Linear and Nonlinear Absorption

For sufficiently low incident intensities /q, the absorption coefficient a is independent of 7o (this implies that the population difference AA is not dependent on 7o ), and the absorbed power AP is linearly dependent on the incident power Pq-Integration of (2.1a) gives Beer s law of linear absorption  

When the absorption is measured through the fluorescence intensity 7h emitted from the upper level A ), which is proportional to the absorbed power, we obtain the linear curve in Fig. 2.1. As the intensity of the incident radiation increases, the population Nj in the absorbing level /) will be depleted when the absorption rate becomes faster than the relaxation processes that can refill it. Therefore, the absorption decreases and the curve 7pi(7o) in Fig. 2.1 deviates from the straight curve until it reaches a constant value (saturation). We must then generalize (2.1a) to 

Example 2.1 In a first approximation we can write a = o o(l — bl), which gives with a cross-section A of the incident radiation 

The first term describes the contribution from linear absorption, and the second from nonlinear quadratic absorption. 

We will now consider the nonlinear absorption in more detail. 

Assume that a monochromatic plane lightwave E = E co cot — kz), with the mean intensity 

If the incident plane wave with the spectral energy density py v) = Iv v)/c [W s /m ] has the spectral width Avl, which is large compared to the halfwidth 3v of the absorption profile, the total intensity becomes 

This absorbed power corresponds to riph — SP/hv absorbed photons. From (2.15) we can deduce 

BikPv gives the net probability for absorbing a photon per molecule and per second within the unit volume dV = 1 m, see (2.15,2.78). 

The absorption of the incident wave causes population changes of the levels involved in the absorbing transition. This can be described by the rate equations for the population densities N, N2 of the nondegenerate levels 11) and 2) with = 2 = 1 (Fig. 7.1)  

Assume that a plane lightwave E = Eg cos(wt - kz) with the mean intensity 

The rate equations for the stationary population densities N, Nj of the nondegenerate levels l) and 2) with gi = g2 = 1 then become (Fig.7.1)  

If the quantities Cj are not noticeably changed by the radiation field we obtain from (7.3) under stationary conditions (dN/dt = 0) the unsaturated population difference for p = 0 

In Sect.2.6 we saw that the intensity decrease dl of a wave with intensity I propagating along the z direction through an absorbing sample is 

As long as the population densities N. and N of the two levels E. and E are not noticeably altered by the interaction with the radiation field (weak-signal approximation of Sect.2.9.2), one can regard them as constant. The absorbed intensity is then proportional to the incident intensity ( Linear absorption). Integration of (2.84a) over the absorption path z gives Beer s law for linear absorption, 

At larger intensities I, the density of the lower state can noticeably decrease while the upper state density Nj increases. This corresponds to the strong field case in Sect.2.9.4 and means that N. (I) and N (I) are functions of I and therefore dl is no longer proportional to I [nonlinear absorption). We shall illustrate this nonlinear absorption by the simple example of a two-level system with population densities and N2 and equal statistical weights = 92 = total number density N = + N2 of the 

The time derivatives of the population densities can be related to the Einstein coefficients using (2.15-17) (E2 is assumed) 

The ratio S = induced-to-spontaneous transition rates is called 

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In terms of beam delivery, the DLW method is based on optical microscopy, confocal microscopy [4,6,13] and laser tweezers [14] (for reviews on laser tweezers see [ 15,16]). These techniques allow for a high spatial 3D resolution of a tightly focused laser beam with optical exposure of micrometric-sized volumes via linear and nonlinear absorption. In addition, mechanical and thermal forces can be exerted upon objects as small as 10 nm molecular dipolar alignment can be controlled by polarization of light in volumes of with submicrometric cross-sections. This circumstance widens the field of applications for laser nano- and microfabrication in liquid and solid materials [17-22]. 

Linear and nonlinear absorption at the frequency, co (ci> = Inkle, where k is the wavelength and c is speed of light) can be described by ... 

TPA chromophore 107 represents, in analogy to its dipolar analog 52, a typical quadrupolar chromophore with both increased linear and nonlinear absorption properties [428]. Thus, the extinction coefficient and 8 are about 1.3 and 1.4 times larger (about /2) in comparison with that of 52. This again shows that incorporation of the electron-deficient triazine results in an increase of 8 for quadrupolar compounds considering both OP and TP quantities. 

Norman. P.. Cronstrand. P.. Ericsson, J. Theoretical study of linear and nonlinear absorption in platinum-organic compounds. Chem. Phys. 285. 207-220 (2002)... 

However, the figure of merit presented in Eq. (78) is not useful in characterizing third-order nonlinear materials because it is not dimensionless and it does not separate between linear and nonlinear absorption. More appropriate dimensionless figures of merit have been proposed by Stegeman [55, 56],... 

Fig. 2.1 Fluorescence intensity /fi(/l) as a function of incident laser intensity for linear and nonlinear absorption...

The book begins with a discussion of the fundamental definitions and concepts of classical spectroscopy, such as thermal radiation, induced and spontaneous emission, radiation power and intensity, transition probabilities and oscillator strengths, linear and nonlinear absorption and dispersion, and coherent and incoherent radiation fields. In order to understand the theoretical limitations of spectral resolution in classical spectroscopy, the next chapter treats the different causes of the broadening of spectral lines. Numerical examples at the end of each section illustrate the order of magnitude of the different effects. 

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