1. Shortening a laser pulse at the focus of a lens Yuelin Li Accelerator Systems Division Argonne National Laboratory ylli@aps.anl.gov

2. CLEO 08, San Jose, May 4-9, 2008 2 Content Introduction –Pulse lengthening at lens focus Fourier formulation of the problem –Axial temporal history –Pulse shortening due to GVDE and SPM –Numerical simulation Experiment outlook Summary

3. CLEO 08, San Jose, May 4-9, 2008 3 Content Introduction –Pulse lengthening at lens focus Fourier formulation of the problem –Axial temporal history –Pulse shortening due to GVD –Numerical simulation Experiment outlook Summary

4. CLEO 08, San Jose, May 4-9, 2008 4 Pulse lengthening due to Group Velocity Delay Z. Bor, Opt. Lett. 15, 119 (1989)

5. CLEO 08, San Jose, May 4-9, 2008 5 Content Introduction –Pulse lengthening at lens focus Fourier formulation of the problem –Axial temporal profile –Pulse shortening due to GVD, with SPM –Numerical simulation Experiment outlook Summary

6. CLEO 08, San Jose, May 4-9, 2008 6 Formula of the problem: Fourier optics Full wave optics (Fresnel diffraction) adapted from Kempe et al. (JOSA B 9, 1158 (1992)) Group velocity dispersion and group velocity delay effect considered up to the second order

7. CLEO 08, San Jose, May 4-9, 2008 7 On axis formula U: field in frequency domain representation at the focus f : is the focal length r: ray location K l, k a : wave vectors in the lens and air, n: refractive index in the lens. A(  ): input filed (homogeneous) in the frequency domain. , Lens transfer function r Inverse Fourier transform a=a(t)=F -1 A(  ) and  =  (r,t)=F -1  (r,  ) Field at focus, on axis d f

8. CLEO 08, San Jose, May 4-9, 2008 8 Pulse with no SPM For a=a 0 exp(-2ln2t 2 /  2 ), no SPM,  =4k  T/  2. Chirping and broadening due to Group Velocity DIspersion (GVDI) Relative delay between pulse slices due to Group Velocity DElay (GVDE) GVDE and GVDI have both been studied.

9. CLEO 08, San Jose, May 4-9, 2008 9 Max phase modulation, over lens of thickness d: Pulse with SPM Assume smaller bandwidth and d 2  /dn 2 =0,  =4k  T/  2. Delay due to GVDE Case not been studied. Interference due to GVDE

10. CLEO 08, San Jose, May 4-9, 2008 10 Pulse shortening at the focus Li and Crowell, Opt. Lett. 32, 92 (2007).

11. CLEO 08, San Jose, May 4-9, 2008 11 Parameter  f /  i Max GVDE to pulse duration ratio Max phase modulation  =2   =3 

12. CLEO 08, San Jose, May 4-9, 2008 12 Numerical simulation Li and Crowell, Opt. Lett. 32, 92 (2007).

13. CLEO 08, San Jose, May 4-9, 2008 13 Content Introduction –Pulse lengthening at lens focus Fourier formulation of the problem –Axial temporal history –Pulse shortening due to GVD –Numerical simulation Experiment outlook Summary

14. CLEO 08, San Jose, May 4-9, 2008 14 An envisaged experiment C AL ZSL SF PP D ODL

15. CLEO 08, San Jose, May 4-9, 2008 15 Summary Pulse shortening demonstrated in simulation, up to 5 time shortening observed Potential for few cycle pulse generation with simpler setup