The Measurement and Analysis of Heterogeneous Emissions by Multifrequency Phase and Modulation Fluorometry

Abstract During the last few decades fluorescence spectroscopy has developed into an important and widely used technique in the physical, chemical, and biological sciences. The increasing sophistication of theory and methodology has permitted us to apply fluorescence techniques profitably to a variety of complex problems. A fundamental consideration in the spectroscopic analysis of any system is the degree of sample heterogeneity. We wish to know, for example, if the teterogeneity arises from a melange of fluorophores (this condition often reduces to the question of sample purity) as a consequence of solvent-fluorophore interactions or from the fluorophore per se. An exact quantitation of the nature and extent of the heterogeneity may be critical to our understanding of the origins of that phenomenon. These data have important implications whether the desired information concerns a compositional description for analytical purposes or whether the fluorophore is designed to probe the molecular environment.

the nature and extent of the heterogeneity may be critical to our understanding of the origins of that phenomenon. These data have important implications whether the desired information concerns a compositional description for analytical purposes o r whether the fluorophore is designed to probe the molecular environment.
The extent of heterogeneity may not be apparent from observation of the spectral distribution o r polarization of the emission. Fluorophores exhibiting very similar emission or excitation spectra may differ markedly in their lifetimes. Hence time domain measurements offer a powerful means of examining and quantitating heterogeneous emissions.
The fluorescence lifetime can be expressed as the reciprocal of the radiative rate process , kr . The observed relative fluorescence intensity, I, of a system will be related to the radiative (k,) and nonradiative (knr) processes according to I = kr/(kr + knr) In this greatly simplified expression we lump together the various mechanisms which may deactivate the excited state such as internal conversion, energy transfer, and chemical quenching. Exact knowledge of the lifetime may be required for the correct interpretation of many observations related to the intrinsic photophysics of optical probes and their behavior in complex systems such as biological milieu. The correct physical interpretation of results on quantum yields , polarizations, and quenching rate constants, for example, depend upon knowledge of the fluorescence lifetime. Several excellent discussions of the pertinence of lifetime data have appeared [ 1-41, B. Measurement of Fluorescence Lifetimes 1. Impulse Response The measurement of fluorescence lifetimes is generally accomplished by either an impulse response o r a harmonic response technique. With the impulse response technique the illumination function is a pulse of light of short duration, ideally a delta function. Such pulses are traditionally generated by flashlamps [ Z ] , pulsed lasers [5], and more recently by synchrotron radiation sources [6, 71. The direct decay curve, that is, the record of fluorescence intensity versus time, is obtained by using detectors with temporal resolution such a s streak cameras, time sampling circuitry or, most commonly, the time correlated single photon counting approach [Z, 31. Lf the fluorescence lifetime is monoexponential and long compared to the exciting pulse duration and if the distortion of the pulse shape by the detecting elements is negligible, then the data analysis is straightforward. In cases where the exciting pulse duration is not negligible compared to the fluorescence lifetime, then the fluorescence decay curve will be distorted by the excitation function and a deconvolution procedure must be used to extract the time decay curve. Furthermore, if the emission is characterized by two or more decay times, then the data must be fit by the appropriate decay rates and their preexponential factors. The observed decay curves will thus take the form -t/'Ji i where I(t) is the fluorescence intensity at time t after excitation, ai is the amplitude of the i-th component, and -ri is the lifetime of the i-th component. Critical analysis of a number of deconvolution and data fitting routines have been given [8-lo].

Harmonic Response
In the traditional harmonic response approach the sample is illuminated by a light source with an intensity which is sinusoidally modulated FIG. 1. Schematic representation of the excitation R(t) and fluorescence S(t) waveforms. The fluorescence is delayed by an angle $I with respect to the excitation. at high frequencies, typically in the megahertz range. In such a case the fluorescence intensity will also be modulated sinusoidally at the excitation frequency but will be delayed by a phase angle, @, due to the finite persistence of the excited state, a situation depicted in Fig. 1. Additionally, the relative modulation, that is, the effective ac/dc ratio of the fluorescence, will be less than that of the exciting light. This demodulation forms the basis of a lifetime measurement independent of the phase shift. In this review we shall consider primarily the instrumentation required to realize the modulation of the exciting light and to effect the phase and modulation measurements a s well as the appropriate data analysis procedures. We shall anticipate our discussion by emphasizing that the availability of true multifrequency instrumentation has added a new dimension to phase and modulation fluorometry and has allowed the method to realize its true potential.

A. Brief History
The literature on phase fluorometry is considerable. A thorough account of the development of phase fluorometry up until 1966 has been written by Birks and Munro [l]. A more recent discussion of phase fluorometry up until 1980 has been penned by Teale [ 113. We shall only outline some of the major developments in this area and refer readers with a sustaining interest to the aforementioned articles and to the original literature for detailed discussions. The first phase fluorometer appears to be that of Gaviola [12], built in 1926, although certain aspects of the problem were anticipated by Abraham and Lemoine [13] in 1899. Gaviola's instrument utilized the Kerr effect to modulate the light intensity and relied upon visual detection of the fluorescence. The following decades brought a number of diverse approaches to the light modulation and detection systems. The ultrasonic diffraction grating was utilized by Maercks [14] and Tumerman and Szymanowski [15] while photomultipliers were introduced in this area of research by Ravilious et al. [16]. Noteworthy advances were also made by Bailey and Rollefson [ 171,Birks and Little [18], Muller et al. [ 191,Schmillen [ZO] , Bauer and Rozwadowski [21], and Spencer and Weber [22].
Modulation measurements really began seriously with the instruments of Hamilton [23], Birks and Dyson [24], and Spencer and Weber [22]. The notion of using cross-correlation techniques to move the measurement to the low frequency domain with subsequent improvement in stability was introduced by Schmillen [ZO] although his particular approach did not fully realize the potential of the method. Birks and Little [18] modulated the photomultiplier's response with the same frequency used to excite the sample. Spencer and Weber [22] extended this cross-correlation approach by adding a variable low frequency component to the main frequency and modulating the photomultiplier's response with this new frequency (see Appendix 3).
The phase fluorometers mentioned above were essentially single (or at best two) frequency instruments, moreover modification of the operating frequency was not generally a simple and rapid procedure. True multifrequency operation has been developed only recently and is characterized by the ability to modulate the light at arbitrary frequencies rapidly without recourse to additional instrument adjustment.
B. State of the Art 1. General descriptions of new phase fluorometers. These instruments include those by Haar and Hauser [25], Saleem and Rimai [26], Gugger and Calzaferri [27,281,Hieftje et al. [29], Menzel and Popovic 1301, Merkelo et al. [31], Lytle et al. [32], Ide et al. [33], and Gratton and Limkeman [34]. These instruments differ in their approach to the strategies for light modulation and signal detection and processing but all utilize a C W laser for the sample excitation. As we shall discuss later, synchrotron radiation is also now being utilized for phase fluorometry.
In the last few years a number of research groups have published

