ON DEMODULATION ANALYSIS OF LIGHT

G. S. Gorelik

1. INTRODUCTION

In acoustics and in radiophysics it is easy to observe experimentally that, upon the superposition of two harmonic oscillations with close frequencies, beats are obtained. In other words—where a resonant device of high selectivity detects two harmonic components, a rapidly responding, i.e. low-selectivity, receiver will show the presence of periodic pulsations of intensity.

Can beats in light be observed in an analogous way in those cases where a spectral instrument of sufficiently high resolving power would detect the presence of a doublet? Or, taking the more general case, will it be possible to detect in light that modulation of intensity which (together with a possible modulation of phase) is associated, by Fourier’s theorem, with the fine structure of a spectral line?

Until recently the undertaking of such experiments was considered hopeless in optics. Mastery of the radio engineering of the microwave range compels one to reconsider this point of view.

In visible light one can obtain doublets, or multiplets, consisting of narrow lines with widths of the order of \(10^8\)—\(10^9\) hertz, separated from one another by distances of the order of \(10^9\)—\(10^{10}\) hertz*).

In such doublets, or multiplets, the beat frequency (intensity modulation) is of the order of those frequencies to which microwave radio-receiving apparatus responds. In wave numbers, \(10^{10}\) hertz corresponds to \(0.3\ \mathrm{cm}^{-1}\), i.e. to a wavelength of about \(3\ \mathrm{cm}\). Therefore, if by means of a nonlinear low-inertia light converter, namely a photoelement of suitable design (in a photoelement the current is proportional to the square of the amplitude of the electric field), the beats are transformed into oscillations of photocurrent of the difference frequency (i.e. a frequency equal to the difference of the frequencies of the optical lines or to the frequency

*) A line width of the order of \(10^8\) hertz, which in wave numbers is approximately \(3 \cdot 10^{-3}\ \mathrm{cm}^{-1}\), can be obtained only in atomic beams (see, for example, 1).

(beats), these oscillations can be isolated, amplified, and studied with the aid of a suitable microwave radio-receiving device. The author of this review has recently drawn attention to this² and, independently of him and somewhat later, Forrester, Parkins, and Gerjuoy³. The basic schematic of an apparatus for carrying out experiments of this kind is shown in Fig. 1.

Fig. 1.

Together with the monochromator, which selects the multiplet of interest to us, the device in Fig. 1 is a complete optical analogue of a radio receiver. The monochromator plays the role of the resonant circuit, the device in Fig. 1 the role of the demodulator. One may say, using the language of radiophysics, that this is a device for the demodulation of light.

For the experiments discussed here, it is in principle immaterial whether we are dealing with neighboring lines of a single source or with two independent sources.

2. BEATS AND INCOHERENCE

The range of questions touched upon here has long since been clarified from the fundamental standpoint. It has been explained more than once, in particular by L. I. Mandelstam (see, for example,⁴). Nevertheless, as experience shows, it continues to remain a source of misunderstandings. One of them is the objection to what was said in § 1, which amounts to the following:

“The theorem according to which, when two oscillations with close frequencies \(f_1, f_2\) are superposed, beats of frequency \(|f_1-f_2|\) are obtained, in optics applies only to the case of coherent oscillations. Such oscillations can be obtained by splitting one initial beam of light into two and changing the frequency of one of them, for example by reflection from a moving mirror. In the cases indicated above, however, the oscillations are incoherent. In these cases the intensities of the oscillations are added, and consequently there will be no beats.”

We shall begin the analysis of this objection somewhat from afar*).

Neither in optics, nor in acoustics, nor in radiophysics do we ever deal with strictly sinusoidal oscillations, i.e. oscillations of the form \(A\cos(\omega t-\varphi)\), where \(A\), \(\omega\), \(\varphi\) are constants. Light perceived by a spectral instrument as a narrow line is a modulated oscillation of the form**)

\[ E = A(t)\cos[\omega t-\varphi(t)], \tag{1} \]

*) There is no need to dwell in detail on another objection sometimes encountered: “beats contradict the existence of photons.” With equal success one could deny the possibility of interference of light.

