Beam Spots (Waists) in White Light Output by a Collimating Lens

Is the beam spot output by a collimating lens an image of the lamp or LED source?

If a lens is placed one focal length away from sources like lamps or light-emitting diodes (LEDs), the lens will output light consisting of overlapping bundles of parallel rays that do not form an image but may form a beam spot (waist). Since each bundle of rays crosses the optical axis at a different angle, the intensity of the output light varies along the axis. The intensity is highest within the beam spot, where it may appear that the light comes to a focus. However, the rays in each bundle remain parallel, preventing the formation of an image. The location and diameter of the beam spot depends on the optical characteristics of the source and collimating lens, and both can be engineered to provide a beam spot that suits the application.

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Figure 2: When the source emits light over a wide range of angles, the light rays from each source spot are incident over the entire lens surface area. The lens outputs bundles of parallel rays, one bundle for each source point, but the beam spot is at the output surface of the lens, instead of being located one focal length away. 

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Figure 1: Several factors provide a beam spot one focal length (f ) away from the lens. These include the symmetry of the emitted rays around the z-axis, and the limited lens surface area over which each cone of rays is incident. The lens outputs a single bundle of rays for each source point, and each bundle has a diameter smaller than the lens diameter. The overlapping ray bundles form the beam spot.

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Figure 3: A system of lenses can be used to form a beam spot wherever it is needed. In this case, the beam spot is located one focal length (f_CD ) from the output side of the condenser lens. Since the rays emitted by each source point are parallel at the beam spot, no image is formed at the beam spot. Note that a scaled image of the source, where the rays from each point on the source intersect, is formed between the field and condenser lenses, one focal length (f_FD ) away from the field lens. In this example, the light source is one focal length (f_CL ) away from the collector lens.

Beam Spot, but No Image
The illustration in Figure 1 shows a light source on the left, which consists of a vertical stack of light-emitting points. While this is not a typical source, a system of lenses can be used to create an image of a source with these characteristics, as is discussed later.

The light rays from each point are spread over a limited range of angles (). This light is incident on a lens placed one focal length (f ) away. The lens outputs the light as bundles of parallel rays, one bundle for every light-emitting point on the source.

Each bundle of rays crosses the optical axis (z-axis) at a different angle. The angle for the light emitted from a particular light-emitting point on the source can be determined by first tracing a ray, from the point on the source, parallel to the z-axis, to the lens. From the endpoint of this ray, a second ray begins and crosses the z-axis one focal length away from the output side of the lens. In the figure, these are the rays at the center of each group and drawn with a thicker line.

The crossing of all of the parallel ray bundles provides a beam spot one focal length away from the lens. Here, the output beam has the smallest diameter and highest intensity. However, the spot does not include an image of the source, since the rays in each bundle are parallel to one another. All of the rays in each bundle would have to come back together, and intersect at a point, in order to form an image of the source point that emitted them.

There Is Not Always a Beam Spot
The lens in Figure 1 provides output light with a beam spot one focal length away from the lens for a few reasons. One is that the source is located one focal length away. Another is that the total group of rays emitted by the source are symmetric around the z-axis. In addition, the ray cone from each point intersects a limited region of the lens' surface area, so that each output bundle of rays has a relatively small diameter.

Figure 2 depicts a more typical white-light, or broadband source. Each point on the source emits rays over an angular range so wide that rays are incident upon the entire input face of the lens. When this occurs, the beam waist is located at the output surface of the lens.

Engineering a Beam Spot
Beam spots are useful for providing uniform illumination, since the light intensity is maximized and the beam spot does not include an image of the light-emitting structure of the source. When each point on the light source emits over a wide range of angles, and / or it is necessary to provide uniform illumination far from the lens, a system of lenses is often used. 

A representative lens system is illustrated in Figure 3. The collector lens outputs bundles of parallel rays, and the beam spot is at the output surface of the lens, as in Figure 2. The field lens focuses each bundle of rays output by the collector lens. An upside-down, scaled image of the source's light emitting structure is formed one focal length away from both the field and condenser lenses. Since the light rays from this image are incident on the condenser lens over a limited region of the lens' input surface, the light output by this lens creates a beam spot one focal length away. The illumination provided by this beam spot is uniform, without an image of the source.