The document discusses self-similar propagation of optical pulses in fiber amplifiers and lasers. It is perturbed by effects like third-order dispersion (TOD) and gain saturation. Theoretical and numerical models are presented to analyze pulse breakup due to TOD and the emergence of hyper-gaussian pulses under high gain saturation. Good agreement is found between analytical predictions of critical distances and numerical results.

1. Claude Aguergaray with: V. I. Kruglov, J. D. Harvey e-mail : [email_address] Perturbations to self-similar propagation in optical amplifiers Physics Research 3

2. Context Sech input pulse in DSF Raman amplifier. Self-similar pulses P arabolic pulses are generated asymptotically in the fiber amplifier independent of the shape or the noise properties on the input pulse , and possess linear chirp . A class of solution of the nonlinear Schrödinger equation (NLSE) with gain. Self-similar evolution of the pulse intensity and chirp the intensity profile retains its parabolic shape and resists the deleterious effects of optical wave breaking .

3. Under normal dispersion , nonlinearity and gain ↪ Any input pulse will evolve asymptotically into a similariton with a parabolic intensity profile and positive linear chirp. Context Finot et al. OE 11 (2003). Self-similar pulses

4. Context Billet et al. OE 13 (2005) Dudley et al. Nature 3 (2007). Self-similar pulses The Self-Similar dynamics : - Peak power - Temporal width increase exponentially with propagation length. - Spectral width

5. The common point between Self-similar and the well-known technique of chirped-pulse amplification (CPA) is that they aim at avoiding the pulse break-up due to excessive nonlinear phase shifts accumulated through the fiber. - When CPA avoids nonlinearity by stretching the pulse before amplification, - Self-similar amplifier actively exploits nonlinearity, (possibility of obtaining output pulses shorter than the initial input pulse). For energies >µJ similariton amplifiers can be limited by the available gain-bandwidth. But they are a good alternative to more complex CPA systems below this limit. Self-similar amplifiers have been demonstrated : - Several types of gain medium: ytterbium, erbium and Raman - For a broad range seed pulses in the range 180 fs–10 ps - Fiber lengths in the range 1.2 m to 5.3 km - Gains varying from 14 to 32 dB. Self-similar pulse in amplifiers : Self-similar pulses Context

6. Amplification to the μJ level in an environmentally stable and polarization-maintaining configuration has been a demonstrated. Self-similar pulse in amplifiers : Compressed duration: 240 fs Repetition rate: 27 MHz Average power: 21 W Peak power: 5 MW Schreiber et al. OL 31 (2006). Self-similar pulses Context

7. Context Billet et al. OE 13 (2005). All fiber compression stage by use of photonic bandgap optical fibre to replace bulk gratings lead to the realization of an all-fiber source delivering pulses in the 100 fs range at 1550nm. Self-similar pulse in amplifiers : Self-similar pulses After 7m of propagation FWHM 136 fs FROG measurement of compressed pulses

8. Context Self-similar pulse in lasers : The net GVD of the cavity can be normal or anomalous. - With large net anomalous GVD, soliton like pulses. These lasers (1st developed) have stringent limitation in energy (nJ) and pulse duration (ps) due to excessive nonlinear phase shift accumulated by the pulse. To overcome this limitation researcher have developed laser cavity with dispersion map. Self-similar pulses

9. Context Self-similar pulse in lasers : Aguergaray et al. OE 18 (2010). - GVD ≈ 0, stretched-pulse operation occurs. The pulse energy can be an order of magnitude higher than in a soliton laser. - GVD >> 0, higher pulse energies can be achieved directly from an oscillator. Among these are the self-similar laser and the so-called chirped pulse oscillator (CPO). Pulse shaping in such a laser is based on spectral filtering of the chirped pulse, which cuts off the temporal wings of the pulse. Laser output pulse energy: 21 nJ Self-similar pulses

10. Motivation Linearly chirped parabolic pulse is an asymptotic class of solution of the GNLSE with constant gain, normal dispersion and in the presence of non-linearities. ↪ Any input pulse with right energy will evolve into a similariton regardless of its shape and duration. Until recently most theoretical descriptions of the self-similar (SS.) propagation had been done assuming that only low-order nonlinear effects and low-order dispersion effects dominate the pulse evolution. But experiment have shown that SS. propagation can be perturbed by effect like: - Gain bandwidth limitation, - Third-order dispersion , - Gain saturation. Self-similar pulses

