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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)