Download Enhanced Femtosecond Optical Pulses Compression

Journal of Soft Computing and Applications 2014 (2014) 1-13
Available online at www.ispacs.com/jsca
Volume 2014, Year 2014 Article ID jsca-00044, 13 Pages
doi:10.5899/2014/jsca-00044
Research Article
Enhanced Femtosecond Optical Pulses Compression Induced by
Negative Third-Order Dispersion in Solid-Core Photonic Crystal
Fibers
Ashkan Ghanbari1*, Mehran Nikoo1
(1) Young Researchers and Elits Club,Islamic Azad University (Tehran North Branch)
Copyright 2014 © Ashkan Ghanbari and Mehran Nikoo. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
Abstract
In this paper, the chirped- free optical pulses compression through propagating in silica – index guiding
photonic crystal fibers will be studied. The results show that the Raman Effect can enhances the quality of
compressed pulses and also the positive third-order dispersion (TOD) can be considered as the main reason
in limiting the optical pulses compression. We will show that, the combined effects of the Raman
scattering and negative TODs can causes generating much further compressed pulses in comparison with
applying the only Raman effect or combination of that with the positive TODs. In this paper, We also
numerically present a novel technique to solve the wave propagation equation, which is known as the
concrete algorithms of split-step fourier method and fourth order Runge-Kutta (RK4) method. This method
has a higher accuracy, although the computational time is longer because the number of calculation steps is
chosen very large.
Keywords: Photonic crystal, dispersion, fiber, optical, wave equation.
1 Introduction
Propagation of optical pulses in the photonic crystal fibers (PCFs) has been always considered by many
photonics researchers. Also many nonlinear applications such as supercontinuum generation (SCG),
optical pulses compression, and etc [1] have been described for this type of waveguides. Since the
nonlinear refractive indexes of silica- guiding PCFs are small and also due to the ability of these
waveguides to perform large anomalous dispersions [1, 2], many researchers prefer them to other different
structures of these fibers for using in higher - power optical pulses compression. The role of Raman
scattering in increasing the quality of the optical pulses compressed by soliton - effect has been specified
in standard fibers [3, 4]. Although the silica index guiding PCFs inherently have low nonlinearity, but if
the input pulse width is small, (femtosecond duration) and a quite high input pulse power is selected, both
* Corresponding Author. Email address: [email protected], Tel: +989359976241
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the TOD and Raman scattering can be effective further compression of optical pulses through propagation
in the waveguides. Some related works are mentioned in the following, for example,the generation of
femtosecond optical pulses by the interaction between the third-order dispersion and the XPM was done in
briefringent fibers [5]. Also, generation of femtosecond optical pulses by the interaction between the
negative third-order dispersion and the Raman effect was observed theoretically in hollow-core badgap
fibers [6] or, Ouzonov proved the effect of TOD on the spectral and temporal changes of optical pulses,
and Gerome had an experimental observation of energy transfer from high frequencies to low frequencies
due to the Raman effect [3]. The reports of Ouzonov and Gerome working groups are common in focusing
on optical pulses deformation caused by TOD occurring at the optical pulses edges in the form of
continuous fluctuations. They also emphasized that TOD is the most important preventing factor in an
ideal pulse compression. Totally, it seems we need small (almost zero) or negative values of TOD in order
to have an ideal compression. Furthermore, according to our researches soliton-effect compression of lowenergy 850nm ultrashort pulses induced by negative third-order dispersion in solid-core PCFs has not been
studied so far. Ultrashort pulses generation at 850 nm can be specifically employed in material processing,
biophotonic sensors, etc [3, 4]. However considering the best existing lasers as the sources, producing such
a short pulse at near visible wavelength is still difficult. Our paper is organized as follows: in section II, we
consider an important algorithm for the approximate solution of the schrodinger equation, namely rhe
fourth-order runge-kutta (RK4) method, section III we investigate the compression of chirped-free
femtosecond optical pulses generation in conventional solid-core PCF when TOD and Raman effect are
neglected. Section IV we consider the wave equation when only the Raman effect is included, section we
investigate the compression of femtosecond optical pulses generation when both TOD and Raman effect
are considered. Section VI contains the role of higher order solitons (corresponding to the input power) on
femtosecond optical pulses generation and finally Section VII contains conclusion.
2 Wave Propagation in Optical Fibers
The wave equation depicting the optical pulses propagation through the optical fibers might be stemmed
from Maxwell equations [4, 5]. Examining the linear and nonlinear effects on optical fibers needs utilizing
femtosecond width optical pulses. Also, when these pulses are propagated through the optical fibers, both
linear and nonlinear effects are applied to both the spectrum and shape of the pulses. With regard to A(z,t)
as the pulses amplitude, it shows that the differential equation for propagation of optical pulses through
PCFs named as General Nonlinear Schrodinger Equation (GNLSE) is generally formed as the following
[5]:
Linear  Effects
3  3A
i  2A
U
2
  22