**) For simplicity it is understood that the light is linearly polarized.

where the amplitude and phase, \(A(t)\), \(\varphi(t)\), are random (irregularly varying) functions, fluctuating slowly in comparison with the variation of the function \(\cos \omega t\). Some idea of such a chaotically modulated oscillation is given by Fig. 2, \(a\). It follows from Fourier’s theorem that the line width \(\Delta f\) is of the order of \(1/T\), where \(T\) is the mean time separating two nearest statistically independent values of \(A(t)\) or \(\varphi(t)\), \((f=\omega/2\pi)\). Approximately the same appearance is exhibited by the oscillogram of oscillations occurring in a weakly damped oscillatory circuit under the action of the anode current of an electron tube operating in the saturation regime (Fig. 3); this current, owing to the independence of electron emission, is a sequence of random impulses (shot effect)*.

Fig. 2

In this case \(f\) is the natural frequency of the circuit, \(T=1/\delta\), where \(\delta\) is its damping coefficient. The amplitude of the oscillation arising in the circuit under the action of an individual impulse decreases as \(e^{-\delta t}\). The “mean fluctuation time” \(T\), together with the “period” \(\tau=1/f\)**, is an essential characteristic of the oscillatory process. In optics \(\tau \sim 10^{-15}\)–\(10^{-14}\) sec, while \(T\) may, for narrow lines, be of the order of \(10^{-9}\)–\(10^{-8}\) sec, which agrees in order of magnitude with the periods of oscillations in the radio range.

Fig. 3

Radiophysics has at its disposal (speaking somewhat schematically) three methods for investigating modulated oscillations, in particular the chaotically modulated voltage \(u\) on the circuit of Fig. 3:

a) The inertia-free method, which makes it possible to follow the instantaneous values of \(u\); this method is based on the use of an electronic oscillograph and makes it possible to see individual oscillations (the time required for deflection of the electron beam is small in comparison with \(\tau\));

* Slow fluctuation modulation of the amplitude and, mainly, of the phase also occurs in self-oscillations (when there is nonlinearity of the “sustained oscillations” in a circuit with feedback). The theory of this phenomenon and an estimate of the associated line width of a tube generator were given by Bernstein\({}^{5}\).

** “Period” is in quotation marks because the oscillation (1) is not periodic.

b) The low-inertia method, in which the measuring instrument responds to \(\overline{F(u)}\), where \(F\) is some function, and the bar denotes the mean value over some time, large in comparison with \(\tau\), but small in comparison with the time over which the amplitude or phase changes appreciably; this method is implemented, for example, in the circuit of Fig. 4 with \(RC \ll T\). Here the oscilloscope indicates, for a quadratic tube characteristic, \(\overline{u^2}\) (Fig. 2b);

Fig. 4.

Fig. 5.

c) The inertial method, in which, for example, the quantity \(\overline{F(u)}\) is measured, where the double bar denotes the mean value over a time large in comparison with the mean time of the fluctuations; this method is implemented, for example, in the circuit of Fig. 5 with \(RC \gg T\). The pointer of the instrument in this case remains almost motionless; for a quadratic tube characteristic it indicates the value \(\overline{u^2}\).

Optics in this respect is in a much worse position: all the usual methods of photometry (by means of the eye, a photographic plate, a photoelement connected to a galvanometer) give only \(\overline{E^2}\), i.e. the mean over a time very large in comparison with the optical \(T\), and therefore smooth out intensity fluctuations. This was emphasized already by Rayleigh\(^6\). Inertia-free observation in optics is in principle impossible, for according to quantum mechanics it is impossible to measure \(E\) over a time small in comparison with the optical \(\tau\). But what is the situation in optics regarding the possibility of low-inertia observation?

