1880 J. Opt. Soc. Am. B/Vol. 20, No. 9 /September 2003 Zhang et al.

All-optical reshaping of light pulsesby using x (2) media

Kuanshou Zhang,* Laurent Longchambon, Thomas Coudreau,† and Claude Fabre

Laboratoire Kastler Brossel, Universite Pierre et Marie Curie, Campus Jussieu, Case 74,75252 Paris Cedex 05, France

Received February 28, 2003; revised manuscript received May 9, 2003

We have developed a new method based on two cavities containing x (2) media to reshape optical pulses by anall-optical technique. The system is entirely passive; i.e., all the energy is brought by the incoming pulse anduses two successive optical cavities with independent thresholds. The output pulse is close to a rectangularshape. We show that this technique could be extended to high bit rates and telecommunication wavelength byusing very small cavities containing current nonlinear materials. © 2003 Optical Society of America

OCIS codes: 190.4410, 190.4970, 320.5540.

1. INTRODUCTIONIn optical telecommunications, multiple amplification andattenuation of information-carrying light pulses leads toan amplification of noise that quickly deteriorates thepulse shape and therefore increases the bit error rate. Toovercome such a signal distortion, one can use an activetechnique of pulse regeneration, which is usually per-formed by optoelectronic techniques. The so-called 3Rregeneration implies a reamplification, reshaping, andretiming of the pulses. An all-optical method to performthese different regeneration functions would be poten-tially faster and with broader bandwidths. Severalmethods, based on third-order nonlinearities or nonlinearamplifiers,1,2 have been proposed so far and are activelybeing studied to implement this function.

Second-order nonlinearities have not so far been stud-ied to implement regeneration functions, in spite of thehigh nonlinear effects that they are likely to produce evenat low input powers and in spite of their intrinsic ul-trashort response time. We propose here a passive andefficient method, which uses two successive optical cavi-ties containing nonlinear x (2) crystals, for reshaping opti-cal pulses. We give the results of a first experiment,which show that our proposed scheme is actually able toreshape low-power light pulses, performed at 1.06 mmand at low bit rates. We also discuss the potentialities ofthe extension of our technique to real optical telecommu-nication conditions.

Our scheme consists of two nonlinear optical cavitiesthat have input–output characteristics with a thresholdbehavior and lead to a reshaping of the input pulse. Thefirst cavity has a minimum threshold: As in a laser or anoptical parametric oscillator (OPO), no significant powerexits the cavity below a certain value, Pthreshold

low of thepump power. As a result, below Pthreshold

low the transmittedpower is zero. The second cavity has a maximum thresh-old Pthreshold

high : An input signal with a power abovePthreshold

high is transmitted with a constant value equal toPthreshold

high . The total transfer function of the two opticaldevices put in series is thus a steep Heaviside step func-

0740-3224/2003/091880-07$15.00 ©

tion: With an incident distorted pulse, the output pulseis close to a square pulse.

We begin in Section 2 by giving the theoretical expres-sions of the transfer functions of the two cavities. Sec-tion 3 is devoted to the description of the experimentalsetup, and in Section 4 we describe the experimental re-sults obtained at the wavelength of 1.06 mm. Finally, inSection 5 we analyze the possibility of applying theseideas to pulses having a wavelength of 1.5 mm and to veryhigh bit rates.

2. THEORETICAL ANALYSISA. First Optical System: Intracavity Second-HarmonicGeneration–Optical Parametric OscillatorThe first cavity contains a type-II x (2) crystal and is illu-minated by a beam at frequency v0 , polarized linearly at145° of the crystal axes. The cavity is assumed to beresonant, at the same time, for the ordinary and extraor-dinary waves at the input frequency v0 and for its secondharmonic 2v0 and to have a single coupling mirror atboth frequencies. The following well-known equations3

describe the nonlinear resonant coupling between the in-tracavity amplitudes of the three interacting fields in thesteady-state regime and in the case of exact triple reso-nance:

~1 2 r !A1 5 gA0A2* 1 t1

A2A in, (1)

~1 2 r !A2 5 gA0A1* 1 t1

A2A in, (2)

