Ultrafast Pulses and GVD John O Hara Created: De. 6, 3 Introdution This doument overs the basi onepts of group veloity dispersion (GVD) and ultrafast pulse propagation in an optial fiber. Neessarily, it goes into some detail about onepts suh as group veloity, phase aumulation, and so on. Initial Conepts When a monohromati wave travels through a medium with index n its phase fronts travel at v p = /n. The term v p is alled the phase veloity. When two monohromati waves of differing frequenies travel together, they beat against one another so as to produe groups. If the two waves travel with the same phase veloity, then the group will also travel the same speed. However, in general the group travels at a different speed than the phase fronts, and this ours when the two separate waves have different phase veloities. Group veloity may be either faster or slower than the phase veloities, and it may even be in the opposite diretion. The speed and diretion in whih the group travels is alled the group veloity, v g. 3 Gaussian Pulses Very often, ultrafast optial pulses take the mathematial form of Gaussians. Even if this is untrue, there are many instanes where you an approximate the pulse as Gaussian. Therefore, it s worthwhile to study the properties of a Gaussian pulse. The main feature of a pulse is that its extent in time is limited. Therefore, pulses are usually quantified by their pulse width. For optial pulses, pulse width desribes the temporal duration of the intensity envelope. This value depends on where you define the edges of the pulse. Two ommon definitions are used. The first pulse width is desribed as the duration between two points where the intensity envelope reahes half its peak value. This is alled the full-width-half-max (FWHM) pulse width. The seond definition is alled the /e (pronouned one over E ) pulse width. It is the duration between two points where the intensity envelope reahes about 37% (or /e, where e is Euler s number) of its peak value. In order to properly study pulses and GVD, however, it s neessary to work with the eletri field, not the intensity. So, begin with the time-domain definition of the eletri field of a Gaussian optial pulse. This pulse has a full-width-half-max (FWHM) pulse width of t. ( ) E(t) = E t exp [ ln()] os(ω t) () t where, ω is alled the arrier frequeny, and E is the maximum field amplitude. Timedomain plots of this funtion are shown in later figures. These all use a arrier frequeny of
about f = 375 THz, whih orresponds to a free-spae wavelength of λ = 8 nm, a ommon ultrafast laser operating wavelength. 3. Relation between t and ω To be ompleted... 3. Frequeny Domain Representation Most materials, inluding glass, are dispersive. This means their refrative index n and phase veloity vary with frequeny, sine v p = /n. Thus the many frequeny omponents that make up an ultrafast pulse travel at different speeds through the material. As disussed earlier, this an ause the phenomenon of GVD. Sine n and v p are funtions of frequeny, and not time, it is usually muh easier to work in the frequeny domain. This requires a frequeny-domain definition of the eletri field. The Fourier transform of a Gaussian in time is a Gaussian in frequeny. One an perform the Fourier transform, shown in Appendix A, to find that ( ) E(ω) = Ê (ω ω ) exp [ ln()] exp ( ) jφ p. () ω Here, Ê represents some version of E normalized through the Fourier transform proess. This omplex equation is in polar form, so it s obvious that the magnitude term is just another Gaussian, as expeted, and the phase of the wave is φ p, whih, importantly, is a funtion of frequeny. In the following setions, this φ p term will be expanded and defined in a manner onvenient to studying GVD. This will ome in very useful during material dispersion alulations. For now, just realize that there is some relationship among the phases of all the many frequeny omponents of a pulse, and that φ p summarizes them. 4 Mathematial Conept of Pulse Reshaping To study GVD it is onvenient to express φ p by a Taylor expansion in terms of (ω ω ). Namely, (ω ω ) (ω ω ) 3 φ p = φ p + φ p (ω ω ) + φ p + φ 3p + (3) 6 The terms φ p, φ p, φ p, et. are all oeffiients of this expansion and have units of radians (rad), radian-seonds (rad-s), rad-s, et., respetively. Putting aside the question of how to alulate these terms, it s important to at least understand what effet they atually have on the ultrafast pulse. The plots in Fig. were generated with the Matlab sript PhaseExpansionDemonstrator.m. The first one, Fig. (a), shows the time-domain version of a fs optial pulse, with λ = 8 nm. These plots are very instrutive. Plot (b) shows the effet of hanging φ p. Notie how it produes almost no hange to the pulse at all. The only hange is the relative position of the yle peaks with respet to the envelope peak. This has no effet at all on the intensity, so the term φ p is not at all a onern in GVD situations, or in most other situations. The next plot () shows how hanges in φ p ause a shift in time of the entire pulse, but it does not hange the envelope. This is beause φ p (ω ω ) is only a linear hange in the phase over frequeny. Any suh hange an only delay the pulse, but annot reshape it. Hene, φ p is also not a onern in GVD situations. However,
E field (a) E field (b) E field () E field (d) 4 4 Figure : Plots of effets of phase terms in expansion. (a) shows the original time-domain pulse. (b) shows the same pulse with φ p shifted by π/. () shows the original pulse but with φ p shifted by 5 5 rad-s. (d) shows the original pulse but with φ p shifted by 8 rad-s. plot (d) finally shows an effet on the envelope. Changing the quadrati term φ p affets the phase aording to the square of frequeny, hene it s a nonlinear effet to the frequeny-dependent phase. Any nonlinear effet like this, whether it is from φ p, φ 3p or higher-order terms, will reshape the pulse. In this example ase, the pulse width is massively inreased. Note that beause the sign of the shift to φ p is positive, the pulse beomes positively-hirped, meaning high frequenies (bluer olors) arrive later than low frequenies (redder olors). This is also alled normal dispersion. If the sign of shift is negated then dispersion also beomes negative or anomalous and the output wave is negatively-hirped, meaning bluer olors arrive earlier than redder ones. Beause of this behavior the term φ p has a speial designation and is alled the group delay dispersion (GDD). The next term φ 3p also has a speial name, third order dispersion (TOD). These two parameters are the main ators in most GVD problems. Glass omponents suh as lenses, beamsplitters, and optial fibers all have linear and nonlinear frequeny-dependent refrative indies. Realizing that phase aumulation is related to the index by φ = nkl, it is obvious that any glass omponent through whih an optial pulse travels has the potential to reshape it. Ideally, GDD and TOD would be zero, but glass is not ideal. For the purposes of this doument, TOD will be ignored, sine it s usually harder to get rid of than GDD and its effet is smaller anyway. Note that this is not always possible for very short pulses going through lots of glass (e.g. optial fiber). 5 Calulating Pulse Reshaping This setion goes over the basi math in determining the extent of pulse reshaping due to GVD. This is general to any optial system, although the implementation might be somewhat varied. For example, a prism pair would impart GVD, but in a muh different way than an optial fiber. Again, this treatment ignores TOD and higher-order effets. 3
To begin, reall that only GDD will affet the normalized pulse shape. This statement assumes optial losses have been ignored, whih would, of ourse, redue the pulse amplitude, possibly even as a funtion of frequeny. But sine GDD is the only element of the phase expansion that matters, it is suffiient to express the optial system as a transfer funtion involving GDD only. This brings up another point. By working in the frequeny domain, the optial system an be treated as a simple transfer funtion H(ω). The transfer funtion is then multiplied by the omplex spetrum of the optial pulse to get the spetrum of the output pulse. Finally, an inverse Fourier transform provides the time-domain pulse oming out of the system. While the pulse itself might have GDD, manifest as φ p, the optial system will also impart GDD. This may be aptured by a separate term φ s. The system will also have the onstant