Fig. 6.

If one disregards the shot effect\(^*\), the instantaneous value of the photocurrent is proportional to the mean value of \(E^2\) over some time characterizing the photoeffect. The experiments of Lawrence and Beams\(^7\) showed that this time is less than \(3\cdot 10^{-9}\) sec., and there are apparently no grounds to expect that it exceeds, say, \(10^{-10}\) sec. or even a still smaller quantity. Thus one may expect that the photocurrent in the case of a narrow optical line \((T \sim 10^{-8}\)—

\(^*\) The shot effect in a photoelement will play an essential role in demodulation analysis of light of low intensity.

\(10^{-9}\) sec.) is proportional not to \(\overline{E^2}\), but to \(\overline{E}^2\), and, consequently, reproduces fluctuations of the intensity of light. It is precisely under this assumption—which still, of course, remains to be experimentally verified—that the objection based on incoherence disappears.

In order to make this clearer, let us first turn to a radio-physical analogy. Suppose there are two independent circuits, excited by the shot effect (Fig. 6), with close but unequal natural frequencies \(f_1, f_2\) and with the same damping coefficient \(\delta \ll |f_1 - f_2|\). What will be obtained by investigating the sum of the voltages \(u_1 + u_2\) across the circuits by each of the three methods indicated above?

Both \(u_1\) and \(u_2\) have the character illustrated by Fig. 2,a, but their amplitude and phase fluctuations are completely independent; this is illustrated by Fig. 7, a and b. An oscillograph to which the summed voltage \(u_1 + u_2\) is fed directly will give a curve of the type shown in Fig. 7, c. It is an oscillation with double modulation: an ordered modulation, in which the “amplitude” changes with period \(T_1\), where \(1/T_1 = |f_1 - f_2|\) (i.e. beats), and superposed upon it a disordered (fluctuational) amplitude modulation

Fig. 7.

and phase, characterized by the same time \(T\) as each of the oscillations \(u_1, u_2\). The curve in Fig. 7, \(c\), is easy to obtain by applying the usual construction of beats over intervals of time small in comparison with \(T\), during which the amplitudes and phases of the component oscillations do not have time to change appreciably.

The oscillogram of the quantity \(\overline{u^2}\), which we obtain by applying \(u_1+u_2\) to the input terminals of the device of Fig. 4, is shown in Fig. 7, \(d\). Here the amplitude and phase of the resultant oscillation change appreciably already during the time \(T_1\), and therefore, for low-inertia observation, the circuit must be adjusted so that \(RC \ll T_1\). The oscillogram represents the sum of a randomly fluctuating quantity and an oscillation of difference frequency \(F=|f_1-f_2|\), randomly modulated in amplitude and phase, with mean fluctuation time \(T\). By means of a circuit tuned to the frequency \(F\), one can isolate the component of frequency \(F\) (Fig. 8). In this case, if the width of the resonance curve of the circuit \(\Delta F\) is large in comparison with \(\Delta f\), i.e., if the time required for the establishment of oscillations in the circuit is small in comparison with \(T=1/\Delta f\), it will reproduce the fluctuations of the amplitude and phase of the quantity \(\overline{u^2}\); if, however, \(\Delta F \ll \Delta f\), the amplitude and phase in it will fluctuate much more slowly, with a mean time equal to the establishment time in the circuit.

Fig. 8.

