Intuition The one core ideaA lens bends light rays , and the whole topic reduces to a single number — the power — that says how hard it bends. Once you can read that one number, stacking lenses becomes plain addition instead of scary fraction arithmetic.
This page assumes you have seen none of the notation in the parent note. We build every letter, every ratio, every sign from a picture, in an order where each idea leans only on the one before it. When you finish, re-read Power of a lens, combination of lenses and it should feel obvious.
Before any symbol, look at the scene everything lives in.
Definition The principal axisThe principal axis is the straight horizontal line drawn through the exact centre of the lens, perpendicular to it. It is our "ground line" — every distance and height is measured from it.
Picture: the dashed gray line running left-to-right through the middle of the lens above.Why we need it: without a reference line we cannot say "how high" a ray is or "how far" a point is. It is the origin of our whole coordinate world.
A ray is a thin straight arrow showing the path light travels. Light comes in from the left, hits the lens, and gets bent (changes direction) as it passes through.
Picture: the blue arrow entering from the left, the orange arrow leaving bent toward the axis.Why we need it: "bending rays" is literally the job of a lens, so a ray is the thing we track.
h h how far above the principal axis a ray strikes the lens, measured straight up.
Picture: the green vertical segment in the figure below, from the axis up to where the ray meets the lens.Why we need it: a lens bends a ray more the further out it hits. So the bending will turn out to depend on h
Definition Focus and focal lengthf parallel to the axis . After the lens bends it, it crosses the axis at a special point called the focus F focal length f
Picture: in the figure above, the orange bent ray meets the axis at the red dot F f Why we need it: f short f long f
f f f f power in Section 6.
δ δ delta ) is the angle by which the lens turns the ray — the angle between the incoming direction and the outgoing direction.
Picture: in the figure below, the shaded wedge between the flat blue "would-have-gone-straight" line and the orange "actually-bent" line.Why we need it: "how hard the lens bends" is exactly this angle. It is the physical quantity that power will measure.
The parent note writes tan δ ≈ δ = h / f
Look at figure s03. The bent ray, the axis, and the vertical height h right triangle :
the opposite side (opposite the angle δ h
the adjacent side (next to δ f
the right angle sits where h
tan tan ( angle ) = adjacent side opposite side
Picture: in s03, tan δ = f h Why THIS tool and not another? We want to connect an angle (δ lengths (h f tan opposite over adjacent — and those are exactly the two sides our triangle hands us. Sine or cosine would need the slanted hypotenuse, which we don't care about here.
Recall Why does small-angle meantan δ ≈ δ tan
The lensmaker's relation in the parent uses R 1 R 2 are .
R radius of curvature R Flatter surface ⇒ bigger R more sharply curved ⇒ smaller R
Picture: imagine blowing up a balloon behind the lens face; R R 1 first surface light meets, R 2 second . The subscripts 1 , 2 Why we need it: the parent's formula P = ( n − 1 ) ( R 1 1 − R 2 1 ) makes power. Sharper curves (R R 1
n n how much the glass slows and bends light compared to air. Ordinary glass has n ≈ 1.5
Why the ( n − 1 ) If the glass were identical to air (n = 1 ( n − 1 ) must multiply the curvature.
Now everything above collapses into one number.
P Power is the bending strength per unit height , and it equals the reciprocal of the focal length:
P = f 1 ( f in metres )
From Section 4: bending per unit height = h δ = f 1 is P Why reciprocal? Small f big P f Unit: the dioptre (D 1 D = 1 m − 1 f metres — the unit is defined per metre.
f f = 20 cm not P = 1/20 20 cm = 0.2 m P = 1/0.2 = 5 D
Every quadrant of behaviour must be covered — here the "cases" are the two lens shapes and their signs.
Definition Sign convention for lenses
Lens
Shape
What it does to parallel rays
f P
Convex (converging) fat middle
pulls them together to a real focus
f > 0 P > 0
Concave (diverging) thin middle
spreads them apart (they seem to come from a focus behind )
f < 0 P < 0
Picture: convex = ")(" reversed, arrow-heads meeting; concave = "()" , arrows fanning out.Why the minus matters: a negative power can cancel a positive one. When we add powers of a combination, the sign is what lets a concave lens undo part of a convex lens. Drop the sign and the physics breaks.
The combination derivation in the parent uses u v v 1 F
u v
u object (the thing you're looking at).v image (where the lens sends the light to a point).They obey the thin-lens equation v 1 − u 1 = f 1 Thin Lens Equation .
Why we need them: the "powers add" proof takes an object at u v 1 between the lenses, then the final image at v
v 1 F
v 1 lens 1 only ; it becomes the object for lens 2. The subscript 1 Capital F combined focal length of the whole stack (do not confuse with the focus point F f P = 1/ F
Principal axis - reference line
tan delta approx delta equals h over f
P equals 1 over f - the core number
Lensmaker P equals n minus 1 times curvature
Convex positive - Concave negative
Combination 1 over F equals sum
Power and Combination of Lenses
Answer each before diving into the parent note; if any stumps you, re-read that section.
What is the principal axis, and why is it needed? The centre line through the lens; it is the reference from which all heights and distances are measured.
What does h The height above the axis at which a ray strikes the lens.
What is f small f The lens-to-focus distance; small f
What is the deviation angle δ The angle by which the lens turns a ray from its original direction.
Why does tan Our triangle gives the opposite side (h f tan
Why is tan δ ≈ δ For small (paraxial) angles the tangent and the radian angle nearly coincide.
What are R 1 R 2 Radii of curvature of the first and second lens surfaces; smaller R
Why the factor ( n − 1 ) If glass matched air (n = 1 ( n − 1 )
Define power P P = 1/ f f 1 D = 1 m − 1
Signs of P Convex P > 0 P < 0
Difference between lowercase f F f F
Power of a lens, combination of lenses — the parent this page prepares you for.Thin Lens Equation — where u v 1/ v − 1/ u = 1/ f Lensmaker's Equation — uses n R 1 R 2 Magnification of Lenses — next quantity after distances.Defects of Vision — spectacle powers in dioptres.