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Contents lists available at ScienceDirect

Optik

jou rn al homepage: www.elsev ier .de / i j leo

ltra-short pulse compression at 1065 nm in nonlinear photonicrystal fiber

. Cherbia, N. Lamhenea, F. Boukhelkhala, Anjan Biswasb,c,∗

Instrumentation Laboratory (LINS), University of Sciences and Technology ‘USTHB’, AlgeriaDepartment of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USADepartment of Mathematics, King Abdulaziz University, PO Box 80203, Jeddah 21589, Saudi Arabia

r t i c l e i n f o

rticle history:eceived 29 January 2013

a b s t r a c t

A pulse compressor has been designed using a 13 mm highly nonlinear photonic crystal fiber to compresspulses centered at 1065 nm from 28 fs to 1.8 fs with a compression factor of 16.2. This compression is

ccepted 3 June 2013vailable online xxx

eywords:olitons

achieved by using a high level of energy and generating different orders of solitons without resorting tolarge values of fiber’s dispersion.

© 2013 Elsevier GmbH. All rights reserved.

umericsntegrability

. Introduction

The appearance of photonic crystal fibers has revolutionized theonlinear optics [1–4] particularly in the field of ultrashort pulse

aser [5,2,6]. Indeed, the interest toward photonic crystal fibersPCF) is growing steadily. The advent of this new fiber provides inac-essible properties with standard fibers. Over the last two decades,any efforts have been made in the generation of ultrashort pulses

n order to improve transmission rates, this was accomplished bysing compression of the higher orders of solitons [7–9,5] thusvoiding the need of post-compression devices [10]. Compressionf pulses down to 5 fs has been demonstrated in several experimen-al studies that required laser pulses with energy level higher than0 nj, proved to be very difficult in order to reach with existing fem-osecond lasers [11,12]. However, the use of small effective areass in the case of PCF, yielded the pulse compression of duration0–35 fs with energy levels of sub-nanojoule, as was demonstrated

n [13,14]. In recent years, in particular with the development ofhotonic crystal fibers [15,16] with adjustable dispersion and non-

inear properties, pulse compression attracted a lot of attention.his effect of pulse compression for solitons in photonic nanowiress shown in [17]. A 30 fs pulse around of 1.55 �m was obtained

Please cite this article in press as: L. Cherbi, et al., Ultra-short pulse compLight Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.06.014

y using a highly nonlinear fiber of 7 cm [18]. In another experi-ent, a pulse of 50 fs was achieved with doped PCF [19]. Recently,

0.357 ps pulse is obtained in a high dispersive PCF (600 ps/nm km)

∗ Corresponding author at: Department of Mathematical Sciences, Delaware Stateniversity, Dover, DE 19901-2277, USA.

E-mail address: biswas.anjan@gmail.com (A. Biswas).

030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved.ttp://dx.doi.org/10.1016/j.ijleo.2013.06.014

[20]. In this work, we came up to realize pulse compression of28.4 fs around 1065 nm by the generation of solitons of differ-ent orders and tuning the power levels and the PCF parameterswithout using high values of dispersion, which enabled us to obtainultrashort pulses. First, we obtained 2.3 fs pulses in a PCF length of2.5 cm with a compression factor of 12.5. Afterwards, we improvedthe pulse compression until 1.8 fs in a PCF length of 1.3 cm with acompression factor of 16.

2. Description of the numerical method

The propagation of a light pulse through the optical fiber isgoverned by the nonlinear Schrödinger equation (NLS), which iswritten as follows [1,2,21]:

∂A

∂z+ j

ˇ2

2∂2A

∂t2− ˇ3

6∂3A

∂t3= j� |A|2A − ˛

2A (1)

where A is the magnitude of the pulse envelope, ˇ2 is the groupvelocity dispersion parameter, ˇ3 is the fiber’s third-order disper-sion and � is the effective Kerr law nonlinearity.

In the case of an anomalous dispersion (ˇ2 < 0) and with a certaindegree of compromise between the input power and the chromaticdispersion, we came to generate solitons of order N such that:

�P0T20

|ˇ2| = LD

LNL= N2 (2)

where T0 is the initial width of the pulse, P0 is the launched power,

ression at 1065 nm in nonlinear photonic crystal fiber, Optik - Int. J.

