Nonsinusoidal phase modulations for high-power laser performance control: stimulated Brillouin...

Nonsinusoidal phase modulations for high-powerlaser performance control: stimulated Brillouin

scattering and FM-to-AM conversion

Steve Hocquet,1,* Denis Penninckx,1 Jean-François Gleyze,1

Claude Gouédard,2 and Yves Jaouën3

1CEA/CESTA, 15 avenue des Sablières, BP 2, 33114 Le Barp, France2CEA/DIF, BP 12, 91680 Bruyères-le-Châtel, France

3Institut Telecom/Telecom Paris Tech, CNRS UMR 8141, 46 rue Barrault, 75634 Paris Cedex 13, France

*Corresponding author: [email protected]

Received 26 October 2009; revised 25 January 2010; accepted 26 January 2010;posted 27 January 2010 (Doc. ID 119012); published 23 February 2010

High-power lasers, such as the Laser MegaJoule (LMJ), have to be phase modulated to avoid stimulatedBrillouin scattering (SBS) thatmay strongly damage optics at the end of the laser chain. Current spectralbroadening on LMJ is performed with a sinusoidal phase modulation. This pure sinusoidal phase mod-ulation leads to inhomogeneous spectral power densities (SPD). Thus, for a same SBS power threshold,the sinusoidal phase-modulated spectrum has to be larger than the equivalent ideal SPD with isoener-getic peaks. We present in this paper a technique to generate energy-balanced Dirac peaks spectrathanks to nonsinusoidal phase modulations. Thus, we can build a narrower spectrum with a nonsinu-soidal phasemodulation that has the same SBS threshold as a sinusoidal phasemodulation, and we showthat FM-to-AM conversion can be strongly reduced, which is of great interest for LMJ laser performance,with reductions up to 40%. © 2010 Optical Society of America

OCIS codes: 060.5060, 070.2615, 140.3518.

1. Introduction

The Laser MegaJoule (LMJ) in France and the Na-tional Ignition Facility (NIF) in the United Statesare high-power laser facilities designed to achieveinertial confinement fusion of a deuterium tritium(D-T) capsule [1,2]. Such energetic lasers (up to2MJ in a few nanoseconds) need broadened spectrafor two reasons: to avoid stimulated Brillouin scat-tering (SBS) and to “smooth” the final focal spot thatreaches the D-T capsule. First, without spectralbroadening, spectral power density (SPD) of ignitionlaser pulses would exceed the transverse Brillouinthreshold in optics at the end of the amplificationchain. Such SBS would strongly damage these optics[3]. Spectral broadening spreads energy to different

wavelengths (frequencies) to reduce the SPD maxi-mum level below the SBS threshold. Second, focalspots on the LMJ require a high spatial homogeneity.As a matter of fact, the focal spot of the laser beam isshaped by a phase plate and looks like a speckle pat-tern with hot spots. The intensity is time averagedthrough a spectral broadening and a dispersion sys-tem (grating) that moves the hot spots during thepulse. This scheme is called smoothing by spectraldispersion [4].

The required spectral broadening on LMJ is gen-erated by a phase modulation [i.e., frequency modu-lation (FM)] in the fibered part of the front end.Detrimental consequences of phase modulation ap-pear during propagation through the laser chain.Ideally, a pure phase modulation does not changebeam intensity. However, propagation through thedifferent components slightly filters the optical

0003-6935/10/071104-12$15.00/0© 2010 Optical Society of America

1104 APPLIED OPTICS / Vol. 49, No. 7 / 1 March 2010

spectrum. The consequences of these filters are anenergy loss and a partial conversion of FM into inten-sity modulation [i.e., amplitude modulation (AM)]:this is FM-to-AM conversion. For fusion experimentsand laser–plasma interaction, the control of thetemporal shape is a key point and, thus, FM-to-AMconversion needs to be reduced [5].In this paper, we only consider the phase modula-

tion used to avoid SBS (the so-called anti-Brillouinfunction). However, the techniques mentioned inthe following can as well be used to perform thesmoothing function. The examples of sinusoidalphase modulation presented in Fig. 1 show thatthe optical spectrum is made of regularly spacedDirac peaks but energies in these peaks are notequal, except for one particular case with only threepeaks [Fig. 1(a)]. Thus, to control the SPD maximumto avoid reaching the SBS threshold, we need to dra-matically enlarge the spectrum. Total bandwidth isconsiderably larger than an equivalent homogeneousdistribution of energy and this is a drawback, consid-ering FM-to-AM conversion.In the following, we describe a method to generate

more homogeneous spectral power distributionsusing nonsinusoidal phase modulations (Section 2).Then, we focus on the anti-Brillouin function andFM-to-AM conversion. We show that we can get anarrower spectrum compared to sinusoidal phasemodulation that can achieve the same anti-Brillouinperformance on LMJ (Section 3). Also, we experimen-tally observe that the impact on FM-to-AM conversi-on is significant on amplitude filters representativeof propagation filtering in the LMJ chain (Section 4).

