RMS WIDTH OF PULSES PROPAGATING IN AN ARBITRARY DISPERSIVE MEDIUM P.- A. Bélaer, A. Gajadharsih COPL, Départemet de Physique, Uiversité Laval. ad C. Paré Istitut Natioal d Optique INO, Québec. Abstract The propaatio law of the RMS width of a arbitrary pulse propaati i a eeral dispersive medium is derived.i particular, for a uiform medium alo the propaatio distace, the square of the RMS width evolves as the square of the distace i a maer similar to the propaatio i a purely secod order dispersive medium. This propaatio law shows the possibility to miimize the spreadi of a pulse by maai differet orders of dispersio. Itroductio For several eieeri applicatios, it is ofte sufficiet to be able to follow the pulse width evolutio i a dispersive medium such as a optical fiber lik. For a arbitrary pulse evelope, a uiversal width defiitio must be a RMS oe. Followi Marcuse ref., the RMS width is related to the square root of the variace of the pulse ad here we defie it as: RMS WIDTH For a pure secod order dispersive medium, it is well kow that the mathematical model of propaatio is aaloous to the optical beam propaatio i space ad oe ca copy from this field the propaatio law of a arbitrary pulse evelope amely:
where is the spectral variace of the pulse, is the secod order dispersio of the medium ref.. Marcuse ref.,, has already show how the variace of Gaussia pulses evolve i a dispersive medium of secod ad third order. I this short ote, we will demostrate that this useful propaatio law ca easily be eeralized for a arbitrary pulse propaati i a eeral dispersive medium. Mathematical derivatio I the spectral domai, the propaatio of the pulse evelope is calculated from: V, V ep i, d [ ] where V is the iitial spectral distributio of the pulse ad, is the characteristic dispersio of the medium alo the propaatio distace. The temporal variace of the pulse is calculated from the evaluatio of the secod ad first order momets: where the momets of the pulse evelope are, T T T t V t, dt V t, dt The propaatio law for the dispersive medium is ive i the spectral domai ad it is advised to cotiue our calculatio i the same domai. This is achieved by usi the relatioship betwee the time ad spectral domai liked by a Fourier trasformatio. It ca also be doe directly after usi this theorem of Fourier aalysis: [ ] [ ] T i V V V d d [ ]
Usi this operator ad the propaatio law, the evolutio of the temporal variace is overed by: [ ] 5,, d V d V d V d V where we have itroduced the accumulated temporal roup delay: [ ] 6, d The eeralized propaatio law Equatio 6 is our mai result : the chae of variace is ive by the differece betwee the quadratic mea spectral time roup delay ad the square of the mea spectral time roup delay. Equatio [5] ca be writte i a more compact form: [ ] 7 Here we have assumed that the iitial pulse evelope was real uchirped. Whe the iitial pulse is chirped, aother term appears i the propaatio law ref. We have used a uchirped iitial pulse for the sake of aalytical simplicity but we stress o the fact that it could be added to the evolutio equatio [5]. A eample As a practical eample, we shall ow cosider the propaatio of a iitial Gaussia pulse i a uiform dispersive medium:!!!
For this eample the iterals ivolved i equatio 5 are easily calculated ad the propaatio of the variace becomes: 5 For a uchirped Gaussia pulse the temporal ad the spectral related by: [ 8] variaces are For vaishi third ad fourth order dispersio, the well-established parabolic law of propaatio of a Gaussia pulse i a secod order dispersive medium is recovered ad is: where we have itroduced the typical dispersio distace: [ 9] [ ] This particular result clearly shows that the variace of the pulse is still parabolic ad its broadei affected by all the orders of dispersio ecepti of course the zero ad first order. By varyi the spectral variace i equatio 8 it miht be possible to desi a eperimetal set up to measure the differet orders of dispersio depedi o how accurate the RMS spectral ad temporal widths ca be obtaied. It is also iteresti to observe from equatio 8 that if the secod ad fourth order dispersio are of opposite si, the broadei of the pulse will be miimised. For eample if we ca make the third order dispersio to vaish, a optimum coditio ca be derived to miimise the broadei for a ive secod order dispersio. This coditio is:. Ad for this optimum coditio, the propaatio law is: [ ] 5 [ ]
The broadei is therefore reduced by a factor of.5 as compared to the propaatio i a dispersive medium with the same secod order dispersio oly. Fiures,, ad show how a Gaussia pulse evolves duri propaatio i a dispersive medium havi differet orders of dispersio ad show also that the predicted law of propaatio is also verified for the variace of the pulse. Fiure shows the case where we have reversed the si of the fourth order dispersio foud above to show how the broadei is affected whe the optimum coditio is ot applied. I this case, the propaatio law is:.8 We ca see that whe the fourth order dispersio is chose for the optimum coditio, the broadei of the variace is miimized as compared to the two other cases ivolvi either secod order dispersio aloe or secod ad fourth order dispersio without optimizatio. The propaatio laws aree perfectly with the umerically foud values for the variace, which was epected. [ ] Summary I coclusio, we have eeralized the propaatio law of the RMS pulse width of secod order dispersio to the most eeral dispersio medium. I particular, for a uiform medium alo the propaatio distace, we have show that the variace of the pulse still evolves as the square of the distace i.e parabolic. We thik that this eeralizatio is ot oly of academic purpose but adds to our physical uderstadi of how the dispersio to all orders modifies a pulse after its jourey i optical fiber commuicatio liks. I practice, it is ot always possible to measure the RMS width of a pulse accurately due to the icrease of the square term i the secod order momet. However, it is possible to obtai adequate results after trucati a very small less tha % part of the eery i the pulse. This was successfully achieved for the RMS width of beams propaati i space ref. 5. We have already verified that for a applicatio i dispersio maaemet, this trucatio is advised. For very short pulses the width is ofte measured throuh a correlatio. Usi some wellkow properties of Fourier Trasformatios it is elemetary to show that the variace of
a secod order correlatio of the itesity of a pulse is twice the temporal variace c c for ay pulse shape. Therefore this correlatio obeys a similar propaatio law 5 ad the lobal characteristic of the dispersio medium could be measured directly from a secod order correlatio. Refereces. D.Marcuse, Appl. Opt., vol. 9, 65-66, 98.. G. P. Arawal, Noliear fiber optics, Academic Press Ed., Sa Dieo, 995.. D.Marcuse, Appl. Opt., vol., 969-97, 98.. D.Marcuse, Appl. Opt., vol., 57-579, 98. 5. C. Paré ad P.- A. Bélaer, Opt. Comm., vol., 679-69, 996.
Fiures 5 X X A.5 5-5 t Fiure. Secod order dispersio aloe. 5 X X A.5 5-5 t Fiure. Secod ad fourth order dispersio chose for optimum coditio.
5 X A.5 X 5 t -5 Fiure. Secod ad fourth order dispersio without optimum coditio. 8 6 ad without optimisatio. 5 aloe. ad with optimum coditio. X X Fiure. Variace evolutio for the three studied cases., o ad š are values foud umerically whereas the other curves are obtaied throuh the derived propaatio laws.