3.3 The Gaussian Optics of a Lens System |
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Hence, in the paraxial region about the optical axis, the radius of the image of a spherical object is independent of the magnification and depends only on the ratio of the refractive indices of the object and image spaces.
3.3 THE GAUSSIAN OPTICS OF A LENS SYSTEM
In 1841, Professor Carl Friedrich Gauss (1777–1855) published his famous treatise on optics (Dioptrische Untersuchungen) in which he demonstrated that, so far as paraxial rays are concerned, a lens of any degree of complexity can be replaced by its cardinal points, namely, two principal points and two focal points, where the distances from the principal points to their respective focal points are the focal lengths of the lens. Gauss realized that imagery of a rotationally symmetric lens system could be expressed by a series expansion where the first order provided the ideal or stigmatic image behavior and the third and higher orders were the aberrations. He left the computation of the aberrations to others.
To understand the nature of these cardinal-point terms, we imagine a family of parallel rays entering the lens from the left in a direction parallel to the axis (Figure 3.8). A marginal ray such as A will, after passing through the lens, cross the axis in the image space at J, and so on down to the paraxial ray C, which crosses the axis finally at F2.
If the entering and emerging portions of all of these rays are extended until they intersect, we can construct an “equivalent refracting locus” as a surface of revolution about the lens axis, to contain all the equivalent refracting points for the entire parallel beam. The paraxial portion of this locus is a plane perpendicular to the axis and known as the principal plane, and the axial point itself is called the principal point, P2. The paraxial image point F2, which is conjugate to an axial object point located at infinity, is called the focal point, and the longitudinal distance from P2 to F2 is the posterior focal length of the lens, marked f 0.
A beam of parallel light entering parallel to the axis from the right will similarly yield another equivalent refracting locus with its own principal point P1
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Figure 3.8 The equivalent refracting locus.
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Paraxial Rays and First-Order Optics |
and its own focal point F1, the separation from P1 to F1 being known as the anterior focal length f. The distance from the rear lens vertex to the F2 point is the back focal distance/length or more commonly the back focus of the lens, and of course the distance from the front lens vertex to the F1 point is the front focus of the lens. For historical reasons the focal length of a compound lens has often been called the equivalent focal length, or EFL, but the term equivalent is redundant and will not be used here.4
3.3.1 The Relation between the Principal Planes
Proceeding further, we see in Figure 3.9 that a paraxial ray A traveling from left to right is effectively bent at the second principal plane Q and emerges through F2, while a similar paraxial ray B traveling from right to left along the same straight line will be effectively bent at R and cross the axis at F1. Reversing the direction of the arrows along ray BRF1 yields two paraxial rays entering from the left toward R, which become two paraxial rays leaving from the point Q to the right; thus Q is obviously an image of R, and the two principal planes are therefore conjugates. Because R and Q are at the same height above the axis, the magnification is þ1, and for this reason the principal planes are sometimes referred to as unit planes.
When any arbitrary paraxial ray enters a lens from the left it is continued until it strikes the P1 plane, and then it jumps across the hiatus between the principal planes, leaving the lens from a point on the second principal plane at the same height at which it encountered the first principal plane (see Figure 3.10).
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Figure 3.9 The principal planes as unit planes.
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Figure 3.10 A general paraxial ray traversing a lens.
3.3 The Gaussian Optics of a Lens System |
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3.3.2 The Relation between the Two Focal Lengths
Suppose a small object of height h is located at the front focal plane F1 of a lens (Figure 3.11). We draw a paraxial ray parallel to the axis from the top of this object into the lens; it will be effectively bent at Q and emerge through F2 at a slope o0. A second ray from R directed toward the first principal point P1 will emerge from P2 because P1 and P2 are images of each other, and it will emerge at the slope o0 because R is in the focal plane and therefore all rays starting from R must emerge parallel to each other on the right-hand side of the lens.
From the geometry of the figure, o ¼ – h/f and o0 |
¼ h/f 0; hence, |
o0=o¼ f =f 0 |
(3-7) |
We now move the object h along the axis to the first principal plane P1. Its image will have the same height and will be located at P2. We can now apply the Lagrange theorem to this object and image, knowing that a paraxial ray is entering P1 at slope o and leaving P2 at slope o0. Therefore, by the Lagrange equation,
hno ¼ hn0o0 or o0=o ¼ n=n0 |
(3-8) |
Equating Eqs. (3-7) and (3-8) tells us that
f =f 0 ¼ n=n0
The two focal lengths of any lens, therefore, are in proportion to the outside refractive indices of the object and image spaces. For a lens in air, n ¼ n0 ¼ 1, and the two focal lengths are equal but of opposite sign. This negative sign simply means that if F1 is to the right of P1 then F2 must lie to the left of P2. It does not mean that the lens is a positive lens when used one way round and a negative lens when used the other way round. The sign of the lens is the same as the sign of its posterior focal length f 0. For a lens used in an underwater housing, n ¼ 1.33 and n0 ¼ 1.0; hence, the anterior focal length is 1.33 times as long as the posterior focal length.
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Figure 3.11 Ratio of the two focal lengths.
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Paraxial Rays and First-Order Optics |
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3.3.3 Lens Power |
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Thus for a lens in air the power is the reciprocal of the posterior focal length. Focal length and power can be expressed in any units, of course, but if focal length is given in meters, then power is in diopters. Note also that the power of a lens is the same on both sides no matter what the outside refractive indices may be.
Applying Eq. (3-2) to all the surfaces in the system and summing, we get
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The quantity under the summation is the contribution of each surface to the lens power. The expression in parentheses, namely, (n0 – n)/r, is the power of a surface which is also called surface power.
3.3.4 Calculation of Focal Length
1. By an Axial Ray
If a paraxial ray enters a lens parallel to the axis from the left at an incidence height y1 and emerges to the right at a slope u0 (see Figure 3.12a), then the posterior focal length is f 0 ¼ y1/u0. The anterior focal length f is found similarly by tracing a parallel paraxial ray right to left, and of course we find that f ¼ –f 0 if the lens is in air. The distance from the rear lens vertex to the second principal plane is given by
lpp0 ¼ l0 f 0
and similarly
lpp ¼ l f
2. By an Oblique Ray
The Lagrange equation can be modified for use with a very distant object in the following way. In Figure 3.12b, let A represent a very distant object and A0 its image. As the object distance l becomes infinite, the image A0 approaches the rear focal point. Then by the Lagrange equation, the following equation applies: