# Time delay of a laser pulse due to 2nd, 3rd and 4th order dispersions

I'll explain my problem :

Given a Gaussian pulse with central wavelength 550 nm and a HMFW of 300 nm, and let's assume the pulse is transform-limited. I need to get the pulse through 1mm of quartz.

I expect group delay dispersion (second order dispersion) to spread in time the wavelengths from each other making the red end arrive sooner then the green then the blue end.

I have correctly calculated the group velocity delay GVD, TOD (third order dispersion) and FOD (fourth order dispersion) for the quartz using the peak wavelength as the central wavelength for the Taylor expansion of the spectral phase.

Now my problem is that I'm having difficulty calculating the arrival time-shift with respect to the peak wavelength for GVD TOD and FOD of the quartz. I'm able to calculate the time-broadening of the pulse but not the time-shift for each wavelength.

Can anybody explain to me how to calculate this? Thanks for the time.

PS: The reason I need to know the time delay for each wavelength it's because I need to theoretically predict a FROG- frequency resolved optical gating.

• Your spectral width is extremely high. Are you sure you have quoted it correctly? – sammy gerbil Feb 1 '17 at 16:33
• It's the shape of the pulse after going through white light generation from saphire and NOPA (non linear optical parametric amplification) , I'm assuming it still retains a gaussian shape... – David jorge Nordmann Feb 1 '17 at 16:43
• What pulse length (time duration) have you estimated? Are you working with a mode-locked laser? A transform-limited spectrum with your FWHM would have a duration of about 0.02 fs, well below the current record of approx. 0.4 fs. ... While the frequency spectrum might be broad, the pulse duration is not necessarily short - ie it is not necessarily transform-limited. What evidence do you have for this assumption? – sammy gerbil Feb 1 '17 at 16:53
• Well the pulses we used are longer than 1fs , can't remember how much now. We're using a Ti:sapphire laser stretched with a grating pair and amplified with 9-10 passes through another Ti:sapphire, then compressed again with a pair of gratings. The supervisor defines the pulse width with different criteria and I'm not going to argue with him that he needs to do it at half maximum...but you're right about it. – David jorge Nordmann Feb 1 '17 at 17:19
• But you're right I'll change the program so that it asks the user for pulse width in time and not wavelength so that I don't have to get into that discussion with him... Thanks!!! – David jorge Nordmann Feb 1 '17 at 17:21

## 1 Answer

The wikipedia article on Dispersion includes the following comment under high-order dispersion :

...The effects [of high order dispersion] can be computed via numerical evaluation of Fourier transforms of the waveform, via integration of higher-order slowly varying envelope approximations, by a split-step method (which can use the exact dispersion relation rather than a Taylor series), or by direct simulation of the full Maxwell's equations rather than an approximate envelope equation.

Your problem is discussed in section 2.2 of the article Effect of high order dispersion in slow-light propagation in photonic crystal waveguides. The difficulty is that dispersion is defined in the phase/frequency domain, whereas time delay is calculated in the space/time domain, so an accurate calculation requires the application of the Fourier Transform (equation 5 in this source, p 1664). The authors comment that :

When third order dispersion is included, the pulse envelope can be approximated using the Airy function [20]. This implies, that for a specific z, the pulses are deformed asymmetrically in time. The temporal asymmetry of the pulses can be a measure for the TOD in the photonic material.

When also higher-order dispersive term are included, Eq. (5) is no longer analytically solvable. Then, Eq. (5) needs to be solved numerically. In such a numerical solution, the full dispersion relation k(ω) is used as input. Hence the calculation is not limited by the accuracy of the Taylor expansion, since all dispersive orders are included, if present.

The effects of high-order dispersion on time delay are illustrated in figure 8. Even with the large effective refractive index of the photonic crystal waveguide ($n=\frac{c}{v_g}=8.62$, bottom of page 1670) the effect on delay is very small. In your case $n$ is significantly smaller (about 1.46).

Quartz is not a highly non-linear material, and a 1mm length of crystal is not particularly long. Even with an ultrashort pulse (less than 10 fs) the effects of high-order dispersion will be incredibly small.

Unless your interest is theoretical, I suggest that inclusion of high-order dispersion in your calculation will have no noticeable impact on your predicted FROG trace, and will be swamped (no pun intended!) by the other approximations you make.

• My problem was exactly how to link the spectral phase in the frequency domain to the time delay in the space-time. Well it's not actually just 1 mm of quartz , it's about 6 meters of air , several mm of quartz , 2 mm of BBO, 1 mm of sapphire , I have made a program in matlab to try to predict the frog using the database from refractiveindex.info website, in fact I'm trying to make a software like vchirp but I need to include the option to add prism pairs and predict how it will affect the delay times. – David jorge Nordmann Feb 1 '17 at 17:08
• I have yet not enough knowledge to do numerical approaches as they suggest with the split-step method. But I can see in the lab the effects of TOD when the pulse is compressed using a prism pair and a grating , and with a different setup using chirped mirrors. – David jorge Nordmann Feb 1 '17 at 17:11
• I presume that you have a supervisor. Have you discussed the problem with him/her? – sammy gerbil Feb 1 '17 at 17:13
• Yes, unfortunately my supervisor is not specialized enough yet to get into these details – David jorge Nordmann Feb 1 '17 at 17:14
• Have you tried contacting groups who are doing (or have done) similar work? I found even the leading experts in the field willing to answer my questions by email. – sammy gerbil Feb 1 '17 at 17:52