In a recent article,^{1} I described a method for determining the refractive index of a lens by first measuring the lens power in air, and then measuring the lens power in a solution of known refractive index, such as water. This method works for positive or negative power lenses, thin or thick lenses, and spherical or aspheric lenses. The refractive index of progressive addition lenses can also be found, as long as the same position on the variable lens surface is measured in both instances. Calculation would then reveal the lens refractive index. It can be shown that measuring the front vertex power of a lens leads to less error in the value of refractive index determined than if back vertex power is used (originally the formula indicated back vertex power).

μ=(nK_{v,1} − K_{v,n})/(K_{v,1} − K_{v,n})

where n is the refractive index of the immersion medium, μ is the refractive index of the lens, K_{v,1} is the lens power determined in air, and K_{v,n} is the lens power determined in the immersion medium liquid.

Recently, a patient presented who wanted their spectacles remade (because one of the lenses was cut irretrievably small). The prescription was plano R and L, with three prism diopters base out (split). I did not know what lens material the lenses were made from. The outside edge thickness of the prism wedge was a concern to me. Because the method outlined in the previous article is useless if the lens power is zero (and of reduced accuracy for very low powers), I attempted using the essence of the previous method but measuring the prism power rather than the vergence power in this instance.

The lenses were measured in a glass-sided cell with a hinged “roof”of glass. Dimensions were ∼75 × 75 × 12 mm. The glass used was standard 3-mm plate glass. No thickness variation across the plate glass was evident. The lensmeter I have used is a Tomey TL 100 (Tomey Corporation, Nishi-ku, Nagoya 451, Japan). This is a digital lensmeter.

As an experiment, I measured the 10 prism diopter lens in my trial lens set and, using the method outlined above, found a value for refractive index (n) of 1.50 (I already knew the material was CR39). I then measured the 1.5 prism diopter plano lens from the spectacles in question. The value for n was calculated to be 1.60. I asked my laboratory to ascertain what material they thought the lens material was. They verified that it was a “mid index” material, which was consistent with my finding (this is obviously not the only possible mid index material, but it was encouraging). I will provide a rationale for why this method is valid below. A thorough examination of results for lenses of different powers and materials can be done elsewhere.

#### METHODS

Consider a prism in air.

For simplicity of calculation, the prism shall be regarded as a right-angled type rather than the symmetrical isosceles type.

Let δ_{a} be the deviation between the initial and final direction of a light ray in air.

It can be shown that for small angles of incidence, the angle of deviation (δ) is independent of the angle of incidence.^{2} Hence, the prism should be placed so that the angle of incidence is small.

If this same thin lens was in air, then the deviation (δ_{a}) could be derived from applying Snell's Law at the back (or second) surface of the prism, because the angle of incidence at the first surface will be arranged to be zero, and therefore can be ignored.

Using Snell's Law,

where δ_{a} is the deviation in air, α is the apex angle of the prism, and n_{2} is the refractive index of the prism.

Consider a prism in solution, as seen in Fig. 1.

If the prism is now immersed in solution of refractive index n_{1} then

Snell's Law can be applied at the rear surface of the prism (again the front surface is aligned so that the angle of incidence is zero and can be ignored).

n_{2}sin α = n_{1}sin(δ + α)

An important point in this application of the equation is to realize that the δ value referred to here is the deviation between the light entering and leaving the prism while still in solution. I will assume that the immersion liquid is water.

I will assign this value of δ the term δ_{w}. So,

When light meets the liquid/glass boundary and the glass/air boundary, it will be refracted. For calculation simplicity, the glass can be ignored, and the interface can be regarded as a water/air interface. By using Snell's Law,

where n_{3} is the refractive index in air and δ_{ia} denotes the immersed prism value of δ, measured in air. This quantity can be measured by a lensmeter. (To be precise, the prismatic power is measured, which is related to the deviation angle by the equation Δ/100 = tan δ, where Δ is the prism, measured in prism diopters.)

Because the refractive index of air is 1.00, we have

From Eqs. 1 and 2 we have two expressions for the same thing, so

Equation 1 Image Tools |
Equation 2 Image Tools |
Equation (Uncited) Image Tools |

Expanding the left-hand side of this equation, it becomes

n_{1}(sinδ_{w} cos α + sinα cos δ_{w})

Substituting sin δ_{w} = sin δ_{ia}/n_{1} and

(because sin^{2}θ +cos^{2}θ = 1) then the left-hand side becomes

So

Dividing by sin α

Rearranging Eq. 6 to make cot α the subject,

Equation (1) can be expanded to become

n_{2}sin α = sin δ_{a} cos α + cos δ_{a} sin α

Dividing both sides by sin α,

n_{2} = sin δ_{a} cot α + cos δ_{a}

Rearranging to make cot α the subject,

Combining Eqs. 7 and 8,

Equation 7 Image Tools |
Equation 8 Image Tools |

Cross multiplying,

Bringing the n_{2} terms together

Hence,

If the complex form in Eq. 9 has paraxial type approximations applied to it, such as sin δ_{a} = δ_{a}, sin δ_{ia} = δ_{ia}, sin δ_{w} = δ_{w},cos δ_{a} = 1, sin δ_{ia}^{2} = 0, then the equation simplifies to

n_{2} = (1δ_{ia} − δ_{a}n_{1})/(δ_{ia} − δ_{a})

which can be tidied up to

where δ_{a} is the deviation the prism induces in air and δ_{ia} is the deviation induced when the prism is immersed in solution n_{1}, measured in air.

