# Gaussian Analysis of Afocal Lenses

This is a continuation from the previous tutorial - rapid estimation of lens performance.

### Introduction

If collimated (parallel) light rays from an infinitely distant point source fall incident on the input end of a lens system, rays exiting from the output end will show one of three characteristics:

1. They will converge to a real point focus outside the lens system.
2. They will appear to diverge from a virtual point focus within the lens system.
3. They will emerge as collimated rays that may differ in some characteristics from the incident collimated rays.

In cases 1 and 2, the paraxial imaging properties of the lens system can be modeled accurately by a characteristic focal length and a set of fixed principal surfaces. Such lens systems might be called focusing or focal lenses, but are usually referred to simply as lenses.

In case 3, a single finite focal length cannot model the paraxial characteristics of the lens system; in effect, the focal length is infinite, with the output focal point an infinite distance behind the lens, and the associated principal surface an infinite distance in front of the lens.

Such lens systems are referred to as afocal, or without focal length. They will be called afocal lenses here, following the common practice of using ‘‘lens’’ to refer to both single element and multielement lens systems. The first afocal lens was the galilean telescope, a visual telescope made famous by Galileo’s astronomical observations. It is now believed to have been invented by Hans Lipperhey in 1608.

Afocal lenses are usually thought of in the context of viewing instruments or attachments to change the effective focal length of focusing lenses, whose outputs are always collimated.

In fact, afocal lenses can form real images of real objects. A more useful distinction between focusing and afocal lenses concerns which optical parameters are fixed, and which can vary in use.

Focusing lenses have a fixed, finite focal length, can produce real images for a wide range of object distances, and have a linear magnification which varies with object distance.

Afocal lenses have a fixed magnification which is independent of object distance, and the range of object distances yielding real images is severely restricted.

Afocal lenses differ from focusing lenses in ways that are not always obvious. It is useful to review the basic image-forming characteristics of focusing lenses before defining the characteristics unique to afocal lenses.

### Focusing Lenses

In this tutorial, all lens elements are assumed to be immersed in air, so that object space and image space have the same index of refraction.

Points in object space and image space are represented by two rectangular coordinate systems $$(x,y,z)$$ and $$(x',y',z')$$, with the prime indicating image space.

The $$z-$$ and $$z'-$$axes form a common line in space, the optical axis of the system. It is assumed, unless noted otherwise, that all lens elements are rotationally symmetric with respect to the optical axis.

Under these conditions, the imaging geometry of a focusing lens can be defined in terms of two principal points $$P$$ and $$P'$$, two focal points $$F$$ and $$F'$$, and a single characteristic focal length $$f$$, as shown in Fig. 1. $$P$$, $$P'$$, $$F$$, and $$F'$$ all lie on the optical axis.

The focal points $$F$$ and $$F'$$, will be the origins for the coordinate systems $$(x,y,z)$$ and $$(x',y',z')$$. If the origins are at $$P$$ and $$P'$$, the coordinates will be given as $$(x,y,s)$$ and $$(x',y',s')$$, where $$s=z-f$$ and $$s'=z'+f$$.

Normal right-hand sign conventions are used for each set of coordinates, and light travels along the $$z$$-axis from negative $$z$$ toward positive $$z'$$, unless the optical system has internal mirrors. Figure 1 a illustrates the terminology for finite conjugate objects.

Object points and image points are assumed to lie in planes normal to the optical axis, for paraxial computations. Object distance is specified by the axial distance to the object surface, $$z$$ or $$s$$, and image distance by $$z'$$ or $$s'$$. The two most commonly used equations relating image distance to object distance are

$\tag{1}\frac{1}{s'}-\frac{1}{s}=\frac{1}{f}$

and

$\tag{2}zz'=-f^2$

For infinitely distant object points, $$z'=0$$ and $$s'=f$$, and the corresponding image points will lie in the focal plane at $$F'$$.

To determine the actual distance from object plane to image plane, it is necessary to know the distance $$sp$$ between $$P$$ and $$P'$$. The value of $$sp$$ is a constant specific to each real lens system, and may be either positive [moving object and image further apart than predicted by Eqs. (1) or (2)] or negative (moving them closer together).

For rotationally symmetric systems, off-axis object and image coordinates can be expressed by the object height $$h$$ and image height $$h'$$, where $$h^2=x^2+y^2$$ and $$h'^2=x'^2+y'^2$$. Object height and image height are related by the linear magnification $$m$$, where

$\tag{3}m=\frac{h'}{h}=\frac{s'}{s}=\frac{z'+f}{z-f}$

Since the product $$zz'$$ is a constant, Eq. (3) implies that magnification varies with object distance.

