Thin lens

A lens may be considered a thin lens if its thickness is much less than the radii of curvature of its surfaces (d ≪ |R₁| and d ≪ |R₂|).

In optics, a thin lens is a lens with a thickness (distance along the optical axis 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 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:

${\displaystyle {\frac {1}{f}}=(n-1)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}+{\frac {(n-1)d}{nR_{1}R_{2}}}\right],}$

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 ^[1]

${\displaystyle {\frac {1}{f}}\approx \left(n-1\right)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right].}$

Derivation using Snell's law

Refraction of a thin planoconvex lens

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

Applying Snell's law, light entering the first surface is refracted according to ${\displaystyle \sin i=n\sin r_{1}}$, where ${\displaystyle i}$ is the angle of incidence on the interface and ${\displaystyle r_{1}}$ is the angle of refraction.

For the second surface, ${\displaystyle n\sin r_{2}=\sin e}$, where ${\displaystyle r_{2}}$ is the angle of incidence and ${\displaystyle e}$ is the angle of refraction.

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

$${\displaystyle {\begin{aligned}e&\approx nr_{2}\\&=n(i-r_{1})\\&\approx n(i-{\frac {i}{n}})\end{aligned}}}$$

Focusing by a thin planoconvex lens

If the incoming ray is parallel to the optical axis and distance ${\textstyle h}$ from it, then $${\displaystyle \sin i={\frac {h}{R}}\implies i\approx {\frac {h}{R}}.}$$

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

This ray crosses the optical axis at distance ${\displaystyle f}$, given by $${\displaystyle \tan e={\frac {h}{f}}\implies e\approx {\frac {h}{f}}}$$

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

It can be shown that if two such lenses of radii ${\textstyle R_{1}}$ and ${\textstyle -R_{2}}$ are placed close together, the inverses of the focal lengths can be added up giving the thin lens formula:

$${\displaystyle {\frac {1}{f}}=\left(n-1\right)\left({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right)}$$

Image formation

Thin lens ray rules

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 ${\displaystyle f_{2}}$ on the other side (red in figure).
• Any ray that arrives at the lens after passing through the focal point ${\displaystyle f_{1}}$ 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 s_o and the image distance s_i (these distances are with respect to the lens) can be shown to be

${\displaystyle {1 \over s_{o}}+{1 \over s_{i}}={1 \over f}}$

which is known as the Gaussian thin lens equation, which sign convention is the following. ^[2]

Sign convention for Gaussian lens equation

Parameter	Meaning	+ Sign	− Sign
so	The distance between an object and a lens.	Real object	Virtual object
si	The distance between an image and a lens.	Real image	Virtual image
f	The focal length of a lens.	Converging lens	Diverging lens
yo	The height of an object from the optical axis.	Erect object	Inverted object
yi	The height of an image from the optical axis	Erect image	Inverted image
MT	The transverse magnification in imaging (= the ratio of yi to yo).	Image upright as object	Image inverted to object

Plots of image distance sᵢ and magnification Mᴛ vs object distance sₒ for the Gaussian lens equation – the dashed curve denotes virtual image

There are other sign conventions such as Cartesian sign convention where the thin lens equation is written as$${\displaystyle {1 \over s_{o}}+{1 \over f}={1 \over s_{i}}.}$$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. ^[3]

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: ^[4]

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

References

1. Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. § 5.2.3. ISBN   0-201-11609-X.
2. Eugene, Hecht (2017). "Finite Imagery". Optics (5th ed.). Pearson. p. 173. ISBN   978-1-292-09693-3.
3. Hecht, Eugene (2017). "Chapter 6.1 Thick Lenses and Lens Systems". Optics (5th ed.). Pearson. p. 257. ISBN   978-1-292-09693-3.
4. Saleh, B.E.A. (2007). Fundamentals of Photonics (2nd ed.). Wiley.