11. Similariton propagation and break-up with third-order dispersion influence

12. Motivation ↪ Asymmetric temporal pulse shape, ↪ Peak shifted towards the edges of the pulse, (direction depends on TOD sign). Lead to generation of shock wave instability and pulse break-up . TOD can have a detrimental effect on parabolic pulse propagation. Similariton break-up with  3

13. Anterior work z = 1 km z = 779 m  3 = 0.025 ps 3 /km  3 = 0 ps 3 /km Wabnitz and Finot have observed pulse break-up in Dispersion Decreasing Fibers (increased TOD influence with distance) Wabnitz and Finot JOSA B 25 (2008). 10 % accuracy for pulse break-up critical distance. z = 755 m  3 = 0.025 ps 3 /km Similariton break-up with  3

14. Bale and Boscolo: ↪ Partial analytical description of the pulse before and at break-up, ↪ No theoretical prediction of critical distance. z/z c = (a) 0.25 , (b) 0.5 , (c) 0.75 , (d) 1 . Bale and Boscolo, J. Opt. 12 (2010). 16% error between analytical and numerical simulations. Similariton break-up with  3 Anterior work

15. Theoretical study. Our Analytical model : Similariton break-up with  3

16. Theoretical study. Numerical model :  (z)   3 controls pulse shape (asymmetry of the pulse) Critical parameter is given by T 2 (z) = T 3 (z) (  3 >0), or T 1 (z) = T 2 (z) (  3 <0). Condition gives critical length z c at which pulse breaks down. Similariton break-up with  3

17. Theoretical study. Yields to the critical distance parameter : where Similariton break-up with  3

18. Numerical results. Analytical expression of critical distance :  2 =0.13 ps 2 /m  3 =10 -3 ps 3 /m g=2 m -1  =2.10 -3 w -1 m -1 Similariton break-up with  3

19. Numerical definition of pulse break-up : ↪ Pulse experiences growth of side peak under  3 influence Numerical results. Similariton break-up with  3

20. Numerical results. Numerical simulations varying energy E 0 :  3 = 0.96x10 -3 ps 3 /m 0.1pJ<E 0 <10pJ <1% error for critical length prediction Similariton break-up with  3

21.  3 < 0  3 > 0 Numerical results. Very good agreement between analytical prediction and numerical solution. Similariton break-up with  3

22. Summary of β 3 study Novel analytical theory for propagating pulses in normal dispersion fiber amplifier with TOD. Found the critical length z c at which the TOD generate pulse break-up for constant gain. Shown numerically the limitations for input value  providing a highly accurate analytical description of the quasi-similariton and the critical length:  ≤ 10 -4 . Critical distance z c does not depend on the sign of TOD. Published in : Optics Letters / Vol. 35 , No. 18, p. 3084 (2010) Physical Review A / Vol. 84 , No. 2, 023823 (2011) Similariton break-up with  3

23. Motivation Until recently most theoretical descriptions of the self-similar (SS.) propagation had been done assuming that only low-order nonlinear effects and low-order dispersion effects dominate the pulse evolution. But experiment have shown that SS. propagation can be perturbed by effect like: - Gain bandwidth limitation , - Third-order dispersion, - Gain saturation . Self-similar pulses

24. Parabolic and Hyper-Gaussian similaritons propagating in fiber with saturation effect.

25. Gain saturation effect is important for the pulse evolution in normal fiber ring lasers. It is negligible over the duration of a single pulse but cannot be neglected for a long pulse train since the amplifier gain will saturate over long time scale (> population relaxation time). We use here the standard model equation for saturation effect obtained by averaging the gain dynamics in the presence of the pulse train. Similariton with gain saturation Gain saturation model Dependence of laser gain on the optical power at the steady state

26. Similariton with gain saturation Analytical solution The analytical solution is an exact asymptotical solution of the NLSE . (From differential equation for the propagating pulses in optical amplifiers with an arbitrary gain function). Slowly varying envelope : Gain function :

27. Similariton with gain saturation Analytical solution Peak power : Pulse duration : Analytical solution with saturation effect takes the form : Dimensionless variables Dimensionless parameters

28. Similariton with gain saturation Parabolic similaritons  = 4000  = 400 Simulation parameters: β 2 = 0.02 ps 2 m -1 /  = 2 10 -5 W -1 m -1 / g 0 = 2 m -1 . Input energy: E 0 = 200 pJ   0 = 0.02. Saturation energy: E S = 20 nJ   S = 2. Temporal profile and the chirp of the pulses for two different propagation distances : Input parameters:

