Feasibility Study of Direct Measurement of Two-Photon Absorption Cross-Sections

Abstract

Background The two-photon absorption (2PA) cross-section, σ ( 2 ) , is one of the most fundamental properties in any experiment that uses 2PA. σ ( 2 ) is the direct analogue of the 1 PA cross-section, a, or extinction coefficient, e. Most methods to measure σ ( 2 ) , including 2P fluorescence, [ 1 - 3 ] z-scan, [ 4 ] and loss modulation, [ 5 ] are either indirect or require tightly focussed laser beams. In this paper, we explore the feasibility of measuring σ ( 2 ) directly, using an unfocused laser, via a similar approach to the measurement of e or a in 1 P experiments. 1 PA is described by the Beer-Lambert law with the light intensity decaying exponentially in the sample, whereas for2PA the light decays in an inverse fashion: 1 I - 1 I 0 =σ ( 2 ) Nl (1) where I 0 and I are the initial and final light intensities respectively, N is the number density of absorbers, and l the pathlength. Commercial, amplified Ti:sapphire lasers can routinely provide an intensity I of 5 ×10 2 7 photons cm - 2 s - 1 . According to Eqn (1), a 10 mM solution of a commercial dye (Rhodamine 6G, σ ( 2 ) = 20GM [ 1 ] ) absorbs 5-10% of the unfocussed laser beam over 10cm. Proper extraction of σ ( 2 ) requires precise measurement of the laser beam's spatial and temporal profiles (with, say, a CCD camera and an autocorrelator respectively). Nonetheless, these calculations suggest that direct measurement of σ ( 2 ) is feasible. Choice of solvent is important when performing these experiments. The two criteria for a suitable solvent are solubility (which is easily quantified), and 1 PA. The significance of 1PA can be demonstrated with methanol, which has a small a of 5 × 10 - 2 5 cm 2 molecule - 1 at 800 nm, yet absorbs 16% of the laser power over 10 cm. Competition between solvent 1 PA and dye 2PA results in an absorption decay that is a convolution of first- and second-order kinetic processes. Also, significant 1PA causes thermal lensing. Thermal lensing is the result of absorption inducing a thermal (and therefore a refractive index) gradient, causing the solvent to act as a lens. This is problematic because Eqn (1) requires the laser beam to maintain a constant spatial and temporal profile.