Angular Magnification





The Optical Principles of Telescopes


The principle of angular magnification is to use an optical system to change the angle formed at the nodal point of the eye by the rays of light from the object. Telescopes are a very effective way of producing this angular magnification, which occurs without changing the object or the viewing distance. The patient can achieve an enlarged retinal image whilst staying at their chosen distance from the task, whether this is at distance (street signs, bus numbers, whiteboard), intermediate (TV, music, playing cards, game console) or near (writing, handicrafts). The disadvantage is the very restricted field-of-view (FOV) allowed by such devices, and they can rarely be used whilst the patient is mobile (although some special designs for this purpose are described in the section on ‘ Telescopes for Mobility ’). Even if the patient is stationary, telescopes are often used to view moving objects (e.g. watching sport) which is challenging because of the FOV, and magnification of the object’s apparent speed.


Telescopes in low-vision work are often required to focus on objects closer than infinity, and they can be modified to correct for the wearer’s refractive error. Their optical principles, however, are those of afocal systems, where parallel rays of light enter the telescope from an infinitely distant object, and parallel rays leaving the telescope form a final image at infinity. The two types of telescope used in low-vision work are illustrated simply in Fig. 9.1A , the astronomical (Keplerian), and Fig. 9.1B , the Galilean telescope. In the astronomical telescope, a ray from the bottom of the object forms the top of the image, and thus the image is inverted (and also laterally reversed). This is obviously unsuitable for low-vision work, so a reflecting system (a combination of prisms or mirrors) is always included to reorient the image. The more correct name for such a telescope is ‘terrestrial’, but ‘astronomical’ has become accepted in the low vision field. In the astronomical telescope, the convex objective lens F O forms an image of the distant object (focuses the incident parallel light) at F′ O , the second focal point of this lens. The distance between the image and the objective lens is obviously the second focal length, f′ O . Light then diverges from this focus and is refracted by the convergent eyepiece lens F E . If this lens is positioned so that its first focal point F E coincides exactly with F′ O and the image, then parallel light will emerge from the system. For the Galilean telescope, the eyepiece lens F E is negative and is positioned so that its first focal point is coincident with F′ O : rays of light converging towards F′ O are intercepted before focusing and emerge parallel from the system. The ray of light which left the top of the object emerges at the top of the image: the image is erect and no additional components are required to make practical use of the system.




Fig. 9.1


A schematic representation of the optical system of (A) an astronomical and (B) a Galilean telescope. In the astronomical telescope, the top-most ray ‘ a ‘ entering the telescope becomes the lowest ray of the exiting bundle, so the image is inverted. In the Galilean telescope, the order of the rays is the same on entering and exiting, so the image is erect.

See text for explanation of variables.


The reflecting system which the astronomical telescope requires to produce an erect image is usually a prism system which takes advantage of total internal reflection. A typical example is illustrated in Fig. 9.2 , where the use of two right-angled (Porro) prisms is illustrated. This also illustrates how the use of prisms allows the optical path length between the objective and eyepiece lenses to be ‘folded’, thus reducing the overall length of the telescope. The separation of the two lenses t = f′ O + f′ E , and as these second focal lengths are both positive in the astronomical telescope, this will be longer than the Galilean telescope of equivalent magnification (where f′ E is negative): folding of the light path as described above may be able to reduce the difference in physical size between the finished telescopes.




Fig. 9.2


The use of two right-angled (Porro) prisms as an example of the way in which the image in an astronomical telescope can be laterally and vertically inverted. The prisms also allow the light path to be ‘folded’ to the reduced distance b shown in (A). The full path length between the objective ( F O ) and eyepiece ( F E ) lenses is shown in (B).


Fig. 9.3 illustrates the extra-axial rays from an infinitely distant object which shows how the angular magnification is produced by the telescopes: there is an increase in the angle made by the rays with the optical axis after passing through the telescope.


