1. 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

2. 2μm Solid-core PCFs Hollow-core PCFs strong nonlinearity weak nonlinearity

3. [ 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 part longwavelength 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.

5. 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

6. 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.

7. 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

8. Dispersion 3) Theoretical framework Propagation constant Effective (refractive) index: Mix of the material and geometry induced dispersions

9. NORMAL Phase Velocity DISPERSION ANOMALOUS P.V. DISPERSION Normal dispersion at the air glass interface

10. Group velocity dispersion and group index Normal GROUP VELOCITY DISPERSION Anomalous G.V.D. group index Anomalous GVDNormal GVD Wavelength,  m

11. 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 pulse With freq. \omega_0

12. Net result on the pulse envelope is spreading for both normal and anomalous GVD Dispersive waveguide

13. Normal GVD: high frequencies are SLOW Anomalous GVD: high frequencies are FAST time 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

14. 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

15. Mathematics and physics of pulse propagation in fibers

16. are the Dispersion coefficients of different orders beta_1 is the inverse group velocity beta_2 is a formal definition of GVD

17. [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

18. 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

19. 2μm Telecom fibers:

20. Numerical method NNNLLL dZ Govind Agrawal: Nonlinear Fiber Optics

21. Nonlinearity without dispersion: Self-phase modulation Net effect of SPM on the pulse time Associated spectral evolution frequency ChirpIntensitySpectrum SPM GVD

23. time Normal GVD Anomalous GVD Solitons SPM Can compensate one another, for a special pulse profiles Positive and negative chirps increase equally over the dispersion length

24. Anomalous GVD and nonlinearity Anomalous GVD only PCFs substantially extended the spectral range of the soliton existence relative to the telecom fibers

25. Impact of Raman effect on solitons: soliton-self-frequency shift

26. Emission of narrow band dispersive waves by a soliton close to the zero GVD point

28. Supercontinuum from fs pulses how does it happen ?

29. [ 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 generation by fs pulses

31. What is essential Dispersion, correctly changing with wavelength Kerr nonlinearity Raman effect What is (can be?) left out Noise Multimode effects Dispersion of nonlinearity

33. Time-domain spectrum Solitons and frequency conversion in the PRE supercontinuum era 1.Multi-soliton effect pulse compression

34. Correlated pairs of femtosecond nondispersive pulses across the zero GVD point with frequencies shifting in the opposite directions

36. 2. Raman only and soliton delay wavelength z group index Anomalous GVDNormal GVD Wavelength,  m Anomalous GVD + Raman == delay (solitons are delayed)

37. Interplay Resonant or Cherenkov radiation from solitons with Raman Backward emissionForward emission

38. For 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)

39. 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 group index Normal GVD Wavelength,  m Why radiation is blue shifted ???

40. Red solitons Blue pulses Why radiation is localised on the femtosecond time scale and does not disperse ???

41. 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

42. Frequency soliton radiation z 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. Group velocities of the trapped radiation mode and of the soliton are matched across the zero GVD point

43. Trapped radiation experiments Recent experimental work: Nishizawa, Goto (Japan) Stone, Knight (Bath, UK) R. Taylor (Imperial, UK) Kudlinski (France) before the first theoretical paper on Cherenkov radiation by fiber solitons

44. 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.

45. 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.

46. Degenerate 4WM in fibers (modulational instability)

47. Odd order dispersion coefficients are irrelevant for 4WM gain Is the condition of the FWM gain

48. 2 pump photons Converted to 2 Side-band photons Modulational instability growth rate, when 2 nd 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

49. Typical nonlinear fibre parameter due to Kerr effect: γ = 10 - 6 1/ [ Wm ] Fs pulse propagation In hollow core PCFs

51. Mode profiles by P.J. Roberts Core nonlinearity 10 - 6 1/ [ Wm ] Surface nonlinearity 10 - 1 1/ [ Wm ]

52. 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

53. 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)

56. 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)