# Keplerian Afocal Lenses

This is a continuation from the previous tutorial - ** a few practical laser examples**.

A simple afocal lens can be made up of two focusing lenses, an objective and an eyepiece, set up so that the rear focal point of the objective coincides with the front focal point of the eyepiece.

There are two general classes of simple afocal lenses, one in which both focusing lenses are positive, and the other in which one of the two is negative. Afocal lenses containing two positive lenses were first described by Johannes Kepler in Dioptrice, in 1611, and are called keplerian. Lenses containing a negative eyepiece are called galilean, and will be discussed separately. Generally, afocal lenses contain at least two powered surfaces. The simplest model for an afocal lens consists of two thin lenses.

## Thin-Lens Model of a Keplerian Afocal Lens

Figure 4 shows a thin-lens model of a keplerian telescope. The focal length of its objective is \(f_o\) and the focal length of its eyepiece is \(f_e\). Its properties can be understood by tracing two rays, ray 1 entering the objective parallel to the optical axis, and ray 2 passing through \(F_o\), the front focal point of the objective. Ray 1 leads directly to the linear magnification \(m\), and ray 2 to the angular magnification \(M\):

\[\tag{12}m=-\frac{f_e}{f_o};\qquad{M}=-\frac{f_o}{f_e}=\frac{\tan{u}_{pe}}{\tan{u_{po}}}\]

Equation (12) makes the relationship of afocal magnification to the Scheimpflug rule of Eq. (4) [refer to the *Gaussian analysis of afocal lenses tutorial*] more explicit, with focal lengths \(f_o\) and \(f_e\) substituting for \(s_a\) and \(s_a'\).

The second ray shows that placing the reference point \(RO\) at \(F_o\) will result in the reference point \(RE\) falling on \(F_e'\), the rear focal point of the eyepiece. The reference point separation for \(RO\) in this location is

\[\tag{13}\text{SF}=2f_e+2f_o=2(1-M)f_e=2(1-m)f_o\]

Equation (13) can be used as a starting point for calculating any other locations for \(RO\) and \(RE\), in combination with Eq. (9) [refer to the *Gaussian analysis of afocal lenses tutorial*].

One additional generalization can be drawn from Fig. 4: the ray passing through \(F_o\) will emerge from the objective parallel to the optical axis. It will therefore also pass through \(F_e'\) even if the spacing between objective and eyepiece is increased to focus on nearby objects. Thus the angular magnification remains invariant, if \(u_{po}\) is measured from \(F_o\) and \(u_{pe}\) is measured from \(F_e'\), even when adjusting the eyepiece to focus on nearby objects makes the lens system depart from being strictly afocal.

The simple thin-lens model of the keplerian telescope can be extended to systems composed of two real focusing lenses if we know their focal lengths and the location of each lens’ front and rear focal points. Equation (12) can be used to derive \(M\), and \(\text{SF}\) can be measured. Equation (9) [refer to the *Gaussian analysis of afocal lenses tutorial*] can then be used to compute both finite and infinite conjugate image geometry.

## Eye Relief Manipulation

The earliest application of keplerian afocal lenses was to obtain magnified views of distant objects. To view distant objects, the eye is placed at \(RE\). An important design consideration in such instruments is to move \(RE\) far enough away from the last surface of the eyepiece for comfortable viewing.

The distance from the last optical surface to the exit pupil at \(RE\) is called the eye relief \(ER\). One way to increase eye relief \(ER\) is to move the entrance pupil at \(RO\) toward the objective. Most telescopes and binoculars have the system stop at the first surface of the objective, coincident with the entrance pupil, as shown in Fig. 5a.

In the thin-lens model of Fig. 5a, \(RO\) is moved a distance \(zo=f_o\) to place it at the objective. Thus \(RE\) must move a distance \(ze=f_o/M^2=-f_e/M\), keeping in mind that \(M\) is negative in this example. Thus for a thin-lens keplerian telescope with its stop at the objective, the eye relief \(ER_k\) is

\[\tag{14}ER_k=\frac{(M-1)}{M}f_e\]

It is possible to increase the eye relief further by placing the stop inside the telescope, moving the location of \(RO\) into virtual object space. Figure 5b shows an extreme example of this, where the virtual location of \(RO\) has been matched to the real location of \(RE\). For this common-pupil-position case, the eye relief \(ER_{cp}\) is

\[\tag{15}ER_{cp}=\frac{(M-1)}{(M+1)}f_e\]

A price must be paid for locating the stop inside the afocal lens, in that the elements ahead of the stop must be increased in diameter if the same field of view is to be covered without vignetting.

