ZONE PLATE

S. M. Raiskii

For eighty years the zone plate has continued to attract the attention of physicists. This is explained in part by the fact that the theory of the zone plate is closely connected with general questions of wave propagation.

Of no lesser importance is also the circumstance that the focusing and dispersive properties of the zone plate serve as an effective illustration of the laws of diffraction and interference[^1].

The calculation of a zone plate, as is well known, is carried out according to the following scheme:

Let us place, between the source and the observer, a surface of arbitrary shape. On this surface let us create transparent and opaque zones. The boundaries of the transparent zones are chosen so that, for points situated inside these zones, the sums of the distances to the source and to the observer differ from one another by an integral number of wavelengths, with an accuracy up to \(\pm \lambda/4\). Then at the point of observation a bright image of the source will be obtained, since in this case, according to the Huygens–Fresnel principle, there is a superposition of the amplitudes of the oscillations produced by the active portions of the front of the incident wave.

The topography of the Fresnel zones on the chosen surface depends on its shape and on the position of the surface relative to the source and the observer.

As for the shape of the zone plate, then, so far as we know, in the investigations published up to the present time only plane plates operating in transmitted waves have been considered.

The theory of a plane zone plate and the characteristics of contemporary examples of such plates are set forth in the first part of the present article. This part is a review of a recently published work[^3].

In the second part of the article are given previously unpublished results of our work concerning a spherical zone plate operating in reflected and transmitted light, in combination with a mirror or a lens.

I. ZONE PLATES OPERATING IN TRANSMITTED LIGHT

1. Optical properties of a zone plate

If, for some point \(P\) lying on the principal optical axis, the distances to neighboring transparent zones differ by one wavelength, then at this point an increase in the intensity of light will be observed.

The distance \(b\) between the zone plate and the point \(P\), when the plate is illuminated by a parallel beam of light, is called the principal focal distance for the given wavelength \(\lambda\). From this it is easy to calculate the radius of the \(k\)-th zone. Let \(R_k\) be the distance from \(P\) to the outer boundary of the \(k\)-th zone (counting both transparent and opaque zones).

Then

\[ R_k=b+\frac{1}{2}k\lambda. \]

For the radius of the \(k\)-th zone we have:

\[ r_k=(R_k^2-b^2)^{\frac{1}{2}} =\left(k\lambda b+\frac{1}{4}k^2\lambda^2\right)^{\frac{1}{2}}. \tag{1} \]

Usually \(b \gg k\lambda\). Then, with high accuracy,

\[ r=(kb\lambda)^{\frac{1}{2}}. \tag{2} \]

This relation is used in the manufacture of zone plates.

Optical zone plates can be obtained by photographing, on a reduced scale, the corresponding drawings. A more perfect method consists in photographing the pattern of Newton’s rings. In this way it is possible to make zone plates of satisfactory quality, having a diameter up to \(30\) mm and a principal focal distance of approximately \(1\) m. The area of the \(k\)-th zone is equal to \(\pi r_k^2-\pi r_{k-1}^2\), or approximately \(\pi b\lambda\).

Since a plane plate with ring radii proportional to the square roots of integers only approximately reproduces the dimensions of Fresnel zones, as the number of rings on the plate increases we shall obtain, beginning from some point, a noticeable deterioration in image quality. Only in the case of a parallel beam of light propagating along the axis of the zone plate does it make sense to increase without limit the number of rings on its surface.

For the case of a point source located at a finite distance \(p\) from the zone plate on its axis, it is easy to calculate

ZONE PLATE

permissible number of transparent zones. From Fig. 1 it follows that the optical path

\[ S=(p^2+r_n^2)^{\frac12}+(q^2+r_n^2)^{\frac12}, \tag{3} \]

where \(q\) is the focal distance of the plate.

Let us take into account that

\[ r_n=(2bn\lambda+n^2\lambda^2)^{\frac12}, \]

where \(n=\frac{1}{2}k\) is the number of transparent zones, and expand \(S\) in powers of \(\lambda\). Restricting ourselves to terms containing \(\lambda^3\), we obtain for the path length corresponding to the \(n\)-th transparent Fresnel zone:

\[ S=p+q+n\lambda b\left(\frac{1}{p}+\frac{1}{q}\right) +\frac{1}{2}n^2\lambda^2\left(\frac{1}{p}+\frac{1}{q}-\frac{b^2}{p^3}-\frac{b^2}{q^3}\right). \tag{4} \]

If we use a zone plate made according to (2), then for the \(n\)-th transparent zone the path length \(S_0\) differs from \(S\) and is equal to

\[ S_0=p+q+ \]

\[ +n\lambda b\left(\frac{1}{p}+\frac{1}{q}\right). \tag{5} \]

Fig. 1. Scheme of the course of rays in the formation of the image of a point source \(A\), located on the principal axis at a finite distance \(p\) from the zone plate. The image is located at point \(B\) at a distance \(q\) from the plate.