Commercial Instrumentation
To our knowledge the only commercially available phase fluorometer is in the SLM 4800 series. Since many researchers in the biophysical and biochemical sciences utilize this instrument, we shall discuss a few of the more important operational principles in some detail. Some additional components a r e described in the section concerning the multifrequency instruments (Sec. II-B-3).
In the SLM 4800A instrument the light from a 450-W xenon arc passes through the excitation monochromator and is then intensity modulated at either 6 , 18, o r 30 MHz before impinging upon the sample. The light modulation principle is the  effect which is realized with an ultrasonics tank placed after the excitation monochromator. Ultrasonic waves are generated in a stainless steel tank containing a liquid (typically 19% ethanol in water to minimize the temperature dependence of the sound velocity) by a transducer, an x-cut piezoelectric quartz crystal, mounted in an aluminum o r stainless-steel cylinder. Proper orientation of a reflector situated at the opposite end of the tank establishes an ultrasonic standing wave in the solution. The standing waves a r e formed when the length of the acoustic path is a multiple of half a wavelength of the ultrasonic wave. These standing waves create planes of higher refractive index which collapse and form with the ultrasonic waves. Incident plane waves of light will respond a s they do toward an ordinary low order diffraction grating and an intermittent interference pattern will form and collapse following the condition of the standing wave. The crystal face and the reflecting surface must be adjusted to parallel orientation with micrometers (on the reflector mount) to insure the standing wave. Additionally, since the wavelength of the acoustic wave in water is short (100 pm at 15 MHz) and since the width of the acoustic resonance peak is on the order of 2-5 kHz (at 15 MHz), the length of the tank must also be slightly adjusted in order to achieve the resonance condition at a constant frequency. The length of the ultrasonics path is approximately 5 cm which corresponds to an acoustic resonance every 15 kHz. In this geometrical condition the width of the resonance is on the same order of magnitude as the difference between two adjacent resonances which means that exciting one mode exclusively is difficult. The tuning of the tank is achieved by successively adjusting the orientation of the reflector with respect to the crystal and then adjusting the distance between these two components until a maximum modulation value is reached. When standing waves are obtained, modulation of the light a t twice the exciting frequency occurs.
The modulation is detected by an ac meter, and when the tank is perfectly tuned the modulation of the exciting light, that is, the a d d c ratio, should, given the particular amplifier gain characteristics utilized, reach a value of about 2. The degree of modulation attained depends strongly, however, on the parallelism of the exciting light passing through the tank and good modulation can be reached only with narrow excitation slits. In addition, more than one mode of modulation may exist in the tank which can result in artifacts which we shall discuss presently.
fundamental, 3rd, and 5th harmonic of the piezoelectric crystal. We should note that in principle a DebyeSears tank permits modulation at many frequencies. In practice the lower frequency is limited by the details of the tank's construction and modulation frequencies less than 1 MHz a r e difficult to achieve. The high frequency limit is imposed by the absorption of sound in the tank medium and by the stringent requirement for parallelism between the crystal and reflector. The practical upper limit at present is less than 100 MHz.
The emitted light (or scattered light from a scattering solution which can be interposed in the same position as the sample) is detected by a photomultiplier, the last dynode of which is modulated at a frequency The modulation frequencies of 6, 18, and 30 MHz result from the equal to the light modulation frequency ( 6 , 18, o r 30 MHz) plus a small additional frequency (typically 25 Hz). This procedure, cross-correlation (see Appendix 3), offers advantages over direct rf measurements in terms of accuracy and noise considerations. Essentially, the phase and modulation information of the signal is transposed to the low frequency range where digital averaging techniques can be easily implemented. The cross-correlation frequency is obtained by an electronic phase shifting method developed by one of us (E. G.), called the digital phase shifter. The digital phase shifter is simple and inexpensive and the version present in the SLM instrument can be used in principle over the modulation range of 1 to 60 MHz (see Appendix 5). The phase delay and demodulation of the emission relative to the scattered light is then calculated a s discussed in the following section. Standard photomultipliers utilized on the SLM instrument include the EMI 9813QB and the Hamamatsu R928. Although the EMI 9813QB is a high sensitivity tube, it has the disadvantage of demonstrating a severe color e r r o r , an artifact which will be discussed later on. (We shall note that on recent models a modified 9813QB photomultiplier with improved characteristics is used.) The precision of lifetime measurements on a standard SLM instrument is on the order of 100 to 200 ps depending upon the signal strength and the experience of the operator. The most important factor which limits the precision of the instrument is undoubtably the stability of the ultrasonics tank. Although the precision of the present commercial phase and modulation instrument is inferior to that of the more sophisticated, homebuilt apparatus described below, its performance compares favorably to that of commercially available pulse lifetime systems. Certainly, the introduction of the commercial phase and modulation fluorometer in the early 1970s was an important step in the general renewal of interest in phase fluorometry.

