Spectacle Lenses

A prescription says nothing about what the lenses are made of, how they are curved or where they sit, yet each of these changes what the patient sees. The material sets the thickness, weight, toughness and colour fringes; the curves and the distance from the eye set the magnification; multifocal designs trade one kind of image quality for another; and a few millimetres of misplacement add prism or astigmatism that nobody prescribed.

#Lens materials

In the United States every spectacle lens must pass a drop-ball test, a ⅝-inch steel ball dropped from 50 inches, unless the prescriber decides that an impact-resistant lens will not meet the patient’s needs; glass lenses are tested one by one, plastic lenses by batch. Occupational safety eyewear must meet ANSI Z87.1, with a heavier ball and, for impact-rated (Z87+) eyewear, high-velocity and high-mass tests; sports eyewear has its own standard (ASTM F803), and polycarbonate lenses in an ordinary frame are not sports protection. Polycarbonate or Trivex is the material of choice for children, for sport, and for anyone with useful vision in only one eye. A material’s Abbe number, $V=(n_d-1)/(n_F-n_C)$, compares its refractive power with its dispersion between blue and red light: the higher it is, the less the colours spread.

Away from its optical centre a lens acts as a prism (Prentice’s rule, Chapter 4), and like any prism it spreads white light into colours. The transverse chromatic aberration, in prism dioptres, is the prismatic effect divided by the Abbe number, where c is the distance from the optical centre in centimetres:

Looking 1 cm from the centre of a −6.00 D lens meets 6 Δ of prism, which spreads into 0.2 Δ of colour in polycarbonate but only 0.1 Δ in CR-39. Wearers of strong, low-Abbe lenses may notice coloured fringes on high-contrast edges when they look through the periphery.

#Thickness and the sagitta

The depth of a curved surface over a given width is its sagitta, or sag. For a surface of radius r, at a distance y from its centre,

A surface’s power is $(n-1)/r$, so the two surfaces of a lens of power F (dioptres) differ in sag, at y millimetres from the optical centre, by about

A minus lens gains this much thickness toward its edge; a plus lens is this much thicker at its centre than at its edge. Thickness grows with the square of the distance from the optical centre, so the frame matters as much as the material. A −6.00 D CR-39 lens is 5.4 mm thicker at the edge than at the centre if it is 60 mm across, and 3.8 mm if it is 50 mm across. If the frame’s own PD (eye size plus bridge) is larger than the patient’s, the optical centres must be decentred nasally. That moves the temporal edge farther from the optical centre, thickening a minus lens’s temporal edge, and forcing a thicker centre in a plus lens. A high-index material helps less than people expect: going from CR-39 to 1.67 reduces the sag difference by only about a quarter (0.498/0.67 = 0.74). For strong prescriptions, the best advice is a small, round frame whose centres lie close to the pupils.

#Measuring a surface: the lens clock

A lens clock, or Geneva lens measure, has three pins: the outer two fixed and the middle one sprung. Pressed against a surface, the middle pin is pushed in or out by the sag between the pins, and the dial converts that sag into dioptres, assuming a refractive index of 1.53 (close to crown glass). For any other material the reading must be corrected:

On polycarbonate a reading of +6.00 D means +6.63 D; on CR-39, +5.64 D. The lens clock also finds the base curve, shows whether the two lenses of a pair are curved alike, and reveals a toric surface, whose reading changes as the clock is turned. Since the power of a thin lens is roughly the sum of its surface powers, the readings of front and back together give a rough check of the lens power.

#Lens form

A lens of a given power can be made in many shapes, or bendings. As the eye turns to look through the periphery of a spectacle lens, it looks through it obliquely, and an oblique pencil of light through a spherical lens is focused as a Sturm’s conoid: it picks up astigmatism (oblique, or marginal, astigmatism) and a mean power error (curvature of field). How much depends on the bending, and a design can remove one or balance the two. Spectacle lenses are menisci, convex in front and concave behind, and best-form or corrected-curve lenses use the front curve (the base curve) that keeps these errors small for an eye rotating about its centre of rotation. Aspheric surfaces allow flatter, thinner lenses with good peripheral optics, especially in high plus. Because the base curve also affects magnification (below), a patient happy with an old pair should keep the same base curve when the lenses are remade.