But in order to obtain good image quality, the optical paths of all rays converging at the focus must differ from an integer number of wavelengths by an amount not exceeding \(\pm\lambda/4\) (Rayleigh criterion). In our case the length of the optical path of the ray \(AC+CB\), going to the focus through the \(n\)-th transparent zone (Fig. 1), must lie within the limits

\[ p+q+n\lambda\pm\frac{\lambda}{4}. \tag{6} \]

Assuming that approximation (4) gives for \(S\) a value sufficiently close to \(p+q+n\lambda\), we must accept that increasing the number of rings on a zone plate satisfying condition (2) makes sense only as long as

\[ |S_0-S|<\frac{\lambda}{4}. \]

Hence, after simple transformations, we obtain for the limiting number of zones:

\[ n<\left[\left|p^3/6\lambda(p-b)\right|\right]^{\frac12}. \tag{7} \]

Equation (7) shows that for the case of a parallel beam,

incident on the zone plate perpendicularly \((p=\infty)\), the number of zones is not limited.

Since approximately \(S=S_0\), then, comparing (5) and (6) and discarding the small quantity \(\lambda/4\), it is easy to see that for a zone plate, as well as for a lens,

\[ \frac{1}{p}+\frac{1}{q}=\frac{1}{b}. \tag{8} \]

Let us now consider the case of a parallel beam incident obliquely on a zone plate, for which \(r_n^2=2n\lambda b\). Let the principal axis of the plate and the incident ray form some angle \(\alpha\). We shall determine the greatest value of the angle \(\alpha\) for which good focusing is still possible for the given wavelength \(\lambda\). To do this, the geometrical path length, calculated to terms containing the first power of \(\lambda\), must be equated to \(b+n\lambda-\lambda/4\). The final result may be represented in the form:

\[ \left|-(2bn\lambda)^{\frac{1}{2}}\cdot\frac{n\lambda}{b}\sin\alpha-n\lambda\sin^2\alpha\right|<\frac{\lambda}{4}. \tag{9} \]

The next important characteristic of a zone plate is its resolving power. As usual, the problem reduces to determining the angle between the principal axis and the line connecting the center of the plate with the first minimum of the annular diffraction pattern obtained from a point source located on the principal axis of the plate. Let the plate be illuminated by a monochromatic beam of light traveling along the principal axis. In the formation of the principal focus lying on the principal axis, all Fresnel zones participate with their entire area. In another position there are points somewhat displaced from the axis of the plate. Rays arriving at these points from some parts of a transparent zone can interfere with rays arriving from other parts of it.

A quantitative calculation of the intensity distribution near the principal focus can be carried out in the following way: the difference in the lengths of the optical paths (Fig. 2) \(AB\) and \(AC\) is equal to

\[ \Delta l \simeq -b\,\tg\beta\cdot\sin\varphi. \tag{10} \]

But for small angles \(\tg\beta=\beta\) and \(r=r_n=b\tg\varphi\simeq b\sin\varphi\). Taking (2) into account, we obtain:

\[ \Delta l\simeq -r\beta\simeq -\beta(2n\lambda b)^{\frac{1}{2}}. \]

Equation (10) is valid for parts of zones lying in the plane of the drawing (Fig. 2). For other parts of the zones the quantity \(r\) must be replaced by \(r\sin\theta\), where \(\theta\) is the angle formed by the straight line connect-

connecting the point \(O\) with a point on the circumference, and a straight line perpendicular to the plane of the drawing.

If the phase of the oscillation at the point \(B\), due to the light coming from the point \(O\), is taken to be zero, then the phases at \(B\) due to the action of other parts of the zone plate will be equal to

\[ -\frac{2\pi}{\lambda}\,\beta r \sin\theta . \tag{11} \]

Denote

\[ \frac{2\pi}{\lambda}\,\beta r=\rho . \]

Then the resultant amplitude at \(B\), arising as a result of the action of the zone of radius \(r\), is proportional to

\[ \frac{2}{\pi}\int_{0}^{\pi/2}\cos(\rho\sin\theta)\,d\theta=I_0(\rho), \tag{12} \]

where \(I_0(\rho)\) is the Bessel function of order zero.

Since the intensities due to each of the rings are approximately identical, the amplitude at the point \(B\) is equal to the sum of the amplitudes:

\[ \sum_{1}^{n} I_0\!\left(\frac{2\pi}{\lambda}\cdot\sqrt{2m\lambda b}\right). \tag{13} \]

Fig. 2. Diagram of the path of rays for the case of an infinitely distant point source located outside the principal axis of the zone plate. \(B\) is the position of the image, \(C\) is the point at which the image would be formed if rays parallel to the principal axis were incident on the plate.

By a series of transformations it can be shown that, approximately, the resultant amplitude first becomes zero when

\[ \frac{2\pi\beta}{\lambda}\cdot[2(n-1)b\lambda]^{1/2}\simeq \frac{2\pi\beta r_n}{\lambda}=3.8 . \]

Hence

\[ \beta=\frac{1.22}{2r_n}=1.22\,\frac{\lambda}{d}. \tag{14} \]

Thus, the resolving power of the zone plate—the angular half-width of the first maximum—is equal to (14), as also for a lens with the same aperture.

