“Axicon” Optics

The term “axicons” unites a broad class of optical elements possessing axial symmetry and producing multiple images of a point source situated along the axis of the element. The theory of such devices has not yet been fully developed, except for certain special cases—for example, a zone grating. At the same time, they have repeatedly been used in practice. The article under review makes

Fig. 1.

an attempt is made to clarify certain general properties of devices of this type and to outline ways of applying them*). The author, however, confines himself to only “crude” axicons, in which interference phenomena are of no significance, leaving aside questions connected with axicons of the zone-plate type. Nevertheless the article is of evident interest, since it directs research thought into a new, still unexplored channel.

Axicon optics consists of refracting or reflecting surfaces formed by rotation about the axis of some, generally arbitrary, curve. The simplest axicon is the annular diaphragm, which has found wide application both in phase microscopy and in a number of other devices. A set of such annular axicons forms, as is well known, a zone plate. The action of this kind of axicon is essentially connected with diffraction phenomena. However, as is not difficult to see, the presence of the latter is not necessary. In particular, a circular aperture may be replaced by a toroidal lens or by a toroidal depression in a transparent plate. In this case the action of the axicon will be caused not by diffraction, but by refraction. Likewise, if circular scratches are made on a glass disk (say, with the aid of emery paper), then a certain analogue of a zone plate will be obtained, but diffraction here will be replaced by scattering at the scratches.

In order to clarify the principle of operation of axicons, let us consider an element of a long toroidal lens, on the axis of which there is placed a small luminous disk \(abcd\) (Fig. 1). An elementary cylindrical lens projects this disk onto a screen in the form of a bright strip \(c'd'\), whose dimensions are determined by the dimensions of the source \(abcd\), the properties of the lens, and the distance to the screen. However, such a bright strip intersects the axis of the lens over some interval of distances \(r\) from \(r_{\min}\) to \(r_{\max}\). Different elements of the toroidal lens will form strips having different directions, but likewise intersecting the axis in the same interval of distances \(r\). The combined action of all the elements reduces to the mutual superposition of the bright strips, i.e. to their mutual reinforcement in the neighborhood of the axis. As a result, along the axis in the region from \(r_{\min}\) to \(r_{\max}\) a brightly illuminated region (column) is obtained—an image of the source, surrounded by a much more weakly illuminated penumbra region. The transverse dimensions of the image are determined by the width of the strip produced by one element of length of the lens, i.e. by the direct projection of the size \(ab\) onto the screen relative to the lens element as the center of the hole.

The action of axicons of the other types shown in Fig. 2 will be analogous. Thus, the most essential point for the action of axicons is that they are figures of rotation.

Let us note that the illumination of the strip created by one element of the axicon changes in proportion to the length of the corresponding cross section of the source. Therefore the brightness of the image decreases from the center toward the periphery.

A peculiarity of the axicon is that, simultaneously with a set of real images of the source on the axis, it forms an annular imaginary image of the source, and the position and dimensions of this annular image depend on the form of the axicon surface. It is not difficult to convince oneself that the illumination on the axis, at a certain point \(P\) (Fig. 3), is proportional to \(2\pi RTB\), where \(B\) is the brightness of the source, \(R\) is the angular radius of its annular image, and \(T\) is the angular width of this image, if it is considered from the point \(P\). It follows from this that the illumination of the image on the axis (at point \(P\)) decreases inversely propor—

*) J. H. McLeod, JOSA 44, No. 8, 592 (1954).

...proportional to the distance of the point \(P\) from the annular image of the source. At the same time, the illuminance increases as the angular magnification of the annular image of the source increases.

If the dimensions of the image of the source exceed the dimensions of the diffraction disk, then the angular width \(T\) of the annular image, and consequently also the illuminance of the image at \(P\), are proportional to the diameter of the source. This conclusion, as the author states, has been verified experimentally.

Fig. 2.

In the case when the dimensions of the image are smaller than the dimensions of the diffraction disk, then, as the author indicates, one should expect the illuminance at the point \(P\) to be proportional to the third power of the diameter of the source.