LD is the dispersion length given by the following shape:

LD = T20

|ˇ2| (3)

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Table 1Results of simulation of the soliton pulse compression for various values of P0, L and N.

P0 (kW) N L = 1.3 cm L = 2.5 cm L = 4.5 cm

6 2 Tcomp = 22 fs Tcomp = 12.7 fs Tcomp = 4.6 fsFc = 1.27 Fc = 2.5 Fc = 6.2Qc = 0. 84 Qc = 0.51 Qc = 0.33Tdiff = T0 − Tcomp Tdiff = 15.7 fs TDiff = 23.8 fsTdiff = 6.4 fs (L ≈ Z0/2)

13.5 3 Tcomp = 7.2 fs Tcomp = 2.3 fs Soliton fission regionFc = 4 Fc = 12.5Qc = 0.35 Qc = 0.2Tdiff = 21.2 fs Tdiff = 26.1 fs

(L ≈ Z0/3)

24 4 Tcomp = 1.8 fs Soliton fission region Soliton fission regionFc = 16

ei

L

p

Z

fcEf

U

w

wpaTglFtt

3

ao

itrpo

obtain the soliton of order 3 (Fig. 3). In Fig. 4, the evolution ofthis soliton is plotted for different lengths given in Table 1 whereone notices that we can further improve the compression pulse of28.4 fs to 2.3 fs with a compression factor of 12.5 in a PCF length of

Qc = 0.15Tdiff = 26.6 fs(L ≈ Z0/8)

LNL is defined as the critical distance at which the nonlinearffects become important for the pulse propagation along the fiber,t is given by the following relationship:

NL1

P0 · �(4)

The higher-order solitons (N > 1) recover their original shape ateriod Z0, which is defined by:

0 = �LD

2= �T2

0

2|ˇ2| (5)

In this work, the compression of femtosecond pulses is per-ormed in a nonlinear PCF by exciting higher orders of solitons thatan propagate. The soliton of order N can be obtained by solving theq. (1), its mathematical expression may be written in the followingorm [22]:

(�, �) = N sech (�) exp

(j�

2

)(6)

here � = t

T0, � = z

LD, and U = A√

P0

We injected a pulse having the form of equation 6 and initialidth of 28 fs centered at 1065 nm, to the input of a single modehotonic crystal fiber having � = 11 W−1 km−1 and zero dispersionround 1040 nm. The diameter of the fiber’s core is about 4.8 �m.he dispersion value is D = 22 ps/nm/km. We simulated the propa-ation of the pulse through the PCF by solving Eq. (1) and neglectingosses and the third-order dispersion in the fiber. The Split Stepourier Method [10], which is a pseudo-spectral method dedicatedo solving the NLS, was used to solve Eq. (1) and to derive theemporal and spectral evolution pulse given by (Eq. (6)).

. Simulations and results

Depending on the injected pulse power into the PCF having thebove opto-geometrical parameters, we generate different ordersf sotitons according to (Eq. (2)).

Subsequently, we study the compression of the injected pulsento the PCF according to the order of the generated solitons and

Please cite this article in press as: L. Cherbi, et al., Ultra-short pulse compLight Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.06.014

he period Z0 which also depends on the length of the fiber. Theesults of different simulations for different values of the injectedower “P0”, different lengths of the fiber “L” and different ordersf the soliton “N” are summarized in Table 1. The obtained pulse

compression is characterized by the compression factor Fc definedby:

Fc = T0

Tcomp(7)

where Tcomp = T0 − Tdiff is the width of the compressed pulse andTdiff is the difference between the width of the initial pulse T0 andthat of the compressed pulse.

The quality of compression is evaluated by the quality factor Qc

that is given by the following relationship:

Qc = Pcomp

Fc(8)

Pcomp is the peak power of the compressed pulse normalized tothe input pulse.

