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6 Image formation by thin lenses
6.1 Definition of the equation and the respective variables
Optical lenses form images of objects by means of light refraction. Light rays passing through
a lens are refracted at both boundaries of the lens since the indices of refraction are different
for the lens material and the surrounding medium. Depending on the position of the object
and the type of the lens images can be either real or virtual. The image is formed in that
location where the refracted rays (or their extensions in the case of virtual images, see later)
A thin lens produces the sharp image of the object, if the reciprocal of its focal distance equals
to the sum of the reciprocals of image and object distances
(thin lens equation):
1 1 1
f o i
where f, o and i are the focal, object and image distances,
respectively (the SI unit is meter for all the quantities, but
for practical reasons, they are mostly expressed in cm or
The equation above is the common Gaussian form of the lens equation. In the case of real
objects, the o object distance is positive. The focal length is positive (f > 0) for converging
lenses, whereas it is negative (f < 0) for diverging lenses. The i image distance is positive (i >
0) for a real image, whereas it is negative for a virtual one (i < 0).
Optical lens: A lens is a transparent optical device that focuses or disperses a light beam by
means of refraction. Their two surfaces are parts of the surfaces of spheres. Both surfaces can
be either convex or concave, and one of the surfaces can be planar (flat) as well. A thin lens is
a lens with a thickness that is negligible compared to the radii of curvature of the lens
Convex lens: thicker at the middle than at the edges. If the lens is made of an optically denser
material than the surrounding medium (typically this is the case) a collimated beam of light
passing through the lens converges at one point (focus or focal point). In this case, the lens is
called a positive or converging lens.
https://en.wiktionary.org/wiki/convex https://en.wiktionary.org/wiki/concave https://en.wikipedia.org/wiki/Collimated_light
Concave lens: thinner at the middle than at the edges. If the lens is made of an optically
denser material than the surrounding medium, a collimated beam of light passing through the
lens is diverged (spread); the lens is thus called a negative or diverging lens.
Principal plane: optical planes are stretched surfaces given by the interception of beams
entering to the lens parallel to the axis and the extension of beams leaving the system. Lenses
have two principle planes: on the side of the object and on side of the image. For thin lenses,
the location of the principle planes is practically the same (i.e. the two planes overlap with
Optical axis: the axis, which passes through the optical center of the lens, perpendicular to
the principal plane(s).
Focal point: The point, where the converging lens focuses the collimated beams of light or
the point where the beams of light starts from, in the case of a diverging lens. All kind of
lenses have two focal points located on the two sides of the lens. For symmetric lenses the
two focal points are located at the same distances measured from the center of the lens. (We
focus on symmetric lenses here.)
Focal length/distance (f): the distance from the principal plane(s) to the corresponding focal
points of the lens (F and F′). Its SI unit is m. (Although it is often given in cm or mm.)
Real image: Outgoing rays from a point converge at a real location. The image of an object is
formed by the interception of rays on the opposite side of the lens. Real images can be
projected onto a screen.
Virtual image: Outgoing rays from a point diverge; therefore not the refracted rays themselves but their backward extensions will intercept and form the image on the object’s
side. These images cannot be projected onto a screen.
Object and image distances: the distance of the object or the image from the principal plane
of the lens. Their SI unit is m. (They are often given in cm or mm.)
6.2 Further relationships
The ratio of image and object sizes (I and O, respectively), which is equal to the absolute
value of the ratio of image and object distances.
I i M
The magnification is often calculated using distances with their actual sign as follows:
If M is negative, as it is for real images, the image is inverted (upside-down) with respect to
the object. For virtual images M is positive, so the image is upright.
Dependence of the focal length on the properties of the lens and the medium:
1 1 1 ( 1)( )n
f R R
where n is the relative index of refraction of the lens with respect to the medium, R1 and R2 are
radii of spheres surrounding the lens. (In case of diverging lens these values are negative.) For
a symmetric lens R1 and R2 are equal.
Refracting power of the lens (D, diopter):
reciprocal of the focal length measured in meter. (Unit: diopter, m-1)
6.3 Image formation by thin converging lenses
6.3.1 Graphical representation of the thin lens equation
Let’s examine the dependence of image distance on the object distance. Rearrange the thin
lens equation and plot the image distance as a function of object distance:
The obtained curve is a hyperbola, which approaches asymptotically the o=f and i=f dashed
lines. The curve illustrates the characteristics of image formation (see 6.3.2):
(o i ) fo
i x y o f
if of (the object is outside the focal length), i is positive; therefore a real image is formed on the opposite side of the lens;
if the object approaches the focal point from either directions, the image distance increases (for virtual images it will be a more negative value);
if the object is located at the focal point (o=f), the image would be formed at infinity, which practically means, that no image is formed;
for an object located at infinity (o=), the image is formed in the focal point (i=f);
It can be also seen from the graph that the location of the image is more sensitive to the position of the object in the vicinity of the focal point than farther away from it.
If we plot the reciprocal of the image distance as a function of the reciprocal of the object
distance, a straight line is obtained:
The slope of the straight line is -1 and it crosses the x-axis at x=1/f. The y-interception of the
straight line is 1/f as well.
Characteristics of image formation can be illustrated using this plot as well (see 6.3.2).
If the object is located inside the focal length (o1/f), the image distance is negative; i.e. a virtual image is formed. Approaching the object to the lens (1/o
increases), the image distance decreases (the absolute value of 1/i increases).
If the object is located exactly at the focal point (o=f, i.e. 1/o = 1/f), the image would
be formed at infinity (i=, i.e. 1/i approaches zero).
If the object is located beyond the focal point (o>f, i.e. 1/o
twice the focal length away from the lens, the image is formed at the same distance
(i.e. twice the focal length) on the opposite side of the lens.
If the object is at infinity (o=), the image would be formed at the focal point on the opposite side of the lens (i=f, i.e. 1/i=1/f).
From the graphs above one can estimate the magnification as well using the corresponding o
and i (or 1/o and 1/i) values.
6.3.2 Construction of images – ray diagrams of thin converging lenses
Characteristics of image formation by lenses can be represented using the so-called ray
diagrams as well. The image formed by a single lens can be located and sized with the three
principal rays given below:
any ray that enters parallel to the optical axis on one side of the lens proceeds towards the focal point on the other side;
any ray that arrives at the lens after passing through the focal point on the front side, comes out parallel to the axis on the other side;
any ray that passes through the center of the lens will not change its direction.
Intersection of any two rays out of the three mentioned above is enough to construct the
image. In the cases of simple objects (such as an arrow, or a section; see below) it is enough
to use the beams starting from their two end points in order to construct the image.
Consequently, if one of the end points is located on the optical axis, the corresponding image
point will be located on the optical axis as well.
As we could see in 6.3.1, the properties of the image (position, size) are determined by the