#Spectacle magnification

A spectacle lens changes the size of the retinal image compared with that of the uncorrected eye. This spectacle magnification is the product of two factors:

The power factor depends on the back vertex power $F_v$ and on h, the distance from the lens to the eye’s entrance pupil (the vertex distance plus about 3 mm). The shape factor depends on the front surface power $F_1$ and the centre thickness t. Plus lenses magnify and minus lenses minify, by roughly 1.5% per dioptre at the usual vertex distance, and a steeper front curve or a thicker lens adds magnification. Contact lenses sit almost at the entrance pupil and hardly magnify: a myope who changes from spectacles to contact lenses sees the world larger, and a hyperope smaller. Aphakic spectacles, about +12 D, magnify by a quarter to a third.

#Aniseikonia and Knapp’s law

When the two eyes need different corrections, their retinal images can differ in size, and the patient may perceive the two images as unequal in size or shape: aniseikonia. A small difference goes unnoticed; a few percent can cause headaches and eyestrain, and differences beyond about 5% are often hard to fuse, although tolerance varies widely. A rough rule is 1% of aniseikonia for every dioptre of anisometropia corrected with spectacles, although measured values vary widely.

Which correction gives the most equal images depends on the cause. In refractive ametropia the eye is of normal length but its optics are too strong or too weak; a correcting lens at the eye, a contact lens, restores the normal image size. In axial ametropia the eye is the wrong length, and Knapp’s law says that a correcting lens placed at the eye’s anterior focal point, about 15 mm in front of the cornea, gives a retinal image the same size as an emmetropic eye’s (a relative spectacle magnification of 1). In theory, then, axial anisometropia is best corrected with spectacles and refractive anisometropia with contact lenses:

In practice, contact lenses are often better tolerated even in axial anisometropia. The stretched retina of a long myopic eye has its receptors farther apart, which offsets some of its larger image, and contact lenses avoid the vertical prism of unequal spectacle lenses (Chapter 4). Aniseikonia can also be reduced with iseikonic spectacle lenses, whose base curve, thickness and vertex distance are chosen to adjust the shape and power factors.

#Bifocals

A bifocal adds a segment of extra power, the add, to the lower part of the lens. Chapter 4 showed that crossing into the segment brings a sudden prism, the image jump. A second effect matters at the reading position: there the eye looks through the distance lens below its optical centre, where that lens has prism of its own, and through the segment some distance from the segment’s optical centre. The sum of the two is the image displacement: how far the print appears to be shifted. Here ruled lines are seen through the lens, with the displacements exaggerated:

The segment’s optical centre lies at its geometric centre in a round segment (11 mm below the top of a 22 mm segment), about 5 mm below the top of a flat-top segment, and on the dividing line of an executive bifocal. So:

• Image jump depends only on the add and the distance from the top of the segment to its optical centre: none with an executive bifocal, little with a flat-top, most with a round segment.
• Image displacement also depends on the distance prescription. A myope’s distance lens gives base-down prism at the reading level, and a round segment adds more base down; a segment with its centre near the top adds little, so myopes do best with flat-top (or executive) segments. A hyperope’s distance lens gives base-up prism there, which a round segment’s base-down prism partly cancels, so round segments suit hyperopes, at the cost of more jump.

The top of a bifocal segment is usually set at the lower lid margin. In a trifocal, an intermediate segment, usually of half the add, sits between the two, with its top near the lower edge of the pupil. Occupational designs add a second segment at the top of the lens for overhead work (a double-D bifocal), or a wide intermediate zone for computer screens.

#Progressive lenses

A progressive addition lens increases power smoothly from the distance zone, down a narrow corridor, to the near zone, with no visible line and no image jump. The price is unwanted astigmatism on either side of the corridor, and it cannot be designed away: Minkwitz’s theorem shows that astigmatism increases sideways from the corridor at twice the rate that power increases along it. Short corridors, needed for shallow frames, and high adds therefore narrow the clear zones. Hard designs concentrate the astigmatism into small lateral areas of high value, leaving wider clear zones; soft designs spread it more thinly, with gentler peripheral blur and easier adaptation.

A progressive lens is fitted with its fitting cross over the centre of the pupil as the patient looks straight ahead, which places the distance zone at the pupil and the near zone where the eyes turn down to read. Most complaints come from fitting errors (a lens set too high, too low or off centre) or from a frame too shallow for the corridor. Wearers learn to point their nose at what they want to see, turning the head rather than only the eyes. The near zone is inset nasally to follow the eyes’ convergence, and many progressive lenses are made with equal base-down prism in both lenses (prism thinning) to reduce their thickness, checked at the prism reference point marked on the lens.

