Frequency-Domain Phase-Shift Measurements

In the frequency-domain, the experimentally measured quantities are the frequency- (w) and wavelength- (X) dependent phase shift (0m(X,a>)) and demodulation factor (MnXX, )). For any assumed decay model (equation 1), these values are calculated from the sine (S(X,o>)) and cosine (C(X,w)) Fourier transforms. If we assume the decay kinetics are described by a simple sum of exponential decay times we have (24) ... 

According to the Fourier method, the measured line integral p r,4>) in a sinogram is related to the count density distribution A(x,y) in the object obtained by the Fourier transformation. The projection data obtained in the spatial domain (Fig. 4.2a) can be expressed in terms of a Fourier series in the frequency domain as the sum of a series of sinusoidal waves of different amplitudes, spatial frequencies, and phase shifts running across the image (Fig. 4.2b). This is equivalent to sound waves that are composed of many sound frequencies. The data in each row of an acquisition matrix can be considered to be composed of sinusoidal waves of varying amplitudes and frequencies in the frequency domain. This conversion of data from spatial domain to frequency domain is called the Fourier transformation (Fig. 4.3). Similarly the reverse operation of converting the data from frequency domain to spatial domain is termed the inverse Fourier transformation. 

At present, two main streams of techniques exist for the measurement of fluorescence lifetimes, time domain based methods, and frequency domain methods. In the frequency domain, the fluorescence lifetime is derived from the phase shift and demodulation of the fluorescent light with respect to the phase and the modulation depth of a modulated excitation source. Measurements in the time domain are generally performed by recording the fluorescence intensity decay after exciting the specimen with a short excitation pulse. 

In frequency-domain FLIM, the optics and detection system (MCP image intensifier and slow scan CCD camera) are similar to that of time-domain FLIM, except for the light source, which consists of a CW laser and an acousto-optical modulator instead of a pulsed laser. The principle of lifetime measurement is the same as that described in Chapter 6 (Section 6.2.3.1). The phase shift and modulation depth are measured relative to a known fluorescence standard or to scattering of the excitation light. There are two possible modes of detection heterodyne and homodyne detection. 

In phase-modulation fluorometry, the pulsed light source typical of time-domain measurements is replaced with an intensity-modulated source (Figure 10.5). Because of the time lag between absorption and emission, the emission is delayed in time relative to the modulated excitation. At each modulation frequency (to = 2nf) this delay is described as the phase shift (0, ), which increases from 0 to 90° with increasing modulation frequency. The finite time response of the sample also results in demodulation to the emission by a factor m which decreases from 1.0 to 0.0 with increasing modulation frequency. The phase angle (Ow) and the modulation (m, ) are separate... 

A proof of this relation may be found in Bracewell (1978). Note that the spectral variable used in this and the next chapter is the same as that defined in Eqs. (7) and (8). Now consider a spatial distribution /(x) and its Fourier spectrum F(w) that come close to satisfying the equality in Eq. (4). We may take Ax and Aw as measures of the width, and hence the resolution, of the respective functions. To see how this relates to more realistic data, such as infrared spectral lines, consider shifting the peak function /(x) by various amounts and then superimposing all these shifted functions. This will give a reasonable approximation to a set of infrared lines. To discuss quantitatively what is occurring in the frequency domain, note that the Fourier spectrum of each shifted function by the shift theorem is given simply by the spectrum of the unshifted function multiplied by a constant phase factor. The superimposed spectrum would then be... 

Deslouis et al. [9] in 1977 measured experimentally with a high performance apparatus the response of the limiting current in a large frequency domain and for a large range of Schmidt number. In particular, they showed experimentally that the phase shift tends towards 180° in the high frequency range. 

In the time domain, the fluorescence acquires a phase shift and a demodulation, while the scattering does not suffer from a delay. In the frequency domain the phase shift always starts at 0° for low frequencies and tends to 90° in the limit for high frequencies, while the demodulation starts at 1 and tends to 0. However, for the immediate scattering, no phase shift and no demodulation is observed. This is exactly what is used in our approach at very high modulation frequencies, the fluorescence is completely "demodulated" and does not contribute to the measurement signal, that is solely comprised of the not-demodulated scattering. Any HRS measurement at high modulation frequencies will reveal an inherent, fluorescence-free, hyperpolarizability value. 

Over a substantial number of years the phase-shift or frequency-domain method has been employed for the measurement of fluorescence lifetimes. The technique requires the continuous excitation of a fluorescent sample with a source of varying intensity. The fluorescence response would normally be expected to increase and decrease to reflect the changes in excitation intensity. However, in a frequency-domain experiment the excitation beam is modulated at a high frequency, (o = 2nf, to produce a sinusoidally changing intensity given by ... 

