Thick Lens Back Vertex Power Calculator
Thick Lens Back Vertex Power Calculator
F₁ ÷ (1 − (t⁄n)F₁) + F₂, the exact form. A +9.40 / −4.70 lens 5.2 mm thick in polycarbonate is +5.00 D, not the +4.70 D its surfaces add to.
Surface powers + thickness → back vertex power
Reduced thicknessFront +9.40 D, back −4.70 D, 5.2 mm centre thickness, polycarbonate at n = 1.586 — the published worked case
Back vertex power, and the reduced thickness in it
- the reduced thickness
- quoted: it “is simply the thickness, in the desired measurement … divided by the index of refraction”. 5.2 mm of polycarbonate is 3.28 mm reduced. That is the only way the thickness and the index enter the calculation at all, which is why a high-index material reduces the thickness effect twice over — once by needing less glass, and once by dividing by a bigger number
- the procedure, as published
- the source read works it step by step as plain text: the front surface’s focal length is 1 ÷ 9.40 = 0.106383 m; subtract the reduced thickness, 0.106383 − 0.00328 = 0.103103 m; invert, 1 ÷ 0.103103 = 9.70 D; then “adding the true power of the back surface”, +9.70 + (−4.70) = +5.00 D. The closed form at the top of this block is that same procedure in one line, and the proof behind this page asserts the two agree over a sweep of powers, thicknesses and indices
- WHY ONLY THE FRONT SURFACE IS DIVIDED
- back vertex power is measured FROM the back surface, so light has already crossed the thickness by the time it reaches the back surface and that surface’s power is added unchanged. Front vertex power is the mirror and divides the BACK surface instead. On the lens above the two are +5.00 D and +4.77 D, which is a quarter of a dioptre apart — and a prescription means the back vertex one
- the approximation, and its error
- the dispensing texts print the first-order expansion, F₁ + F₂ + (t⁄n)F₁², which the source read gives as “Deff = Df + Db + t(Df)²/n”. It is ALWAYS low, because the exact form is a geometric series whose later terms are all positive, and it is 0.009 D low on the lens above. It reaches 0.10 D at a +15.00 D front surface 8 mm thick and 0.33 D at a +20.00 D front surface 10 mm thick, so it is adequate for dispensing and not for lens design
- the denominator is the shape factor
- 1 ÷ (1 − (t⁄n)F₁) is printed identically by three independent sources as the SHAPE FACTOR of spectacle magnification, and one of them works it at t = 4 mm, n = 1.50 and F₁ = +10.00 D to 1.027, which this page’s arithmetic reproduces as 1.0274. That three-way agreement is the corroboration this page rests on, because NO SOURCE READ FOR THIS BATCH PRINTED THE EXACT BACK-VERTEX EXPRESSION IN CLOSED FORM: four documents that should have carried it — two textbook chapters and two encyclopaedia articles — rendered their equations as images. What was read is the worked procedure, and this page reproduces its answer exactly. See spectacle magnification for the same term used the other way
Worked example
Front +9.40 D, back −4.70 D, 5.2 mm centre thickness, polycarbonate at n = 1.586 — the published worked case
Reduced thickness. 5.2 ÷ 1.586 = 3.279 mm, or 0.003279 m. The source read rounds it to 3.28 mm
Front surface focal length. 1 ÷ 9.40 = 0.106383 m
0.106383 − 0.003279 = 0.103104 m
1 ÷ 0.103104 = +9.699 D — the front surface seen from the back of the lens
Add the back surface. +9.699 + (−4.70) = +5.00 D, which is exactly the figure the published case reaches
What the thickness was worth. The two surfaces add to +4.70 D, so the thickness is worth +0.299 D — more than a quarter of a dioptre, on a lens whose thickness is 5 mm. That is why a plus lens cannot be specified by its surface powers
The approximation. 9.40 + (−4.70) + 0.003279 × 9.40² = 4.990 D, which is 0.0092 D low. Adequate here, and 0.33 D low on a +20.00 D front surface 10 mm thick
And the clocked curves. (1.586 − 1) ÷ 0.530 = 1.1057, so a lens measure calibrated for 1.53 would read these surfaces as +8.50 and −4.25 — which add to +4.25 D, three quarters of a dioptre below the real back vertex power. Two separate corrections, in the same direction
Where it refuses. Set the front surface to +60.00 D, the index to 1.498 and the thickness to 30 mm and no answer appears: 1 − (t⁄n)F₁ has gone negative, the front surface's focal point is inside the lens, and there is no back vertex power to report
What the centre thickness is worth, and what the approximation misses
| Front (D) | Back (D) | t (mm) | n | Thin-lens sum (D) | Exact BVP (D) | Thickness worth (D) | Approximation error (D) |
|---|---|---|---|---|---|---|---|
| +9.40 | −4.70 | 5.2 | 1.586 | +4.70 | +5.00 | +0.299 | 0.0092 |
| +15.00 | 0.00 | 8.0 | 1.498 | +15.00 | +16.30 | +1.304 | 0.1043 |
| +20.00 | −2.00 | 10.0 | 1.67 | +18.00 | +20.72 | +2.721 | 0.3259 |
| +6.00 | −10.00 | 1.5 | 1.498 | −4.00 | −3.96 | +0.036 | 0.0002 |
| +2.00 | −8.00 | 1.2 | 1.74 | −6.00 | −6.00 | +0.003 | 0.0000 |
| 0.00 | −6.00 | 2.0 | 1.586 | −6.00 | −6.00 | 0.000 | 0.0000 |
Back vertex, front vertex and the thin-lens sum compared
| Quantity | Value on the default lens | What it is for |
|---|---|---|
| Clocked surface sum, on a 1.53 lens measure | +4.25 D | Nothing — it is a reading, not a power |
| True surface sum | +4.70 D | The thin-lens approximation, and the starting point |
| First-order approximation | +4.990 D | What the dispensing texts print |
| BACK vertex power | +5.00 D | What a focimeter reads and a prescription specifies |
| FRONT vertex power | +4.77 D | The neutralising power, used when verifying from the front |
| Equivalent power | Not computed here | A thick-lens property used in lens design, not dispensing |
A lens is not the sum of its surfaces, and on a plus lens the gap is a quarter of a dioptre
A thin lens has the power of its two surfaces added together. A real lens does not, because light diverging from the front surface’s focus has further to travel before it reaches the back surface, and the extra distance changes the vergence that arrives there. The quantity that captures it is the REDUCED THICKNESS, the centre thickness divided by the refractive index: 5.2 mm of polycarbonate behaves optically like 3.28 mm of air. The exact back vertex power is then the front surface power divided by (1 − (t⁄n)F₁), with the back surface power added unchanged.