<SPAN role=presentation tabIndex=0 id=MathJax-Element-1-Frame class=MathJax style="POSITION: relative" data-mathml='Magnification(M)=anglesubtendedateyebyimageanglesubtendedateyebyobject=θ′θ’>Magnification(𝑀)=𝑎𝑛𝑔𝑙𝑒𝑠𝑢𝑏𝑡𝑒𝑛𝑑𝑒𝑑𝑎𝑡𝑒𝑦𝑒𝑏𝑦𝑖𝑚𝑎𝑔𝑒𝑎𝑛𝑔𝑙𝑒𝑠𝑢𝑏𝑡𝑒𝑛𝑑𝑒𝑑𝑎𝑡𝑒𝑦𝑒𝑏𝑦𝑜𝑏𝑗𝑒𝑐𝑡=𝜃′𝜃Magnification(M)=anglesubtendedateyebyimageanglesubtendedateyebyobject=θ′θ
Magnification(M)=anglesubtendedateyebyimageanglesubtendedateyebyobject=θ′θ



Fig. 9.3


The path of rays from the extra-axial ( solid lines ) and axial ( dashed lines ) portions of an infinitely distant object through (A) an astronomical and (B) a Galilean afocal telescope. In each case the objective lens ( F O ) and the eyepiece lens ( F E ) are a distance t apart and are assumed to be thin. The object subtends an angle θ at F O and the image subtends an angle θ′ at the eye. h′ is the size of the primary image formed by F O . The position of the exit pupil ( EP ) is also shown.


since the object would subtend an angle θ at the eye without the telescope.


From the shaded triangles in Fig. 9.3 :


a. Astronomical


<SPAN role=presentation tabIndex=0 id=MathJax-Element-2-Frame class=MathJax style="POSITION: relative" data-mathml='θ=−h′f′O’>𝜃=−ℎ′𝑓′𝑂θ=−h′f′O
θ=−h′f′O


and


<SPAN role=presentation tabIndex=0 id=MathJax-Element-3-Frame class=MathJax style="POSITION: relative" data-mathml='−θ′=h′f′E’>−𝜃′=ℎ′𝑓′𝐸−θ′=h′f′E
−θ′=h′f′E


So


<SPAN role=presentation tabIndex=0 id=MathJax-Element-4-Frame class=MathJax style="POSITION: relative" data-mathml='θ′=h′f′E’>𝜃′=ℎ′𝑓′𝐸θ′=h′f′E
θ′=h′f′E


b. Galilean


<SPAN role=presentation tabIndex=0 id=MathJax-Element-5-Frame class=MathJax style="POSITION: relative" data-mathml='θ=−h′−fO=−h′f′O’>𝜃=−ℎ′−𝑓𝑂=−ℎ′𝑓′𝑂θ=−h′−fO=−h′f′O
θ=−h′−fO=−h′f′O


and


<SPAN role=presentation tabIndex=0 id=MathJax-Element-6-Frame class=MathJax style="POSITION: relative" data-mathml='θ′=−h′−f′E=h′f′E’>𝜃′=−ℎ′−𝑓′𝐸=ℎ′𝑓′𝐸θ′=−h′−f′E=h′f′E
θ′=−h′−f′E=h′f′E


Thus for either afocal telescope,


<SPAN role=presentation tabIndex=0 id=MathJax-Element-7-Frame class=MathJax style="POSITION: relative" data-mathml='M=θ′θ=h′f′E−f′Oh′=−f′Of′E’>𝑀=𝜃′𝜃=ℎ′𝑓′𝐸−𝑓′𝑂ℎ′=−𝑓′𝑂𝑓′𝐸M=θ′θ=h′f′E−f′Oh′=−f′Of′E
M=θ′θ=h′f′E−f′Oh′=−f′Of′E


or


<SPAN role=presentation tabIndex=0 id=MathJax-Element-8-Frame class=MathJax style="POSITION: relative" data-mathml='M=−FEFO’>𝑀=−𝐹𝐸𝐹𝑂M=−FEFO
M=−FEFO


As both component lenses are positive in the astronomical telescope, the resultant magnification value is negative, showing that the image is inverted and that an erecting system is required: magnification for the Galilean telescope is positive indicating an erect image. In both systems, to obtain magnification values numerically greater than one requires that the eyepiece lens must be the more powerful. In purely practical terms, the powers of both components should be high, so the corresponding focal lengths will be short and the overall length of the telescope minimised. The use of more powerful lenses will inevitably cause aberrations to affect the final image quality. Thus, in high-magnification astronomical systems (up to 14× is available) the eyepiece and objective lenses each consist of up to four components to minimise aberrations. This large number of air/glass interfaces (added to those of the erecting system) inevitably causes a loss of image brightness, even with antireflection-coated lenses. By contrast, Galilean systems are not generally available beyond 3.5× distance magnification due to the poor image quality associated with the higher powers. This means that the objective and eyepiece lenses are generally of lower power than in the astronomical designs, and in the interest of producing compact and lightweight aids, each component may be reduced to a single aspheric lens.


