Two-Photons Absorption

Two-photon absorption can be formally described by a two-step process from the initial level i) via a virtual level v) to the final level / (Fig. 2.30b). This fictitious virtual level is represented by a linear combination of the wave functions of all real molecular levels kn) that combine with i) and f) by allowed one-photon transitions. The excitation of w) is equivalent to the sum of all off-resonance excitations of these real levels kn). The probability amplitude for a transition i) v) 

For a molecule moving with a velocity v, the probability A,/ for a two-photon transition between the ground state Et and an excited state E/ induced by the photons fuoi and fuo2 from two light waves with the wave vectors k, k2, the polariza- 

Because the transition probability (2.66) is proportional to the product of the intensities I I2 (which has to be replaced by in the case of a single laser beam), pulsed lasers, which deliver sufficiently large peak powers, are generally used. The spectral linewidth of these lasers is often comparable to or even larger than the Doppler width. For nonresonant transitions o)ki —coi v-ki, and the denominators (coki — — A v) in the sum in (2.66) can then be approximated by coki — coi). 

The second factor in (2.66) describes the transition probability for the two-photon transition. It can be derived quantum mechanically by second-order perturbation theory (see, for example, [240, 241]). This factor contains a sum of products of matrix elements Di Dk/ for the transitions between the initial level i and intermediate molecular levels k or between these levels k and the final state /, see Vol. I, (2.110). The summation extends over all molecular levels k that are accessible by allowed one-photon transitions from the initial state /. The denominator shows, however. 

Often the frequencies twi and W2 can be selected in such a way that the virtual level is close to a real molecular eigenstate, which greatly enhances the transition probability. It is therefore generally advantageous to excite the final level / by two different photons with a i + u 2 = (Ef - Ei)/h rather than by two photons out of the same laser with 2co = Ef — , )/h. 

The TPA rate for a two-parabolic band model may be obtained from the Fermi s golden rule as 

Hint is the optical interaction Hamiltonian given by Hi t=—er-E, where r is the position vector and E is the optical field. c) and v) represent the states in the conduction and valence bands, respectively, and i) is the virtual state within the transparency region. A simple model with a parabolic conduction band and a parabolic valence band gives for zinc blende semiconductors [226] 

Ep = 2 edcv /wto is approximately 21 meV for most of the III-V and II-VI semiconductors, no is the linear refractive index, and Kpi, is a material independent constant (1940 cm (eV) for a two-parabolic-band model). It should be mentioned that the above model does not correctly account for the degeneracy of the valence band (heavy hole, light hole, and spin-orbit/crystal-field split-off bands) and assumes single parabolic conduction and valence bands. Using the Kane band structure with three valence bands and including excitonic effects have been shown to produce larger TPA coefficients [227]. 

When photons of two different energies are present, that is, for the nondegenerate case, the absorption of the two light beams of intensities li and I2 is described by a 

TPA coefficient is often overestimated when the freeorrier absorption effects are 

4 and 10.6) exhibits transition rates far smaller than those of El-allowed one-photon emission [1], and has not been detected. It is likely to contribute to decay in astrophysical systems in which one-photon decay is El-forbidden. 

To make the last integral on the right converge, we may replace o)k by (cOk + is), where e is small and positive, and then let e - 0 after the integration  

This is more than just a mathematical artifice. This substitution is tantamount to replacing the energy E by E — ihs), so that the intermediate state 1 exhibits the time dependence exp — iE t/h — st) and hence physically decays with lifetime l/2e. The constant e can be identified with y/4, where y is the Lorentzian linewidth (Chapter 8). Such linewidths are generally much smaller than level energies E , so that dropping s at the end of the integration yields good approximations to cf t) in Eq. 10.10. Next, we have 

Since the delta function in Eqs. 10.13-10.14 is proportional to — E + h (Oi + ( 2)] it yields the obvious energy-conserving criterion ( — = h a i + CO2) relevant to TP A. It is thus clear that the terms included 

Equation 10.14 deseribes the TPA transition amplitude for a molecule subjected to two light beams with arbitrary electric field vectors and propagation vectors. A particularly useful application of TPA in gas phase spectroscopy employs two counterpropagating laser beams with k2 = — ki k2. In this case, a molecule traveling with velocity parallel to k2 will experience Doppler shifts 

The quantity which describes a process of simultaneous absorption of two-photons ofdifferent energy ( icci fitC2) with different polarization (ii 12) at the molecular level is given by the following Eq. [66]  

Since in most experiments one source of photons is used, one can substitute the angular frequencies uj and cc2 for 0.5 ccf. In the case of isotropic media the averaged two-photon absorption cross section is given by  

