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Optimal Chirp Parameter Determination for Soliton Pulse Compression by Cascaded Quadratic Nonlinearities                                                                                                      Author

Kai Xiong

Supervisor Prof. Morten Bache, DTU Fotonik

© September 5, 2011 by Kai Xiong

Technical University of Denmark © Kai Xiong DTU Fotonik

September 5, 2011     

                                      © Kai Xiong DTU Fotonik

September 5, 2011     

Contents 1. Simple Pulse-Propagation Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 2. Introduction to Split-Step Fourier Method . . . . . . . . . . . . . .. . . . ….. 4 2.1 Split-Step Method 2.1.1 Unsymmetrized Split-Step Scheme . . . . . . . . . . . . . . . . . . .. . .4 2.1.2 Symmetrized Split-Step Scheme . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Runge-Kutta-Fehlberg Method. . . . . . . . . . . . . . . . . . . . . . . . . . . .8

 

3. Introduction to Soliton and its formation . . . . . . . . . . . . . . . . . . . . . . 9 3.1 Group-Velocity Dispersion (GVD) . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Self-Phase Modulation (SPM) . . . . . . . . . . . . . . . . . . . . . . . . ….. . . 13 3.3 Soliton Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4. Chirp Parameter Determination and its impact on Solitons . . . . . .15 4.1 Soliton Compression Process with 0.03 TOD. . . . . . . . . . . . . . . . . . . 16 4.2 Soliton Compression Process with more TOD. . . . . . . . . . . . . . . . . . .19 4.3 Soliton Compression with different soliton numbers. . . . . . . . . . . . . . 24 4.4 Soliton Compression with varying GVD. . . . . . . . . . . . . . . . . . . . . . . 26 4.5 Relationship between soliton compression distance and chirp value. .28 4.6 Relationship between soliton compression distance and soliton order 29 4.7 GVD’s impact on the optimal chirp value . . . . . . . . . . . . . . . . . . . . . . 30 4.8 Poor compression cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 References

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List of Figures    Fig 2.1 Schematic illustration of the symmetrized split-step Fourier method used for numerical simulations…………..……………………………………………………………… …7 Fig 4.1.1 the relationship between the Full Width at Half Maximum and the Chirp parameter ……..………..………..………..………..………..………..………..……… 16 Fig 4.1.2 the relationship between soliton formation distance and the Chirp parameter……..………..………..………..………..………..………..………..………. 17 Fig 4.1.3 Compression factor versus chirp ……..………..………..………..………..…18 Fig 4.1.4 the time plot of the input soliton and output soliton ……..………..………..18 Fig 4.1.6 Soliton compression distance versus time, frequency ……..………..………..19 Fig 4.2.1 Full Width at Half Maximum versus Chirp with TOD 0.05……..……….. …19 Fig 4.2.2 Soliton compression distance versus Chirp with TOD 0.05……..……………20 Fig 4.2.3 Compression factor versus Chirp with TOD 0.05………………………… …20 Fig 4.2.4 the time plot of the input soliton and output soliton with soliton order 2, TOD 0.05 , chirp 1.24…………………………………………………………………………21 Fig 4.2.5 Soliton compression distance versus time, frequency, respectively, with soliton order 2, TOD 0.05 , chirp 1.24…………………………………………………… ……21 Fig 4.2.6 Full Width at Half Maximum versus Chirp with TOD 0.1……………………22 Fig 4.2.7 the relationship between soliton formation distance and the Chirp parameter with TOD 0.1……………………………………………………………………………22 Fig 4.2.8 compression facto versus Chirp with TOD 0.1……………………………… 23 Fig 4.2.9 the time and frequency plot of the input soliton and output soliton with soliton order 2, TOD 0.1 , chirp 1.26…………………………………………………… … … 23 Fig 4.2.10 Soliton compression distance versus time, frequency, respectively, with soliton order 2, TOD 0.1, chirp 1.26……………………………………………………24 Fig.4.3.1 FWHM versus Chirp under varying soliton numbers with TOD 0.03… ……24 Fig.4.3.2

