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FRESNEL DIFFRACTION AT LARGE DISTANCES
A. F. Panasenkov
Fresnel diffraction phenomena may at the present time be considered sufficiently well studied, both theoretically and experimentally. However, despite numerous investigations in this field, the question of diffraction at large distances, of the order of hundreds and thousands of meters—not to mention astronomical distances—remained in the shadows until the very last years. Only in 1930 did papers on this question by Williams1 and Whitford2 appear in America.
In Williams’s work a theoretical analysis is given of the diffraction of the rays of a star by the disk of the Moon, an analysis experimentally carried out by Whitford.
The problem arose in connection with the determination of the angular diameters of stars. Although the linear diameters of stars are enormous, because of the immense distances of the stars their angular diameters prove to be smaller than the resolving power of modern telescopes. Therefore indirect methods are used to measure the angular diameters of stars.
In 1909 MacMahon3 proposed that the angular diameter of a star could be determined from the time during which it is occulted by the Moon. This may be shown as follows.
Suppose a star is located behind the dark disk of the Moon and we consider the stationary pattern of the distribution of illumination on the Earth. Since the star has finite dimensions, from the edge of the Moon we obtain a penumbra, which is shown once in Fig. 1 by hatching. Calculation of the distribution of illumination in the penumbra on the Earth gives:
\[ I(y)=i_0\frac{\pi R^2}{2}+i_0\left(\frac{a}{b}y-R\right) \sqrt{R^2-\left(\frac{a}{b}y-R\right)^2} +i_0R^2\arcsin\frac{\frac{a}{b}y-R}{R}. \tag{1} \]
Fig. 1. Distribution of illumination on the Earth in the penumbra of the Moon from the rays of a star. For \(\delta=0'',001\), the distance \(b\delta=1.9\ \text{m}\). For \(\delta=0'',004\), the distance \(b\delta=7.6\ \text{m}\).
Here \(a\) is the distance from the star to the Moon, \(b\) the distance from the Moon to the Earth, and \(i_0\) the brightness of the disk of the star.
The graphical distribution of illumination on the Earth, calculated from formula (1), is presented in Fig. 1. Denoting the angular diameter of the star \(\frac{2R}{a}\) by the letter \(\delta\), we write formula (1) in the form
\[ I(y)=\frac{I_0}{2}+\frac{I_0}{\pi}\left(\frac{2y}{b\delta}-1\right) \sqrt{1-\left(\frac{2y}{b\delta}-1\right)^2} +\frac{I_0}{\pi}\arcsin\left(\frac{2y}{b\delta}-1\right), \tag{1'} \]
where \(I_0=i_0\pi R^2\) is the luminous intensity of the star.
FRESNEL DIFFRACTION AT LARGE DISTANCES
If, at a definite moment during the passage of the Moon, one measures \(I\) at various points along the line \(Oy\), then one can choose such a parameter \(\delta\) that it will make it possible to represent the obtained curve of the distribution of illumination \(I(y)\) by formula (1′). Conversely, placing at some point \(A\) on the line \(Oy\) an apparatus that records at various instants of time \(t\) the illumination at this point \(A\), one obtains \(I(t)\); since \(y=vt\), where \(v\) is the velocity of the shadow’s motion over the Earth, \(I(vt)=I(y)\) is readily found from \(I(t)\). In all other respects the procedure is as before.
Eddington\(^4\) showed that, when angular diameters are measured by this method, it is necessary to take into account diffraction from the edge of the Moon; for measurements to be possible, according to Eddington, a star must have an angular diameter of not less than \(0''.008\); therefore the approach to the problem of determining the angular diameters of stars from the standpoint of geometrical optics is, according to Eddington, completely hopeless, except in the case of stars of very large angular diameter. Analyzing the problem of diffraction of the rays of a star by the disk of the Moon, Williams assumes that the diffraction from the edge of the lunar disk will be the same as from a plane screen perpendicular to the line of sight and situated so that its edge is tangent to the lunar disk at the point where diffraction appears. Then this problem is reduced to the simple problem of diffraction at a straight edge, which was solved theoretically and experimentally by Fresnel\(^5\).