The larger the magnitude of \(M\), the smaller the gain in \(ER\) yielded by using an internal stop. To increase the eye relief further, it is necessary to make the objective and/or the eyepiece more complex, increasing the distance between \(F_o\) and the first surface of the objective, and between the last surface of the eyepiece and \(F_e'\). If this is done, placing \(RO\) at the first surface of the objective will further increase \(ER\).

Figure 6 shows a thin-lens model of a telephoto focusing lens of focal length \(f_t\). For convenience, a zero Petzval sum design is used, for which \(f_1=f\) and \(f_2=-f\). Given the telephoto’s focal length \(f_t\) and the lens separation \(d\), the rest of the parameters shown in Fig. 6 can be defined in terms of the constant \(C=d/f_t\). The component focal length \(f\), back focal length \(bfl\), and front focal length \(ffl\), are given by

\[\tag{16}f=f_tC^{1/2};\qquad{bfl}=f_t(1-C^{1/2});\qquad{ffl}=f_t(1+C^{1/2})\]

and the total physical length \(ttl\) and focal point separation \(sf\) are given by

\[\tag{17}ttl=f_t(1+C-C^{1/2});\qquad{sf}=f_t(2+C)\]

The maximum gain in eye relief will be obtained by using telephoto designs for both objective and eyepiece, with the negative elements of each facing each other. Two cases are of special interest. First, \(ttl\) can be minimized by setting \(C=0.25\) for both objective and eyepiece. In this case, the eye relief \(ER_{ttl}\) is

\[\tag{18}ER_{ttl}=1.5\frac{(M-1)}{M}f_e=1.5ER_k\]

Second, \(sf\) can be maximized by setting \(C=1.0\) for both objective and eyepiece. This places the negative element at the focal plane, merging the objective and eyepiece negative elements into a single negative field lens. The eye relief in this case, \(ER_{sf}\), is

\[\tag{19}ER_{sf}=2.0\frac{(M-1)}{M}f_e=2.0ER_k\]

Placing a field lens at the focus between objective and eyepiece can be problematical, when viewing distant objects, since dust or scratches on the field lens will be visible.

If a reticle is required, however, it can be incorporated into the field lens. Equations (14), (18), and (19) show that significant gains in eye relief can be made by power redistribution. In the example of Eq. (18), the gain in \(ER\) is accompanied by a reduction in the physical length of the optics, which is frequently beneficial.

## Terrestrial Telescopes

Keplerian telescopes form an inverted image, which is considered undesirable when viewing earthbound objects. One way to make the image erect, commonly used in binoculars, is to incorporate erecting prisms. A second is to insert a relay stage between objective and eyepiece, as shown in Fig. 7.

The added relay is called an image erector, and telescopes of this form are called terrestrial telescopes. (The keplerian telescope is often referred to as an astronomical telescope, to distinguish it from terrestrial telescopes, since astronomers do not usually object to inverted images. Astronomical has become ambiguous in this context, since it now more commonly refers to the very large aperture reflecting objectives found in astronomical observatories. Keplerian is the preferred terminology.) The terrestrial telescope can be thought of as containing an objective, eyepiece, and image erector, or as containing two afocal relay stages.

There are many variants of terrestrial telescopes made today, in the form of binoculars, theodolites, range finders, spotting scopes, rifle scopes, and other military optical instrumentation. All are offshoots of the keplerian telescope, containing a positive objective and a positive eyepiece, with intermediate relay stages to perform special functions.