~1 2 r0!A0 5 2gA1A2 , (3)

where A1 and A2 are the intracavity ordinary and ex-traordinary wave envelope amplitudes at the fundamen-tal frequency v0 , A0 is the intracavity second-harmonicamplitude, A in is the input field amplitude, r and t are theamplitude reflection and transmission coefficients of the

2003 Optical Society of America

Zhang et al. Vol. 20, No. 9 /September 2003 /J. Opt. Soc. Am. B 1881

cavity’s coupling mirror at the fundamental frequency,and r0 and t0 are the same quantities at the second-harmonic frequency. These equations can be simplified ifone introduces the rotated basis,

A6 51

A2~A1 6 A2!,

which represents the amplitude of the fundamental fieldalong the two directions at 645° from the crystal’s ex-traordinary and ordinary directions. The steady-stateequations can now be written in a decoupled form:

~1 2 r !A1 5 2gA0A1* 1 tA1in, (4)

~1 2 r !A2 5 1gA0A2* , (5)

~1 2 r0!A0 5 2g

2~A1

2 2 A22!. (6)

These equations can be readily solved.4 One finds thatthere are two different regimes, depending on the inputintensity P in 5 uA inu2, separated by a ‘‘pitchfork bifurca-tion’’ occurring at an intensity Pthreshold

low (Ref. 5) (see Fig.1):

1. For P in , Pthresholdlow , one has

A2 5 0, (7)

F1 2 r 2g2

2~1 2 r0!uA1u2GA1 5 tA1

in, (8)

A0 5 2g

2~1 2 r0!A1

2.

(9)

This is the usual behavior of an intracavity doublyresonant frequency doubler, in which more and morepump power is converted into the second harmonic as thepump power increases.

2. For P in . Pthresholdlow , one has A2 Þ 0 and

Fig. 1. Intracavity pump power along the 1 polarization, uA1u(solid curve), intracavity pump power along the 2 polarization,uA2u2 (dotted line), and intracavity second-harmonic power, uA0u2

(dashed–dotted curve), for the first cavity as a function of the in-put pump power.

A1 5t

2~1 2 r !A1

in, (10)

A22 5

1 1 r

4~1 2 r !~A1

in!2 22~1 2 r !~1 2 r0!

g2,

(11)

A0 51 2 r

g. (12)

The intracavity second-harmonic power is sufficient togenerate an oscillation on the A2 mode. The behavior ofthe system is similar to an OPO: The second-harmonicfield is now clamped to its threshold value while the A2

power increases linearly with the input power. If oneuses as the output of the device the output field orthogo-nally polarized to the input field, and at the same fre-quency, A2

out 5 tA2 , one will have a zero output belowthreshold and a value rapidly growing with the inputabove threshold, which is the behavior that we need inour reshaping device. It is worthy to note that thepresent threshold Pthreshold

low , equal to T2T0/2 g2 for smallintensity transmission coefficients at both frequencies,Tt2 and T0 5 t0

2, is the threshold of a triply resonantOPO, which can be in the milliwatt range in optimizedconditions.6

Two difficulties must be solved to operate this device.First, we have assumed perfect triple resonance to solvethe equations. This is not an easy condition to fulfill, asthe fields A0 , A1 , and A2 see three different indices in thetype II nonlinear crystal, corresponding to three differentoptical paths. As all the frequencies are fixed, we mustthen adjust precisely three parameters of the system tofulfill the triple-resonance conditions. Two parameters,the cavity length and the crystal temperature, are easilycontrollable, but a third one is needed. This last onecould be the setting of a variable birefringent system inthe cavity. We chose another approach, which is to inserta quarter-wave plate in the linear cavity, adjusted so thatit induces a 90° rotation of the polarization plane for thefundamental wave when it is crossed twice by the beaminside the cavity and does not change the second-harmonic field polarization. In this configuration, forany crystal temperature, the eigenmodes of the cavity areautomatically (A1 , A2). We need then to adjust onlytwo parameters to ensure the triple-resonance condition:the cavity length and the crystal temperature. Second,we have assumed a degenerate operation for the OPO.This is one possibility for the system. But a nondegener-ate operation of the OPO can also take place, with signaland idler modes oscillating at different frequencies v1and v2 such that v1 1 v2 5 v0 . This regime of cascad-ing has been theoretically studied7,8 and observed.9 Ac-tually, the system will oscillate in the regime (degenerateor nondegenerate) that has the lowest threshold. We willsee in Subsection 4.A that one can find conditions forwhich it is indeed possible to observe the stable degener-ate operation that is needed for reshaping without fre-quency change, which is necessary for practical applica-tions. A forthcoming publication will give more preciseinsight into the operation of this kind of optical device

1882 J. Opt. Soc. Am. B/Vol. 20, No. 9 /September 2003 Zhang et al.

that contains, at the same time, second-order nonlinearelements and birefringent elements.

