Paraxial Ray Tracing Through Concave Spherical Lenses
In the examples we have used thus far, the lenses have been convex, or positive. Light emerges from a convex lens more convergent—or at least less divergent—than it entered. By contrast, a concave, or negative, lens makes light more divergent.
The principles of paraxial ray tracing are the same for concave spherical lenses as for convex spherical lenses. Consider a –2 D lens. Its Fa is (1/–2 D) = 50 cm behind the lens. By definition, a ray of light directed through Fa will exit the lens parallel to the optical axis (Fig 1-32A). Similarly, a virtual object in the anterior focal plane of a concave lens will image to plus infinity. A ray of light entering the lens parallel to the optical axis will pass through Fp after exiting the lens (Fig 1-32B). Similarly, a real object at minus optical infinity will produce a virtual image in the posterior focal plane of a concave lens.
Figure 1-32 A, Incoming light directed through the anterior focal point, Fa, of a concave spherical lens exits the lens collimated. B, Collimated incoming light parallel to the optical axis leaves the lens as if it had come through the posterior
focal point, Fp. (Illustration developed b y Kevin M. Miller, MD, and rendered b y C. H. Wooley.)
Now consider an object placed 100 cm in front of the lens. The 3 usual rays are drawn (Fig 1- 33). A virtual image is formed 33 cm in front of the lens. By similar triangles, the transverse magnification is +0.33×. No matter where a real object is placed in front of a minus lens, the resulting image is upright, minified, and virtual.
Figure 1-33 No matter where a real object is placed in front of a concave (negative) spherical lens, the image is upright,
minified, and virtual. (Illustration developed b y Kevin M. Miller, MD, and rendered b y C. H. Wooley.)
Objects and Images at Infinity
If an object is placed 50 cm in front of a +2 D thin lens in air, where is the image? Light emerges from the lens with a vergence of zero. A vergence of zero means that light rays are neither convergent nor divergent but parallel; thus, the light is collimated. In this example, light rays emerge parallel to one another, neither converging to a real image nor diverging from a virtual image. In this case, the image is said to be at infinity.
Objects can be located at infinity as well. If a second lens is placed anywhere behind the first one, light striking the second lens has a vergence of zero; the object is at infinity. As a practical matter, a sufficiently distant object may be regarded as being at infinity. Clearly, an object like the moon, which is 400 million meters away, has a vergence of essentially zero. For clinical work, objects more than 20 ft (6 m) distant may be regarded as being at optical infinity. An object 20 ft away has a vergence of about –0.17 D; clinically, this is small enough to be ignored. When a refractive correction is being determined, few patients can notice a change of less than 0.25 D.
Some people think that objects in the anterior focal plane are imaged in the posterior focal plane. This is not true. Objects in the anterior focal plane image at plus infinity; objects at minus infinity
image in the posterior focal plane.
Principal Planes and Points
If an object’s position changes in front of a lens, both the location and magnification of the image change. Most optical systems have one particular object location that yields a magnification of 1. In other words, when an object is located in the correct position, the image will be upright and the same size as the object. The principal planes are perpendicular to the optical axis and identify the object and image locations that yield a magnification of 1. The principal planes are also called the planes of unit magnification and are geometric representations of where the bending of light rays occurs.
Consider an optical system consisting of 2 thin lenses in air (Fig 1-34). The first lens is +6 D, the second lens is +15 D, and the 2 lenses are separated by 35 cm. An object located 50 cm in front of the first lens is imaged 25 cm behind the first lens with a magnification of –0.5. The real image becomes a real object for the second lens, which produces a real image 20 cm behind the second lens with a magnification of –2. The anterior principal plane of this system is 50 cm in front of the first lens; the posterior principal plane is 20 cm behind the second lens. Often, both the anterior and posterior principal planes are virtual; in some cases, the posterior principal plane is in front of the anterior principal plane.
Figure 1-34 These 2 thin lenses in air produce an image that is upright, real, and the same size as the object. (Illustration
developed b y Kevin M. Miller, MD, and rendered b y C. H. Wooley.)
The intersection of the anterior and posterior principal planes with the optical axis defines the corresponding anterior and posterior principal points. Like the nodal points, the principal points are an important pair of reference points.