Supercontinuum Generation In Optical Fibres · Nonlinear Fibre ... nonlinearity compared with other...
Transcript of Supercontinuum Generation In Optical Fibres · Nonlinear Fibre ... nonlinearity compared with other...
Supercontinuum Generation InOptical Fibres
Ben Chapman
MRes Project Report
September 2010
Femtosecond Optics Group
Photonics Group, Department of Physics
Imperial College London
i
Abstract
In this work, a high power Ytterbium pump laser was used in a quasi-
continuous wave (CW) pump scheme to generate supercontinua in two dif-
ferent optical fibres, a depressed cladding, graded index highly nonlinear fibre
(HNLF), and a solid core photonic crystal fibre (PCF). Pumping a 300 m
length of the HNLF with a peak pump power of 140 W, a continuum with
a spectral flatness of 5 dB over 1000 nm with an average spectral power of
0.33 mW/nm was achieved. Pumping a 28 m length of the PCF gave a spec-
trum with a spectral flatness of 6 dB over 740 nm with an average spectral
power of 1.7 mW/nm.
The PCF exhibited two zero dispersion wavelengths (ZDW) and, in addi-
tion to the continuum, a broad (80 nm FWHM) high power (330mW) spectral
component was generated at 1.98 µm, in the normal dispersion region past
the second ZDW. Through computer simulations and an understanding of
the physical mechanisms, this was unambiguously understood to be generated
through the interaction of solitons and dispersive waves, first the emission
of ‘Cherenkov’ radiation by solitons at the ZDW, and then through the in-
teraction of the Cherenkov radiation with solitons in the continuum through
soliton FWM. This is the first experimental demonstration of soliton FWM
in the context of CW pumped supercontinuum generation.
ii
Acknowledgements
I owe a debt of gratitude to Prof. JR Taylor and Dr. SV Popov for their
support and supervision this year, but also for letting me back in the lab in
the first place and giving me the opportunity of studying for a PhD. I would
also like to thank my fellow students EJ Kelleher and C Schmidt Castellani
for their company, ideas and after-work beers. I am also grateful of Dr. JC
Travers, for letting me tap his knowledge of supercontinua and particularly
for all his help with numerical simulations.
iii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Guiding Mechanism in Optical Fibres . . . . . . . . . . . . . 2
2.2 Linear and Nonlinear Propagation . . . . . . . . . . . . . . . 4
2.3 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Self-Phase Modulation . . . . . . . . . . . . . . . . . . . . . . 7
2.5 The Non-Linear Schrodinger Equation . . . . . . . . . . . . . 7
2.6 Four-Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . 9
2.7 Modulation Instability . . . . . . . . . . . . . . . . . . . . . . 10
2.8 Optical Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.9 Raman Scattering and Soliton Self Frequency Shift . . . . . . 15
2.10 Solitons and Dispersive Waves . . . . . . . . . . . . . . . . . . 16
2.11 Solitons and the Evolution from MI to Supercontinuum . . . 17
3 Design of a Fibre Fuse Protector for use in High Power Fibre
Laser Applications . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Fibre Fuse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 The Fibre Fuse Protector . . . . . . . . . . . . . . . . . . . . 20
4 CW Continuum Generation in a Depressed Cladding Highly
Nonlinear Fibre . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1 Sumitomo Depressed Cladding Highly Nonlinear Fibre . . . . 24
4.2 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Monochromating Scanning Spectrometry . . . . . . . . . . . . 26
4.3.1 J10D Indium Antinomide Detector . . . . . . . . . . . 28
4.3.2 Lock-In Amplification . . . . . . . . . . . . . . . . . . 28
4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 29
4.4.1 Modal Content at Short Wavelength Edge . . . . . . . 32
4.5 Power Scaling of Continuum Output . . . . . . . . . . . . . . 33
5 Supercontinuum Generation in Photonic Crystal Fibre . . . . 36
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5.1 Fibre T606-D . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2 Experimental Set-Up . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 39
5.3.1 Numerical Simulations . . . . . . . . . . . . . . . . . . 42
5.3.2 Cherenkov Radiation . . . . . . . . . . . . . . . . . . . 44
5.3.3 Soliton Four Wave Mixing . . . . . . . . . . . . . . . . 47
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
v
List of Figures
1 Example cross sections and refractive index profiles for stan-
dard, graded index and photonic crystal fibre . . . . . . . . . 4
2 Diagram of Self-Phase Modulation inducing a chirp on a pulse 7
3 The effects of a fibre fuse observed in a single mode fibre. . . 19
4 A fibre fuse propagating through a length of coiled fibre . . . 21
5 Schematic of fuse protector circuit. . . . . . . . . . . . . . . . 23
6 Dispersion curve and refractive index profile of Sumitomo HNLF 25
7 Experimental set-up used to generate CW supercontinua in
HNLF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8 Diagram of Spex 500M scanning spectrometer . . . . . . . . . 27
9 Transmittance of FEL1400 filter . . . . . . . . . . . . . . . . 28
10 Specified detectivity of J10D Indium Antinomide (InSd) de-
tector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
11 Spectrum of continuum output generated in 100, 200, 300 and
600 m lengths of HNLF for 87 or 140 W pump power at 1.07 µm 30
12 Output beam from HNLF at wavelength of 1.07 µm . . . . . 33
13 Output beam from HNLF at wavelength of 1.5 µm . . . . . . 33
14 Continuum output power from 300m length of Sumitomo
HNLF as a function of pump power. . . . . . . . . . . . . . . 34
15 Ouput spectrum from 300 m length of Sumitomo HNLF pumped
to 140 W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
16 SEM image of fibre T606D . . . . . . . . . . . . . . . . . . . 36
17 Dispersion curves for Fibre T606-D . . . . . . . . . . . . . . . 37
18 Experimental set-up used to generate CW supercontinua in
PCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
19 Cutback along 32 m length of T660D pumped at 1.07 µm to
133 W. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
vi
20 Total power in continuum output with fibre length for T606-D
and spectrum of output from 28 m of fibre . . . . . . . . . . . 41
21 Simulated evolution of spectrum in T606-D, averaged from
an ensemble of 100 individual simulations . . . . . . . . . . . 43
22 Computed phasematching curves between solitons and Cherenkov
radiation across the second ZDW in fibre T606D . . . . . . . 45
23 Shedding of dispersive radiation through the Cherenkov pro-
cess by a high power (4.5 kW) soliton across the ZDW . . . . 46
24 Matching curve for power-independent four wave mixing of
solitons in the presence of a CW dispersive line at either
1.07 µm or 1.98 µm . . . . . . . . . . . . . . . . . . . . . . . 48
25 Spectrogram showing generation of new frequency compo-
nents generated through power independent Soliton FWM . . 49
26 2.5 kW soliton inserted at 1.25 µm after 0.5 m propagation
distance with various simulation parameters . . . . . . . . . . 51
27 Spectrograms showing the generation of new a frequency com-
ponent around 2.2 µm due to soliton FWM . . . . . . . . . . 52
1 Introduction 1
1 Introduction
Optical fibres present an attractive platform for the study on nonlinear op-
tics. Although fused silica (even with dopants) has a relatively low material
nonlinearity compared with other platforms for nonlinear optics, nonlinear-
ities can build up over long propagation distances as fibres are able to guide
high intensity fields with low attenuation.
This work discusses the generation of spectrally broad (100s of nm) and
bright (10s of mW/nm) outputs, generated in optical fibres from high-power
(> 100 W), narrow (∼ 3 nm) continuous wave (CW) inputs. Broad spectra
such as these are referred to as supercontinua, and have found applications
in fields such as optical coherence tomography (OCT) [1, 2], fluorescence
imaging of cellular processes [3], spectroscopy [4] and telecommunications
[5] to name but a few. The main features of a supercontinuum are its
spectral width (generally 100s of nm), and high degree of directionality, due
to its spatial coherence. CW supercontinua (that is, supercontiuum sources
generated through continuous wave, rather than pulsed pumping) are also
notable for their relatively easy experimental realisation (essentially a CW
pump source and a length of fibre) and high spectral power.
A variety of nonlinear processes are involved in the broadening of a nar-
row CW input to a broad supercontinuum output, enabling the observation
of interesting and varied nonlinear processes in experimentally simple set-up.
The process is initiated by the break up of the CW pump field into a train of
solitons through modulation instability (MI, see Section 2.7). These solitons
will be seen to ‘shift’ their frequency through Raman scattering (see Section
2.9), broadening the spectrum. The interaction of solitons with dispersive
radiation can give rise to new frequency components. This was observed in
both experimental and numerical results, the details and analysis of which
are given in Section 5.
2 Background 2
2 Background
This section gives an overview of of the physical mechanisms involved in
the generation of supercontinua in fibre. Initially some background is given
on the structure and guiding mechanism in optical fibre (Section 2.1), de-
scribing the graded-index and photonic crystal fibre structures which form
the nonlinear fibres examined in Sections 4 and 5 respectively. Sections 2.2,
2.3 and 2.4 give an overview of the dispersive and nonlinear properties of
optical fibre, which leads on to the discussion of the nonlinear Schrodinger
equation (NLSE) in Section 2.5 and four-wave mixing (FWM) is in Section
2.6. Section 2.7 details the derivation of modulation instability (MI) as ei-
ther a FWM process or as an instability in the NLSE. MI is integral to the
continuum formation process as it causes the breakdown of the CW field
temporally, which in turn leads to the formation of solitons (discussed in
Section 2.8). These solitons then shift to new wavelengths due to Raman
scattering (Section 2.9) and can also interact with dispersive radiation to
generate new frequency components (Section 2.10).
2.1 Guiding Mechanism in Optical Fibres
The profile of a standard step-index fibre, shown in Figure 1, consists of a
core with a slightly higher refractive index than that of the cladding. The
difference in refractive index between the core and the cladding in conven-
tional step-index fibre is usually achieved through doping of the core with
germanium oxide, so as to raise its refractive index relative to the pure silica
cladding. Alternatively the cladding can be doped with fluoride to lower its
refractive index, or indeed a combination of the two.
The guiding of light down the fibre can be roughly thought of in general
terms as total internal reflection of the light at the interface between the core
and cladding. More accurately, the fibre forms a waveguide, where standing
wave solutions of the transverse field can be found from the cross-sectional
2 Background 3
structure of the fibre, giving a finite number of modes. The number and
form of modes available in the fibre is dependent on the wavelength of the
light propagating through the fibre. In the case where only one transverse
mode is available, the fibre is said to be a single mode fibre.