2. Principle, Optimization and Experimental resultsof Nonsinusoidal Phase Modulation

In this section, we show limitations of sinusoidalphasemodulations andwe propose to use nonsinusoi-dal phase modulations to satisfy the same anti-Brillouin requirements on LMJ. First, we justifythe current parameters of the sinusoidal phase mod-

ulation used to avoid SBS onLMJ. Second,we presentthe principles of nonsinusoidal phase modulationsand a strategy for a user-friendly implementation.Finally, we measure anti-Brillouin function perfor-mance to compare both phase modulations.

A. Sinusoidal Phase Modulation

On LMJ, the anti-Brillouin function is performedthanks to a sinusoidal phase modulation that is char-acterized by two parameters: the frequency f m andthe modulation index m. The optical field is given by

aðtÞ ¼ffiffiffiffiffiffiffiffiIðtÞ

p· expði½2πf 0tþm sinð2πf mtÞ�Þ; ð1Þ

where IðtÞ is the pulse intensity, and f 0 is the opticalfrequency of the laser. The contribution of the phasemodulation, amod, can be expressed by

amodðtÞ ¼ expðim sinð2πf mtÞÞ: ð2Þ

So the optical spectrum of the phase-modulated sig-nal is the convolution of the initial temporal shapespectrum and the phase modulation spectrum.

On LMJ, the temporal pulse shape bandwidth isabout 0:5GHz (>95% of the energy). Thus, in the fol-lowing, the optical spectrum is given in the base bandand is deduced from Eq. (2). amodðtÞ is a periodic sig-nal and the corresponding optical spectrum is com-posed of uniformly separated f m spaced Dirac peaks:

~amodðf Þ ¼Xþ∞

n¼−∞

JnðmÞδðf − nfmÞ; ð3Þ

where JnðmÞ is the nth Bessel function [6].Phase modulation spreads the pulse energy on dif-

ferent wavelengths. Energy in the spectral domain isgiven by the spectral power distribution defined as

Fig. 1. Different optical spectra obtained by sinusoidal phase modulation. (a) Energy-balanced spectra with three peaks form ¼ 1:43 rad(f m ¼ 2GHz). (b) Normalized spectral power of density of LMJ anti-Brillouin function sinusoidal phase modulation at 3ω: f m ¼ 2GHz andm ¼ 21 rad.

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SPDðf Þ ¼ j~amodðf Þj2 ¼Xþ∞

n¼−∞

jJnðmÞj2δðf − nfmÞ: ð4Þ

Without phase modulation, 100% of the energy is inthe central frequency. With a sinusoidal phase mod-ulation, energy in each peak is thus given by thefunctions jJnðmÞj2. These functions are odd for thevariable n and. thus, for any sinusoidal phase mod-ulation, the spectrum barycenter remains the lasercarrier frequency and, thus, no change of the carrierfrequency is required when phase modulation is seton and off.Note that, on LMJ, amplification of the laser signal

is generated in the IR domain (1053nm or 1ω) butlaser–target interaction is made in the UV (351nmor 3ω). Simple considerations on the frequency con-version system show that the optical field a3ω at 3ω islinked to the optical field a1ω at 1ω and a3ωðtÞ ∝a1ω

3ðtÞ. Considering Eq. (2), one can see that phasemodulation remains sinusoidal with frequency con-version, but the modulation index at 3ω is m3ω ¼3m1ω [7].

B. Anti-Brillouin Requirements

Considering the anti-Brillouin function, SBS can besuppressed in high-power lasers if energy is spreadon different wavelengths for the photons to excite in-dependently different Stokes modes. This conditionis satisfied if the frequency difference between Diracpeaks is larger than the considered FWHM Brillouinlinewidth of gain spectrumΔνB. For silica at 351nm,the transverse Brillouin linewidth is ΔνB ∼ 200MHz[8]. Thus, broadening is efficient if SPD peaks areseparated by more than ΔνB. On LMJ, the anti-Brillouin modulation frequency has been chosen atf m ¼ 2GHz (on NIF, f m ¼ 3GHz).For the LMJ frequency modulation, each of the

SPD frequency peaks excites one different Stokesmode. The only condition to avoid SBS is, then, thatall these SPD peaks have to be below the SBS powerthreshold. The current anti-Brillouin function has amodulation index at 3ω m3ω ¼ 21 rad. Therefore, inthe following, we consider that the anti-Brillouin

function is performed for 2GHz phase-modulatedsignals if the SPD’s most energetic peak is less than6% of the total energy at 3ω, as in the sinusoidal mod-ulation case [presented in Fig. 1(b)], by taking toaccount the LMJ requirement.