 i ( A A 
2
3
3
z
2T 0 
6T 0 
SPM
i
0T 0
( A A )


2
Self steepening
TR
T0
A
A

2
(2.1)
Raman effect
Nonlinear effects
Where  
T0 
T
is normalized time, T is reduced time,T 0 is the initial pulse width corresponding to
T0
T FWHM
where T FWHM is the half maximum pulse width,  2 and  3 are Group Velocity Dispersion
1.76
(GVD) and Third-Order Dispersion(TOD) respectively. 0 is the central angular frequency, and  is the
nonlinear coefficient defined by the equation below,
n
  2 0
Aeff c
(2.2)
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Where n 2 ( m
2
W
) is the nonlinear refractive index having the value of 3.2  1020 m
W
for silica [4, 6],
z
) and the normalized
LD
and c refers to the speed of light. Utilizing the normalized distance (  
amplitude (U (z, ) 
2
A (z, )
) parameters, equation (2.1) can be normalized as the folllowing [3, 5],
P0
Linear  Effects
2
2
U
i sgn(  2 )  2U
 3  3U
i ( U U ) T R  U
2


 i( U U 
 U
)
2
3

2
6  2 T 0 
0T 0

T0


(2.3)
Nonlinear effects
In which P0 is the peak power of the input pulse,T R is the Raman gain slope(3fs for silica [5]) and L D is
the dispersion length defined by equation (2.4),
LD 
T 02
(2.4)
2
The input pulses are supposed to be as the following form,
T
U (0,t )  N sec h ( )
T0
(2.5)
In this equition, N shows the soliton-order (related to the input power of initial optical pulses [5, 6])
defined by the following relation,
N 
LD
L NL
(2.6)
Where L NL is he nonlinear length, defined by equation (2.7),
L NL 
1
 P0
(2.7)
Having mentioned the existing relations and taking a pulse width of T 0  0.3 ps (300fs) in to account, the
current study tries to investigate the simulation steps for third-order soliton (N=3) propagation through the
2
ps
)
proposed PCF. Moreover, the value of GVD is calculated as,  2  18.15(
km for all the simulation
steps in this study. Further, the authors consider a novel algorithm, the concrete algorithms of Split-Step
Fourier Method and the fourth order Runge-Kutta (RK4) Method, for the approximate calculation of
solutions of the pulse propagation equation. The followings are the procedures:
Initially, Phase 1: we offer the Split-Step Algorithm to finding approximate solutions of the pulse
propagation equation. Thus, we can write Equation (2.3) as:
U
(2.8)
 (L N) U

Where L and N are the linear and nonlinear operator, respectively, acting on the envelope function
[3,7,9],
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L
i 2
3 3

2  2 6  2 T 0  3
(2.9)
3
2
2
( U )
1 (U U )
N(U )  iN ( U  iS
 R
)
U


2
2
Integrating equation (2.8) over the variable
(2.10)
 in the interval      , we obtain,
U(   , )  exp(A B) U( , )
(2.11)
With
 
A

 
L    L

   L
(2.12)
 