Compression factor VS Chirp under varying soliton numbers………… … 24

Fig.4.4.1 TFWHM versus varying GVD with optimal chirp………………… …… … 25 Fig.4.4.2 TFWHM versus varying GVD with constant chirp……………… … ………26 Fig.4.5.1 compression distance versus varying GVD………………… …… …………28 Fig.4.6.1 compression distance versus soliton order……………………………………29 Fig.4.6.2 FWHM versus soliton order………………………………………… … ……29 Fig.4.7.1 Full width at half maximum versus varying Chirp with constant GVD(0.05) …………………………………………………… ……………… ………30 Fig. 4.7.2 Compression Factor versus varying Chirp……………………… ……… …31 © Kai Xiong DTU Fotonik

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Fig.4.7.3 FWHM versus Chirp with varying GVD……………………………………31 Fig.4.7.4 Intensity versus time, frequency……………………………………… … …33 Fig.4.7.5 Propagation distance versus time, frequency……………………… … ……33 Fig.4.8.1 Intensity versus time, frequency……………………… … …………………34 Fig.4.8.2 compression distances versus time, frequency………………………………34 Fig.4.8.3 Intensity versus time, frequency………………… …………………………35 Fig.4.8.4 Compression distance versus time,frequency … ……………………………35

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    Preface            This report covers the special course “Optimal Chirp Parameter Determination for  Solion Pulse Compression by Cascaded Quadratic Nonlinearities . This project was  carried out in August, 2011 at DTU‐Fotonik at the Technical University of Denmark.  The course as a whole was rated to 5 ECTS points.                         The project is about determining the optimal chirp parameter in the soliton pulse  compression, which serves the experimental work of the Femto‐VINIR project.                                            

Acknowledgement 

          My acknowledgement goes towards my supervisor, Associate Professor, Ph.D.  Morten Bache (DTU Fotonik), for the supervision and constructive dialogue during  the progress of this project.                            

Responsible author of this report 

                  Kai Xiong            © Kai Xiong DTU Fotonik

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                                                                                                          Chapter 1 

Introduction to Pulse‐Propagation Equation      As it is well known, the propagation of soliton in optical fibers is governed by Maxwell’s equations. To master the properties of solitons, we shall start from the Maxwell’s equations and deduce a specific equation for solitons. It is a set of partial differential equations, which have the form:

∇× E = −

dB , dt

∇× H = J +

dD , dt

(1.0.0)

(1.0.1)

∇× D = ρf ,

(1.0.2)

∇× B = 0,

(1.0.3)

where E and H are the electric and magnetic field vectors and D and B are electric and magnetic flux densities. J is current density vector, ρ f represents the charge density. It is noteworthy to mention that the above equations are all in International System of Units. The wave propagation is derived by joining the above four Maxwell’s equations. If we simplify the equations in the way that canceling the electric and magnetic field vectors D and B, a foundational equation is achieved:

∇×∇× E = −

1 ∂2E ∂2P μ − 0 c 2 ∂t 2 ∂t 2

(1.0.4)

Normally, to evaluate the polarization P, we may have to use quantum-mechanical approach . Yet, if only taking third-order nonlinear effects-which are governed by

χ (3) -into account, we can split the polarization into two parts: linear part PL and the © Kai Xiong DTU Fotonik

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non-linear part PNL. Thus, Equation 1.0.4 can be rewritten with the substitution of PL and PNL:

∇×∇× E = −

2 ∂（P 1 ∂2E L +PNL） μ − 0 2 2 ∂t 2 c ∂t

(1.0.5)

In the real world, the nonlinear polarization changes in the refractive index are always >1, SPM dominates. The value of GVD just means when the SPM will be compensated and when the soliton forms. As the simulation shows, the higher soliton order is, the smaller GVD value will be.

4.5 Relationship between soliton compression distance and chirp value As shown above, the TFWHM of different GVD may be varying. It raises our curiosity that how’s the compression distance change. The simulation shows that different soliton order really matters while GVD doesn’t affect so much the compression distance.