C. Nonsinusoidal Phase Modulations: Interestand Choices

Asmentioned inEq. 3 and the examples inFig. 1, SPDpeaks evolve with Bessel functions and do not carrythe same energy for sinusoidal phase modulations.It is not optimal if we want to minimize spectralbroadening. In the following, we consider the spec-trum bandwidth as the frequency range correspond-ing to 95% of the total energy. For sinusoidal phasemodulation, the bandwidth, Δf , is given by the Car-son rule that links the bandwidth to the phase mod-ulation parameters m and f m: Δf ¼ 2 · ðmþ 1Þ · f m[9]. For example, at 3ω, LMJ anti-Brillouin functionbroadening is 88GHz. Note that the ideal spectrumto achieve an anti-Brillouin function is a perfectly en-ergy-balanced SPD, which means with the same en-ergy in every peak. Thus, with only 17 peaks of lessthan 6% of the total energy, we have the same anti-Brillouin performance with a broadening of only32GHz, as shown in Fig. 2. However, this ideal spec-trum cannot be obtained with a pure phase modula-tion, as shown in Appendix A. We are looking forsolutions close to this ideal case and we want anotherphase modulation that keeps the anti-Brillouin func-tion with a narrower spectral bandwidth. The 2GHzspacedpeaks requirement is still valid, soweare look-ing for a 500ps periodic signal. As the signal is peri-odic, it can be developed in Fourier series as

aoptðtÞ ¼ exp�iX∞n¼1

mn sinð2πnfmtþ φnÞ�; ð5Þ

withmn as the differentmodulation indexes andφn asthe related phases.

However, in practice, the number of harmonicsis limited by phase modulator bandwidth (about40GHz). Therefore, the number of harmonics islimited to N:

Fig. 2. (Color online) SPD to achieve anti-Brillouin function. (a) Sine phasemodulation SPD at 3ω (m ¼ 21, f m ¼ 2GHz). (b) Ideal energy-balanced spectrum with same value of maximum of SPD (about 6% of the total energy).


aoptðtÞ ¼ exp�iXNn¼1

mn sinð2πnfmtþ φnÞ�: ð6Þ

The limited value of N prevents us from reachingperfectly energy-balanced spectra. However, non-sinusoidal phase modulation [or multisinusoidal-phase-locked phase modulation considering Eq. (6)]can provide better spectral shapes than sinusoidalphase modulation. In the following, we study spec-tral performance for different values of N.Phase values φn are important. Indeed, Eq. (6)

shows that φn values determine how the spectra fromeach harmonic interferes. In particular, nonsinusoi-dal phase-modulation SPDs may not be symmetricaland barycenter may not be the carrier frequency.Note that we can set φ1 ¼ 0 as a time referencechoice. Then, the number of freedom degrees is equalto 2N − 1.To quantify the spectrum quality, we simply con-

sider an error function εðSPDÞ that is the norm ofthe difference between the nonsinusoidal phase mod-ulation SPD and an ideal energy-balanced spectrum(SPDideal):

εðSPDÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXþ∞

k¼−∞

ðSPDðkfmÞ − SPDidealðkfmÞÞ2vuut : ð7Þ

The definition used in Eq. (7) is illustrated inFig. 3(a).Optimization is performed thanks to a gradient al-

gorithm with a random starting point in the (2N − 1)dimension solution space. For each value of N, thealgorithm has been running more than 100,000times. An adaptation of the Gerchberg–Saxton algo-rithm has also been tested: for each iteration, thetemporal signal is filtered to keep only the N first

harmonics [10]. It is faster (a few minutes insteadof hours), but error function values remain higherthan with the gradient algorithm. Figure 3(b) repre-sents error function optimization for different valuesof the number of harmonics N, considering an arbi-trary ideal spectrum of 25 peaks. ForN ¼ 15, the bestspectrum is very close to the ideal energy-balancedspectrum. However, one can see that important gainon the error function is already obtained for N ¼ 3,for which cost and complexity are still low (five pa-rameters). Furthermore, if we want to use an equiva-lent system to smooth the beam (i.e., FM f m ¼14:25GHz), only the first three harmonics are trans-mitted with a commercially available 40GHz phasemodulator.

Among all the spectra obtained with nonsinusoidalphase modulations, symmetrical SPDs are particu-larly interesting. Indeed, they avoid shifting the la-ser carrier frequency, considering that the amplifierlaser cavities are set for a particular wavelength. ForN ¼ 2, SPD is an odd function if φ2 ¼ π=2 [π]. In thecase of N ¼ 3, symmetrical positions are limited bythe phase choices and φ2 ¼ π=2 [π] and φ3 ¼ 0 [π].Demonstration of these assertions is given in Appen-dix B. Hopefully, these phase choices for symmetricalSPDs also correspond to minimum values of εðSPDÞ,as shown in Fig. 4 for the particular choice of a spec-trum with 25 peaks.

