1.2 Imaging

1.2.1 Spherical Surfaces

Because a simple lens consists of a piece of glass with,in general,two spherical surfaces,we will find it necessary to examine some of the properties of a single,spherical refracting surface.We will for brevity call such a surface,as shown in Fig.1.5,a“len”.Two of these form a lens.To avoid

confusion,we will always place“len”in quotes.

We are interested in the imaging property of the‘len”.We consider a bright point A and define the axis along the line AC,where C is the center of the spherical surface.We examine a particular ray AP that strikes the“len”at P.We shall be interested in the point A′ where this ray intersects the axis.

Before proceeding any further,we must adopt a sign convention.The choice of convention is, of course,arbitrary,but once we choose a convention,we shall have to stick with it.The convention we adopt appears,at first,quite complicated.We choose it at least in part because it is universally applicable;with it we will not need to derive a special convention for spherical mirrors.

To begin, imagine a set of Cartesian coordinate axes centered at O. Distances are measured from O.Distances measured from O to the right are positive;those measured from O to the left are negative.Thus,for example, OA′ and OC are positive,whereas OA is negative.Similarly,distances measured above the axis are positive;those below are negative.This is our first sign convention.

We now adopt a convention for the signs of angles such as OAP or OA′P.We determine their signs by trigonometry.For example,the tangent of angle OAP is approximately

where y is the distance indicated between P and the axis.Our previous convention shows that y is positive,and OA,negative.Thus,tan OAP is negative and so is OAP itself.Similarly, OA′P and OCP are positive.

This is our second sign convention.An equivalent statement is that angle OA′P (for example) is positive if it opens clockwise from the axis, or negative otherwise. It is probably simplest, however,merely to remember that angle OAP is negative as drawn in Fig.1.5.

Finally, we deal with angles of incidence and refraction, such as angle CPA′. It is most convenient to define CPA′ to be positive as shown in Fig.1.5.The angle of incidence or refraction is positive if it opens counterclockwise from the normal (which is, in this case, the radius of the spherical surface）.

Unfortunately, when the last convention is expressed in this way, the statement differs from that which refers to angles(such as OAP)formed by a ray crossing the axis.It is best to learn the sign convention by remembering the signs of all of the important angles in Fig.1.5.Only angle OAP is negative.

Let us now assign symbols to the more important quantities in Fig.1.5.The point A′ is located a distance l′ to the right of O, and the ray intersects the axis at A′ with angle u′. The radius R through the point P makes angle α with the axis.The angles of incidence and refraction are i and i′, respectively.

We must be careful of the signs of OA and angle OAP,both of which are negative according to our sign convention.This is indicated in Fig.1.5 with parenthetical minus signs.We shall later find it necessary,after a derivation based on geometry alone,to go through our formulas and change the signs of all quantities that are algebraically negative.This is so because our sign convention is not used in ordinary geometry. To make our formulas both algebraically and numerically correct, we must introduce our sign convention,which we do as indicated,by changing signs appropriately.

1.2.2 Object-Image Relationship

We now attempt to find a relationship between the quantities l and l′ for a given geometry. First,we relate angle u and i to angle α.The three angles in triangle PAC are u,α and π−i.Because the sum of these angles must be π,we have

At this point,it is convenient to make the paraxial approximation,namely,the approximation that the ray AP remains sufficiently close to the axis that angles u, u′,i and i′ are so small that their sines or tangents can be replaced by their arguments;that is

where θ is measured in radians.

It is difficult to draw rays that nearly coincide with the axis, so we redraw Fig.1.5 by expanding the vertical axis a great amount,leaving the horizontal axis intact.The vertical axis has been stretched so much that the surface looks like a plane.In addition,because only one axis has been expanded,all angles are greatly distorted and can be discussed only in terms of their tangents. Thus,for example,

in paraxial approximation.Note also that large angles are distorted.Although the radius is normal to the surface,it does not look normal in the paraxial approximation.