B. Second Optical System: Nondegenerate OpticalParametric OscillatorThe second system consists of a standard, nondegenerate,triply resonant OPO. It is well known that, in a laserabove threshold, the gain is clamped to its thresholdvalue by the condition that the saturated gain must equalthe losses in the steady-state regime. The same behavioroccurs in an OPO, as can be seen very easily from itssteady-state equations, which are very similar to Eqs.(1)–(3), except that now the input power is put in the cav-ity on the high-frequency mode:

~1 2 rs!As 5 gApAi* , (13)

~1 2 ri!Ai 5 gApAs* , (14)

@1 2 ~rp!2#Ap 5 2gAsAi 1 tpApin, (15)

where As and Ai are the intracavity signal and idler modeamplitudes (these two modes oscillating at different fre-quencies); Ap is the intracavity pump amplitude; Ap

in isthe input pump amplitude; and rs , ri , and rp are the am-plitude reflection coefficients of the coupling mirror at thesignal, idler, and pump frequencies. For the pump field,one has assumed that the cavity has two identical cou-pling mirrors, of amplitude reflection and transmissioncoefficients rp and tp , one used as the input and the sec-ond as the output of our optical device.

Solving these very well-known equations,3 one findsthat there are two different regimes, depending on thepump intensity Ppump 5 uAp

inu2, separated by a thresholdPthreshold

high (see Fig. 2):

1. For Ppump , Pthresholdhigh 5 @tp

2(1 2 rs)(1 2 ri)#/g2,one has As 5 Ai 5 0. No parametric oscillation takes

Fig. 2. Transmitted power I transm by the second cavity as a func-tion of the input pump power I in .

Fig. 3. Transmitted power through the total system as a func-tion of the input pump power.

place. The OPO cavity is a pure passive, resonant,Fabry–Perot cavity with input and output mirrors ofequal transmission. Its transmission is therefore 1, andthe device is exactly transparent.

2. For Ppump . Pthresholdhigh , one has As Þ 0 and As Þ 0,

which occurs only when uApu2 5 @(1 2 rs)(1 2 ri)#/g2,whatever the input pump field is: The intracavity pumppower is therefore clamped to a value independent of theinput, and the output pump field is then clamped to itsvalue at threshold. The excess power brought by thepump is then transferred to the signal and idler beams.

Let us stress that here also the pump thresholdPthreshold

high is the threshold of a triply resonant OPO,which can be very low. But now the system is much sim-pler to operate than the previous one, as the frequenciesof the signal and idler modes are not a priori given (exceptthat their sum is equal to the pump frequency). There isone more degree of freedom than in the first device, andthe cavity length and crystal temperature are the onlytwo parameters that need to be adjusted to get the triplyresonant condition.

C. Total SystemIf the two previously described devices are put in series,one obtains overall input–output characteristics that aresketched in Fig. 3. This curve is close to the one we needfor pulse reshaping, except for the intermediate regionPthreshold

low , Ppump , Pthresholdhigh , for which the response is

linear. The best reshaping will be obtained when thiscentral part is as steep as possible.

3. EXPERIMENTAL SETUPThe input beam is produced by a Nd:YAG laser at 1.06mm (Lightwave 126-1064-700). To produce a light pulse,we use an acousto-optic modulator to modulate the inten-sity of the transmitted beam (modulation frequency '3kHz). To mimic the high-frequency noise existing on thepulse, we superimpose on the envelope a high-frequencymodulation ('100 kHz) and simulate the high-frequencynoise [Fig. 4(a)].