Nonsinusoidal phase modulation use for the anti-Brillouin function has been studied before, but it onlyappears in [11,12]. Both of them are related to theparticular operating point of the sinusoidal phasemodulation when m ≈ 1:4 rad. As presented inFig. 1(a), SPD form ¼ 1:43 rad is made of three equalpeaks. The principle in [11,12] is to add a secondphase modulation at the third harmonic with thesame value of m: convolution of spectra leads to aspectrum made of nine energy-balanced peaks. This

Fig. 3. (Color online) (a) Illustration for the definition of the error function ε used when we are looking for an ideal 25-peak energy-balanced spectrum. (b) Results of gradient algorithm optimization for nonsinusoidal phase modulation for different values of the numberof harmonics: the higher N is, the lower the error function is. However, most of the gain is obtained for N ¼ 3, which is simple enough forpractical realization.


idea is generalized to spectra composed of any powerof three peaks. However, as it appears in Fig. 5(a), theenergy is not totally concentrated in the three centralpeaks. The impact of the remaining energy (about10%) in the other peaks is important and limitsthe performance of such systems. Furthermore,[11,12] do not consider any phase choice betweenthe harmonics, which leads to systems that do notwork and are judged by the authors as not stable,which is logical with a lack of phase control and aspresented in Fig. 6. Indeed, [11] uses different fre-quency oscillators to generate the harmonics and[12] uses one generator, but does not control therelative phases that are affected by amplifiers, at-tenuators, low-pass filters. or time delay before re-combination. Phase control is the key point of ourmethod, as well as the choice of any number of peaksand not only the power of three.

For optimized solutions, values of m1 and m3 arealmost constant for spectra with a bandwidth be-tween 10 and 40 peaks (larger spectra have not beenstudied). These values are m1 ¼ 1:4� 0:1 and m3 ¼0:4� 0:1 and, to set the spectrum bandwidth, onlym2 has to be modified, as illustrated in Fig. 6. It isinteresting, since it means that, even with a morecomplex signal to generate the phase modulation,all the parameters except one are fixed and the lastparameter (m2) set the spectrum bandwidth: opti-mized nonsinusoidal phase modulation with a N ¼3 tuneable bandwidth becomes as simple as sinusoi-dal phase modulation to set [13].

Recently, another team has published results onthe same topic [14]. Since they want to avoid SBSin fiber, the frequency modulation used (f m ¼50MHz) is much lower than our frequency modula-tions (2 and 14:25GHz). So they are not limited bythe frequency modulator bandwidth and the cost ofcomponents remains low and the use of many harmo-nics remains possible. However, the drawback is the2N − 1 degrees of freedom that might be hard to con-trol, instead of our solution with only one degree offreedom. In their solutions, they use only odd harmo-nics, where the optical spectrum remains symmetri-cal whatever the relative phase. However, to getefficient spectra, relative phase relations close tomultiples of π=2 are necessary. However, they haveother constraints on modulation indices that we donot have: they consider m < 4 for each harmonic,whereas we can use our modulators to reach m >10, keeping a linear response. Thus, the proposedsolutions are different.

In the next section, we present experimental re-sults for nonsinusoidal phase modulation. We showthat, with our system, we can perform better en-ergy-balanced spectra and we measure the SBSthreshold in particular, through representative con-figurations, and we show that anti-Brillouin functioncan be performed with a narrower spectrum with oursystem.

Fig. 4. Error function for nonsinusoidal phase modulation withN ¼ 3 for any values of phases φ2 and φ3. In that example, theideal spectrum considered is a 25-peak spectrum and m1 ¼ 1:4,m2 ¼ 5:6, and m3 ¼ 0:4. Minima of error function are symmetrypositions for SPD.

Fig. 5. (Color online) Importance of phase control. Particular operating point for a nonsinusoidal phase modulation composed of twoharmonics: f m ¼ 2GHz and 3 f m ¼ 6GHz with the same modulation index m ¼ 1:4. Two U.S. patents [11,12] assume that the SPD asso-ciated is composed of nine energy-balanced peaks. However, (a) if φ3 ¼ 0, as presented in patent from [12], SPD is not energy balanced.(b) Energy-balanced spectrum is performed only for a good choice of the relative phase: φ3 ¼ π=2 [π]. (c) Evolution of error function in thecase of an ideal energy-balanced spectrum with nine peaks for different values of φ3.


