Rigid Lens Optics


Unlike soft lenses, which drape to fit the cornea so that on the eye the geometry of the back surface closely conforms to that of the anterior cornea, the back surface of a rigid lens (corneal, corneoscleral or scleral) maintains its shape. As a result, a tear lens of predictable form and power is generated between the contact lens and the cornea. The overall optical system producing the retinal image therefore effectively contains three elements: the rigid lens, the tear lens and the eye itself. We can imagine each of these elements as being separated from its neighbour by an infinitely thin film of air ( Fig. 13.1A ). This chapter is largely concerned with the role of the tear lens and the factors that influence its properties. Aberration will also be discussed. Questions of effectivity, spectacle magnification and related matters are addressed in Chapter 3 .

Fig. 13.1

(A) The basic tear lens concept. Although exaggerated for clarity, the air gaps between the contact and tear lenses and tear lens and cornea are effectively negligible. See text for definitions of r 1 , r 2 and r C . (B) Geometry of the tear lens as a function of fit.

Appendix A details conventions for specifying the essential design features of contact lenses (soft or rigid) as well as the terms, symbols and abbreviations used to describe these features.

Basic Tear Lens Properties

The power of the tear lens, sometimes called the liquid, fluid or lacrimal lens, depends on the relative geometry of the optic zone of the back surface of the rigid lens and the anterior surface of the cornea. Fig. 13.1B shows possible variations in the basic tear lens form. The tear lens may contribute negative, zero or positive power to the overall lens–eye system, depending on whether the fitting of the rigid lens is flat, in alignment or steep, respectively.

From a clinical perspective, it is important to determine the likely magnitude of the power of the tear lens and how it varies as the back optic zone radius (BOZR) of the lens is changed. If r 1 and r 2 are the front optic zone radius (FOZR) and BOZR of the rigid lens and r C is the radius of the anterior cornea, then the front and back radii of the tear lens are r 2 and r C , respectively. With corneal lenses it is reasonable to consider that the thickness of the tear lens can be neglected, so that its power, F T , will therefore be:

<SPAN role=presentation tabIndex=0 id=MathJax-Element-1-Frame class=MathJax style="POSITION: relative" data-mathml='FT=nT−11/r2−1/rc=nT−1rc−r2/r2rc’>𝐹T=(𝑛Tβˆ’1)(1/π‘Ÿ2βˆ’1/π‘Ÿc)=(𝑛Tβˆ’1)(π‘Ÿcβˆ’π‘Ÿ2)/π‘Ÿ2π‘ŸcFT=nTβˆ’11/r2βˆ’1/rc=nTβˆ’1rcβˆ’r2/r2rc
where n T is the refractive index of the tears (1.336). Both r 2 and r C will normally be about 8 mm. This gives, with adequate accuracy,
<SPAN role=presentation tabIndex=0 id=MathJax-Element-2-Frame class=MathJax style="POSITION: relative" data-mathml='FT≈1.336−1rc−r2/64×10−6′>𝐹Tβ‰ˆ(1.336βˆ’1)(π‘Ÿcβˆ’π‘Ÿ2)/(64Γ—10βˆ’6)FTβ‰ˆ1.336βˆ’1rcβˆ’r2/64Γ—10βˆ’6
where the radii are in metres. Setting, as an example, r C – r 2 = 0.05 mm, we find
<SPAN role=presentation tabIndex=0 id=MathJax-Element-3-Frame class=MathJax style="POSITION: relative" data-mathml='FT≈0.336⋅0.05×10−3/64×10−6≈+0.25D’>πΉπ‘‡β‰ˆ(0.336)β‹…(0.05Γ—10βˆ’3)/64Γ—10βˆ’6β‰ˆ+0.25DFTβ‰ˆ0.336β‹…0.05Γ—10βˆ’3/64Γ—10βˆ’6β‰ˆ+0.25D

Thus as an approximate rule of thumb, for a rigid lens, the tear lens power increases by about +0.25 D for each 0.05 mm and the BOZR of the lens is steeper than the corneal radius. Correspondingly, on any cornea the back vertex power (BVP) of the rigid contact lens needs to be changed by –0.25 D for each 0.05 mm that the BOZR is made steeper, to compensate for the extra positive power of the liquid lens. If the lens BOZR is made flatter by 0.05 mm, the BVP needs to be changed by +0.25 D.