The theory of the zone plate set forth above is approximate. Therefore the optical properties of real plates, however well the plates may have been made, may differ somewhat from the properties predicted by the elementary theory.

2. Experiments with zone plates

In the work³ the zone plate used was a photographic image of Newton’s rings, taken on spectroscopic plates resolving approximately 1000 lines per millimeter. To obtain the rings, a biconvex lens of diameter 10 cm, with radius of curvature 164 cm, was used. Illumination was produced by a parallel beam of monochromatic rays with wavelength 5461 Å (a mercury lamp with a filter).

In this way it was possible to make zone plates having more than 300 transparent zones.

To verify the dependence of the principal focal distance on wavelength:

\[ b=\frac{r_n^2}{2n\lambda} \tag{15} \]

a plate with an effective aperture of 17.5 mm and \(n=100\) transparent zones was used. Measurements by means of a comparator showed that the radii of the zones of this plate are inversely proportional to the square roots of successive integers. The focal distance was determined for various wavelengths both in the visible and in the near ultraviolet regions of the spectrum. It turned out that, with strict parallelism of the incident beam, equation (15) is confirmed within the limits of experimental error. In addition to the intense focus of the first order, the zone plate must give less intense secondary foci of the third, fifth, seventh, and other odd orders, corresponding to the coincidence of three, five, seven, etc. Fresnel zones with each of the transparent rings on the surface of the plate.

The actions of two, four, six, etc. Fresnel zones in these cases mutually annihilate one another. One uncompensated Fresnel zone participates in producing the illumination. The secondary foci are located respectively at distances \(\frac{b}{3}\), \(\frac{b}{5}\), \(\frac{b}{7}\), \(\frac{b}{9}\), etc. from the zone plate. The intensities of the images in these foci are, obviously, as the reciprocals of the ordinal numbers of the secondary foci. The difference in the lengths of the optical paths for rays going to the secondary foci through neighboring transparent rings of the plate is equal to \(3\lambda\), \(5\lambda\), \(7\lambda\), etc. (for the focus of the first order the path-length difference is equal to \(\lambda\)). Along the main optical axis, at distances \(-b\), \(-\frac{b}{3}\), \(-\frac{b}{5}\), etc., there are virtual foci of the zone plate.

Theoretically, foci of even order should not arise in a zone plate, since the actions of two, four, six, etc. Fresnel zones coinciding with each transparent ring of the plate mutually annihilate one another. In the experiment, however, for all the zone plates investigated, foci of the second

order, located approximately halfway between the plate and the focus of the first order. This phenomenon is connected in part with the inequality of the areas of the transparent and opaque rings on the surface of the plate. The total width of two adjacent Fresnel zones differs somewhat from the width of the corresponding transparent ring. As a result, in the focus of the second order the actions of adjacent Fresnel zones are not completely compensated and produce a partial increase in intensity. To compare the light intensity of the zone plate and the lens, an image of a slit was photographed in roughly monochromatized light. A plane diffraction grating was used to isolate a portion of the spectrum. The grating was illuminated by a parallel beam coming from a collimator with a quartz lens. The zone plate replaced the camera lens. The diameters of the optical components being compared—the lens and the plate—were 30 mm, and the focal lengths were, respectively, 85 cm and 90 cm. The photographs obtained are shown in Figs. 3 and 4. The exposures were 8 sec. for the plate and 0.5 sec. for the lens. It turned out that a zone plate with a relative aperture of 1:28 is, in its action, approximately equivalent to a lens of 1:120.

A considerably better image quality was obtained when the diffraction grating was replaced by a filter. This was done in experiments on the investigation of aberration. The light source was a mercury lamp with a filter for the 5461 Å line. The collimator was equipped with two slits, the images of which were photographed by means of a zone plate with an aperture of 17.6 mm (100 transparent zones) and a focal length of 66 cm. Photographs were obtained with the double slit in the central position and with it displaced in a direction perpendicular to the principal optical axis. The photographs showed that good focusing is observed for a displacement from the axis not exceeding 3°18′. At larger angles the image of the slits becomes noticeably blurred (Figs. 5 and 6). The theoretical limiting angle, calculated from formula (9), is 3°15′.

A comparison of the properties of the zone plate and of a converging lens is given in Table I (see p. 524).

Here it should be noted that the chromatism of a zone plate is connected with the method used for making the zone diaphragm—formula (1). It is possible to indicate conditions under which a zone diaphragm will be free of chromatic aberration for several specified wavelengths. Let us consider two plates with the same focal length, of which one is calculated for the wavelength \(\lambda_1\), and the second for \(\lambda_2\). Let us place the plates together, aligning their optical axes, and determine the positions of the transparent portions common to both plates. This problem is readily amenable to calculation if one takes into account that the distances from the center of the plates to any points lying within the transparent zones

Fig. 3.

Fig. 4. Photographs of the triplet of mercury lines with wavelengths 3663, 3654, and 3650 Å. The images were obtained using a zone plate (Fig. 3) and a concavo-convex lens of the same aperture (Fig. 4).

Fig. 5.

Fig. 6.