#Fitting: centration and tilt

Horizontally, each optical centre should lie on the line of sight (the monocular PD); otherwise Prentice’s rule adds prism that was never prescribed (Chapter 4). Frames also sit with a pantoscopic tilt, the lower rims closer to the cheeks, typically by about 8 to 12°, so that the lenses stay more nearly perpendicular to the line of sight when the eyes look down to read. The optical centre is then lowered by about 1 mm for every 2° of tilt, so that the optical axis of each lens passes through the centre of rotation of the eye, about 27 mm behind the lens. As the eye turns, it then looks along lines through the centre of rotation, as a best-form lens is designed for. If the optical centre were left in front of the pupil, the eye would look through it at an angle to the optical axis.

Looking through a lens along a line tilted away from its optical axis adds sphere and cylinder, as Martin’s formula describes. For a lens of power S, index n and tilt θ:

The induced cylinder has the same sign as the sphere, and its axis lies along the axis of tilt: 180 for pantoscopic tilt, 90 for the wrap of a face-form frame. The effect is small for small tilts but grows quickly: a +10.00 D lens tilted by 20° acts as +10.39 / +1.38 × 180.

#Aphakic spectacles

Before intraocular lenses, aphakic patients wore spectacles of about +10 to +14 D. Besides magnifying by a quarter to a third, these lenses bend straight lines into a pincushion shape, restrict the field, and have a peculiar effect at their edge. Near the edge a strong plus lens acts as a strong prism, base toward its centre, which bends rays toward the axis: the view through the lens ends short of the view past it, leaving a ring of directions that cannot be seen at all, the ring scotoma:

The scotoma is tied to the lens edge, so it moves relative to the visual field as the eyes move, in the direction opposite to the eye movement. Objects in the periphery vanish and then reappear as the patient looks toward them: the jack-in-the-box phenomenon, alarming when the object is a car. Strong minus lenses have the opposite effect at their edge, a ring in which objects are seen twice. Contact lenses and intraocular lenses avoid all of this.

#Tints, filters and coatings

• Ultraviolet: polycarbonate, Trivex and most high-index plastics absorb almost all ultraviolet light; CR-39 needs an added UV absorber to do the same. Lenses labelled UV400 block wavelengths up to 400 nm.
• Photochromic lenses darken in ultraviolet light and fade indoors, more slowly and to a darker tint in the cold. Because windscreens absorb ultraviolet, most darken only a little in a car.
• Polarising sunglasses cut glare from roads, water and snow. Light reflected from a horizontal surface is partly polarised horizontally, and the lenses transmit only vertically polarised light (Chapter 19).
• Anti-reflective coatings use thin-film interference to reduce reflections (Chapter 19). Each uncoated surface reflects ${\left(\frac{n-1}{n+1}\right)}^2$ of the light: 4% for CR-39 but 7% for a 1.74 lens, so high-index lenses benefit most.
• Tints for night driving do not help: any tint removes light when there is least of it. Blue-light-filtering lenses have not been shown to reduce digital eye strain.

#Recap

• Polycarbonate and Trivex for children, sport and one-eyed patients. Transverse chromatic aberration = prism ÷ Abbe number.
• Sag difference ≈ $y^{2}F/2000(n-1)$ mm: thickness grows with the square of the lens radius, so frame size and decentration matter.
• A lens clock assumes n = 1.53; multiply its reading by (n − 1)/0.53.
• Spectacle magnification = power factor $1/(1-h{F}_v)$ × shape factor $1/(1-(t/n)F_1)$; about 1.5% per dioptre.
• Knapp’s law: a lens at the anterior focal point gives an axial ametrope a normal image size. Contact lenses for refractive anisometropia (and often for axial anisometropia too).
• Image jump = add × distance from segment top to segment centre (none for executive). Myopes: flat-top; hyperopes: round segments reduce displacement.
• Progressive lenses: astigmatism grows sideways at twice the rate of the power gradient (Minkwitz).
• Lower the optical centre about 1 mm per 2° of pantoscopic tilt. Tilt adds $S^{\prime }{\tan }^2\theta$ of cylinder, axis along the axis of tilt.
• Aphakic spectacles: 25–33% magnification, pincushion distortion, a roving ring scotoma and the jack-in-the-box phenomenon.

#Test yourself

#Further reading

• American Academy of Ophthalmology, Basic and Clinical Science Course, Section 3: Clinical Optics and Vision Rehabilitation, chapters on spectacle lenses and multifocal lenses.
• M. Jalie, Principles of Ophthalmic Lenses. Association of British Dispensing Opticians.
• C. W. Brooks and I. M. Borish, System for Ophthalmic Dispensing. Butterworth-Heinemann.
• T. E. Fannin and T. Grosvenor, Clinical Optics. Butterworth-Heinemann.