In Table 1, one notes that the soliton of order 2 is obtained byinjecting a power P0 = 6 kW. Fig. 1 shows the 3D evolution of thesoliton of order 2 depending on the length and time of propaga-tion, where we note that the soliton recovers its initial shape atZ0 ≈ 9.5705 cm.

Fig. 2 shows the pulse compression based on the three fiberlengths given in Table 1. This figure depicts that the best compres-sion is obtained for the length of 4.5 cm (Fig. 2c) which correspondsapproximately to a length of Z0/2. The 28 fs pulse is compressed to4.60 fs with a compression factor of 6.2 and a quality factor of 0.33.

By launching the PCF with a power of 13.5 kW (Table 1), we

ression at 1065 nm in nonlinear photonic crystal fiber, Optik - Int. J.

Fig. 1. Simulated 3D plot of the evolution of the 2order-soliton according to thedistance and time of the propagation in nonlinear PCF.

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Fig. 2. Simulated temporal evolution of a secant hyperbolic pulse of 6 kW and initialwidth of 28.7 fs at (a) 1.3 cm, (b) 2.5 cm and (c) 4.5 cm.

Fd

2tt

ioicip

Fo

Fig. 5. Temporal evolution of a secant hyperbolic pulse of 13.5 kW and initial widthof 28.7 fs at 4.5 cm.

Fig. 6. Simulated 3D plot of the evolution of the 4order-soliton according to thedistance and time of the propagation in nonlinear PCF.

Fig. 7. Temporal evolution of a secant hyperbolic pulse of 24 kW and initial width

ig. 3. Simulated 3D plot of the evolution of the 3order-soliton according to theistance and time of the propagation in nonlinear PCF.

.5 cm, which corresponds to L ≈ Z0/3 (Fig. 4b). At length of 4.5 cm,he pulse compression cannot be performed because at this length,he soliton undergoes fission rather than compression (Fig. 5).

To generate the fourth order soliton (Table 1), a power of 24 kWs injected at the input of PCF. Fig. 6 illustrates the 3D evolutionf this soliton according to the distance and propagation time. Thenjected pulse could be compressed from 28.7 fs until 1.8 fs with aompression factor of 16 over a PCF length of 1.3 cm correspond-ng to L ≈ Z0/8, which leads to a better compression compared to

Please cite this article in press as: L. Cherbi, et al., Ultra-short pulse compression at 1065 nm in nonlinear photonic crystal fiber, Optik - Int. J.Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.06.014

revious cases (Fig. 7).

ig. 4. Temporal evolution of a secant hyperbolic pulse of 13.5 kW and initial widthf 28.7 fs at (a) 1.3 cm, (b) 2.5 cm.

of 28.7 fs at 1.3 cm.

Fig. 8. Temporal evolution of a secant hyperbolic pulse of 24 kW and initial widthof 28.7 fs at 2.5 cm (b) 4.5 cm.

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In the case of the fourth order of soliton, the compressionecomes impossible for lengths of 2.5 cm and 4.5 cm which are

ncluded in the fission region of this soliton as shown in Fig. 8.One can conclude from the various simulations that the com-

ression rate varies proportionally to the order of the soliton. Ifne is willing to get a better pulse compression, the fiber lengthhould be choosen carefully such that it should be included in theompression zone of the soliton of order N without reaching itsssion region.

. Conclusion

A pulse compressor based on a Nonlinear PCF fiber is devised,t compresses 28.4 fs pulses to 4.6 fs and 2.5 fs for lengths of 4.5 cmnd 2.5 cm, respectively. This work can be applied in a laser basedn mode-locking at 1065 nm.

The compression factor of pulses and the quality factor can bemproved by adjusting the length of the PCF, its dispersion valuend launched pulse power and therefore the order of the soliton.

The results can be also applied in the transmission of informa-ion to improve their bit rate while providing a tradeoff betweenhe quality factor and the compression factor.

For example, if one wants to send a video via the Internet,ne can send it full despite its size while losing a little in theuality. In this case we use a large value of compression factor.owever, if one is interested in the quality of transmission as

n the case of the transmission of images, it becomes necessaryo reduce the order of soliton and use a maximum quality fac-or.

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