Thin lens

Focal length

The focal length, f, of a lens in air is given by the lensmaker's equation: :\frac{1}{f} = (n-1) \left[ \frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)d}{n R_1 R_2} \right], where n is the index of refraction of the lens material, R1 and R2 are the radii of curvature of the two surfaces, and d is the thickness of the lens. Here R1 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: R2 is positive if the surface is concave, and negative if it is convex. This is an arbitrary sign convention; some authors choose different signs for the radii, which changes the equation for the focal length.

For a thin lens, d is much smaller than one of the radii of curvature (either R1 or R2). 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{1}{f} \approx \left(n-1\right)\left[ \frac{1}{R_1} - \frac{1}{R_2} \right]. CAUTION TO EDITORS: This equation depends on an arbitrary sign convention (explained on the page). If the signs don't match your textbook, your book is probably using a different sign convention.

Derivation using Snell's law

Consider a thin lens with a first surface of radius R and a flat rear surface, made of material with index of refraction n.

Applying Snell's law, light entering the first surface is refracted according to \sin i = n \sin r_1, where i is the angle of incidence on the interface and r_1 is the angle of refraction.

For the second surface, n \sin r_2 = \sin e , where r_2 is the angle of incidence and e is the angle of refraction.

For small angles, \sin x \approx x. The geometry of the problem then gives:

\begin{align} e &\approx n r_2 \ &= n (i - r_1)\ &\approx n (i - \frac in) \end{align}

If the incoming ray is parallel to the optical axis and distance h from it, then \sin i = \frac hR \implies i \approx \frac hR.

Substituting into the expression above, one gets e \approx \frac {h}{R} (n-1).

This ray crosses the optical axis at distance f, given by \tan e = \frac hf \implies e \approx \frac hf

Combining the two expressions gives \frac 1f = \frac 1R (n - 1) .

It can be shown that if two such lenses of radii R_1 and -R_2 are placed close together, the inverses of the focal lengths can be added up giving the thin lens formula:

\frac 1f = \left(n-1\right)\left( \frac 1{R_1} - \frac 1{R_2} \right)

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 f_2 on the other side (red in figure).
• 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 (blue).
• Any ray that passes through the center of the lens will not change its direction (green).

If three such rays are traced from the same point on an object in front of the lens (such as the top), their intersection will mark the location of the corresponding point on the image of the object. By following the paths of these rays, the relationship between the object distance so and the image distance si (these distances are with respect to the lens) can be shown to be :{1\over s_o} + {1\over s_i} = {1\over f} which is known as the Gaussian thin lens equation, which sign convention is the following.

There are other sign conventions such as Cartesian sign convention where the thin lens equation is written as{1\over s_o} + {1\over f} = {1\over s_i}.For a thick lens, the same form of lens equation is applicable with the modification that parameters in the equation are with respect to principal planes of the lens.

Physical optics

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

:\exp\left(\frac{2\pi i}{\lambda} \frac{r^2}{2f}\right).

References

References

1. (2024). "Exact Equations for the Back and Effective Focal Lengths of a Plano-Concave Thick Lens". University Siegen.
2. Eugene, Hecht. (2017). "Optics". Pearson.
3. (1976). "Physics". Worth Publishers, Inc..
4. Hecht, Eugene. (2017). "Optics". Pearson.