Building a Progressive Lens
A progressive lens gives you distance, in-between and reading power in one lens with no visible lines. Here we build one from a single curved surface, the way the idea developed, to show why every progressive has soft zones at the sides and how modern designs manage them. Every picture is calculated live from the formulas beneath it.
One curve, one power
A lens bends light because its surface is curved. The more steeply it curves, the more strongly it bends light. A single-vision lens has the same curve everywhere, like a thin slice cut from the surface of a ball.
Reading takes extra power. Move the slider: more reading power needs a steeper curve, and a steeper curve is a slice of a smaller ball.
- F
- Power of the surface, in dioptres (D).
- n
- Refractive index of the lens material: 1.50 for standard plastic. Higher-index materials bend light more for the same curve.
- r
- Radius of the curve, in metres. Halve the radius and you double the power.
- s, h
- Sag: how far the surface curves back from flat at a height h from the centre. The approximation is very close for spectacle lenses.
In 3DEvery point of the surface sits on one sphere of radius r. The power is the same at every point and in every direction, so there is no blur anywhere in the lens. Reading power A focuses at 1/A metres for an eye that can no longer focus on its own: +2.50 D gives 40 cm.
Stack bands of more power
The simplest way to get more power lower in the lens is to stack bands. Each band is a slice of a slightly smaller ball than the band above it, so the power climbs as your eye looks down. One step gives you a bifocal. Two steps give you a trifocal.
Turn the lens and look along the sides. Neighbouring bands meet perfectly down the centre line, but a steeper band curves away faster, so a ledge opens up that gets taller toward the edge. That ledge is the line you can see on a bifocal, and it is why the image jumps when your eye crosses it.
More steps make each ledge smaller, but there are more of them.
This is the reading-power layer of the lens on its own, drawn as a solid block: the distance curve has been flattened out so only the added power shows, and the depth is magnified so the steps are visible. Each band is a slice of a sphere, and all of them touch along the vertical centre line. Orange faces are the ledges. Add +2.00 D, built over 15 mm below the fitting cross. The depth setting here applies to every 3D view on the page.
- z
- Sag of the surface: how far it curves back at a point x mm across and y mm up the lens.
- k(y)
- Curvature of the horizontal slice at height y. Its power is (n − 1)·k. For stacked bands, k jumps up by Add ÷ N at each seam.
- f(y)
- The shape of the vertical centre line. Making its curvature equal k keeps every band a true sphere and keeps them all touching along the centre.
The ledge is zero on the centre line and grows with the square of the distance out: twice as far from the centre, four times as tall.
In 3DEach horizontal slice of the surface is a circle (in the small-angle form used here, a parabola). The circles get smaller down the lens. With sharp seams the lens is a stack of spherical bands with steps between them. Blending replaces the jump in k with a smooth climb.
Smooth the ledges, and the surface twists
Now slide Blend up to smooth the ledges away. The surface becomes one continuous piece, but the mismatch that made the ledges is still there. It still grows toward the sides, and when it is spread smoothly down the lens the surface has to twist, a little like a towel being gently wrung.
A twisted surface bends more along one diagonal than the other. Your eye sees that as astigmatism, blur that smears in one direction, with its axis at an angle: near 45° on one side of the lens and 135° on the other. It is zero on the centre line and grows steadily toward the edge.
The 3D view below is the same lens as in step 2, coloured by blur. Watch the orange ledge faces as you blend: they melt into the surface, and blur appears in exactly the places they were. The thin lines across it are its horizontal slices, the stacked circles, now infinitely many.
Tap the map to read the lens at any point, the way a lensometer would. The same point is marked in 3D, with a green bar along the direction the surface curves most and a grey bar along the direction it curves least.
The map is the front view, as you would look at the lens on the patient. The 3D view and the map use the same colours, with bands 0.50 D apart. Short ticks show the cylinder axis. The green marks are the ones printed on an uncut progressive: distance circle, fitting cross, prism dot and reading circle. Dashed lines are ledges.
- k′(y)
- How fast the curvature changes per millimetre down the lens. Times (n − 1), it is how fast the reading power climbs, in D per mm.
