Practical Optical System Layout- And Use of Stock Lenses.pdf

Source: Practical Optical System Layout

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Chapter

1.1 Introduction, Assumptions, and

Conventions

This chapter is intended to provide the reader with the tools neces-

sary to determine the location, size, and orientation of the image

formed by an optical system. These tools are the basic paraxial equa-

tions which cover the relationships involved. The word “paraxial” is

more or less synonymous with “first-order” and “gaussian”; for our

purposes it means that the equations describe the image-forming

properties of a perfect optical system. You can depend on well-correct-

ed optical systems to closely follow the paraxial laws.

In this book we make use of certain assumptions and conventions

which will simplify matters considerably. Some assumptions will

eliminate a very small minority* of applications from consideration;

this loss will, for most of us, be more than compensated for by a large

gain in simplicity and feasibility.

Conventions and assumptions

1. All surfaces are figures of rotation having a common axis of sym-

metry, which is called the optical axis.

2. All lens elements, objects, and images are immersed in air with

an index of refraction n of unity.

*Primarily, this refers to applications where object space and image space each has a

different index of refraction. The works cited in the bibliography should be consulted in

the event that this or other exceptions to our assumptions are encountered.

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Chapter One

3. In the paraxial region Snell’s law of refraction ( n sin I

′

sin I

)

are the angles between

the ray and the normal to the surface which separates two media

whose indices of refraction are n and n

, where i and i

′

4. Light rays ordinarily will be assumed to travel from left to right

in an optical medium of positive index. When light travels from

right to left, as, for example, after a single reflection, the medium

is considered to have a negative index.

5. A distance is considered positive if it is measured to the right of a

reference point; it is negative if it is to the left. In Sec. 1.2 and

following, the distance to an object or an image may be measured

from ( a ) a focal point, ( b ) a principal point, or ( c ) a lens surface,

as the reference point.

6. The radius of curvature r of a surface is positive if its center of

curvature lies to the right of the surface, negative if the center is

to the left. The curvature c is the reciprocal of the radius, so that

′

1/ r.

7. Spacings between surfaces are positive if the next (following) sur-

face is to the right. If the next surface is to the left (as after a

reflection), the distance is negative.

8. Heights, object sizes, and image sizes are measured normal to

the optical axis and are positive above the axis, negative below.

9. The term “element” refers to a single lens. A “component” may be

one or more elements, but it is treated as a unit.

10. The paraxial ray slope angles are not angles but are differential

slopes. In the paraxial region the ray “angle” u equals the dis-

tance that the ray rises divided by the distance it travels. (It

looks like a tangent, but it isn’t.)

1.2 The Cardinal (Gauss) Points and Focal

Lengths

When we wish to determine the size and location of an image, a com-

plete optical system can be simply and conveniently represented by

four axial points called the cardinal, or Gauss, points. This is true for

both simple lenses and complex multielement systems. These are the

first and second focal points and the first and second principal points.

The focal point is where the image of an infinitely distant axial object

is formed. The (imaginary) surface at which the lens appears to bend

the rays is called the principal surface. In paraxial optics this surface

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′

becomes simply ni

′

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is called the principal plane. The point where the principal plane

crosses the optical axis is called the principal point.

Figure 1.1 illustrates the Gauss points for a converging lens sys-

tem. The light rays coming from a distant object at the left define the

second focal point F 2 and the second principal point P 2 . Rays from an

object point at the right define the “first” points F 1 and P 1 . The focal

length f (or effective focal length efl ) of the system is the distance

from the second principal point P 2 to the second focal point F 2 . For a

lens immersed in air (per assumption 2 in Sec. 1.1), this is the same

as the distance from F 1

to P 1 . Note that for a converging lens as

F 2

P 2

bfl

f=efl

F 1

P 1

ffl

f=efl

Figure 1.1 The Gauss, or cardinal, points are the first and second focal

points F 1 and F 2 and the first and second principal points P 1 and P 2 . The

focal points are where the images of infinitely distant objects are formed.

The distance from the principal point P 2 to the focal point F 2 is the effective

focal length efl (or simply the focal length f ). The distances from the outer

surfaces of the lens to the focal points are called the front focal length ffl

and the back focal length bfl .

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Chapter One

shown in Fig. 1.1, the focal length has a positive sign according to our

sign convention. The power

of the system is the reciprocal of the

1/ f. Power is expressed in units of reciprocal length,

e.g., in 1 or mm 1 ; if the unit of length is the meter, then the unit of

power is called the diopter. For a simple lens which converges (or

bends) rays toward the axis, the focal length and power are positive; a

diverging lens has a negative focal length and power.

The back focal length bfl is the distance from the last (or right-hand)

surface of the system to the second focal point F 2 . The front focal

length ffl is the distance from the first (left) surface to the first focal

point F 1 . In Fig. 1.1, bfl is a positive distance and ffl is a negative dis-

tance. These points and lengths can be calculated by raytracing as

described in Sec. 1.5, or, for an existing lens, they can be measured.

The locations of the cardinal points for single-lens elements and

mirrors are shown in Fig. 1.2. The left-hand column shows converg-

ing, or positive, focal length elements; the right column shows diverg-

ing, or negative, elements. Notice that the relative locations of the

focal points are different; the second focal point F 2 is to the right for

the positive lenses and to the left for the negative. The relative posi-

tions of the principal points are the same for both. The surfaces of a

positive element tend to be convex and for a negative element concave

(exception: a meniscus element, which by definition has one convex

and one concave surface, and may have either positive or negative

power). Note, however, that a concave mirror acts like a positive, con-

verging element, and a convex mirror like a negative element.

1.3 The Image and Magnification Equations

The use of the Gauss or cardinal points allows the location and size of

an image to be determined by very simple equations. There are two

commonly used equations for locating an image: (1) Newton’s equa-

tion, where the object and image locations are specified with refer-

ence to the focal points F 1 and F 2 , and (2) the Gauss equation, where

object and image positions are defined with respect to the principal

points P 1 and P 2 .

Newton’s equation

′

f 2

(1.1)

gives the image location as the distance from F 2 , the second

focal point; f is the focal length; and x is the distance from the first

′

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focal length f ;

where x

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f 1

P 1

f 2

P 1

f 1

P 2

Biconvex

Biconcave

f 1

P 1

P 2

f 2

P 1

P 2

f 1

Plano convex

Plano concave

f 1

P 1

P 2

f 2

P 1

P 2

f 1

Positive meniscus

Negative meniscus

f 2

P 2

f 2

Concave mirror

(converging)

Convex mirror

(diverging)

Figure 1.2 Showing the location of the cardinal, or Gauss, points for lens elements. The

principal points are separated by approximately ( n 1)/ n times the axial thickness of

the lens. For an equiconvex or equiconcave lens, the principal points are evenly spaced

in the lens. For a planoconvex or planoconcave lens, one principal point is always on

the curved surface. For a meniscus shape, one principal point is always outside the

lens, on the side of the more strongly curved surface. For a mirror, the principal points

are on the surface, and the focal length is half of the radius. Note that F 2 is to the right

for the positive lens element and to the left for the negative.

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