Other fibre geometries, beyond the normal step index fibre, can be desir-
able, to increase the nonlinearity of the fibre, or to tailor the fibre’s dispersive
properties. For example, in Sections 4 and 5, supercontinua are produced in
a graded-index fibre and photonic crystal fibre respectively. Graded-index
fibres (also shown in Figure 1) have parabolic index profiles across the core
region. Photonic crystal fibres are microstructured fibres which incorporate
airholes to form an effective index difference between the core and cladding
regions. Solid core PCFs (such as the one used in Section 5) are usually
formed of pure silica, with air holes running along the length of the fibre,
arranged in a regular pattern around the core (shown in Figure 1). The
index of the cladding region around the core is now effectively dependent on
both the glass and air parts of the cladding. This gives rise a large refractive
index contrast between the core and cladding regions, meaning PCFs will
have a strong optical confinement, resulting in high optical intensity in the
core. Because of this, the nonlinearity of PCF (see Section 2.2) can also be
orders of magnitude higher than conventional fibre. Tailoring the structure
of the PCF can allow for the fabrication of fibres with unique properties;
for example, zero dispersion wavelengths (ZDW, see Section 2.3) less than
1.27 µm are impossible for single mode conventional (step-index) fibres, but
this can be readily achieved in PCF. Exotic dispersion properties such as
strong high order dispersion and the presence of multiple ZDWs are also pos-
sible in PCF. Because of their high nonlinearity, and the ability to tailor the
geometry of the fibre to achieve dispersion curves impossible in conventional
fibre, PCF has found particular application in supercontinuum generation.
2 Background 4
Cro
ss-S
ectio
nR
efra
ctiv
eIn
dex
Step-Index Graded Index PCF
Fig. 1: Example cross sections and refractive index profiles for standard, graded indexand photonic crystal fibre . Grey represents silica, with darker grey representing greaterrefractive index. White represents air holes in the PCF structure.
2.2 Linear and Nonlinear Propagation
From Maxwell’s equations, it can be shown that an electromagnetic wave
with electric field, E, propagating in a medium will be governed by the wave
equation:
∇2E− µ0ε0∂2E
∂t2= µ0ε0
∂2P
∂t2(1)
where ε0 and µ0 are the permittivity and permeability of free space respec-
tively, and P is the polarisation induced in the material by the incident elec-
tric field, which leads to re-radiation and propagation of the wave through
the medium. The relationship between E and P is give by the expansion
P = ε0
(χ(1)E + χ(2)E2 + χ(3)E3 + . . .
)(2)
where χ(n) is the material’s nth order susceptibility. For low intensity elec-
tric fields, where the medium’s response is well approximated as harmonic,
only the first order term is important, and the polarisation is linear with
the electric field. This is the realm of linear optics. With higher intensity
electric fields, the material’s response will not be well approximated as har-
monic, and higher order terms will become important. In silica fibre, due
to the symmetry of the SiO2 molecule, even order terms vanish, and the
2 Background 5
polarisation, P, will be given by
P = ε0
(χ(1)E + χ(3)EEE
)= ε0
(χ(1) +
3
4χ(3)|E|2
)E. (3)
This will lead to an intensity-dependent refractive index. Where the normal
(linear) refractive index is defined as
n0 =√
1 + χ(1), (4)
the total refractive index will be given by
n =
√1 + χ(1) +
3
4χ(3)|E|2
= n0
√1 +
3
4n20
χ(3)|E|2. (5)
As χ(3) χ(1), Equation 5 can be written as
n = n0 +3χ(3)|E|2
8n0
= n0 + n2I, (6)
where I is the optical intensity, and n2 is the nonlinear refractive index
coefficient, given by
n2 =3χ(3)
4n20cε0
. (7)
This means that an intense optical field in a dielectric medium will result
in a local shift in the refractive index, proportional to the optical intensity.
This is known as the optical Kerr Effect, and its effects are discussed in
Section 2.4.
2 Background 6
2.3 Dispersion
As the refractive index (and hence phase velocity) is frequency dependent,
pulses with significant spectral width will be subject to chromatic dispersion.
This may be considered by expanding the propagation constant,
β = n(ω)ω
c(8)
(where n(ω) is the effective index of at frequency ω), about the central
frequency of the pulse, ω0:
β(ω) = β(ω0) +∂β
∂ω[ω − ω0] +
1
2
∂2β
∂ω2[ω − ω0]2 +
1
6
∂3β
∂ω3[ω − ω0]3 + . . .
= β(ω0) + β1[ω − ω0] +1
2β2[ω − ω0]2 +
1
6β3[ω − ω0]3 + . . . (9)
where
βi =∂iβ
∂ωi
∣∣∣∣ω=ω0
. (10)
The first order term contains β1, the inverse of the group velocity, vg, of
the pulse. This accounts for the overall delay on a pulse. The second order
term contains β2, the derivative of inverse group velocity with respect to fre-
quency, and specifies the variation in group velocity for different frequency
components within the pulse, resulting in group-velocity dispersion (GVD).
GVD results in temporal broadening of unchirped pulses as different fre-
quency components travelling with different velocities walk off with respect
to each other. The GVD of a fibre at a given frequency is said to be normal
when β2 > 0 (group velocity decreasing with increasing optical frequency)
or anomalous for β2 < 0 (group velocity increasing with increasing optical
frequency). The wavelength corresponding to a value of β2 = 0 is referred
to as the zero-dispersion wavelength (ZDW).
2 Background 7
!"#$%%&'()%
*+,$+(",-%&./0)%
12$30$+4-%(5"6%
&./0)%
Fig. 2: SPM induces an instantaneous frequency shift (chirp) on the pulse with the front(left hand side on graph) of the pulse is red-shifted and the back of the pulse blue-shifted
2.4 Self-Phase Modulation
Self-Phase Modulation (SPM) is caused by the optical Kerr effect, where a
high optical intensity will cause a change in the local refractive index. A
high intensity pulse will induce a varying shift in the local refractive index
along the pulse, and hence induce a time dependent phase-delay. As the
instantaneous frequency is the first derivative in time of the phase delay, the
result will be an instantaneous frequency shift, or chirp, along the pulse
(provided the response time of the medium is much less that the pulse
duration). This causes the leading edge of the pulse to be red-shifted (its
instantaneous frequency being shifted downwards) and the back of the pulse
to be blue-shifted (as shown in Figure 2).
Closely related to SPM is cross-phase modulation (XPM). In the case
of two co-propagating optical fields, for example - two pulses with different
central frequencies, though temporally local to each other, the optical in-
tensity of one field will alter the local refractive index. This will give rise to
a nonlinear phase-shift across the co-propagating field.
2.5 The Non-Linear Schrodinger Equation
Starting with Equation 1, it can be shown [6, Ch. 2.3] that pulses propa-
gating in an optical fibre with amplitude A and central frequency ω0 will
2 Background 8
obey
∂A
∂z+α
2A+
iβ2
2
∂2A
∂T 2− iβ3
6
∂3A
∂T 3+ . . . =
iγ
(|A|2A+
i
ω0
∂
∂T
(|A|2A
)− TRA
∂|A|2
∂T
)(11)
where T = t− z/vg = t− β1z is the time in the retarded frame - the frame
of reference travelling at the group velocity of the central wavelength, vg,
β2 is the second order coefficients in the expansion of the wave number, α
is the attenuation coefficient and γ is the nonlinear coefficient, given by
γ =n2(ω0)ω0
cAeff(ω0)(12)
where Aeff is the effective mode area in the fibre.
The ellipsis on the left hand term represents further higher order terms
in the expansion of the dispersion which have been truncated. This is not
always a valid approximation, however, for example where β2 ' 0 for some
frequencies ω0 and higher order dispersion terms become significant. The
three terms on the right hand side relate to different nonlinear processes in
the fibre. The first, proportional to |A|2 relates to self-phase modulation
(SPM) discussed in Section 2.4.
The second term, term proportional to ω−10 relates to ‘self steepening’.
Noting that γ is a function of frequency, for short pulses (i.e. with sufficient
spectral width), γ may vary significantly across the pulse. This variation in
γ is accounted for by introducing a first order correction, which is found to
be proportional to ω−10
The final term relates to intrapulse Raman scattering which can lead to
an overall frequency shift for a suitably short pulse (See section 2.9), where
TR is the characteristic time for delayed response of the Raman scattering
process.
2 Background 9
In the special case where only the linear effect is group velocity dispersion
(i.e. the β2 term), attenuation is disregarded and the only nonlinear effect
considered is self phase modulation (the iγ|A|2A term), Equation 11 reduces
to
i∂A
∂z+β2
2
∂2A
∂T 2+ γ|A|2A = 0 (13)
which is referred to as the nonlinear Schrodinger equation (NLSE). Similarly,
Equation 11 is often refereed to as the generalised nonlinear Schrodinger
equation (GNLS).
2.6 Four-Wave Mixing
Four wave mixing (FWM) is the parametric interaction of four co-propagating
light waves, dur to the Kerr nonlinearity. The process arises due to the mix-
ing of the four fields through the third order (χ(3)) term in the polarisation
(see Equation 2):
P3 = ε0χ(3)EEE, (14)
where P3 is the third order polarisation and E is the electric field.
If the electric field is made up of four copropagating waves with frequen-
cies ω1, ω2, ω3 and ω4, the expansion of the right hand side of equation will
contain cross terms proportional to (ω1 + ω2 − ω3 − ω4), which quantum
mechanically relate to the annihilation of photons at ω1 and ω2, and the
generation of photons at ω3 and ω4. Energy conservation dictates that
ω1 + ω2 = ω3 + ω4, (15)
whilst momentum conservation provides the phase-matching condition, ∆β =
0, where
∆β = β(ω1) + β(ω2)− β(ω3)− β(ω4), (16)
where β(ωi) is the propagation constant, β at the frequency ωi.
2 Background 10
Of particularly interest is the degenerate case where ω1 = ω2. In this case
the four wave mixing process can be initiated by an intense pump, leading
to the formation of spectrally symmetric sidebands at ω3 and ω4 = 2ω1−ω3.
2.7 Modulation Instability
Modulation instability (MI) is an important process in the field of nonlinear
fibre optics, particularly in CW supercontiuum generation and has been the
focus of much experimental and theoretical investigation [7, 8, 9, 10, 11, 12].
MI is a noise seeded process which leads to the temporal breakdown of
continuous wave (CW) field in a fibre into a train of pulses.