The principal surfaces of a focusing lens intersect the optical axis at the principal points $$P$$ and $$P'$$. In paraxial analysis, the principal surfaces are planes normal to the optical axis; for real lenses, they may be curved.

The principal surfaces are conjugate image surfaces for which $$m=+1.0$$. This property makes the raytrace construction shown in Fig. 1a possible, since a ray traveling parallel to the optical axis in either object or image space must intersect the focal point in the conjugate space, and must also intersect both principal surfaces at the same height.

In real lenses, the object and image surfaces may be tilted or curved. Planes normal to the optical axis are still used to define object and image positions for off-axis object points, and to compute magnification.

For tilted object surfaces, the properties of the principal surfaces can be used to relate object surface and image surface tilt angles, as shown in Fig. 1b. Tilted object and image planes intersect the optical axis and the two principal planes. The tilt angles with respect to the optical axis, $$u$$ and $$u'$$, are defined by meridional rays lying in the two surfaces. The points at which conjugate tilted planes intercept the optical axis are defined by $$s_a$$ and $$s_a'$$, given by Eq. (1).

Both object and image planes must intersect their respective principal surfaces at the same height $$y$$, where $$y=s_a\tan{u}=s_a'\tan{u'}$$. It follows that

$\tag{4}\frac{\tan{u'}}{\tan{u}}=\frac{s_a}{s_a'}=\frac{1}{m_a}$

The geometry of Fig. 1b is known as the Scheimpflug condition, and Eq. (4) is the Scheimpflug rule, relating image to object tilt. The magnification $$m_a$$ applies only to the axial image.

The height off axis of an infinitely distant object is defined by the principal ray angle $$u_p$$ measured from $$F$$ or $$P$$, as shown in Fig. 1c. In this case, the image height is

$\tag{5}h'=f\tan{u_p}$

A focusing lens which obeys Eq. (5) for all values of $$u_p$$ within a specified range is said to be distortion-free: if the object is a set of equally spaced parallel lines lying in an object plane perpendicular to the optical axis, it will be imaged as a set of equally spaced parallel lines in an image plane perpendicular to the optical axis, with line spacing proportional to $$m$$.

Equations (1) through (5) are the basic Gaussian imaging equations defining a perfect focusing lens. Equation (2) is sometimes called the newtonian form of Eq. (1), and is the more useful form for application to afocal lens systems.

### Afocal Lenses

With afocal lenses, somewhat different coordinate system origins and nomenclature are used, as shown in Fig. 2.

The object and image space reference points $$RO$$ and $$RE$$ are at conjugate image points. Since the earliest and most common use for afocal lenses is as an aid to the eye for viewing distant objects, image space is referred to as eye space.

Object position is defined by a right-hand coordinate system $$(x_o,y_o,z_o)$$ centered on reference point $$RO$$. Image position in eye space is defined by coordinates $$(x_e,y_e,z_e)$$ centered on $$RE$$.

Because afocal lenses are most commonly used for viewing distant objects, their imaging characteristics are usually specified in terms of angular magnification $$M$$, entrance pupil diameter $$D_o$$, and total field of view.

Figure 2a models an afocal lens used at infinite conjugates. Object height off axis is defined by the principal ray angle $$u_{po}$$, and the corresponding image height is defined by $$u_{pe}$$. Objects viewed through the afocal lens will appear to be magnified by a factor $$M$$, where

$\tag{6}\tan{u_{pe}}=M\tan{u_{po}}$

If $$M$$ is negative, as in Fig. 2a, the image will appear to be inverted. [Strictly speaking, since $$RO$$ and $$RE$$ are separated by a distance $$S$$, the apparent magnification seen by an eye at $$RE$$ with and without the afocal lens will differ slightly from that indicated by Eq. (6) for nearby objects.]

The imaging geometry of an afocal lens for finite conjugates is illustrated in Fig. 2b. Since a ray entering the afocal lens parallel to the optical axis will exit the afocal lens parallel to the optical axis, it follows that the linear magnification $$m$$ relating object height $$h_o$$ and image height $$h_e$$ must be invariant with object distance.