More exact calculation of F T can, of course, be carried through using Equation 13.1. The approximation is less acceptable for corneas that are much steeper or flatter than the normal value of about 7.8 mm. This can be seen if we note that, for small changes Ξ΄r 2 in r 2 , we can write for the change, Ξ΄F T , in tear lens power,

<SPAN role=presentation tabIndex=0 id=MathJax-Element-4-Frame class=MathJax style="POSITION: relative" data-mathml='δFT=(∂FT/∂r2)δr2=−[(nT−1)/r22]δr2′>𝛿𝐹T=(βˆ‚πΉT/βˆ‚π‘Ÿ2)π›Ώπ‘Ÿ2=βˆ’[(𝑛Tβˆ’1)/π‘Ÿ22]π›Ώπ‘Ÿ2Ξ΄FT=(βˆ‚FT/βˆ‚r2)Ξ΄r2=βˆ’[(nTβˆ’1)/r22]Ξ΄r2

That is, for a given change in r 2 , the change in tear lens power is actually inversely proportional to r 2 2 .

The Tear Lens During Trial Lens Fits

Trial or diagnostic lenses are often used to find the BOZR that gives the required fit with a particular lens design – an overrefraction then being carried out to determine any additional power needed in combination with the trial lens used to give the patient clear vision. In this case, the ordered lens power is simply the sum of the BVP of the trial lens and the overrefraction (assuming that the power of the latter is small enough for effectivity to be ignored; see Chapter 3 ). This is because the BOZR and corneal radii, r 2 and r C , with the final lens will be exactly the same as with the trial lens so that the tear lens has equal power in both cases ( Fig. 13.2 ).

Fig. 13.2

Geometry of the rigid lens and the tear lens when a trial lens with the same BOZR as the final lens is used. BOZR , Back optic zone radius; BVP , back vertex power.

Required BVP When the Lens to Be Ordered Has a Different BOZR From the Trial Lens Used

The situation may arise where a trial lens with a given BOZR is not available, in which case it may be necessary to order a lens with a BOZR that differs from that of the trial lens actually used. Since the BOZR, r 2 , in the two cases differs, so also will the power of the tear lens, and this, in turn, will influence the required BVP of the ordered lens. This effect is best illustrated by an example ( Fig. 13.3 ).

Fig. 13.3

Change in tear lens geometry and required back vertex power BVP when the BOZR of the ordered lens differs from that of the trial lens. BOZR , Back optic zone radius; BVP , back vertex power.

Suppose that the trial lens giving a fit that is closest to that desired has a BOZR of 8.10 mm and a BVP of –3.00 D. With this lens, an overrefraction of +1.00 D is found. Consideration of fluorescein patterns and the corneal radius, however, suggests that the trial lens fit is too steep and that the ordered lens should have a BOZR ( r 2 ) = 8.25 mm. Qualitatively, it can be seen from Fig. 13.3 that this flattens the front surface of the tear lens and hence tends to make its power less positive. To compensate for this, the final contact lens must have relatively more positive power. What is the required BVP for the ordered lens?

Evidently, taking into account the BVPs of the trial lens and the overrefraction, the ideal BVP for a lens of the original 8.10 mm BOZR would be

<SPAN role=presentation tabIndex=0 id=MathJax-Element-5-Frame class=MathJax style="POSITION: relative" data-mathml='F′v=−3.00+1.00=−2.00D’>𝐹′v=βˆ’3.00+1.00=βˆ’2.00DFβ€²v=βˆ’3.00+1.00=βˆ’2.00D

Thus light would leave the back surface of the correcting lens of 8.10-mm BOZR with a vergence of –2.00 D. The power of the anterior surface of the original tear lens (in air) is

