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Optics & Laser Technology 39 (2007) 1120–1124

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Shaping ability of all fiber coherent pulse stacker

Feng Lia,�, Fan Jia, Xinjie Lub, Zhan Suib, Jianjun Wangb, Kun Gaoa,Jianping Xiea, Hai Minga

aDepartment of Physics, University of Science and Technology of China, Hefei 230026, ChinabLaser Fusion Research Center of CAEP, Mianyang 621900, China

Received 7 November 2005; received in revised form 13 September 2006; accepted 19 September 2006

Available online 7 November 2006

Abstract

Pulse stacking is an effective method to generate a long-shaped pulse from short pulses. In this paper, we study all-fiber coherent pulse

stacking systematically; we show that the time delays and phase differences between the short pulses are the key parameters of the

stacked pulse. The permitted variation of the time delay and phase difference are obtained. The ability of the stacker to produce arbitrary

pulse shapes is discussed.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Pulse stack shaping; Parameter choose; Shaping ability

1. Introduction

Pulse shaping is widely used in applications includinghigh-speed optical communication, laser fusion and ultra-fast nonlinear optics. Different techniques have been usedto satisfy different requirements. Amplitude modulation isusually used to realize the shaping of long pulses from ns toms and limited by the modulator and signal generator [1].For ultra-short pulses as fs, spectral shaping usingtechniques such as liquid crystal modulator arrays or otherspectral modulators is always performed, a technique thatprofits from the wide optical spectrum of ultra-short pulses[2,3]. But for the pulses in the ps to sub-ns range, both theamplitude and spectral modulations were limited by theirresolving power. The concept of optical pulse stacker wasproposed to generate long-shaped pulses by Soures [4] andHughes [5] independently in 1974. It is an effective methodto generate a long pulse with desired pulse shape from anumber of short pulses, and avoids the problems of theelectronic-based techniques, which are limited by theelectrical shaping ability or lack of shaping freedom. Thepulse stacker was developed to produce arbitrary shapingpulse by using a pair of F–P etalons instead of reflectors in

e front matter r 2006 Elsevier Ltd. All rights reserved.

tlastec.2006.09.011

ing author. Tel.: +86551 3606952; fax: +86 551 3601745.

ess: lf@ustc.edu (F. Li).

1976 by Thomas [6]. But Thomas neglected the interferencebetween adjacent pulses, which was discussed by Martin [7]in the same year. To estimate the efficiency of pulsestacking, Bates [8] reported the analysis of Gaussian pulsestacking. When the pulse stacking was used in Gekko-XII,the coherence between adjacent chirped pulses was cited asthe main reason of pulse instability [9]. To avoid thecoherence problem of pulse stacking, multimode fiber andpartially coherent light has been used in a fiber-based pulsestacking system in Gekko-XII [10].In this paper, we study the all-fiber coherent pulse

stacking systematically; we show that the time delays andphase differences between the short pulses are the keyparameters to determine the characteristics of the stackedpulse. The permitted variation regions of the time delayand phase difference are obtained. The pulse shapingability of the stacker to produce varying shapes is alsodiscussed.

2. Method of all fiber pulse stacker

A schematic diagram of coherent pulse stacking is shownin Fig. 1. The stacker actually is an interferometer formedby a 1�N splitter and an N� 1 coupler connected by N

optical fibers in different lengths. The differences of fiberlengths provide different time delays as an arithmetical

ARTICLE IN PRESS

1

2

N

Pulse In Pulse Out

S pl

itte

r (1

¡ÁN

)

Cou

pler

(N

¡Á1)

Fig. 1. The schematic diagram of pulse coherent stacking.

F. Li et al. / Optics & Laser Technology 39 (2007) 1120–1124 1121

progression. To obtain the ability of arbitrary pulseshaping, variable optical attenuator should be insertedinto every branch. In the different target pulses, the longsmooth flat-top pulse is the most primary one.

The incident optical pulse is obtained from the mode-locked laser after narrow band-pass filter. The lightpropagating in fiber spectrally matches the conditionDo5o0.

We assume the time delay between two neighboringbranches as t. The total amplitude of the output light fromthe stacker is a sum of the complex amplitudes correspond-ing to the N branches

EðN; tÞ ¼XN�1

m¼0

Aðt�mtÞ e�io0ðt�mtÞ, (1)

where o0 is the center angular frequency. A(t) is theamplitude profile of the input light wave as a functionof time. A(t�mt) is the amplitude of the mth branch,which is a slowly varying function with t because thepulse width is larger than 10 ps in most cases. Let thevariation of t be denoted as Dt. With the condition Dt�col0, the variation of A(t�mt) can be neglected. The slightvariation of t will only affect the phase differences. Thenthe time delay and phase difference can be dealt withindependently. We suppose the phase differences betweenneighbor pulses a ¼ 2pn0t ¼ o0t and the correspondingintensity is