The linear magnification $$m$$ is the inverse of the angular magnification $$M$$:

$\tag{7}m=\frac{h_e}{h_o}=\frac{1}{M}$

The axial separation $$\Delta{z_e}$$ of any two images $$h_{e1}$$ and $$h_{e2}$$ is related to the separation $$\Delta{z_o}$$ of the corresponding objects $$h_{o1}$$ and $$h_{o2}$$ by

$\tag{8}\Delta{z_e}=m^2\Delta{z_o}=\frac{\Delta{z_o}}{M^2}$

It follows that any convenient pair of conjugate image points can be chosen as reference points $$RO$$ and $$RE$$. Given the location of $$RO$$, the reference point separation $$S$$, and the magnifications $$m=1/M$$, the imaging geometry of a rotationally symmetric distortion-free afocal lens can be given as

$\tag{9}x_e=mx_o=\frac{x_o}{M};\qquad{y_e}=my_o=\frac{y_o}{M};\qquad{z_e}=m^2z_o=\frac{z_o}{M^2}$

Equation (9) is a statement that coordinate transformation between object space and eye space is rectilinear for afocal lenses, and is solely dependent on the afocal magnification $$M$$ and the location of two conjugate reference points $$RO$$ and $$RE$$.

The equations apply (paraxially) to all object and image points independent of their distances from the afocal lens. Any straight line of equally spaced object points will be imaged as a straight line of equally spaced image points, even if the line does not lie in a plane normal to the optical axis.

Either $$RO$$ or $$RE$$ may be chosen arbitrarily, and need not lie on the axis of symmetry of the lens system, so long as the $$z_o$$- and $$z_e$$-axes are set parallel to the axis of symmetry.

A corollary of invariance in lateral and axial linear magnification is invariance in angular magnification. Equation (6) thus applies to any ray traced through the afocal system, and to tilted object and image surfaces. In the latter context, Eq. (6) can be seen as an extension of Eq. (4) to afocal lenses.

The eye space pupil diameter $$D_e$$ is of special importance to the design of visual instruments and afocal attachments: $$D_e$$ must usually be large enough to fill the pupil of the associated instrument or eye.

The object space pupil diameter $$D_o$$ is related to $$D_e$$ by

$\tag{10}D_e=\frac{D_o}{M}=mD_o$

### Subjective Aspects of Afocal Imagery

The angular magnification $$M$$ is usually thought of in terms of Eq. (6), which is often taken to indicate that an afocal lens projects an image which is $$M$$-times as large as the object.

Equation (9) shows that the image height is actually $$1/M$$-times the object height (i. e., smaller than the object when $$|M|\gt1$$). Equation 9 also shows, however, that the image distance is reduced to $$1/M^2$$-times the object distance, and it is this combination of linear height reduction and quadratic distance reduction which produces the subjective appearance of magnification.

Equation (6) can be derived directly from Eq. (9).

$\tan{u_{pe}}=\frac{y_e}{z_e}=\frac{y_o/M}{z_o/M^2}=M\tan{u_{po}}$

Equation (9) is therefore a more complete model than Eq. (6) for rotationally symmetric, distortion-free afocal lenses.

Figure 3 illustrates two subjective effects which arise when viewing objects through afocal lenses.

In Fig. 3a, for which $$M=+3\times$$, Eq. (9) predicts that image dimensions normal to the optical axis will be reduced by 1/3, while image dimensions along the optical axis will be reduced by 1/9. The image of the cube in Fig. 3a looks three times as tall and wide because it is nine times closer, but it appears compressed by a factor of 3 in the axial direction, making it look like a cardboard cutout.

This subjective compression, most apparent when using binoculars, is intrinsic to the principle producing angular magnification, and is independent of pupil spacing in the binoculars.

Figure 3a assumes the optical axis is horizontal within the observer’s reference framework. If the axis of the afocal lens is not horizontal, the afocal lens may create the illusion that horizontal surfaces are tilted.

Figure 3b represents an $$M=+7\times$$ afocal lens whose axis is tilted 10° to a horizontal surface. Equation (6) can be used to show that the image of this surface is tilted approximately 51° to the axis of the afocal lens, creating the illusion that the surface is tilted 41° to the observer’s horizon.

This illusion is most noticeable when looking downward at a surface known to be horizontal, such as a body of water, through a pair of binoculars.

### Afocal Lenses and the Optical Invariant

Equations (6) and (7) can be combined to yield

$\tag{11}h_e\tan{u_{pe}}=h_o\tan{u_{po}}$

which is a statement of the optical invariant as applied to distortion-free afocal lenses.

Neither $$u_{po}$$ nor $$u_{pe}$$ is typically larger than 35°–40° in distortion-free afocal lenses, although there are examples with distortion where $$u_{po}\rightarrow90°$$.

Given a limit on one angle, Eq. (11) implies a limit on the other angle related to the ratio $$h_o/h_e=D_o/D_e$$. Put in words, the ratio $$D_o/D_e$$ cannot be made arbitrarily large without a corresponding reduction in the maximum allowable field of view. All designers of afocal lens systems must take this fundamental principle into consideration.

The next tutorial introduces fiber nonlinearities