Only the front surface is divided, and the reason is in the name. Back vertex power is measured from the back of the lens, so light has already crossed the glass by the time it meets the back surface and that surface acts on it directly. Front vertex power is the mirror image and divides the back surface instead, and the two are different numbers: on the lens this page opens with, +5.00 D and +4.77 D. A prescription, and a focimeter, mean the back vertex one.
The size of the effect is the useful half. On the published worked case reproduced here — a +9.40 front, a −4.70 back and 5.2 mm of polycarbonate — the surfaces add to +4.70 D and the real power is +5.00 D, so the thickness is worth +0.299 D. On a +15.00 D front surface 8 mm thick it is worth +1.30 D. On a minus lens it is worth almost nothing: a +6.00 front with a −10.00 back and a 1.5 mm centre moves by 0.036 D. The asymmetry is structural, because the term goes as the front surface power SQUARED and a minus lens has a flat front and a thin middle. That is also why the first-order approximation the dispensing texts print — F₁ + F₂ + (t⁄n)F₁² — is good enough for a workshop and not for a design: it is always low, by 0.009 D here and by 0.33 D on a +20.00 D front surface 10 mm thick.
Two honest notes about sources. The exact expression is printed in none of the documents read for this page; what was read is a continuing-education article that works the whole computation step by step in plain text, and this page reproduces its +5.00 D answer exactly. Four documents that should have carried the closed form rendered their equations as images instead. What corroborates the denominator is that three independent sources print 1 ÷ (1 − (t⁄n)F₁) as the shape factor of spectacle magnification, and one of them works it to a value this page reproduces to four decimal places. The second note is about the input: the figures a lens measure clocks are not surface powers unless the material happens to be index 1.53, and the correction — multiply by (n − 1) ÷ 0.530 — is worth more than the thickness term on a high-index lens. For the power a lens must have at a different distance from the eye rather than a different thickness, see vertex distance compensation.
Frequently asked questions
How do I calculate the back vertex power of a thick lens?
Divide the front surface power by (1 − (t⁄n)F₁), with the thickness in metres, then add the back surface power. Equivalently: take the front surface’s focal length, subtract the reduced thickness t⁄n, invert, and add the back surface. A +9.40 / −4.70 lens 5.2 mm thick at n = 1.586 is +5.00 D.
What is reduced thickness?
The centre thickness divided by the refractive index, which is the thickness the glass is worth optically. 5.2 mm of polycarbonate at 1.586 is 3.28 mm reduced. It is the only form in which the thickness and the index enter the calculation.
Why isn’t the lens power just the sum of its surface powers?
Because that is the thin-lens approximation and real lenses have thickness. On a plus lens the gap matters: a +9.40 / −4.70 lens 5.2 mm thick measures +5.00 D, not +4.70 D. On a minus lens it barely does, because the thickness term goes as the front surface power squared and a minus lens has a flat front and a thin centre.
What is the difference between back and front vertex power?
Which surface the thickness term divides. Back vertex power divides the front surface and adds the back one unchanged, because it is measured from the back of the lens; front vertex power does the reverse. On the lens above they are +5.00 D and +4.77 D. A prescription and a focimeter mean back vertex power.
Is F1 + F2 + (t/n)F1 squared accurate enough?
For dispensing, usually. It is the first-order expansion of the exact form and is always slightly LOW: 0.009 D low on a +9.40 / −4.70 lens 5.2 mm thick, 0.10 D low on a +15.00 D front surface 8 mm thick, and 0.33 D low on a +20.00 D front surface 10 mm thick. The exact form costs one division.
Related calculators
References
- Musladin MG. Reduced Thickness. 20/20 Magazine continuing education, 2020mag.com, October 2023. (Works the back vertex power of a +9.40 / −4.70 lens, 5.2 mm thick, n = 1.586, step by step as plain text to +5.00 D; every step was recomputed here and the result reproduced exactly.)
- Optician’s Friend. Optics Study Guide. opticiansfriend.com. Accessed 10 October 2026. (The page states that its information may contain mistakes and should be independently verified, so every formula taken from it here is corroborated from a second source.)
- Open Exam Prep. Section 3.2: Lens Thickness, Magnification and Effective Power, ABO Advanced study guide. open-exam-prep.com. Accessed 10 October 2026.
- Aniseikonia, Binocular Vision Anomalies course notes hosted at umsl.edu/~garziar. University of Missouri-St. Louis. Accessed 10 October 2026.
Not medical advice. For healthcare professionals and education. Reference intervals vary by laboratory and assay — always use your own laboratory's. Never base a dose or a treatment decision on this page alone. Full disclaimer at calcengines.com/disclaimer/