IðN; tÞ ¼ EE� ¼XN�1

m¼0

Aðt�mtÞ e�iðo0t�maÞ

�����

�����

2

¼XN�1

m¼0

Aðt�mtÞ eima

�����

�����

2

. ð2Þ

To get a flat-top pulse at the output port, we assume thepeak amplitudes of the pulses in every branch are identical.According to the symmetric law, the maximum andminimum intensities of the stacked long pulse will appearat or close to the peaks of each pulse or the center betweenneighboring pulses. So we can use the intensity ripple ratioin the profile of the stacked long pulse as our criterion tostudy the effects of the variations in t and a. As generaldesign, the tolerable allowance of the ripple ratios of the

flatness should not be more than 10%. The limitation canbe written as follow:

� 0:1pRmn ¼DImn

Ip0:1; man,

m; n 2 0; 1; 2; . . . ; 2ðN � 1Þ, ð3Þ

where Rmn is the intensity ripple ratio of between m and n.The intensity difference between m and n is defined as

DImn ¼ IðN;mt=2Þ � IðN ; nt=2Þ,

I ¼1

2N � 1

X2ðN�1Þ

m¼0

IðN;mt=2Þ. ð4Þ

According to the condition �0.1pRp0.1, the tolerablevalues of t and a can be obtained by solving the inequationas shown in Eq. (3). There are NðN � 1Þ=2 inequations atmost with the relations

IðN; nt=2Þ ¼ IðN; ð2ðN � 1Þ � nÞt=2Þ,

Rmn ¼ �Rnm. ð5Þ

3. The requirement of coherent pulse stacking

As a typical example of pulse interference stack shaping,we first calculate the variable region of time delay andphase difference of four pulses for flat-top pulse output.In our calculation, we use Gaussian pulse as the profile

of input pulses as follow:

EðtÞ ¼ e�ðt=TÞ2 e�iot. (6)

The phase difference varies from 01 to 1801, and time delayfrom 0 to 2T, where T is the width of the Gaussianfunction. Consider the symmetry of the pulse. There areonly three points related to the intensity ripple ratio: I(4,t/2),I(4,t), and I(4,3t/2). Then there are three ripple ratios left:R12, R13, and R23. At the same time, the correspondingaverage intensity is

I ¼1

N � 1

XN�1

m¼1

IðN;mt=2Þ. (7)

To solve Eq. (3), we can plot Rmn vs. t/T as shown in Fig. 2with fixed phase difference a of adjacent pulses. Theintervals A and B in Fig. 2 give the values of Rmn between[�0.1, +0.1], which indicate the valid values of tdetermined by Eq. (3) corresponding to a ¼ 301.With the different spans of t/T corresponding to various

a, we can draw the figure of valid regions of t/T and a asshown in Fig. 3. As indicated by Fig. 3, a should be lessthan 901 and t/T less than 1.5 in most conditions.From the diagram, we can find that t ¼ 1.1T is suitable

to use in four pulses interference stacking. The spans of afor t ¼ 1.1T is about 1801 (from �901 to 901) for thesymmetry of a. On the other hand, for 561oao681, thetime delays can vary from 0 to 1.3T under the limitation ofEq. (3). The typical pulses under the conditions are shownin Fig. 4.

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For multi-pulse stacking, the method of analysis issimilar to four pulses stacking. With multi-pulses stackingespecially for more than eight pulses, the middle part of thepulse will be repetitive and the solving of Eq. (5) at thatsection will be very simple. But the rise and fall edges willbe complicated. Since the pulses near the edges are affectedby other pulses on single side without symmetry, the peaks

Fig. 2. Rmn vs. t/T with fixed a ¼ 301. Intervals A[0,0.25] and B[0.96,1.43]

indicate the valid values of t/T determined by Rmn.

0 30 60 90

0.0

0.5

1.0

1.5

τ /

T

α (Deg.)

Fig. 3. The slash-hatched region is the valid region of t–a determined by

the smoothing requirement for four pulses stacking. The dash line

t ¼ 1.1T and 561oa o681 indicate the most suitable time delays and

phase differences for stacking.

Fig. 4. The pulses examples corresponding to the optimized conditions. (a)

near the edges will not appear at the point of mt/2. Theyshould be treated accurately. The sampling in Eqs. (3) and(4) should be densified on the region between the first peakand the repetitive part of the pulse. The valid region of 32pulses stacker to produce long flat-top pulse is shown inFig. 5. To get flat-top square pulse by multi-pulse stacking,the parameters should be selected from the cross-hatchedregion from Fig. 5. The slash-hatched region indicates thevalid values only fit requirement of flat-top of the middlesection, but the fluctuation near the edges will exceed the10% limitation for the unsymmetrical pulse distribution onthe edges.

