amplitude approximation, satisfies aödinger-like equation, with a potential
the periodic modulation (kicks):
2c
T 2 1 AcosTXN
d�z� 2 N�c , (1)
umber of kicks, z� � z�z0 is the spa-coordinate normalized to the length oft is the internal time variable relative
to the center of the pulse and multiplied by modulationfrequency V, and g � 1/2b2z0V2. We do not includeabsorption in Eq. (1), as it can be compensated for byan amplif ier.
For weak dispersion, g ø p, we can write thestandard discrete time mapping for the optical kickedrotor as
pN � p0 2
NXM�1
k sin TM ,
TN � T0 1
N21XM�0
pM , (2)
where pN � 2gnN is the angular momentum, nN �njz��N1 is the sideband number, TN � T jz��N1 , andk � 2gA.
It can be seen that from Eqs. (2) we can obtain thefollowing relation for pN :
pN11 2 pN21 � 2k sin�TN 1 pN�2�cos�pN�2� . (3)
When pN is an odd product of p, the changes to pcancel each other out, resulting in a classical barrier.Otherwise, p is uniform (at resonance). We can findthe border between the uniform spectrum section andthe classical barrier by first approximating Eqs. (2)naïvely:
pN � p0 2 Nk sinT0 2 1/2N2kp0 cosT0,
TN � T0 1 Np0 . (4)
This approximation is valid only when the followingassumptions hold:
Light pulses in a dispersive medium that undergorepeated phase modulation upon propagation at equaldistances were shown to behave as quantum-kickedrotors1 with localization properties in their frequencydomains. This behavior is related to Anderson lo-calization for electrons in one-dimensional disorderedsolids.2,3 The long-term localization behavior inthe kicked optical system was demonstrated withmode-locked dispersive fiber lasers.4 These laserswere shown to exhibit conf ined exponential spectra,which are typical of localization, besides special reso-nances (dispersion modes) in some regions. As inmany other cases, such experimental studies skip thebuildup step that commonly endures a few kicks andis in general diff icult to follow. Study of this buildupstage is our aim in the present research.
We present an experimental demonstration of theevolution of localization in frequency of the opticalkicked rotor in dispersive single-mode fibers. Theexperiment was performed with a long recirculatingfiber loop such that we could follow every round tripof propagation of the light in the loop. The opticalsystem provides a unique opportunity to track thebuildup of localization, almost an impossible task inthe usual quantum-kicked rotor. This system alsoprovides the opportunity for following the buildup ofpulses in the time and frequency domains.
The localization received here occurs after propaga-tion in a dispersive f iber of broad light pulses that arerepeatedly kicked by sinusoidal phase modulation atequally spaced locations along the fiber. The naïveexpectation concerning the evolution of the spectrumand the buildup of sidebands (harmonics) is that theirnumber will diffusively increase with the number ofkicks, such that the spectrum will continually broadenwith propagation. However, because of localizationthe spectrum is confined, usually with an exponentialsignature. The focus of this Letter is on the transitionbetween the broadening and the localization regimesand on the number of kicks needed for it to occur.The experiment was performed with a recirculatingfiber loop system that enabled the pulse to be trackedafter each round trip of the loop.