and linear phase omponents. The transfer funtion an be desribed quite aurately as ( ) (ω ω ) H(ω) = H (ω ω ) exp jφ s jφ s (ω ω ) jφ s. (4) The term H (ω ω ) aptures any frequeny-dependent loss fators one may wish to inlude. This term would be neessary for a grating-pair dispersion ompensation system, for example, sine gratings are always well below % effiient. With this transfer funtion the output spetrum is alulated E out (ω) = E(ω)H(ω) (5a) ( ) { = H (ω ω (ω ω ) )Ê exp [ ln()] exp j(φ p + φ s ) ω j [ (φ p + φ s )(ω ω ) ] [ j (φ p + φ s ) (ω ω ) ] }. (5b) As long as one is onerned only with GVD, this output spetrum an be simplified by letting φ s = φ s =. Quik Note: In alulating the effet of GVD on optial pulses it is usually wise to let the linear phase term φ s =. The GDD ontribution to total phase in most systems is quite small ompared to the linear term. This is true, for example, in a system like an optial fiber. It takes a lot of preision to aurately keep trak of the phase of a wave having λ = 8 nm propagating over a distane of meters of glass. Just by a quik estimation, the total phase aumulated in m of silia fiber is around φ s = nk l = (.45)(π/8 9 )() =, 776, 546.74 radians. Now, if you must aurately represent eah yle of the wave by 4 data points (the Nyquist riterion), then the number of points needed in every alulation is 4.5 million! Some triks an redue this number by orders of magnitude, but the best trik is just to not use φ s. 5. Original Pulse Parameters For a typial transform-limited, Gaussian optial pulse, two of the phase terms may generally also be ignored. That is, φ p = φ p =. The plots in Fig. show the exat phase of a transform-limited pulse with t = 7 fs. In addition, they show the fit of an equivalent phase funtion generated by using φ p only. Atually, the FFT funtion used by Matlab produes a φ p term, but it is 4
Phase (rad) 4 Phase of original pulse as funtion of wavelength FFT linear fit Field Magnitude (V/m).8.6.4. Normalized spetrummagnitude FFT analyti fit 4 75 8 85 (nm) 75 8 85 (nm) Figure : Plot of phase of transform-limited pulse along with linear phase fit where only φ p is non-zero. manually zeroed out. The Matlab sript is alled LinearPhaseOnlyFit.m. This plot shows some odd phase features from the original pulse at the outer edges. These are simply beause there is very little spetral amplitude at these wings, and Matlab annot intelligently extrat phase from a near-zero-amplitude signal. The sript will also plot the time-domain pulse and the Gaussian spetral envelope, though these are not shown here. In fat the spetral amplitude is almost zero even as near as 5 nm from λ, so it s not surprising that the phase is not reliable way out at 8±4 nm. The numerial value used to ahieve this fit was φ p = 3 rad-s. This term ultimately will have to be inluded in any alulations to maintain the position of the pulse at t = in any inverse Fourier transform alulations. A different value ould be used but will result in a time-domain pulse being shifted in the alulation window and possibly wrapping onto itself. This phenomenon should be familiar to anyone austomed to doing time and frequeny-domain alulations in Matlab. Now, letting φ p = φ p =, and letting φ s = φ s = beause they don t matter for GVD, the output field is finally expressed as ( ) { E out (ω) = H (ω ω (ω ω ) )Ê exp [ ln()] exp jφ p (ω ω ) ω } (ω ω ) jφ s. (6a) 5. Solving for Optial System Parameters Any optial system we re onerned will produe dispersion by some nonzero φ s term. All of the other terms should be apparent from the original optial pulse. Therefore, the physial system behavior must analyzed to find out what φ s is. One that s done, the output field and intensity an be easily found. To proeed, reognize that φ s is a term in a Taylor expansion of the phase aumulation imparted to a wave by the optial system. In general the phase is expressed φ s = ñkl (7) 5