B

N (U ( , ))   N (U ( , ))
(2.13)
When the step-size  of the propagation distance is very small, using the Baker – Campbell formula for
the exponential operator involved in equation (2.10) Further, its approximate form can be calculated as [510],
A
A
exp( A  B )  exp( )exp(B)exp( )
(2.14)
2
2
In so doing, it is supposed that operators A and B are commutative between themselves when  is small.
The error of the formula (2.14) is of the order  2 . By replacing above the expression into equation (2.10)
we will reach the following formula,


 exp(
L)exp(  N (U ( , )))  exp(
L) U( , )
2
2
(2.15)
This expression allows us to specify the almost value of the envelope function in the location
  
from its value in the  .
In order to estimate the value of the envelope function by using equation (2.15), we need to know the
calculating procedure for the action of the envelope function. As these operators include time partial
derivatives, we can estimate them by Fourier transform. In doing so, we include the value of the time
variable in the finite interval [a, b] which is also large. Moreover, its borders don’t have any effect on the
final results of the calculations. Then, we consider the periodic condition on borders that
U ( ,a)  U ( , b) for   [0, 0 ] . Further, to facilitate the procedures, we change the variable in equation
(2.15) in a way that it normalizes the interval [a, b] in to the interval [0,2 ] , and then we divide this
interval into N points with distance between them   2
We represent these points as  j  2 j
N
N
.
, j=0,1,2..,N. Thus, we have the discrete Fourier Transform of
the series U ( ,  j ) as follows,
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U ( , K )  FK [U ( , J )] 
N 1
U ( ,
1
N
J
)exp( i K  J )
j 0
(2.16)
N
N
  K 
1
2
2
The definition of The Inverse Fourier Transform is,
N
U ( , J ) 
F j1[U
( , K )] 
2
1
 U ( ,
K
k N
)exp(i K  J )
(2.17)
2
j  0,1,2...N
F here shows Fourier Transform and F 1 signifies its inverse transform. By utilizing the Calculations
mentioned in (2.16) and (2.17) by the fast algorithm FFT [5-10], the procedures would be fast. In both
linear and nonlinear operator (2.9) and (2.10), the time partial derivatives of the envelope function in the
can be easily estimated by multiplying the Fourier coefficients U ( , k ) by powers of i k
corresponding to the order of derivative and then taking the Inverse Fourier Transform. Furthermore, we
can solve Equation (2.3) approximately by using the Runge- Kutta (RK) algorithm in which the time
discretization and calculations of time partial derivatives are the same as the previous description.
However, the spatial derivatives are calculated by Runge-Kutta (RK) algorithm. As very popular for
solving the differential equations [9], fourth order, Runge-Kutta (RK) algorithm can be applied here. By
using using Fourier Transform for calculating the time partial derivatives, the equation (2.3) becomes,

i
2
( F [U ])  (( i  )2  ( i  )3 3 ) F [U ]  iN 2 [(1  iS ( i  ) F[ U U ]  ..

2
1
(2.18)
..   R F [UF [( i  ) F [U ]]]]
2
Showing
V  exp((
i 2
 i  3 3 ) ) F [U ]
2
(2.19)
We can reform equation (2.3) as,
V
 f ( ,U )

(2.20)
In which
f ( ,U )  iN 2 exp((
i  2
2
2
 i  3 3 ) )  [(1  S  ) F [ U U ]   R F [UF 1[(i ) F[ U ]]]]
2
We calculate the value of the function V in the location
Runge-Kutta (RK) algorithm for equation (2.20)
1
V(   )  V( )   [ K 1  2( K 2  K 3 )  K 4 )]
6
(2.21)
   [8-10] by the use of the fourth order
(2.22)
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Where the coefficients K i are calculated as follows,
K1   .f ( ,U ( , ))
(2.23)

1
,U ( , )  K1 )
2
2

1
K3   .f ( 
,U ( , )  K 2 )
2
2
K4   .f (   ,U ( , )  K3 )
K 2   .f ( 
(2.24)
(2.25)
(2.26)
From equation (2.19) and equation (2.22) we reach the value of the envelope function in the location
   [10, 11],
U(   )  F 1[V(   )exp(((
i  2
 i  3 3 )(   ))]
2
(2.27)
We have to consider the errors in applying equation (2.27) (  5 ). In compare to the procedures taken in
equation (2.15), equation (2.27) has a higher accuracy, even though the computational time is longer
because the number of calculation steps is chosen very large [9-12].
3 TOD and ISRS are Neglected
Regardless of the TOD and Raman effect, equation (2.3) will come in the form below,
Linear  Effect
U
i sgn(  2 )  2U