Focal Telescopes


The telescopic systems used in low-vision work are therefore basically afocal, and the formula derived for their magnification (M = − F E / F O ) is only applicable when they are used in that way. However, telescopes are rarely used by emmetropic eyes to view infinitely distant objects: in normal circumstances, ametropic eyes viewing objects at less remote distances need to be considered. When the telescopes are used in this way, the magnification will be changed, but it is possible to calculate the extent to which this occurs.


The ‘Vergence Amplification’ Effect


It will be realised by anyone using a telescope that it needs to be refocused for each different object distance and that the observer’s own accommodation cannot be used to create clear retinal images of near objects. The reason for this is that the amount of accommodation required for viewing objects at a finite distance through the telescope is greatly in excess of the expected value. The incident vergence has in fact been ‘amplified’ by its passage through the telescope. derived a precise formula for the actual emergent vergence:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-9-Frame class=MathJax style="POSITION: relative" data-mathml='U′=M2U(1−tMU)’>𝑈′=𝑀2𝑈(1−𝑡𝑀𝑈)U′=M2U(1−tMU)
U′=M2U(1−tMU)


where U’ is the emergent vergence as it leaves the telescope eyepiece; U is the actual incident vergence at the telescope objective; M is the magnification of the telescope (positive for Galilean, negative for astronomical) and t is the optical path length of the telescope.


An approximate formula has become commonly used:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-10-Frame class=MathJax style="POSITION: relative" data-mathml='U′=M2U’>𝑈′=𝑀2𝑈U′=M2U
U′=M2U


In words, this becomes:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-11-Frame class=MathJax style="POSITION: relative" data-mathml='emergentvergence=(magnification)2×incidentvergence’>emergentvergence=(magnification)2×incidentvergenceemergentvergence=(magnification)2×incidentvergence
emergentvergence=(magnification)2×incidentvergence


or in terms of its effect on the patient:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-12-Frame class=MathJax style="POSITION: relative" data-mathml='actual​accommodationrequired=(magnification)2×expectedaccommodation’>actualaccommodationrequired=(magnification)2×expectedaccommodationactual​accommodationrequired=(magnification)2×expectedaccommodation
actual​accommodationrequired=(magnification)2×expectedaccommodation


In fact, this approximate formula is only accurate for a limited range of conditions, but it does illustrate the general problem that patients rarely possess the amount of accommodation required for even modest viewing distances. For example, a 3× telescope used for viewing at 1 m may require approximately 9.00 DS of accommodation rather than the expected 1.00 DS.


Compensating for Ametropia


There are three ways in which either astronomical or Galilean telescopes could be adapted to compensate for spherical ametropia:



  • 1.

    Adding the full refractive correction to the eyepiece


  • 2.

    Adding a partial refractive correction to the objective


  • 3.

    Changing the telescope length



Considering the practicalities of each of these methods in turn:


Adding the Full Refractive Correction to the Eyepiece


This is a very simple strategy. In practice, it is realised by the patient holding the telescope up against their spectacles; or the telescope being clipped over the spectacles; or a small auxiliary lens being attached behind the eyepiece lens of the telescope. The magnification of the telescope is unchanged, because the telescope is still afocal: parallel light leaves the telescope and it is only then that its vergence is changed before entering the eye. The method allows both spherical and cylindrical refractive errors to be corrected.


Adding a Partial Refractive Correction to the Objective


It is theoretically possible to place a partial correction in front of the objective lens, to slightly alter the vergence of light entering the telescope. This vergence would then be amplified by its passage through the telescope, to the extent that the vergence of light leaving the telescope would be appropriate to correct the refractive error. This method is never used practically because the degree of correction required, and its influence on the magnification, are difficult to calculate, and it offers no practical advantage.


Changing the Telescope Length


The uncorrected myope could shorten the telescope by the amount required to make the previously parallel light leaving the afocal telescope into light which is divergent to the correct extent to correct the refractive error and be clearly focused on the retina. The hypermetrope would lengthen the telescope to create a convergent emergent beam. This is a useful practical strategy in cases of low (in Galilean telescopes) to moderate/high (in astronomical telescopes) degrees of ametropia. Cylindrical ametropia cannot be corrected in this way, but it has been suggested that uncorrected astigmatism up to 2.00 DC does not influence acuity. Larger spherical ametropias may also create problems because of physical restrictions in the change of telescope length that the housing of the telescope lenses will permit. The telescope is clearly no longer afocal, but a method of calculating the change in magnification can easily be determined by using an example:


Consider a Galilean telescope consisting of an eyepiece lens F E = −50.00 DS and an objective lens F O = +20.00 DS. The magnification, if the telescope is operating as an afocal telescope, is M = − F E / F O = −(−50.00)/+20.00 = +2.5×. Fig. 9.4A shows the relative position of the lenses, separated by the algebraic sum of their focal lengths: t = f′ O + f′ E = 50 mm + (−20 mm) = 30 mm.