Note that all of these terms (ñ, k, and L) may be funtions of frequeny. It is obvious that k = ω/ is a funtion of frequeny by its very definition. And if the optial pulse is traveling through some dispersive material, whih is very ommon, then the dependene of ñ on ω is also quite apparent. It may be harder to visualize a system in whih the propagation length L is a funtion of frequeny, but this is, in fat, exatly how grating- and prism-based dispersion ompensation modules work. So, for the most general ase, all three terms are frequeny dependent. If any two or more are linear funtions of frequeny, or if any one is a nonlinear funtion of frequeny, then φ s is nonlinear in ω and via the Taylor expansion it s apparent that there will be dispersion. Of ourse, k is always linear in ω so it only takes a linear hange in ñ or L to produe a quadrati φ s and dispersion. It is not often trivial to determine the dependene on ω for ñ or L in real systems. So, for the sake of demonstrations, a ouple examples are now provided where ñ is given a simple mathematial desription. Example : A fs optial pulse propagates through a slab of glass that is m thik. It s index of refration, around the optial arrier frequeny, ω, is defined as: ñ =.5 + (6 as)(ω ω ) (8) where as is attoseonds. Inidentally, this is a pretty good approximation to fused silia. The total phase aumulation of an optial wave passing through this slab of glass, in general, may onsist of propagation phase and phase due to boundary strutures. Assume there are no boundary strutures, so the total phase is due only to propagation. Also neglet amplitude effets. All frequenies propagate through the same slab thikness, L, whih is frequeny-independent. Assume λ = 8 nm, then k = 7.854 µm. Now, to keep things in the proper form to assign the phase parameters, it is neessary to express k as a funtion of (ω ω ). By definition k = ω/ so k k = (ω ω ) k = (ω ω ) + k (9a) (9b) and the total phase aumulation is: φ s = ñkl (a) [ ] [ ] =.5 + (6 8 (ω ω ) )(ω ω ) + k L (b) =.5k L +.5L(ω ω ) + k L(6 8 )(ω ω ) + L(6 8 ) (ω ω ) () Colleting terms gives { } { }.5L φ s =.5k L + + k L(6 8 L(6 8 ) (ω ω ) ) (ω ω ) + (ω ω ) = φ + [φ a + φ b ] (ω ω ) + φ (a) (b) 6
.5 (a) Original E( ).5 (b) a ( )E( ) E field.5 E field.5 5 5 b ( )E( ) 5 5 ( )E( ).5 ().5 (d) E field.5 E field.5 4 6 8 5 5 Figure 3: Plot of phase of transform-limited pulse and the effets of the various phase ontributions. (a) original pulse. (b) pulse affeted by φ a. () pulse affeted by φ b, (d) pulse affeted by φ. Note that temporal shifts in the pulse enter positions are not labeled aurately beause the timedomain pulse will wrap through the available time window numerous times as phase aumulates to great values. Notie how the term of ñ that is linear in (ω ω ) now ontributes to both linear and quadrati terms in the phase expansion. For the purposes of understanding GVD in an optial fiber this quadrati term is relevant, meaning the linear variability in index of refration ontributes signifiantly to GVD. It s important to keep this in mind. This quadrati term in phase is due to the linear term in index. Figure 3 shows plots of the original pulse, and the pulse after applying the three ontributions of Eq., namely φ a, φ b, and φ. Note that Figs. 3-7 were all generated by a sript alled QuadratiIndexEffet.m. Figure 3 illustrates how signifiant GVD an be to a short pulse. It s worthwhile to re-iterate that the variation in refrative index is ausing this effet. The FWHM bandwidth of this pulse is about f = ln()/(π t) = THz, or ± THz. Inidentally, this orresponds to a wavelength band of about 8 ± 4 nm, or 48 nm of bandwidth. Figure 4 shows how little the index atually varies over that bandwidth. Again, this is a pretty lose approximation to fused silia around 8 nm. The refrative index is varying by less than.% over the whole bandwidth. This very small variation has a very large effet on short pulses. In addition it affets shorter pulses more than longer ones. By the last term in Eq. one an see that the phase shift is most signifiant for frequeny omponents far from ω. Suh omponents simply don t exist for longer pulses. The next example shows what happens when the quadrati index term is inluded. This term is 7