The geometry optimizations of molecular structures were performed using the semi-empirical PM3 Hamiltonian [67,68] and the B3LYP/6-3 lG(d) method. All stationary points were confirmed to be true minima by evaluation of hessian. In order to speed up HE and DFT calculations, the fast multipole method (EMM) [47, 48, 69] has been used as implemented in Gaussian suite of programs [56]. We also used linear scaling approaches for calculations of nonlinear optical properties as implemented in ADF package [58]. All the properties are expressed in atomic units. Conversion factors can be found elsewhere [28-31], 

From (2)-(7) one can deduce the properties of two-photon spectroscopy. First and foremost is the fact that selection rules are different from those pertaining to one-photon absorption. For instance, in centrosymmetric molecules the selection rules are u - u and gv u, that is, the opposite of the one-photon selection rules. Thus, two-photon spectroscopy in this case is complementary to one-photon spectroscopy much in the same way as infrared and Raman spectroscopy are in the domain of molecular vibrations. Another property is the fact that 8 depends on the polarization of the radiation even in fluid samples, that is, for complete orientation randomization. The form of 8 for all point groups at various polarization combinations of the two photons has been derived and tabulated. Consequently, polarization studies may be used to obtain structural information. Another unique feature of two-photon absorption is the possibility to observe spectra free of Doppler broadening by using two counter propagating beams.  

Power requirements for experimental observation of two-photon absorption may be estimated by rewriting (2) as (deleting subscripts from 8gf for brevity)  

A/ is the intensity decrease on traversing unit distance in a sample with ground-state population density of A typical value of 5 is 10 ° cm s photon molecule and at 1 torr pressure Ng is about 10 molecules cm . In this case a relative attenuation A/// of 10 is obtained at 7=10 photons cm s , i.e., about 10 watt cm s for visible radiation. Powers of these orders of magnitude, at narrow frequency bandwidths, are practical only with laser sources. For TPE applications, lasers are often (not always) operated in a pulsed mode, allowing easy 

This is conceptually the most straightforward technique. In view of the small relative change in laser intensity, it is mainly used in cases where two different laser beams are crossed in the sample. Often one of the lasers is a high-power, fixed-frequency device attenuation is measured on the second, weaker laser appropriately termed the probe, or monitor, laser. The method was used in early days of two-photon absorption spectroscopy and applied mostly to high-density and liquid samples. Its obvious drawback is poor sensitivity, as the desired quantity is obtained as a small difference between two large numbers. 

A major disadvantage of thermal methods for kinetic applications is their poor time resolution—determined by the rate of gas density changes. This usually limits the resolution to 10 -10 s, washing out many interesting temporal effects. 

Another nonlinear absorption mechanism in dye molecules that becomes important at high light intensities is the simultaneous absorption of two photons of energy E = hv to reach a stationary energy level, situated at 2E, in the dye. This effect 

For a numerical calculation of the two-photon absorption one can employ a simple phenomenological theory which describes very well the experimental facts at least in the case of an excitation of the lowest lying excited singlet state Si 48 . 

The factor of 2 takes into account the fact that two photons disappear in each act of absorption. Integration over the length L of the cell gives the absorption law for the two-quantum absorption  

As can be seen from Eq. (25), (nc)-p is proportional to the square of the photon flux nc. It should also be proportional to the product of the maximum absorption cross section ffmax and the cross section oi at wavelength . This relation has been checked experimentally in 48 . The relation between the fluorescence output (wc)F and the excitation power nc for an aqueous solution of rhodamine resulted in a straight line in a double-logarithmic plot with a slope of 2.05 0.1, thus verifying the square law of two-photon absorption. 

A theoretically well-founded theory of two-photon absorption using the free electron-gas model of dye molecules 50 is to be found in 49 . 

---

We will now look at two-photon processes. We will concentrate on Raman scattering although two-photon absorption can be handled using the same approach. In Raman scattering, absorption of an incident photon of frequency coj carries... 

One very important aspect of two-photon absorption is that the selection ndes for atoms or synnnetrical molecules are different from one-photon selection ndes. In particular, for molecules with a centre of synnnetry, two-photon absorption is allowed oidy for g g or u u transitions, while one-photon absorption requires g-f u transitions. Therefore, a whole different set of electronic states becomes allowed for two-photon spectroscopy. The group-theoretical selection ndes for two-photon spectra are obtained from the synnnetries... 

Monson P R and McClain W M 1970 Polarization dependence of the two-photon absorption of tumbling molecules with application to liquid 1-chloronaphthalene and benzene J. Chem. Rhys. 53 29-37... 

Strickler S J, Gilbert J V and McClanaham J E 1984 Two-photon absorption spectroscopy of molecules Lasers and Applications eds H D Bist and J S Goela (New Delhi Tata McGraw-Hill) pp 351-61... 