In the case of a point source of light, the intensity is zero inside the geometrical shadow of the screen, then, near the edge, it increases, reaching at the edge of the geometrical shadow \(1/4\) of the intensity of the free wave, and outside the geometrical shadow oscillations of intensity appear, which become smoothed out, and far from the shadow’s edge the intensity is equal to the intensity of the free wave. The indicated distribution of intensity is expressed by the following formula:
\[ I=\frac{I_0}{2}\left(C^2+S^2\right), \tag{2} \]
where
\[ C(v)=\int_0^v \cos\frac{\pi v^2}{2}\,dv'=\frac{1}{2}\pm\int_0^v \cos\frac{\pi v^2}{2}\,dv, \]
\[ S(v)=\int_0^v \sin\frac{\pi v^2}{2}\,dv=\frac{1}{2}\pm\int_0^v \sin\frac{\pi v^2}{2}\,dv. \]
Here the parameter \(v\) is related to the arc \(s\) of the wave front by the formula
\[ v=s\sqrt{\frac{2(a+b)}{ab\lambda}}, \]
where \(\lambda\) is the wavelength of the light source; the upper signs correspond to points outside the shadow, the lower signs to points inside the shadow; the intensity of the free wave is \(I_0=1\).
These Fresnel integrals have been calculated and are given in tables for values of the argument \(v\) differing by \(\Delta v=0.1\). The curve of the distribution of light intensity calculated by formula (2) is shown in Fig. 2, which represents a modification of Fresnel’s well-known experiment with diffraction at the sharp and blunt edges of a razor.
Fig. 2 is taken from the work of V. K. Arkadiev\(^6\). Presented here are four strips cut out from diffraction photographs obtained from the edge of a screen with different radii of curvature. The upper photograph \(2a\) represents the diffraction shadow of the sharp blade of a razor with a radius of curvature less than \(1\mu\). Photograph \(2b\) refers to the edge of a screen formed by a glass rod …
...with a radius of curvature of 3.83 mm. The following strips, Figs. 2c and 2d, represent the shadow of a screen whose radius of curvature was equal to 40 m. In photograph 2c the surface of the screen was smooth and shiny (glass), while in photograph 2d it was covered with soot.
It is not difficult to see that the arrangement and relative brightness of the diffraction bands in all four cases is one and the same; neither depends either on the radius of curvature or on the material of the edge. Therefore Williams’ simplification, consisting in replacing a part of the Moon’s disk by the edge of a flat screen, is quite possible.
Fig. 2. Diffraction from the straight edge of a screen of various radii of curvature
Fig. 3. Diffraction curve from the edge of the Moon’s disk, calculated for a point source and for a star of angular diameter \(\delta = 0'',004\)
For the case under consideration, schematically represented in Fig. 3, we have:
\(b = 3.8 \cdot 10^8\) cm — the distance from the Earth to the Moon, \(\lambda = 4.3 \cdot 10^{-7}\) m \(= 4300\) Å — the wavelength of light corresponding to the maximum photographic action on the plate.
The distance from the star to the Moon is many times greater than the distance from the Moon to the Earth, so that one may write
\[ a \gg b. \]
If by means of \(x\), connected with the parameter \(v\) by the formula
\[ x = v \sqrt{\frac{b(a+b)}{2a}} \cong v \sqrt{\frac{b\lambda}{2}}, \tag{3} \]
we denote the distance in meters from the geometrical edge of the shadow, then for the positions of the first maxima and minima we obtain the following values of \(x\):
| maximum | 11.0 | 21.2 | 27.8 m |
| minimum | 16.9 | 24.7 m |
Plotting these values on a graph, Williams obtains the diffraction curve from the edge of the Moon’s disk. On this curve, shown in Fig. 3, the width of the diffraction bands is of the order of 10 m.