Aberration observed when the source (two slits) is shifted from the principal axis of the zone plate. The photographs were obtained with the slits illuminated by monochromatic light (5461 Å). Fig. 5 corresponds to a shift of the slits from the axis by an angle of 2.9°, Fig. 6—to a shift of 4.1°.

S. M. RAISKIY

Table I

Comparison of the properties of a zone plate and a lens

of the diaphragms under consideration are respectively equal to:

$$ r_n^2(\lambda_1)=bn\lambda_1,\qquad r_n^2(\lambda_2)=bn\lambda_2, $$

where \(n\) may take all values from 0 to 1, from 2 to 3, from 4 to 5, etc., or all values from 1 to 2, from 3 to 4, from 5 to 6, etc.

Let us now make a plate with such an arrangement of transparent regions as was obtained for our system consisting of two superposed zone diaphragms. This plate will evidently be achromatic for the wavelengths \(\lambda_1\) and \(\lambda_2\). By the method described one can obtain a plate with a common focus for several wavelengths. Of course, the achromatization of the plate is associated with a decrease in its luminosity.

II. SPHERICAL ZONE PLATE COMBINED WITH A CONCAVE MIRROR OR WITH A CONVERGING LENS

1. Concave reflecting zone plate

Turning to Fig. 7, we see that for a plane zone plate operating in transmission, the path difference of the central ray \(AOM\) and the noncentral ray \(ALCKM\) is determined by the sum of the lengths of the segments \(LC\) and \(CK\). Because of the different sign of curvature of the wave fronts for the source \(A\) and the observer \(M\), an increase of the optical path by an amount equal to \(\lambda/2\) is achieved by a very small increase

angle \(\alpha\). Therefore the path difference, expressed in \(\lambda/2\) (the number of Fresnel zones), assumes very large values at comparatively small angles \(\alpha\), if, of course, the distances \(a\) and \(b\) are not very great (lie within tens of centimeters).

To obtain a high-aperture plate having wide Fresnel zones at a large aperture and a small focal distance, it is natural to resort to such an arrangement of the source and observer in which the curvature of the fronts \(A\) and \(M\) has the same sign (Fig. 8). This is easily accomplished by means of a reflecting concave mirror whose center \(3\) is located

Fig. 7. \(A\) — source, \(M\) — point of observation, \(O\) — middle of a plane zone plate, \(C\) — point on the plate situated at a distance \(r\) from the middle of the plate, \(a\) and \(b\) — distances of the source and the point of observation from the plate.

Fig. 8. \(A\) — source, \(M\) — point of observation, \(O\) — middle of the concave mirror \(3'\), having its center at \(3\), \(A'\) and \(M'\) — spherical surfaces with radii \(OA\) and \(OM\), \(r\) — distance of the point \(C'\) on the surface of the mirror from the axis \(OA3M\), \(q\) — projection of \(OC'\) onto the axis \(OA3M\).

between \(A\) and \(M\). Obviously, now the path difference for the central and noncentral rays \(\Delta S = AOM - AL'C'K'M\) is determined by the difference of the lengths of the segments \(S = L'C\) and \(S_1 = C'K\), each of which is shorter than the segments \(LC\) and \(CK\) corresponding to the case of a plane zone plate. As \(A\) and \(M\) approach the center of the mirror \(3\), the path difference \(\Delta S\) tends to zero and the central Fresnel zone gradually spreads over the entire surface of the mirror.

With the aid of the drawing given in Fig. 8, it is easy to obtain the relations necessary for determining the path difference \(\Delta S = S - S_1\):

\[ S = -a + \sqrt{a^2 - 2q(a - R)}; \]

\[ S_1 = b_1 - \sqrt{b_1^2 - 2q(b_1 - R)}, \tag{16} \]

where \(q = R - \sqrt{R^2 - r^2}\) (\(R\) is the radius of the spherical mirror).

For a concave mirror with a small relative aperture*) one may use the approximate formula, putting \(S^3\), \(S_1^2\), and \(q^2\) equal to zero. Then

\[ \Delta S=\pm \frac{r_m^2}{2}\left(\frac{1}{a}+\frac{1}{b}-\frac{2}{R}\right). \tag{17} \]

The positive sign of \(\Delta S\) corresponds to an arrangement in which the optical path from the source to the zonal focus is greater for the central ray than for a noncentral one. In the opposite case \(\Delta S\) has a negative value. The number of Fresnel zones \((m)\) fitting on a mirror of radius \(r\) can be determined by expressing the path difference \(\Delta S\) for the central and marginal rays in units of \(\lambda/2\):

\[ m=\frac{\Delta S}{\lambda/2}. \]

Hence, finally:

\[ \pm \frac{m\lambda}{r_m^2}=\frac{1}{a}+\frac{1}{b}-\frac{2}{R}. \tag{18} \]

When the left-hand side is zero \((r_m=\infty)\), we obtain the formula for a mirror with its geometrical focus at the point \(b\). For finite \(r_m\) there are two additional bright foci \(b_1\) and \(b_2\), for which

\[ \frac{1}{b_1}-\frac{1}{b}=\frac{1}{b_2}-\frac{1}{b}=\frac{\lambda m}{r_m^2}. \]