Lee D and Albrecht A C 1985 A unified view of Raman, resonance Raman, and fluorescence spectroscopy (and their analogues in two-photon absorption) Advances in Infrared and Raman Spectroscopy vo 12, ed R J H Clark and R E Hester (New York Wiley) pp 179-213... 

Lindek St, Cremer Chr and Stelzer E H K 1996 Confocal theta fluorescence microscopy using two-photon absorption and annular apertures Optik 02 131-4... 

Figure Bl.22.7. Left resonant seeond-hannonie generation (SHG) speetnimfrom rhodamine 6G. The inset displays the resonant eleetronie transition indueed by tire two-photon absorption proeess at a wavelength of approximately 350 mn. Right spatially resolved image of a laser-ablated hole in a rhodamine 6G dye monolayer on fiised quartz, mapped by reeording the SHG signal as a fiinetion of position in the film [55], SHG ean be used not only for the eharaeterization of eleetronie transitions within a given substanee, but also as a mieroseopy tool.

Plakhotnik T, Walser D, Pirotta M, Renn A and Wild U P 1996 Nonlinear spectroscopy on a single quantum system two-photon absorption of a single molecule Science 271 1703-5... 

The similarity between a two-photon absorption and a Raman scattering process is even closer. Figure 9.27(a) shows that a Raman transition between states 1 and 2 is really a two-photon process. The first photon is absorbed at a wavenumber to take the molecule from state 1 to the virtual state V and the second photon is emitted at a wavenumber Vj,. 

In a two-photon absorption process the first photon takes the molecule from the initial state 1 to a virtual state V and the second takes it from V to 2. As in Raman spectroscopy, the state V is not an eigenstate of the molecule. The two photons absorbed may be of equal or unequal energies, as shown in Figures 9.27(b) and 9.27(c). It is possible that more than two photons may be absorbed in going from state 1 to 2. Figure 9.27(d) illustrates three-photon absorption. 

Two-photon absorption has been observed in the microwave region with an intense klystron source but in the infrared, visible and ultraviolet regions laser sources are necessary. 

Because Raman scattering is also a two-photon process the selection rules for two-photon absorption are the same as for vibrational Raman transitions. For example, for a two-photon electronic transition to be allowed between a lower state j/" and an upper state... 

Figure 7-13. (a) Linear absorption of DOO-PPV, (b) imaginary part of ) (proportional to two-photon absorption), and (c) real part ol (proportional to the nonlinear index of refraction. .). 

The first-generation dendrimer 51 was directly observed by transmission electron microscopy (TEM). The TEM image showed that the dimensions of individual molecules are about 50 A, which is consistent with the calculated one [36]. Third-order NLO measurements showed a significant enhancement of two-photon absorption upon proceeding from the constituent molecules to the dendritic complex [35]. 

Photoluminescence from gold nanoplates induced by near-field two-photon absorption. Appl. Phys. Lett., 88, 023104 (3 pages). 

For the application of QDs to three-dimensional biological imaging, a large two-photon absorption cross section is required to avoid cell damage by light irradiation. For application to optoelectronics, QDs should have a large nonlinear refractive index as well as fast response. Two-photon absorption and the optical Kerr effect of QDs are third-order nonlinear optical effects, which can be evaluated from the third-order nonlinear susceptibility, or the nonlinear refractive index, y, and the nonlinear absorption coefficient, p. Experimentally, third-order nonlinear optical parameters have been examined by four-wave mixing and Z-scan experiments. 

Figure 9.4 Two-photon absorption-induced luminescence of CdTe QDs in D2O and H2O obtained from the luminescence spectrum as a function of time. The diameter and solvent are (a) 4.5 nm in D2O, (b) 3.7 nm in D2O, (c) 4.5 nm in H2O,...

Third-order nonlinear optical properties of CdTe QDs were examined by Z-scan and FWM experiments in the nonresonant wavelength region. We found that the two-photon absorption cross section, a, is as high as 10 GM, although this value decreases with decreasing size. In addition, the nonlinear response is comparable to the pulse width of a fs laser and the figures of merit (FOM = Re Xqd/ Xqd)... 

Kamada, K., Matsunaga, K., Yoshino, A. and Ohta, K. (2003) Two-photon-absorption-induced accumulated thermal effect on femtosecond Z-scan experiments studied with time-resolved thermal-lens spectrometry and its simulation. J. Opt. Soc. Am. B, 20, 529-537. 

Two-Photon Absorption in Near-IR Conjugated Molecules Design Strategy and Structure-Property Relations... 

Keywords Cyanine dyes Excited state absorption Polymethine dyes Pump-probe Two-photon absorption Z-scan... 

Brief Historical Account of Two-Photon Absorption and TT-Conjugated Systems... 

---