Graph 7, \(d\), is easily obtained directly from graph 7, \(c\), by first constructing the oscillogram of the quantity \(u^2\) and then drawing a line passing through half the height of each peak. It is also easy to investigate the problem analytically. We have

\[ u=u_1+u_2=A(t)\cos[\omega_1t-\varphi(t)]+B(t)\cos[\omega_2t-\psi(t)], \]

whence

\[ \begin{aligned} u^2={}&\frac{1}{2}A^2(t)+\frac{1}{2}B^2(t)+A(t)B(t)\cos\{\Omega t-[\varphi(t)-\psi(t)]\}\\ &+\frac{1}{2}A^2(t)\cos 2[\omega_1t-\varphi] +\frac{1}{2}B^2(t)\cos 2[\omega_2t-\psi]\\ &+A(t)B(t)\cos\{(\omega_1+\omega_2)t-[\varphi+\psi]\}. \tag{2} \end{aligned} \]

Here

\[ \omega_1=2\pi f_1,\qquad \omega_2=2\pi f_2,\qquad \Omega=2\pi F,\qquad F=f_1-f_2. \]

On averaging over a time small in comparison with \(T_1\), but embracing a large number of “periods” \(\tau\), all the terms written in the second and third lines disappear, and we obtain

\[ \overline{u^2}=\frac{1}{2}A^2(t)+\frac{1}{2}B^2(t)+A(t)B(t)\cos\{\Omega t-[\varphi(t)-\psi(t)]\}. \tag{3} \]

The first two terms represent a randomly fluctuating quantity; the third is an oscillation of the “frequency” \(F\) with fluctuating amplitude \(A(t)B(t)\) and phase \(\varphi(t)-\psi(t)\).

Finally, with the inertial method of investigation (i.e., if we apply \(u_1+u_2\) to the input terminals of the device in Fig. 5, where \(RC \gg T\)), we obtain simply the sum of the intensities of the two oscillations. Indeed, by virtue of the statistical independence of \(A\), \(B\), \(\varphi-\psi\), we have

\[ \overline{AB \cos\{\Omega t-[\varphi-\psi]\}} = \overline{AB}\cdot \overline{\cos\{\Omega t-[\varphi-\psi]\}} =0, \]

since the cosine assumes positive and negative values equally often, and its mean value is therefore zero. Consequently,

\[ \overline{u^2} = \frac{1}{2}\,\overline{A^2} + \frac{1}{2}\,\overline{B^2} = \overline{u_1^2} + \overline{u_2^2}. \]

The oscillations corresponding to the two lines of the optical doublet are analogous, up to the time scale, to our two oscillations \(u_1\), \(u_2\). We may write for them

\[ E_1=A(t)\cos[\omega_1t-\varphi(t)], \qquad E_2=B(t)\cos[\omega_2t-\psi(t)] \]

and repeat, with respect to the independence of the fluctuations of the quantities \(A\), \(B\), \(\varphi\), \(\psi\), what was said above for the radio-physical case. It remains to apply the superposition principle, according to which the resultant field is obtained as

\[ E=E_1+E_2, \tag{4} \]

and to investigate how such an oscillation acts on the measuring apparatus, in exactly the same way as was done for the radio-physical case.

Fig. 9.

Let a galvanometer be connected after the photoelement (Fig. 9). It averages the photocurrent over a time of at least tenths of a second, which exceeds \(T\) by \(10^8\)–\(10^9\) times. Therefore the pointer will remain practically motionless and indicate

\[ \overline{E^2}=\overline{E_1^2}+\overline{E_2^2}. \]

But if the instantaneous value of the current in the photoelement is proportional to the mean value of \(E^2\) over a time small in comparison with \(T_1\), and moreover \(T_1 \ll T\), then the photocurrent itself varies in the same way as \(\overline{u^2}\) in Fig. 7*). If the pulsation frequency \(F=|f_1-f_2|\) is of the order of \(10^9\)–\(10^{10}\), then at the present level of experimental technique it is difficult to oscillogra-

*) Here it is assumed that the dimensions of the light source and of the photocathode are sufficiently small for \(\overline{E^2}\) to have practically the same value throughout the latter. The corresponding estimate is given in note\(^2\).

…filter them, but they can be picked out by a resonator tuned to this frequency and the amplitude of the oscillations in the resonator can be measured. The equivalent circuit is shown in Fig. 10. In reality, of course, at the frequencies in question a cavity resonator must be used. One of the possible variants is shown in Fig. 11.