- x
- Distance sideways from the centre line.
- k″(y)
- How fast that climb itself changes. It matters only where the power starts and stops climbing, so where the climb is steady the shorter second form holds.
In 3DAt a point x mm from the centre, the surface bends by k + k′x along one diagonal and k − k′x along the other. The average is the same as on the centre line, but the two directions differ. That difference is the cylinder, and because the twist reverses sign across the centre line, the axis mirrors from one side to the other.
The price of a clear channel
This blur is not a manufacturing flaw. In 1963 the mathematician Günter Minkwitz proved that on any smooth lens, the blur beside a clear centre line grows sideways twice as fast as the power grows down it.
That sets the width of the clear channel, or corridor, through every progressive. Two things decide it: how much reading power you need, and how much height the lens has to build it up in. A stronger add, or a shorter corridor for a shallow frame, means a narrower channel.
The 3D view is the fully smoothed lens from step 3. The pale strip running down its middle is the corridor, and the dark bar marks its width where the power is climbing fastest.
Close-up of the smooth stacked-circle lens from step 3, around the corridor. The bracket marks where blur reaches 0.50 D, measured on this surface.
- C, P
- Cylinder (blur) and mean power. Sideways, blur grows at twice the rate that power grows downward.
- L
- Corridor length: the height, in mm, over which the add builds up.
- w
- Corridor width: the band where blur stays under 0.50 D. It comes straight from step 3: set 2 × (Add ÷ L) × x equal to 0.50 and solve for x.
In 3DAlong a blur-free centre line the surface cannot twist, but one millimetre to the side it must twist enough to carry the change in power. The narrower the strip over which the power changes, the harder the twist. The eye's centre of rotation sits about 27 mm behind the lens, so 1 mm on the lens is about 2° of eye turn.
Let the computer reshape it
The stacked-circle lens from step 4 already has the right power down the centre line. What it gets wrong is everywhere else. Every horizontal slice is a full circle, so the twist, and the blur, keep growing all the way to the edge of the lens.
A staircase with the same problem
Picture a staircase that turns a corner. Along its walking line, the path down the middle, every step is the same comfortable 25 cm deep. But the inside of the turn is a shorter path than the walking line, and the outside is a longer one. So toward the inside the steps get shallower, toward the outside they get deeper, and the farther you walk from the walking line, the worse the mismatch. That is the pattern on the lens: exactly right along the corridor, with blur growing steadily toward each side.
No design can make the total mismatch go away. On this staircase the inside edge is 78.5 cm shorter than the walking line: half the stair's width times the angle of the turn. The simplest layout cuts every step in the turn square to the walking line, so four steps absorb all of it and are barely 5 cm deep at the inside edge.
Stair builders fix this by balancing the winders, also called dancing steps. Each step line pivots about its fixed point on the walking line, so steps before and after the turn take a share of the shortfall. Press play to watch the step lines swing. The walking line stays exactly 25 cm per step, the worst step at the inside grows to about 16 cm, and the straight flights give up a little of their perfection. Let more steps share the work and every step gets closer to right.
A quarter-turn staircase 1 m wide, seen from above, with 18 steps each rising 18 cm. Steps are coloured by how far their depth is from 25 cm, in 2 cm bands. Violet dots are the fixed points on the walking line that every step line must pass through. Violet outlines are steps kept square. Dark dots are your footsteps along the path you choose.
- d(u)
- Depth of a step at a distance u toward the inside of the walking line. Negative u is toward the outside, where steps are deeper.
- g
- Depth along the walking line: 25 cm.
- rw
- Radius of the walking line through the turn, 63.7 cm here. A tighter turn means a smaller radius, so the mismatch grows faster.
- fixed
- Every step line passes through its point on the walking line, and steps outside the allowed range stay square.
- λ term
- Keeps neighbouring steps similar, so the depth changes gradually from one step to the next.
Compare the lensOn the staircase the mismatch is g × u ÷ rw. On the lens the blur is 2 × x × Add ÷ L. Both grow in proportion to the distance from the centre line, and both grow faster when the centre line changes faster: a tighter turn, or a stronger add over a shorter corridor. Neither can be removed, only shared out.
