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Evolution of light trapped by a soliton in

a microstructured fiber

S. Hill, C. E. Kuklewicz, U. Leonhardt, F. K¨ onig

School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, Fife,

KY16 9SS, UK.

fewk@st-andrews.ac.uk

http://www.st-andrews.ac.uk/ qinfo/

Abstract: We observe the dynamics of pulse trapping in a microstructured

fiber. Few-cycle pulses create a system of two pulses: a Raman shifting

soliton traps a pulse in the normal dispersion regime. When the soliton

approaches a wavelength of zero group velocity dispersion the Raman

shifting abruptly terminates and the trapped pulse is released. In particular,

the trap is less than 4ps long and contains a 1ps pulse. After being released,

this pulse asymmetrically expands to more than 10ps. Additionally, there is

no disturbance of the trapping dynamics at high input pulse energies as the

supercontinuum develops further.

© 2009 Optical Society of America

OCIS codes: (060.7140) Ultrafast processes in fibers; (190.5530) Pulse propagation and tem-

poral solitons; (190.4370) Nonlinear optics, fibers

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1. Introduction

When a short and intense pulse of light is launched into a microstructured fiber (MF), non-

linear effects can significantly broaden the spectrum to much more than an octave [1]. These

supercontinua are used in a variety of applications such as ultrafast optical switching, spec-

troscopy, optical coherent tomography, optical clocks, etc. Routinely, around 100-fs pulses

are used at a wavelength close to a zero-group velocity dispersion wavelength (ZDW) of the

fiber. Using nearly octave–spanning input pulses [2, 3, 4], the phenomenon of ‘pulse trapping’

[5, 6, 7, 8, 9, 10] can be observed. In this effect a non-dispersing pulse can form at the short

wavelength end of the broad spectrum. This is surprising because the pulse exists in a region

of normal dispersion in the fiber, where the dispersion-induced chirp cannot be cancelled by

self-phase modulation. The non-dispersing pulse is trapped behind a fundamental soliton in the

anomalous dispersion regime that was generated by soliton fission. Soliton fission can create

multiple fundamental solitons, which shift to longer wavelengths via the soliton self-frequency

shift (SSFS) [11].

Nishizawa and Goto [6] were the first to demonstrate that a soliton undergoing the SSFS can

trap light behind it. The trapped light adjusts to a wavelength that is group velocity matched

to the soliton. Because it is forced to travel with the soliton and lies in the normal dispersion

regime, it has to shift to shorter wavelengths in order to keep the same group-velocity as the

soliton. Gorbach and Skryabin provided an alternative view of pulse trapping [8]: the light is

trapped because the soliton accelerates. In simulations they turned off the (negative) accelera-

tion induced by the SSFS and the trapping ceased. The acceleration of the soliton provides a

‘gravity-like’ potential to the trapped pulse.

In this paper we study experimentally the phenomenon of pulse trapping in a fiber with two

ZDWs. We use intense few-cycle pulses to generate a soliton and a trapped pulse and focus in

particular on the trap dynamics as the soliton reaches the longer zero-dispersion wavelength and

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decays. We observe how the light escapes the trap, expanding to a few times the trap length. For

higher input pulse energies, a further component in the spectrum at even shorter wavelengths is

formed, which dominates the blue end of the evolving supercontinuum.

2.Pulse Trapping

As previously shown, pulse trapping can be explained by an effective potential that is produced

by an accelerating pulse – the fundamental soliton undergoing SSFS [8]. The discussion below

follows this analysis. The soliton induces a nonlinear modification of the refractive index, n,

via the optical Kerr effect [12]:

n(I,ω) = n0(ω)+n2I,

(1)

where I is the intensity and n2the nonlinear index coefficient, assuming an instantaneous

response, and n0is the linear index.

Any light field A in the fiber that interacts with the soliton sees the nonlinear contribution

to the refractive index, n2I = n−n0, and experiences cross-phase modulation (XPM) [13]. The

evolution of the slowly varying envelope A(z,t) of this light is governed by [12]:

∂A

∂z+β1∂A

∂t+iβ2

2

∂2A

∂t2= iVA.

(2)

z and t are space and time in the laboratory frame. β1and β2are the dispersion parameters,

where βn=

∂ωn

c

dispersion regime, β2is positive. iVA is the XPM-term and V(z,t) = 2γ|As(z,t)|2, where Asis

the envelope of the soliton and γ is the nonlinear parameter.

