Temperature grating

The direct coupling mechanism between the optical grating and the sample is the absorption of energy from the light field, which results in a periodic heating with the same -vector as the optical grating. To describe this temperature grating, we start with the heat equation... 

T0 is the initial sample temperature. The amplitude Tq(t) of the temperature grating is expressed as a linear response for arbitrary excitations Sq(t) = a(p cp) ... 

Concentration grating Due to the Ludwig-Soret effect, the temperature grating is the driving force for a secondary concentration grating, which starts to build up and is superimposed upon the thermal one. Its temporal and spatial evolution is obtained from the one-dimensional form of the extended diffusion equation... 

Now, all steps can be combined to calculate the heterodyne diffraction efficiency from Eqs. 9,16,20,21, and 23. After normalization to the diffraction efficiency of the steady state amplitude of the temperature grating, one arrives at... 

The 5-function accounts for the fast contribution from the temperature grating. The normalization to the amplitude of the temperature grating in Eq. (24) takes away the need for difficult absolute intensity measurements for the determination of Dt and S7-, which otherwise would be necessary. The only quantities that must be obtained from separate measurements are the two contrast factors (d n/d T)cp and (d n/d c)Tp. All the transport coefficients Dlh,D,Dv and ST can be extracted from the sample response to suitably chosen excitation patterns. 

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) ... 

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). 

Now, the effective linear response function h(t) can be identified with g(t) as defined in Eqs. (25) and (29) h(t) = g(t). The primary sample response is the heterodyne diffraction efficiencyy (t) = Chet(t)- The instantaneous contribution of the temperature grating to the diffraction efficiency is expressed by the 5-function in g(t) [Eq. (25)]. After the sample, an unavoidable noise term e(t) is added. The continuous yff) is sampled by integrating with an ideal detector over time intervals At to finally obtain the time-discrete sequence y[n]. 

In Ref. [75], it is discussed in more detail why it is advantageous to convolute the response of the temperature grating into the excitation and how to treat systematic errors arising from this approximation and from imperfections of the components in the setup. Especially the switching properties of the Pockels cell require careful analysis, since the switching number increases from 2 in case of pulsed excitation to approximately N in case of pseudostochastic binary sequences. 

The lower half of the insert shows the heterodyne diffraction efficiency as seen by the detector. It has a random character, but the influence of the memory function is obvious when compared to the excitation. The main part of Fig. 26 shows the concentration part of the memory function g(t) h(t) after deconvolution according to Eq. (61). The amplitude of the contribution from the temperature grating is normalized to unity and contributes only to the very first data point. 

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