6.3 - Image Formation by Lenses

Types of Lenses and Optical Terms

A lens is transparent glass, perspex or plastic with two surfaces, at least one of which is curved.
A convex lens is a converging lens: parallel rays converge at a real focal point; its focal length is positive.
A concave lens is a diverging lens: parallel rays appear to diverge from a virtual focal point; its focal length is negative.

Term	Meaning
Optical centre, $O$	Centre of the lens; a ray through it travels without being refracted.
Principal axis	Straight line through $O$ and the centres of curvature of both lens surfaces.
Axis of lens	Line through $O$ , perpendicular to the principal axis.
Focal point, $F$	Point on the principal axis where initially parallel rays converge, or appear to diverge from.
Object distance, $u$	Distance from object to $O$ .
Image distance, $v$	Distance from image to $O$ .
Focal length, $f$	Distance from $F$ to $O$ .

Convex lenses with different thicknesses have different focal lengths.
Light from a distant object forms a real image near the focal point. Adjusting a white screen behind the lens until this image is sharp allows the lens-to-screen distance to estimate $f$ .
A concave lens has a virtual focus, so this screen method is unsuitable.

Convex lens	Concave lens
A ray through $O$ continues straight without refraction.	A ray through $O$ continues straight without refraction.
A ray parallel to the principal axis refracts through the far focal point.	A ray parallel to the principal axis refracts as if from the near focal point.
A ray through the near focal point refracts parallel to the principal axis.	A ray directed towards the far focal point refracts parallel to the principal axis.

The intersection of refracted rays locates a real image. The intersection of their backward extensions locates a virtual image.

Object position for a convex lens	Image position and characteristics
At infinity	At the far $F$ , $v = f$ ; real, inverted, diminished.
Beyond $2 F$ : $u > 2 f$	Between the far $F$ and $2 F$ , $f < v < 2 f$ ; real, inverted, diminished.
At $2 F$ : $u = 2 f$	At the far $2 F$ , $v = 2 f$ ; real, inverted, same size.
Between $F$ and $2 F$ : $f < u < 2 f$	Beyond the far $2 F$ , $v > 2 f$ ; real, inverted, magnified.
At $F$ : $u = f$	Refracted rays are parallel; image at infinity.
Between $F$ and $O$ : $u < f$	On the object’s side, farther from the lens than the object; virtual, upright, magnified.

For a concave lens, objects beyond $2 F$ or between $F$ and $O$ produce a virtual, upright, diminished image between $O$ and the near $F$ , on the object’s side.
A magnifying glass produces an upright, magnified image when the object is closer than the convex lens’s focal length.

m = \frac{h _{i}}{h _{o}} = \frac{v}{u}

$h_{i}$ : image height; $h_{o}$ : object height. Use the magnitudes of distances when calculating the size ratio, as in the virtual-image examples.
$m$ has no unit: $m < 1$ means diminished; $m = 1$ means the same size; $m > 1$ means magnified.