Formulas (2) and (3) are valid for a point source of light. A star, however, has although a small angular diameter, nevertheless a finite one. Consequently, the law of variation of the light intensity must change. The corresponding
curve for a light source with an angular diameter of \(0''.004\) is given in the same Fig. 3. Comparing the curves of Fig. 3, one may note that the positions of the maxima and minima in the diffraction pattern coincide, but the difference in intensity is considerable.
The intensity distribution for a point source was studied by Lyot[^8]. With the aid of a microphotometer he measured the blackening on a photograph of the diffraction pattern and then calculated from these data the ratio of the intensity of the first maximum to the intensity of the following maxima and minima. The results proved to be in agreement with Fresnel’s theory.
Experimental observation of the diffraction of the rays of a star by the lunar disk should consist in measuring and recording the varying intensity of the light from the star on the Earth at the moment of eclipse; at this time the calculated velocity of the diffraction fringes is about \(500\ \text{m/sec}\).
It is clear that this observation can be made only, first, by using a sufficiently sensitive instrument, such as a photocell, and, second, by using an instrument sufficiently inertia-free for recording oscillations, such as a cathode oscillograph.
Having used such a fast-acting photoelectric system, Whitford in 1938 carried out this experiment at the Mount Wilson Observatory. As light sources there were taken the stars \(\beta\)-Capricorni and \(\gamma\)-Aquarii, which in September 1938 were subject to eclipse by the Moon.
Fig. 4. Diagram for observing the diffraction of the rays of a star by the edge of the lunar disk
Fig. 5. Change in the intensity of light at the moment of the beginning of the eclipse
According to the description, the observational arrangement may be represented as shown in Fig. 4. Here the light of the star \((S)\), diffracted at the edge of the lunar disk, was collected by the 100-inch (258 cm in diameter) reflector \(R\) and directed onto the photocell \(Ph\). The voltage of the resulting photocurrent was taken from the resistance \(r\), amplified by a four-stage amplifier \(M\), and fed to the vertically deflecting plates of a 3-inch (7.5 cm) cathode oscillograph \(Os\). The vertical oscillations of the luminous spot were photographed on a moving film, which was wound on a drum rotating on an axis with a screw thread. The speed of the film was \(450\ \text{mm/sec}\).
The observed curve of the change in light, obtained by photographing the oscillograph screen, is shown in Fig. 5 as a segment of 5 cm from a continuous strip 90 cm long, wound on the rotating drum. Time increases from right to left and from bottom to top.
From this picture it is seen that before the beginning of the eclipse the spot traces on the oscillograph a uniform illumination with small distortions, which may be attributed to the action of atmospheric waves with a 100-inch reflector. A sharp distortion of the pattern appears at the moment when the star’s eclipse begins, i.e. when the star is behind the dark disk of the Moon. The points of the diffraction curve of the star’s rays from the lunar disk observed in this way are presented in Fig. 6 in the form of circles.
For comparison of the experimental result with theory, two more curves are plotted in the same figure. The solid line represents the diffraction-
curve for a point source of light. It was computed by the method of Cornu’s spiral for the broad range of wavelengths emitted by the star. The dotted line represents the diffraction curve from the star β Capricorni. Its computed angular diameter was equal to \(0''.001\). It is evident that the difference between these curves is insignificant. Comparison of the points of the observed curve with the theoretical one gives complete satisfactory agreement. The width of the diffraction bands, as in Williams’s calculations, proved to be of the order of \(10\) m.
Fig. 6. Comparison of the computed and observed diffraction curve from the edge of the Moon’s disk. Motion of the telescope from right to left.