MI can be considered in terms of a degenerate four-wave mixing process
with phase-matching effected through the Kerr nonlinearity. In the phase-
matching condition for FWM (Equation 16), it is important to note that for
sufficiently intense fields, the propagation constant β will contain a nonlin-
ear part, proportional to power. The linear propagation constant, β, for a
wave at ωi with power Pi will be replaced by
β(ωi) + γPi. (17)
Assuming that the only the pump fields (ω1,2) are sufficiently intense to
contribute significantly to the nonlinear phase shift, Equation 16 becomes
∆β = β(ω1) + β(ω2)− β(ω3)− β(ω4) + γ[P1 + P2]. (18)
In the case of degenerate FWM, the generated frequencies, ω3,4 will be
located symmetrically about the pump frequency, ωp = ω1 = ω2, with a
frequency detuning of Ω, i.e. ω3,4 = ωp ± Ω. The propagation constants at
the generated wavelengths can be approximated using the Taylor expansion
2 Background 11
of β around the pump frequency:
β(ωp ± Ω) = β(ωp)± Ωβ1 +1
2Ω2β2 ±
1
6Ω3β3 + . . . (19)
where βi is the ith derivative of the propagation constant at the pump
frequency, ωp. Hence Equation 18 becomes
∆β = 2βωp − β(ωp + Ω)− β(ωp − Ω) + 2γP0
= Ω2β2 + 2γP0 (20)
where P0 is the power of the pump field, and assuming no higher than third
order dispersion. Now, in the case of anomalous dispersion, β2 < 0, there
will be resonant values of Ω (where ∆β = 0) for
Ω =
√2γP0
|β2|. (21)
In essence, then, this shows that in the case of a strong pump field in
the case of anomalous dispersion, the phase matching of degenerate FWM
can be achieved through the nonlinear refractive index change induced by
the pump field. This process is referred to as modulation instability (MI). It
is so called because in the temporal domain, the new frequency components
result in temporal modulation of the CW line at the characteristic frequency
defined by Ω.
Although MI can be readily understood in terms of degenerate FWM
phasematched through the Kerr effect, it is also possible to derive Equation
21 from the NLSE (Equation 13) through linear stability analysis. This
analysis is based on that outlined by Agrawal [6, p. 121].
Starting with the NLSE (Equation 13) and assuming that for a CW
input, A is constant at z = 0 (the fibre input), the NLSE is solved to give
2 Background 12
the steady state solution
A =√P0 exp(iφNL), (22)
where φNL is the nonlinear phase shift induced by SPM. The steady state
solution is subject to a small perturbation such that
A = [√P0 + a] exp(iφNL), (23)
where a is the amplitude of the perturbation. Substituting the perturbed
solution into Equation 13 yields:
i∂a
∂z=β2
2
∂2a
∂T 2− γP0[a+ a∗]. (24)
which can be solved to find the form of the perturbation. Due to the a∗
term, solutions of the form
a(z, T ) = a1 exp(i[Kz − ΩT ]) + a2 exp(−i[Kz − ΩT ]) (25)
are considered, where K and Ω are the wave number and frequency of the
perturbation. Combining equations 24 and 25, a non-trivial solution is found
only when K and Ω obey the dispersion relation
K = ±1
2|β2Ω|
[Ω2 +
4γP0
β2
]1/2
. (26)
In the case of β2 < 0 (i.e. in the case of anomalous GVD), for
|Ω| < Ωc =
(4γP0
|β2|
)1/2
, (27)
K becomes purely imaginary in equation 26 and the perturbation will grow
exponentially. As a result the CW solution for a beam propagating in a fibre
2 Background 13
is intrinsically unstable in the case of anomalous dispersion. The power gain
spectrum, g(Ω) for the growth of the modulation instability is obtained from
equation 26:
g(Ω) = 2Im(K) = |β2Ω|(Ω2c − Ω2)1/2. (28)
Thus it can be shown that the gain is maximised for frequency detuning of
Ω =Ωc√
2=
√2γP0
|β2|, (29)
which is the same result given by Equation 21.
In this equivalent picture of MI, then, it can be seen that in the case of
anomalous dispersion, small temporal noise will lead to perturbation of a
CW field resulting in temporal modulation at the characteristic frequency as
given in Equation 29 or 21, or equivalently the generation of new frequency
components detuned by the characteristic frequency.
This temporal modulation will lead to local enhancement of the opti-
cal field. As the instability relies on anomalous dispersion, these local en-
hancements can become enhanced by the interplay of SPM and anomalous
dispersion to form optical solitons.
2.8 Optical Solitons
Solitons, or solitary waves, are particular pulses which maintain their tem-
poral and spectral shape over arbitrarily long propagation distances with-
out dispersing, and are resilient to perturbation in many cases. Intuitively,
soliton propagation in optical fibre can be thought of as a result of the si-
multaneous action of SPM and anomalous dispersion on an arbitrary pulse.
A pulse’s intensity will, through SPM, give rise to new frequency compo-
nents, inducing a chirp along the pulse, with the leading edge frequency
down-shifted, and the trailing edge up-shifted. In the case of anomalous
dispersion, where group velocity increases with increasing optical frequency,
2 Background 14
the back of the pulse will now have a higher velocity than the front, and the
pulse will compress. This in turn leads to enhancement of the peak power,
and further spectral broadening of the pulse. Over long propagation dis-
tances these two processes will act to compress the pulse to a solitary wave
state (i.e. propagating without dispersion, with constant pulse shape).
The exact form of optical solitons can be found, again using the NLSE as
a starting point, through the inverse scattering method (See [6, Ch. 5] and
references therein). The form of the fundamental optical soliton is found to
be a transform limited sech shaped pulse, with the soliton amplitude profile
given by
A(z, T ) =√P0sech
(T
T0
)exp
(iγP0z
2
)(30)
where the soliton’s peak power, P0, and characteristic time, T0, are related
P0 =|β2|γT 2
0
. (31)
The exponential part of Equation 30 represents the overall phase shift result-
ing from the combination of dispersion and nonlinearity. This is constant
across the pulse, and modifies the soliton’s propagation constant such that
all spectral components propagate without being dispersed, causing the soli-
ton to propagate without chirp. The propagation constant across a soliton
with central frequency ωsol is given by [13]
βsol(ω) = β(ωsol) + β1(ωsol)[ω − ωsol] +γP0
2. (32)
The fundamental soliton will propagate without dispersing and with con-
stant spectral profile over arbitrarily long distances. Furthermore, the fun-
damental soliton is robust against perturbations (e.g. loss, higher order
dispersion), adiabatically adjusting itself to return to a valid soliton shape,
for example by shedding energy to co-propagating dispersive waves.
2 Background 15
The fundamental soliton is part of a larger set of solutions to the NLSE.
Higher order solitons are characterised by input (z = 0) pulse shapes at
given by
A = N√P0sech(T/T0) (33)
where N , the soliton order, is an integer. These solitons do not have a
constant shape with propagation, but instead have a evolve periodically
with propagation distance, whose periodicity is defied by the soliton period
z0 =π
2
T 20
|β2|. (34)
2.9 Raman Scattering and Soliton Self Frequency Shift
Raman scattering is a process by which new frequency components can be
generated through the interaction of photons with optical phonons (quanta
of vibrational energy) in the fibre. In the case of Stokes Raman Scattering,
an incident photon can excite the material into a virtual energy state, which
will then relax, emitting a photon and an optical phonon.
With regard to the optical field, this scattering process is obviously
inelastic, with energy lost to the phonon. This results in the re-emitted
photon being frequency down-shifted from the incident photon. For two
co-propagating fields, the lower frequency field can stimulate this process,
leading to transfer of energy from the high frequency to the low frequency
components. As the frequency detuning between the incident and re-emitted
fields is governed by the frequencies of the emitted phonons, the gain profile
for this process is independent of the pump frequency and is instead a func-
tion of the frequency detuning between the two fields, with the peak gain
corresponding to a detuning of ∼13 THz.
In the case of solitons with sufficient spectral width, the high frequency
components of the pulse will act as a pump for low frequency components
through Raman gain, shifting the central frequency of the soliton to longer
2 Background 16
wavelengths [14, 15]. This process is referred to as soliton self-frequency
shift, and was observed experimentally as early as 1986 [16]. The total shift
in the soliton central frequency is linear with the total propagation distance
and proportional to T−40 [15].
2.10 Solitons and Dispersive Waves
The interaction of fibre solitons with non-solitonic (dispersive) radiation has
been the subject of much theoretical and experimental study [13, 17, 18, 19].
As mentioned above, a soliton, under the perturbative influence of higher
order dispersion, can emit dispersive radiation [19]. For this to occur, there
must be phasematching between dispersive radiation and solitonic radiation
for some frequency which lies within the spectrum of the soliton, i.e.
β(ω) = βsol(ω) (35)
where the higher order dispersive terms effect phasematching between the
solitonic and dispersive radiation. This can lead to the transfer of energy
from the soliton to the dispersive wave, which can be thought of as the
soliton adiabatically adjusting itself to the perturbation caused by higher
order dispersion through the shedding of energy. This process is normally
referred to as ‘Cherenkov’ radiation by solitons due to its equivalence with
classical Cherenkov radiation [19].
The presence of higher order dispersion and co-propagating dispersive ra-
diation at a discrete, separate frequency brings about further phase-matching
conditions between solitons and dispersive waves, leading to four-wave mix-
ing of the soliton and the CW pump to form new frequency components
[18].
Soliton FWM and Cherenkov radiation were observed in experimental
and numerical results and are discussed further in Sections 5.3.3 and 5.3.2
2 Background 17
respectively.
2.11 Solitons and the Evolution from MI to Supercontinuum
Modulation instability, then, can cause a CW input to undergo temporal
modulation. SPM and anomalous dispersion will cause then cause local
intensity variations to increase, leading to the formation of a train of solitons.
Each soliton will undergo soliton self frequency shift to longer wavelengths.
As MI is fundamentally a noise seeded process, the solitons will be created
with a range of peak powers, and hence temporal widths, and so will shift
to a range of wavelengths, leading to the formation of a broad, smooth
continuum.
If the soliton formation occurs close the the zero dispersion wavelength
of the fibre (i.e. the wavelength for which β2 = 0), then dispersive waves
may be excited in the normal dispersion regime, and can be ‘trapped’ by a
soliton in the anomalous dispersion regime and go on to form a blue-shifted
continuum. Dispersive wave trapping has been the focus of much recent
study, both experimental [20, 21, 22] and theoretical [23, 24, 25, 26]. In
short, for dispersive radiation temporally local to a soliton, the soliton’s
intensity will induce a phase shift on the dispersive wave through XPM,
causing a shift in the dispersive wave’s spectrum. The dispersive wave will
be blue-shifted, and hence delayed temporally with respect to the soliton.
The soliton will simultaneously undergo self frequency shift, being shifted
to longer wavelengths. As the soliton is located in the anomalous dispersion
regime it will decelerate and fall back on the dispersive wave, and as the
soliton is constantly decelerating the dispersive wave is effectively ‘trapped’
behind its parent soliton so that it is prevented from dispersing temporally.
