Ordered dispersion

A Formulas for the sixth- through tenth-order dispersion coefficients for 7 and 7 4... 

P5.08.08. FIRST ORDER DISPERSION REACTION. GENERAL SOLUTION... 

First order dispersion reaction, using P5.08.06 or the method of P5.08.08,... 

FIGURE 8.1 Time-domain effects of the second- and third-order dispersion. A TL pulse is one that is as short as possible given the available bandwidth. GDD causes time separation between different wavelengths of the pulse, and this broadens its duration. Third-order dispersion breaks the laser pulse into different sub-pulses in the time domain. 

The term that depends on the third power of the frequency shift is known as third-order dispersion (TOD). When a TL pulse acquires a significant amount of TOD, the pulse envelope is distorted and a series of sub-pulses is produced, as shown in Figure 8.1. Unlike a pulse with SOD, a pulse with TOD leads to two-photon excitation with the same efficiency as a TL pulse but only for a particular two-photon frequency. At other frequencies, the amount of excitation is suppressed. The control over TOD would allow for preferential excitation in different spectral regions, while its correction would lead to efficient two-photon excitation over the whole accessed spectral range. Unfortunately, measuring and correcting TOD is not a simple task. 

The ability to cancel all orders of phase distortion gives us an opportunity to evaluate the effect of partial dispersion correction on TPM. In particular, we focus on comparing SOD correction, which can be achieved with a prism pair arrangement, and correction of all orders of phase dispersion using MIIPS. For these measurements we used a pair of prisms in addition to our pulse shaper. With the aid of the pulse shaper, we found the condition for which SOD at the center wavelength was fully eliminated by the prism pair, and only higher-order dispersion was compensated by the pulse shaper. 

To study the implications of spectral phase correction for two-photon depth-resolved imaging, we imaged a thick section of monse kidney tissne, stained with DAPl (cell nnclei), Mitotracker-488 (mitochondria), and Phalloidin-568 (actin). The resnlts, snmmarized in Fignre 8.5, demonstrate increased penetration depth when higher-order dispersion is compensated via MllPS, compared to GDD-only compensation. The images show the collagen wall components of a blood vessel in the monse kidney at a depth of 40 am. The collection dnct region above it, at a depth of 50 jam, is seen only when MllPS is applied. 

We have identified high-order dispersion as the main reason why ultrashort, 10 fs, pulses have rarely been used for nonlinear optical imaging. We discussed the MllPS method for automated measurement and elimination of high-order dispersion. We provided quantitative analysis for the advantage of high-order dispersion as compared with correction limited to SOD. This enhancement was confirmed experimentally in fixed and living cells, as well as in depth imaging. Finally, we demonstrated that the broad bandwidth of ultrashort pulses can be used for selective two-photon excitation when appropriate phase or amplitude modulation is used. 

It is important to remember that often, especially for the He-Ar system, the empirical B(7> coefficients given in Table 4.1 are not necessarily the same as the lowest-order theoretical dispersion coefficients, Dy. An accurate determination of the true dispersion coefficient would require the inclusion of higher-order dispersion terms in the analysis and, moreover, experimental data of a quality that is presently not available. We note that for He-Ar the ab initio dipole specified below is probably superior to the empirical model of Table 4.1. 

The results given in the last column of Table 4.3 are well represented by the analytical form, Eq. 4.30, with B °) = 0.0386 a = 1.371, Rq = 4.5, b = 0.04832, and B(7 = -290, in atomic units. We note that the B 7 coefficient was determined from a fit of the long-range distributions, ptot — pmlrd, for separations from 6.5 to 7.5 bohr. It must not be identified with a Dj dispersion coefficient because, at such separations, contributions from higher-order dispersion terms are not negligible, albeit non-discernible in the data given in the table. 

The comparison of spectral line shapes computed on the basis of the ab initio dipole surface of He-Ar with absorption measurements has demonstrated the soundness of the data. The agreement indicates that exchange effects due to intra-atomic correlation and higher-order dispersion terms contribute significantly to the induced dipole. However,... 

Fig. 7.9. Dependence of the fragment ion yield ratio of m2g/m3i on third-order dispersion coefficient...

The second-order dispersion energy is defined as the difference between the second-order polarization and induction energies, E = E j — E J. One can also use the following direct definition... 

---