Shaping ability of all fiber coherent pulse stacker

ARTICLE IN PRESS

Optics & Laser Technology 39 (2007) 1120–1124 www.elsevier.com/locate/optlastec

Shaping ability of all fiber coherent pulse stacker Feng Lia,, Fan Jia, Xinjie Lu¨b, Zhan Suib, Jianjun Wangb, Kun Gaoa, Jianping Xiea, Hai Minga a

Department of Physics, University of Science and Technology of China, Hefei 230026, China b Laser 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 including high-speed optical communication, laser fusion and ultrafast nonlinear optics. Different techniques have been used to satisfy different requirements. Amplitude modulation is usually used to realize the shaping of long pulses from ns to ms and limited by the modulator and signal generator [1]. For ultra-short pulses as fs, spectral shaping using techniques such as liquid crystal modulator arrays or other spectral modulators is always performed, a technique that profits from the wide optical spectrum of ultra-short pulses [2,3]. But for the pulses in the ps to sub-ns range, both the amplitude and spectral modulations were limited by their resolving power. The concept of optical pulse stacker was proposed to generate long-shaped pulses by Soures [4] and Hughes [5] independently in 1974. It is an effective method to generate a long pulse with desired pulse shape from a number of short pulses, and avoids the problems of the electronic-based techniques, which are limited by the electrical shaping ability or lack of shaping freedom. The pulse stacker was developed to produce arbitrary shaping pulse by using a pair of F–P etalons instead of reflectors in Corresponding author. Tel.: +86 551 3606952; fax: +86 551 3601745.

E-mail address: [email protected] (F. Li). 0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.09.011

1976 by Thomas [6]. But Thomas neglected the interference between adjacent pulses, which was discussed by Martin [7] in the same year. To estimate the efficiency of pulse stacking, Bates [8] reported the analysis of Gaussian pulse stacking. When the pulse stacking was used in Gekko-XII, the coherence between adjacent chirped pulses was cited as the main reason of pulse instability [9]. To avoid the coherence problem of pulse stacking, multimode fiber and partially coherent light has been used in a fiber-based pulse stacking system in Gekko-XII [10]. In this paper, we study the all-fiber coherent pulse stacking systematically; we show that the time delays and phase differences between the short pulses are the key parameters to determine the characteristics of the stacked pulse. The permitted variation regions of the time delay and phase difference are obtained. The pulse shaping ability of the stacker to produce varying shapes is also discussed. 2. Method of all fiber pulse stacker A schematic diagram of coherent pulse stacking is shown in Fig. 1. The stacker actually is an interferometer formed by a 1  N splitter and an N  1 coupler connected by N optical fibers in different lengths. The differences of fiber lengths provide different time delays as an arithmetical

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

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

1

N

Coupler (N¡Á1)

S plitter (1¡ÁN)

2

Pulse In

1121

DI mn p0:1; man, I m; n 2 0; 1; 2; . . . ; 2ðN  1Þ,

 0:1pRmn ¼

ð3Þ

Pulse Out

where Rmn is the intensity ripple ratio of between m and n. The intensity difference between m and n is defined as DI mn ¼ IðN; mt=2Þ  IðN; nt=2Þ,

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

I¼

progression. To obtain the ability of arbitrary pulse shaping, variable optical attenuator should be inserted into every branch. In the different target pulses, the long smooth flat-top pulse is the most primary one. The incident optical pulse is obtained from the modelocked laser after narrow band-pass filter. The light propagating in fiber spectrally matches the condition Do5o0. We assume the time delay between two neighboring branches as t. The total amplitude of the output light from the stacker is a sum of the complex amplitudes corresponding to the N branches EðN; tÞ ¼

N 1 X

Aðt  mtÞ eio0 ðtmtÞ ,

(1)

To get a flat-top pulse at the output port, we assume the peak amplitudes of the pulses in every branch are identical. According to the symmetric law, the maximum and minimum intensities of the stacked long pulse will appear at or close to the peaks of each pulse or the center between neighboring pulses. So we can use the intensity ripple ratio in the profile of the stacked long pulse as our criterion to study the effects of the variations in t and a. As general design, the tolerable allowance of the ripple ratios of the