This results in the dispersive wave remaining temporally local to its parent
soliton and being continuously blue shifted as the soliton is red-shifted.
This can lead to the formation of a blue-shifted (with respect to the pump)
3 Design of a Fibre Fuse Protector for use in High Power Fibre Laser Applications 19
3 Design of a Fibre Fuse Protector for use in High Power Fibre
Laser Applications
In situations with high power (10s of Watts average power) being guided in
an optical fibre, small local imperfections can lead to the triggering of a so-
called ‘fibre-fuse’ which can lead to the catastrophic and rapid destruction
of fibre up-stream of the point at which the fuse instigates. The fuse can
propagate back into fibre lasers and amplifiers spliced directly into a set-up,
leading to irreparable damage.
To prevent fibre fuses causing damage to important components, a ‘fibre
fuse protector’ was designed.
3.1 Fibre Fuse
The fibre fuse was first reported in the literature as early as 1988 [27, 28],
although had been observed and discussed informally for years beforehand.
Although silica fibre can reliably guide very high intensity light without
being subject to damage, local heating or imperfections in the fibre can
initiate a self-focussing process to propagate towards the input end of the
fibre, resulting in a periodic damage structure to the fibre core. An example
of the damage caused by a fibre fuse is shown in Figure 3.
Fig. 3: The effects of a fibre fuse observed in a single mode fibre.
Imperfections, discontinuities or environmental effects can result in lo-
calised heating within the fibre. This leads to local vaporisation of the fibre
3 Design of a Fibre Fuse Protector for use in High Power Fibre Laser Applications 20
core and increased backreflection. The backreflected radiation is simulta-
neously focussed resulting in localised intensity enhancement and heating
immediately upstream of the previous location of the fibre fuse. This results
in the fuse propagating backwards from its point of initiation, resulting in
permanent catastrophic damage of the fibre waveguide. Typically, fuses can
only propagate where the intensity within the core is above ∼ 2 MW cm−2
[29], although often a higher intensity is required to spontaneously initiate
a fuse due to, for example, localised heating due to splice losses (a common
cause of fibre fuse initiation in experimental situations). If after a fuse has
been initiated, the laser input power is reduced to under the threshold for
propagation, or switched off entirely, the propagation of a fuse will halt. As
shown in Figure 3, when the fibre is viewed under a microscope, the periodic
damage along the path of the fuse can be seen. The ‘bullet’ shaped damage
structure along the centre of the fibre’s core is a characteristic effect of the
fibre fuse.
Macroscopically, the fuse is characterised by bright white emission at the
location of the fuse due to the intense heat within the core at the point of
vaporisation. This is shown in Figure 4, where a fibre fuse was allowed to
propagate through a short length of coiled fibre. The fuse was initiated by
‘roughing’ the end of a fibre and pushing it up against an abrasive surface
(specifically, a defunct thermal power meter). The fibre was then pumped
up to ∼ 8 W using an Ytterbium fibre laser. The end of the fibre then
heated, initiating a fibre fuse along the coil of the fibre.
3.2 The Fibre Fuse Protector
Various methods have been suggested to avoid damage by the fibre fuse
effect. As there is a characteristic threshold power below which the fuse
cannot occur, if power levels are kept below the threshold value, a fuse
cannot occur. Methods have been suggested to monitor intensity levels
3 Design of a Fibre Fuse Protector for use in High Power Fibre Laser Applications 21
Fig. 4: A fibre fuse propagating through a length of coiled fibre
in the fibre to ensure they do not exceed this threshold [30]. This has
obvious drawbacks, however, and is not possible in situations where high
power inputs are required. The initiation of the fuse will result in a drop in
output power at the output end of the fibre system, and so it has also been
suggested that a fuse protection system may involve monitoring the output
power of the fibre system, and shutting down the source if a sudden drop in
output power is detected [31]. Again, this method has its drawbacks. It is
not always possible to constantly measure the power at the output end of
a fibre, or doing so may impede the ease of realisation of an experimental
set-up.
A fibre fuse propagating through a fibre is visible to the eye as a bright
white emission propagating along the length of the fibre at a speed on the
order of metres per second. It was felt that the detection of the white light
flash would be the least disruptive method to detect the occurrence of a fibre
fuse. An electronic fuse protector based on this principle was designed, the
schematic of which is shown in Figure 5. The output of a photodiode located
against the fibre, shielded from ambient light, is passed to an operational
3 Design of a Fibre Fuse Protector for use in High Power Fibre Laser Applications 22
amplifier (op-amp), the output of which is then passed to a comparator
which will output a digital high signal when the op-amp output reaches a
threshold. The gain across the amplifier is proportional to the resistance of
the feedback resistor R1. A 470Ω resistor was found to provide suitable gain
so that the comparator was triggered as the leading edge of the fuse passed
under the photodiode. In parallel to R1 is a decoupling capacitor C1 (33pF)
which acted to prevent the internal capacitance of the photodiode causing
the fuse protector to trigger. The output of the comparator is passed to the
reset input of a set-reset (S-R) latch, while the set input of the latch was
connected to a switch which could be closed to provide high voltage to the set
input. The output of the latch could be set to high by momentarily closing
the switch and would remain high until a high input was detected on the reset
input (from the comparator). At this point the output of the latch would
be set to low and held there until the switch was once again momentarily
closed. The output of the S-R latch was passed to an optoisolator, which
would act as an open switch across the interlock of the laser unless it was
provided with a high input from the S-R latch.
This fuse protector design can be easily integrated into experimental set-
ups. The photodiode was attached to the rest of the circuit by a ∼1m cable,
and held in place over the fibre by a bespoke PTFE casing. The photodiode
is placed downstream of the component to be protected, so that a fuse
would be detected before it reached the component. The fuse protector was
the connected in series across the safety interlock of the laser used. The
protector could then be ‘primed’ by momentarily depressing the switch. If a
fibre fuse did occur and pass under the photodiode the fuse protector would
latch open across the laser’s safety interlock, shutting down the output, and
terminating the propagation of the fuse.
Several fuse protectors were constructed so that they could be connected
in series to provide simultaneous protection for multiple components.
3 Design of a Fibre Fuse Protector for use in High Power Fibre Laser Applications 23
1
2
5
4
4N25
220
IL +
IL −0.47M
33p
4
3
8
5
21
67
MAX931
21
6.8
K
+5V
+5V
3
2
67
4
CA3140
QR
SHEF4043
CA3140 - Op-AmpMAX931 - ComparatorHEF4043 - S-R Latch4N25 - Opto-isolatorIL± - Interlock Connector
Fig. 5: Schematic of fuse protector circuit.
4 CW Continuum Generation in a Depressed Cladding Highly Nonlinear Fibre 24
4 CW Continuum Generation in a Depressed Cladding Highly
Nonlinear Fibre - Normally Dispersive Pumping of
Supercontinua
A depressed cladding highly nonlinear fibre was pumped using a high power
Ytterbium laser operating at 1.07 µm. A broad, flat continuum (5 dB band-
width of over 900 nm) was formed.
The CW supercontinnum formation process, as described in Section 2.11
relies on pumping in the anomalous dispersion region of the fibre, as this
is a pre-requisite for the temporal break-up of the pump line through MI
and subsequent soliton formation and self-frequency shift. In this case, how-
ever, the fibre was pumped far into the normal dispersion regime. As the
field propagates through the fibre, a Raman cascade initially forms. As the
cascade extends past the zero dispersion wavelength, into the anomalous
dispersion region, a continuum is seen to form, intially on the long wave-
length side and subsequently on the short wavelength side. It is thought that
the dispersive radiation in the Raman cascade across the normal dispersion
regime seeds soliton-dispersive wave dynamics and enhances the short wave-
length side of the continuum.
Continuum formation in normally-dispersive pump schemes through Ra-
man shifting of the pump has been previously demonstrated with both pulse-
[32, 33] and CW- [34] pumped schemes.
4.1 Sumitomo Depressed Cladding Highly Nonlinear Fibre
The fibre examined in this Section is a depressed cladding highly nonlinear
fibre (HNLF) produced by Sumitomo [35]. The index profile of the fibre is
shown in Figure 6b. The fibre has a parabolic index profile across its core,
which is doped with a high Germanium dioxide (GeO2) concentration. The
cladding of the fibre is doped with fluoride, which acts to lower the material
4 CW Continuum Generation in a Depressed Cladding Highly Nonlinear Fibre 25
index relative to pure silica (hence ‘depressed cladding’). The large index
difference and parabolic profile results in a small mode area (mode field
diameter of 3.7 µm), which combined with its high core Germanium doping
concentration, results in its high nonlinearity (γ ' 21 km−1W−1).
The index profile of the fibre was used to compute the dispersion curve
for the fundamental mode using freely available software (MIT Photonic
Bands [36]). The computed curve is shown in 6a. The ZDW was calculated
to be 1.55 µm.
1.0 1.2 1.4 1.6 1.8 2.0 2.2Wavelength (µm)
60
50
40
30
20
10
0
10
20
GVD
ps
nm−
1 k
m−
1
(a) Dispersion Curve for FundamentalMode of Sumitomo HNLF
4 2 0 2 4Radius (µm)
1.445
1.447
1.449
1.451
1.453
Refr
activ
e In
dex
GeO2−SiO2F−SiO2 F−SiO2
(b) Index Profile of Sumitomo HNLF
Fig. 6
4.2 Experimental Set-up
A diagram of the experimental set-up is shown in Figure 7 The collimated
output of a high power Ytterbium fibre laser (operating at 1.07 µm was
coupled into a large mode area fibre (LMA) by focussing onto the cleaved
face of the fibre with a singlet lens. The LMA was fusion spliced to a length
of single mode fibre (SMF), which in turn was spliced onto a coupler with
splitting ratio measured as 99.6:0.4. This provided a low power reference, so
that the power coupled into the HNLF could be monitored. The high power
arm of the coupler was then spliced onto a further length of SMF which was
in turn spliced onto the sumitomo HNLF, with a splice loss of -0.7 dB.
4 CW Continuum Generation in a Depressed Cladding Highly Nonlinear Fibre 26
LMA HNLFSMF SMF
Ref.
99.6:0.4 -0.7dB
To OSA/Spectrometer
Collimated OutputFrom Yb Laser
Fig. 7: Experimental set-up used to generate CW supercontinua in HNLF
To avoid thermal damage to the reference coupler, or at the lossy splices
onto the fibre, the average power output of the laser was reduced by on-off
modulating the laser at a repetition rate of 34 Hz and a switched-on time
of 0.64 ms (i.e. duty factor of 46). This also facilitated lock-in amplification
of the signal from the detector on the output side of the spectrometer.