A. Intracavity Second-Harmonic Generation–OpticalParametric OscillatorThe characteristics of the two cavities are summarized inTable 1. The first cavity has highly reflecting mirrors forboth 1.06 and 0.53 mm. The pump at 1.06 mm is sent ata 45° angle with respect to the crystal axis. The outputbeam is separated from the input by use of a polarizingbeam splitter in front of the optical device. At low inputpower, it is first converted into green light by a standarddoubling process. When the intracavity green power issufficient, parametric downconversion occurs, whichtransfers the power back to the pump wavelength but onthe orthogonal polarization. However, this system has ahigh threshold, above 600 mW in our experimental condi-tions. To reduce this threshold, we have added a quarter-wave plate at 1.06 mm inside the cavity. In that case, weobserved a much more efficient intracavity second-harmonic generation, and a parametric oscillation thresh-old could be as low as 300 mW. This fairly large thresh-old is mainly due to the fact that the reflection coefficients

Zhang et al. Vol. 20, No. 9 /September 2003 /J. Opt. Soc. Am. B 1883

were not optimized (in particular, the green reflection co-efficients should be maximum, which is not the case).

B. Nondegenerate Optical Parametric OscillationThis system has been described extensively in a previouspublication.6,10 Let us briefly mention here its main fea-tures. A periodically poled lithium niobate (PPLN) crys-tal with an inversion period of 30 mm is placed inside asymmetric cavity that has a large finesse (over 200) forthe wavelengths around 2 mm and a lower finesse at 1 mm(around 40). The crystal temperature is kept close to thedegeneracy temperature (parametric conversion between1.06 and 2.12 mm), so that parametric oscillation occurredfor all cavity lengths within a pump resonance because ofthe overlap between the oscillation ranges of nearby sig-nal and idler pairs of modes of wavelengths close to 2mm.6 The threshold was of the order of a few milliwatts.This threshold was chosen to obtain a transmitted field asclose as possible to a square function: Its value is muchlower than the maximum intensity at the output of thefirst cavity to ensure a steep transmission function. Let

Fig. 4. Experimental results for cavity 1: (a) input power as afunction of time and (b) output power as a function of time.

us also mention that in our experiment the input and out-put mirrors for the 1.06-mm beam were not of equal trans-mission. As a result, the power of the transmitted beamwas very small compared with the input one and notequal, as in the theoretical approach of Section 2. This isdue to the fact that we used, instead of an optimized cav-ity, an existing one as used in the experiment described inRef 6.

4. EXPERIMENTAL RESULTSA. Intracavity Second-Harmonic Generation–OpticalParametric OscillatorFigure 4 shows the input and output intensities of thefirst device as a function of time. The maximum inputpower on this cavity was 350 mW in order to be above thethreshold Pthreshold

low mentioned above. It can be seen inFig. 5 that the effect of the cavity is close to a perfecthigh-pass filter, as far as intensities are concerned: Pow-ers below Pthreshold

low are not transmitted, whereas thoseabove this value are linearly transmitted.

A crucial point of the device is that the output beam (or-thogonally polarized reflected pump beam) is at the samefrequency as the input beam. In a first experiment welooked at the interference pattern between the input andthe output beams and observed that there were no fringeswhen the second-harmonic beam was not TEM00 . Clearfringes appeared only when, by a careful alignment andcrystal temperature tuning, the green output was TEM00 .The frequency-degenerate operation was corroborated bya second experiment, in which we used a confocal Fabry–Perot cavity to monitor the frequency of the output beam,orthogonally polarized with respect to the input beam.The confocal Fabry–Perot cavity is formed by two curvedmirrors with radius of 50 mm and reflectivity of 95% for1.064 mm so that the free spectral range is 1.5 GHz. Fig-ure 6 shows the output beam and the pump beam trans-mitted intensities through the confocal Fabry–Perot cav-ity when scanning the analysis cavity at 60 Hz andscanning the self-pumped OPO at 650 Hz. This figureshows that, when the system is not perfectly tuned up, itoscillates in a nondegenerate regime and generates side-bands around the pump frequency (Fig. 6, left), whereasone can find experimental conditions for which the down-converted output has the same frequency as the pump,within the experimental uncertainties (Fig. 6, right).