If \(b_1\) and \(b_2\) differ little from \(b\), then we obtain an almost symmetric arrangement

\[ b_1-b \simeq b_2-b \simeq b^2\cdot \frac{\lambda m}{r_m^2}. \]

A comparison of concave and plane zone plates is most conveniently made for the special case \(a=R\), in which the source is placed at the center of the mirror. Then

\[ \frac{r_m^2}{\lambda_m}=\pm \frac{ab}{a-b}, \]

in contrast to the formula for a plane plate:

\[ \frac{r_{om}^2}{\lambda_m}=\frac{ab}{a+b}. \]

For equal source and observer distances \(a\) and \(b\) for both plates, we have:

\[ \frac{r_m^2}{r_{om}^2}=\frac{a+b}{a-b}. \]

Let, for example, \(a=100\ \text{cm}\), \(b=98\ \text{cm}\), \(\lambda=5\cdot 10^{-5}\ \text{cm}\), \(R=100\ \text{cm}\), and let the diameter of the plates \(D\) be \(10\ \text{cm}\). Then the radius of the first zone will be \(0.5\ \text{mm}\) for the plane plate and \(5\ \text{mm}\) for the concave one. On the surface of the plane plate of the chosen diameter it is necessary to inscribe 10,000 zones, whereas on the surface of the concave one—only...

*) See criterion (7).

100 zones. The width of the last ring for the plane plate will be equal to \(2.5\mu\), and for the concave one, \(0.25\) mm.

A concave plate with data corresponding to the example given was, at our request, manufactured by Yu. N. Efimov. A concave aluminized mirror 10 cm in diameter was mounted in the chuck of a lathe. A cutter, whose cutting edge was \(0.25\) mm wide, was fixed in the tool holder. The readouts of the cutter displacement over the surface of the mirror during removal

Fig. 9. External appearance of the rings on the surface of a concave zone plate. The diameter of the central zone is 10 mm.

of the aluminum layer were made by means of a vernier attached to the lathe tool holder. The external appearance of the rings of the resulting zone plate is shown in Fig. 9.

Testing of the zone plate gave results that should be regarded as satisfactory.

With the aid of this plate, all three bright foci are clearly observed—the geometrical one and two additional ones.

In Fig. 10 are shown photographs of the image of an illuminated scale behind the scale, taken with a light filter in the geometrical focus (a) and in one of the additional foci (b). The photographs were obtained at \(a = 80\) cm, \(b = 137.5\) cm, \(b_1 = 133.7\) cm.

The image quality in the additional foci testifies to the satisfactory agreement of the dimensions of the rings on the plate with the calculated Fresnel zones.

The result (18) may also be arrived at in another way.

Let us regard a concave reflecting zone plate as an optical system consisting of a converging mirror and a zone diaphragm located close to it, operating in transmitted light. For such a diaphragm, as for a plane zone plate, the principal focal lengths of the first order \(F_{\text{pl}}\) (real and virtual) are approximately equal to

\[ \pm \frac{r_m^2}{\lambda m}. \]

Let us recall that a zone plate acts simultaneously as both a converging and a diverging lens, producing convergent and divergent beams of light.

Fig. 10. Photographs of the image of the backside-illuminated milk glass in the geometrical focus \((a)\) and in one of the additional foci \((b)\).

The principal focal length of the system \(F_c\) is connected with the principal focal lengths of the elements composing the system—the mirror \((F_z)\) and the plate \((F_{\text{pl}})\)—by the well-known relation

\[ \frac{1}{F_c}=\frac{1}{F_z}+\frac{1}{F_{\text{pl}}} \quad \text{or} \quad \frac{1}{F_c}=\frac{2}{R}\pm \frac{m\lambda}{r_m^2} \; *). \]

The system has two real foci of the first, third, fifth, etc. orders. These foci lie on both sides of the mirror focus, corresponding respectively to the positive and negative signs of \(F_{\text{pl}}\).

Knowing the focus of the system \(F_c\), it is easy, by the usual method, to relate the position of the source and the image for a concave reflecting zone plate:

\[ \frac{1}{a}+\frac{1}{b}=\frac{1}{F_c}. \]

Hence we obtain, as before,

\[ \frac{1}{a}+\frac{1}{b}-\frac{2}{R}=\pm \frac{m\lambda}{r_m^2}. \]

Let us further take into account that a zone diaphragm introduced into a beam of light does not change the optical path between the closed—

*) To determine the principal focal lengths of higher orders, in the last equation one should replace \(F_{\text{pl}}\) by \(\frac{F_{\text{pl}}}{3}\), \(\frac{F_{\text{pl}}}{5}\), \(\frac{F_{\text{pl}}}{7}\), etc.