Slight defocus onto the face of the LMA could lead to coupling of light
into the cladding mode of the fibre, which for Watt level pump powers could
lead to the shedding of the radiation from the fibre and the plastic buffer
of the fibre burning, or damage at the LMA-SMF splice. To avoid this, the
cladding mode radiation was decoupled from the fibre by stripping back the
plastic buffer, using index-matching optical glue to fix the fibre to a glass
slide which was then passively cooled by being submerged in water.
To prevent back-reflection from effecting the continuum dynamics, the
output end of the HNLF was angle cleaved, so that any radiation reflected
at output face would not be directed back down the fibre. The output was
collimated so that it could be projected onto the input slit of a spectrometer
to measure the output spectrum.
4.3 Monochromating Scanning Spectrometry
A computer controlled monochromating scanning spectrometer was used to
measure the spectrum of the fibre output. The Spex 500M is a commer-
cial grating monochromator, a diagram of which is shown in Figure 8. The
beam is projected onto the input slit of the spectrometer. This slit is at the
effective focus of a curved mirror (via a folding mirror), so that the beam
is collimated and directed onto a diffraction grating. The beam, now dis-
4 CW Continuum Generation in a Depressed Cladding Highly Nonlinear Fibre 27
persed, is then focussed onto the plane of the output slit, beyond which is
a detector. As the angle of reflection from the grating is frequency depen-
dent, by scanning the grating angle, it is possible to scan different spectral
components of the input beam across the output slit.
Diffraction Grating
InputBeam
To Detector
Fig. 8: Diagram of Spex 500M scanning spectrometer (not to scale).
The spectrometer was used to measure spectra which extended from
1 µm to greater than 2.2 µm. As the spectrometer would be used to make
measurements across a range greater than an octave, it was necessary to
consider second order diffraction. To this end a low-band pass filter was
used. Spectra were taken in two parts, the first ranging from 1 - 2 µm. The
second part, ranging from 1.7 µm up to 2.4 µm, was recorded with a low
band pass filter across the detector, blocking all wavelengths below 1.4 µm
with >40 dB extinction. Figure 9 shows the transmittance of the filter used
to block short wavelength components.
The two parts of the spectra could then be ‘stitched’ together, matching
the relative power levels of the overlapping section of the spectrum. The
total output power from the end of the fibre was measured using a thermal
power meter, so that the spectra could be normalised to the total output
power and plotted in terms of spectral power density (in units of dBm/nm).
All spectra plotted below have been measured in two parts and stitched
together in this way.
4 CW Continuum Generation in a Depressed Cladding Highly Nonlinear Fibre 28
Fig. 9: Transmittance of FEL1400 filter used to filter out second order diffraction ofshort wavelength signals
4.3.1 J10D Indium Antinomide Detector
An Indium Antinomide (InSd) detector was used in conjunction with the
Spex monochromater. Its specified detectivity curve is shown in Figure 10
- the detectivity of the detector varies by only 2.5dB over the 1 - 2.5 µm
wavelength range. In all spectra below, this change in detectivity across the
measurement range is corrected for.
Fig. 10: Specified detectivity of J10D Indium Antinomide (InSd) detector (after [37]).
4.3.2 Lock-In Amplification
The InSd detector was used with a lock-in amplifier (EG&G 5205 Lock-In
Amplifier) to make high dynamic range measurements across the spectrum.
In essence, lock-in amplification works through the multiplication the input
4 CW Continuum Generation in a Depressed Cladding Highly Nonlinear Fibre 29
(which contains a periodic input signal along with noise) with a reference
signal with the same frequency as the input signal. The result of this will
contain a DC term proportional only to signal at the reference frequency. A
low pass filter can then be used to filter out all but this DC term.
This enables the measurement of periodic signals even with high levels
of noise. The pump laser was on-off modulated, and so the continuum out-
put was similarly modulated. The output of the signal generator used to
provide a modulation signal to the pump laser was split, so that it could
be used as the reference signal for the lock-in amplifier. The lock-in ampli-
fier also provided analogue to digital conversion of the output signal, which,
combined with computer control of the SPEX 500M Monochromator, fa-
cilitated convenient, computer controlled acquisition of spectra. Using this
set-up a dynamic range of ∼ 30 dB was achievable over the entire 1.0−2.4µm
range.
4.4 Results and Discussion
The HNLF was pumped with the high power Ytterbium fibre laser source
to produce an octave-spanning supercontinuum with flatness of ∼5 dB. The
output of the fibre pumped to both 87 and 140 W for fibre lengths of 100,
200, 300 and 600 m were measured using the monochromating spectrometer
with the J10D InSd detector, as described above. This way the continuum
generation process as a function of the propagation distance along the fibre
could be established. These spectra are shown in Figure 111.
Modulation instability, soliton formation and hence the CW supercon-
tinuum generation process requires pumping in the anomalous dispersion
regime, in this case the zero-dispersion wavelength of the fibre was calcu-
lated to be 1.55 µm, while the wavelength of the pump source is 1.07 µm.
1 All spectra in this section and the section after are plotted in terms of spectral power(dBm/nm) normalised to the time averaged output power, while pump powers are the‘switched-on’ power of the laser. The spectral power of the output while the pump isswitched on is therefore a factor of 46 (16 dB) greater.
4 CW Continuum Generation in a Depressed Cladding Highly Nonlinear Fibre 30
z = 100m
z = 200m
z = 300m
z = 600m1000 1350 1700 2050 2400
Wavelength(nm)
40
30
20
10
0
10
Spec
tral
Pow
er (d
B/nm
)
40
30
20
10
0
10Sp
ectr
al P
ower
(dB/
nm)
(a) 87 W Peak Pump Power
z = 100m
z = 200m
z = 300m
z = 600m1000 1350 1700 2050 2400
Wavelength(nm)
40
30
20
10
0
10
Spec
tral
Pow
er (d
B/nm
)
40
30
20
10
0
10
Spec
tral
Pow
er (d
B/nm
)
(b) 140 W Peak Pump PowerFig. 11: Spectrum of continuum output generated in 100, 200, 300 and 600 m lengthsof HNLF for 87 or 140 W pump power at 1.07 µm
MI and breakdown of the CW pump line is therefore not expected. The
process by which the continuum is formed is most clearly shown in Figure
11a, the 87 W pump power case.
As can be seen in the spectrum for a 100m length of fibre, a CW Raman
cascade initially forms, i.e. power is transferred through Stokes Raman Scat-
4 CW Continuum Generation in a Depressed Cladding Highly Nonlinear Fibre 31
tering to successive Raman orders. This is evident as a comb of intensity
peaks up to 1480 nm, separated in frequency by 13 THz, which corresponds
to the peak in the Raman gain for fused silica fibre. The next Raman order
would be expected at 1584 nm, and indeed a slight peak can be seen at this
wavelength, although a full 10dB down from the line at 1480 nm. To longer
wavelengths, a continuum is seen to form. This is readily understood as
the 1584 nm line is the first Raman order located in the anomalous disper-
sion regime, and so is subject to modulation instability. The CW Raman
generated line then breaks down into solitons, which undergo self-frequency
shift and form the continuum at the long wavelength end. With increasing
propagation distance, power continues to be transferred through the Ra-
man cascade, into the anomalous dispersion region, feeding the continnum
generation.
With increasing fibre length, the long wavelength edge of the continuum
continues to extend up to ∼2100 nm, forming a steep edge on the spectrum.
This is understood as the increasing loss for wavelengths longer than ∼ 2 µm
in silica causing shifting solitons to loose power, restricting their spectral
bandwidth such that they no longer are subject to self frequency shift.
As the continuum broadens to the long wavelength edge, there is also
simultaneous continuum formation in the normal dispersion regime. As the
solitons generated by the breakdown of the 1584 nm CW line are generated
close to the ZDW, their spectrum overlaps with normally dispersive wave-
lengths whose propagation constants are matched to that of the solitonic
radiation. This will cause the solitons to shed normally dispersive radiation,
short of the ZDW, through the ‘Cherenkov’ radiation process. More detail is
given on this process in Section 5.3.2. As solitons shift to longer wavelengths
they will simultaneously cause dispersive radiation to be blueshifted through
cross-phase modulation. It is thought that residual dispersive radiation from
the Raman cascade further seeds this process, leading to an enhanced short
4 CW Continuum Generation in a Depressed Cladding Highly Nonlinear Fibre 32
wavelength part of the continuum, and better spectral flatness.
4.4.1 Modal Content at Short Wavelength Edge
A strong undepleted pump component can be seen at 1.07 µm, even after
600 m fibre length, for both pump power cases. Examination of the beam
profile lead to the conclusion that this was due to coupling of the pump into
a higher order transverse mode.
The cut-off wavelength for single-mode propagation in the Sumitomo
HNLF is located at 1.55 µm. This means that light was coupled into the
fibre at a wavelength where multiple modes are available. It is unsurprising,
therefore, that some radiation may be coupled into higher order modes.
This was confirmed by examining the beam profile of the fibre output
for different wavelengths. A lens was used to control the expansion of the
beam from the angle cleaved end of the fibre, which was then reflected from
a wavelength selective mirror and projected onto a screen. The projected
beam profile was then recorded using an InGaS camera.
Using a mirror reflective at ∼ 1.07 µm, the beam profile of the pump
line was recorded for pump powers of 5 W and of 140 W. The resulting
beam profiles are shown in Figure 12. These beam profiles indicate that
significant amounts of power have been coupled into higher order modes.
The high power pump case shows the beam profile is to some extent ‘donut
shaped’ with a dip in intensity in the centre of the beam. With low power
pumping the beam appears more flat-topped, likely a superposition of the
fundamental mode with one or more higher order modes. There is also a
slight asymmetry to the beam profile, though this is due to the fact that the
end of the fibre is angle cleaved.
The mirror was then replaced by one reflective around 1.5µm. The beam
profiles were again recorded for low and high pump power cases, and are
shown in Figure 13. In both cases these profiles appear correspond to the
4 CW Continuum Generation in a Depressed Cladding Highly Nonlinear Fibre 33
fundamental mode.
These profiles support the conclusion that the undepleted pump line at
1.07 µm visible in the spectra in Figure 11 is in a higher order mode.
P = 5 W P = 140 W
Fig. 12: Output beam from HNLF at wavelength of 1.07 µm
P = 5 W P = 140 W
Fig. 13: Output beam from HNLF at wavelength of 1.5 µm
4.5 Power Scaling of Continuum Output
The total output power from the continuum in the 300 m length of HNLF
was measured as a function of the pump power. This is shown in Figure
14. It is evident that the maximum pump power (140 W) does not result
in the maximum continuum output power. This is to be expected due to
the increased loss with increasing pump power. The Raman scattering re-
sponsible for the formation of the cascade, and then for the shifting of the
solitons generated past the ZDW to longer wavelengths is a lossy process,
4 CW Continuum Generation in a Depressed Cladding Highly Nonlinear Fibre 34
and so for high powers, where a greater proportion of the pump power is
converted to longer wavelengths, a lower input-output efficiency is expected.