Fig. 9.4


(A) A diagram showing the distances ( t ) between the component lenses ( F O and F E ) in a 2.5× Galilean telescope, focussed for infinity (afocal). (B) shows the same telescope focussed for use by an uncorrected −10.00 DS myope. (C) shows the equivalent component lens positions for a 2.5× astronomical telescope, focused for infinity. (D) shows the astronomical telescope focussed for use by an uncorrected −10.00 DS myope. F O ’ and F E are the focal points of the objective and eyepiece lenses respectively.


If a −10.00 DS myope were to use the telescope, and the ametropia were corrected by a spectacle correction placed behind the eyepiece, no focusing of the telescope (i.e. no change in length) would be required, and the magnification would remain at 2.5×. If the uncorrected −10.00 DS myope is now to look through the telescope, refocusing would be required. The situation can be thought of as shown in Fig. 9.4B . Consider the −50.00 DS eyepiece to be made of two components, a −10.00 DS element which is being ‘borrowed’ to correct the ametropia, and a −40.00 DS element which is the eyepiece of an afocal telescope. In order to make the ‘new’ Galilean telescope afocal, the length of the telescope must be altered to t = f′ O + f′ E = 50 + (−25) mm = 25 mm, so it is shortened by 5 mm in the focusing process. In addition, the magnification changes: the effective eyepiece power is now F E = −40.00 DS (the original −50.00 DS − (−10.00 DS)) which has been used to correct the ametropia). Magnification M = − F E / F O = −(−40.00)/+20.00 = +2.0×. Therefore, the myope obtains less magnification by using the telescope in this way.


If the telescope were an astronomical one of equivalent power, but with a positive eyepiece this time, the analogous calculations can be made ( Figs. 9.4C and D ):


Assume that the eyepiece lens F E = +50.00 DS and the objective lens F O = +20.00 DS. The magnification, if the telescope is operating as an afocal telescope, is


<SPAN role=presentation tabIndex=0 id=MathJax-Element-13-Frame class=MathJax style="POSITION: relative" data-mathml='M=−FEFO=−+50.00+20.00=−2.5×’>𝑀=−𝐹𝐸𝐹𝑂=−+50.00+20.00=−2.5×M=−FEFO=−+50.00+20.00=−2.5×
M=−FEFO=−+50.00+20.00=−2.5×


If the ametropia were corrected by a spectacle correction placed behind the eyepiece, no focusing of the telescope (i.e. no change in length) would be required, and the magnification would remain at 2.5×. This is exactly analogous to the situation when using the Galilean telescope. Fig. 9.4C shows the relative position of the lenses, separated by the algebraic sum of their focal lengths: t = f′ O + f′ E = 50 + 20 mm = 70 mm. Consider the +50.00 DS eyepiece to be made of two components, a −10.00 DS element which is being ‘borrowed’ to correct the ametropia, and a +60.00 DS element which is the eyepiece of an afocal telescope. In order to make the ‘new’ astronomical telescope afocal, the length of the telescope must be altered to t = f′ O + f′ E = 50 + 16.7 mm = 66.7 mm, so it is shortened by 3.3 mm in the focusing process. In addition, the magnification changes: the effective eyepiece power is now F E = +60.00 DS (the original +50.00 DS − (−10.00 DS) which has been used to correct the ametropia). Magnification M = − F E / F O = −(+60.00)/+20.00 = −3.0×. Therefore, the myope gets more magnification by using the telescope in this way, rather than placing the telescope over their spectacle lens. The equivalent argument applied to the hypermetropic wearer shows that this patient would get more magnification if using the Galilean telescope and lengthening it to correct the ametropia, but less magnification if using the astronomical telescope. The effects in the astronomical and Galilean telescopes are opposite because it is the apparent power of the eyepiece lens which is changing, and this is of opposite sign in the two instruments.