B. Nondegenerate Optical Parametric OscillationWe have plotted in Fig. 7 the relevant intensities for thesecond cavity as a function of time: Figure 7(a) shows

Table 1. Characteristics of the Cavitiesa

Cavity CrystalCavity Length

(mm)

Input Mirror End Mirror

Rc

Rl

(%)R2l

(%) Rc

Rl

(%)R2l

(%)

1l 5 532 nm

KTP 50 50 .99.9 95 50 90 .99.9

2l 5 1064 nm

PPLN 65 30 87 99.8 30 99.8 99

a Rc radius of curvature in millimeters; R(2)l reflection coefficient at (2)l.

1884 J. Opt. Soc. Am. B/Vol. 20, No. 9 /September 2003 Zhang et al.

the incident intensity, and Fig. 7(b) shows the transmit-ted intensity. The output intensity displays a very clearclamping of the power above the threshold Pthreshold

high , at avalue equal to the transmitted pump power at threshold,typically a few milliwatts. In Fig. 8 the low-pass filter(for intensities) effect of this cavity is shown via its trans-fer function.

The peak that one observes at the beginning of the flattop of the transmitted intensity is due to a dynamical ef-

Fig. 5. Transfer function of the first cavity.

Fig. 6. Downconverted beam and pump beam transmitted in-tensities through the confocal Fabry–Perot cavity as a function ofthe confocal Fabry–Perot cavity length. The sharp and lightpeaks are the transmission of the downconverted output, and theblunt and dark peaks are the transmission of the pump beam.Left, the nondegenerate case; right, the degenerate case.

Fig. 7. Experimental results for cavity 1: (a) input power as afunction of time and (b) output power as a function of time.

fect of delayed bifurcation that has already been observedin OPOs when their pump power is scanned with time.11

As the incoming pump power increases above threshold,the onset of the oscillation is delayed by a time intervalthat is larger than the characteristic evolution times ofthe cavity.

C. Total Cascaded SystemThe complete experimental setup is shown in Fig. 9. Thebeam reflected by the intracavity second-harmonicgeneration–OPO is separated from the input beam by apolarizing beam splitter and sent to the nondegenerateOPO. The output beam intensity and the experimentaltransfer function for the total system are shown in Fig.10. One observes that the time dependence of the outputbeam intensity is close to a rectangular pulse and, accord-ingly, that the transfer function shows the desired behav-ior with a response very close to a step function.

5. POSSIBLE IMPLEMENTATION TO VERYHIGH BIT RATES AT 1.5 mmIn all-optical pulse-reshaping systems, two parametersplay an important role: the operating power and the re-

Fig. 8. Transfer function of the second cavity.

Fig. 9. Experimental setup with the two cavities in series show-ing the intracavity second-harmonic generation–OPO (low-passfilter) and nondegenerate OPO (high-pass filter).

Fig. 10. Total transfer function of the system.

Zhang et al. Vol. 20, No. 9 /September 2003 /J. Opt. Soc. Am. B 1885

sponse time. In the present demonstration experiment,the incident power on the first cavity is approximately350 mW, which is beyond the current powers of opticaltelecommunication pulses. This is mainly due to the factthat the first cavity is not optimized. Thresholds in therange of tens of milliwatts could be obtained by usinghigh-quality materials and coatings and optimizing thecavity parameters. The response time of our present sys-tem is rather long, of the order of 200 ns: It is related tothe cavity build-up time of the two cavities that are longcavities with rather high finesses. The most importantquestion for the future of our proposed technique iswhether our system can be modified and optimized to beable to reach very high bit rates, namely, 40 Gbit/s, and atthe telecommunication wavelength of 1.5 mm. We willaddress this question in the present section.

Several nonlinear materials are compatible with thetelecommunication wavelength, namely, gallium ars-enide, aluminum gallium arsenide, or zinc selenide.These materials possess very large nonlinearities, aslarge as 120 pm/V in the case of gallium arsenide, butthey cannot be phase matched by use of birefringence.

The rising time issue is the most difficult to solve. Toreduce this time, one must use small cavities and there-fore use shorter nonlinear crystals or lower reflectivitiesor both for the mirrors, two methods that have the detri-mental effect of increasing the thresholds at the sametime.

