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

iv

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)

2 Background 18

extension of the continuum.

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


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


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


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.


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


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.


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-


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.


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


(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.


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-


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


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


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,


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-


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


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.


5.2 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-


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


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.


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.


5.3.1 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-


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


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


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


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.


5.3.3 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


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-


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


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-


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


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


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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