Research Instrumentation
We shall now discuss in some detail a variable frequency crosscorrelation phase fluorometer. This instrument, recently described [34], operates over the frequency range of 1 to 160 MHz with a time resolution of several picoseconds. The basic instrument layout is shown in Fig. 2 Before entering the Pockels cell the laser beam passes through a calcite prism polarizer which assures complete polarization of the light and establishes a polarization angle of 55" to the vertical laboratory axis. This angle corresponds to the "magic angle" of Spencer and Weber [36] which eliminates e r r o r s in the measured lifetime due to Brownian diffusion of the fluorophore. A polarizer placed after the Pockels cell is oriented at 90" to the polarization plane of the first polarizer while the x-axis of the Pockels cell is set at 45' with respect to both polarizers. The Pockels cell is set in a gimbal mounting to facilitate positional adjustments. Application of a voltage across the Pockels cell results in rotation of the plane of polarization of the incident light which may then be passed by the second polarizer. The biasing voltage for the Pockels cell is chosen to insure that the light modulation frequency is equal to the rf driving frequency.
The sample compartment and part of the optical path for this apparatus consists of the Model DP450 optical module from SLM-AMINCO, Urbana, Illinois. The lenses in the excitation path of this module were removed for operation with the laser. The incident light is split by a quartz plate and a fraction of the light is sent to a cuvette containing a scattering solution which is monitored by a reference photomultiplier (Hamamatsu R928). The intensity of the reference signal may be adjusted using neutral density filters. The fluorescent sample and a scattering sample o r reference compound are placed in a thermostated rotating turret assembly. The emitted or scattered light is collected and focused on the entrance slit of a Jarrel-Ash 1/4 meter grating monochromator mounted vertically to align the slits parallel to the propagation direction of the incident laser beam to enhance the light collection.
The emission photomultiplier generally utilized is the Hamamatsu R928 which has the advantage of exhibiting no sensible color error (see Section IV-B) and a modest voltage requirement for rf modulation. The photomultiplier output can be varied by 90% with a variation of about 10 V on the last dynode which corresponds to a power of 0.25 W on 50 ohms. An rf amplifier (model 502C from RF Power Labs) supplies this voltage. The frequency of the modulated voltage sent to the last dynode of the photomultiplier is set at 31 Hz above the frequency sent to the Pockels cell. This small frequency difference accomplishes the cross-correlation discussed in Appendix 3.
Two identical detection electronic systems are used to analyze the outputs of the reference and sample photomultipliers. Each channel consists of an ac and a dc amplifier (SLM-AMINCO, Urbana, Illinois). The dc amplifier provides two output signals: an integrated signal proportional to the average intensity (the dc voltage) monitored by a digital voltmeter and a low impedance sinusoidal signal which is input to the ac amplifier. The ac amplifier, tuned at 31 Hz with a Q of 200, has three outputs. One output corresponds to the sinusoidal filtered signal at 31 Hz which can be used with a digital phase sensitive detector (see Section V). Another output is the rectified value of the ac signal (the ac voltage) which is monitored by a digital voltmeter. The third output consists of a square wave used for phase measurements which is generated by the zero crossing of the ac signal. An integrating digital voltmeter displays either the dc signal, the ac signal, or the a d d c ratio which represents the modulation of the signal.
In a typical measurement the phase delay and modulation ratio for scattered light (from glycogen or a suspension of latex particles) and the fluorescence are obtained relative to the reference photomultiplier or internal reference signal. The absolute phase delay of the fluorescence, @, is then given as where 9~ represents the phase of the internal electronic reference signal (or the signal from the reference photomultiplier) and $F and 9s represent the measured phase readings for the fluorescent and scattered signals, respectively, The phase angle is measured in degrees with a resolution of 0.01" over the entire frequency range; typically an integration period of 10 s is utilized. The modulation of the fluorescence, M, is defined as where the suffixes F and S refer to fluorescence and scattering and the ac/dc ratio has been described earlier.
The frequency may be changed in seconds merely by entering the desired frequency into the synthesizers-no additional adjustments on the Pockels cell are required. The phase and modulation data at a proptoporphyrin-IX in dioxan. Modulation (*) and phase (+) . Solid curves correspond to a single exponential decay, r = 14.14 ns.
number of frequencies may be entered into a program written for an Apple 11 computer which plots the data points and various curve fits a s shown in Figs. 3 and 4 , for example. Details of the heterogeneity analysis typically used will be presented in a following section.

EI. DATA ANALYSIS
A. Single Exponential Decay The theory of the phase fluorometer has been given in detail by Dushinsky [37] who demonstrated that a fluorescent species can be illuminated by light modulated at circular frequency o according to the expression where ME is a modulation factor. In this case the fluorescence due to a population decaying according to a single exponential, e-t/7, where r is the lifetime, is given by where M is defined a s the relative modulation and corresponds to the ratio of the modulation of the fluorescence to the modulation of the excitation (Eq. 4 ) . One may thus determine the lifetime either by measurement of the phase delay or the relative modulation of the exciting light a s illustrated in Fig. 1. For a homogeneous emitting population these two measurements of the lifetime must be equal and independent of the modulation frequency of the measurement. We give in Appendix 1 a derivation of the Dushhsky relations.

General
Should the emission be characterized as multiexponential or nonexponential, then the expressions for the single exponential decay will assign only some sort of average lifetime to the system, and the details of the decay rates and their component amplitudes are not immediately apparent. A recurring criticism of the phase technique by those only casually acquainted with the methodology is that multiexponential decays cannot be resolved by phase fluorometry. This outdated criticism is true only if the phase lifetime is taken at a single modulation frequency. As we shall see, when phase and modulation lifetimes are obtained at several frequencies, then multiexponential decays can indeed be resolved. population in the absence of excited state reactions will display an apparent lifetime by phase which is less than the apparent lifetime by modulation. Moreover, the higher the modulation frequency, the shorter will be these measured lifetimes. The relationships for the phase angle, 4, and square amplitude, M2, observed for a mixture of sinusoidally modulated components of phases $1, modulation Mi, and fractional intensities f, are given by [38] One can demonstrate (Appendix 2) that a heterogeneous emitting the photocurrent). One notes that the phase angle increases as the modulation frequency increases while the relative modulation declines.
Several approaches have been taken to derive the component lifetimes from the apparent phase and modulation lifetimes obtained at different modulation frequencies. These approaches are discussed in some detail in the following section.

Early Approaches
The possibility of using multifrequency phase and modulation measurements to analyze for lifetime heterogeneity has been recognized for some time [l, 221 but few examples of such analysis appear in the literature. Schuldiner et al. [39] used phase and modulation data obtained at two frequencies (14.2 and 28.4 MHz) to analyze data for a biochemical preparation (discussed in Section 6). Their analysis procedure consisted essentially of fitting the two frequencies phase and modulation data to Eqs. (9) and (10) utilizing an iterative procedure to find the best fit for the two components lifetimes, T , and T , , and their respective fractional contributions, f, and f,.

