Spectroscopy homodyne

We will illustrate homodyne spectroscopy (self-beating spectroscopy) with some examples. 

The intensity spectrum of the elastically scattered radiation from N scatterers is 

When monochromatic light is scattered by moving particles that show thermal motion, the field amplitudes E oo) show a Gaussian distribution. The experimental arrangement for measuring the homodyne spectrum is shown in Fig. 7.31. The power spectrum P (jo) of the photocurrent (7.68), which is related to the spectral distribution I oo), is measured either directly by an electronic spectrum analyzer, or with a correlator, which determines the Fourier transform of the autocorrelation function C(r) a i t)) i t -f- r)). According to (7.63), C(r) is related to the intensity correlation function G (r), which yields (7.64), and I co). 

A famous example for the application of intensity-correlation interferometry in astronomy is the Hanbury Brown-Twiss interferometer sketched in Fig. 7.33. In its original form it was intended to measure the degree of spatial coherence of starlight (Vol. 1, Sect. 2.8) [945] from which diameters of stars could be determined. In its modern version it measures the degree of coherence and the photon statistics of laser radiation in the vicinity of the laser threshold [946]. 

An example of an application of homodyne spectroscopy is the measurement of the size distribution of small particles in the nanometer range that are dispersed in liquids or gases and fly through a laser beam. The intensity 4 of the scattered light depends in a nonlinear way on the size and the refractive index of the particles. For small homogeneous spheres with diameters d, which are small compared to the wavelength (d A), the relation oc d holds. In Fig. 12.25 the measured intensity distribution of laser light scattered by a mixture of latex spheres with d = 22.8 nm and d = 5.7 nm (small squares) is compared with the size distribution obtained from electron microscopy, which can be used for calibration [12.91]. 

For heterodyne correlation spectroscopy the scattered light that is to be analyzed is superimposed on the photocathode by part of the direct laser beam (Fig. 12.27). Assume the scattered light has the amplitude E, at the detector and a frequency distribution around cos, whereas the direct laser radiation 

El = 0 exp(—icooO acts as monochromatic local oscillator with constant amplitude Eq. The total amplitude is then 

Inserting (12.69) into (12.70a) we note that for E Eq the terms with El can be neglected. Furthermore, the time average ( l s) is zero. Therefore (12.70a) reduces to 

The output of a laser does not represent a strictly monochromatic wave (even if the laser frequency is stabilized) because of frequency and phase fluctuations (Sect.5.6). Its intensity profile I(w) with the linewidth Aw can be detected by homodyne spectroscopy. The different frequency contributions inside the line profile I(w) interfere, giving rise to beat signals at many different frequencies Wj-Wjj Aw [12.75]. If a photodetector is irradiated by the attenuated laser beam, the frequency distribution of the photocurrent (12.66) can be measured with an electronic spectrum analyser. This yields according to the discussion above the spectral profile of the incident light. In case of narrow spectral linewidths this correlation technique represents the most accurate measurement for line profiles [12.82]. 

The correlation function C(r) = i(t))((t+r))/(i)2 can directly be measured with a digital correlator which measures the photoelectron statistics. A simple version of several possible realizations is depicted in Fig. 12.24. By an internal clock the time is divided into equal sections At. If the number N-Atj of photoelectrons measured within the ith time interval Atj exceeds a given number the correlator gives a normalized output pulse, counted as one . For NAtj the output gives zero . The output pulses are transferred to a shift register and to AND gatters which open for one and close for zero , and are finally stored in counters (Malvern correlator [12.79,84]). 

A further example is the light scattering by a liquid sample when the temperature is changed around the critical temperature and the sample undergoes a phase transition [12.86]. 

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Fig. 7.31 Schematic experimental setup for measuring the autocorrelation function of scattered light (homodyne spectroscopy), with a correlator as an alternative to an electronic spectrum analyzer...

Tanaka et al. (1973) have compared the elasticity moduli of 5% and 2.,5% polyacrylamide gels in water, obtained by meaisuring the deformation of their cylindrical samples and by means of dynamic light scattering. The time correlations of scattered light were meeisured on a set of the homodyne spectroscopy of mixing, and the squared correlation function of the field was determined (see Equation 2.3 33). 

Correlation spectroscopy is based on the correlation between the measured frequency spectrum S(co) of the photodetector output and the frequency spectrum I((jo) of the incident light intensity. This light may be the direct radiation of a laser or the light scattered by moving particles, such as molecules, dust particles, or microbes (homodyne spectroscopy). In many cases the direct laser light and the scattered light are superimposed on the photodetector, and the beat spectrum of the coherent superposition is detected (heterodyne spectroscopy) [12.81,12.82]. 

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