The simplest from the standpoint of manufacture and, in the author’s opinion, the most promising of the axicons is a cone with rectilinear...

forming them (see Fig. 3). The image given by the cone extends from zero to a certain maximum, determined both by the dimensions of the cone and by the position of the source. This image (apart from its end) is free from chromatic aberration. If the source is at infinity, then the illumination of the image does not depend on the position of the point \(P\). At the same time, the dimensions of the image increase as one moves away from the cone, and therefore the total amount of energy collected in the image also increases.

Diffraction phenomena make themselves felt only in the case when the cone is made sufficiently carefully. In this case the diffraction pattern surrounding the image proves to be very sharp and is observed at exceptionally large distances, measured in many meters.

Fig. 3.

Figure 4 shows the optical scheme of a telescope with an objective consisting of two conical axicons and an ordinary lens eyepiece. The presence of the second axicon cone leads to a change in the angular diameter \(R\) of the annular image from \(\beta\) to \(\gamma\), but is not connected with a change in the angular thickness \(T\) of this image (if a lens were in the place of the second axicon, then \(T\) would change proportionally to \(R\)). Therefore, in contrast to a lens objective, the decrease in illumination is proportional not to the square, but to the first power of the linear magnification of the second axicon. The authors report that this conclusion has been confirmed experimentally.

Fig. 4.

If both axicons are identical, then the angular magnification given by such a system of two axicons is always equal to unity, independently both of the distance to the object and of the distance to the point of observation—the entering and exiting rays are parallel to one another. However, at excessively large distances to the source (in one case about \(6\ \mathrm{m}\)) the image degenerates owing to the influence of the vertex of the cone. The field of view at all distances from the object is about \(2\ \mathrm{mm}\).

A feature of a telescope with a conical axicon instead of an objective is the possibility of simultaneous vision of several sources located on the axis at different distances.

A conical axicon can also be used as a projector. For this purpose the light source should be placed near the largest

focal length of the axicon objective. The result is an image in the form of a thin column extending to infinity. If one axicon is used, the image near it will have a very small cross-section, gradually increasing with distance. With two identical axicons the cross-section of the image will not vary with distance, but will degenerate at distances exceeding a certain limit. However, if the two conical axicons differ little from one another, and the axicon placed closer to the source has the larger apex angle, then the ray can extend over large distances, increasing only slightly in cross-section.

If the conical or plane surfaces of a glass conical axicon are metal-coated, then the source and its image can be superposed by means of a semitransparent mirror or a mirror with an aperture (Fig. 5) at any distances from the axicon. In this case the dimensions

Fig. 5.

of the diffraction pattern likewise do not depend on distance. The illuminance of the image is inversely proportional to the first power of the distance, owing to the fact that the angular diameter of the annular image remains constant and only its angular thickness depends on distance. An image obtained by means of a conical reflecting axicon 8.75 cm in diameter with a maximum focal length of 18 m could, throughout this entire range of distances, be localized with an accuracy of up to 1–2 wavelengths. Like an ordinary autocollimator, such a device can be used to fix the perpendicularity of the mirror to the direction of the beam.

As follows from the foregoing, axicon optics possesses a number of interesting features that deserve attention and may prove useful in solving certain optical problems. At the same time it is obvious that, before it can find application, extensive investigations are needed, as well as the creation of a full-fledged theory, which is still lacking. The theoretical considerations advanced by the author are merely of an exceptionally primitive and roughly qualitative character. It should be noted that the author passes over in silence a whole series of most important applications of axicon optics, in particular, for example, reflecting hollow cones and spheres of the type shown in Fig. 2, е, ж, 3, и, reflecting cones for examining the inner spherical surfaces and, most importantly, all devices of the zone-plate type.

Without considering all these diverse devices from a unified point of view, one can hardly expect serious successes. However, the very fact that interest has been aroused in devices of this type deserves attention.

R. G.