ZONE PLATE

...by those parts of the wavefront blocked by the diaphragm and points lying on the optical axis. Therefore, in the optical system mirror—zone plate, not only do new “zone” foci appear (at points on the optical axis for which the difference of distances \(\Delta S\) to the open zones is successively equal to \(\lambda \pm \lambda/4\), \(2\lambda \pm \lambda/4\), \(3\lambda \pm \lambda/4\), etc.), but the former focus of the mirror, observed in the absence of the diaphragm (at the point for which \(\Delta S = 0\)), is also preserved. Considering a concave reflecting zone plate as a mirror—zone-diaphragm system, it can be shown that, in order to obtain an appreciable difference \(\Delta F\) between the quantities \(F_c\) and \(F_3\), it is sufficient to introduce into the path of the rays a diaphragm with a considerably larger focal length \((F_{\mathrm{pl}})\) than \(F_3\). This follows from the expression for the relative change of focus

\[ \frac{\Delta F}{F_3} = \frac{F_c}{F_{\mathrm{pl}}}. \]

From this point of view, a concave reflecting zone plate makes it possible to replace the study of the foci of the diaphragm by the study of the foci of the system.

It is significant that, in this version of the experiment, it is not required that the zone diaphragm have a short focal length.

To obtain a high-aperture system, one must use a high-aperture mirror. As for the zone plate, it is necessary only to ensure sufficiently large dimensions for it, not smaller than the diameter of the mirror.

The manufacture of such a plate with broad Fresnel zones is a comparatively simple task. It is advisable, however, to realize the mirror—zone-plate system not by assembling it from separate parts (a concave mirror and a zone diaphragm), but in the form of the concave reflecting zone plate described above.

If the relative aperture of the mirror is large, then the determination of the path difference \(\Delta S\) must be performed according to the formula (16) given above, taking into account the previously omitted terms containing \(S^2\), \(S_1^2\), and \(q^2\). Then the restriction (7) is no longer necessary.

Since, for large mirror apertures, the geometrical focus of a point is a segment (for all positions except \(a + b = R\)), a significant number of Fresnel zones can fit on the surface of the mirror even in the case when the observation point lies inside the geometrical focus.

It deserves attention that, for a well-made concave zone plate with a large aperture, the “zone” monochromatic image of a point, in principle free from spherical aberration, can have considerably better quality than the geometrical image. Therefore, in particular, one can manufacture a reflecting zone plate with such a calculation that, at one of the points of the aberrational segment

(geometrical focus), an additional bright monochromatic “zonal” image of the source was obtained. To assess the technical difficulties in manufacturing such plates, let us consider several particular examples.

a) A luminous plate with a zonal focus located at the geometrical focus of the central rays

Table II gives the results of calculating, by formula (16), the path difference \(\Delta S\) for the central ray and for a ray reflected from a point on the mirror surface located at a distance \(r\) from its axis.

The calculations were carried out for the following data: the radius of the mirror aperture \(\sigma = 10\ \text{cm}\), the principal focal length \(\dfrac{R}{2} = 20\ \text{cm}\). Suppose further that the point source is on the main optical axis and is \(30\ \text{cm}\) from the mirror. In this case the geometrical

Fig. 11. \(\sigma\) — radius of the mirror aperture with center at \(3\), \(A\) — source, \(O\) — center of the mirror, \(M_0\) — focus of the central rays \(ANM_0\), \(M_\sigma\) — focus of the marginal rays \(APM_\sigma\).

Table II

focus of the central rays \(M_0\) (Fig. 11) is located at a distance \(b = 60\ \text{cm}\) from the mirror, and the length of the aberration segment \(M_0M_\sigma\) is \(7\ \text{mm}\). The first column of the table gives the values of \(r\), and the second gives the path difference, expressed in \(\lambda/2\). Figure 12 shows graphically the dependence between \(\Delta S\) and \(r\). Observation is carried out at the point \(M_0\). The wavelength is taken to be \(\lambda = 5000\ \text{Å}\). As the calculation results show, the radius of the first Fresnel zone is approximately \(3\ \text{cm}\), the width of the marginal zone is \(\simeq 0.15\ \text{mm}\). Thus 178 zones fit over the entire surface of the mirror. The Fresnel zones are therefore sufficiently wide.

However, the difficulties increase substantially if we move the source to infinity. For the limiting case, a plane front of the incident wave, the path difference of the central and marginal rays is

\[ \Delta S=\sqrt{\sigma^{2}+\left(\frac{R}{2}-\sqrt{R^{2}-\sigma^{2}}\right)^{2}}-\frac{3}{2}R+\sqrt{R^{2}-\sigma^{2}} . \tag{19} \]

The quotient \(\dfrac{\Delta S}{\lambda/2}\) gives the number of rings on the surface of the mirror with relative aperture \(\dfrac{2\sigma}{R}\), when observing from the geometrical focus of the central rays. Let, as before, \(R=40\ \text{cm}\), \(\sigma=10\ \text{cm}\), \(\lambda=5\cdot10^{-5}\ \text{cm}\). Then the number of Fresnel zones will be approximately 1500. We note that, for a plane plate with the same aperture diameter and focal length, the number of zones reaches 100,000.