The lens does the same thing
The lens optimizer works like the stair builder, moving surface heights instead of step angles:
| Staircase | Lens |
|---|---|
| Walking line, every step exactly 25 cm | Corridor, power climbing exactly as prescribed |
| The turn | The climb from distance power to reading power |
| How tight the turn is | Add ÷ corridor length |
| Steps too shallow inside, too deep outside | Blur, with its axis mirrored left and right |
| Inside edge always 78.5 cm short | Twist that cannot be removed, only moved (step 4) |
| Straight steps kept square | Distance and reading zones kept clear |
| Step lines pivot on fixed points | Heights move while the corridor power stays fixed |
| How many steps share the shortfall | Hard or soft design (step 6) |
A modern design starts from a wish list instead of a construction. Keep the distance zone clear. Keep the reading zone clear, and move it a little toward the nose because the eyes turn inward to read. Keep the power climbing down the corridor exactly as before. Then make the leftover blur as small as possible. The computer turns that list into a single score and is free to raise or lower every point of the surface to bring the score down.
Drag the slider, or press play, to move every point from its stacked-circle height to its optimized height. The score falls the whole way. The band of blur running across the lens breaks up and settles into two pockets at the lower sides, and the reading zone slides toward the nose. Switch to height change to see which parts of the surface the computer moved: mostly the lower sides, lifted by a fraction of a millimetre.
Right lens seen from the front, add +2.00 D, 15 mm corridor, balanced design. Violet lines are the wish list: the edge of the protected distance zone at the top, the protected reading zone at the bottom, and the dotted corridor line where the power must climb exactly as in the stacked-circle lens. The dashed outline is a typical frame.
- w C²
- Blur everywhere, squared, so a little blur spread wide costs less than a lot packed tight. The weight w is highest along the corridor and in the middle of the lens.
- zones
- Inside the protected zones, power must hit its target (the Rx above, Rx + Add below) and blur must be near zero. The corridor line has its own power target: the same climb as the stacked-circle lens.
- λ |∇P|²
- Keeps the power changing smoothly, so the surface has no ripples.
- J0, J*
- The score of the stacked-circle lens, and the lowest score possible. The optimized surface is the one that reaches J*.
- why it falls
- J is a sum of squares and the optimized surface sits at its lowest point, so along this straight path the score drops steadily and never rises again.
In 3DThe heights change by fractions of a millimetre. The biggest change is low on each side, beside and below the corridor, where the full-width circles curved steeply away: the optimizer lifts the surface back by up to 0.65 mm inside the frame. The centre and the top of the lens barely move. Lifting the sides takes out most of the twist the circles had forced on them, and that is where the blur was. With about 2,800 free heights on a 1 mm grid, this is a least-squares problem a laptop solves in under a second.
Deciding where the blur goes
The computer only follows the wish list. Writing the list is the designer's real job, and the biggest choice on it is how large to make the protected zones.
A hard design asks for large clear zones, which leaves the blur nowhere to go but steep-sided pockets at the sides. A soft design asks for smaller clear zones, which gives the computer more surface to spread the blur over, so it is milder and changes more gradually. Many people find soft designs easier to get used to, because the view swims less when they move their head.
Move the design slider and watch the violet zones of the wish list grow or shrink, and the blur rearrange around them. Switch to stacked circles to compare with the lens you started from.
The dashed outline is a typical frame shape, so the numbers above count only what ends up in the frame. Violet lines are the protected zones on the wish list. Optimized surfaces were solved for this page on a 1 mm grid. They show the real trade-offs, not any manufacturer's actual design.
- s
- Softness, from 0 (hardest) to 1 (softest). It is the only thing that changes between the five designs.
- c
- How sharply the edge of the distance zone curves up toward the sides. A soft design lets the blur reach a little into the outer distance zone.
- h0, m
- Half-width of the reading zone at its top, and how fast it widens below. Measured from 2.5 mm toward the nose; yN is the height of the reading circle.