 i(

2
 2
2
U U
)
(3.28)
SPM
Nonlinear effect
Fig.1a and b shows the pulse evolution in a soliton period and the optimum compressed pulse when pulse
propagation in silica solid-core PCFs.
From Fig.1a we can see the high order soliton temporal evolution over one soliton period, which is given
by the equation,
Z
(3.29)
Z  1.570L D 
1
Z0
Z0
Fig.1b shows the optimum compressed pulse at Z  0.25Z 0 ,the temporal full width at half maximum
(FWHM) of the pulse is about 100fs,so the compressed factor is FC  T FWHM
TComp
 3 , where T FWHM
and TComp are the FWHM of the input pulse and the optimum compressed pulse width respectively.
Although the compressed pulse has a larger compression factor and higher peak power, The Normalized
peak Intensity (P.I) of the pulse is, P . I  6 . It is noted, however, that a part of the input pulse energy is
contained in the pedestal surrounding the compressed pulse (small peaks on the edges of the optical
pulse).This means that the temporal change result from an interplay between self-phase modulation (SPM)
and group velocity dispersion (GVD). The SPM creates a positive frequency chirp in such a way that the
leading edge is red shifted and the trailing edge is blue shifted from the central frequency. Simultaneously,
anomalous GVD contracts the pulse as the pulse is positively chirped. On the other hand, as the chirp is
approximately linear only over the central part, only the central portion of the pulse is compressed.
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(a)
(b)
Figure 1: (a) Pulse evolution in a soliton periode when TOD and ISRS are neglected;
(b) The optimum compressed pulse.
4 ISRS is Considered
If TOD is neglected while ISRS is considered, equation (2.3) changes into,
Linear  Effects
i  2  2U
TR  U
U
2


i(
U
U

U
)

2  2
T0

2
(4.30)
Nonlinear effects
Fig.2a shows the pulse evolution under such conditions for third-order soliton propagation in a soliton
period. In this case, due to the soliton self-frequency shift, a narrow spike separates from its main pulse
and moves toward the right. As can be seen from Fig.2b, the pulse spectrum shifts toward longer
wavelength side. For the input pulse widths about 1ps or shorter, the spectral width of the input pulse is
very large. Therefore, the Raman gain can increase the low-frequency spectral components of the pulse.
Also, high frequency components of the same pulse acts as a pump source In this case. Fig.2b depicts the
frequency shift increases with propagation distance, and further simulation indicates that the narrower of
the input pulse duration, the larger of the frequency shift. And, as the red-shifted components travel slower
than other elemnts, they are delayed with respect to the input pulse.
Fig.2c depicts the FWHM of pulses as a function of Z
Z 0 .Initially, the pulse is compressed and the width
of the main peak reaches a minimum value of 85fs when propagates into Z  0.43L D . After that a period
of oscillations and the temporal FWHM of pulses become smooth with slow increase, showing the
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formation of red shifting solitons. The oscillations can be described as following: The Raman effect is seen
stronger around the maximum compression point, and the considerable amount of optical pulses spectra is
shifted towards the higher wavelengths, stemming from the beating and vibrations of the separated parts of
the spectrum. Then, this reduced amount of beatings and vibrations will help to decrease the fluctuations,
resulting in a slow increase in the pulse intensity. Furthermore, comparing the obtained results from Figs.
1a & 1b, it is clear that the peaks appearing in pulse edges (due to Pedestal phenomenon) have been greatly
improved because of the Raman effect. This causes the combination of the pulse's main energy on the side
peaks rather than the main peak; finally it makes an increase in the pulse quality compared to Fig.1.
(a)
(b)
(c)
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(d)
Figure 2: (a) The pulse evo;ution under conditions identical to those of Fig.1 except that ISRS is considered;
(b) The pulse spectrum corresponding to (a);
(c) Temporal FWHM of pulses as a function of normalized distance;
(d) The compressed pulse shape at Z  0.43Ld .
5 ISRS and TOD are Both Considered
When considering both TOD and ISRS effects, we fully employ equation (2.3) in investigating their
simultaneous impacts on the optical pulses propagated in PCFs. It is noteworthy that, since the TOD (slope
of GVD [6]) in solid core PCFs is relation to the fiber geometry parameters, such as the hole pitch (  )
and normalized hole diameter (
d
) (the definition of all the parameters see Fig.3 [14, 15]). So by changing

the fiber geometry, we can change the value and the sign of the TOD. According to Ref. [13], by applying
empirical relations method (ERM), in the simulation, we take the parameters of  3 ( ps
and -0.03 with using   (1,0.95,0.9) and
3
km
) are 0.03,0
d
 0.65,0.68,0.7 at 850nm in our proposed PCF

respectively. It should be noted that, as we already described in Section II, due to slight changes in the
GVD values of proposed PCF with variation of  and
d
(as mentioned above), a certain value of,