## Field of View Limitations in Keplerian and Terrestrial Telescopes

The maximum allowable eye space angle \(u_{pe}\) and magnification \(M\) set an upper limit on achievable fields of view, in accordance with Eq. (11) [refer to the *Gaussian analysis of afocal lenses tutorial*]. MIL-HDBK-141 lists one eyepiece design for which the maximum \(u_{pe}\) = 36°. If \(M=7\times\), using that eyepiece allows a 5.9° maximum value for \(u_{po}\). It is a common commercial practice to specify the total field of view \(\text{FOV}\) as the width in feet which subtends an angle \(2u_{po}\) from 1000 yards away, even when the pupil diameter is given in millimeters. \(\text{FOV}\) is thus given by

\[\tag{20}\text{FOV}=6000\tan{u_{po}}=\frac{6000}{M}\tan{u_{pe}}\]

For our \(7\times\) example, with \(u_{pe}\) = 36°, \(\text{FOV}\) = 620 feet at 1000 yard. For commercial \(7\times50\) binoculars (\(M=7\times\) and \(D_o\) = 50 mm), \(\text{FOV}\) = 376 feet at 1000 yard is more typical.

## Finite Conjugate Afocal Relays

If an object is placed in contact with the front surface of the keplerian telescope of Fig. 5, its image will appear a distance \(ER_k\) behind the last surface of the eyepiece, in accordance with Eq. (14). There is a corresponding object relief distance \(OR_k= M^2ER_k\) defining the position of an object that will be imaged at the output surface of the eyepiece, as shown in Fig. 8.

\(OR_k\) and \(ER_k\) define the portions of object space and eye space within which real images can be formed of real objects with a simple keplerian afocal lens.

\[\tag{21}OR_k=M(M-1)f_e\]

Object relief is enlarged by the power redistribution technique used to extend eye relief. Thus there is a minimum total length design corresponding to Eq. (18), for which the object relief \(OR_{ttl}\) is

\[\tag{22}OR_{ttl}=1.5M(M-1)f_e\]

and a maximum eye relief design corresponding to Eq. (19), for which \(OR_{sf}\)

\[\tag{23}OR_{sf}=2.0M(M-1)f_e\]

is also maximized.

Figure 9 shows an example of a zero Petzval sum finite conjugate afocal relay designed to maximize \(OR\) and \(ER\) by placing a negative field lens at the central infinite conjugate image.

Placing the stop at the field lens means that the lens is telecentric (principal rays parallel to the optical axis) in both object and eye space. As a result, magnification, principal ray angle of incidence on object and image surface, and cone angle are all invariant over the entire range of \(OR\) and \(ER\) for which there is no vignetting.

Magnification and cone angle invariance means that object and image surfaces can be tilted with respect to the optical axis without introducing keystoning or variation in image irradiance over the field of view. Having the principal rays telecentric means that object and image position can be adjusted for focus without altering magnification. It also means that the lens can be defocused without altering magnification, a property very useful for unsharp masking techniques used in the movie industry.

One potential disadvantage of telecentric finite conjugate afocal relays is evident from Fig. 9: to avoid vignetting, the apertures of both objective and eyepiece must be larger than the size of the associated object and image.

While it is possible to reduce the diameter of either the objective or the eyepiece by shifting the stop to make the design nontelecentric, the diameter of the other lens group becomes larger. Afocal relays are thus likely to be more expensive to manufacture than focusing lens relays, unless object and image are small.

Finite conjugate afocal lenses have been used for alignment telescopes, for laser velocimeters, and for automatic inspection systems for printed circuit boards. In the last case, invariance of magnification, cone angle, and angle of incidence on a tilted object surface make it possible to measure the volume of solder beads automatically with a computerized video system. Finite conjugate afocal lenses are also used as Fourier transform lenses.

## Afocal Lenses for Scanners

Many optical systems require scanners, and if the apertures of the systems are large enough, it is preferable to place the scanner inside the system.

Although scanners have been designed for use in convergent light, they are more commonly placed in collimated light. A large aperture objective can be converted into a high magnification keplerian afocal lens with the aid of a short focal length eyepiece collimator, as shown in Fig. 10, providing a pupil in a collimated beam in which to insert a scanner.

For the polygonal scanner shown, given the desired scan angle and telescope aperture diameter, Eq. (11) [refer to the *Gaussian analysis of afocal lenses tutorial*] will define the combination of scanner facet size and number of facets needed to achieve the desired scanning efficiency.

