Nonlinear effects and pulse propagation in PCFs

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. 1) Examples of nonlinear effects in small glass core photonic crystal fibers Prime example of nonlinear optics in PCF is supercontinuum generation Tapered fibers Photonic crystal fibers (PCF) [ 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.

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. 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.

7. 2) Physics of nonlinear effects in fibers • 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

8. 3) Theoretical framework Dispersion 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. Normal GVD Anomalous GVD group index Wavelength, m Group velocity dispersion and group index Normal GROUP VELOCITY DISPERSION Anomalous G.V.D.

11. time GVD and pulse propagation Let’s take a Gaussian pulse With freq. \omega_0 Z=0 the front and trailing tails of the pulse are symmetric in terms of their frequency content

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

13. time time After some propagation distance Z=L Normal GVD: high frequencies are SLOW Anomalous GVD: high frequencies are FAST The positive t part arrives to the point z after the negative t part This is called frequency chirping

14. Fig. 1. (A) GVD plots for the telecommunication fiber (SMF 28) and PCF used in our experiments. Zero GVD points, can be moved around by design D V Skryabin et al. Science 2003;301:1705-1708

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. 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

19. 2μm Telecom fibers:

20. Numerical method dZ dZ dZ N L N L N L Govind Agrawal: Nonlinear Fiber Optics

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

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

24. Anomalous GVD only Anomalous GVD and nonlinearity 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. Classic experiments on supercontinuum generation by fs pulses Photonic crystal fibers (PCF) Tapered fibers [ J.K. Ranka et. al., OL 25, 25 (2000) ] [ T.A. Birks et. al., OL 25, 1415 (2000) ] ‘blue’ edge ‘infrared’ edge

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

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

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

37. Interplay Resonant or Cherenkov radiation from solitons with Raman Backward emission Forward 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. Normal GVD group index Wavelength, m Why radiation is blue shifted ??? 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

40. Blue pulses Red solitons 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. radiation Frequency soliton 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 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)

44. 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.

45. 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.

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. 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

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