The threshold of a triply resonant OPO is given by

Pthreshold 5T0T1T2

64x2Lc2

, (16)

where Ti denotes the transmission coefficient of the cav-ity for mode i, Lc is the crystal length, and x is the non-linear coupling strength depending on the nonlinearity,geometry, and optical indices through the relation

x 5 deff

w0w1w2

w02w1

2 1 w02w2

2 1 w12w2

2 S \v0v1v2

pe0c3n0n1n2D 1/2

,

(17)

where deff is the crystal nonlinearity in meters per volt.wi is the waist size, v i is the pulsation, and ni are the in-dices of refraction of the three interacting modes. As-suming that the system is operated close to frequency de-generacy, that is, v1,2 5 v0/2, and that the pump, signal,and idler indices and Rayleigh ranges are identical, thenonlinear coupling strength can be expressed in terms ofwavelength and Rayleigh length as

S xexp

xD 2

5 F ~deff!exp

deffG2 zR

~zR!expF l0

~l0!expG4S n

nexpD 2

, (18)

where the index exp denotes experimental values and l0 ,(l0)exp denote the pump wavelength in the vacuum; thesetwo wavelengths are different because our experimentwas performed with a (l0)exp 5 532-nm pump, whereas atelecom implementation would require l0 ' 1550 nm.

If we also assume that the cavity length is equal to thecrystal length, one can use the expression for the finesseas a function of the cavity rise time:

F 5pct

2nLc. (19)

We now obtain the following formula for the ratio of thethresholds:

Pthreshold

Pexp5 ~F0F1F2!expS 2n

pctD 3F ~deff!exp

deffG2 zR

~zR!exp

3 F l0

~l0!expG4S n

nexpD 2

Lexp2Lc . (20)

One obtains a counterintuitive result, namely, that thethreshold becomes lower as the crystal/cavity length di-minishes. This is due to the fact that the finesse is fixedby the value of the cavity length in order to keep the cav-ity lifetime t constant. A small cavity is thus desirable toobtain a low threshold as well as a short rise time. Theexperimental values correspond to an optimized cavitycontaining a KTP crystal pumped by a frequency-doubledNd:YAG laser at 532 nm generating signal and idleraround 1064 nm, yielding Pexp 5 20 mW with nexp ' 1.8,(deff)exp 5 3 pm/V, Lexp 5 1 cm, and (F0,1,2)exp 5 115.When one sets a cavity/crystal length of 10 mm, a risetime t of 10 ps (compatible with an operating bit rate of 40Gbit/s), and a Rayleigh length of 10 mm compatible withthe crystal length, one obtains a finesse F ' 140 andthreshold Pthreshold ' 110 mW with the values corre-sponding to gallium arsenide (n 5 3.4 and deff5 120 pm/V). This value of the threshold is compatiblewith the values used in optical telecommunications. Thevery short length of the crystal is also an advantage asone can have a crystal length equal to or shorter than thecoherence length of the material, which partly removesthe problem of phase matching. It is important to notethat the variation of the threshold with t is very fast sothat a small increase of the rise time up to 15 ps leads toa similar threshold by use of lithium niobate (n 5 2.2 anddeff 5 20 pm/V).

We have shown in this section that the realization ofvery short rise-time systems for all-optical reshaping iswithin reach by developing current techniques and usingavailable materials. However, because of the presence ofthe resonance cavity, this system is able to reshape onlyoptical pulses centered around a given wavelength.

6. CONCLUSIONWe have demonstrated that optical cavities containingx (2) media can be used for all-optical passive reshaping ofoptical pulses. We have experimentally obtained reshap-ing with a threshold compatible with optical powerspropagating in optical fibers. We have shown that veryfast response times can be realized using very shortmonolithic cavities made of high nonlinearity crystals.

ACKNOWLEDGMENTSLaboratoire Kastler Brossel, of the Ecole Normale Super-ieure and the Universite Pierre et Marie Curie, is associ-ated with the Center National de la Recherche Scienti-

1886 J. Opt. Soc. Am. B/Vol. 20, No. 9 /September 2003 Zhang et al.

fique. This research was supported by France-Telecom(project CTI 98-9.003). The authors thank V. Berger forfruitful discussions.

T. Coudreau’s e-mail address is coudreau@spectro.jussieu.fr.

*On leave from the Key Laboratory for Quantum Op-tics, Shanxi University, Taiyuan, China.

†Also with the Pole Materiaux et Phenomenes Quan-tiques, FR, Centre National de la Recherche Scientifique2437.

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