Weber's Algorithm
The closed form, analytical solution for determining N lifetime components and their respective fractional contributions to the total intensity given phase and modulation data at N modulating frequencies has been presented by Weber [38]. The basic operational principles of this procedure are outlined below; the reader is referred to the original article for the rigorous derivation of the algorithm. linear modulating frequency), one can determine an apparent lifetime by phase, T r p 9 and an apparent lifetime by modulation, Tr' , for a heterogeneous emitting population. In the nomenclature already established we may express the measured phase shift, & .
, and relative modulation, Mr, as Given a circular frequency, c + (where w = 2nf and f represents the 4r = tan-' (Sr/Gr) We see then that the Gr and S, terms are directly related to the components of the system according to the relations Gr = fi(l -k u r 2~i 2 ) -1 i where f i is the fraction of the intensity of the fluorescence detected due to the i-th component and ~i is the lifetime of the i-th component (note that z: f i = 1). In fact, f i represents the contribution of the i-th component to the detector photocurrent. If fi is to be related to the chemical composition, i. e., the mole fraction, then one must take into account the relative quantum yields of the various species as well as the relative response characteristics of the detection system to the different emissions. We should also note that the fractional contributions to the steady-state intensity, fi, is related to the preexponential factors, q , commonly given in pulse analysis (Eq. 2) by the relation where the j index runs over all the contributions to the emission.
S,, and S, from the phase (~1f: and T,P) and modulation ( T~M and T,M) lifetimes at two modulation frequencies, o1 and w2. One thus obtains with w , chosen as a convenient base frequency such as one megahertz.
Weber defines the symmetric functions of the lifetimes as The component lifetimes a r e then given by where the two roots of the quadratic expression are wo T~ and wo T~.
example : The fractional contributions may be calculated in many ways, for The solution for N components using N modulating frequencies proceeds along similar lines. The first study to utilize Weber's algorithm was designed to verify the applicability of the method and to explore the precision and sources of errors inherent in the instrumentation then available [40]. This study concerned the fluorescence of tryptophan a s a function of the pH which, under the conditions of the experiments, could be treated a s a twocomponent system. The results indicated that quite good precision was needed to effectively utilize the analytical solution. The results further indicate that standard e r r o r s of a few tens of picoseconds in the measured phase and modulation values lead to uncertainties of a few hundred picoseconds in the resolved component lifetimes. The effect is indicated in Fig. 6 which depicts the standard e r r o r s expected in the resolved lifetimes for a two-component system ( T~ = 3.1 ns; T , = 8 . 7 ns) versus f,/f,, the ratio of the fractional intensities. The figure shows the calculated deviations resulting from e r r o r s in the phase and modulation lifetimes (AT' = A+) of 100, 50, and 25 ps; the frequencies utilized were 6 and 30 MHz.
A recent study of Weber's algorithm [41] explored the origins of the stringent precision requirements of the method. An interesting result of this study was the finding that the use of more than two.frequencies to solve a two-component system can Iead to an increase in the uncertainty of the resolved lifetimes. This result was unexpected since one may suppose that knowledge of phase and modulation values at many different frequencies would reduce the uncertainty in the calcu- solution the moments of the distribution of the lifetimes a r e related to the experimentally determined functions Gr and Sr through coefficients Yrk (see Eqs. 13 and 14 of Weber's article [38]). The values of the coefficients, however, depend upon the set of modulation frequencies chosen, and one observes that the coefficients for the higher frequencies a r e weighted more than for the lower frequencies. This weighting effect is negligible if the experimental data a r e of very high precision. In practice, however, the result can be an amplification of the experimental statistical e r r o r which may render the closed form solution practically inapplicable for the case of three or more frequencies given the accuracy of present day instrumentation. Weber's algorithm was an important step, however, since it represents the true analytical solution of the problem. We should also emphasize that the analytical method was developed for applications wherein only a few modulation frequencies were available.

Least-Squares Analysis
A different approach was presented by Jameson and Gratton [41]. The basic idea is to use well-known statistical methods for the derivation of the component lifetimes and fractional intensities. A nonlinear least-squares routine was written for an Apple I1 computer closely following the procedure outlined in Ref. 42. The function to be minimized is defined a s where GrC and MrC a r e the calculated values of phase and modulation, respectively, according to Eqs. 15 and 16 using the relations of Eqs.
(17) and (18), @rm and Mrm a r e the measured values of phase and modulation at frequency wr, and ur@ and urM a r e the experimental e r r o r s in the measurements of phase and modulation respectively. The parameter w is a weighting factor. For w = 1 only modulation data a r e computed while for w = 0 only phase data a r e considered. Generally the values of ur@ and urM a r e independent of the frequency and equal for each measurement, so they factorize out. A normalized quantity used to judge the "goodness" of the fit is defined a s  Table 1.
The program written for the Apple 11 computer can resolve up to four independent components. Also, the program allows one to f i x the  values of some of the parameters and solve for the others. The procedure usually employed consists in first trying a fit using a single component. If very large values of x and systematic deviations of the calculated phase and modulation values with respect to the measured data a r e obtained, then a fit using two components is attempted. In some particular cases the lifetime of one component is known independently. The fitting problem is thus simplified and in these cases as many a s three components can be fit successfully.
Recently, an extensive study has been carried out using a mixture of two fluorophores of known lifetime in order to test the ability of the method to resolve a double exponential emission. In all cases studied the values of the component lifetimes and fractional intensities determined were in very close agreement with the expected results. In Table 2 we report some representative examples of this unpublished study. We believe that these experiments demonstrate convincingly the capability of phase and modulation fluorometry to analyze heterogeneous emission.
A slightly different least-squares approach has been presented by Klausner et al. [43]. These researchers used least-squares analysis, but instead of minimizing the direct observable quantities (phase and modulation), they chose to use the derived quantities of phase and aLifetimes in nanoseconds. Expected values of lifetime were taken from multifrequency measurements on the pure compounds and steadystate intensities were calculated from measured intensities of the pure compound prior to the mixing. Derived values a r e preliminary results of a detaiIed forthcoming analysis (Gratton, Lakowicz, Laczko, Limkeman, Cherek, and Maliwal, manuscript in preparation). modulation lifetime. Although in principle this approach should be equivalent to the one discussed above, in practice different results can be obtained because of the presence of systematic errors. For example, as discussed in Section IV-B, color errors can add a small angle to the expected phase measurement. Under particular circumstances the color artifact can translate in a large deviation of the phase lifetime rp with respect to the expected value giving an incorrect fit.

Other Methods
When phase and modulation values are measured at two frequencies only, a simplified approach can be used. For example, Visser et al.
[44], using only three of the four measured values (two phase angles and one modulation ratio), have inverted the expressions given in Eqs. (17) and (18) with respect to the unknown lifetimes values and fractional intensities. This approach is clearly an exact solution to be used for two-component cases when only three experimental data are available.

A. General
We may distinguish between several types of instrument-related artifacts which can affect fluorescence lifetime measurements in general and heterogeneity analysis in particular. We shall not discuss here the more obvious correction factors which take account of the relative intensity response characteristics of monochromators and photomultipliers. Rather we shall concentrate on those effects which are peculiar to time domain measurements. Such considerations are important since these systematic errors are generally much larger than the noise in the measurements. As we shall see in our discussion of heterogeneity analysis, the presence of small systematic errors in phase or modulation data can have profound effects upon the results.

