Fresnel lenses

The Fresnel lens is a complex composite lens. It does not consist of a solid polished piece of glass with spherical or other surfaces (like ordinary lenses), but of separate concentric rings of small thickness adjacent to each other, which in cross section have the shape of prisms of a special profile. Proposed by Augustin Fresnel.

This design provides a small thickness (and hence weight) of the Fresnel lens even with a large angular aperture. The sections of the rings near the lens are constructed in such a way that the spherical aberration of the Fresnel lens is small, the rays from a point source placed at the focus of the lens, after refraction in the rings, come out in an almost parallel beam (in the annular Fresnel lenses).

Fresnel lens calculation

The Fresnel lens is one of the first devices based on the physical principle of light diffraction.

This device has not lost its practical significance to this day. The general scheme of the physical model on which its operation is based is shown in (Fig. 1).

Rice. 1 Scheme for constructing Fresnel zones for an infinitely distant observation point (plane wave)

Let us assume that at point O there is a point source of optical radiation of wavelength l. Naturally, like any point source, it emits a spherical wave, the wave front of which is shown in the figure as a circle. Let us set the condition to change this wave to a plane one, which will propagate along the dashed axis. Several wavefronts of this variable wave lagging behind each other by l/2 are shown in (Fig. 1). To begin with, we note that we are considering a variable plane wave from an existing spherical wave in free space. Therefore, in accordance with the Huygens-Fresnel principle, only electromagnetic oscillations in the existing one can serve as the “sources” of this variable wave. And if this does not suit the spatial distribution of the phase of these oscillations, then there is a wave front (spherical) of the original wave. Let's try to correct it. Let's go through the steps.

Action one: note that from the point of view of the secondary Huygens-Fresnel waves (which are spherical), a spatial displacement of an entire wavelength in any direction does not change the phase of the secondary sources. Therefore, we can afford, for example, to “break” the wavefront of the original wave as shown in (Fig. 2).

Rice. 2 Equivalent phase distribution of secondary radiators in space

Thus, we have “disassembled” the original spherical wavefront into “ring parts” number 1, 2... and so on. The boundaries of these rings, called Fresnel zones, are determined by the intersection of the wave front of the original wave with a sequence of wave fronts of the “projected wave” displaced relative to each other by l/2. The resulting picture is already significantly “simpler”, and represents 2 slightly “rough” flat secondary radiators (green and red in Fig. 2), which, however, cancel each other out due to the mentioned half-wave mutual displacement.

So, we see that the Fresnel zones with odd numbers not only do not contribute to the fulfillment of the task, but even actively harm. There are two ways to deal with this.

The first method (amplitude Fresnel lens). You can simply geometrically close these odd zones with opaque rings. This is how it is done in large-sized focusing systems of sea lighthouses. Of course, this may not achieve ideal beam collimation. It can be seen that the remaining, green, part of the secondary emitters is, firstly, not quite flat, and secondly, it is discontinuous (with zero dips in the place of the former odd Fresnel zones).

Therefore, the strictly collimated part of the radiation (and its amplitude is nothing more than the zero two-dimensional Fourier component of the spatial distribution of the phase of green emitters along a plane wavefront with zero offset, see (Fig. 2) will be accompanied by wide-angle noise (all other Fourier components except Therefore, the Fresnel lens is almost unrealistic to use for imaging - only for collimating radiation. However, nevertheless, the collimated part of the beam will be significantly more powerful than in the absence of the Fresnel lens, since we have at least got rid of the negative contribution to the zeroth Fourier component from odd Fresnel zones.

The second method (phase Fresnel lens). It is possible to make the rings covering the odd Fresnel zones transparent, with a thickness corresponding to the additional phase shift l/2. In this case, the wave front of the “red” secondary radiators will shift and become “green”, see fig. 3.

Fig.3 Wavefront of secondary emitters behind the phase Fresnel lens

Actually phase Fresnel lenses have two versions. The first one is a flat substrate with deposited half-wave layers in the regions of odd Fresnel zones (a more expensive option). The second is a three-dimensional turning part (or even polymer stamping on a once-made matrix, like a gramophone record), made in the form of a “stepped conical pedestal” with a step of half the wavelength of the phase shift.

Thus, Fresnel lenses make it possible to cope with the collimation of beams with a large transverse aperture, while at the same time being flat parts of low weight and relatively low manufacturing complexity. An equivalent lighthouse glass lens weighs half a ton and costs slightly less than an astronomical telescope lens.

Let us now turn to the question of what happens when the light source is displaced along the axis relative to the Fresnel lens, which was originally designed to collimate the source radiation in position O (Fig. 1). The initial distance from the source to the lens (that is, the initial curvature of the wavefront on the lens) we agree in advance to call the focal length F by analogy with a conventional lens, see (Fig. 4).

Rice. 4 Building an image of a point source with a Fresnel lens

So, in order for the Fresnel lens to continue to be a Fresnel lens when the source is shifted from position O to position A, it is necessary that the boundaries of the Fresnel zones on it remain the same. And these boundaries are the distances from the axis at which the wave fronts of the incident and “projected” waves intersect. The initially incident one had a front with a radius of curvature F, while the “projected” one was flat (red in Fig. 4). At a distance h from the axis, these fronts intersect, setting the boundary of one of the Fresnel zones,

where n is the number of the zone starting at this distance from the axis.

When the source moved to point A, the radius of the incident wavefront increased and became R1 (blue in the figure). So we need to come up with a new wavefront surface, such that it intersects with the blue one at the same distance h from the axis, giving the same MN on the axis itself. We suspect that such a surface of the projected wave front can be a sphere with radius R2 ( green color on the image). Let's prove it.

