optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...

, a thin lens is a

lens A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements ...

with a thickness (distance along the

optical axis An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight. The optical axis is an imaginary line that defines the path along which light pro ...

between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are sometimes called ''thick lenses''. The thin lens approximation ignores optical effects due to the thickness of lenses and simplifies ray tracing calculations. It is often combined with the

paraxial approximation In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray which makes a small angle (''θ'') to the optical ...

in techniques such as ray transfer matrix analysis.

Focal length

The focal length, ''f'', of a lens in air is given by the lensmaker's equation: :

\frac = (n-1) \left \frac - \frac + \frac \right

where ''n'' is the

index of refraction In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...

of the lens material, and ''R''₁ and ''R''₂ are the radii of curvature of the two surfaces. For a thin lens, ''d'' is much smaller than one of the radii of curvature (either ''R''₁ or ''R''₂). In these conditions, the last term of the Lensmaker's equation becomes negligible, and the focal length of a thin lens in air can be approximated by :

\frac \approx \left(n-1\right)\left \frac - \frac \right

Here ''R''₁ is taken to be positive if the first surface is convex, and negative if the surface is concave. The signs are reversed for the back surface of the lens: ''R''₂ is positive if the surface is concave, and negative if it is convex. This is an arbitrary

sign convention In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly describ ...

; some authors choose different signs for the radii, which changes the equation for the focal length.

Image formation

Certain rays follow simple rules when passing through a thin lens, in the paraxial ray approximation: *Any ray that enters parallel to the axis on one side of the lens proceeds towards the

focal point Focal point may refer to: * Focus (optics) * Focus (geometry) * Conjugate points, also called focal points * Focal point (game theory) * Unicom Focal Point UNICOM Focal Point is a portfolio management and decision analysis tool used by the p ...

f_2

on the other side. *Any ray that arrives at the lens after passing through the focal point

f_1

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. If three such rays are traced from a point on an object in front of the lens, they will intersect at the point where an image of that object point will be formed. By such ray tracing, the relationship between the object distance ''s'' and the image distance ''s′'' can be shown to be :

+  =

which is known as the thin lens equation.

Physical optics

In scalar wave optics a lens is a part which shifts the phase of the wave-front. Mathematically this can be understood as a multiplication of the wave-front with the following function: :

\exp\left(\frac \frac\right)

References

{{Reflist Lenses