B. Color Errors
Lifetime measurements are complicated by the well-known [19,45] color error which results as a consequence of the systematic increase in the kinetic energy of the photoelectrons with the energy of the generating photons. The result of this systematic increase is that the fluorescence generates a photocurrent that i s delayed by a small time interval, ArEL, relative to the photocurrent generated by the scattered photon 1401. For phase measurements this delay translates into an increase, A@EL, in the phase angle difference between the two photocurrents. AC#IEL and AT^^ are related by One thus finds that the measured o r technical phase delay, @, and corresponding lifetime, T~, for a homogeneous emitting population will depend upon this delay angle a s well as on the molecular o r electronically undistorted phase delay, G o , and the associated molecular lifetime, T~, according to the relation Assuming that A@EL is small, we have These equations are the same as those given in Ref. 40 although the starting point is slightly different. Hence we note that a fixed delay, ATEL, results in a phase lifetime which increases rapidly with T,, and w. This fact owes, of course, to the nature of the tangent function, i.e., an increase of 1 ' at a nominal phase angle of 45" is not nearly so deleterious as the same 1 ' increase at a nominal phase angle of 80". Table 3, using data from Ref. 40, shows the effect of various electronic delays on the apparent phase lifetime for three frequencies assuming a molecular lifetime of 30 ns. In actual measurements using the EMI 9813QB photomultiplier tube and a fluorophore having a 30-ns lifetime in water (DENS), phase lifetimes corresponding to ATEL = 0 . 5 ns were found. The value of ATEL depends, of course, upon the wavelength of excitation and emission. The A T~L = 0 . 5 ns situation corresponded to excitation at 350 nm and observation through a Corning  a A~E L and lifetimes in nanoseconds. T = 30 ns.
We may note that if the color e r r o r is due to a time delay in the photomultiplier, then the modulation lifetime will not be affected. A s regards color effects, the modulation measurements offer a distinct advantage over both the phase and the pulse method. One may circumvent these color e r r o r s in several ways such a s by using, in place of the scattering solution, a reference compound of known lifetime and spectral distribution similar to the fluorophore . This procedure has been followed for a number of years by the pulse lifetime community. Alternatively, one can determine the ATEL for the photomultiplier under the particular conditions of excitation and emission using a plot of the reciprocal of the measured phase lifetime versus the square of the circular modulation frequency which yields ATEL a s the slope, a procedure outlined in Ref. 40. The best approach, however, would be to utilize a photomultiplier with negligible color e r r o r , such as the Hamamatsu R928 o r equivalent.

C . Geometrical Effects
Another type of artifact can arise in phase and modulation fluorometry if the modulation of the exciting light is nonhomogeneous with regard to phase and amplitude over the cross section of the beam. The effects of these artifacts in both phase and modulation measurements a r e discussed in detail in Ref. 34. These effects can be quite dramatic and we illustrate the consequences with the following brief study performed on a conventional SLM 4800A instrument.
After traversing the ultrasonics tank the light is focused on a narrow slit approximately 0.2 mm wide and 1 cm high. With the excitation monochromator set at 500 nm and using 30 MHz modulation frequency, the exciting beam was divided in half along the vertical axis at the position of the cuvette holder using an opaque mask. When the tank was retuned (using the micrometer adjustment procedure described earlier), different values of @ and M were obtained demonstrating that the modulation of the beam was nonhomogeneous along the vertical axis.
The lifetime of a solution of dimethyl POPOP in ethanol at 30 MHz modulation frequency was measured using the upper part of the beam for the scatter reference and the lower part for the fluorescence, then repeating the measurements with the masks reversed and removed altogether. The results a r e presented in Table 4. The tank was then  These results indicate an improvement in the data as a result of the tuning. Similar results for quinine sulfate lifetimes are reported in Table 5. These results also indicate that the measured lifetime values may depend upon the tuning of the ultrasonics tank but that if the criteria of spatial homogeneity of the phase and modulation of the exciting beam is applied, accurate results may be obtained.

V. PHASE-SENSITIVE DETECTION
A. History One of the first examples of phase-sensitive detection must be the original experiment of Armand Hippolyte Louis Fizeau [46] who, in 1849, effected terrestrial measurements of the speed of light by observing the occlusion of light using a rotating toothed wheel as a phase sensitive detector. Light passed through the notches of the rapidly rotating toothed wheel and reflected from a mirror located 8633 m away; at the appropriate wheel rotation speed the reflected light could be extinguished by a tooth which meant that the signal was 180" out of phase with its source.
have been applied to fluorescence lifetime measurements for many years (see Ref. 1, for example) but it remained for Veselova et al.
When +D -+i = 90" o r 270", the intensity contribution of the i-th component to the signal PSD is zero and the observed intensity will be proportional to the contribution of the remaining species. Figure 7 illustrates the operational principle. Without phasesensitive detection the electronics will normally isolate the ac component of the signal, rectify it, and determine the average voltage of the complete and rectified signal. The phase-sensitive detector, however, examines only a 180" intervaI of the unrectified sine wave. Thus, the average voltage detected will be a function of that portion of the sinusoidal signal examined. When the detector "window" is centered around the waveform's maximum and minimum (Fig. 7a), then the average voltage signal consists of a set of component sinewaves originating from different emitting species. Selection of a detector angle which leads to nulling of one component (Fig. 7b) will yield a net voltage corresponding to the contribution of other components out of phase with the nulled component.
In order to select the appropriate detector angle for nulling a specific component of the emission, the lifetime of that component must be known. In some cases one can determine the detector angle which nulls the emission signal at wavelengths where only one component emits. Alternatively, the mixture can be scanned at several phase angle settings and the phase-detected signal can be compared with spectra of the pure components.
Examples of the resolving power of the technique are shown in Figs. 8 and 9. Figure 8 shows the steady-state spectrum of a 1:l mixture of anthracene and perylene in methanol. The individual components of the mixture were resolved (solid lines) when the detector phase angle was set to be 90' out of phase with one component o r the other. The lifetimes of the individual components were 4 . 5 and 5.1 ns for anthracene and perylene, respectively (the modulation frequency was 30 MHz). Figure 9 shows a similar experiment using POPOP and dfmethyl-POPOP in ethanol which have lifetimes of 1.0 and 1.2 ns, respectively. Returning to Eq. (37), we emphasize a limitation of the technique which has received scant attention in the literature. We have written the equations for phase-sensitive detection as the sum of many components in order to focus on the fact that eliminating one intensity component from the total emission intensity does not guarantee that the remaining signal in demonstrating the resolution capabilities of the technique, the application of phase-sensitive detection to more complex chemical and biological systems threatens to be much more problematic.
be the correspondence of the resolved spectra to the spectra expected for the pure components. When even three species contribute to the overall intensity, the results of phase-sensitive detection are difficult to assess and may, in fact, be entirely misleading. In such cases alternative approaches may be utilized. For example, Moya and Garcia [70] and Sebban and Moya [71] have described a method for decomposing fluorescence emission spectra into their elementary components which is based on the simultaneous recording of the fluorescence intensity and apparent lifetime versus the emission wavelength. Phase lifetimes at a particular modulation frequency were obtained; in one case up to seven components in the emission from algae cells at liquid nitrogen temperatures were resolved using a Gaussian fitting procedure.
A demonstration that a system consists of two components would