<SPAN role=presentation tabIndex=0 id=MathJax-Element-6-Frame class=MathJax style="POSITION: relative" data-mathml='1.336−1/8.10×10−3=+41.48D’>(1.336βˆ’1)/(8.10Γ—10βˆ’3)=+41.48D1.336βˆ’1/8.10Γ—10βˆ’3=+41.48D

Using the general vergence equation,

<SPAN role=presentation tabIndex=0 id=MathJax-Element-7-Frame class=MathJax style="POSITION: relative" data-mathml='L′−L=F’>πΏβ€²βˆ’πΏ=𝐹Lβ€²βˆ’L=F
where L and L β€² are the object and image vergences and F is the surface power, we find that, after passing through the anterior surface of the tear lens, the vergence of the light would be
<SPAN role=presentation tabIndex=0 id=MathJax-Element-8-Frame class=MathJax style="POSITION: relative" data-mathml='−2.00+41.48=+39.48D’>βˆ’2.00+41.48=+39.48Dβˆ’2.00+41.48=+39.48D

If we assume that the tear lens has a negligible thickness, this vergence incident upon the rear surface of the tear lens will correct the eye.

Now consider the ordered lens with a BOZR of 8.25 mm. Since the rear surface of the tear lens is unchanged, we again require that the vergence of the light leaving the anterior surface of the tear lens be +39.48 D. The power of the anterior surface of the new tear lens is

<SPAN role=presentation tabIndex=0 id=MathJax-Element-9-Frame class=MathJax style="POSITION: relative" data-mathml='1.336−1/8.25×10−3=+40.73D’>(1.336βˆ’1)/(8.25Γ—10βˆ’3)=+40.73D1.336βˆ’1/8.25Γ—10βˆ’3=+40.73D

Thus the vergence, L , incident upon the anterior surface of the tear lens must be

<SPAN role=presentation tabIndex=0 id=MathJax-Element-10-Frame class=MathJax style="POSITION: relative" data-mathml='L=L′−F=39.48−40.73=−1.25D’>𝐿=πΏβ€²βˆ’πΉ=39.48βˆ’40.73=βˆ’1.25DL=Lβ€²βˆ’F=39.48βˆ’40.73=βˆ’1.25D

Since this vergence is provided by the correcting lens, the required BVP for the ordered lens of BOZR 8.25 mm is –1.25 D (as compared with the value of –2.00 D for a lens of BOZR 8.10 mm). This agrees well with our earlier rule of thumb, which suggests that, since the BOZR is 0.15 mm (or 3 Γ— 0.05 mm) flatter, the required BVP must be +0.75 D (or 3 Γ— 0.25 D) more positive than its original value of –2.00 D; that is the required BVP is –1.25 D, as also found by the more exact approach.

Calculation of Required Surface Radii From a Trial Lens Fit

Assuming that the BOZR that gives the desired fit has been determined by the use of trial lenses, it is of interest to determine what value of FOZR, r 1 , is needed to give the required BVP, given values for the centre thickness and refractive index for the lens. Again, it is helpful to give an illustrative example to demonstrate the thick lens calculation involved.

Let us suppose that it is found that a good fit is achieved with a trial lens of BVP –7.00 D and a BOZR of 8.00 mm. The overrefraction is –1.00 D at a vertex distance of 13 mm. The lens index is 1.49 and its centre thickness is to be 0.30 mm.

We are justified in ignoring tear lens effects, since these are identical for the initial and final lens. As the overrefraction is small, effectivity can be ignored and the required BVP for our final lens is –8.00 D.

The power of the back surface of the final lens is

<SPAN role=presentation tabIndex=0 id=MathJax-Element-11-Frame class=MathJax style="POSITION: relative" data-mathml='F2=1−1.49/8.00×10−3=−61.25D’>𝐹2=(1βˆ’1.49)/(8.00Γ—10βˆ’3)=βˆ’61.25DF2=1βˆ’1.49/8.00Γ—10βˆ’3=βˆ’61.25D

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Aug 6, 2023 | Posted by in OPHTHALMOLOGY | Comments Off on Rigid Lens Optics

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