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 Solid-core PCFs Hollow-core PCFs strong nonlinearity weak nonlinearity 2µm
1) Examples of nonlinear effects in small glass core photonic crystal fibers Prime example of nonlinear optics in PCF is supercontinuum generation Photonic crystal fibers (PCF) Tapered fibers [ J.K. Ranka et. al., OL 25, 25 (2000) ] [ T.A. Birks et. al., OL 25, 1415 (2000) ] shortwavelength part longwavelength part 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. Cundiff 15. Biophotonics applications of supercontinuum generation C. Dunsby and P. M. W. French 16. Fiber sources of tailored supercontinuum in nonlinear microspectroscopy and imaging A. M. Zheltikov Parametric four-wave mixing in solid-core PCF W. Wadsworth et al 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 a) Ultrafast (fs) Kerr nonlinearity, related to the oscillations of the electron cloud time 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 3) Theoretical framework Dispersion 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.
GVD and pulse propagation Let s take a Gaussian pulse With freq. \omega_0 Z=0 time the front and trailing tails of the pulse are symmetric in terms of their frequency content Net result on the pulse envelope is spreading for both normal and anomalous GVD Dispersive waveguide
After some propagation distance Z=L Normal GVD: high frequencies are SLOW Anomalous GVD: high frequencies are FAST time time The positive t part arrives to the point z after the negative t part 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 Generalised nonlinear Schrodinger equation 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
2µm Telecom fibers: Numerical method dz dz dz N L N Govind Agrawal: Nonlinear Fiber Optics L N L
Nonlinearity without dispersion: Self-phase modulation Net effect of SPM on the pulse Associated spectral evolution 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 -10-8 -6-4 -2 0 2 4 6 8 10 time SPM GVD Chirp Intensity frequency Spectrum
Solitons SPM Normal GVD 1 0.8 0.6 0.4 time 0.2 0-0.2-0.4-0.6-0.8-1 -10-8 -6-4 -2 0 2 4 6 8 10 Can compensate one another, for a special pulse profiles Anomalous GVD time Positive and negative chirps increase equally over the dispersion length Anomalous GVD and nonlinearity Anomalous GVD only
Multi-soliton evolution in ideal NLS Impact of Raman effect on solitons: soliton-self-frequency shift
Emission of narrow band dispersive waves by a soliton close to the zero GVD point
Supercontinuum from fs pulses how does it happen? Classic experiments on supercontinuum generation by fs pulses Photonic crystal fibers (PCF) [ J.K. Ranka et. al., OL 25, 25 (2000) ] Tapered fibers [ T.A. Birks et. al., OL 25, 1415 (2000) ] blue edge infrared edge
^ 2 2 [ iβ( i )] A = iγ(1 θ) A A+ iγθa dtr ( t ) A( t t, z) z t 0 What is essential Dispersion, correctly changing with wavelength Kerr nonlinearity Raman effect What is (can?) be left out Noise Multimode effects Dispersion of nonlinearity
Solitons and frequency conversion in the PRE supercontinuum era 1. Multi-soliton effect pulse compression Time-domain spectrum
Correlated pairs of femtosecond nondispersive pulses across the zero GVD point with frequencies shifting in the opposite directions
2. Raman only and pulse delay/acceleration z wavelength Anomalous GVD == delay Normal GVD == acceleration group index Normal GVD β >0 2 Anomalous GVD β 2 <0 Wavelength, µm Interplay Resonant or Cherenkov radiation from solitons with Raman Backward emission Forward emission
For β >0 3 repeated soliton-radiation collisions lead to the sequence of the sadden jumps of the radiation frequency Gorbach et al, Opt. Express, vol 14, 9854 (2006) Why radiation is blue shifted??? Backward reflection from the slowing 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 group index Normal GVD β >0 2 β 2 <0 Wavelength, µm
Why radiation is localised on the femtosecond time scale and does not disperse??? Blue pulses Red solitons 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 elevator Blue balls are the radiation
Frequency of the trapped radiation is continuously blue shifted, which is dictated by the fact the radiation is trapped by the soliton and hence slowed down together with it. radiation Frequency soliton z Group velocities of the trapped radiation mode and of the soliton are matched across the zero GVD point Gorbach, Skryabin, Physical Review A (2007) Onset of strong recoil
Trapped radiation experiments before the first theoretical paper on Cherenkov radiation by fiber solitons Recent experimental work: Nishizawa, Goto (Japan) Stone, Knight (Bath, UK) R. Taylor (Imperial, UK) Kudlinski (France) 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. 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.
Parametric four-wave mixing in solid-core PCF W. Wadsworth et al 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)
Instab. Cond. Expressed via mismatch of the total (lin+nonlin) wavenumbers Instab. Cond. Expressed via mismatch of the linear wavenumbers Odd order dispersion coefficients are irrelevant for 4WM gain Is the condition of the FWM gain
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 2 pump photons Converted to 2 Side-band photons Fs pulse propagation In hollow core PCFs Typical nonlinear fibre parameter due to Kerr effect: γ = 10-6 1/ [ Wm ]
Core nonlinearity 10 Mode profiles by - 6 1/ [ Wm ] Surface P.J. Robertsnonlinearity 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 mode is not coupled to the core mode, but the core mode still overlaps with the glass, therefore there are 2 nonlinearities 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 of Operation, 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)