ð4Þ

According to the condition 0.1pRp0.1, the tolerable values of t and a can be obtained by solving the inequation as shown in Eq. (3). There are NðN  1Þ=2 inequations at most 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 and phase difference of four pulses for flat-top pulse output. In our calculation, we use Gaussian pulse as the profile of input pulses as follow:

m¼0

where o0 is the center angular frequency. A(t) is the amplitude profile of the input light wave as a function of time. A(tmt) is the amplitude of the mth branch, which is a slowly varying function with t because the pulse width is larger than 10 ps in most cases. Let the variation of t be denoted as Dt. With the condition Dt col0, the variation of A(tmt) can be neglected. The slight variation of t will only affect the phase differences. Then the time delay and phase difference can be dealt with independently. We suppose the phase differences between neighbor pulses a ¼ 2pn0t ¼ o0t and the corresponding intensity is 2   N 1 X  iðo0 tmaÞ  IðN; tÞ ¼ EE ¼  Aðt  mtÞ e   m¼0 2   N 1  X ¼ Aðt  mtÞ eima  . ð2Þ  m¼0

2ðN1Þ X 1 IðN; mt=2Þ. 2N  1 m¼0

2

EðtÞ ¼ eðt=TÞ eiot .

(6)

The phase difference varies from 01 to 1801, and time delay from 0 to 2T, where T is the width of the Gaussian function. Consider the symmetry of the pulse. There are only 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 corresponding average intensity is I¼

X 1 N1 IðN; mt=2Þ. N  1 m¼1

(7)

To solve Eq. (3), we can plot Rmn vs. t/T as shown in Fig. 2 with fixed phase difference a of adjacent pulses. The intervals A and B in Fig. 2 give the values of Rmn between [0.1, +0.1], which indicate the valid values of t determined 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 as shown in Fig. 3. As indicated by Fig. 3, a should be less than 901 and t/T less than 1.5 in most conditions. From the diagram, we can find that t ¼ 1.1 T is suitable to use in four pulses interference stacking. The spans of a for t ¼ 1.1 T is about 1801 (from 901 to 901) for the symmetry of a. On the other hand, for 561oao681, the time delays can vary from 0 to 1.3 T under the limitation of Eq. (3). The typical pulses under the conditions are shown in Fig. 4.

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1122

For multi-pulse stacking, the method of analysis is similar to four pulses stacking. With multi-pulses stacking especially for more than eight pulses, the middle part of the pulse will be repetitive and the solving of Eq. (5) at that section will be very simple. But the rise and fall edges will be complicated. Since the pulses near the edges are affected by other pulses on single side without symmetry, the peaks

near the edges will not appear at the point of mt/2. They should be treated accurately. The sampling in Eqs. (3) and (4) should be densified on the region between the first peak and the repetitive part of the pulse. The valid region of 32 pulses stacker to produce long flat-top pulse is shown in Fig. 5. To get flat-top square pulse by multi-pulse stacking, the parameters should be selected from the cross-hatched region from Fig. 5. The slash-hatched region indicates the valid values only fit requirement of flat-top of the middle section, but the fluctuation near the edges will exceed the 10% limitation for the unsymmetrical pulse distribution on the edges. 4. The shaping ability of the stacker

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.

To estimate the shaping ability of the stacker in theory, we use correlation function of target and fit waveforms by multi-pulse stacking as the evaluation. Assume the target function as Y ¼ F(X), to evaluate the correlation, we should sample the target function as Yi ¼ F(Xi), i ¼ 1 y M. The mean value of the target 1.5

1.0

1.0 τ/T

τ/T

1.5

0.5

0.5

0.0

0.0 0

30

60

90

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 1.1 T and 561oa o681 indicate the most suitable time delays and phase differences for stacking.

0

30

60

90

120

150

180

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.

Fig. 4. The pulses examples corresponding to the optimized conditions. (a) t ¼ 1.1 T, a vary from 0 to 801. (b) a ¼ 601, t vary from 0.5 to 1.3 T.

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the partial derivation of R2,

function is M 1 X Y i. Y¯ ¼ M i¼1

(8)

Another two required variables are regressive sum of squares (RGS) and corrected sum of squares (CSS), which are defined as RGS ¼

M X

ðY^ i  Y¯ Þ2 ,

(9)

i¼1

CSS ¼

M X

1123

ðY i  Y¯ Þ2 .