2
 2  18.15 ( ps km ) , is estimated for all calculations process in this paper at 850nm.
Figure 3: The proposed PCF schematic [15].
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Fig.4 depicts the temporal FWHM of third –order soliton pulses as a function of Z
Z 0 .It shows that the
aggregated influence of the positive TOD and the ISRS increases the width of the compression pulse (as
the intensity decrease), and more of the positive TOD, the more of the pulse width (the Green curve),
thought the combined effects of the zero and negative TODs and the ISRS are on the contrary (Red and
blue curves). Fig.5.a&b illustrates the corresponding shapes at Z
Z0
 1 . As the figs show the combined
effects of the negative TOD and the ISRS increases the relative peak intensity of the compression pulses in
compare to the positive TOD condition. This is due to the role of the ISRS inducing red shifting of the
soliton wavelength. According to the TOD definition [3, 6, 8, 13, 14],

3  2
(5.31)

By considering the negative TOD (  3  0 ), at the longer wavelength (   0 ), then we have,  2  0 ,
this indicates that  2 decreases in the anomalous dispersion region. That is, the nonlinear effect increases
because of the decrease in  2 , and in this way then the compression of the pulse is enhanced.
The merit of this method is achieving a higher compression factor of propagated optical pulses in optical
fibers using much less energy. In soliton- effect method the less power of in optical pulses (corresponding
to order of soliton(N)) we have, the less physical destruction of fiber occurs.In this novel method for
achieving ultrashort optical pulses in addition to increasing the input power, an appropriate design can be
applied to gain higher negative third-order dispersion and combining that with Raman-effect will result in
sharper optical pulses generation with less physical destruction of fiber in comparison with other methods.
3
Figure 4: The peak intensities of the optical pulses for different TOD values ( 0.03 ps km (Green), 0(red) and
0.03 ps
3
km
(blue)) in the presence of the ISRS.
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(a)
(b)
Figure 5: The compressed pulse at Z  Z 0 ,
(a) For negative TOD,
(b) For positive TOD, both in presence of the ISRS.
6 Impact of Higher Oder Soliton
In order to study the effects of higher –order solitons (N) in the optical pulse compression factor, we
3
ps
)
simulate equation (2.3) for  3  0 and 3  0.03(
km in the presence of the ISRS and propagating
of soliton order of N=4 and the assumed input pulse width of T 0  300fs respectively. Fig.5a and Fig.6 a
& b, including the simulation results, show that as the order of soliton goes higher the optical pulse starts
narrowing in shorter distances of the fiber. This expresses that, the optical pulse is more compressed. In
this case, the presence of the negative TOD and the combination of that with the ISRS results in a
significant increase (nearly 2 times more than before) in main peak of the optical pulse in comparison with
the zero TOD and also lower – order solitons.
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(a)-(b)
Figure 6: The compressed pulse in PCF for:
(a) The combination of the negative TOD and ISRS;
(b) The combination of the zero TOD and ISRS.
7 Conclusion
In conclusion, we investigated numerically the soliton formation during compression of unchirped secant
optical pulse in solid-core photonic crystal fibers. It was shown that by adjusting the geometrical
parameters of the PCF such as pitch (  ) and normalized air holes diameter ( d ), we are able to make

dispersions negative, zero or positive and totally, optimized its value as needed for further compression of
optical pulses; Also The results indicate that, as in the modified hollow core band gap PCFs, the positive
TOD will limitate the pulse compression. however, the combined effect of negative TOD (or even zero
TOD) and ISRS can improve the performance and quality of compression pulse, and with the increase of
the peak power of the input pulse (corresponding to use of the higher order solitons), the compression
factor will increase in the solid core PCFs with different Values of compression factors.
References
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