Of course, the effects on magnification were quite dramatic in the example given, because the degree of ametropia to be corrected was large—more limited effects will be experienced with low refractive errors. There are also occasions when the choice presented here is not available for practical reasons—the patient may, for example, be a contact lens wearer who cannot remove the correction just to use the telescope intermittently. As is clear from the diagrams in Fig. 9.4 , the length changes required seem practical, and they are even less if the eyepiece lens is of higher power. However, as suggested previously, Galilean telescopes are often not manufactured to have an extensive focusing ability. The typical range of ametropia compensated may be ±3.00 DS, compared to ±15.00 DS or more in an astronomical system.


Intermediate and Near Viewing


As discussed earlier, the telescope user will not be able to accommodate to view at these distances. In an analogous manner to the way in which ametropic correction is provided, there are three ways in which the afocal telescope can be adapted to intermediate or near viewing:



  • 1.

    Adding full correction for the viewing distance to the objective


  • 2.

    Adding increased correction for the viewing distance to the eyepiece


  • 3.

    Focusing the telescope by changing (increasing) the telescope length



Considering the practicalities of each of these methods in turn:


Adding Full Correction for the Viewing Distance to the Objective


This is the simplest practical solution, because the power of the correcting lens, the working space, and the ‘new’ magnification of the system are all easily determined. If, for example, the patient wishes to view a near object at a distance of 50 cm, the light entering the telescope would be divergent (vergence = −2.00 DS), and the telescope would no longer be afocal. Addition of a +2.00 DS lens in front of the objective lens would neutralise this divergence: parallel light would now enter the telescope, which would once again be afocal. This positive lens power can be incorporated into the objective lens or can be clipped over the objective as a reading cap . The use of a reading cap is the more versatile option, as the cap can be removed or changed so that the telescope can be used for a variety of purposes. This modified system is often called a telemicroscope ( Fig. 9.5 ).




Fig. 9.5


A schematic representation of the optical principles of a telemicroscope—a telescope with full correction for the near viewing distance added over the objective lens ( F O ) as a reading cap ( RC ). The object is placed at the anterior focal point ( F RC ) of RC, so parallel light enters the (still afocal) telescope (Galilean in this case). F E is the telescope eyepiece lens.


The magnification of the system is the product of that provided by its individual components, sototal magnification = afocal telescope magnification × plus-lens reading cap magnification


<SPAN role=presentation tabIndex=0 id=MathJax-Element-14-Frame class=MathJax style="POSITION: relative" data-mathml='MTOTAL=MTEL×FRC4′>𝑀𝑇𝑂𝑇𝐴𝐿=𝑀𝑇𝐸𝐿×𝐹𝑅𝐶4MTOTAL=MTEL×FRC4
MTOTAL=MTEL×FRC4


Such a formula can also be used to calculate the plus-lens reading cap which will be required to produce a particular total magnification. For example, if the patient required 6× magnification at near, and the telemicroscope was to be formed using a 2× telescope as its base, then


<SPAN role=presentation tabIndex=0 id=MathJax-Element-15-Frame class=MathJax style="POSITION: relative" data-mathml='MTOTAL=MTEL×FRC4′>𝑀𝑇𝑂𝑇𝐴𝐿=𝑀𝑇𝐸𝐿×𝐹𝑅𝐶4MTOTAL=MTEL×FRC4
MTOTAL=MTEL×FRC4


And substituting gives


<SPAN role=presentation tabIndex=0 id=MathJax-Element-16-Frame class=MathJax style="POSITION: relative" data-mathml='6=2×FRC4′>6=2×𝐹𝑅𝐶46=2×FRC4
6=2×FRC4

<SPAN role=presentation tabIndex=0 id=MathJax-Element-17-Frame class=MathJax style="POSITION: relative" data-mathml='FRC4=3,’>𝐹𝑅𝐶4=3,FRC4=3,
FRC4=3,


so


<SPAN role=presentation tabIndex=0 id=MathJax-Element-18-Frame class=MathJax style="POSITION: relative" data-mathml='FRC=+12.00DS’>𝐹𝑅𝐶=+12.00DSFRC=+12.00DS
FRC=+12.00DS


The working space is not affected by the afocal telescope, so (as for any plus-lens system) it is simply the anterior focal length of the plus lens:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-19-Frame class=MathJax style="POSITION: relative" data-mathml='fRC=1FRC=1+12.00=+0.0833m=8.33cm’>𝑓𝑅𝐶=1𝐹𝑅𝐶=1+12.00=+0.0833m=8.33cmfRC=1FRC=1+12.00=+0.0833m=8.33cm
fRC=1FRC=1+12.00=+0.0833m=8.33cm


