KAHIBARO
Table of Contents
A thin lens is a transparent optical element that bends light by refraction at two surfaces. It is called thin when its thickness is small compared with the radii of curvature of its surfaces and with the object and image distances being studied. This approximation lets us describe the lens as if all refraction happens in one plane, which makes ray tracing and image calculations much simpler.
Thin lenses are central tools in optics because they can form images, magnify objects, and focus or spread light. Glass lenses in cameras, microscopes, eyeglasses, and telescopes are often treated as thin lenses in basic physics.
Types of thin lenses
There are two main kinds of thin lenses, converging lenses and diverging lenses. A converging lens is thicker at the center than at the edges. It bends parallel incoming rays so that they move toward one another. A diverging lens is thinner at the center than at the edges. It bends parallel incoming rays so that they spread apart.
A converging lens is also called a convex lens in many simple cases. A diverging lens is often called a concave lens. These names describe the general shape, but the more important physical idea is what the lens does to light.
| Lens type | General shape | Effect on parallel rays | Focal length |
|---|---|---|---|
| Converging lens | Thicker in middle | Brings rays together | Positive |
| Diverging lens | Thinner in middle | Spreads rays apart | Negative |
Principal axis and optical center
To describe image formation, we use a few standard parts of a lens system. The principal axis is the straight line that passes through the center of the lens and is perpendicular to the lens surfaces in the idealized picture. The optical center is the point near the middle of the lens through which a light ray can pass without significant change in direction in the thin lens approximation.
This does not mean the lens has no effect at all, but in basic geometrical optics, a ray through the optical center is drawn as undeviated.
Focal points and focal length
A focal point is the point where rays initially parallel to the principal axis either meet, or appear to come from, after passing through the lens. Every thin lens has two focal points, one on each side, at equal distances from the lens center in a lens surrounded by the same medium on both sides.
The focal length, written $f$, is the distance from the optical center of the lens to a focal point.
For a converging lens, parallel rays actually meet at the focal point on the far side of the lens, so $f$ is positive.
For a diverging lens, parallel rays spread out after the lens, but if we extend those rays backward, they appear to come from a focal point on the same side as the incoming light. In this case, $f$ is negative.
For thin lenses, the sign of the focal length is:
$$f > 0 \quad \text{for a converging lens}$$
$$f < 0 \quad \text{for a diverging lens}$$
The three principal rays
A very useful way to find images formed by a thin lens is to draw special rays whose paths are easy to predict. For a point on an object, any two of these rays are enough to locate the corresponding image point.
For a converging lens, the common principal rays are these. A ray parallel to the principal axis refracts through the far focal point. A ray that passes through the near focal point emerges parallel to the principal axis. A ray through the optical center continues approximately straight.
For a diverging lens, the rules are slightly different. A ray parallel to the principal axis emerges as if it came from the near focal point. A ray aimed toward the far focal point emerges parallel to the axis. A ray through the optical center is again drawn straight.
Principal ray rules for thin lenses are the key tool for ray diagrams.
For a converging lens:
$$\text{parallel ray} \to \text{through far focus}$$
$$\text{through near focus} \to \text{parallel ray}$$
$$\text{through optical center} \to \text{straight line}$$
For a diverging lens:
$$\text{parallel ray} \to \text{appears from near focus}$$
$$\text{toward far focus} \to \text{parallel ray}$$
$$\text{through optical center} \to \text{straight line}$$
Image formation by converging lenses
A converging lens can produce different kinds of images depending on where the object is placed relative to the focal length.
If the object is farther from the lens than one focal length, the refracted rays actually meet on the other side of the lens. The image is real. Real images can be projected onto a screen.
If the object is placed exactly at the focal point, the outgoing rays are parallel, and no image forms at a finite distance.
If the object is placed closer to the lens than the focal length, the rays spread after leaving the lens, but their backward extensions meet on the same side as the object. The image is virtual, upright, and magnified.
Image formation by diverging lenses
A diverging lens behaves more simply in introductory optics. For a real object placed in front of the lens, the outgoing rays diverge, and their backward extensions meet on the same side as the object. So the image is virtual, upright, and smaller than the object.
This means a single diverging lens does not produce a real image from a real object in the standard situation.
Real and virtual images
A real image forms where actual light rays meet. Because light really passes through that location, a screen placed there can capture the image.
