Sinusoidal modulation

For fluorescent compounds and for times in die range of a tenth of a nanosecond to a hundred microseconds, two very successftd teclmiques have been used. One is die phase-shift teclmique. In this method the fluorescence is excited by light whose intensity is modulated sinusoidally at a frequency / chosen so its period is not too different from die expected lifetime. The fluorescent light is then also modulated at the same frequency but with a time delay. If the fluorescence decays exponentially, its phase is shifted by an angle A([) which is related to the mean life, i, of the excited state. The relationship is... 

Theory. If two or more fluorophores with different emission lifetimes contribute to the same broad, unresolved emission spectrum, their separate emission spectra often can be resolved by the technique of phase-resolved fluorometry. In this method the excitation light is modulated sinusoidally, usually in the radio-frequency range, and the emission is analyzed with a phase sensitive detector. The emission appears as a sinusoidally modulated signal, shifted in phase from the excitation modulation and partially demodulated by an amount dependent on the lifetime of the fluorophore excited state (5, Chapter 4). The detector phase can be adjusted to be exactly out-of-phase with the emission from any one fluorophore, so that the contribution to the total spectrum from that fluorophore is suppressed. For a sample with two fluorophores, suppressing the emission from one fluorophore leaves a spectrum caused only by the other, which then can be directly recorded. With more than two flurophores the problem is more complicated but a number of techniques for deconvoluting the complex emission curve have been developed making use of several modulation frequencies and measurement phase angles (79). 

In the thermal bath modulation, a thermocouple is used as a weak link to the thermal bath, and the temperature of the bath is modulated sinusoidally in time. This configuration eliminates the need for a separate thermometer and heater on the sample, while retaining the ability to make measurements with minimal addenda. 

A kinetic technique for determining a fluorophore s excited state lifetime by using a light source whose intensity is modulated sinusoidally at a certain frequency, such that the intensity of the fluorescence emission likewise varies sinusoidally but with an added delay from the finite relaxation constant for fluorescence decay. The period of the sinusoidal modulation is chosen to be in the neighborhood of the magnitude of the fluorescence lifetime. 

The coincidence measurements discussed in the previous section were concerned with the total coincidence signal, i.e. the signal obtained when the decay of a particular ensemble of states is integrated over. These states are produced in a very short time ( 10 s) in electron impact excitation, and can sometimes evolve in a complicated way. In the absence of internal fields (e.g. the n P states of helium) each of the fm) states decays with the same exponential time dependence exp(—yt), and the coincidence technique can be used to yield the decay constant y of the excited state (see Imhof and Read, 1977, and references therein). However, if the excited state is perturbed by an internal (or external) field before decay, then the exponential decay is modulated sinusoidally giving rise to the phenomenon of quantum beats (Blum, 1981). 

In the so-called pulsed force mode AFM technique developed by Marti and coworkers [124], the data acquisition rates are increased. In this mode, as outlined in Sect. 3.2, the sample is modulated sinusoidally ( 1 kHz) during a conventional contact mode AFM scan to cause that the tip contacts, indents, and breaks free from the surface with a frequency of 1 kHz during scanning. Instead of recording the complete f-d curve, pull-off forces and stiffness are acquired parallel to sample topography at useful scan rates that become possible (Fig. 3.54). 

If m2j = 0 the magnetic structure corresponds to a modulated sinusoid of amplitude A = my. 

Now consider the case where the incident photon flux is modulated sinusoidally ... 

Frequency modulation relies on modulating the frequency of a simple periodic waveform with another simple periodic waveform. When the frequency of a sine wave of average frequency(called the carrier) is modulated by another sine wave of frequency (called the modulator), sinusoidal sidebands are created at frequencies equal to the carrier frequency plus and minus integer multiples of the modulator frequency. FM is expressed as ... 