4. The shaping ability of the stacker

To estimate the shaping ability of the stacker in theory,we use correlation function of target and fit waveforms bymulti-pulse stacking as the evaluation.Assume the target function as Y ¼ F(X), to evaluate the

correlation, we should sample the target function asYi ¼ F(Xi), i ¼ 1yM. The mean value of the target

t ¼ 1.1T, a vary from 0 to 801. (b) a ¼ 601, t vary from 0.5 to 1.3T.

0 30 60 90 120 150 180

0.0

0.5

1.0

1.5

τ/T

α (Deg.)

Fig. 5. The hatched region is the valid region of t–a determined by the top

smoothing requirement for multi pulses stacking. The cross-hatched

region is determined by both the long repeating top and the rise edge.

ARTICLE IN PRESSF. Li et al. / Optics & Laser Technology 39 (2007) 1120–1124 1123

function is

Y ¼1

M

XM

i¼1

Y i. (8)

Another two required variables are regressive sum ofsquares (RGS) and corrected sum of squares (CSS), whichare defined as

RGS ¼XM

i¼1

ðY i � Y Þ2, (9)

CSS ¼XM

i¼1

ðY i � Y Þ2. (10)

Then the correlation function can be written as

R2 ¼RGS

CSS. (11)

The fitting will be better as R2 closer to 1.To fit the target pulses by multi-pulse stacking, variable

optical attenuators should be inserted into the delay linesto adjust the amplitudes of the corresponding pulses. Theamplitudes of the pulses are independent variables of R2.Find the peaks of R2 by solving the system of equations of

0 2 31 4 5 6 7

0

1

2

3

4

5

6TargetPulseFittedPulse

Time(a) (

Inte

nsity

Fig. 6. Fitted and target pulses. The fitting used 32 pulses. The delays between

exponential pulse. R2 is 0.997. (b) Fitting of Haan pulse. R2 is 0.996. The corre

target pulses.

Fig. 7. The experimental results of pulse-stack shaping. The stacker has 32 bra

(b) Haan pulse. The correlations indicate that the pulse stacker has good shap

the partial derivation of R2,

qR2

qAi

¼ 0; i ¼ 1; . . . N, (12)

where N is the amount of pulses to stack the target pulse.The best fitting of the target pulse will be determined by Ai.As examples, we calculated several target pulses to show

the shaping ability of the pulse stacker. In our calculation,the delays between adjacent pulses are equal to the initialGaussian pulse width. In each simulation 32 pulses wereused to fit the target pulse. The fitting results of exponentialand Haan pulses [11] are shown in Fig. 6.In other fittings, except for the targets that have rise

times less than the rise time of the initial pulse, thecorrelation can always be lager than 0.95. The simulationsresult of pulse stacking can fit the shaping requirements intheory.

5. Experimental results and discussion

In our experiment, the initial pulse is generated by acompound ‘‘8’’ cavity passively mode-locked Yb3+-dopedfiber ring laser followed by a narrow band-pass filter and afiber amplifier. The pulse is Gaussian shape of a pulse

0 2 31 4 5 6 7

Time

0

1

2

3

4

5

6

Inte

nsity

Target Pulse

Fitted Pulse

b)

adjacent pulses are equal to the initial Gaussian pulse width. (a) Fitting of

lations indicate that the pulse stacker has good shaping ability to generate

nches. The delays between adjacent pulses are 80 ps. (a) Exponential pulse.

ing ability to generate required pulses.

ARTICLE IN PRESSF. Li et al. / Optics & Laser Technology 39 (2007) 1120–11241124

width of 90 ps. The stacker is formed as shown in Fig. 1. Toget stable delays, we put the whole stacker into solidifiedglue that has some flexibility. The delays are adjusted toabout 80 ps to fit the pulse width. The selection of delaysfits the requirement shown in Fig. 5. The experimentalresults of pulse stacking corresponding to the two pulses inFig. 6 are shown in Fig. 7. The waveforms are captured byoscilloscope model TDS6604 of Tektronix cooperatingwith photonic detector model SD43 of Tektronix.

The experimental results of pulse stacking have intensityjitter in the waveform for the control precision of fiberlength. Nevertheless the pulses shown in the figure haveilluminated the shaping ability of the stacker accordant toour analysis.

6. Conclusion

We have studied the all-fiber coherent pulse stackingsystematically. To satisfy the requirement of flat-top pulseshape, we got the valid regions of t and a by separating thevariables of the time delays into phase a and large timedelay t to solve the transcendental equation. The long–smooth pulse can be obtained from the pulse-interferencestacker by selecting suitable time delays and phasedifferences for the adjacent pulses from the regions shownin Figs. 3 and 5. The optimized output pulse examples areillustrated by simulation. The shaping ability of all-fibercoherent pulse stacker is estimated by the correlationfunction. The experimental results of the pulse stackingagree with our analysis.

Acknowledgment

The work in this paper is supported by the NationalHigh Technology Research and Development Program ofChina (863-804) under Grant no. 2003AA845061 andGrant no.2004AA849070.

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