1. Why combine lenses
Sometimes a single lens cannot give the required image size, brightness or focus. By placing two thin lenses close together, I can change the overall bending of light. The combined effect behaves like a single lens with a different focal length and power.
This idea is used in cameras, microscopes, telescopes and eyeglass corrections.
2. How two thin lenses placed in contact behave
When two thin lenses are placed very close to each other (their separation is negligible), light passes through them one after another. Each lens bends the rays a little, and the combined bending is the sum of the two effects.
The entire combination behaves like a single lens whose focal length depends on the focal lengths of the individual lenses.
3. Equivalent focal length of two lenses in contact
If two thin lenses with focal lengths \(f_1\) and \(f_2\) are placed in contact, the equivalent focal length \(F\) of the combination is given by:
\( \dfrac{1}{F} = \dfrac{1}{f_1} + \dfrac{1}{f_2} \)
This formula works for convex and concave lenses as long as I use the correct signs for focal lengths.
3.1. Meaning of signs
- Convex lens → positive focal length.
- Concave lens → negative focal length.
This sign rule helps identify whether the combination strengthens or weakens the overall focusing power.
4. Power of lens combinations
Because power is the reciprocal of focal length, it becomes even easier to add the effects of two lenses. The total power of lenses in contact is simply:
\( P_{\text{total}} = P_1 + P_2 \)
This is very convenient when dealing with eyeglass prescriptions or optical instruments.
4.1. How signs affect total power
- Convex lens → positive power.
- Concave lens → negative power.
A positive and a negative power can cancel each other partially or completely depending on their magnitudes.
5. Understanding the combined effect with examples
Combining lenses can produce interesting results depending on their focal lengths.
5.1. Two convex lenses
Both have positive focal lengths. Their powers add up, so the overall system becomes stronger (shorter equivalent focal length). The combination bends rays more than either lens alone.
5.2. Convex + concave lens
The convex lens has positive power, the concave has negative. Depending on their strengths:
- If convex power > concave power → overall converging system.
- If concave power > convex power → overall diverging system.
- If powers cancel → equivalent focal length becomes very large (system becomes weak).
5.3. Two concave lenses
Both have negative powers, so they add to form a more diverging lens. Rays spread out even more.
6. Effect of separation between lenses
The simple formulas above assume the lenses are in contact. If the lenses are separated by a distance, the calculation becomes more complex because the image formed by the first lens becomes the object for the second.
For most ray-diagram cases, I handle this by:
- Finding the intermediate image using the first lens.
- Treating that image as the object for the second lens.
- Using the lens formula twice.
7. A simple numerical example
Suppose two lenses are in contact:
- Lens 1: Convex with \( f_1 = +20\,\text{cm} \)
- Lens 2: Convex with \( f_2 = +10\,\text{cm} \)
Using the formula:
\( \dfrac{1}{F} = \dfrac{1}{20} + \dfrac{1}{10} = \dfrac{1}{20} + \dfrac{2}{20} = \dfrac{3}{20} \)
\( F = \dfrac{20}{3} \approx 6.67\,\text{cm} \)
The combination behaves like a single strong convex lens with a much shorter focal length.
8. Where lens combinations are used
I often see combinations of lenses in:
- Eyeglasses (to fine-tune power)
- Microscope objectives (compound lenses)
- Telescope eyepieces
- Cameras (to correct distortions and improve focus)
By cleverly combining lenses, optical designers can achieve effects that a single lens cannot produce.