Periodics TDFRS

Before stochastic TDFRS is treated in detail, periodic amplitude modulation of the grating in combination with phase-sensitive lock-in detection, similar to the procedure proposed by Bloisi [73], will be briefly discussed. With periodic amplitude modulation with a single frequency, which is slowly scanned through the frequency range of interest, the Fourier transform of the TDFRS response function, G( y), is measured more or less directly. 

If the retardation of the temperature grating is neglected for the moment, which is valid if A t T, a sinusoidal amplitude modulation of the optical grating translates directly into a sinusoidal amplitude modulation of the temperature grating with amplitude 5 T = % al0 (pc ) 1 according to Eq. (16)  

From an experimental point of view, a sinusoidal amplitude modulation of the optical grating can be achieved with the setup shown in Fig. 23 in one of the writing beams. 

The real and imaginary part of G((o) Fig. 24 has been measured in for an almost monodisperse PS (M = 49 kg/mol) in toluene. The slow mode at (o 2n 5 Hz corresponds to mass diffusion. The fast mode at oo/2n 5 kHz stems from 

The corresponding time domain experiment with a long exposure pulse is shown in the insert. Both measurements have been normalized to the amplitude of the signal from the temperature grating. The amplitudes of the concentration signal and the diffusion time constant % agree between both experiments within the experimental error (Table 2). 

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The main problem with periodic TDFRS is, that the different frequencies are measured at different times. This requires a long-time stability, especially of the heterodyne reference, lasting about as long as the entire experiment. Time domain experiments, on the other hand, are frequency multiplexed, and stability of the heterodyne background is only required for one homodyne/heterodyne separation cycle as described in the experimental section, which is only of the order of seconds, not hours. No stability of the signal amplitude is required for the averaging of C lel(t) over arbitrary times. 

Frequency domain functions are denoted by upper case letters. Of importance for the TDFRS experiment is the discrete Fourier transform of an array of N data points within a period of N At ... 

The response y(t) of a linear system to an excitation x(t) is a convolution of x(t) with the response function h(t). The TDFRS experiments are performed with periodic boundary conditions and discrete sampling, asking for the discrete periodic convolution... 

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