- corridor
- The corridor's weight also eases off as s rises, so a soft design may let a trace of blur into the corridor in exchange for gentler sides.
In 3DEvery design here comes from the same score as step 5; only the zone sizes differ. Smaller zones leave more of the surface free to bend gradually, so the twist is spread over a wider area and its peak falls. Commercial software does the same with full ray tracing and far finer detail.
Fitted to how you wear it
Everything so far describes the surface. What your eye actually receives also depends on how the lens sits: tilted in the frame, wrapped around your face, at some distance from your eye, and looked through at an angle. Tilting a lens on its own adds unwanted cylinder.
Freeform, or digital, lenses are cut point by point by computer-controlled machines. That means each lens can be calculated for your prescription, your frame and the way you wear it, with effects like this one built out before the lens is made.
- F
- The lens power as prescribed.
- θ
- Tilt angle. Pantoscopic tilt rotates the lens about a horizontal line, so the extra cylinder has its axis at 180 (plus-cylinder form).
- S′, C
- The sphere and cylinder the eye actually gets (Martin's formula for a thin tilted lens).
In 3DLight meets a tilted lens at an angle, so it crosses the curve at a slant and sees it as steeper in one direction than the other. A freeform design computes the surface the eye should look through at every point, for this tilt, wrap and fitting, and cuts the difference into the lens.
What a premium lens pays for
Every progressive, at every price, obeys the rules in steps 2 to 7. No lens beats Minkwitz: the blur beside the corridor is always there. What changes with price is how much of the lens is calculated for you, how it is made, and how precisely it matches the way you wear your glasses.
How progressive lenses got here
- 1907First patent for a progressive lens, by Owen Aves in Britain. It was never sold.
- 1959Varilux, the first modern progressive. Its surface was close to the stacked-circle lens of steps 3 and 4: spherical distance and reading zones joined by a corridor of circles, with the blur crowded into the sides.
- 1972Varilux 2. A fully aspheric surface lets the blur spread more gently, the first move toward the soft designs of step 6.
- 1980s–90sDesigns optimized by computer, as in step 5. Zeiss's Gradal HS (1983) matched what the two eyes see when looking to the side. Later came designs that change with the add, and lenses for computer and desk work.
- 2000The first progressive calculated for one wearer: prescription, pupil distance, how the frame sits and reading distance, cut by a freeform machine (Zeiss Gradal Individual).
- TodayFreeform cutting is routine in modern labs. Premium designs also fit the corridor and the blur to your frame, balance the two eyes together, and some use measurements of your eyes or of how you move your head and eyes.
Moulded or made for you
A moulded progressive is mass-produced. The progressive design is moulded into the front of a lens blank in a factory, in a set of standard curves and add powers, and it cannot be changed. The lab only grinds your prescription onto the back. Everyone with the same blank gets the same layout, with the same corridor length, designed for an average face and frame.
A freeform progressive starts as a plain blank. Software calculates the whole back surface for your order, ray tracing your prescription, your frame and your measurements. A computer-controlled diamond lathe then cuts it point by point, and a soft pad polishes it. Nothing is moulded in advance, so every lens can be different.
Below, one frame is fitted both ways. The moulded lens has a standard 18 mm corridor. The made-for-you lens uses a corridor that fits the frame, and its blur score from step 5 only counts the part of the lens that stays in the frame. The rest, shaded here, is cut away when the lens is edged, so the optimizer is free to push blur into it.
Right lens seen from the front, inside a frame 48 mm wide. Both lenses are the balanced design from step 6, made with the same method; only the corridor and the frame-fitting differ. Reading area counts the part of the frame within 0.25 D of the full add and with blur under 0.50 D. In a shallow frame the shorter corridor raises the peak blur, as step 4 predicts: fitting the frame is a trade, not a free gain, but it keeps the reading zone inside the frame.
The second difference is in the prescription itself. Your prescription is measured with the trial lens straight in front of your eye, about 12 mm away. Real frames tilt, wrap around the face and sit at their own distance, and each of these changes the power your eye receives (step 7). A moulded lens is ground to the prescription as written. A freeform lens can be calculated for the prescription as worn: the lab deliberately alters the lens so that, tilted and wrapped in your frame, it gives your eye exactly what was prescribed. That is why a personalized lens can read slightly differently from your prescription on a lensometer.