Fig. 12. Graph of the dependence of the path difference \(\Delta S\) (in units of \(\lambda/2=2.5\cdot10^{-5}\ \text{cm}\)) on \(r\), the distance of the point (in cm) on the mirror surface from the main optical axis of the zone plate.

b) A plate with a zonal focus located inside the geometrical focus. The case of coincidence of parallel rays

To calculate this case it is convenient to use the fact that the distance \(\tau\) between the sphere 1 and the paraboloid 2 (Fig. 13) is, with great accuracy, equal to:

\[ \tau=r^{2}(r_{0}^{2}-r^{2})\cdot\frac{1}{8R^{3}}, \]

where \(r\) is the ordinate, and \(R\) is the radius of curvature of the sphere; \(r_{0}\) is the ordinate of the point of intersection of the circle and the parabola. The distance \(\tau\) is equal to zero at \(r=0\) and at \(r=r_{0}\). The maximum value of \(\tau\) is obtained at \(r=\dfrac{r_{0}}{\sqrt{2}}\). The path difference \(\Delta S\) of two rays converging at the focus of the paraboloid—one reflected from the spherical mirror at the point \(\vartheta\) \((r=r_{0})\), and the other reflected at a point with ordinate \(r\ne r_{0}\)—is equal to:

\[ \Delta S=2\tau=r^{2}(r_{0}^{2}-r^{2})\cdot\frac{1}{4R^{3}} . \]

The maximum value is \(\Delta S_{\max}=2\tau_{\max}=\dfrac{r_0^4}{16R^3}\). Over the entire surface of the mirror, when observed from the focus of the paraboloid, there will fit

\[ n=\frac{2\Delta S_{\max}}{\lambda/2} \]

Fresnel zones:

\[ n=\frac{r_0^4}{4R^3\lambda}. \tag{21} \]

For example, with \(R=40\ \text{cm}\) and \(r_0=10\ \text{cm}\) we obtain \(n=800\).

In the case of interest to us, a mirror operating under oblique incidence of a parallel beam may be regarded as an off-axis part \(3\) (in Fig. 13) of another spherical mirror \(1\) (in Fig. 13), considerably larger in dimensions, illuminated by a parallel beam of light traveling along the principal optical axis. In this case the Fresnel zones on the surface of mirror \(3\) have the form of bands directed along arcs drawn with radius \(r\) from the point \(O\) (Fig. 14). Therefore, the manufacture of a zone plate corresponding to oblique incidence of a parallel beam can be carried out by the same method that we use for the case of a plate operating in the central beam.

Fig. 13. \(R\) — radius of sphere \(1\); \(2\) — paraboloid; \(O\) — midpoint of the spherical surface \(1\); \(3\) — spherical mirror; \(a\) — angle between the directions to the focus and to the midpoint \(O\) of the spherical surface \(1\); \(\tau\) — distance between the sphere and the paraboloid; \(r_0\) — ordinate of the point of intersection of the sphere and paraboloid; \(r\) — ordinate of a point on the surface of mirror \(3\).

The width, in the projection of the Fresnel zones onto the plane \(rZ\), when observing the surface of sphere \(1\) from the focus of the paraboloid, will be equal to:

\[ l=1:\frac{\Delta S}{\Delta r\cdot \lambda/2} =\frac{\lambda R^3}{r\left(r_0^2-2r^2\right)}. \tag{22} \]

It should be noted that the distance between the zones on the surface of mirror \(3\) depends on the position of the mirror on sphere \(1\) (Fig. 13). Let \(r_0/\sqrt{2}=\xi\). If the extreme ordinates \(r_1\) and \(r_2\) are greater or less than \(\xi\), then the zones will become denser or more rarefied

with increasing \(r\) from \(r_1\) to \(r_2\). If, however, \(r_1<\xi\) and \(r_2>\xi\), then the zone width has a maximum near \(r=r_0/\sqrt{2}\).

In the first case the number of Fresnel zones on the surface of the mirror is equal to:

\[ n=\frac{2(\tau_1-\tau_2)}{\lambda/2} =\frac{1}{2R^3\lambda}\left[r_1^2\left(r_0^2-r_1^2\right)-r_2^2\left(r_0^2-r_2^2\right)\right]. \tag{23} \]

In the second case:

\[ n=\frac{4(\tau_{\max}-\tau_1)}{\lambda} +\frac{4(\tau_{\max}-\tau_2)}{\lambda} = \frac{1}{2R^3\lambda} \left[ \frac{r_0^4}{2} -r_1^2\left(r_0^2-r_1^2\right) -r_2^2\left(r_0^2-r_2^2\right) \right]. \tag{24} \]

A zonal plate made in accordance with the above calculation will have an additional zonal focus, located at the focus of the paraboloid

\[ f=\frac{1}{4}\left(R+\sqrt{R^2-r_0^2}\right). \]

Fig. 14. View of the Fresnel zones for mirror 3. \(O\) is the middle of the spherical surface, \(r\) is the ordinate of a point on the mirror surface.