(10)

i¼1

Then the correlation function can be written as RGS . (11) CSS 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 lines to adjust the amplitudes of the corresponding pulses. The amplitudes of the pulses are independent variables of R2. Find the peaks of R2 by solving the system of equations of R2 ¼

qR2 ¼ 0; qAi

5. Experimental results and discussion In our experiment, the initial pulse is generated by a compound ‘‘8’’ cavity passively mode-locked Yb3+-doped fiber ring laser followed by a narrow band-pass filter and a fiber amplifier. The pulse is Gaussian shape of a pulse 6

TargetPulse FittedPulse

5

Target Pulse Fitted Pulse

5

4

4 Intensity

Intensity

(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 initial Gaussian pulse width. In each simulation 32 pulses were used to fit the target pulse. The fitting results of exponential and 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, the correlation can always be lager than 0.95. The simulations result of pulse stacking can fit the shaping requirements in theory.

6

3 2

3 2

1

1

0

0 0

(a)

i ¼ 1; . . . N,

1

2

3

4 Time

5

6

7

0 (b)

1

2

3

4

5

6

7

Time

Fig. 6. Fitted and target pulses. The fitting used 32 pulses. The delays between adjacent pulses are equal to the initial Gaussian pulse width. (a) Fitting of exponential pulse. R2 is 0.997. (b) Fitting of Haan pulse. R2 is 0.996. The correlations indicate that the pulse stacker has good shaping ability to generate target pulses.

Fig. 7. The experimental results of pulse-stack shaping. The stacker has 32 branches. The delays between adjacent pulses are 80 ps. (a) Exponential pulse. (b) Haan pulse. The correlations indicate that the pulse stacker has good shaping ability to generate required pulses.

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

width of 90 ps. The stacker is formed as shown in Fig. 1. To get stable delays, we put the whole stacker into solidified glue that has some flexibility. The delays are adjusted to about 80 ps to fit the pulse width. The selection of delays fits the requirement shown in Fig. 5. The experimental results of pulse stacking corresponding to the two pulses in Fig. 6 are shown in Fig. 7. The waveforms are captured by oscilloscope model TDS6604 of Tektronix cooperating with photonic detector model SD43 of Tektronix. The experimental results of pulse stacking have intensity jitter in the waveform for the control precision of fiber length. Nevertheless the pulses shown in the figure have illuminated the shaping ability of the stacker accordant to our analysis. 6. Conclusion We have studied the all-fiber coherent pulse stacking systematically. To satisfy the requirement of flat-top pulse shape, we got the valid regions of t and a by separating the variables of the time delays into phase a and large time delay t to solve the transcendental equation. The long– smooth pulse can be obtained from the pulse-interference stacker by selecting suitable time delays and phase differences for the adjacent pulses from the regions shown in Figs. 3 and 5. The optimized output pulse examples are illustrated by simulation. The shaping ability of all-fiber coherent pulse stacker is estimated by the correlation function. The experimental results of the pulse stacking agree with our analysis.

Acknowledgment The work in this paper is supported by the National High Technology Research and Development Program of China (863-804) under Grant no. 2003AA845061 and Grant no.2004AA849070. References [1] Burkhart SC, Wilcox R, Browning D, et al. Amplitude and phase modulation with waveguide optics. Proc SPIE 1997;3047:610–7. [2] Weiner AM. Femtosecond optical pulse shaping and processing. Prog Quant Electron 1995;19:161–237. [3] Jiang Z, Seo DS, Leaird DE, Weiner AM. Spectral line-by-line pulse shaping. Opt Lett 2005;30(12):1557–9. [4] Soures J, Kumpan S, Hoose J. High power Nd:glass laser for fusion applications. Appl Opt 1974;13:2081–94. [5] Hughes JL, Donohue PJ. Pulse tailoring system for laser fusion. Opt Commun 1974;12:302–3. [6] Thomas CE, Siebert LD. Pulse shape generator for laser fusion. Appl Opt 1976;15:462–5. [7] Martin WE, Milam D. Interpulse interference and passive laser pulse shapers. Appl Opt 1976;15:3054–61. [8] Bates HE, Henderson BJ. Theoretical maximum efficiencies for passive optical pulse shaping systems. J Opt Soc Am 1978;68:919–24. [9] Kanabe T, Nakatsuka M, Kato Y, Yamanaka C. Coherent stacking of frequency-chirped pulses for stable generation of controlled pulse shapes. Opt Commun 1986;58:206–10. [10] Shinichi M, Noriaki M, Akinobu A, Shinji A, Masahiro N, Sadao N. Flexible pulse shaping of partially coherent light on Gekko XII. Proc SPIE 1995;2633:627–33. [11] Rothenberg JE. Ultrafast picket fence pulse trains to enhance frequency conversion of shaped inertial confinement fusion laser pulses. Appl Opt 2000;39:6931–8.