For higher pump powers (greater than ∼80 W),the total output power ac-
tually decreases with increasing pump power. This is due to the opacity
of silica at longer wavelengths. With increasing pump power, solitons will
be generated with greater peak powers, shorter temporal width, and so for
higher pump powers and a fixed fibre length, z, the average spectral shift of
the solitons will be greater, as the shift of a soliton is proportional to T−40 z.
The solitons shifting to longer wavelengths will eventually be attenuated due
to the rapidly increasing opacity of silica with wavelength from ∼ 2 µm, and
so an increase in pump power will act to shift greater amounts of power to
the spectral region of increasing opacity at shorter propagation lengths.
0 20 40 60 80 100 120 140Pump Power (W)
0
50
100
150
200
250
300
350
400
Cont
inuu
m O
utpu
t Pow
er (m
W)
Fig. 14: Continuum output power from 300m length of Sumitomo HNLF as a functionof pump power.
Higher pump powers will generally result in improved spectral flatness,
and this is indeed observed for all lengths of the HNLF. The optimum pa-
rameters to form a continuum with the best spectral flatness, but still with
a high total output power (and therefore high spectral power) was found to
be 140 W of pump power in the 300 m length of the HNLF. The output
spectrum for these parameters is shown in Figure 15. The spectrum dis-
4 CW Continuum Generation in a Depressed Cladding Highly Nonlinear Fibre 35
plays a spectral flatness of 5 dB over a region of 1000 nm extending from
1.15 µm to 2.15 µm. Across this region there is an average spectral power
of 0.33 mW/nm (i.e. while the pump is switched on, an average spectral
power of 15 mW/nm).
1.0 1.2 1.4 1.6 1.8 2.0 2.2Wavelength (µm)
20
15
10
5
0
5
10
15
20
Spec
tral
Brig
htne
ss (d
B/nm
)
Fig. 15: Ouput spectrum from 300 m length of Sumitomo HNLF pumped to 140 W .Black dotted lines show maximum and minimum power levels in the region 1.15 - 2.15 µm,indicating a spectral flatness in this region of 5 dB
5 Supercontinuum Generation in Photonic Crystal Fibre 36
5 Supercontinuum Generation in Photonic Crystal Fibre -
Observations of soliton-dispersive wave interaction
A 32 m length of photonic crystal fibre (PCF) was pumped using the high
power Ytterbium fibre laser. This resulted in a supercontinnum output
spanning 900 nm with a flatness of 5 dB. Due to higher order dispersion,
the fibre exhibited a second zero dispersion wavelength, which resulted in
the creation of a strong dispersive wave component centred around 1980 nm.
The interaction of solitons in the continnum with the generated dispersive
component led to further broadening of this component out to ∼ 2.15 µm.
A cut-back was performed on the fibre, with spectra taken for various
fibre lengths, so that the evolution of the continuum along the fibre length
could be evaluated. Simulations were performed to verify the mechanisms
involved in the generation of various spectral features.
5.1 Fibre T606-D
The continuum was generated in a solid core PCF (Fibre T606-D). A scan-
ning Electron Microscope (SEM) image of the fibre’s cross-section is shown
in Figure 16.
Fig. 16: SEM image of fibre T606D
Dispersion curves for the fibre were generated using this SEM image and
freely available software (MIT Photonic Bands [36]). The fibre was found
5 Supercontinuum Generation in Photonic Crystal Fibre 37
to have slight birefringence. The computed dispersion curves are shown in
Figure 17. The fibre exhibits two zero dispersion wavelengths (ZDW) on
each axis, 818 nm and 1838 nm on the fast axis and 826 nm and 1842 nm
on the slow axis, with anomalous dispersion between the two ZDWs.
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2Wavelength (µm)
100
50
0
50
100
GVD
ps
nm−
1 k
m−
1
Fast AxisSlow Axis
Fig. 17: Dispersion curves for Fibre T606-D
The computation of dispersion curves from SEM images is susceptible to
errors in higher order terms which define the location of the second zero. It
was found that slightly changing the parameter which governed the thresh-
old level between glass and air in the analysis of the SEM image could cause
the location of the second ZDWs to shift by 10s of nms. A suitable threshold
level was attained through trial and error by comparing the phasematching
conditions between solitons and dispersive radiation about the second zero
derived from the computed dispersion curves with the wavelengths of spec-
tral features observed in experimental results. Specifically, the threshold
level in the analysis was tailored so that the spectral features caused by
Cherenkov radiation past the second ZDW shed solitons just short of the
second ZDW in simulations using the generated dispersion curve matched
the experimentally observed features. Using this software it was also con-
firmed that the fibre had only a single mode solution (with two polarisations)
at the pump wavelength of 1.07 µm.
5 Supercontinuum Generation in Photonic Crystal Fibre 38
5.2 Experimental Set-Up
The experimental set-up used is shown in Figure 18. A singlet lens was used
to focus the collimated output of the high power Ytterbium laser onto the
flat-cleaved face of a short (∼1m) length of large mode area fibre (LMA).
This was spliced to a length of Flexcore SMF which was in turn spliced
onto a fibre couple with a coupling ratio at the pump wavelength of 1.07µm
measured as 99.4:0.6. This coupler was used to provide a low power reference
arm, so that the coupling into the PCF could be monitored. The high power
output of the coupler was spliced onto another length of Flexcore, which in
turn was spliced onto a length of small mode area Nufern fibre (mode field
diameter ∼ 3µm). The gradual stepping down from large mode area fibre
to small mode area fibre meant that at the splice onto the PCF, the mode
field diameter was relatively well matched to the mode field diameter of the
PCF (2 µm), and overall splice losses could be minimised.
LMA HF30SMF(Flexcore)
SMF(Flexcore)
Ref.
99.4:0.6 -0.2dB
Spectrometer InputAperture Plane
Collimated OutputFrom Yb Laser
SMF(Nufern)
-0.7dB
Fig. 18: Experimental set-up used to generate CW supercontinua in PCF
Again, to avoid damage to components or splices, the laser was on-
off modulated with a duty factor of 46. As before, the monochromating
spectrometer was used in conjunction with the J10D InSd detector (with
lock-in amplification of the signal) to record the spectrum of the fibre output
Spectra were taken at 2 m intervals for total fibre length of 32 m cut back
to 6 m, and then in 1 m intervals for the remaining 6 m. At each interval
the total output power from the PCF was measured using a thermal power
meter so that each spectrum could be properly scaled and spectral powers
plotted in dBm/nm. The results of this are shown in Figure 19 and discussed
in Section 5.3.
To ensure consistency in the measurement of the spectrum between cut-
5 Supercontinuum Generation in Photonic Crystal Fibre 39
backs, the output end of the PCF was placed in the plane of the input
aperture of the spectrometer, with the beam diverging from the output to
fill the collimating mirror. To ensure uniform illumination of the grating,
the end of the fibre could not be angle cleaved, so the core collapse method
was used to prevent backreflection from the fibre end affecting the contin-
uum dynamics. The end of the fibre was inserted into a fusion splicer, and
subjected to a long (3s) arc which caused the air-filled holes to collapse in
the last few 10s of µm of fibre. This meant that the waveguide structure of
the fibre was destroyed and the radiation unguided, hence any backreflection
from the end would not return along the fibre.
To prevent detector saturation, neutral density filters were used at the
output slit of the spectrometer to reduce the signal level. Extra neutral
density filters were added for shorter fibre lengths (< 6 m) as the continuum
narrowed, resulting in increased spectral power density and hence higher
signal levels at the output slit of the spectrometer. This is apparent in
Figure 19 - as the dynamic range of the detector set-up was constant at
∼30dB, reducing the absolute signal levels for measurements of the output
from shorter lengths led to an increased noise floor for these measurements.
5.3 Results and Discussion
The spectra taken at different fibre lengths are shown in Figure 19 in the
form of a false colour ‘cutback’ plot.
The continuum is seen to quickly broaden to long wavelengths. At a fibre
length of 5 m, a spectral component forms at 2 µm, beyond the second ZDW.
This component is attributed to Cherenkov radiation from self-shifting soli-
tons approaching the second ZDW, and is discussed in Section 5.3.2. Up
to 10 m, the spectrum around the second ZDW is ‘filled-in’, as the soliton
population at the ZDW increases, so that the peak of the Cherenkov feature
settles at 1.98 µm. From ∼10 m onwards, the Cherenkov feature is seen to
5 Supercontinuum Generation in Photonic Crystal Fibre 40
1.0 1.2 1.4 1.6 1.8 2.0 2.2Wavelength (µm)
0
5
10
15
20
25
30
Fibr
e Le
ngth
(m)
-20
-16
-12
-8
-4
0
4
8
12
16
20
Aver
age
Spec
tral
Pow
er (d
Bm/n
m)
1.0 1.2 1.4 1.6 1.8 2.0 2.25
515
Fig. 19: Cutback along 32 m length of T660D pumped at 1.07 µm to 133 W.
broaden to longer wavelengths still. Numerical modelling and consideration
of the phase matching conditions for interactions between solitons and dis-
persive waves indicates that this broadening to longer wavelengths is due to
soliton four wave mixing [18]. This is discussed in Section 5.3.3
Figure 20a shows the total continuum output power is shown plotted
against fibre length. With increasing fibre length, the total output power
decreases due to both the linear absorption of the fibre, but also due to loss
intrinsic in the Raman self-frequency shift, which increases with propagation
distance. A significant drop off in power is seen over the last 4 m of fibre
length, this is assumed to be partly due to the uptake of water into the air
holes along the fibre increasing the attenuation. Indeed, in the spectrum at
the full 32 m fibre length, as shown in Figure 19, a significant dip in the
spectrum can be seen around 1.4 µm, corresponding to the first overtone of
the vibrational resonance in the O-H bond.
