel separation, bit rate and dispersion which makes FWM the dominant non-linear penalty.  Phase modulation schemes such as PSK and QAM are more directly impacted by phase crosstalk, so it will be more difficult to ignore XPM for these.

Amplifiers are typically specified in terms of noise figure.  3 dB is the theoretical high gain limit, corresponding to nsp = 1.0.  6 dB is a reasonably conservative real world noise figure (nsp = 2), after including insertion losses from isolators, pump multiplexers, monitor tap couplers and connectors and other annoyances.  However, if pushed, you could justify intermediate values (nsp  1.2 - 2).

Linear OSNR increases with launch power.  Non-linear Kerr crosstalk (FWM and XPM) scale as the power cubed, so nonlinear OSNR varies as P-2.  There will be an optimum power which maximises the effective SNR, but which will depend on the number of spans.

Hope this helps.  Good luck with the thesis.

Alan

I just investigate fwm in fibre only and wouldn't consider the effect of modulation.  So i will stick with my choice. Are all formulas ok? What should i choose between power penalty or osnr or ber?

again, many thanks to u, sir

Power penalty is not generally a concern with FWM unless you can guarantee that none of the FWM products falls within the electrical bandwidth of the receiver.  For uniformly spaced optical frequencies some of the FWM products are guaranteed to have identical optical frequency to the target channel, and will interfere strongly, so you don't meet this criterion. 

What optical bandwidth do you choose for the ASE?  Do you understand the different contributions of signal-ASE and ASE-ASE beat noise terms?

OSNR is OK as a metric, but you need to normalise the FWM and ASE noise terms appropriately.  ASE is distributed continuously across the EDFA bandwidth, and can be specified as a power spectral density, often referenced to 0.1 nm bandwidth (e.g. as dBm/(0.1 nm).)  FWM is generated at discrete frequencies, convolved with the modulation bandwidths of the contributing signals.  Both contribute beat noise after square law detection by a photodiode, but the magnitude of the ASE beat noise term will depend on the receiver electrical bandwidth, and to a lesser extent on the optical bandwidth of the DWDM DMUX filter.

I don't recognize the exact form of your BER equation, but it looks broadly similar to calculations based on "Q", estimated from noise standard deviation and eye opening.  Generally, it is not very useful to estimate BER for non-linear crosstalk  independently of the contribution from ASE noise.  At the optimum launch power, the ASE noise term will be larger, and the combined BER much greater than that from either ASE or FWM alone.

Another measures sometimes used is "Q penalty", in which the Q is estimated with and without the non-linear crosstalk, keeping linear OSNR constant.  This should be broadly similar to the OSNR penalty, but is potentially more accurate as it can take account of waveform and receiver-specific details such as DMUX filter, electrical bandwidth and eye distortion.

but they didn't give the value of G. what is the typical value of G?

Looks like a reasonable starting point, though I have reservations about some  parts.

Equation 3 is similar to the esu Hill formula, and needs different factors of (4 pi), not to mention the permeability of free space (epsilon0) when using SI units.  The 3rd order susceptibility relates electric displacement to electric field, and as far as I recall, does not have units m3/W.  Having said that, if you plug in their value for the susceptibility, you get a sensible non-linear index, including a polarisation averaging factor not incorporated in the original Hill analysis.  I suspect they have worked backwards from published values of n2.

The phase mismatch formula (3) is wrong.  The modulus operations are not appropriate when calculating the dispersion slope correction.  I seem to recall a typo (sign error?) in either or both of the Inoue and Shibata papers (refs 19,20), but not the modulus terms.  I would use instead:

{D(lambdak) + dD/dlambda (lamdak2 (2fk - fi - fj)/(2c) )}

You can easily check this by a 2nd order Taylor expansion of the propagation constant (beta) in powers of optical frequency offset.

I haven't looked closely at their analysis and it is an academic exercise, but anyone quoting OSNR to 4 decimal places has probably not had much hands-on experience of optical transmission systems.

Good luck.

Alan.

thanks sir. u save my life