As stated earlier, the δ_{ia} and δ_{a} terms are similar to but not exactly the same as the prismatic deviation as measured by a lensmeter. Hence, the equation could be altered by substituting δ_{ia} = atan(Δ_{2}/100) and δ_{a} = atan(Δ_{1}/100), where Δ_{1} is the lensmeter prism, measurement in air and Δ_{2} is the prism, measurement in air of the immersed prism. By using a simplification akin to the paraxial approximation of sin θ = θ, we can say tan θ = θ and so δ = Δ/100. Then the prism measurement can be substituted for the deviation, so the equation becomes

This means that by measuring the prismatic power of any lens in air and again in a solution of known refractive index, the refractive index of the lens can be calculated.

An alternative derivation of this formula using paraxial optics is given in Appendix (Supplemental Digital Content 1, http://links.lww.com/OPX/A14).

#### RESULTS

If we devise some likely lens prisms using different refractive index materials values, we can find how different the values for refractive index generated by the equations (paraxial and non-paraxial) are (Table 1).

Because of the method of calculation, the paraxial method (Eq. 10) is shown as the true result.

Table 1 shows the small difference in values arising from the use of the paraxial form of the equations, in comparison to the other methods. Another method involved non-paraxial formulae being used for both the deviation being converted to prism, and for refraction. The last column used the paraxial approximation regarding prism, and non-paraxial refraction formula. In all four variants, the error in refractive index value calculated is <0.003, even with very large amounts of prism. The difference decreases as the refractive index of the prism increases. This is probably because a higher refractive index prism will have smaller refraction angles and hence be more “paraxial” in its nature for the same amount of prism.

The use of prism values directly into the paraxial equation does increase the error of the method, but Table 1 shows the effect is very minor, with a maximum error in the refractive index calculated being <0.003.

One implication of this prismatic method of assessing refractive index of a lens is that there are now two separate ways to calculate the refractive index of a lens. That is, either the lens power in air and lens power in, for example, water or the prismatic power in air and in, for example, water can be used to derive a refractive index value. The prismatic power will vary across the surface of the lens when a lens has dioptric power, which may limit its usefulness. Yet, in instances where the lens power is very small or zero, the prism power method appears to be a useful adjunct to the method previously shown for lens power. Thin lens theory limits accuracy when the power of thick lenses is being measured. Being inside a wetcell also alters the lens power measured slightly. These factors do not affect the prismatic power.

I am not aware of any lens type for which these methods of refractive index determination would be invalid. Of course, multifocal lenses would require special care to ensure that the same location was measured on both occasions when the lens is measured.

A method to assist with this might be to use a plastic overlay on the lens (such as those that come with some multifocal lenses) with an aperture allowing lensmeter assessment only at that location.

The error of measurement the lensmeter manufacturer accepts is that of ISO 8598, which allows for an error of 0.1 prism diopters up to 10 prism diopters and 0.2 prism diopters for powers above 10 prism diopters (C. Loveless, personal communication, 2007). The refractive index calculation hinges on the accuracy of both measurements of prism.

For a sample prism of 3^{Δ} measured in air, made from refractive index 1.498, the permitted error in measurement (0.1^{Δ}) would have an error associated with it of 0.0062 and 0.021 coming from the first and second (immersed prism) measurements, respectively. Errors at this magnitude are quite significant and diminish the relevance of this method of refractive index determination. It is evident that non-digital lensmeters would be of less usefulness. It would also appear useful to have a digital lensmeter that provides a readout with smaller increments than 0.1^{Δ}. Note that a greater portion of the error is due to the lensmeter reading for the immersed prism.

The maximum error will decrease as the prism power increases.

I believe the accuracy of digital lensmeters for prism is greater than that claimed. Some further examples of the maximum error of this method follow.

If, for example, a 3^{Δ} lens made from polycarbonate was measured with a lensmeter of accuracy 0.05^{Δ}, the maximum error in refractive index determination from each prism reading would be 0.0072 and 0.0177.

This calculation involves differentiating the equation for refractive index, first with respect to one prism measurement and then the other.

A third example is for a 5^{Δ} lens made from 1.498 material where the lensmeter accuracy was 0.01^{Δ}. The maximum error from each prism measurement is 0.0005 and 0.0015, respectively. If we assume the two prism measurements are independent, then it is reasonable to regard the error in the same way as the variance of two independent variables being added. That is Var(X + Y) = Var(X) + Var(Y).

In the above example, max

There is an argument against regarding these machine-based errors as random. A source of error in these machines arises from their use of Abbe numbers to extrapolate from measured values of prism power or vergence power at their reference wavelength (660 nm in this case) back to the reference standard wavelength for these measurements. This reference standard can be either 588 or 546 nm. Whatever lens material is being measured will suffer from a systematic source of error introduced by this factor. I found this was a minor source of error when measuring the vertex power of spectacle lenses. I have not investigated the effect with prism measurement.

I expect the issue of which surface of the prism is at the front will be irrelevant, except that the paraxial assumptions benefit from the angles of incidence being as small as possible. If one configuration creates a more paraxial light path, this would improve the accuracy of the refractive index value calculated.

#### CONCLUSIONS

Another method of calculating the refractive index of a material is described. This method has similarities to an earlier method, and is complementary to it, allowing some lenses that could not otherwise be assessed to be assessed. I am unaware of any alternative clinical method for calculating the refractive index of a plano prismatic spectacle lens.

#### ACKNOWLEDGMENTS

I thank Dr David Atchinson for reviewing a number of drafts of this paper. I also thank Ellen Buckley for assistance with the derivation of formulae.

John G. Buckley

Unit 14, Craigieburn Plaza

Craigieburn, Victoria 3064

Australia

e-mail: jgb@netspace.net.au