To return to the problem at hand,the law of refraction is

in paraxial approximation,from which we write

Because OC=R,we write α as

The last equation therefore becomes

A factor of y is common to every term and therefore cancels.We rewrite this relation as

At this point, we have made no mention of the sign convention. We derived the proceeding equation on the basis of geometry alone.According to our sign convention,all of the terms in the equation are positive, except l, which is negative. To make the equation algebraically correct, we must,therefore,change the sign of the term containing l.This change alters the equation to

which we refer to as the“len”equation.

There is no dependence on y in the“len”equation.Thus,in paraxial approximation,every ray leaving A(and striking the surface)crosses the axis at A′.We therefore refer to A′ as the image of A.A and A′ are called conjugate points,and the object distance l and image distance l′ are called conjugates.

Had we not made the paraxial approximation,the y dependence of the image point would not have vanished.Rays that struck the lens at large values of y would not cross the axis precisely at A′.The dependence on y is relatively small,so we would still refer to A′ as the image point.We say that the image suffers from aberrations if all of the geometrical rays do not cross the axis within a specified distance of A′.

1.2.3 Use of the Sign Conventions

A word of warning with regard to the signs in algebraic expression: Because of the sign convention adopted here, derivations based solely on geometry will not necessarily result in the correct sign for a given term.There are two ways to correct this defect.The first,to carry a minus sign before the symbol of each negative quantity,is too cumbersome and confusing for general use. Thus,we adopt the second,which is to go through the final formula and change the sign of each negative quantity.This procedure has already been adopted in connection with the“len”equation and is necessary,as noted,to make the formula algebraically correct.It is important,though,not to change the signs until the final step,lest some signs be altered twice.

1.2.4 Lens Equation

A thin lens consists merely of two successive spherical refracting surfaces with a very small separation between them. Fig.1.6 shows a thin lens in air. The index of the lens is n. The two refracting surfaces have radii R1 and R2,both of which are drawn positive.

We can derive an equation that relates the object distance l and the image distance l′ by considering the behavior of the two surfaces separately. The first surface alone would project an image of point A to a point .If is located at a distance to the right of the first surface,the“len”equation shows that,in paraxial approximation,

because n is the index of the glass(second medium)and 1,the index of the air.

The ray does not ever reach ,because it is intercepted by the second surface.The second surface, however, behaves as if an object were located at . The object distance is , if we neglect the thickness of the lens. In applying the“len”equation to the second surface, we must realize that the ray travels across the interface from glass to air. Thus n is the index of the first medium and1,that of the second.The final image point A′ is also the image projected by the lens as a whole.If we call the corresponding image distance l′,then the“len”equation yields

for the second surface.

If we add the last two equations algebraically,we find that

which is known as the lens-maker’s formula.The lens-maker's formula was derived from the“len”equation by algebra alone. There are no signs to change because that step was included in the derivation of the“len”equation.

We may define a quantity f ′ whose reciprocal is equal to the right-hand side of the lens-maker’s formula,

The lens-maker’s formula may then be written as

where f ′ is the focal length of the lens.We call this equation the lens equation.

We may see the significance of f ′ in the following way.If the object is infinitely distant from the lens, then l=−∞. The lens equation then shows that the image distance is equal to f ′. If the object is located along the axis of the lens,the image also falls on the axis.We call the image point in this case the secondary focal point F′. Note that any ray that travels parallel to the axis is directed by the lens through F′,an observation that we will later find particularly useful.

We define the primary focal point F in a similar way.The primary focal length f is the object distance for which l′=∞.Thus,the lens equation shows that

the primary and secondary focal lengths have equal magnitudes.Any ray that passes through F will be directed by the lens parallel to the axis.