Index.5.5.5.499 Index.5.5.5.499.498 36 365 37 375 38 385 39 Freq. (THz).498 76 78 8 8 84 (nm) Figure 4: Index of refration aording to Eq. 8 as a funtion of f (left) and λ (right). quite small in atual fused silia, so that index is quite aurately fit over a large frequeny range using the linear term only. Nevertheless, atual fused silia fibers are modeled quite well around λ = 8 nm by ñ fs =.4533 + (5.8 as)(ω ω ) + (. as) (ω ω ). () This quadrati index term will be inluded in the next example. Example : Using the same onstant and linear terms from the last example, the index is modified by adding a quadrati index term, thus the total index is Then the phase aumulation beomes ñ =.5 + (6 as)(ω ω ) + ( as) (ω ω ). (3) φ s = ñkl (4a) [ =.5 + (6 8 )(ω ω ) + ( 34 )(ω ω ) ] [ ] (ω ω ) + k L (4b) =.5k L +.5L(ω ω ) + k L(6 8 )(ω ω ) + L(6 8 ) (ω ω ) + k L( 34 ) (ω ω ) + T.O.D. + (4) Colleting terms and throwing away TOD and higher-order terms gives { }.5L φ s =.5k L + + k L(6 8 ) (ω ω ) { } L(6 8 ) + + k L( 34 (ω ω ) ) = φ + [φ a + φ b ](ω ω ) + [φ a + φ b ] (ω ω ) (5a) (5b) where the new index term is manifest in the phase expansion as φ b = k L( 34 ). Finally, Fig. 5 shows the effet of this term along with the original pulse. In this ase, the φ b term has about / th of the effet of φ a, if that. Clearly, the effet of this term is pretty muh nil. However, this ould play a muh larger role in a fiber, where the path length is muh longer. Finally a omment 8
E field.5.5 (a) Original E( ) E field.5.5 (b) b ( )E( ) 5 5 5 5 Figure 5: Original pulse (left) and pulse after effet of quadrati term in index (right). about T.O.D.. The terms of T.O.D. beome signifiant in some ases of short pulses, long fibers, or even high powers. Peak optial power an get very large with ultrashort pulses, partiularly in the onfines of an optial fiber. Very high peak powers an atually alter a fiber s refrative index. The effet must be inluded when optial pulses get strong enough and often these terms are manifest as T.O.D.. For example, if strong pulses affeted the index, the index would then impart dispersion, whih would inrease the pulse width, whih would drop the peak optial power, whih would turn off the dispersion! Therefore, it s self-limiting. And the phase relationship between frequeny omponents determines whether it s on or off. Clearly this beomes harder to model. 5.3 Conversion to Standard Notation The GVD-relevant phase expansion terms are now ready to be put into some type of standard notation that is not dependent on the length of propagation. Reall from the last setion that φ s = L(6 8 ) + k L( 34 ) (6) and that this is alled the GDD (group delay dispersion). If this is normalized to length by dividing out L, then it s alled the group veloity dispersion (GVD) parameter. It is typially given a speial designation of k or k. In this ase k = φ s (6 as) = + (7.854 µm )( as ) = 46 fs /m (7) L This value is pretty lose to tabulated values for fused silia. An important point about this parameter is that it s frequeny-dependent, meaning it depends on where the bandwidth is entered, or the fit is performed. In the example ase it was found by a fitting proedure around 8 nm. However, it would be different if found by fitting around some other wavelength. In fat the GVD parameter an be derived in another fashion as ( ) ( ) k = d = ω d ñ dω dω + dñ dω = λ3 d ñ π dλ. (8) v g where v g = dω/dk. This shows that this parameter an be determined pretty simply so long as the frequeny-dependent index is known. The derivation of Eq. 8 is shown in the appendix. Figure 6 below shows that this equation orretly predits k at ω = π 375 THz (λ = 8 nm). The form of v g helps to larify what k atually is. The term /v g is the rate of hange of k with respet to ω. The term k inludes multiple possible system parameters suh as the material 9
k" (fs /m) 45 4 45 4 45 3 35 4 45 Freq. (THz) Figure 6: Plot of k versus frequeny from Eq. 8. properties (refrative index) as a funtion of frequeny. Therefore /v g aptures the rate of hange of system properties with respet to frequeny. In alulating k another derivative WRT ω is performed, meaning k is related to the rate of hange, of the rate of hange of the system properties. In the examples, this amounts to the seond derivative of the index of refration, as is apparent in the end of Eq. 8, although this form is given as a funtion of wavelength instead. Another parameter frequently used to quantify GVD is the Dispersion Parameter, (D), usually expressed in ps/(nm km). The only differene between