CEP stabilization and measurement in the highly

nonlinear regime

S. B. P. Radnor, P. Kinsler and G.H.C. New.Blackett Laboratory, Imperial College London, Prince Consort Road, London

SW7 2BW, United Kingdom.samuel.radnor@imperial.ac.uk; +44-20-7594-7641; Fax: +44-20-7594-7714

Abstract: CEP stabilization of few-cycle pulses experiencing strong nonlinearities is investi-gated. Phase stable difference frequency generation is numerically demonstrated in the presenceof SPM. Absolute CEP measurement via harmonic interference and sub-cycle CEP are alsostudied.

Control and measurement of the Carrier Envelope Phase (CEP) is one of the most important challengesfacing current nonlinear optics. The stabilization that precedes any absolute CEP measurement is generallydone using f-2f self-referencing techniques. We numerically investigate a novel 0-f self-referencing techniquethat harnesses Difference Frequency Generation (DFG) taking place in a periodically poled MgO:LN crystal[1, 2]. A complete understanding of this complex nonlinear process is of paramount importance, given therange of possible applications.

The stabilization scheme hinges on DFG occurring between opposite wings of the fundamental. This meansthat any phase slip experienced by the initial pulse is canceled out in the difference frequency signal, providingmany advantages over more traditional self-referencing methods. The benefits include: elimination of extraphase noise (generally added in the spectral broadening stage of f-2f), and the availability of the full laserpower for applications.

Our tool of choice for exploring these effects is the Pseudo-Spectral Spatial Domain (PSSD) technique, whosenatural use of the frequency domain is fully exploited [3]. The model includes χ(2) and χ(3) nonlinearitiesreplicating the nonlinear response of the medium, and allowing us to verify the accuracy and limits ofthe scheme. We map the evolution of the EM field through the crystal, which suggests optimizations ofthe output signal based on alternative crystal lengths. Given the power of these pulses and the materialresponse, the regime is highly nonlinear; SPM effects are present and other strong second order nonlineareffects dominate. For this reason, a stable difference frequency signal emerging from the system is perhapsan unexpected triumph.

2 2.5 3 3.5 4x 10

15

2

4

6

8

10x 107

ω (rad s−1)

|E(ω

)|2 [A

rb.]

CEP=π/2

CEP=0

Fig. 1. Interference region between the 1st and 3rd harmonic: An enlargement of the intensity spectrumin a χ(3) material for different distances of propagation . We can see that CEP=π/2 always producesmaximum destructive interference between adjacent harmonic lobes (dotted curves). This phenomenon mightbe exploited in the measurement of absolute CEP.

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We further pursue other ways of increasing the DFG signal that would be tracked in a typical experiment. Onepossibility is to use SPM to broaden the fundamental spectrum. This provides a two-fold advantage: reducingthe peak intensity so that less energy is frequency doubled, and increasing the intensity at the differencefrequencies of interest. It turns out that the interplay between the rate of SPM induced broadening andthe DFG plays a crucial role in the determination of the final signal. This is important considering thatphase stable signals at low frequencies are very useful for seeding other optical parametric processes. Otheraspects of phase cancelation have been demonstrated elsewhere, involving optical parametric amplificationon a shot-to-shot basis [4].

Even more challenging in the characterization of few-cycle pulses is the determination of the absolute CEP.One approach involves measuring the interference regions between harmonics, where the level of interferencecan then be mapped to a particular value of CEP [5]. We investigate the feasibility of extracting the absoluteCEP using this approach. We also show that the method is insensitive to the distance traveled through thenonlinear crystal, if the harmonics are sufficiently well defined. In principle, this enables a direct measurementof CEP to be made (see fig. 1).

In other areas of nonlinear optics half-cycle terahertz pulses have been produced [6]. The duration of theenvelope compared to the period of the carrier in a sub-cycle pulse poses fundamental questions relatingto the central wavelength, envelope and energy content of such waveforms. Is there a difference betweenchanging the CEP of a time domain picture and adding a phase in the frequency domain? For sub-cyclepulses, we show how the envelope is not necessarily constant. The effect of adding correction terms from thevector potential associated with the net-force condition can also play a role. We discuss some of these issuesand suggest how they can be addressed.

References

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2. J. Rauschenberger, T. Fuji, M. Hentschel, A.-J. Verhoef, T. Udem, C. Gohle, T. W. Hansch and F. Krausz, Laser Phys.Lett., 3 (2006) 37.

3. J. C. A. Tyrrell, P. Kinsler and G. H. C. New, J. Mod. Opt. 52 (2005) 975.4. X. Fang and T. Kobayashi, Opt. Lett., 29 (2004) 1282.5. M. Mehendale, S. A. Mitchell, J.-P. Likforman, D. M. Villeneuve and P. B. Corkum, Opt. Lett., 25 (2000) 1672.6. J. Bromage, S. Radic, G. P. Agrawal, C. R. Stroud, Jr., P. M. Fauchet and R. Sobolewski, J. Opt. Soc. Am. B, 15 (1998)

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