D. Experimental Results

For experimental purpose, we have built a practicalsystem able to generate the first three harmonicsfrom a 2GHz single oscillator. We present the systemconfiguration in Fig. 7. Controlling the relativephases between harmonics at these frequencies iseasy because a 2π phase shift corresponds to 100psat 10GHz, i.e., 1–2 cm in a coax cable. We decided tocombine the harmonics in the high frequency signalpart and so use only one phase modulator. Other con-figurations can be developed and we could imagine asystem with a phase modulator for each harmonic.Figure 8 shows different spectra measured with a

Fabry–Perot interferometer that can scan the opticalspectrum domain with a free spectral range of40GHz (equivalent to a maximum of twenty 2GHzspaced peaks; in practice, we consider spectra witha maximum of 15 peaks to avoid aliasing): for anyof these spectra, m1 and m3, as well as φ2 and φ3,are set as presented in Subsection 2.C: there is a goodagreement between measured spectra and numeri-cally simulated spectra. Energy in peaks is betterbalanced for nonsinusoidal than for sinusoidal phasemodulators.

In the following, we check that, because energy isbetter spread among the optical spectrum, the anti-Brillouin function can be performed with a narrowerspectrum by comparing SBS thresholds for sinusoi-dal and nonsinusoidal phase modulations. Forpractical reasons, we choose to measure the SBSthreshold for backscattering in fiber at 1053nm.The only change, considering the SBS parameter,are the Brillouin linewidth and the power threshold:the Brillouin linewidth for backscattering in bulk si-lica at 1053nm is ΔυB ∼ 50MHz and is confirmed inoptical fiber [8]. Thus, the choice of f m ¼ 2GHz be-tween each peak will stimulate different Stokes.We measure the required injected power, Pinj, toget the same stimulated Brillouin backscattering(SBB) level. The spectra are characterized by themaximum value of their spectral power SPD,SPDmax, expressed in fraction of the total power:SPDmax ¼ 1 for a nonbroadened spectra and de-creases with enlargement. Then, if each peak stimu-lates a different Stokes mode, Pinj follows the rule

Pinj ¼PThreshold

SPDmax; ð8Þ

with PThreshold as the required power to reach the SBSthreshold level without broadening. And so Pinj incr-eases when SPDmax decreases for a given PThresholdvalue.

As presented in Fig. 9(a), energy-balanced spectrareduce SPDmax values compared to sine phase mod-ulation spectra for a given broadening. So, for a giventhreshold, one can use narrower nonsinusoidal phasemodulation spectra instead of sinusoidal phasemodulation.

We verified Eq. (8) experimentally. For a single-mode fiber, evaluation of the power threshold is givenby [15]

kgbPThresholdLeff=Aeff ≈ 21; ð9Þ

with gb as the peak value of the Brillouin gain. Forbulk silica, gb is about 5 × 1011 m=W. If we consider a

Fig. 6. (Color online) Modulation indexes for optimized three-harmonic-signal solutions. ðm1;m3Þ are almost constant and m2

increases with the spectrum width. Exceptions are presented incontrasting shades.

Fig. 7. (Color online) (a) Nonsinusoidal phasemodulation scheme: the harmonic combination and phase control is performed directly withthe HF signal. Optical components remain the same. (b) HF multiharmonic generator at 2–4–6GHz used in our different setups.


non polarization maintaining fiber, k is between 1=3to 2=3 [16]. Aeff is the effective core area of the fiber(Aeff ¼ 30 μm2 for a single-mode fiber at 1053nm)and Leff is the effective fiber length linked to the fiberlength and to the linear loss α by [15]

Leff ¼ ½1 − expð−αLÞ�=α: ð10Þ

As presented in Fig. 10, in our setup Leff ¼ 1:6kmsince we used a 2km optical fiber length with mea-sured loss of 0:9dB=km. The laser is pulsed butshape controlled: gain of the amplifier is constantto prevent temporal distortion that would modifythe SBB signal shape; the power level is set with avariable attenuator before injection into the long fi-ber. SBB measurements are made with two photo-

diodes synchronized for both original and backSBB signals: we consider that the SBS thresholdis reached when SBB appears over the Rayleigh scat-tering with always the same level. Criterion on theexperimental detection threshold is arbitrary but,because SBS evolves exponentially, any criterionleads to similar power levels. According to Eq. (4),the power threshold should be equal to 11dBm, closeto the experimental value of 12:1dBm. As shown inFig. 9(a), when the spectrum is broadened, the max-imum of SPD peaks becomes less energetic andtherefore, the more powerful the pulse has to be toget back the same SBB level. Thus, influence ofenergy-balanced spectra is important: narrowerspectra reached same anti-Brillouin function perfor-mance as sinusoidal phase modulation spectra.

Fig. 8. (Color online) Top line: different nonsinusoidal phasemodulation SPD [(a)m ¼ 2:4, (b)m ¼ 5:1, (c)m ¼ 7]. Bottom line: Sinusoidalphase modulation SPD with equivalent bandwidth [(d) m2 ¼ 1:8, (e) m2 ¼ 2:05, (f) m2 ¼ 3:3, m1 ¼ 1:4� 0:1, m3 ¼ 0:4� 0:2, φ2 ¼ 90° andφ3 ¼ 180°]. Vertical scale is the same for both lines: nonsinusoidal phase modulation SPD maxima are below equivalent sinusoidal phasemodulation SPD maxima.