Fig.4.5.1 compression distance versus varying GVD shown that the change in the compression distance is trivial. It almost remains constant. However, the higher soliton order will have a shorter compression distance under the optimal chirp value. This shows that the soliton compression process only concerns the soliton order rather than the transmission material(GVD). Different transmission material will roughly give the same compression distance. The reason is that the soliton number decides the effective distance of SPM and GVD. so if the soliton number is determined, the compression distance will be roughly constant.

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4.6 Relationship between soliton compression distance and soliton order While GVD doesn’t have an apparent impact on the compression distance, the soliton order does play a significant role in the compression process. With constant chirp and GVD, the soliton compression distance and full width at half maximum(FWHM) are varying with soliton order, as follows:

Fig.4.6.1 compression distance versus soliton order shown that compression distance lessens considerably with increasing soliton order.

Fig.4.6.2 FWHM versus soliton order shown that high soliton orders will have © Kai Xiong DTU Fotonik

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a better compression performance. Within an acceptable range, the higher soliton orders, the shorter the compression distances and the smaller the full width at half maximum. It is because that the intensity is proportional to soliton order. A high order soliton usually indicates high intensity. So if the soliton is compressed, in order to keep the total area consistent, the soliton will be therefore thinner. That is the reason why high order solitons have a better compression performance. Regarding the compression distance, high soliton order also implies more SPM effect. Normally, GVD will first affect the pulse. However, as the pulse propagates, SPM will gradually compensate GVD. Finally the two effects will reach a balance. The soliton is therefore formed. Since for high order solitons, the SPM effect is enhanced. So the soliton will be formed earlier. Thus, the higher soliton orders are, the shorter the compression distance will be.

4.7 GVD’s impact on the optimal chirp value In 4.3, a set of simulation seeking the optimal chirp value for different solitons is presented. While it is done with GVD 0.03, this time, a similar simulation will be repeated but the GVD is increased to 0.05.

Fig.4.7.1 Full width at half maximum versus varying Chirp with constant GVD © Kai Xiong DTU Fotonik

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(0.05) shown that there is an optimal initial chirp value for soliton compression process. Comparing to the simulation with GVD 0.03, the cases with GVD 0.05 have different optimal chirp values

Fig. 4.7.2 Compression Factor versus varying Chirp shown that the curve of compression factor is almost parabolic and it peaks at certain chirp value. for varying GVD, the FWHM of second order soliton can be shown as follows:

Fig.4.7.3 FWHM versus Chirp with varying GVD shown that solitons of different GVD will have similar plot of FHWM versus Chirp. However, GVD makes the optimal chirp value a little shifted. © Kai Xiong DTU Fotonik

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Soliton Order

Optimal Chirp Value

TFWHM (fs)

2

1

39.88

3

0.45

20.18

4

0.36

12.22

5

0.21

9.42

Since the Chirp will compensate GVD, the optimal chirp will be different if the GVD is varying. In the end, the GVD and SPM will reach a new balance, where the soliton will be formed. In addition, High GVD will lead to soliton phase shift. Regarding the third order dispersion, the phase shift is

LD =

T02

β3

. So

δ3 =

ΔΩdw =

β3 6T0 β 2

3 β2

β3

. And the third order dispersion is

δ3 =

LD β 3 , 6T03

.

Thus, we can derive the relationship between nonlinear phase shift and third order dispersion:

ΔΩ dw =

1 2δ 3T0

(4.7.1)

As for dispersion of order 4 or above, the relationship between phase shift and it is rather complex, which is out of the topic of this paper. So we don’t discuss it here.

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TOD generated soliton phase shift:

Fig.4.7.4 Intensity versus time, frequency shown that the soliton is compressed and nonlinear phase shift is also generated. The calculation can be done using formulas above. Sidelobes also appeared because of the high TOD.

Fig.4.7.5 Propagation distance versus time, frequency shown that the soliton is formed at 0.2. and after that, its direction is more inclined to right-hand side due to the phase shift.