It is not difficult to see that, in the case of incoherent oscillations \(T \ll T_1\), it will be impossible to detect any traces of beats. Indeed, in this case the term \(\Omega t\) under the cosine sign in formula (3) plays no role in comparison with the more rapidly varying random quantity \(\varphi(t)-\psi(t)\). Thus, the difference frequency \(F\) can be singled out only if it exceeds the line width \(\Delta f\). This is possible only for \(F\) of the order of \(10^9—10^{10}\) hertz or more. That is why, before the advent of microwave apparatus, the experiments discussed here were impossible.

Fig. 10.

Fig. 11.

The situation may be characterized as follows.

1. Incoherence cannot spoil beats whose period is small in comparison with the mean fluctuation time.

2. In optics two types of apparatus are still used: a) instruments with high spectral resolving power and high inertia (a long settling time); the latter is a fundamentally inevitable consequence of the former*); b) instruments with low spectral resolving power and likewise with high inertia (for example, a combination of a photocell with a galvanometer). However, low spectral resolving power is compatible with low inertia. Owing to this circumstance, still insufficiently used in optics, there exists in principle the possibility of observing, also for light, the quantity \(\overline{\mathbf E}^2\), and not \(\overline{E^2}\), as is usual, which may make it possible to detect beats.

*) For example, in the \(m\)-th principal maximum of a grating, the oscillation from each slit appears \(m\tau\) sec. later than the oscillation from the preceding one; therefore the settling of the oscillation lasts \(Nm\tau = Rt\) sec., where \(N\) is the number of slits, \(R\) the resolving power. If the photocell reacted only to one of the doublet frequencies, i.e. possessed a large resolving power, observation of pulsations with its aid would be impossible.

3. RESOLVING POWER IN DEMODULATION ANALYSIS

The transformation of light into radio-frequency oscillations, discussed here, is analogous to demodulation in a radio receiver—the transformation of a radio-frequency oscillation, modulated by an audio frequency, into an audio-frequency oscillation. (Demodulation in a radio receiver is carried out, as is well known, for example, by means of the device shown in Fig. 4, with the oscillograph replaced by an audio-frequency amplifier and a loudspeaker, with \(RC\) small in comparison with the modulation period.) We may therefore speak of demodulation of light and call the analysis of light by means of apparatus reacting to \(\overline{E^2}\) demodulation analysis. Demodulation analysis makes it possible to transfer into the radio-frequency region the investigation of processes which, in ordinary spectral analysis, manifest themselves through the width and fine structure of lines.

Fig. 12.                  Fig. 13.

In this connection, as we shall show, the use of demodulation analysis may in principle be equivalent to increasing the resolving power of an optical spectral instrument by many orders of magnitude in comparison with existing records. The gain in resolving power (selectivity) possible as a result of demodulation is well known and widely used in radiophysics.

Let us first consider the simplest (ideal) case:

\[ E = A_1 \cos(\omega_1 t - \varphi_1) + A_2 \cos(\omega_2 t - \varphi_2), \tag{5} \]

where \(A_1, A_2, \omega_1, \omega_2, \varphi_1, \varphi_2\) are constants. On the basis of (3) we have

\[ \overline{E^2} = \frac{A^2 + B^2}{2} + AB \cos\{(\omega_1-\omega_2)t - (\varphi_1-\varphi_2)\}. \]

Figure 12 shows the spectra of the functions \(E(t)\) and \(\overline{E^2(t)}\). The latter consists of a line of frequency 0 and a line of frequency \(F = |\omega_1-\omega_2|/2\pi\).