Fig. 9. (Color online) (a) Experimental measurements. (b) Power Pinj required to observe SBS in function of SBS threshold for anot-broadened spectrum using Eq. (8). Discrepancies between (a) experiments and (b) simulations are small and equivalent betweensinusoidal and nonsinusoidal cases. The letters (a) to (f) in the figure refer to the spectra in Fig. 8.


Figures 8(c) and 8(e) presented two measured spec-tra that have similar SPDmaximum (12%); we verifythat the SBS thresholds are about the same in ourexperiment and we notice that the nonsinusoidalphase modulation spectrum is 25% narrower thatthe sinusoidal one (24 instead of 32GHz).Experimental measurements were limited by

broadening capabilities (we use a 1ω design, largespectra as on LMJ at 3ω were not reachable withour setup). Since we agreed that the SBS thresholdis directly linked to the maximum of SPD peaks,and we extend this results to larger spectra fromFig. 9(b), where the benefits of nonsinusoidal phasemodulation are even more important. In the case ofthe LMJ anti-Brillouin function at 351nm, m ¼21 rad and has a broadening of 88GHz. An ideal,

perfectly energy-balanced spectrum would havethe same efficiency for a broadening of 32GHz. Withour nonsinusoidal phase modulation with a variablem2, we have the same efficiency for a broadening of48GHz: same anti-Brillouin efficiency is obtainedwith a gain of 45% of the spectrum bandwidth.

In the next section, we assess FM-to-AM conver-sion performance when reducing the bandwidththanks to nonsinusoidal phase modulations.

3. Reduction of FM-to-AM Conversion withNonsinusoidal Phase Modulations

Phase modulation does not change the intensityshape. But, if any optical device during propagationchanges the optical spectrum (so the optical field byinverse Fourier transform), the intensity will bemodulated; this is FM-to-AM conversion. To quantifythe FM-to-AM conversion, the distortion criterion αdefined by the following expression is used:

α ¼ 2 ·Imax − Imin

Imax þ Imin: ð11Þ

α ranges from 0 to 200% and is ideally equal to 0 (nointensity variation).

On LMJ, FM-to-AM conversion is mostly due totwo different effects: symmetrical amplitude filtersand phase filtering (chromatic dispersion). Ampli-tude filters that are symmetrical for the central spec-trum wavelength have different origins in a LMJchain: gain narrowing in the preamplifier module,as well as in the amplification section and the limitedspectral acceptance of the frequency conversion.

Fig. 10. (Color online) Setup of SBS measurement experiment.Laser is pulsed but shape controlled: gain of the amplifier is con-stant to prevent temporal deformation; power lever is set with avariable attenuator before injection in the long fiber. The phasemodulation used is either sinusoidal or nonsinusoidal. SBS mea-surements are made with two photodiodes synchronized for bothoriginal and back SBS signals: we consider that the SBS thresholdis reached when back SBS appears with always the same level.

Fig. 11. (Color online) Parametrical optical filter using birefringence in PMF: impacts of the different parameters on the transferfunction. The second wave plate is useful to get back the linear polarization on a fiber axis.


Chromatic dispersion has two origins: long fibers inthe source (about 200m) and the propagation be-tween the plane grating and the focusing gratingin the frequency conversion system. All these effectshave already been studied in [7].

Simple considerations predict that the impact of aphase modulation change is different for these twotypes of transfer function. Chromatic dispersionstudy from [7] shows that distortion criterion, α,evolves with m:f m2 for a sinusoidal phase modula-tion, and so is not directly linked to the broadeningΔf ¼ 2ðmþ 1Þf m. In the case of a nonsinusoidalphase modulation, as m2 is usually higher than m1and m3 (see Fig. 6) and a first-order m:f m2 becomes4m2:f m2, the impact of the chromatic dispersion iseven more detrimental. Nonsinusoidal phase modu-lations are not interesting for minimizing the impactof chromatic dispersion. However, compensation ofdispersion is possible [17]. On the other hand, analy-tical results for symmetrical amplitude filters showthat the associated distortion criterion evolves withðm · f mÞ2 in the case of a sinusoidal phase modula-tion, which means that α ∝ Δf 2, with Δf as the op-tical bandwidth. Thus, bandwidth reduction is ofinterest in minimizing FM-to-AM conversion fromsymmetrical amplitude filters.

To experimentally check this result, we built aparametrical optical filter as presented in Fig. 11.The transfer function can be rewritten as follows:

H0ðf Þ ∝ 1þ A expðið2πfΔτ þΨÞÞ: ð12Þ

Fig. 12. (Color online) Distortion criterion for different symmetri-cal amplitude filters. Comparison between sinusoidal (squares)and nonsinusoidal (circles) phase modulation with the sameanti-Brillouin efficiency. We added numerical simulations withMATLAB.