It should be noted that the occultation took place on a comparatively smooth portion of the lunar surface, since neither diffraction nor irregularities of mountains or other inequalities of the Moon took part in it. For stars with diameters greater than \(0''.005\), this method can be applied for direct measurement of their diameters. For this it is sufficient to compare the observed diffraction curve with curves computed for various angular diameters. The identity of the curves being compared will indicate the size of the star.
The calculation of diffraction phenomena for large distances of several kilometers was carried out in the above-mentioned article by V. K. Arkadiev in 1912 with the aid of the law of similarity of diffraction figures.
Fig. 7. Sharp shadow of a disk, with diameter \(d_1 = 21\) cm, at a distance \(a + b = 3\) m.
Starting from Lommel’s theory\(^9\) of diffraction by a disk, one may conclude that geometrically similar diffraction patterns occur in the case when the functions
\[ Y = 2\pi n = \frac{2\pi}{\lambda}\,\frac{a+b}{ab}\,r^2 \]
and
\[ Y_1 = 2\pi n_1 = \frac{2\pi}{\lambda_1}\,\frac{a_1+b_1}{a_1 b_1}\,r_1^2 \tag{4} \]
FRESNEL DIFFRACTION AT LARGE DISTANCES
will be identical. Here \(n\) is the number of selected Huygens–Fresnel zones, \(d=2r\) is the diameter of the disk, and the remaining quantities are the above-mentioned parameters.
Equating these functions, we have
\[ \frac{1}{\lambda}\frac{a+b}{ab}\,r^2 = \frac{1}{\lambda_1}\frac{a_1+b_1}{a_1 b_1}\,r_1^2 . \]
Imposing the condition
\[ \frac{a_1}{a}=\frac{b_1}{b}=K \quad \text{and} \quad \lambda_1=\lambda, \tag{5} \]
we obtain:
\[ \frac{a_1}{a}=\frac{b_1}{b} = \frac{a_1+b_1}{a+b} = \frac{d_1^2}{d^2}, \tag{6} \]
i.e., under the condition that the wavelength of the light source is preserved, the diffraction patterns will be similar if the distances are in the same ratio as the squares of the diameters of the circles.
Fig. 8. Diffraction from a disk of diameter \(d_1=21\ \text{cm}\),
at a distance \(a+b=150\ \text{km}\)
To illustrate this conclusion, in Fig. 7 a photograph from the cited article by V. K. Arkadiev is presented, showing the shadow of a disk of diameter \(d_1=21\ \text{cm}\) at a distance \(a+b=3\ \text{m}\). If, however, one takes the distance \(a_1+b_1=150\ \text{km}=15\cdot 10^4\ \text{m}\), then instead of a sharp shadow we obtain the diffraction pattern shown in Fig. 8.
The latter photograph was obtained by enlarging a photograph taken at a distance \(a+b=30.6\ \text{m}\) from a small model of a hand with a disk of diameter \(d=3\ \text{mm}\), cut out of thin tin. According to relation (6), the picture was enlarged 70 times, since the model of the disk was smaller than the natural one by
\[ \frac{d_1}{d}=70 \]
times.
Now, after Wightford’s observations of diffraction at astronomical distances, the reality of photographs such as Fig. 8 cannot be in doubt.
References
- J. L. Williams, Astroph. Journ., 89, 4, 1939.
- A. E. Whitford, Astroph. Journ., 89, 4, 1939.
- Mac Mahon, Monthly Notices, London, 63, 126, 1909.
- A. S. Eddington, Monthly Notices, London, 69, 178, 1909.
- A. Fresnel, Oeuvres complètes, 1, 232, 239, 1866.
- V. K. Arkad’ev, ZhRFO, 44, 145, 1912; Phys. Z., 14, 832, 1913.
- Rayleigh, The Wave Theory of Light, Moscow–Leningrad, GTTI, 1940, p. 94.
- Th. Lyman, Proc. Nat. Acad. Sci., 16, 71, 1930.
- E. v. Lommel, Abh. d. Bayr. Ak. d. Wiss., 15, 231, 1886.