It is now possible to see the advantage of such a system over that which uses a plus-lens alone. To continue with the same example, if 6× magnification was to be produced using only a plus lens, then


<SPAN role=presentation tabIndex=0 id=MathJax-Element-20-Frame class=MathJax style="POSITION: relative" data-mathml='M=Feq4′>𝑀=𝐹𝑒𝑞4M=Feq4
M=Feq4


So F eq = 4 × M = 4 × 6 = +24.00 DS


The working space is the anterior focal length of the plus lens, so


<SPAN role=presentation tabIndex=0 id=MathJax-Element-21-Frame class=MathJax style="POSITION: relative" data-mathml='feq=1Feq=1+24.00=+0.0417m=4.17cm’>𝑓𝑒𝑞=1𝐹𝑒𝑞=1+24.00=+0.0417m=4.17cmfeq=1Feq=1+24.00=+0.0417m=4.17cm
feq=1Feq=1+24.00=+0.0417m=4.17cm


So using the telescopic system has allowed the working space to be increased by 2×; that is, by a factor equal to the magnification of the telescope used. If a 3× telescope had been used to form the basis of the system, the working space would have been increased by 3×.


This can be stated mathematically as:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-22-Frame class=MathJax style="POSITION: relative" data-mathml='fRC=MTEL×feq’>𝑓𝑅𝐶=𝑀𝑇𝐸𝐿×𝑓𝑒𝑞fRC=MTEL×feq
fRC=MTEL×feq


where f RC is the anterior focal length of the near telescope (i.e. the anterior focal length of the reading cap), MTEL is the telescope magnification, and f eq is the anterior focal length of the plus-lens of equal magnification.


Whilst the working space of a plus-lens is fixed once a particular level of magnification has been chosen, the near telescopic system could be any one of a number of combinations of afocal telescope and reading cap. Taking the example of 2.5× magnification, the plus-lens which would provide this would have power +10.00 DS, and a working space of 10 cm ( F eq = 4 M , f eq = 1/ F eq ). A variety of near telescopes (telemicroscopes) could give the same magnification:




  • A 1.5× afocal telescope with a +6.75 DS reading cap, working space ~15 cm



  • A 2.0× afocal telescope with a +5.00 DS reading cap, working space 20 cm



  • A 4.0× afocal telescope with a +2.50 DS reading cap, working space 40 cm



The anterior focal length of an unknown system, and hence the working space of a near telescope, can be determined by measurement of its front vertex power (the reciprocal of this distance) using a focimeter. Despite the increased working space, there are situations where it is so small that it has no clinical advantage. For example, even if the working space is 2× larger with a telescope than with a plus lens, this will not be useful in functional terms if the increase is only from 2 cm to 4 cm: it has been suggested that the increase must be at least 5 cm to be practically worthwhile. In fact if the full working distance is considered, measured from the eye to the task, this includes the length of the magnifying system, as well as the working space. As the telescopic system is longer than a plus-lens, this will further increase the advantage of such a device ( Fig. 9.6 ).




Fig. 9.6


A comparison of the degree of convergence required for near vision in (A) plus-lens and (B) telescopic systems of equivalent magnification. u is the distance of the magnifier from the eye’s centre of rotation: working space in (A) is the focal length of the plus lens magnifier ( f M ) but in (B) it is the anterior focal length of the reading cap ( f RC ). In the latter case, the working distance is also increased by the length of the telescope ( t ). The telescope tubes must be correctly centred for the patient’s pupillary distance and correctly angled for the convergence of the visual axes. This position is achieved when the view of the patient’s eyes from directly in front appears as in (C).


As well as the obvious advantage of increased space in which to perform manipulative tasks, or more comfortable working postures, the more remote position of the task renders binocular viewing easier to achieve. Whereas binocular magnification is almost impossible beyond 2.5× with a plus-lens system ( F eq = +10.00 DS, working space = f eq = 10 cm), telescopic systems are commercially available up to 5× magnification. Binocular magnification at near is not without its problems, however, because a way must be found to accurately convergence the two telescope tubes to match the convergence of the patient’s visual axes. This requires a way of compensating for the interpupillary distance of the patient, and the particular working distance employed.


