Nonlinear effects and pulse propagation in PCFs

--Examples of nonlinear effects in small glass core photonic crystal fibers--Physics of nonlinear effects in fibers--Theoretical framework--Solitons and soliton effect pulse compression--Raman effect--Soliton-self frequency shift--Dispersive waves emitted by solitons--Supercontinuum generation--Modulational instability, degenerate and nondegenerate four-wave mixing--Short pulses in hollow core

2μm

Solid-core PCFs Hollow-core PCFs strong nonlinearity weak nonlinearity

[ J.K. Ranka et. al., OL 25, 25 (2000) ]

Photonic crystal fibers (PCF)

[ T.A. Birks et. al., OL 25, 1415 (2000) ]

Tapered fibers

shortwavelength partlongwavelength part

Prime example of nonlinear optics in PCF is supercontinuum generation

1) Examples of nonlinear effects in small glass core photonic crystal fibers

Abstract: We demonstrate experimentally for what is to our knowledge the first time that air–silica microstructure optical fibers can exhibit anomalous dispersion at visible wavelengths. We exploit this feature to generate an optical continuum 550 THz in width, extending from the violet to the infrared, by propagating pulses of 100-fs duration and kilowatt peak powers through a microstructure fiber near the zero-dispersion wavelength.

14. Supercontinuum generation for carrier-envelope phase stabilization of mode-locked lasers S. T. Cundiff15. Biophotonics applications of supercontinuum generation C. Dunsby and P. M. W. French16. Fiber sources of tailored supercontinuum in nonlinear microspectroscopy and imaging A. M. Zheltikov

W. Wadsworth et al

Parametric four-wave mixing in solid-core PCF

Abstract: Photonic crystal fibres exhibiting endlessly single-mode operation and dispersion zero in the range 1040 to 1100 nm are demonstrated. A sub-ns pump source at 1064 nm generates a parametric output at 732 nm with an efficiency of 35%, or parametric gain of 55 dB at 1315 nm. A broad, flat supercontinuum extending from 500 nm to beyond 1750 nm is also demonstrated using the same pump source.

2) Physics of nonlinear effects in fibers

time

a) Ultrafast (fs) Kerr nonlinearity, related to the oscillations of the electron cloud

b) Raman nonlinearity, related to vibrations of glass molecules (10s of fs)

Interplay of nonlinearity and dispersion is the key to understand nonlinear optical processes in PCFs

Dispersion

3) Theoretical framework

Propagation constant

Effective (refractive) index:Mix of the material and geometry induced dispersions

NORMAL Phase Velocity DISPERSION

ANOMALOUS P.V. DISPERSION

Normal dispersion at the air glass interface

Group velocity dispersion and group index

Normal GROUP VELOCITY DISPERSION

Anomalous G.V.D.

grou

p in

dex 02 02

Anomalous GVDNormal GVD

Wavelength, m

time

the front and trailing tails of the pulse are symmetric in terms of their frequency content

Z=0

GVD and pulse propagation

Let’s take a Gaussian pulseWith freq. \omega_0

Net result on the pulse envelope is spreading

for both normal and anomalous GVD

Dispersive waveguide

Normal GVD: high frequencies are SLOW

Anomalous GVD: high frequencies are FAST

timetime

The positive t part arrives to the point z after the negative t part

After some propagation distance Z=L

This is called frequency chirping

Fig. 1. (A) GVD plots for the telecommunication fiber (SMF 28) and PCF used in our experiments.

D V Skryabin et al. Science 2003;301:1705-1708 Zero GVD points, can be moved around by design

Mathematics and physics of pulse propagation in fibers

are the Dispersion coefficients of different orders

beta_1 is the inverse group velocity

beta_2 is a formal definition of GVD

[n2]=m^2/W

we scale intensity with the area S

and get an equation for the amplitude measured in the units of power

at the same time we switch into the reference frame moving together with the pulse

T is usually scaled with the duration of the input pulse and

Z with the dispersion length,

where the pulse intensity profile (in the linear case) is twice as broad as the one of the initial unchirped Gaussian pulse

Generalised nonlinear Schrodinger equation

2μm

Telecom fibers:

Numerical method

N NN LLL

dZ

dZdZ

Govind Agrawal: Nonlinear Fiber Optics

Nonlinearity without dispersion: Self-phase modulation

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Net effect of SPM on the pulse

time

Associated spectral evolution

frequency

Chirp Intensity Spectrum

SPM

GVD

time

time

Normal GVD

Anomalous GVD

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Solitons

SPM

Can compensateone another, fora special pulse profiles

Positive and negative chirps increase equally over the dispersion length

Anomalous GVD and nonlinearity

Anomalous GVD only

PCFs substantially extended the spectral range of the soliton existence relative to the telecom fibers

Impact of Raman effecton solitons:soliton-self-frequency shift

Emission of narrow banddispersive waves by a solitonclose to the zero GVD point

Supercontinuum from fs pulseshow does it happen ?