Let us pass to the real case of an oscillation of the form (4), whose spectrum is continuous and has, for example, the form shown in Fig. 13. The mathematical theory of the transformation of a continuous spectrum under quadratic demodulation is contained in the works of Bernstein\(^8\) and Bunimovich\(^9\). Here it will suffice to restrict ourselves to rough estimates. Just as, when expression (5) is squared, there appears an oscillation—

... of the difference frequency \(|\omega_1-\omega_2|/2\pi\), when the Fourier expansion is squared

\[ E=\int A(\omega)\cos[\omega t-\varphi(\omega)]\,d\omega \]

there will appear oscillations of the form

\[ A(\omega_m)A(\omega_n)\cos\{(\omega_m-\omega_n)t-[\varphi(\omega_m)-\varphi(\omega_n)]\}, \]

where \(\omega_m,\ \omega_n\) are any two values of \(\omega\). As a result, a new continuous spectrum is formed, covering the frequency region from 0 up to frequencies of the order of \(\Delta f\), where \(\Delta f\) is the width of the initial continuous spectrum. If it is double-peaked (as in Fig. 13, \(a\)), then \(A(\omega_m)A(\omega_n)\) will have a maximum for \(\omega_m,\ \omega_n\) close respectively to the values of \(\omega\) in the peaks, and, consequently, the spectrum of the demodulated oscillation will have a maximum near the frequency equal to the distance \(F\) between the peaks (Fig. 13, \(b\)).

In order to recognize the double-peaked character (splitting) of the line, an optical apparatus with resolving power

\[ R>\frac{f}{F}. \]

is required.

In order to reveal this double-peaked character distinctly by means of analysis of the demodulated oscillation, a resonator is needed whose natural frequency can be varied near \(F\), and whose resonance-curve width \(\Delta F\) is small compared with \(F\), say equal to \(0.1F\). Even a very poor circuit with quality factor

\[ Q=\frac{F}{\Delta F}=10 \]

is thus equivalent, for \(F=10^9\) hertz and \(f=10^{15}\) hertz, to a very good spectral instrument with \(R=10^6\).

Let us pose the question differently. In order to detect directly a change \(\delta F\) in the splitting \(F\) (Fig. 13), an optical spectral instrument with resolving power

\[ R>\frac{f}{\delta F}. \]

is needed.

In order to detect this same change after demodulation, a resonator with a resonance curve narrower than \(\delta F\) is sufficient, i.e. with quality factor

\[ Q>\frac{F}{\delta F}. \]

Thus, for \(F=10^9\) hertz and \(f=10^{15}\) hertz, a cavity resonator with \(Q=10^4\) (which is easily attainable) is equivalent to an optical spectral instrument with

\[ R=\frac{f}{F}Q=10^{10}, \]

which is a thousand times greater than the resolving power of the best existing optical spectral instruments.

We are not speaking here of the considerable difficulties connected, for example, with the low intensity of fine lines, etc. But the estimate given shows that, in principle, there exists the possibility of studying the structure of lines in greater detail than is accessible at the present time.

Of course, a demodulated oscillation can be investigated not only by means of spectral decomposition by a resonator. In principle, an oscillographic study of the voltage obtained as a result of demodulation is possible. Then the statistics of the processes determining the width of the lines will appear as the statistics of fluctuations of the demodulated voltage. The study of these fluctuations may be carried out analogously to the study of fluctuations arising in electrical circuits themselves, by the method used by E. Ya. Pumper[^9].

Fig. 14.

A few words on the connection between demodulation analysis and radiospectroscopy (see, for example, the review by V. L. Ginzburg[^11]). The latter is based on the existence in radiating atoms of close energy terms \((A_1, A_2,\) Fig. 14), such that the difference of energies, divided by Planck’s constant, falls in the radio-frequency region. In this case the splitting \(A_1, A_2\) determines the carrier frequency \(f\) of the oscillations. If there are transitions \(BA_1, BA_2\), where \(B\) is a term far from \(A_1, A_2\), the same splitting \(A_1, A_2\) determines the modulation frequency \(F\) of an oscillation whose carrier frequency \(f\) is set by the distance \(BA\). Thus, one and the same frequency appears in radiospectroscopy as the carrier, and in demodulation analysis as the modulation frequency.