A virtual image forms where light rays do not actually meet, but appear to come from a common point when extended backward. A virtual image cannot be projected onto a screen, but it can still be seen by the eye.
| Image type | Rays actually meet? | Screen image possible? | Typical orientation |
|---|---|---|---|
| Real | Yes | Yes | Often inverted |
| Virtual | No | No | Often upright |
The thin lens equation
The object distance, image distance, and focal length are related by the thin lens equation:
$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$
Here, $d_o$ is the object distance and $d_i$ is the image distance. This equation works for both converging and diverging lenses when a consistent sign convention is used.
The thin lens formula is
$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$
This is one of the most important equations in geometrical optics.
Sign convention
A standard sign convention helps us use the lens equation correctly. In a common introductory convention for lenses, a real object in front of the lens has $d_o > 0$. A real image on the opposite side of the lens has $d_i > 0$. A virtual image on the same side as the object has $d_i < 0$. As stated earlier, converging lenses have $f > 0$ and diverging lenses have $f < 0$.
| Quantity | Positive when | Negative when |
|---|---|---|
| $d_o$ | Object is real and in front of lens | Rare special cases |
| $d_i$ | Image is real, opposite side from object | Image is virtual, same side as object |
| $f$ | Converging lens | Diverging lens |
Magnification
The linear magnification of a thin lens tells us how large the image is compared with the object. It is given by
$$m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$$
Here, $h_o$ is the object height and $h_i$ is the image height.
If $|m| > 1$, the image is larger than the object. If $|m| < 1$, the image is smaller. If $m$ is positive, the image is upright. If $m$ is negative, the image is inverted.
Magnification for a thin lens is
$$m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$$
Interpretation:
$$m > 0 \Rightarrow \text{upright image}$$
$$m < 0 \Rightarrow \text{inverted image}$$
Common image cases for a converging lens
A converging lens produces several standard cases that students often memorize, but it is better to understand them from the ray diagram and lens formula.
| Object position | Image type | Orientation | Size |
|---|---|---|---|
| Beyond $2f$ | Real | Inverted | Smaller |
| At $2f$ | Real | Inverted | Same size |
| Between $f$ and $2f$ | Real | Inverted | Larger |
| At $f$ | No finite image | Not applicable | Not applicable |
| Inside $f$ | Virtual | Upright | Larger |
For a diverging lens with a real object, the image is always virtual, upright, and smaller.
Lens power
The strength of a lens is often described by its power, especially in vision correction. Lens power is the reciprocal of focal length measured in meters:
$$P = \frac{1}{f}$$
The SI unit of power is the diopter, written $\text{D}$, where
$$1 \, \text{D} = 1 \, \text{m}^{-1}$$
A converging lens has positive power. A diverging lens has negative power.
For example, a lens with focal length $f = 0.50 \, \text{m}$ has power
$$P = \frac{1}{0.50} = 2.0 \, \text{D}$$
A lens with focal length $f = -0.25 \, \text{m}$ has power
$$P = \frac{1}{-0.25} = -4.0 \, \text{D}$$
Simple example
Suppose a converging lens has focal length $f = 10 \, \text{cm}$ and an object is placed at $d_o = 30 \, \text{cm}$.
Using the lens equation,
$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$
we get
$$\frac{1}{10} = \frac{1}{30} + \frac{1}{d_i}$$
so
$$\frac{1}{d_i} = \frac{1}{10} - \frac{1}{30} = \frac{2}{30} = \frac{1}{15}$$
therefore
$$d_i = 15 \, \text{cm}$$
The image distance is positive, so the image is real.
The magnification is
$$m = -\frac{d_i}{d_o} = -\frac{15}{30} = -0.5$$
So the image is inverted and half the size of the object.
Physical meaning of lens behavior
A lens works because light changes direction when it enters and leaves a material with a different refractive index. The curved surfaces cause different parts of the wavefront to bend by different amounts. In the thin lens model, we do not track all the detailed refraction at each surface. Instead, we summarize the whole effect using focal length and principal rays.
This approximation is powerful because it captures the essential imaging behavior while staying mathematically simple.
Limits of the thin lens model
The thin lens model is an idealization. Real lenses have thickness, imperfections, and optical aberrations. Rays far from the axis may not focus exactly as the model predicts. Also, systems with multiple lenses can require more advanced treatment.
Still, for many introductory problems, the thin lens approximation gives very accurate and very useful results.
Core formulas for thin lenses:
$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$
$$m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$$
$$P = \frac{1}{f} \quad \text{with } f \text{ in meters}$$
Final picture
A thin lens is a simple but powerful model for image formation by refraction. Converging lenses can create real or virtual images depending on object position. Diverging lenses usually create virtual, upright, reduced images for real objects. With the focal length, the lens equation, and magnification, we can predict where an image forms and what it looks like.