The distance h is easily calculated from the “red” part of the figure:

Here we neglect the small square of the wavelength compared to the square of the focus, an approximation that is completely analogous to the parabolic approximation in deriving the usual formula thin lens. On the other hand, we want to find a new border of the nth Fresnel zone as a result of the intersection of the blue and green wavefronts, let's call it h1. Based on the fact that we require the same length of the segment MN:

Finally, requiring h=h1, we get:

This equation is the same as the usual thin lens formula. Moreover, it does not contain the number n of the considered boundary of the Fresnel zones, and therefore, it is valid for all Fresnel zones.

Thus, we see that the Fresnel lens can not only collimate beams, but also build images. True, it must be borne in mind that the lens is still stepped, and not continuous. Therefore, the image quality will be markedly degraded by the admixture of the higher Fourier components of the wavefront discussed at the beginning of this section.

That is, the Fresnel lens can be used to focus radiation to a given point, but not for precision imaging in microscopic and telescopic devices.

All of the above referred to monochromatic radiation. However, it can be shown that by careful choice of the diameters of the rings discussed, a reasonable quality of focus can be achieved for natural light as well.

Unlike prismatic and other diffusers, lenses in lighting fixtures are almost always used for spot lighting. As a rule, optical systems using lenses consist of a reflector (reflector) and one or more lenses.

Converging lenses direct light from a source located at the focal point into a parallel beam of light. As a rule, they are used in lighting structures together with a reflector. The reflector directs the light flux in the form of a beam in the right direction, and the lens concentrates (collects) the light. The distance between the converging lens and the light source is usually variable, allowing the angle to be obtained to be adjusted.

A system of both a light source and a converging lens (left) and a similar system of a source and a Fresnel lens (right). The angle of the light flux can be changed by changing the distance between the lens and the light source.

Fresnel lenses consist of separate concentric ring-shaped segments adjacent to each other. They got their name in honor of the French physicist Augustin Fresnel, who first proposed and put into practice such a design in lighthouse lighting fixtures. The optical effect of such lenses is comparable to that of traditional lenses of similar shape or curvature.

However, Fresnel lenses have a number of advantages due to which they are widely used in lighting designs. In particular, they are much thinner and cheaper to manufacture than converging lenses. Designers Francisco Gomez Paz and Paolo Rizzatto did not fail to take advantage of these features in their work on a bright and magical model range.

Made from lightweight and thin polycarbonate, the "sheets" of Hope, as Gomez Paz calls them, are nothing more than thin and large diffusing Fresnel lenses that create a magical, sparkling and voluminous glow by coating with a polycarbonate film textured with microprisms.

Paolo Rizzatto described the project as follows:
“Why have crystal chandeliers lost their relevance? Because they are too expensive, very difficult to handle and manufacture. We have decomposed the idea itself into components and modernized each of them.”

Here is what a colleague had to say about it:
“A few years ago, the marvelous possibilities of Fresnel lenses caught our attention. Their geometric features make it possible to obtain the same optical properties as conventional lenses, but on a completely flat surface of the petals.

However, the use of Fresnel lenses to create such unique products that combine a great design project with modern technological solutions is still rare.

Such lenses are widely used in stage lighting with spotlights, where they allow you to create an uneven light spot with soft edges, blending perfectly with the overall light composition. Nowadays, they have also become widespread in architectural lighting schemes, in cases where individual adjustment of the angle of light is required, when the distance between the illuminated object and the lamp can change.

The optical performance of a Fresnel lens is limited by the so-called chromatic aberration that forms at the junctions of its segments. Because of it, a rainbow border appears on the edges of the images of objects. The fact that a seemingly flawed feature of the lens was turned into a virtue once again underlines the power of the authors' innovative thought and their attention to detail.

Lighting design of a lighthouse using Fresnel lenses. The ring structure of the lens is clearly visible in the picture.

Projection systems consist of either an elliptical reflector or a combination of a parabolic reflector and a condenser directing light onto a collimator, which can also be supplemented with optical accessories. After that, the light is projected onto a plane.

Spotlight systems: A uniformly illuminated collimator (1) directs the light through a lens system (2). On the left is a parabolic reflector with a high light output, on the right is a condenser that allows for high resolution.

The size of the image and the angle of light is determined by the features of the collimator. Simple curtains or iris diaphragms form light beams of different sizes. Contour masks can be used to create different contours of the light beam. You can project logos or images using a gobo lens with drawings printed on them.

Different angles of light or image size can be selected depending on the focal length of the lenses. Unlike lighting fixtures using Fresnel lenses, here it is possible to create light beams with clear contours. Soft contours can be achieved by shifting the focus.

Examples of optional accessories (from left to right): a lens to create a wide beam of light, a sculpted lens to give the beam an oval shape, a grooved deflector and a "honeycomb lens" to reduce glare.

Stepped lenses convert light rays in such a way that they are somewhere between the "flat" light of a Fresnel lens and the "hard" light of a plano-convex lens. The convex surface is preserved in stepped lenses, however, stepped recesses are made on the side of the flat surface, forming concentric circles.

The front parts of the steps (risers) of concentric circles are often opaque (either painted over or have a chipped matte surface), which makes it possible to cut off the scattered radiation of the lamp and form a beam of parallel rays.

Fresnel spotlights form an uneven light spot with soft edges and a slight halo around the spot, making it easy to mix with other light sources, creating a natural light pattern. That is why Fresnel spotlights are used in cinema.

Projectors with a plano-convex lens, compared to projectors with a Fresnel lens, form a more uniform spot with a less pronounced transition at the edges of the light spot.

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