k and D is that k expresses the temporal delay vs. angular frequeny over a normalized propagation length. The D parameter expresses the temporal delay vs. wavelength over a normalized propagation length. It s really the same information but one is expressed in terms of frequeny and the other in terms of wavelength. The dispersion parameter is alulated as follows. First, D is produed by multiplying k by dω/dλ and effetively hanging its dependene on frequeny to a dependene on wavelength by the Chain Rule. D = dω dλ k (9) This new derivative term an be easily found by the relationship ω = π/λ. Thus, by substitution dω dλ = π λ () D = π λ k () The parameter D, plotted in Fig. 7 below, orresponds to the k plotted in Fig. 6. 6 Example: Optial Fiber In this setion the dispersion of.5 m of fused-silia optial fiber is alulated. The index used for the fiber is given in Eq.. However, the value of the quadrati oeffiient has been zeroed, whih results in a slightly better fit (see below). The linear and zero terms were also refined. The new equation is thus
D [ps/(nm*km)] 8 4 6 8 3 35 4 45 Freq. (THz) Figure 7: Plot of D versus frequeny from Eq.. ñ =.45336 + (5.87 as)(ω ω ) () (ω ω ). () Two pulses will sent down the fiber, the first (P) having a pulsewidth t = 7 fs, and the other (P) having t = fs. The shape of the output pulse will be plotted. Both pulses are entered around λ = 8 nm. All of these results are alulated in the sript entitled ExamplePulsesThruFiber.m. First it s useful to alulate the bandwidth so that the index an be plotted over the relevant spetrum. The bandwidth is f = ln()/(π t). For P the bandwidth is thus f = 3.45 nm and for P, f = 9.4 nm. The index, alulated by Sellmeier parameters for atual fused silia, is plot below in Fig. 8. The plots show about 3 more bandwidth than is neessary. Index.454.4535.453 atual fit Index.454.4535.453 atual fit 36 365 37 375 38 385 39 Freq. (THz) 77 78 79 8 8 8 83 (nm) Figure 8: Plot of index derived from Sellmeier parameters, along with fit. Next, it is neessary to alulate the GVD parameter. This is done aording to Eq. 8 and is shown, along with the resulting dispersion parameter below in Fig. 9. Now the transfer funtion of the fiber an be alulated. First the spetrum of the original pulse E(ω) is alulated. The transfer funtion is then alulated ( ) H(ω) = exp jk (ω=ω )L (ω ω ) (3) Note that even though k is frequeny-dependent, only one value is used in this alulation, the value at ω. This is onsistent with the way k was first derived in previous setions. Sine k is basially onstant over the relevant bandwidth it wouldn t matter in this ase. However it might in other ases. With k the output is derived, E out (ω) = E(ω)H(ω). The original pulses are plotted below with their GVD-altered versions from propagating through the fiber in Fig.. Intensity envelopes, normalized in vertial sale to the field envelopes, are overlaid in red.
k" (fs /m) 396 394 39 39 388 D [ps/(nm*km)] 5 5 386 36 365 37 375 38 385 39 Freq. (THz) 5 77 78 79 8 8 8 83 (nm) Figure 9: Plot of k and D derived from fiber s refrative index. There are two important points to raise from these plots. First, both pulses have spread out signifiantly. Pulse spread to about 3.4 ps (FWHM). Pulse spread out to about. ps. Seond, the pulse that started shorter ends up longer in duration. The atual spread pulse duration an be alulated in the following equation. It is derived in the Appendix A. t t out = 4 + 6(ln ) φ t where φ = k L. This example shows how signifiant pulse propagation through an optial fiber an be. The next setion shows a different system, optial gratings, whih have the opposite effet in GVD. With the fiber, blue wavelengths experiened a higher index of refration, and therefore arrived as an output of the system later than reds. For the grating system, blue wavelengths will (4) Figure : Plot of (a) original pulse, (b) original pulse, () GVD dispersed pulse, (d) GVD dispersed pulse. The lower pulses looks solid beause individual yles are too lose together to distinguish. The intensity profile is also shown in red.
experiene a shorter path length, and therefore arrive at the output earlier than the reds. In this way the grating system will ompensate GVD from the fiber. 3