Scanning efficiency is the time it takes to complete one scan divided by the time between the start of two sequential scans. It is tied to the ratio of facet length to beam diameter, the amount of vignetting allowed within a scan, the number of facets, and the angle to be scanned.

Two limitations need to be kept in mind. First, the optical invariant will place an upper limit on \(M\) for the given combination of \(D_o\) and \(u_{po}\), since there will be a practical upper limit on the achievable value of \(u_{pe}\). Second, it may be desirable in some cases for the keplerian afocal relay to have enough barrel distortion so that Eq. (6) [refer to the *Gaussian analysis of afocal lenses tutorial*] becomes

\[\tag{24}u_{pe}=Mu_{po}\]

An afocal lens obeying Eq. (24) will convert a constant rotation rate of the scan mirror into a constant angular scan rate for the external beam. The same property in ‘‘f-theta’’ focusing lenses is used to convert a constant angular velocity scanner rotation rate into a constant linear velocity rate for the recording spot of light.

The above discussion applies to scanning with a point detector. When the detector is a linear diode array, or when a rectangular image is being projected onto moving film, the required distortion characteristics for the optical system may be more complex.

## Imaging in Binoculars

Most commercial binoculars consist of two keplerian afocal lenses with internal prismatic image erectors. Object and image space coordinates for binoculars of this type are shown schematically in Fig. 11.

Equation (9) [refer to the *Gaussian analysis of afocal lenses tutorial*] can be applied to Fig. 11 to analyze their imaging properties. In most binoculars, the spacing \(S_o\) between objectives differs from the spacing \(S_e\) between eyepieces, and \(S_o\) may be either larger or smaller than \(S_e\).

Each telescope has its own set of reference points, \(ROL\) and \(REL\) for the left telescope, and \(ROR\) and \(RER\) for the right. Object space is a single domain with a single origin \(O\). The object point at \(z_o\), midway between the objective axes, will be \(x_{oL}\) units to the right of the left objective axis, and \(x_{oR}\) units to the left of the right objective axis.

In an ideal binocular system , the images of the object formed by the two telescopes would merge at one point, \(z_e\) units in front of eye space origin \(E\). This will happen if \(S_o=MS_e\), so that \(x_{eL}=x_{oL}/M\) and \(x_{eR}=x_{oR}/M\).

In most modern binoculars, however, \(S_o\ll{M}S_e\), and separate eye space reference points \(EL\) and \(ER\) will be formed for the left and right eye. As a result, each eye sees its own eye space, and while they overlap, they are not coincident. This property of binoculars can affect stereo acuity and eye accommodation for the user.

It is normal for the angle at which a person’s left-eye and right-eye lines of sight converge to be linked to the distance at which the eyes focus. Eyes focused for a distance \(z_e\) normally would converge with an angle \(\beta\), as shown in Fig. 11. When \(S_o\ll{M}S_e\), as is commonly the case, the actual convergence angle \(\beta'\) is much smaller.

A viewer for whom focus distance is strongly linked to convergence angle may find such binoculars uncomfortable to use for extended periods, and may be in need of frequent focus adjustment for different object distances.

A related but more critical problem arises if the axes of the left and right telescopes are not accurately parallel to each other. Misalignment of the axes requires the eyes to twist in unaccustomed directions to fuse the two images, and refocusing the eyepiece is seldom able to ease the burden.

Jacobs is one of the few authors to discuss this problem. Jacobs divides the axes misalignment into three categories: (1) misalignments requiring a divergence \(D\) of the eye axes to fuse the images, (2) misalignments requiring a convergence \(C\), and (3) misalignments requiring a vertical displacement \(V\).

The tolerance on allowable misalignment in minutes of arc is given by Jacobs as

\[\tag{25}D=7.5/(M-1);\qquad{C}=22.5/(M-1);\qquad{V}=8.0/(M-1)\]

Note that the tolerance on \(C\), which corresponds to convergence to focus on nearby objects, is substantially larger than the tolerances on \(D\) and \(V\).

The next tutorial introduces ** galilean and inverse galilean afocal lenses**.