4. On Rigi’s “Light Beats”

Let us dwell briefly on the case of coherent oscillations. Coherence means that the ratio \(A(t)/B(t)\) and the difference \(\varphi(t)-\psi(t)\) do not depend on \(t\). Let, for simplicity, \(A(t)=B(t)\). Then formula (2) gives

\[ E^2=A^2[1+\cos(\Omega t-\varepsilon)]+\text{rapid terms}, \]

where

\[ \varepsilon=\varphi-\psi=\text{const}. \]

If \(T_1=2\pi/\Omega\) is small in comparison with \(T\), i.e. with the mean fluctuation time of \(A(t)\), one may repeat what was said in § 2. But let us suppose,

that \(T_1 \gg T\) and that the apparatus averages over a time small in comparison with \(T_1\), but large in comparison with \(T\). Such an apparatus gives

\[ \bar A^2[1+\cos(\Omega t-\varepsilon)], \]

i.e. pulsations of intensity with frequency \(\Omega\).

For example, in the case indicated in the objection which we discussed in § 2, if the mirror moves sufficiently slowly, such pulsations can simply be seen. This assertion is trivial: it means only that, if the path difference between the interfering beams is changed, the interference fringes are displaced. (Here \(T_1\) is the time during which the path difference changes by one wavelength.) To this reduces, as Wood\({}^{12}\) also noted, the idea of Righi’s experiments (1883), which entered many large expositions of optics, and with which the term “light beats” is often associated. Righi observed that in a certain optical arrangement, where light enters through a rotating polarizer, interference fringes moving in space are obtained. On the basis of the kinematic theorem that a linearly polarized oscillation with a uniformly rotating plane of polarization can be represented as the superposition of two oscillations polarized circularly and having different frequencies, this phenomenon can be interpreted as the result of beats of two light oscillations of different frequency. But in essence Righi’s experiment is devoid of content, since it adds nothing to what can be learned with a stationary polarizer: at each instant the position of the interference fringes is the same as if the polarizer had stopped.

Wood (loc. cit.) considers Righi’s experiments “not very instructive.” An exhaustive criticism of such experiments is given in the dissertation of S. M. Rytov\({}^{13}\). As Rytov wittily remarks in connection with the interpretation of one experiment concerning diffraction of light by a traveling ultrasonic wave, any quasi-static experiments serve no more as a test of the existence of light beats than does the everyday observed fact that the image of a moving object moves: one may say, considering the image according to Abbe, that we see a moving object moving as a result of beats between waves converging on the retina and diffracted by the object, which have undergone the Doppler effect upon diffraction.

This criticism does not apply to the experiments whose possibility has been discussed in this article.

5. CONCLUDING REMARK

If it proves possible, by means of demodulation, to detect pulsations of light intensity due to atomic or molecular phenomena (for example, the precession of atoms in a magnetic field), then such experiments will be on an equal footing with the ordinary “Fourier-conjugate” experiments ...

there, which consists in observing the splitting of lines in the spectral apparatus. One can say even somewhat more. The spectral apparatus gives the intensities, but not the phases, of the individual Fourier components. Therefore, decomposition into a spectrum gives less information about the course of the amplitude modulation in time than can be obtained by a direct investigation of the function \(\overline{E^2(t)}\).

This is especially clearly seen in the following example.

Suppose that we artificially interrupt light consisting of a single line of width \(\Delta f\), with a frequency \(F \ll \Delta f\) so small that spectral analysis reveals neither splitting nor even broadening of the line. Of course, an instrument with sufficiently small inertia will show that the light intensity is pulsating. From the point of view of Fourier decomposition, the pulsation of the intensity in time is connected here not with a change in the distribution of intensity over the spectrum, but with a change—due to the interruption—in the phase relations between the Fourier components.

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