VI. APPLICATIONS
The use of multifrequency phase and modulation data to resolve heterogeneous emission has not received much attention to date. This dearth of effort owes partly to the general lack of knowledge about the technique, to the fact that rigorous analytical procedures have only recently appeared, and to the very limited availability of true multifrequency instrumentation. Until recently the heterogeneity analysis using phase and modulation fluorometry could only be applied in particular favorable cases, i.e., two-component systems with wellseparated lifetimes or systems in which one or more of the component lifetimes are known independently.
We have already alluded to the early work by Schuldiner et al. [39] who studied the binding of 2-(N-dansyl)aminoethyl-l-thio-p-D-galactopyranoside (DG2) to the lac c a r r i e r protein of "energized" Escherichia coli membrane vesicles. A homogeneous lifetime of 3.0 ns was observed for the free DG2; after binding to the membrane, however, the lifetime increased and became heterogeneous. The iterative analysis was consistent with a lifetime of 18 ns for the bound DG2 contributing 18% to the total intensity. This analysis was supported by the observed polarization upon binding which was analyzed according to the polarization additivity relationship [54].
A study by Klausner et al. [43] concerned the distribution of the membrane probe DPH in model membrane systems. The results indicated that the DPH lifetime was monoexponential in single-phase vesicles but became heterogeneous in mixed-phase vesicles; these observations were interpreted in terms of a lipid domain model. The data, obtained on a standard SLM instrument at 6 , 18, and 30 MHq, were analyzed with a least-squares procedure which we discussed previously. The limited precision of their instrument (standard e r r o r s of several hundred picoseconds), coupled with the apparent closeness of the twocomponent lifetimes, precluded a truly rigorous analysis, and they chose to constrain the two components to 11.1 and 7.4 ns (which correspond to the DPH lifetime in the two isolated single phase systems) and solve for the fractional intensities. A similar study on a more stable and wide range multifrequency instrument would allow, in principle, a more rigorous analysis of such a system but in practice would put a great demand on the present methodology due to the closeness of the component lifetimes.
Another membrane -related study using the same least-squares procedure was carried out by Matayoshi and Kleinfeld [55]. They found that the lifetime of 9-anthroyl-fatty acid membrane probes in egg phosphotidyl vesicles and in nonpolar solvents varied in magnitude and heterogeneity across the emission band. These variations, supported by polarization studies, were attributed to solvent relaxation effects, Again the relatively poor precision of the data coupled with the narrow frequency range studied precludes a completely rigorous analysis of the system, but their general conclusions indicate that caution must be exercised when such complex systems are studied by fluorescence techniques.
A complete spectrofluorimetric characterization, including phase and modulation lifetime analysis, was performed by Herron and Voss [ 561 on 2 -dimethylaminonaphthalene-5-sulfonate (DNP) covalently conjugated to rabbit IgG antibodies. The lifetime data, taken at 6 and 18 MHz with the fluorometer of Spencer and Weber [22] and analyzed using Weber's algorithm, indicated near homogeneity of the unconjugated fluorophore (T = 29 ns) and the fluorophore conjugated to leucine (T = 22 ns) yet the existence of two components ( T~ = 30 ns; T~ = 3 ns) upon conjugation of the fluorophore to normal IgG o r purified anti-DNP antibodies. In both antibody systems the long lifetime component contributed about 87% to the emission. Herron and Voss recognized that the large errors they obtained for the values of the short lifetime component could reflect the closed form nature of Weber's algorithm coupled with the precision of their data. The correspondence between the fractional intensities and the molar ratios was not clear in the absence of data on the relative quantum yields of the two forms.
A phase and modulation lifetime study [40] on the pH-dependent heterogeneous emission of tryptophan using Weber's algorithm was discussed earlier. We should emphasize that this study was designed to test the precision of the methodology and not to probe the photophysics of tryptophan. The results helped to establish that quite high levels of precision are required for the optimum application of Weber's algorithm and also served to demonstrate that the precision of an instrument based on the Debye-Sears effect could be in the range of 30 ps. These data were subjected to additional scrutiny [41] in a comparison of Weber's algorithm to a least-squares procedure. In that report a multifrequency Extensive studies on flavins and flavoproteins have been carried out by Visser and co-workers 1441. Of particular interest is their careful study on pig heart lipoamide dehydrogenase wherein lifetime data obtained by the pulse method (both flashlamp and picosecond laser excitation was used) were compared with that obtained by phase and modulation at 15 and 60 MHz. Both techniques indicated a heterogeneous emission and upon two-component analysis (the phase data were analyzed by a procedure discussed in Section m) gave good agreement in the magnitude of the long component (T = 3.1 ns at 20°C) whereas the phase method gave a shorter second component than the pulse (0.11 ns compared to 0 a 54 ns at 20' C ) with a greater relative weight (f = 0.76 (phase) compared to f = 0.54 (pulse)).
A phase and modulation study on the quenching by oxygen and acrylamide of the tryptophan fluorescence from horse liver alcohol dehydrogenase was presented by Eftink and Jameson 1581. This dimeric protein contains two tryptophans in each identical subunit. An elegant pulse fluorometry study by Brand and co-workers 1591 assigned values of 4.0and 7.4 ns to the two residues. Since one tryptophan (the long component) was accessible to quenching by acrylamide while the other one was buried and inaccessible, Eftink and Jameson were able to assign the lifetime (3.6 ns) and fractional intensity of the buried residue by observing the lifetime of the protein in the presence of 1 acrylamide. They were then able to calculate the lifetime of the quenchable component which was consistent with the phase and modulation data in the absence of quencher. The reported lifetimes were 3.6 and 6.9 ns, in reasonable agreement with the pulse results. Simulations were presented in this work of the effect of quenchers on the observed intensity and phase lifetime for a two-component system in which only one component is accessible to the quencher.
An extensive multifrequency phase and modulation fluorometric study was performed on the quenching by oxygen of porphyrin emission from iron-free hemoglobin and myoglobin samples [60, 611. The results indicated that the emission from these proteins were homogeneous (T = 20.2 ns for HBdes F*, for example) in the absence of quencher but became markedly heterogeneous as the quencher concentration increased. The data were obtained in the instrument described in Section II and were taken at 6, 12, 18, 24, 30, and 36 MHz modulation frequency. The appearance of heterogeneity in the presence of the quencher was explained in terms of a theory [61, 621 based on the consideration that the quenching process in proteins requires the penetration and diffusion of the quencher in the protein matrix as well as a distribution of quenchers among the protein ensemble. Ide and Engelborghs [63] have studied the quenching by iodine of colchicine bound to tubilin using the multifrequency phase fluorometer previously mentioned [33]. Phase lifetime data were obtained at a number of modulation frequencies between 1 and 50 MHz. They reported an homogeneous lifetime of 1.14 + 0.02 ns for colchicine bound to tubilin at an ionic strength of 0.1 5 and 1.2 ns at 1 5. The lifetime data indicated that the quenching by iodine was dynamic with a Stern-Volmer constant of 5.0 X lo* M-' s-l. The intensity quenching data showed a marked downward curvature which was interpreted in terms of screening effect due to the specific ion binding in the neighborhood of the fluorophore. They also observed that iodine and other chaotropic ions increased the rate of colchicine release from the complex. Thus lifetime measurements in the presence of iodine were possible with the phase fluorometer which permitted data acquisition at rates faster than the kinetics of the system. Dalbey et al. [73] recently used phase and modulation measurements to determine the distance between two reactive thiols in the heads of the contractile protein, myosin. Their experiments utilized the energy transfer between a fluorescent donor (the adduct between a reactive myosin thiol with 1,5-IAEDANS (N-(iodoacetylaminoethyl)-5-naphthylamine-1-sulfonic acid) and a chromophoric acceptor thiol adduct, (N-(4-dimethylamino-3,5-dinitrophenyl)maleimide). The donor emission became heterogeneous in the presence of the acceptor, and the phase and modulation data obtained on an SLM 4800 at three modulation frequencies, 6, 18, and 30 MHz, were analyzed using an iterative, Monte Carlo approach. The results were inte reted in terms of a model which of Mg ATP. This result is consistent with the author's earlier studies in which chemical cross-linking of the two thiols was dramatically enhanced by Mg ATP. The present study demonstrated the possibility of analyzing for the fractional intensities of a ternary system which has an appropriate spread of lifetime values, in this case 20.6, 9.3, and 2.8 ns, when one or two of the lifetime values can be fixed.
postulates that the two thiols move 6-7 7 closer together upon addition

