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LETTERS TO THE EDITOR
ON THE REVIEW “MODULATION INTERFEROMETRY”
In issue 4 of volume XLVII of UFN a review by G. Rozenberg was published under the above title*). In this review, in particular, an experiment carried out by Brusin, Gorelik, and Pikovsky¹ is discussed, in which oscillatory displacements of a membrane were observed. The authors indicate that the minimally detectable displacements of the membrane with their very imperfect apparatus were approximately equal to 1 Å.
It may perhaps be of some interest to add to the above-mentioned review a theoretical estimate of the minimally detectable mechanical displacements (more generally: changes in the optical path length of a light beam) when two-beam interference and a photomultiplier with the corresponding radio apparatus are used.
Let one of the interfering oscillations be \(\sqrt{2A_1}\sin\omega t\), and the other be
\[ \sqrt{2A_2}\,\sin\left[\omega t+\frac{\omega}{c}(L+l)\right]. \]
Here \(c\) is the speed of light, \(L\) is the path difference of the two light beams, and \(l\) is its very small (in comparison with the wavelength) change. Let \(l=l_0\sin\Omega t\) \((\Omega\ll\omega)\).
The intensity of the resulting oscillation is equal to
\[ A_1+A_2+2\sqrt{A_1A_2}\cos\left[\frac{\omega}{c}(L+l)\right]. \tag{1} \]
The value of the photocurrent is obtained by multiplying the light intensity by a certain coefficient \(\alpha\).
Let us choose the parameters of the setup so that
\[ \frac{\omega L}{c}=2\pi\left(n+\frac14\right),\quad |n|=0,1,2,\ldots \]
Then the mean value of the photocurrent will be
\[ I_0=\alpha A_1+\alpha A_2=I_1+I_2, \tag{2} \]
where \(I_1\) and \(I_2\) are the photocurrents produced by each light beam separately. The amplitude of the variable component of the photocurrent is then equal to
\[ (\Delta I)_{\max}=2\sqrt{I_1 I_2}\cdot\frac{\omega}{c}\,l_0. \tag{3} \]
*) In the named review, through an oversight by the author, the surname of Comrade I. L. Bershtein was distorted, for which the editors apologize.
The intensity of the useful effect is, thus, proportional to the quantity
\[ \overline{(\Delta I)^2}=\frac{(\Delta I)^2_{\max}}{2}=2 I_1 I_2 \left(\frac{\omega l_0}{c}\right)^2 . \tag{4} \]
Let us consider how the effect changes as a result of the nonmonochromaticity of the light. For simplicity, suppose that the spectral density is constant in the frequency band \(\Delta\omega\), with mean frequency \(\omega_0\), and is equal to zero outside this band. Elementary calculations show that in this case the intensity of the resultant oscillation will, instead of (1), be determined by the expression
\[ A_1+A_2+2\sqrt{A_1A_2}\, \frac{\sin \dfrac{\Delta\omega(L+l)}{2c}} {\dfrac{\Delta\omega(L+l)}{2c}} \cos\left[\frac{\omega_0}{c}(L+l)\right], \tag{5} \]
where \(A_1\) and \(A_2\) are still the intensities of each of the interfering beams of light.
If the equality
\[ \frac{\omega_0 L}{c}=2\pi\left(n+\frac14\right) \]
is satisfied, and assuming that the sensitivity of the photocathode (i.e. the quantity \(a\)) is constant within \(\Delta\omega\), we obtain for the mean value of the photocurrent the previous expression (2), and for the amplitude of the variable component of the photocurrent, instead of (3),
\[ (\Delta I)_{\max} = 2\sqrt{I_1I_2}\, \frac{\sin \dfrac{\Delta\omega L}{2c}} {\dfrac{\Delta\omega L}{2c}} \cdot \frac{\omega_0}{c}\,l_0 . \tag{6} \]
If one ensures that
\[ \frac{\Delta\omega L}{2c}\ll \pi \]
(the condition of “interference in white light”), then (6) practically coincides with (3).
The minimum value \(l_0\) accessible to observation is determined by the level of fluctuations in the circuit. This level in amplifier systems with a photomultiplier at the input can be calculated (see, for example, the review\(^2\)) by the formula
\[ \overline{I_{\text{sh}}^{\,2}}=2(1+B)eI_0\Delta f . \tag{7} \]
Here \(I_{\text{sh}}\) is the fluctuation of the photocathode current, \(e\) is the electron charge, \(I_0\) is the mean value of the photocathode current, \(\Delta f\) is the amplifier pass band, and \(B\) is a quantity that may be taken equal to 1.5. We shall neglect the dark current.
Equating the useful effect, determined by expression (4), to the fluctuation intensity (7), we find the minimum detectable value of \(l_0\). Having done this, substituting the value of the electron charge and also introducing the wavelength
\[ \lambda=\frac{2\pi c}{\omega_0}, \]
we obtain:
\[ l_{0\min}=10^{-10}\sqrt{\frac{I_0\Delta f}{I_1I_2}}\,\lambda . \tag{8} \]
Putting, for simplicity, \(I_1=I_2=\frac12 I_0\), we shall have:
\[ l_{0\min}=2\cdot10^{-10}\sqrt{\frac{\Delta f}{I_0}}\,\lambda . \tag{9} \]
The minimally detectable change in the path difference of the interfering beams depends on their intensity. With ordinary optical apparatus and modern photocathodes, it is not difficult to obtain a value \(I_0\) of the order of \(10^{-8}a\) (and even greater).
If, at the output of the radio apparatus, one applies a narrow-band aperiodic circuit (an output pointer instrument with a time constant of 1 sec may serve as such), then this will correspond to an effective frequency band \(\Delta f = 0.25\) cps.
Substituting, for example, in (9) \(I_0 = 10^{-8}a\), \(\Delta f = 0.25\) cps, and \(\lambda = 5000\) Å, we obtain:
\[ l_{0\min} = 5 \cdot 10^{-3}\ \text{Å}. \]
In the usual arrangement of interference experiments, displacement of the mirror by a certain amount causes a change in the path of the beam of light by twice that amount. Therefore the minimally detectable displacement of the mirror will be half the value determined above.
I. L. Bershtein
References Cited
- I. Ya. Brusin, G. S. Gorelik, S. A. Pikovsky, DAN 83, 4, 553 (1952).
- N. O. Chechik, UFN 37, issue 1, 74 (1949).