Finally,we note that,in the general case,a lens may have different media on opposite sides.In this case,the lens equation may be shown to be

where n and n′ are the indices in the first and second media, respectively. The primary and secondary focal lengths are not equal,but are related by

1.2.5 Classification of Lenses and Images

A positive lens is a lens that will cause a bundle of parallel rays to converge to a point. Its secondary focal point lies to the right of the lens and f ′ is therefore positive.It may be regarded as a lens that is capable of projecting an image of a relatively distant object on a screen.An image that can be projected on a screen is called a real image. In general, a positive lens projects a real, inverted image of any object located to the left of its primary focal point F. When an object is located at F,the image is projected to∞.The lens is not strong enough to project an image when the object is inside F. In that case, an erect image appears to lie behind the lens and is known as a virtual image.

A positive lens need not have two convex surfaces.It may have the meniscus shape of Fig.1.6. If the lens is thickest in the middle,the lens-maker’s formula will show it to be a positive lens.

A negative lens has its secondary focal point located to the left.Its secondary focal length f ′is negative,and it cannot project a real image of a real object.Rather,it displays an erect,virtual image of such an object.In only one instance can a negative lens display a real image.This is the case when a positive lens projects a real image that is intercepted by a negative lens located to the left of the image plane. Because the rays are cut off by the negative lens, the real image never appears,but behaves as a virtual object projected by the negative lens.

Like a positive lens, a negative lens need not be concave on both surfaces, but may be a meniscus. If the lens is thinnest in the center, f ′will prove to be negative and the lens, also negative.

1.2.6 Spherical Mirrors

Our formalism allows mirror optics to be developed as a special case of lens optics.We notice first that the law of reflection i′ = −i can also be written

which is precisely analogous to the law of refraction, with n′ = −1. We may therefore regard a

mirror as a single refracting surface,across which the index changes from+1 to−1.It is left as a problem to apply the“len”equation to this case.We find that the focal length of a mirror is

where R is the radius of curvature.In addition,the focal points F and F′ coincide.The formula that relates the conjugates for a curved-mirror system is

Mirrors are usually classified as concave and convex.A concave mirror usually projects a real, inverted image,whereas a convex mirror forms an erect,virtual image.

1.2.7 Aberrations

The aberrations of simple, single-element lenses can be quite severe when the lens is comparatively large(with respect to image or object distance)or when the object is located far from the lens axis.When a simple lens is incapable of performing a certain task,it will be necessary to employ a lens,such as a camera lens,whose aberrations have been largely corrected.For specially demanding functions,special lenses may have to be designed and built.

All real lenses made from spherical surfaces may display spherical aberration.Additionally,if the object point is distant from the axis of the lens, or off-axis, the image may display other aberrations, such as astigmatism, coma, distortion, and field curvature. Furthermore, the index of refraction of the lens is a function of wavelength,so its focal length varies slightly with wavelength;the resulting aberration is called chromatic aberration.

Spherical aberration appears both on the axis and off the axis, and does not depend on the distance off-axis. Astigmatism occurs because an off-axis bundle of rays strikes the lens asymmetrically.This asymmetry causes a pair of line images to appear:one behind the plane of best focus and the other in front of it.Coma gives rise to a cometlike image;the head of the comet is the paraxial image point,and the aberration manifests itself as the tail.The tail points away from the axis of the lens and is 3 times longer than its wide. The length of the comatic image, from the paraxial image point to the end of the tail,increases in proportion to the square of the lens diameter and to the distance of the image point from the axis of the lens.The image projected by a lens does not truly lie on a plane but rather on a curved surface, even if other aberrations are zero. This aberration is called field curvature.If the magnification is function of the distance of an image point from the axis,then the image will not be rectilinear.The resulting aberration is called distortion.

Aberrations may be reduced by adjust the radii of the curvature of lens elements so that,for example, angles of incidence are minimized; this process is sometimes called bending the lens. Astigmatism, however, is only weakly influenced by bending the elements. Similarly, one aberration can sometimes be balanced against another. For example, spherical aberration can be partially compensated by moving the image plane from the paraxial image plane to the waist,that is, by compensating spherical aberration by defocusing. Similarly, coma, distortion, and astigmatism can be reduced by adjusting the axis position of the aperture stop.