29. Similariton with gain saturation  = 600  = 100 Input energy: E 0 = 10 pJ   0 = 0.001. Saturation energy: E S = 10 µJ   S = 100. Results for increased saturation energy : Input parameters: Parabolic similaritons

30. Similariton with gain saturation Hyper-Gaussian similaritons For low amplification regimes ,  S < 0.3 (in our case E S < 3 nJ), the gain seen by the pulse goes very quickly to zero along the fiber. The input pulses evolve into a different similariton regime with a linear chirp but a non parabolic shape. The pulse develops an Hyper-Gaussian shape. It propagates through the fiber self-similarly with a linear chirp !

31. Similariton with gain saturation Hyper Gaussian similaritons This function is a product of a Gaussian and a super-Gaussian therefore we named it Hyper-Gaussian pulse (HG pulse). Two asymptotic non-linear attractors : which route depends on  S

32. Similariton with gain saturation Hyper-Gaussian similaritons The spectral density of differs significantly parabolic shape. HG pulses undergoe small spectral broadening due to weak non-linear effects what leads to a very smooth spectral shape (interest for fiber based amplification systems). Linear chirp proving the self-similar aspect of the HG pulse propagation.

33. Similariton with gain saturation Hyper-Gaussian similaritons Test for different input pulse shape : All the pulses converge towards a HG shape pulse with linear chirp ! HG pulse is a local asymptotic attractor.

34. Similariton with gain saturation Our analytical solution for similariton pulses in a fiber amplifier with gain saturation allows an accurate predictions of the pulse temporal shape and chirp for a wide range of the saturation energy parameter. A limit of  s > 0.3 setting the lower boundary has been found No upper limit on  s … (computation time is restrictive). A new local non-linear attractor leading to self-similar HG pulses has been identified . Summary of E SAT study

35. Conclusion Two analytical solutions able to predict accurately the perturbations to the self-similar propagation caused by the TOD and the gain saturation. Since low amplification is required for a pulse to evolve into a HG pulse , Could be implemented in low energy laser systems delivering linearly chirped pulses. Potential for pre-amplification stage of ultra-short pulse CPA systems to obtain linearly chirped pulses with no spectral structure .

38. Bric a brac

40. Numerical simulations The HG similaritons may form when : - The energy E(z) of the pulse is a slowly growing function of distance, - The peak power of the pulse is a constant or decreasing function of z.

41. Overview Motivations Theoretical study Numerical results Conclusion

42. Motivation How to predict accurately the critical distance and the pulse shape? Similariton break-up with  3

43. Analytical pulse energy coincides with exact energy Numerical results.

44. OWN1 / Finot OSA OFC 2009 Context

45. Theoretical study. Renormalisation procedure :

46. Motivation However self-similar propagation is severally affected by Third Order Dispersion (TOD). Novel features observed due to TOD in fiber amplifiers . Φ NL accumulated in amplifier (SPM) compensated by TOD of fiber stretcher + grating compressor Zhou et al. (Wise) OE 13, 4869 (2005) Grating stretcher and compressor best result. Φ NL = 1.9 π Φ NL = 0.4 π Φ NL = 1.9 π

47. Theoretical study. No renormalisation With renormalisation procedure applied

48. Motivation Novel features observed due to TOD in mode-locked lasers . Logvin et al. OE 15, 985 (2007)

49. Motivation Net cavity GVD= 0.005 ps 2 Logvin et al. OE 15, 985 (2007) SMF and Yb fibers TOD (Negligible TOD) Similariton regime: symmetric pulse, top spectrum tilted. PBF TOD = 500 fs 3 /mm Cubicon-like features: asymmetric pulse, triangular shape spectrum. PBF TOD = 1200 fs 3 /mm. Stretched Pulse regime: narrower pulse, broader spectrum with asymmetric sidebands.

50. Numerical results. Numerical model : with Normalised variable E0 3.5 3.5 8  0.029 0.1 0.1

51. Theoretical study. Respecting  < 10 -4 condition :

52. Theoretical study. Pulse propagation in fibers with TOD is described by the following NLSE : Analytical solution found using a first order perturbation theory :

53. Motivation Latkin et al. OE 32, 331 (2007) Intensity Time [ps] Distance [km] Intensity Wavelength [nm] Distance [km]

54. The Australian Optical Society