Adding Increased Correction for the Viewing Distance to the Eyepiece


If the afocal telescope is used to view a near or intermediate object, the divergent light from the object which enters the telescope objective will have its vergence amplified by passage through the telescope. Thus, it would require a much stronger plus-lens placed over the eyepiece in order to make the light parallel than would be required if positioned over the objective. The power of such a lens, and the resultant magnification of the ‘new’ system, are difficult to calculate unless the system parameters are known in detail, and the high power of the lens makes the method practically unprofitable. It may be worth considering if the patient already possesses a very high-power reading addition in spectacle-mounted form to provide near magnification, but the final effect on magnification and working space would be unpredictable.


Focusing the Telescope by Changing (Increasing) the Telescope Length


Changing the length of the telescope by increasing the separation of the objective and eyepiece is a practical and often-used method of adapting the afocal telescope for finite working distances. The only limit is in the physical restriction on practical tube lengths allowed by the particular device, with astronomical telescopes in general allowing a greater range of focus than Galilean devices. A consideration of some examples will show why this is the case. The effect of focussing for intermediate or near distances by changing the telescope length can be found in an analogous way to that used when considering the effects of correcting for ametropia by changing the telescope length. In the case of correcting for ametropia, this was considered as ‘borrowing’ some of the power of the eyepiece to do this, whereas when focusing for near objects, this will instead borrow some of the objective lens power. The aim is the same: it is necessary to be able to consider the system as an afocal telescope, as the optical characteristics of such a device are well-known.


Consider two 3× telescopes, one Galilean and one astronomical, which have component lenses of equivalent powers, as illustrated in Table 9.1 .



Table 9.1

A Step-by-Step Calculation of Magnification for Example Telescopes Focussed for Near Viewing Using Two Alternative Methods: Adding a Reading Cap and Increasing the Length

























Galilean Astronomical
Afocal telescope components


  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-23-Frame class=MathJax style="POSITION: relative" data-mathml='FE=−45.00f′E=−22mm’>𝐹𝐸=−45.00𝑓′𝐸=−22mmFE=−45.00f′E=−22mm
    F E = − 45.00 f ′ E = − 22 mm



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-24-Frame class=MathJax style="POSITION: relative" data-mathml='FO=+15.00f′O=−67mm’>𝐹𝑂=+15.00𝑓′𝑂=−67mmFO=+15.00f′O=−67mm
    F O = + 15.00 f ′ O = − 67 mm



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-25-Frame class=MathJax style="POSITION: relative" data-mathml='t=f′E+f′O=(−22)+67=45mm’>𝑡=𝑓′𝐸+𝑓′𝑂=(−22)+67=45mmt=f′E+f′O=(−22)+67=45mm
    t = f ′ E + f ′ O = ( − 22 ) + 67 = 45 mm



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-26-Frame class=MathJax style="POSITION: relative" data-mathml='MTEL=−(−45)/+15=3×’>𝑀𝑇𝐸𝐿=−(−45)/+15=3×MTEL=−(−45)/+15=3×
    M T E L = − ( − 45 ) / + 15 = 3 ×


  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-27-Frame class=MathJax style="POSITION: relative" data-mathml='FE=+​45.00f′E=22mm’>𝐹𝐸=+45.00𝑓′𝐸=22mmFE=+​45.00f′E=22mm
    F E = + ​ 45.00 f ′ E = 22 mm



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-28-Frame class=MathJax style="POSITION: relative" data-mathml='FO=+​15.00f′O=67mm’>𝐹𝑂=+15.00𝑓′𝑂=67mmFO=+​15.00f′O=67mm
    F O = + ​ 15.00 f ′ O = 67 mm



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-29-Frame class=MathJax style="POSITION: relative" data-mathml='t=f′E+f′O=22+67=89mm’>𝑡=𝑓′𝐸+𝑓′𝑂=22+67=89mmt=f′E+f′O=22+67=89mm
    t = f ′ E + f ′ O = 22 + 67 = 89 mm



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-30-Frame class=MathJax style="POSITION: relative" data-mathml='MTEL=−45/15=−3×’>𝑀𝑇𝐸𝐿=−45/15=−3×MTEL=−45/15=−3×
    M T E L = − 45 / 15 = − 3 ×

Add separate reading cap to focus for 20 cm (+5.00 required)