Right lens, standard plastic (n = 1.50), at the fitting cross. The prescription was measured 12 mm from the eye with no tilt. The compensated lensometer reading is a first-order estimate; real lens software ray traces the whole lens.
- hfit
- Fitting height: from the fitting cross down to the bottom of the frame. 18, 20 and 23 mm for the three frames here, so the corridors are 12, 15 and 18 mm.
- Δd
- How much farther from the eye the lens sits than the trial lens did, in metres.
- α, β
- Pantoscopic tilt and wrap. Together they tilt the lens by θ about an oblique line, which sets the axis of the extra cylinder.
- compensated
- To first order, the lab makes the lens the prescription minus the error, so that wearing it adds the error back.
The ladder of progressive lenses
- MouldedOne layout for everyone with the same blank. Your prescription is ground onto the back.Measured: pupil distance, fitting height.
- Digital standardA standard design, but cut by freeform on the back and calculated with your prescription, so stronger prescriptions and astigmatism stay sharper toward the sides.Measured: pupil distance, fitting height.
- PersonalizedCalculated for your frame and how it sits: tilt, wrap, lens-to-eye distance, frame size and shape, a corridor that fits, and often your reading distance.Also measured: tilt, wrap, lens-to-eye distance, frame shape.
- Newest premiumAll of the above, with the two eyes balanced together. Some designs also use measurements of your eyes, or of how you move your head and eyes when you read and look around.Sometimes also: specialized eye or behaviour measurements, or a questionnaire about how you use your eyes.
Where the money goes
- A calculation for one person. A personalized lens is a one-off design, ray traced from your prescription and measurements. Nothing comes off a shelf.
- The machinery. Freeform generators, diamond tools, soft polishers and inspection equipment, run in specialized labs, with each lens made to order.
- The design behind it. Years of optical research and trials with real wearers. Labs often pay the lens designer a fee for each lens they cut.
- Measurements and time. Extra measurements at fitting, more careful centring, and follow-up visits.
- What comes with it. Premium lenses usually come with better materials and coatings, and many include an adaptation guarantee: a remake if you cannot get used to them.
- The brand. As with any product, part of the price is the name. The optical gains are real, but how much you notice them depends on you.
Who notices the difference most
The gains are largest when the trade-offs in this page are hardest: stronger prescriptions, astigmatism or a big difference between the two eyes; higher adds, where the corridor is narrowest; shallow, large or wrapped frames; people who struggled to get used to progressives before; and anyone who spends long hours at a computer. With a mild prescription in a regular frame, a good standard design can come close to a premium one. Your optometrist can show you where you fall.
- Powers are surface powers in the small-angle (paraxial) approximation, with n = 1.50 and a plano distance prescription. Real lenses are ray-traced through both surfaces and the eye.
- Only the progressive part of the surface is shown. The distance curve is subtracted so the added power is visible.
- Minkwitz's rule is exact right at a blur-free centre line. Real designers often allow a trace of blur into the corridor and shape how it grows, so measured corridors can be a little wider than the simple estimate.
- The optimized designs minimize a simple blur score on a 1 mm grid. They are teaching models of hard and soft designs, not reproductions of any product.
- Step 8's made-for-you lenses were solved on the same 1 mm grid with the score's weights cut outside the frame. Its as-worn numbers use Martin's thin-lens formula for the combined tilt and a simple lens-to-eye distance correction at the fitting cross; real freeform software ray traces the whole lens through both surfaces.
- The staircase is an analogy for the trade-off, not the same geometry. On the stair the mismatch comes from the inside of the turn being a shorter path; on the lens it comes from the surface having to twist. Both grow in proportion to the distance from the centre line, which is why the comparison holds. The balanced stair here is a least-squares teaching model; real stairs are laid out to building codes.
- Step 5's slider moves each point in a straight line from its starting height to its optimized height. A real optimizer takes a less direct route through many trial surfaces, but it ends at the same surface.