In phase-modulation fluorometry, the sample is excited by a sinusoidally modulated light at high frequency. The fluorescence response, which is the convolution product (Eq. (7.6)) of the d-pulse response by the sinusoidal excitation function, is sinusoidally modulated at the same frequency but delayed in phase and partially demodulated with respect to the excitation. The phase shift and the modulation ratio M (equal to m/mo), that is the ratio of the modulation depth m (AC/DC ratio) of the fluorescence and the modulation depth of the excitation mg, characterize the harmonic response of the system. These parameters are measured as a function of the modulation frequency. No deconvolution is necessary because the data are directly analyzed in the frequency domain. 

Phase shift fluorimetry, the other important method for measuring fluorescent lifetimes, also continues to be developed and improved. The effects of Inaccurate reference lifetimes on the interpretation of frequency domain fluorescence data can be removed or minimized by a least squares analysis method.The direct collection of multi-frequency data for obtaining fluorescence lifetimes can be achieved by the use of digital parallel acquisition in frequency domain fluorimetry. Frequency domain lifetime measurement has been used for on-line fluorescence lifetime detection of eluents in chromatography. An unusual use of frequency domain measurement which has been reported is for the examination of photon migration in living tissue. Photons in the... 

Tvvo vidcl used approaches are used for lifetime measurcnienis. ilie lime-domain approach and the frt i/iu niy-domain approach. In tinte-domain measurements. a pulsed source is employed and the time-depcndcnr decay of fluorescence is measured. In the frequency-domain method, a sinusoidallv modulated source is used to excite the sample. The phase shift and demodulation of the fluorescence emission relative lo the excitation waveform provide the lifetime information. ( onimercial instrumentation is available to implement both techniques. ... 

A comprehensive overview of frequency-domain DOT techniques is given in [88]. Particular instraments are described in [166, 347, 410]. It is commonly believed that modulation techniques are less expensive and achieve shorter acquisition times, whereas TCSPC delivers a better absolute accuracy of optical tissue properties. It must be doubted that this general statement is correct for any particular instrument. Certainly, relatively inexpensive frequency-domain instruments can be built by using sine-wave-modulated LEDs, standard avalanche photodiodes, and radio or cellphone receiver chips. Instruments of this type usually have a considerable amplitude-phase crosstalk". Amplitude-phase crosstalk is a dependence of the measured phase on the amplitude of the signal. It results from nonlinearity in the detectors, amplifiers, and mixers, and from synchronous signal pickup [6]. This makes it difficult to obtain absolute optical tissue properties. A carefully designed system [382] reached a systematic phase error of 0.5° at 100 MHz. A system that compensates the amplitude-phase crosstalk via a reference channel reached an RMS phase error of 0.2° at 100 MHz [370]. These phase errors correspond to a time shift of 14 ps and 5.5 ps RMS, respectively. 

An overview of frequency-domain detection techniques is given in [88]. Frequency-domain techniques compare the phase shift and the modulation degree of the fluorescence with the modulated excitation. Modulation of the excitation is achieved either by actively modulating the light of a continuous laser or by using pulsed lasers of high repetition rate. With pulsed lasers, phase and modulation can be measured at the fundamental repetition frequency or at its harmonics. 

The measurements are different in the frequency-domain. In this case we measure the phase shift between the parallel and perpendicular components of the emission, and a frequency-dependent anisotropy, which is analogous to the steady state anisotropy. These two types of data are used to determine the decay law for the anisotropy (equation 15). 

The experimental procedures and the form of the data are different for FD measurements of the anisotropy decays. The sample is excited with amplitude-modulated light which is vertically polarized (Figure 11.2). As for the TD measurements, the emission is observed through a polarizer, which is rotated between the parallel and the perpendicular orientations. In the frequency domain, th are two observable quantities which characterize the anisotropy decay. These are the phase shift A, at the modulation frequency ca, between the perpendicular and parallel (( > ) con nents of the emission. 

Measured Performance. Under the conditions of space invariance and incoherence, an image can be expressed as the convolution of the object irradiance and the point-spread function, Eq. (26.15). The corresponding statement in the spatial frequency domain, Eq.(26.28), is obtained by taking the Fourier transform of Eq. (26.15). This states that the frequency spectrum of the image irradiance equals the product of the frequency spectrum of the object irradiance distribution and the transform of the point-spread function. In this manner, optical elements functioning as linear operators transform a sinusoidal input into an undistorted sinusoidal output [Eq. (26.33)]. Hence the function that performs this service is the transform of the point-spread function 3 A(x, y), known as the optical transfer function 0 u, v] (OTF). This is a spatial frequency-dependent complex function with a modulus component called the modulation tranter function M u. v] (MTF) and a phase component called the phase tranfer function 4>[ , v] (PTF). The MTF is the ratio of image-to-object modulation, while the PTF is a measure of the relative positional shift from object to image. 

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