4.8 Poor compression cases As we know, if the normalized GVD coefficient

β2

is negative, then the chirp shall

be positive in order to compensate GVD. Thus, the soliton will be formed earlier. However, the chirp can also be negative, which will postpone the soliton formation and the compression will be affected as well. Additionally, If the absolute value of the negative chirp is not too big, then the pulse can still be compressed, and it will not be worse than that with some positive chirp. On the other way round, If it is too great, the noise generated would be unacceptable. © Kai Xiong DTU Fotonik

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To make it clear, the initial chirp is set to -6, and the soliton order is 2. The outcomes are as follows: N=2，C= -6：

Fig.4.8.1 Intensity versus time, frequency, respectively shown that the compressed soliton is around 200 fs broad and the so is the FWHM. It proves that too much negative chirp could lead to bad compression performance.

Fig.4.8.2 compression distances versus time, frequency, respectively shown that there is some noise during the soliton propagation and the soliton forms at a very long distance. Additionally, even if the chirp is positive, if it exceeds certain value, the soliton could not be compressed, either, because it over compensates GVD. The chirp in this simulation is set to 25 and the 2nd order soliton is chosen. N=2，C= 25： © Kai Xiong DTU Fotonik

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Fig.4.8.3 Intensity versus time, frequency, respectively shown that the initial pulse is almost flat and the pulse is compressed eventually. However, it is not as good as the one with the optimal chirp value.

Fig.4.8.4 Compression distance versus time,frequency, respectively shown that the soliton formation also takes a long distance. It is all because that the chirp induced too much SPM so it breaks the balance between SPM and GVD. The pulse is also affected by the extra chirp.

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Conclusion  In the report, the soliton formation process is introduced. More importantly, the chirp mechanism is analyzed. We also determined the optimal chirp value for soliton compression by cascaded quadratic nonlinearities. Chapter 1 introduced the simple pulse propagation equation, which originates from the Maxwell’s equation. It shall be noted that higher-order effects is neglected. Chapter 2 outlined methods for solving NLS equation. It obtains an approximate solution to the NLS equation. It assumes that within a small distance h, there is no interplay between the nonlinear and dispersive effects. Additionally, the symmetrized scheme owns the leading local error which is of third order in the step size while in the unsymmetrized split scheme, it is only of second order in the step size. The Runge-Kutta-Fehlberg Method is also proposed as an alternative and direct way to achieve the solution of NLS equation. Chapter 3 described the mechanism that how the soliton is formed by introducing the Group-Velocity Dispersion (GVD) and Self-Phase Modulation (SPM). Soliton is formed by the interplay of the above two factors. In addition, the soliton formation process is also described in detail. Chapter 4 outlined the chirp and its impact on the soliton formation. If the nonlinearities is positive (self-focusing) and dispersion anomalous, we have solitons. Since the nonlinear refractive index change is positive, the pulse will become positively chirped. The GVD will compensate for the chirp because anomalous GVD gives a negative chirped pulse. If C>0, then the soliton will compress more and earlier because the GVD will be compensated earlier. However, if the Chirp is arbitrary large, it will mess up the soliton as well. Additionally, we investigated the relationship among soliton compression distance, chirp value and soliton order. Eventually, We figured out the optimal chirp value of solitons of different orders and GVD and analyzed their inter-relationship. This report gives a general view of the soliton and its formation. More specifically, it achieves the optimal chirp value for experimental utility.

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Reference      [1] Govind P. Agrawal, Nonlinear Fiber Optics, Third Edition, Academic Press, 2001. [2] Govind P. Agrawal, Nonlinear Fiber Optics, Fourth Edition, Academic

2007

[3] M.Bache, J.Moses, F.W.Wise. “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities”[J], Optical Society of America. [4] Bo.Krag Esbensen. “Analytical and Numerical Methods in Nonlinear Optics”. Report [5] Michael Frosz and Ole Bang, “Implementing a Split-Step Fourier Scheme to solve the Nonlinear Schrödinger Equation”, DTU, 2009. [6] Oleg V. Sinkin, Ronald Holzlohner, Curtis R.Menyuk, “Optimization of the split-Step Fourier Method in Modeling Optical Fiber Communications Systems”[J]. Journal of Lightwave Technology, VOL.21, NO.1 [7] JOHN H.MATHEWS, KURTIS D.FINK, Numerical Methods using Matlab, Fourth Edition, Prentice-Hall Inc.

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