The electric-f ield amplitude c of a pulse that ispropagating in dispersive single-mode fibers, in the
However, because of the dependence of the assump-tions on N , after enough kicks the assumptions willalways fail. When inequality (6) fails first, relativechanges in p appear before T changes. If p canchange freely, then p0 is in the uniform spectrumsection. When inequality (5) fails first, the changesin p are canceled before any relative changes occur;thus p0 is in the classical barrier section. The borderbetween these sections is established when both of theassumptions fail together, or 2gDn � p0 �
pk. This
gives us the number of sidebands:
Dn �qA�g . (7)
When we combine inequalities (5) and (6) and requirethat they fail together, we receive an approximation forthe number of kicks required for the transition fromspectral broadening to localization to occur:
N2k � 1 ) N �q1�k �
q1�g . (8)
The experimental system, shown schematically inFig. 1, consisted of a recirculating loop composed ofan optical f iber, an erbium-doped fiber amplifier, aLiNbO3 phase modulator, a chirped fiber Bragg grat-ing, polarization controls, and an electro-optic switchto control the input and output of the light in the loop.The input to the system was a light pulse with a lowrepetition rate obtained by a LiNbO3 amplitude modu-lator. We used 3 km of dispersion-shifted f iber (DSF;b2 � 1 ps2�km for l � 1550 nm). The purpose of us-ing the filter was to minimize the possibility of the sys-tem’s lasing and consisted of a circulator along with achirped fiber Bragg grating. The electro-optic switchallowed the loop to be opened or closed; when the loopwas open, the input pulse entered the loop and the cir-culating pulse exited the loop, and when the loop wasclosed there was a broad pulse circulating in it.
For high dispersion (large g) localization occursalmost immediately, so to observe the evolution asopposed to the long-term behavior of localization weoperated near the resonance regime, where g ø 1.The total dispersion of one round trip of the loop isapproximated as b2z0 � 212 ps2, determined by thefiber and the chirp of the Bragg grating, leaving gdependent only on the frequency. We require thatthe phase modulation be synchronized to the phaseof the pulses propagating in the loop. Otherwise wereceive destructive interference among the differentmodes, so no localization of the spectrum will occur.In addition, operation at modulation frequencies thatcorrespond to the Talbot length or to fractions of theTalbot length will result not in localization but ratherin good-quality pulses after mode locking is achieved.5
In Figs. 2 and 3 we present experimental results anda numerical simulation for the evolution of the localiza-tion. In Fig. 2 spectra can be seen for different num-bers of kicks, where f � 4.5 GHz and g � 0.0048. Itcan be seen that there is spectral broadening for lower
kicks and that for higher kicks the spectral width re-mains the same, whereas the different sidebands havedifferent intensities, which we observed to be peri-odic. The spectral envelope is not exponential, as isexpected for localization behavior, because of opera-tion near resonance.6,7 At resonance the dispersion iseffectively eliminated, leaving only modulation, whichcauses broadening of the spectrum.
Figure 3 shows the simulation and the experi-mental results, where the spectral width for each kickwas calculated with the average standard deviationand then represented as a function of the number ofkicks. The experimental results versus numericalsimulation of the localization buildup can be observedfor f � 4.5 GHz and f � 7 GHz (where g � 0.0048,and g � 0.011 respectively), where there is diffusivebroadening and then confinement. The localization
Fig. 1. Schematic of experimental system consisting of op-tical f iber, LiNbO3 phase and amplitude modulators, polar-ization controllers (PCs), a circulator, a fiber Bragg grating,erbium-doped f iber amplifiers (EDFAs), a tunable-diodelaser, and an electro-optic switch.
Fig. 2. Spectra obtained from (a) the experiment and (b) anumerical simulation for various numbers of kicks �N�,showing localization in frequency for L � 3 km DSF, wheref � 4.5 GHz and g � 0.0048. The axis is the power [dBm]versus wavelength [nm] in all cases.
Fig. 3. Evolution of localization for L � 3 km DSF, where(a) f � 4.5 GHz, g � 0.0048 and (b) f � 7 GHz, g � 0.011;�, experimental values; solid curves, numerical simulation.
Fig. 4. Experimental results for (a) the number of side-bands and (b) the number of kicks for the transition to lo-calization, both as functions of
p1�g; (c) spectral width as
a function of modulation amplitude, before localization, forf � 4 GHz (g � 0.0038). In all cases, L � 3 km DSF.
is characterized by oscillations about some averagespectral width. It can be seen that localization occurssooner for larger g. The agreement between theexperimental and the theoretical results is very good.