W. CONCLUSIONS
We have discussed the theoretical and practical aspects of phase and modulation measurements as well as the analysis of lifetime heterogeneity, and have demonstrated that the multifrequency approach represents a powerful methodological advance which permits u s to realize measurements previously unattainable. Since so much attention in recent years has been focused on the impulse response approach, we may inquire more deeply into the similarities and differences of the impulse and harmonic response techniques. We shall distinguish between the mathematical equivalence of the two methods and the practical differences.
The impulse response and the harmonic response are strictly related for systems which do not exhibit nonlinear (saturation effects, etc.) behavior. The frequency response of the system can be obtained from the pulse response through the use of Laplace transforms. Complete equivalence requires knowledge of the system's response over a wide time and frequency range. The time response is completely determined when the amplitude of the fluorescence signal is measured from times immediately after the pulse to infinitely long times. The frequency response is completely defined when the phase delay and the modulation ratio are measured at all frequencies. The expressions relating the time to the frequency response in terms of the S and G functions are where I(t) is the pulse response. The relationship between S and G and the observable phase and modulation values has been given in Eqs. (11) $and (12). Knowledge of only one portion of the frequency and time domain renders impossible a direct comparison of the data obtained for the impulse and harmonic response since the integrals in Eqs. (41) and (42) cannot be exactly evaluated. Such a calculation would, however, be an unnecessary exercise since we require to know only the parameters associated with the time response of the system, e.g., the characteristic exponential decay time, 7, in the case of a single exponential decay. The same parameter can be obtained directly from the frequency response without transformation to the time domain. Our purpose, then, is to obtain the parameters of the decay rather than the decay itself.
With this strategy in mind we can implement a procedure in the frequency domain in order to analyze a given fluorescence system. First, we obtain the values of phase and modulation over a significant frequency interval. Second, we calculate the analytical response of the system (the model) directly in the frequency domain. Third, we perform a fit, in the frequency domain, of the desired parameters to the experimentally determined values of phase and modulation. Only at this stage can we make a meaningful comparison of the parameters obtained from the impulse and harmonic methods.
The mathematical equivalence of impulse and harmonic response data does not, however, mean that practical differences between the methods do not exist. One important difference which is seldom mentioned concerns the fact that the impulse response technique does not permit the analysis of photons originating over the complete time domain. The time domain data are in essence truncated and obliged to be between somewhat arbitrary limits in addition to falling into time bins of finite width. In the harmonic response, regardless of the modulating frequency, the time information content of the emitted photons is not sacrificed. Moreover, in the single photon counting pulse technique one is permitted to observe at best only one photon per exciting pulse (to avoid statistical pileup effects this limit is, in fact, reduced) whereas in the harmonic response approach all the photons contribute to the measured signaI. This later fact confers a sensitivity to the harmonic method which compares favorably with that of single photon counting, contrary to popular belief.
We may note another misconception about phase fluorometry: that very high frequencies are required for the measurement of nanosecond decays [ 641. Although the errors in the phase and modulation measurements show a minimum at a given frequency [65] which depends upon the lifetime, e.g., 1 ns may be most accurately measured near 150 MHz, high precision can still be obtained at much lower frequencies. An arbitrary increase of the frequency will eventually lead to an increase in the error.
Another aspect of the comparison of the impulse and harmonic approach concerns the time required for the measurement. With traditional flashlamp light sources, collection of pulse data of reasonable precision may take many minutes while the actual computer analysis of the results requires an additional period (this point, however, has been recently discussed by Brand et al. [72] who studied the decomposition of a two-component system using a 30-s acquisition period). In contradistinction, phase and modulation data at a single frequency can be obtained in a few seconds while data at several frequencies can be collected in a minute or so. Data analysis is sufficiently rapid to be performed virtually in real time with the data collection which allows the experimentalist to monitor a situation closely and to contemplate kinetic lifetime studies on systems with rate constants in the range ofseconds.
Furthermore, as shown in Section V, the use of phase-sensitive detection permits one to record directly the individual spectra of two components with different lifetimes and spectra. Such direct recordings are not obtainable at present using the impulse method (note that "time resolved" spectra obtained using single-channel-pulse-height analyzers to set time windows roughly approximate the true time resolved spectra but are not equivalent to the direct-phase sensitive recordings).
The phase and modulation approach can, in fact, be coupled with a high repetition rate pulsed source [ 66, 671. Recently a multifrequency phase and modulation fluorometer has been coupled to the synchrotron radiation source at the Institute for Nuclear Physics in Frascati, Italy.
This instrument is a virtual replica of the one described in Section 11-C with the exception of the light source. The synchrotron radiation consists of quasi-Gaussian pulses with halfwidths on the order of a nano-second and a fundamental repetition rate of 8.568 MHz. Data were obtained at the fundamental frequency and each harmonic up to 94.253 MHz. A typical result is shown in Fig. 10. Note that one cannot distinguish between such a measurement done with a pulsed source and that performed using a sinusoidally modulated source. At the present time multifrequency-phase fluorometry is still at an early stage in development and use. In the near future one may confidently predict the application of the method to a wide range of problems in the physical, chemical, and biological sciences. The ratio MF/ME is called the modulation ratio o r simply the modulation of the fluorescent emission.
Finally, using the tangent expression for the cosine, cos $ = 1 / ( 1 + tan2 Cp)', we can write the two basic relations in terms of phase and modulation: where T = l/r.