[ J.K. Ranka et. al., OL 25, 25 (2000) ]

Photonic crystal fibers (PCF)

[ T.A. Birks et. al., OL 25, 1415 (2000) ]

Tapered fibers

‘blue’ edge ‘infrared’ edge

Classic experiments on supercontinuum generationby fs pulses

What is essential

Dispersion, correctly changing with wavelength

Kerr nonlinearity Raman effect

What is (can be?) left out

Noise

Multimode effects Dispersion of nonlinearity

^ 2 2

0

[ ( )] (1 ) ( ) ( , )z ti i A i A A i A dt R t A t t z

Time-domain

spectrum

Solitons and frequency conversion in the PRE supercontinuum era

1. Multi-soliton effect pulse compression

Correlated pairs of femtosecond nondispersive pulses across the zero GVD pointwith frequencies shifting in the opposite directions

2. Raman only and soliton delay

wavelength

z

grou

p in

dex 02 02

Anomalous GVDNormal GVD

Wavelength, m

Anomalous GVD + Raman ==delay (solitons are delayed)

Interplay Resonant or Cherenkov radiation from solitons with Raman

Backward emission Forward emission

For repeated soliton-radiation collisions lead to the sequence of the sadden jumps of the radiation frequency

03

Gorbach et al, Opt. Express, vol 14, 9854 (2006)

Backward reflection from the soliton means radiation delay, i.e. decrease in the group velocity, which has to be accompanied by the corresponding change in frequency dictated by the dispersion of the fibre

grou

p in

dex 02 02

Normal GVD

Wavelength, m

Why radiation is blue shifted ???

Red solitons

Blue pulses

Why radiation is localised on the femtosecond time scale and does not disperse ???

IF YOU ARE STANDING IN THE ELEVATOR WITHOUT WINDOWS YOU CAN NOT TELL WHETHER THE LIFT IS IN THE FIELD OF GRAVITY OR

YOU ARE PULLED UP WITH A CONSTANT ACCELERATION

Soliton is the floor of the elevatorBlue balls are the radiation

F

requ

ency

soliton

radiation

z

Frequency of the trapped radiation is continuously blue shifted,which is dictated by the fact the radiation is trapped by the solitonand hence slowed down together with it.

Group velocities of the trapped radiation mode and of the soliton are matched across the zero GVD point

Trapped radiation experiments

Recent experimental work:Nishizawa, Goto (Japan)Stone, Knight (Bath, UK)R. Taylor (Imperial, UK)Kudlinski (France)

before the first theoreticalpaper on Cherenkov radiationby fiber solitons

Skryabin, D.V. & Gorbach, A.V. (2010), "Looking at a soliton through the prism of optical supercontinuum", Reviews of Modern Physics., April, 2010. Vol. 82, pp. 1287-1299.

Gorbach, A.V. & Skryabin, D.V. (2007), "Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres", Nature Photonics., November, 2007. Vol. 1(11), pp. 653-657.

W. Wadsworth et al

Parametric four-wave mixing in solid-core PCF

Abstract: Photonic crystal fibres exhibiting endlessly single-mode operation and dispersion zero in the range 1040 to 1100 nm are demonstrated. A sub-ns pump source at 1064 nm generates a parametric output at 732 nm with an efficiency of 35%, or parametric gain of 55 dB at 1315 nm. A broad, flat supercontinuum extending from 500 nm to beyond 1750 nm is also demonstrated using the same pump source.

Degenerate 4WM in fibers(modulational instability)

Odd order dispersion coefficients are irrelevant for 4WM gain

Is the condition of the FWM gain

2 pump photons

Converted to 2 Side-band photons

Modulational instability growth rate , when 2nd order dispersion dominates

n2 is positive in fibers, therefore gain can exist only if \beta_2 is negative, i.e. GVD is anomalous. If GVD is normal, then there is no gain, and signal+idler are not amplified

Typical nonlinear fibre parameter due to Kerr effect:

γ = 10 - 6 1/ [ Wm ]

Fs pulse propagationIn hollow core PCFs

Mode profiles byP.J. Roberts

Core nonlinearity 10 - 6 1/ [ Wm ]Surface nonlinearity 10 - 1 1/ [ Wm ]

If you are close to the crossing with the ‘surface’ mode,you need account for 2 modes

If you are far from the crossing, then the surface modeis not coupled to the core mode, but the core modestill overlaps with the glass, therefore there are 2nonlinearities involved with one mode

F. Luan, J. Knight, P. Russell, S. Campbell, D. Xiao, D. Reid, B. Mangan, D. Williams, and P. Roberts, "Femtosecond soliton pulse delivery at 800nm wavelength in hollow-core photonic bandgap fibers," Opt. Express 12, 835-840 (2004)

Which Raman and nonlinearity are more important,Depends not only on the fiber design and wavelength ofOperation, but also on the pulse duration !!!

Andrey V. Gorbach and Dmitry V. Skryabin, "Soliton self-frequency shift, non-solitonic radiation and self-induced transparency in air-core fibers," Opt. Express 16, 4858-4865 (2008)