5 Supercontinuum Generation in Photonic Crystal Fibre 41
0 5 10 15 20 25 30Fibre Length (m)
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
Cont
inuu
m O
utpu
t Pow
er (W
)
(a) Total power output from fibre T606-D pumped to 133 W with fibre length
1.0 1.2 1.4 1.6 1.8 2.0 2.2Wavelength (µm)
30
25
20
15
10
5
0
5
10
Spec
tral
Brig
htne
ss (d
B/nm
)
1.0 2.20
2
4
mW
/nm
(b) Spectrum of output for fibre lengthof 28 m in terms of spectral power onboth logarithmic and linear (inset) scale
Fig. 20
Figure 20b shows the ouput from 28 m of fibre T606-D. This length of
fibre was found to give the best balance between spectral flatness, bandwidth
and spectral power. The output has a spectral flatness of 6 dB over 740 nm,
from 1.08 µm to 1.82 µm, with an average spectral power of 1.73 mW/nm
over this region. Taking into account the on-off modulation of the pump,
this means that the average spectral power over this region while the output
is on will be 80 mW/nm.
The linear scale plot of the spectrum shown as an inset in Figure 20b
illustrates that a significant amount of power has been transferred to the
broad Cherenkov feature at 1.98 µm. The total power integrated past the
second ZDW (i.e. the total power in the Cherenkov feature) is 330 mW,
with a FWHM bandwidth of 80 nm. Noting that the pump (and hence
the continuum output) was on-off modulated, this represents a conversion
efficiency of 11% from the narrow 1.07 µm pump line to the broad 1.98 µm
line.
5 Supercontinuum Generation in Photonic Crystal Fibre 42
5.3.1 Numerical Simulations
Numerical simulations were used to gain insight into the continuum gener-
ation process and underpin the conclusions in Sections 5.3.2 and 5.3.3. The
evolution spectral envelope of the field was modelled as it propagated along
the fibre using pre-existing code ([38, Ch. 3]). The generalised nonlinear
Schrodinger equation (GNLSE, Equation 11) was solved using the split-step
Fourier method ([6, Ch. 2], [38, Ch.3], [39]) where with each step the field
is propagated along the fibre; with the linear part of the GNLSE solved in
the frequency domain and the nonlinear part then evaluated in the time
domain, switching between the two domains using the Fourier transform.
The field was solved for a single polarisation, which implicitly assumes
that cross-polarisation effects were not significant. It was assumed that the
133 W pump was coupled evenly into each polarisation, and hence a 66.5 W
sech shaped CW input with a bandwidth of 3 nm and random spectral
phase was used as the model of the pump source. As a long temporal
window for the simulation of the field would quickly become computationally
intractable, a 55 ps temporal window of the field was modelled which, due
to the nature of the Fourier transform, had periodic boundaries. Due to
the stochastic nature of the continuum generation process, it is necessary
to perform numerous simulations of the propagation of the field along the
fibre, and average out the results. The two polarisation axes were treated
separately, with 50 separate shots simulated for each axis, with each shot
instigated with random phase along the pump line and quantum noise over
the simulation window. Each shot was simulated propagating along the full
32 m length of the fibre. The ensemble of 100 shots were then averaged and
the resulting evolution of the spectrum over the length of the fibre is shown
in Figure 21 in the form of a false colour plot to enable direct comparison
with the experimental results from the cut-back along the fibre.
There is good agreement between these numerical results and the exper-
5 Supercontinuum Generation in Photonic Crystal Fibre 43
1.0 1.2 1.4 1.6 1.8 2.0 2.2Wavelength (µm)
0
5
10
15
20
25
30
Prop
agat
ion
Dis
tanc
e (m
)
-40
-36
-32
-28
-24
-20
-16
-12
-8
-4
0
Spec
tral
Pow
er (a
rb. d
B)
1.0 1.2 1.4 1.6 1.8 2.0 2.2
352515
Fig. 21: Simulated evolution of spectrum in T606-D, averaged from an ensemble of 100individual simulations
iment. The most visible difference between the spectrum derived from the
simulation and the experimental results is the clearly visible first Raman
order line in the experimental results. A strong spectral component can
clearly be seen at the first Raman order from the pump (1.12 µm). This is
not seen in the numerical simulation results.
A likely source of this discrepancy is the fact that the numerical sim-
ulations only simulate the forward propagating field. In actuality Raman
scattering of the CW pump line may lead to backward propagating CW
radiation at the first Raman order wavelength. The input splice onto the
fibre is also relatively lossy (∼0.7 dB), meaning that potentially a significant
amount of the back propagating radiation may be reflected back down the
fibre, seeding the formation of the Raman line.
Also, the absolute level and spectral extent of the soliton FWM gen-
erated extension of the long wavelength dispersive component is greater in
5 Supercontinuum Generation in Photonic Crystal Fibre 44
the numerical results than the experimental. This is due to the fact that
attenuation was not included in the simulations. The transmission of silica
in actuality rapidly decreases with wavelength beyond 2 µm.
Beyond this, however, there is good agreement between the experimental
and numerical results. Both the Cherenkov and soliton FWM components
of the spectra agree in their relative power, spectral location and initial
propagation distance for formation.
5.3.2 Cherenkov Radiation
In both the experimental and numerical results, a distinct spectral feature
can be seen to form for propagations distances greater than 5m, initially
around 2µm with its peak at 1.98µm. Consideration of phase matching
conditions for interaction processes between solitons and dispersive radiation
unambiguously indicates that this is due to so-called ‘Cherenkov’ radiation
[19].
If the spectrum of a soliton overlaps with phase-matched dispersive radi-
ation across a zero dispersion wavelength, there can be effective transfer of
power from the soliton to the dispersive wave. The phase-matching condition
for this is that the propagation constant for the dispersive radiation must
match the propagation constant for solitonic radiation at that frequency, i.e.
β(ω) = βsol(ω) (36)
where the soliton propagation constant, βsol is, as defined in Equation 32,
the propagation constant of the soliton at the frequency of the dispersive
wave, ω:
βsol(ω) = β(ωsol) + β1(ωsol)[ω − ωsol] +γP0
2
Transfer of power from the soliton to dispersive radiation in the normal
dispersion regime will occur at roots of Equation 36, or equivalently when
5 Supercontinuum Generation in Photonic Crystal Fibre 45
the phase mis-match, ∆β is minimised
∆β = βsol(ω)− β(ω) (37)
A simple algorithm was written to find the roots of Equation 37 for a range
of soliton wavelengths using the dispersion curve, calculated from the SEM
images, shown in Figure 17. The resulting phase matching curves for solitons
with peak powers of 1, 5 and 10 kW are shown in Figure 22. Also shown
are the location of the observed spectral peaks in the output from 32 m of
the PCF.
1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85Soliton Wavelength (µm)
1.6
1.8
2.0
2.2
2.4
2.6
Cher
enko
v Ra
diat
ion
Wav
elen
gth
(µm
)
1.8µm
1.98µm
1kW5kW10kW
Fig. 22: Computed phasematching curves between solitons of various peak powers (1,5 and 10 kW) and Cherenkov radiation across the second ZDW in fibre T606D. Blackdashed lines show wavelength of experimentally observed peaks in spectrum.
An intensity peak is in the spectrum at seen at 1.8 µm, just short of the
second ZDW at 1.84 µm. Soliton propagation requires anomalous dispersion,
and so it is not possible for solitons to be spectrally located past the second
ZDW. The self-frequency shift of the solitons across the ZDW is therefore not
permitted, causing them to remain spectrally located just short of the second
ZDW. As there will still be slight overlap of the soliton’s spectrum into the
normally dispersive regime, there will be phase-matching between solitonic
and dispersive radiation, allowing energy to effectively ‘tunnel’ out of the
5 Supercontinuum Generation in Photonic Crystal Fibre 46
soliton and form the broad spectral feature beyond the ZDW at 1.98 µm.
Due to the stochastic nature of the soliton formation, solitons in the 1.8
µm line will have a range of peak powers. The solitons with the highest
power will shift up to the second ZDW through self-frequency shift at the
shortest propagation distances, hence why the dispersive component on the
long wavelength side of the ZDW initially forms past 2 µm. As lower power
solitons reach the second ZDW, phase-matched to dispersive radiation closer
to the ZDW, the region across the ZDW ‘fills in’, and the peak of the dis-
persive component settles at 1.98 µm. This leads to the observed formation
of a broad line (∼70 nm -3dB width) on the long wavelength edge of the
continuum.
The shedding of dispersive radiation by solitons across the second ZDW
can be seen clearly in spectrograms from the numerical simulations in the
fibre. Figure 23 shows a soliton shedding Cherenkov radiation which is then
dispersed so that it is temporally advanced with regard to the soliton.
14 12 10 8 6Delay (ps)
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Wav
elen
gth
(µm
)
(a) z = 2.6 m
4 6 8 10 12Delay (ps)
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Wav
elen
gth
(µm
)
(b) z = 2.9 m
18 20 22 24 26Delay (ps)
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Wav
elen
gth
(µm
)
(c) z = 3.2 m
Fig. 23: Shedding of dispersive radiation through the Cherenkov process by a high power(4.5 kW) soliton across the ZDW at propagation distance z. Green dashed line indicateslocation of ZDW.
5 Supercontinuum Generation in Photonic Crystal Fibre 47
5.3.3 Soliton Four Wave Mixing
For fibre lengths beyond ∼10 m, further broadening of the long wavelength
dispersive component of the spectrum occurs. Consideration of the phase-
matching conditions for four-wave mixing processes between solitons and
dispersive waves indicate that this broadening is due to the interaction of
solitons in the continuum with the Cherenkov radiation generated by soli-
tons at the second ZDW. This is further supported by spectrograms from
simulations of the continuum generation.
The theory of mixing processes between solitons and dispersive waves
and the generation of new frequency components was first presented by
Skryabin and Yulin [18], and later demonstrated experimentally by Gorbach
et al. [13]. In addition to the Cherenkov matching condition (Equation
36), in the presence of a weak dispersive field at ωcw two further matching
conditions can be found for the generation of a new frequency component,
ω:
β(ω) = β(ωcw) + βsol(ω)− βsol(ωcw) (38)
β(ω) = −β(ωcw) + βsol(ω) + βsol(ωcw) (39)
where βsol is, as defined in Equation 32, a function of the soliton frequency
ωsol.
When expanded, Equation 38 is seen to be independent of soliton power:
β(ω) = β(ωcw) + βsol(ω)− βsol(ωcw)
β(ω) = β(ωcw) +
(β(ωsol) + β1(ωsol)[ω − ωsol] +
γP0
2
)−(β(ωsol) + β1(ωsol)[ωcw − ωsol] +
γP0
2
)β(ω) = β(ωcw) + β1(ωsol)[ω − ωcw]. (40)
New frequency components will be created at resonances defined by roots
5 Supercontinuum Generation in Photonic Crystal Fibre 48
of Equation 40. An algorithm was written to find resonances for a set CW
wavelength and soliton wavelength. This was used to return the wavelength
resonant to soliton wavelengths across the range 1.07 µm to 1.84 µm for
a set CW wavelength of either 1.07 µm or 1.98 µm (the wavelength of the
initial CW pump line or the dispersive line generated by Cherenkov radiation
of solitons at the second ZDW). The resulting matching curve is shown in
figure 24.