In the co-moving frame given by T = t −β1z and Z = z, eqn. (2) is transformed into the

nonlinear Schr¨ odinger equation

∂n

?n(ω)ω

?evaluated at the frequency of A. Since this frequency lies in the normal

∂A

∂Z+iβ2

2

∂2A

∂T2= iVA.

(3)

In analogy to the ordinary Schr¨ odinger equation in quantum mechanics, the XPM contribu-

tionV induced by the soliton plays the part of a potential barrier.

Equation (3) is transformed into an accelerating frame [8]:

τ = T +αZ2/2

ζ = Z.

(4)

Here α describes the rate of the SSFS, which determines the soliton acceleration. For a

soliton,α =−8TRβ2

velocity dispersion at the soliton wavelength, and the minus sign indicates that the accelerating

soliton is redshifting and slowing down. The transformation fixes the peak of the accelerating

soliton at τ = 0. In this frame, the propagating light A acquires a phase φ(ζ,τ), so A(Z,T) is

replaced with ψ(ζ,τ) [14]:

2,s/15T4

0[12].TRistheRamantime;thesolitonlengthisT0.β2,sisthegroup

ψ = Aeiφ

φ = −αζτ

β2

+α2ζ3

6β2

.

(5)

The propagation equation (3) becomes

∂ψ

∂ζ+iβ2

2

∂2ψ

∂τ2= iUeff(τ)ψ,

(6)

where

Ueff(τ) = (V(τ)−ατ/β2).

(7)

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Effective?potential?U ( )

eff?

V(0)

Time

Fig. 1. The potential created by the accelerating soliton. The induced nonlinear index

change forms a barrier at τ = 0. The soliton acceleration causes the slope in the poten-

tial.

Therefore, in a frame that accelerates with the soliton, other waves propagate in the presence

of an effective potential Ueff. The first term in equation 7 is the barrier due to XPM and the

second term is a linear potential due to the (negative) acceleration α. Ueffis sketched in figure

1. The barrier is located at τ = 0 and at τ = τ?the linear increase reaches the height of the

barrier.

Gorbach and Skryabin pointed out that the potential is ‘gravity-like’, i.e. similar to the grav-

itational potential on earth. Standing in a lift, the floor acts as a strong potential barrier against

the gravitational force. Without gravity, the same potential would be created if the lift had a

constant upward acceleration. So a uniformly accelerating barrier can trap objects just as a

barrier in a gravitational field does.

As seen in figure 1, the potential can trap light between τ =0 and τ =τ?. We can estimate the

temporal length of the trap by calculating τ?usingUeff(τ?) =Ueff(0). ApproximatingV(τ?) ≈ 0

and inserting α we obtain for an accelerating soliton

τ?=15T2

0

4TR

β2

|β2,s|.

(8)

τ?depends on the SSFS as well as the ratio of group velocity dispersions for the trapped light

and the soliton. In our experiments the trap length is relatively constant, so that the trapped light

is expected to have a well defined temporal length. The trap is positioned immediately behind

the soliton center at τ ≥ 0. In consequence, the trapped light travels at the group velocity of

the soliton and slows down with it. The spectrum of the trapped light shifts to a wavelength

associated with the same group velocity as the soliton. Accordingly, the potential determines

both the spectral and temporal properties of the trapped light.

3.Experiment

To create the trapping soliton as well as the trapped light we couple a few-cycle pulse into a

microstructured fiber. We use pulses from a <7fs Ti:Sapphire laser with a repetition rate of

78MHz (Rainbow, Femtolasers GmbH). A reflective-optics telescope expands the beam and

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-40

-30

-20

-10

10

20

30

40

600700 8009001000 11001200 1300

Wavelength?(nm)

Dispersion

(ps /km)

?2

2

0

Fig. 2. The group velocity dispersion, β2, of the fiber NL-PM-760 [15] and the initial spec-

trum of the few-cycle pump pulses (inset). The dispersion zeroes at 760nm and 1160nm.

GDD?=?x?-?30.0?fs

GDD?=?x?-?22.5?fs

GDD?=?x?-?15.0?fs

GDD?=?x?-?7.5?fs

GDD?=?x?-?0.0?fs

2

2

2

2

2

1000

Fig. 3. Output spectra for varying amounts of chirp on the input pulse. The Group-Delay-

Dispersion (GDD) before the fiber is varied over 30fs2. The pulse energy is 39pJ. A GDD

of 7.5fs2broadens an unchirped 7-fs pulse by 10% in time.