Fig. 13. (Color online) Left column: temporal shape measured after filtering (f 3dB ¼ 20GHz) for both sinusoidal (bottom line) and non-sinusoidal (upper line) phase modulation. Right column: AM spectra associated. Data points concentrated at the frequency axis representthe measurement noise filtered for the results. In insert, we give numerical simulations obtained with MATLAB.


The interest of this filter is that it can be consideredas a symmetrical amplitude filter when we setψ ¼ 0°. Indeed, the phase transfer function is negli-gible since H0ðf Þ is then an odd function: the second-order term of the phase transfer function is null.Thus, we may consider that we have the followingamplitude transfer function:

jHoðf Þj2 ∝ 1þ C cosð2πfΔτÞ; ð13Þwhere C ¼ 2A=ð1þ A2Þ is the contrast, set to 0.9 inour experiment. With this value, the 3dB cutofffrequency is equal to f 3dB ¼ 1:67=πΔτ. Δτ is propor-tional to the fiber length, with a measured propor-tionality coefficient of 1:5ps=m, and set the filterbandwidth: the longer the fiber, the more severethe filtering. Our experiment was to set this filterat ψ ¼ 0° for different fiber lengths. Then we mea-sured AM modulations for both sinusoidal and non-sinusoidal phase modulation. The measurementswere made with a high-speed photodiode and an os-cilloscope (12GHz 3dB bandwidth). Thus, we usespectra from Figs. 8(c) and 8(e) for comparison sincewe proved in Section 2 they have same anti-Brillouinefficiency. Results with incertitude estimations aregiven in Fig. 12 and compared with analytical simu-lations made with MATLAB. Incertitude origins aremultiple, and we considered measurement noise, ψcontrol, and spectrum control to give the incertitudelevel. However, experimental results are close to si-mulations and reduction of FM-to-AM conversion byusing nonsinusoidal phase modulation is important:FM-to-AM conversion reduction is better than 40% oftotal AM modulation.Another interest of nonsinusoidal phase modula-

tion on FM-to-AM conversion is that the measuredtemporal shape presented AM modulation at higherfrequencies, as it is presented in Fig. 13. Most of AMmodulation for sinusoidal phase modulations is in-cluded in a 2f m bandwidth. Since nonsinusoidal isa combination of the first three harmonics of f mand not only f m, a consequence is that the AM spec-trum is more complex and has significant contri-bution at higher frequencies than 2f m. This isinteresting because high-frequency AM is less detri-mental in a high-power laser. Indeed, frequenciesabove 7GHz will by filtered by the focalization tothe D-T capsule [7] and, so, FM-to-AM conversiondistortion reduction with nonsinusoidal phase mod-ulation will be improved after the final focalization tothe target.

4. Conclusions

In high-power lasers, phase modulation is requiredto avoid SBS that would damage optics. Sinusoidalphase modulation is used, but associated SPD doesnot have an optimized shape. To minimize enlarge-ment with the same anti-SBS efficiency, homoge-neous SPD is an ideal solution. We have shownthat it is possible to generate spectra close to thisideal case with a nonsinusoidal phase modulation

made of a combination of three harmonics with con-trolled relative phases. This solution is easy to imple-ment because we have proven that we can havevariable broadening spectra by changing only oneparameter. Experimental tests are satisfying andwe proved that, on LMJ, we can have the sameanti-Brillouin efficiency with a 45% narrower spec-trum with our solution. Bandwidth reduction isinteresting for reducing FM-to-AM conversion. FM-to-AM conversion is an important problem in control-ling the temporal shape of pulses. We have shownthat, with a nonsinusoidal phase modulation withthe same anti-SBS efficiency, the distortion criterionof symmetrical amplitude filters can be reducedup to 40%.

Appendix A

In this appendix, we prove that no pure phase mod-ulation leads to perfectly energy-balanced spectra.

Let us consider a perfectly energy-balanced spec-trum with an arbitrary odd number of peaks notedas 2M þ 1. Its SPD is then

SPDðf Þ ¼ 12M þ 1

XMk¼−M

δðf − kfmÞ: ðA1Þ

All the optical spectra, ~A, that have this SPD are

~Aðf Þ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2M þ 1

pXMk¼−M

eiϕkδðf − kfmÞ; ðA2Þ

whereΦk are the relative phases between peaks thatare degrees of freedom in that proof. The associatedoptical field is then

AðtÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2M þ 1

pXMk¼−M

expðið2πkfmtþ ϕkÞÞ: ðA3Þ

The optical field from Eq. (A3) is a pure phasemodulation if IðtÞ ¼ jAðtÞj2 ¼ 1. One can find that