Table III

To estimate the number and sizes of the Fresnel zones that may occur in commonly used mirrors, let us consider an example in which \(R=80\) cm and \(\alpha=15^\circ\). Let \(r_0\sqrt{2}\) be the ordinate of the middle of mirror 3 in Fig. 13. Then \(r_0=14\) cm. Let, further, the dimensions of the mirror be bounded by the ordinates \(r_1=4\) cm and \(r_2=13\) cm. The number of Fresnel zones on the surface of such a mirror will be equal to 230 (the results of calculating the zone width \(l\) near certain values of the ordinates \(r\) are given in Table III). Obviously, in this case, as in the other examples considered above, the manufacture of the zonal plate should not encounter serious difficul-

... For the particular example considered, the focus of the paraboloid (39.69 cm) lies within the geometrical focus of the mirror (39.43 cm—39.96 cm).

It is easy to show that a zone plate made in accordance with the calculation given above always has a zonal focus situated within the aberration segment of the geometrical image of the point. Indeed, from the general formula for a spherical mirror

\[ \frac{1}{a}+\frac{1}{b}+\frac{2R}{ab}\cdot\frac{1-\cos\varphi}{2\cos\varphi-1} = \frac{2}{R}\cdot\frac{\cos\varphi}{2\cos\varphi-1}, \]

where \(\varphi=\arcsin \frac{r}{R}\), it follows that, for a parallel beam \((b=\infty)\), the focal distances \(F_1\) and \(F_2\) for the central rays \((\varphi=0)\) and for the marginal rays \(\left(\varphi=\arcsin \frac{r_0}{R}\right)\), respectively, are equal to

\[ F_1=\frac{R}{2};\quad F_2=R-\frac{1}{2}\sqrt{R^2-r_0^2}, \]

where \(r_0\) is the radius of the mirror aperture. On the other hand, in our case the distance \(f\) between the midpoint of the mirror and the focus of the paraboloid is related to \(R\) and \(r_0\) by the relation

\[ f=\frac{R}{4}+\frac{1}{4}\sqrt{R^2-r_0^2}. \]

Hence

\[ F_1>f>F_2. \]

2. Spherical zone plate working in transmitted light together with a lens

As already noted above, to obtain a high-aperture zone plate with wide Fresnel zones it is necessary to use an arrangement in which the curvature of the wave front incident on the plate has the same sign as the part of the spherical surface facing the source, with its center at the observation point (Fig. 8). This problem can be solved not only with the aid of a concave mirror, but also by preliminary focusing of the light by a converging lens. Consider an optical system consisting of a lens and a zone plate.

For the focal length of the system \(F_c\), composed of two closely spaced elements—a lens and a zone plate

with focal lengths \(F_l\) and \(F_{\mathrm{pl}}\), one may write, as for two lenses:

\[ \frac{1}{F_c}=\frac{1}{F_l}+\frac{1}{F_{\mathrm{pl}}}. \]

Supplementing the lens with a zone plate, we shall therefore obtain a change in focal length \(\Delta F=F_c-F_l\), for which

\[ \Delta F=-\frac{F_cF_l}{F_{\mathrm{pl}}}. \tag{25} \]

If the change is small, then

\[ \Delta F=\frac{F_l^2}{F_{\mathrm{pl}}}. \]

As in the case of the system concave mirror—zone plate, in order to obtain a noticeable displacement of the focus of a lens it is sufficient to place in series with it a zone plate with a considerably greater focal length than that of the lens being used.

It is convenient to combine a meniscus converging lens with a concave zone diaphragm operating in transmitted light (Fig. 15). The manufacture of the diaphragm is easily accomplished, for example, by applying to the concave surface of the lens a continuous layer of an opaque reflecting coating and subsequently removing it with a cutter from the corresponding portions of the surface. Such a device can operate both in transmitted and in reflected light.

Fig. 15. Meniscus lens with a zone diaphragm.

The optical properties of a zone diaphragm applied to a lens coincide with the properties of the concave reflecting diaphragm considered above. Unlike the system of two lenses, the lens—zone plate system has, in addition to the geometrical focus, also additional zone foci.

We made one specimen of a combined spherical zone plate. The focal length of the diaphragm for the middle of the visible spectral region was \(50\ \mathrm{m}\). The diameter of the lens was \(10\ \mathrm{cm}\). The radius of curvature of the concave silvered surface was \(1\ \mathrm{m}\). The radius of curvature of the convex surface was chosen so that, according to the calculation, the principal focal length of the lens \(F_l\) would be equal to \(2\ \mathrm{m}\). Then, according to formula (25), the focal length of the system \(F_c\) should differ by 4% from \(F_l\).

Under these conditions it is still easy to observe three bright foci of the system, if, as the light source, one chooses an object whose structure is favorable for sharply focusing the image. Such an object may be, for example, the spiral filament of an incandescent lamp.

Experiments with a combined spherical zone plate consisted in observing and photographing the image in the “geometrical” and two “zonal” foci of the first-order system. The results of measuring the focal distances of the system proved to be in good agreement with formula (25).

References

1. G. S. Landsberg, Optics, p. 92, Gostekhizdat, 1947; R. W. Wood, Physical Optics, p. 52, ONTI, 1936.
2. O. A. Myers, American Journal of Physics 19, 359—365 (1951).