  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-31-Frame class=MathJax style="POSITION: relative" data-mathml='MTOTAL=MTEL×MRC=MTEL×(FRC/4)=3×(5/4)=3×1.25=3.75×’>𝑀𝑇𝑂𝑇𝐴𝐿=𝑀𝑇𝐸𝐿×𝑀𝑅𝐶=𝑀𝑇𝐸𝐿×(𝐹𝑅𝐶/4)=3×(5/4)=3×1.25=3.75×MTOTAL=MTEL×MRC=MTEL×(FRC/4)=3×(5/4)=3×1.25=3.75×
    M T O T A L = M T E L × M R C = M T E L × ( F R C / 4 ) = 3 × ( 5 / 4 ) = 3 × 1.25 = 3.75 ×



  • (no change in length of telescope)

Refocus telescope for 20 cm


  • ‘Borrow’ +5.00 DS from objective; ‘new’ telescope has



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-32-Frame class=MathJax style="POSITION: relative" data-mathml='FO=+15.00−(+5.00)=+10.00′>𝐹𝑂=+15.00−(+5.00)=+10.00FO=+15.00−(+5.00)=+10.00
    F O = + 15.00 − ( + 5.00 ) = + 10.00

Telescope focussed for near


  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-33-Frame class=MathJax style="POSITION: relative" data-mathml='FE=−45.00f′E=−22mm’>𝐹𝐸=−45.00𝑓′𝐸=−22mmFE=−45.00f′E=−22mm
    F E = − 45.00 f ′ E = − 22 mm



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-34-Frame class=MathJax style="POSITION: relative" data-mathml='FO=+​10.00f′O=100mm’>𝐹𝑂=+10.00𝑓′𝑂=100mmFO=+​10.00f′O=100mm
    F O = + ​ 10.00 f ′ O = 100 mm



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-35-Frame class=MathJax style="POSITION: relative" data-mathml='t=F′E+f′O=(−22)+100=78mm’>𝑡=𝐹′𝐸+𝑓′𝑂=(−22)+100=78mmt=F′E+f′O=(−22)+100=78mm
    t = F ′ E + f ′ O = ( − 22 ) + 100 = 78 mm



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-36-Frame class=MathJax style="POSITION: relative" data-mathml='MTEL=−(−45)/+10=4.5×’>𝑀𝑇𝐸𝐿=−(−45)/+10=4.5×MTEL=−(−45)/+10=4.5×
    M T E L = − ( − 45 ) / + 10 = 4.5 ×


  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-37-Frame class=MathJax style="POSITION: relative" data-mathml='FE=+​45.00f′E=22mm’>𝐹𝐸=+45.00𝑓′𝐸=22mmFE=+​45.00f′E=22mm
    F E = + ​ 45.00 f ′ E = 22 mm



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-38-Frame class=MathJax style="POSITION: relative" data-mathml='FO=+​10.00f′O=100mm’>𝐹𝑂=+10.00𝑓′𝑂=100mmFO=+​10.00f′O=100mm
    F O = + ​ 10.00 f ′ O = 100 mm



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-39-Frame class=MathJax style="POSITION: relative" data-mathml='t=F′E+f′O=22+100=122mm’>𝑡=𝐹′𝐸+𝑓′𝑂=22+100=122mmt=F′E+f′O=22+100=122mm
    t = F ′ E + f ′ O = 22 + 100 = 122 mm



  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-40-Frame class=MathJax style="POSITION: relative" data-mathml='MTEL=−45/10=−4.5×’>𝑀𝑇𝐸𝐿=−45/10=−4.5×MTEL=−45/10=−4.5×
    M T E L = − 45 / 10 = − 4.5 ×

Borrow power from objective to focus for near


  • <SPAN role=presentation tabIndex=0 id=MathJax-Element-41-Frame class=MathJax style="POSITION: relative" data-mathml='MTOTAL=MTEL×MRC=MTEL×(FRC/4)=4.5×(5/4)=4.5×1.25=5.6×’>𝑀𝑇𝑂𝑇𝐴𝐿=𝑀𝑇𝐸𝐿×𝑀𝑅𝐶=𝑀𝑇𝐸𝐿×(𝐹𝑅𝐶/4)=4.5×(5/4)=4.5×1.25=5.6×MTOTAL=MTEL×MRC=MTEL×(FRC/4)=4.5×(5/4)=4.5×1.25=5.6×
    M T O T A L = M T E L × M R C = M T E L × ( F R C / 4 ) = 4.5 × ( 5 / 4 ) = 4.5 × 1.25 = 5.6 ×



  • (increased overall magnification, but increased length)

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Jul 15, 2023 | Posted by in OPHTHALMOLOGY | Comments Off on Angular Magnification

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