In this article, we will explain and show how our Thin Lens Equation Calculator can help you learn how a lens can magnify something in an object and the properties of the simple lenses. We can use this tool for thin lens, which means the ones with a lens thickness that is significantly less than the distances between the object and the image. 

While you are here, check other physics-related posts on our site, such as Power to Weight Ratio or Friction Calculator.

Thin Lens Definition

If we take a glass piece with a finite thickness and two spherical boundaries is a lens. In physics, the definition of a thin lens is an optical device whose thickness allows rays of light to refract but does not allow properties such as dispersion and aberration.

They can be diverging and converging, and we will go into more detail about that below. A simpler definition of them would be when the thickness of the lens is much smaller than the circumference of the lens; we call it a thin lens.

Thin lens equation

The formula that physics introductory textbooks are also called¬†the Gaussian form¬†of the thin lens equation. When we want to know how the equation for thin lenses is calculated, we need to know the¬†three¬†terms, or in this context,¬†distances¬†connected to it, d_o ‚Ästobject distance, which is a distance of an object to the center of a lens.

Next is d_i ‚Ästimage distance, the length from the image to the center of a lens. The last one is the¬†focal length¬†of a lens, and it is the distance from a lens to the¬†focal point,¬†which we will explain in more detail below. We express it in length units.

The equation for the thin lens calculates as: 

\dfrac{1}{d_o} + \dfrac{1}{d_i}=\dfrac{1}{f}

Focal points

In simple terms, it is a focus or a point where sunlight or other forms of radiation concentrate. We can find it by letting parallel rays of light enter the lenses, and the point of contact where they converge (meet) is the focal point or point of convergence.

In contrast to the focal length, which tells us how much the sunlight will bend, if new think of light rays as a cone, for example, ice cream cone, the point of that cone is the focal point, and every lens has two points, one on each side. That is because light can pass through them from both sides and create two points of convergence. They are located on the optical axis.

When you need optimal parameters of a lens given the focal length, our Lens Maker Calculator is perfect for you.

Converging and diverging lenses

Converging or positive lenses have a focal length above 0. Therefore, they cause exiting light bundles to converge more when they enter. On the other hand, diverging or negative lens, while having negative focal length, cause the rays to diverge outward than when entering. 

Besides this, we also have two images, and we determine them by the convergence of rays that enter the lens. The point that we call a real image is where exiting rays converged to one point. Contrary to the term before, the virtual image is defined whenever rays emerge divergent from a common point.

When we see an image through a screen in sharp detail, it is a real image. When the image can only be seen with a lens back toward the light source, we call it a virtual image. 

Examples of both are:

  • The image of seeing yourself in a mirror is virtual, as is looking trough contact lens.
  • The image a camera casts on a film is real.

Magnification lens equation

Our Thin Lens Equation Calculator also has an advanced mode. In it, we can calculate the magnification of the created image. All we need to know is the distance of the object and image. The formula for the magnification of a lens is as follows: 

m= \dfrac{d_i}{d_o} 

We refer to both this and the first formula mentioned as the thin lens equation. Also see this Distance Calculator, to learn more about distance.

Images in the converging lens

You can check and calculate the examples we will show further below in our calculator to familiarize yourself with them. We will check five cases where the focal length of converging lenses is larger than zero (f>0). 

  1. For a case where d_o>2f, the image is real, inverted and reduced in size, which means M<1.
  2. For d_o=2f image is real, and it is magnified M>1
  3. For the case where d_o is located between f and 2f, it is real and magnified.
  4. When the distance of the object to the center of the lens and focal length of the lens is of the same size (d_o=f) image never appears;  d_i -> infinity
  5. When an object is smaller than the focal point, the image is virtual and magnified. 

Thin-Lens Equation: Cartesian Convention

By using the Cartesian sign convention, the Gaussian form of the lens will become:

\dfrac{1}{d_o}+ \dfrac{1}{f}= \dfrac{1}{d_i}

This happens because, in that convention, a quantity’s positive direction is always the same as the direction in which light travels. Because you must travel in the opposite direction of light from the lens to the object, the object distance o is a negative quantity.

Since we mentioned the Cartesian sign convention, we need to mention that it has several points that a part of it, and they are:

  1. Light is drawn as travelling from left to right.
  2. We measure all distances from the reference surface, and distances to the left of it are negative. 
  3. The refractive power of a surface that makes light rays more convergent is positive. The focal length of such a surface is positive.
  4. The distance of an object is negative.
  5. Heights above the optic axis are positive.


What is the power of a lens and how is it calculated?

We define the power of a lens as a reciprocal of the focal length. Therefore the formula for it is:
P= \dfrac{1}{f}

Do different thin lenses have different formulas?

No, the thin lens equation is the same for different kind of lens.