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Old November 10th, 2010, 07:47 AM
Doug Kerr Doug Kerr is offline
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Join Date: May 2006
Location: Alamogordo, New Mexico, USA
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Default Eyeglass lenses and "vertex power"

In most optical work, including, in the photographic realm, the analysis of multi-element lenses and the "rating" of auxiliary closeup lenses, we deal with the refractive power (often called just "power") of a lens, the reciprocal of its focal length. The focal length is the distance from the lens' second principal point to its rear focal point.

But in the field of vision correction (eyeglass lenses, contact lenses, etc.) the power specification of a lens is in terms of the reciprocal of the lens' back focal length, the distance from the rear face of the lens to its rear focal point.

The rationale for this convention is very interesting, but sadly it is rarely clearly explained. The authors who write of it either already know how it works and assume that the reader does as well, or they have no real grasp of it, but heard about it in optometry school, along with some ineffectual catch phrases used to "explain" it ("well, that's really the effective power of the lens, you know").

Here is a "short" version of the story.

*******

By way of background, we note that if we have a "compound" lens made of two simple lens elements, the power of the compound lens (we might want to know that so we can reckon its focal length) is the sum of the powers of the two elements minus a term depending on the distance between the lenses (reckoned between their adjacent principal points).

We find the same situation affecting vision correction lenses. The effect on vision correction of a lens depends both on its power and on its distance from the eye. Conversely, the power of a lens needed to correct for a certain near- or farsightedness condition depends on the distance we place the lens from the eye.

For various practical reasons, for most people, the lens is placed at a standardized distance from the eye, and the lens prescription issued by the ophthalmologist or optometrist contemplates that.

Based on familiar optical theory, we might expect that distance to be defined as to the second principal point of the lens, and the power specified by the prescription to be in the familiar optical terms (the reciprocal of the familiar focal length).

But standardizing the distance from the eye to the second principal point of the lens turns out to not be practical. The reason is that the location of the second principal point varies dramatically with the shape of the lens. In a lens that is equally convex on both surfaces, it is inside the lens, about 1/3 of the lens' thickness from the rear surface. For the commonly-used meniscus lens (where the entire lens is convex toward the eye), the second principal point may be some distance in front of the entire lens.

Thus, if a small standard distance is adopted, a meniscus lens would have to be flat against the face or worse. If a larger distance is adopted to prevent that, a "biconvex" lens would be an unattractive distance out in front of the face.

Thus the distance that is standardized is the one to the rear vertex of the lens (the point on its rear surface on the axis).

It turns out that if we know this distance, then (regardless of the shape of the lens), a certain "power" will always be required to correct a certain vision deficiency, where that power is the reciprocal of the back focal length of the lens, that is, the distance from the rear vertex to its focal point. (This is called the vertex power of the lens.) This is the key to the convention of the use the vertex power to "specify" eyeglass lenses.

The prescriber does not ordinarily know the shape of the lens that will be dispensed. That depends on such tactical factors as the style of the chosen frame, whether the patient elects the "extra light weight" design or not, and so forth.

But he determines the vertex power of the lens that, placed so its rear vertex is the standard distance from the eye (usually 15 mm for adults, by the way), will best correct the patient's vision, and prescribes on that basis.

Now, at the "in about an hour" optical shop, the thousands of "almost finished" lenses they have in stock (needing only to be ground to the perimeter shape to fit the specific frame selected) are all marked in terms of their vertex power. And the frames are made so that, very nearly, they will place the rear vertex of the lens 15 mm from the patient's eye (we hope).

Thus this whole scheme works out properly.

Effective power

"Explanations" of this matter in potometric texts and articles generally say that, in fact, the vertex power of a lens should be considered its "effective power". This of course is a nice acceptance of its pivotal role in the scheme for prescribing vision correction lenses. Indeed, that number tells us the effect of a lens on vision correction if it is assumed to placed at a certain (standardized) position defined in term of the location of its rear vertex. And , in vision correction practice we (usually) do that, so . . .

This notion can also be related to a real optical theory concept. We can say that any lens, having a certain power (in the usual optical terms), will, at any arbitrary plane (perhaps partway on the way to its rear focal point) "exert a certain effective power".

The vertex power of a lens turns out to be its effective power (in that sense) at the specific plane containing the rear vertex of the lens. But sadly, this qualification is never mentioned when saying that the vertex power is "the effective power" of a lens.

In fact those who deal with this notion often go so far as to say that the vertex power is the "true power" of the lens. C'mon guys - the vertex power plays an important practical role in the scheme for prescribing vision correction lenses, but the "true power" of a lens? Really!

*******

I am just finishing an extensive article on this matter, with thorough theoretical background and lots of fussy figures. I'll let you know when it's available.

Best regards,

Doug
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