The index of modulation, 7, is defined as A /. Carson s rule (a rule of thumb) states that the number of significant bands on each side of the carrier frequency (sidebands) is roughly equal to 7+2. For example, a carrier sinusoid of frequency 600 Hz, a modulator sinusoid of frequency 100 Hz, and a modulation index of three would produce sinusoidal components of frequencies 600, 700, 500, 800,400, 900, 300, 1000,200, 1100, and 100 Hz. Inspecting these components reveals that a harmonic spectrum with 11 significant harmonics, based on a fundamental frequency of 100 Hz, can be produced by using only two sinusoidal generating functions. Figure 10.12 shows the spectrum of this synthesis. 

To develop a new method by which we can study the mixing of two components in each domain in the patterned surfaces, we studied CFM of patterned surfaces prepared by fi-CP methods. Figure 5 shows a comparison between the patterned samples prepared by two different fi-CP methods, that is, the wet-inking and the contact-inking methods [57]. All adhesive force maps in Fig. 5 were observed in a 0.1 mM NaFICOs aqueous solution using a CH3 tip on three samples. The z-piezo was modulated sinusoidally at a frequency of 0.5 kHz with amplitudes of... 

Since ASK and PSK involve square-pulse modulated sinusoidal carriers, this is the null-to-null main lobe bandwidth for these modulation schemes with T replaced by Ty. For coherent FSK, note that the minimum frequency spacing between cosinusoidal bursts at frequencies and -F A/ is 1/27 , Hz in order to maintain orthogonahty of the two signals. The first null of the sine function at frequency fc must have 1 / Ti, Hz below it, and the one at frequency fc -F A/ must have 1/ T , Hz above it, giving a total bandwidth for FSK of... 

The mean fluorescence lifetime may also be determined by continuous intensity measurements, if the exciting light intensity is modulated at a high frequency. Fluorescence is excited by light modulated sinusoidally at a known frequency (ajln Hz). The emission is a forced response to the excitation, and is therefore modulated at the same frequency, but with a phase shift, due to the time-lag between absorption and emission. The intensities of the two beams are monitored by photomultipliers. The difference in phase (0) between the two intensities is determined electronically. The lifetime r is given by cox = tan<. The modulation frequency must be made comparable to the decay rate, e.g., around 30 MHz for a mean lifetime of 30 ns. Such frequencies can be achieved by using a hydrogen lamp actuated by a suitably modulated current source. Commercial equipment is available. The method has been applied to quinine sulphate, fluorescein, and acridine, for example, with a precision of 1-2%. It is especially useful for very short (sub-nanosecond) lifetimes. 

Fluorescence decay kinetics also can be measured by exciting the sample with continuous light whose intensity is modulated sinusoidally at a frequency (m) on the order of 1/t, where t again is the fluorescence lifetime. The fluorescence oscillates sinusoidally at the same frequency, but the amplitude and phase of its oscillatirais relative to the oscillations of the excitation light depend on the product of oo and t (Fig. 1.16 and Appendix A4). If mr is much less than 1, the fluorescence amplitude tracks the excitation intensity closely if an is larger, the oscillations are delayed in phase and damped (demodulated) relative to the excitation [28-30]. Fluorescence with multiexponential decay kinetics can be analyzed by measuring the fluorescence modulation amplitude or phase shift with several different frequencies of modulated excitation. 

APT experiment is that a large number of FIDs (e.g., 800), are recorded at different values of ti which is increased gradually (for instance, from zero to one second) over the 800 spectral recordings. Fourier transformation affords a corresponding number of spectra in which the intensity of each spectral line varies (or is modulated ) sinusoidally, and the arrangement of the spectra... 

In the constant speed mode the detector signal is modulated sinusoidally at a frequency equal to the product of optical mirror speed, v, and wavenumber, v. 

Fig. 10 shows a single phase half-bridge inverter. The switches Si and 2 are modulated sinusoidally to... 

For modulated differential scanning calorimetry (MDSC), the sample temperature (T) is modulated sinusoidally about a constant temperature ramp given by ... 

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