IðtÞ ¼ AðtÞ · A�ðtÞ ¼ 12M þ 1

�ð2M þ 1Þ

þ�X2M−1

k¼1

ak cosðð2πkfmtþ γkÞÞ�

þ cosðð2π · 2M · f mtþ ϕ−N þ ϕNÞÞ

�: ðA4Þ

Expression of ak and γk are not necessary, but canbe found analytically. The important point is that wehave a pure phase modulation if

�X2M−1

k¼1

ak cosðð2πkfmtþ γkÞÞ�

þ cosðð2π · 2M · f mtþ ϕ−N þ ϕNÞÞ ¼ 0: ðA5Þ


This expression cannot be verified because it wouldmean that the left part of Eq. (A5) is the Fourier ser-ies of the null function. Then, all of the coefficients ofthe Fourier series have to be null. However, accord-ing to Eq. (A5), the coefficient of the 2M · f m beat isequal to 1 in all the pure phase modulation cases. Asa consequence, there is no possibility of finding apure phase modulation that leads to a perfectlyenergy-balanced spectrum.

Appendix B

First, we consider a nonsinusoidal phase modulationwith N ¼ 2:

amodðtÞ ¼ expði½m1 sinðωntÞ þm2 sinð2ωntþ φ2Þ�Þ:ðB1Þ

We are looking for the values of φ2 that lead to sym-metric SPD. The general expression of SPD isSPDðωÞ ¼ j~aoptðωÞj2. Thus, SPD is symmetrical if,for any value of ω, ~amodð−ωÞ is equal eiθðωÞ~amodðωÞ,with θðωÞ as a real function (condition 1). Accordingto Eq. (3), ~amodðωÞ is obtained with Bessel functionsas follows:

~amodðωÞ ¼�X

n

Jnðm1Þδðω − nωnÞ�

��X

k

Jkðm2Þeikφ2δðω − 2kωnÞ�: ðB2Þ

Equation (B2) can be rewritten as

~amodðωÞ ¼�X

n

�Xk

Jn−2kðm1Þ · Jkðm2Þeikφ2

�δðω

− nωnÞ�: ðB3Þ

Using Bessel function propriety, J−nðmÞ ¼ ð−1Þn

JnðmÞ, for any integer l, we have expression of eachspectral peak:

~amodðlωnÞ ¼�X

k

Jl−2kðm1Þ · Jkðm2Þeikφ2

�and

~amodð−lωnÞ ¼ ð−1Þl�X

k

Jl−2kðm1Þ · Jkðm2Þð−1Þke−ikφ2

�:

ðB4ÞCondition 1 is equivalent to the following expres-

sion for any integer l:

~amodð−lωnÞ ¼ eiθðlωnÞ~amodðlωnÞ: ðB5ÞUsing (B4), condition (B5) is satisfied for any m1 andm2 if, for any k,

ð−1Þlð−1Þke−ikφ2 ¼ eiθðlωnÞeikφ2 : ðB6ÞFor k ¼ 0, we have eiθðlωnÞ ¼ ð−1Þl, and so Eq. (B6) issimplified into ð−e−iφ2Þk ¼ ðe−iφ2Þk. It is verified forany k if −e−iφ2 ¼ e−iφ2. Thus, cosðφ2Þ ¼ 0, which meansthat φ2 ¼ π

2 ½π�.For a symmetrical SPD, we have established the

condition on φ2 for any nonsinusoidal phase modula-tion with two harmonics: φ2 ¼ π

2 ½π�. The same kind ofproof can be made with a three-harmonic phase-modulation signal. Thus, we consider the followingphase-modulated signal:

~amodðtÞ ¼ expði½m1 sinðωntÞ þm2 sinð2ωntþ φ2Þþm3 sinð3ωntþ φ3Þ�Þ: ðB7Þ

Analytically calculated spectrum ~amodðωÞ leads to thefollowing condition to get a symmetrical SPD:

ð−1Þhð−e−iφ2Þkðe−iφ3Þl ¼ eiθðhωnÞðeiφ2Þkðe−iφ3Þl: ðB8ÞFor k ¼ l ¼ 0, we have eiθðhωnÞ ¼ ð−1Þh. Then, con-

sidering l ¼ 0, we find a condition for the relativephases of the second harmonic: φ2 ¼ π

2 ½π�. k ¼ 0 leadsto ðe−iφ3Þl ¼ ðeiφ3Þl, which is true for any l ife−iφ3 ¼ eiφ3 . Thus, sinðφ3Þ ¼ 0, which means φ3 ¼ 0½π�.

For a symmetrical SPD, we have established theconditions on φ2 and φ3 for any nonsinusoidal phasemodulation with three harmonics: φ2 ¼ π

2 ½π� andφ3 ¼ 0½π�.

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