APPENDIX 2. PHASE AND MODULATION LIFE TIME RELATIONS
We have asserted that a heterogeneous emitting population, in the absence of excited state reaction, will demonstrate a phase lifetime which is always less than the modulation lifetime. The algebraic demonstration of this fact is somewhat cumbersome [ 11, 681. We present here a brief and more intuitive demonstration of the phenomenon.
One may make a simple geometrical representation of the phase delay and relative modulation as shown in Fig. 11. Here we depict a vector of length M making an angle 4 with the x-axis where 4 represents the phase delay and M the relative modulation. Since for a single exponential decay we have the relation M = cos 4, the endpoint of the vector is constrained to be on the circle of radius 1/2 with a center at (1/2,0). The intercept of the extension of this vector with the line through x = 1 equals at (since tan 4 = at). This circle is universal for single exponential systems irrespective of the lifetime or modulation frequency. We note that the X and Y intercepts of the vector correspond to our previously defined G and S functions (since G = M cos $ and S = M sin 4). which has a right angle, is congruent to the triangle ODB, which also has a right angle. Hence, from the ratios of the sides we have: (-413) M2) t and OA is equal we see that DB equals UTM. Therefore, it is easily recognized in Fig. 12 that in a heterogeneous emitting population T~ will always be less than TM.

APPENDIX 3. CROSS-CORRELATION
The experimental operating principles of cross-correlation phase fluorometry have been discussed in Section II and we present here a mathematical treatment. When a fluorescent sample is illuminated by light modulated at an angular frequency, w = 2 r f , where f is the linear modulation frequency, then the emission can be described by the relation where F(t) is the average fluorescence intensity, MF the relative modulation, and @F the phase delay. In the cross<orrelation method the emission function is multiplied by a sinusoidal signal of frequency wc which is just slightly different from w, namely: where C, is the average value of the multiplying function, MC its relative modulation, and @C its phase delay. The resulting product signal is the new function V(t) = F C [ 1 + MF COS (at + @) + M cos (act + 4 ) where Aw = wcw and A @ = @C -4. If w c is chosen to be very close to w, then Eq. (A18) contains a constant term plus terms of frequency w, 20, and b. The Aw term contains all the phase and modulation information contained in the original emission signal and can be totally filtered electronically from the remaining terms. The shape of the ac filter which filters the cross-correlation frequency is very important. In fact, as discussed in Ref. 34, the intrinsic nonlinear response characteristics of the modulation system (dynode characteristic, light modulator, amplifiers, etc.) leads to harmonics at 2Aw, 3Aw, etc. in the low frequency signal. The important consideration is the extent to which these unwanted harmonics influence the measured phase and modulation values. In Fig. 13 we report a calculation of the values of phase and modulation in the presence of a second and third harmonic of an amplitude equal to 1/10 of the fundamental as a function of the phase angle between the fundamental and the harmonic. Deviations from the expected values of the phase as large as 6" and from the modulation values as large as 0.03 can result. As a consequence, if an accuracy of 0.1" is required, then the second and the third harmonic must be reduced by the filter by at least a factor of 500 with respect to the fundamental. Depending on the value of the modulation frequency and lifetime of the fluorophore, these deviations can produce dramatic artifacts.

APPENDIX 4. MISCELLANEOUS USEFUL FORMULAS
A. Assume that an emitting system has two forms in equilibrium having lifetimes T~ and T~ (for example, the case of free and proteinbound ligand). At frequency w apparent lifetimes by phase and modulation, T~ and T~ will be measured. One may derive an expression for the fractional contribution to the photocurrent, f, and f, (which must then be independently related to the molar quantities a s previously B. As discussed in Section ID, when one examines the phase and modulation lifetimes (for a two-component system) a s a function of the modulation frequency, one observes that in the high and Iow frequency regions phase and modulation lifetimes will reach a constant divergence (Fig. 14). The limiting values of phase and modulation lifetimes are given a s C. An interesting effect Ln the apparent lifetime by modulation has been noted [69] for the case of two components when the fractional contributions and one lifetime are fixed but the other lifetime is varied. Specifically, one can observe an initial decrease and then an increase in T M when the lifetime of one component is increased from zero. The value of T , which gives a minimum in any particular case is given by the derivative of Eq. (12) as A signal complementary to the ck, clock is generated by an inverter (11). The two complementary outputs of the inverter, at 180"with respect to one another, a r e sent to the input of a pair of identical flipflops F1. Because the flip-flops a r e triggered with the positive-going edge of the clock, after division by two the four outputs of the flip-flops a r e shifted by go", one with respect to the other. This operation is repeated by the four identical flip-flops F2 and the 8 corresponding outputs a r e at 45" with respect to one another. At this point the input clock ck, has been divided by 4. The output of any one of the flip-flops F2 is at a frequency F. The 8 outputs of the F2 flip-flops are multiplexed by a digital multiplexer in such a way that the multiplexer output advances by 45" at each step of the multiplexer clock ck, . The multiplexer output is the cross-correlation frequency F + Af. Because in the cross-correlation method only the difference Af is used, the stability of ck, is relatively unimportant. A stability of 1/105 is adequate for ck, . Using this technique, the frequency shifting is discrete. Observing the form of the signal after the product F * (F + Af) is obtained using a mixer is instructive. Two ranges of frequencies a r e generated, one around 2F and one around Af. The residual low frequency signal after removal of the signal at 2F is shown in Fig. 16. The low frequency signal is composed of a frequency Af plus harmonics a t 8Af and higher frequencies. The signal at Af must be filtered from the harmonics, an operation accomplished by the AC filter already discussed in Section 11-33-3. The design of the digital phase shifter can be further simplified if the F2 flip-flops a r e removed. The operation of the circuit remains the same. Four steps only are produced, but the circuit operates at twice the previous frequency, a clear advantage if higher modulation frequencies a r e desired.