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8Soliton Wavelength (µm)
1.0
1.5
2.0
2.5
3.0
Reso
nant
Wav
elen
gth
(µm
)
λcw =1.98µm
λcw =1.07µm
Fig. 24: Matching curve for power-independent four wave mixing of solitons in thepresence of a CW dispersive line at either 1.07 µm (CW pump for the continuum) or1.98 µm (The cw dispersive element generated by Cherenkov radiation of solitons at thesecond ZDW)
The matching curve for the case of a dispersive CW line at 1.07 µm shows
that the FWM interaction between solitons whose wavelength ranges from
1.07 - ∼ 1.35 µm will generate new dispersive components in the anomalous
dispersion region (below 1.85 µm). This was observed in computer simu-
lations, and is clearly shown in the spectrogram shown in Figure 25a, on
which the generated FWM components are highlighted. Figure 25b shows
the wavelength of the highlighted FWM generated components against the
wavelength of the corresponding soliton, along with the analytically derived
matching curve. These data show good agreement with the matching curve,
indicating these components are indeed generated through the power inde-
5 Supercontinuum Generation in Photonic Crystal Fibre 49
pendent soliton FWM process described above.
20 10 0 10 20Delay (ps)
1.0
1.2
1.4
1.6
1.8
2.0
Wav
elen
gth
(µm
)
(a) Spectrogram of simulated spectrumin T606D after a propagation distance of2.26 m. spectral components generatedthrough power-independent soliton FWM pro-cess are highlighted with white dashed ellipses.
1.15 1.20 1.25 1.30 1.35Soliton Wavelength (µm)
1.3
1.4
1.5
1.6
1.7
Reso
nant
Wav
elen
gth
(µm
)
Calculated Matching CurveObserved in Simulation
(b) Wavelengths of resonant frequen-cies observed in Spectrogram withcalculated matching curve.
Fig. 25
To investigate the process further, a simplified case was simulated. A
single soliton was inserted at 1.25 µm with 2.5 kW peak power (FWHM
duration of 55fs). These paramters correspond to the soliton located at -9 ps
delay in the spectrogram shown in Figure 25a. The soliton was propagated
with a CW beam at 1.07 µm, with a 3 nm wide sech shaped spectrum (as
per the previous simulations of the continuum generation), but with a lower
peak power of only 10 W. In this simulation, background shot noise was
not included, but the full dispersion expansion, self-steepening term and
Raman scattering were included. A lower power CW beam than used in the
continuum simulations was necessary to prevent a CW continuum forming
from the pump line.
A spectrogram of the resulting field after 0.5m propagation distance is
shown in Figure 26a. The soliton has undergone Raman self-frequency shift
to 1.33 µm. The FWM component is clearly visible, centred around 1.73 µm,
which agrees with the wavelength predicted from the matching conditions.
There is also a significant dispersive component which has been shed from
5 Supercontinuum Generation in Photonic Crystal Fibre 50
the soliton, though this is not due to the interaction between the soliton and
the CW pump line. As Raman scattering does not conserve optical energy,
and causes the soliton to be shifted along the dispersion landscape, the
soliton will constantly have to undergo self adjustment to maintain itself. It
is because of this self-adjustment that low-level dispersive radiation is shed.
This was confirmed by running the simulation again, but without the
CW pump line. The resulting spectrogram after 0.5 m of propagation is
shown in Figure 26b. The same distinctive shedding of dispersive radiation
due to the perturbative effects of Raman self-frequency shift can be seen.
These dispersive ‘tracks’ are also clearly visible ahead of the solitons in
simulations of the continuum generation process in the fibre, for example in
the spectrogram in figure 25a.
As sech-shaped pulses are soliton solutions of the NLSE, which includes
only second order dispersion terms, and no Raman terms, it is expected that
higher order dispersion terms should perturb a fundamental soliton and have
a role to play in the shedding of dispersive radiation from the soliton. Figure
26e shows the soliton after 0.5 m of propagation with only the second order
dispersion (i.e. β2) and no Raman scattering. Figure 26d shows the effect of
the third order dispersion (TOD) term. Due to the preturbative effect of the
TOD, the sech shaped pulse is no longer a valid solitary wave solution and
the soliton must initially shed energy. Comparing Figures 26c (The Raman
scattering only case) and 26d, the Raman scattering evidently has a more
pronounced perturbative effect on the soliton propagation.
The low power level nature of the FWM component generated through
the mixing of solitons with the pump line means that using the described
experimental set-up, where ∼30 dB of dynamic range is available, the long
wavelength FWM component, over 50 dB down from the soliton peak in-
tensity, would not be resolvable. Furthermore, it is hard to envisage any
practical application for this process, and is more presented here as a cu-
5 Supercontinuum Generation in Photonic Crystal Fibre 51
10 5 0 5 10Delay (ps)
1.0
1.2
1.4
1.6
1.8
2.0
Wav
elen
gth
(µm
)
(a) Soliton co-propagating with a CW pumpline at 1.07 µm.
10 5 0 5 10Delay (ps)
1.2
1.4
1.6
Wav
elen
gth
(µm
)
(b) Soliton propagating in isolation with fullexpansion of propagation constant.
10 5 0 5 10Delay (ps)
1.2
1.4
1.6W
avel
engt
h (µ
m)
(c) Soliton propagating in isolation with onlyup to second order terms in the expansion ofthe propagation constant.
10 5 0 5 10Delay (ps)
1.0
1.2
1.4
1.6
Wav
elen
gth
(µm
)
(d) Soliton propagating without Raman scat-tering with up to third order terms in propa-gation expansion.
4 0 4Delay (ps)
1.0
1.2
1.4
1.6
Wav
elen
gth
(µm
)
(e) Soliton propagating without Raman scat-tering with only second order in the expansionof the propagation constant.
Fig. 26: 2.5 kW soliton inserted at 1.25 µm after 0.5 m propagation distance with various simulationparameters
riosity.
As it is also not possible to measure the temporal properties of the
continuum in such a way that the dispersive component of the continuum
power could be discerned from the solitonic with the experimental set-up,
and so it would be hard to experimentally confirm the shedding of dispersive
5 Supercontinuum Generation in Photonic Crystal Fibre 52
radiation located within the spectral bandwidth of continuum solitons.
The soliton four wave mixing process was, however, clearly observed
in the experimental results at the long wavelength end of the continuum,
beyond the second ZDW, where the extension of the Cherenkov-generated
dispersive component to longer wavelengths is a clear observation of new
frequency generation through soliton four wave mixing.
15 10 5 0Delay (ps)
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Wav
elen
gth
(µm
)
Soliton
Cherenkov
(a) z = 6.8 m
5 10 15 20Delay (ps)
1.0
1.2
1.4
1.6
1.8
2.0
2.2W
avel
engt
h (µ
m)
Soliton
Dispersive FWM
(b) z = 7.1 mFig. 27: Spectrograms showing the generation of new a frequency component around2.2 µm due to soliton FWM
In both the experimental results, and computer simulations, after prop-
agation distances of 10 m, the Cherenkov radiation component is seen to
broaden to long wavelengths, extending up to ∼ 2.18µm. Examining the
matching curve for power independent FWM with the dispersive CW line
at 1.98µm, as shown in Figure 24, solitons in the continuum which are
temporally located near the Cherenkov radiation emitted by solitons at the
second ZDW will be phasematched to dispersive radiation beyond 2µm. It
is this process which results in the broadening of the continuum to longer
wavelengths beyond the Cherenkov radiation. This was confirmed by exam-
ining spectrogram plots from computer simulations. A clear example of this
process is shown in Figure 27. Figure 27a shows a soliton initially stopping
short of the ZDW, resulting in the generation of a dispersive element around
5 Supercontinuum Generation in Photonic Crystal Fibre 53
2µm due to the Cherenkov process. Figure 27b shows the initial soliton and
its associated Cherenkov radiation, has been delayed past a second soliton
at a shorter wavelength. As this soliton is now temporally collocated to the
Cherenkov radiation emitted by the initial soliton, a new dispersive element
beyond the Cherenkov component has been generated at 2.18 µm due to
the soliton FWM process between the second soliton and the dispersive ra-
diation shed by the first. The wavelength of the newly generated dispersive
radiation matches the wavelength predicted by Equation 40.
The correlation between the numerical results with the analytical pre-
diction for the wavelength of the generated FWM component, along with
the strong agreement between the numerical model and the experimental
results in terms of the evolution of the spectrum with propagation distance
(Figures 19 and 21), strongly suggests that the further extension of the con-
tinuum from the Cherenkov generated component up to ∼ 2.15µm is due to
the soliton four wave mixing process. This is the first experimental obser-
vation of FWM between solitons and dispersive radiation in the context of
CW supercontinuum generation.
6 Conclusion 54
6 Conclusion
In conclusion, a high power Ytterbium fibre laser was used two pump two
different fibres, a depressed cladding HNLF and a PCF. Their starkly differ-
ent index profiles gave the fibres significantly different dispersive properties,
allowing a wealth of different continuum dynamics to be investigated with
relatively simple experimental set-up.
In the HNLF, the pump wavelength was well within the regime of normal
dispersion. A Raman cascade transferred power through concurrent Raman
orders past the ZDW, leading to the generation of a broad continuum which
extended to both long and short wavelengths relative to the ZDW. Pumping
a 300 m length of the fibre with a pump power of 140 W, a continuum with
a spectral flatness of 5 dB over 1000 nm with an average spectral power of
0.33 mW/nm over this region.
In the case of the PCF, computer simulations facilitated exploration
of the temporal as well as spectral dynamics of specific soliton-dispersive
wave interactions, which are unresolvable experimentally in the case of CW
pumped supercontinuum. The initial generation of radiation beyond the
second ZDW was understood as the emission of Cherenkov radiation by
solitons located spectrally just short of the ZDW, while further extension
of this spectral feature to longer wavelengths was understood to be a result
of soliton four-wave mixing between self-shifting solitons approaching the
ZDW and the Cherenkov radiation.
Pumping a 28 m length of the PCF gave a spectrum with a spectral flat-
ness of 6 dB over 740 nm with an average spectral power of 1.7 mW/nm. In
addition to this was a broad (80 nm FWHM) high power (330mW) spectral
component at 1.98 µm caused by the generation of dispersive radiation by
solitons across the ZDW.
These experiments illustrate the relative ease by which broad, high power
supercontinua can be experimentally realised using optical fibres. It also
6 Conclusion 55
highlights the wealth of nonlinear processes involved in formation of super-
continua from CW inputs.
6 Conclusion 56
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