To verify the linear relations presented in expres-sions (7) and (8) we performed a series of experimentsin which the cavity length and the modulation am-plitude remained constant. The only change was inthe modulation frequency, which varies the value of g.The frequencies were varied from 3 to 8 GHz, in stepsof 0.5 GHz, except at 5 GHz. For each g, the spec-trum was measured for every kick as long as measure-ment was possible (until lasing of the system or lossof synchronization). The results calculated from theexperimental data are presented in Fig. 4. The num-ber of sidebands was calculated as Dn � B�V, whereB is the average spectral width of the localization. Itcan be seen [Fig. 4(a)] that there is good agreement be-tween the theory and the experiment; the experimentalinaccuracies for small g can be explained as being dueto an insuff icient number of kicks (spectral widths) be-ing available for averaging. As for large g, the linearrelationship is expected to fail (assumptions made areno longer valid), as can be observed in the deviationfrom linearity as
p1�g goes to zero.
In considering the relationship between the numberof kicks necessary for localization to
p1�g [Fig. 4(b)],
the criteria that we used was selection of the kickfound by intersection between the spectral broadeningregion and the average spectral width, defined above.This average width represents the long-term localiza-tion behavior, meaning that overall localization actu-ally occurs when this average is reached and that anyadditional spectral broadening is due to oscillations.Here, also, the averaging is inaccurate.
It can be seen that there is good correlation betweenthe results and the theory. However, note that thelinear line does not intersect the zero; there is a shiftof two kicks. This shows that the criterion used is notexact.
It can be deduced from Fig. 4(c) that Dn should alsobe linear with
pA for a specific g. To verify this ex-
perimentally, g and the number of kicks must be keptconstant and the parameter varied is A, the modula-
tion amplitude. Verifying this relationship proved tobe impossible in the experiment because of the large os-cillations about the average spectral width, which arecharacteristic of the localization behavior. These os-cillations depend on g but also on A, thus producingtoo many f luctuations in the number of sidebands.
The dependence of Dn onpA in the localization
regime [relation (7)] could not be verified experimen-tally because of the large oscillations that depend onA. However, before localization occurs, we achievespectral broadening without oscillations; thus the de-pendence of n on A is linear according to the classicaltheory. For a small number of kicks, the assumptionsmade in approximations (4) are valid, and it can beseen that the angular momentum is proportional to k
and that the spectral width is proportional to A:
2gDn � k ) B � Vk � A . (9)
This linear relationship was verified in the experi-ment, as shown in Fig. 4(c): The number of kicks(loops) was 10 (before localization occurred), the fiberwas the same as before, f � 4 GHz, and g � 0.0038.The linear relationship is clear; the slight deviationsare a result of the instability of the experimentalsystem and the small shift of the broadening behavioras a function of A. The linear relationship can beseen to fail for large A, where it cannot be assumedthat we are in the region before localization.
In conclusion, we have presented experimental veri-fication of the evolution of localization in optical f ibersand shown additional behavior for weak dispersion.
This research was partially supported by the Di-vision for Research Funds of the Israel Ministry ofScience and by the Fund for Promotion of Researchat the Technion. B. Fischer’s e-mail address [email protected].
References
1. B. Fischer, A. Rosen, and S. Fishman, Opt. Lett. 24,1463 (1999).
2. P. W. Anderson, Phys. Rev. 109, 1492 (1958).3. For reviews see D. J. Thouless, in Critical Phenomena,
Random Systems, Gauge Theories, K. Osterwalder andR. Stora, eds., Proceedings of the Les-Houches summerschool (North-Holland, Amsterdam, 1986), p. 681.
4. B. Fischer, B. Vodonos, S. Atkins, and A. Bekker, Opt.Lett. 27, 1061 (2002).
5. B. Fischer, B. Vodonos, S. Atkins, and A. Bekker, Opt.Lett. 25, 728 (2000).
6. B. Fischer, A. Rosen, A. Bekker, and S. Fishman, Phys.Rev. E 61, R4694 (2000).
7. A. Rosen, B. Fischer, A. Bekker, and S. Fishman, J. Opt.Soc. Am. B 17, 1579 (2000).
