9.5.4. Combination of thin lenses in contact

Consider two lenses A and B of focal length f₁ and f₂ placed in contact with each other. Let the object be placed at a point O beyond the focus of the first lens A (Fig. 9.19). The first lens produces an image at I₁.

In this section, we're discussing how two lenses can work together when they are placed in contact. The first lens (A) takes light from an object positioned at point O. When light passes through this lens, it refracts the light rays and creates an image at point I₁. This image is referred to as a 'real image' because the light rays actually converge at this point, allowing a screen to capture the image.

Think of the first lens like a projector lens showing a movie on a screen. Just as the projector lens forms a clear image on the screen by collecting and focusing light from the film reel, the first lens forms an image from the object placed before it.

Since image I₁ is real, it serves as a virtual object for the second lens B, producing the final image at I₂.

After the first lens forms the image at I₁, this image now acts as the object for the second lens (B). A 'virtual object' means that rather than light emanating directly from the physical object, we are now using the light coming from I₁ to create a new image. The second lens adjusts the direction of rays from this image to create a final image at point I₂.

Consider it like using a magnifying glass after looking into a projector. The image you see on the screen (like I₁) can be viewed again using another lens (like lens B) that magnifies or alters that image further for a better view.

Since the lenses are thin, we assume the optical centres of the lenses to be coincident. Let this central point be denoted by P.

Thin lenses are treated as if they have negligible thickness, which simplifies calculations and assumptions in optical systems. By assuming that both lenses share the same central point—denoted as P—mathematical relationships regarding distances and image formation can be easily derived.

Imagine stacking two pieces of thin glass together; if they fit snugly, you could see through both at once as if they were one single lens. This represents how the optical centers are treated in calculations—simplifying their interaction.

For the image formed by the first lens A, we get 1/f₁ = 1/v₁ + 1/u₁. For the image formed by the second lens B, we get 1/f₂ = 1/v₂ + 1/v₁.

Each lens has its own focal length (f₁ for lens A and f₂ for lens B). The lens formulas indicate relationships between object distance (u), image distance (v) and focal length (f). Rearranging and adding these equations illustrates how the image formed by one lens serves as the object for the next lens, allowing us to understand the combined optics of multiple lenses.

Think of two mirrors in a funhouse setup where reflections occur. The light reflecting off the first mirror becomes the object for the second mirror. Similarly, the images from both lenses interact in a structured way that can be calculated mathematically.

If the two lens-system is regarded as equivalent to a single lens of focal length f, we have 1/f = 1/f₁ + 1/f₂.

This formula expresses the effective focal length of a system comprising multiple lenses. By taking the reciprocal of the individual focal lengths, we derive a single equivalent focal length for the combination, simplifying the analysis of the overall system. This means we can treat two lenses as if they were one effective lens when determining how light is focused and where images form.

Imagine combining two magnifying glasses for reading small print. Instead of calculating how each glass affects the vision separately, we can compute how they work together as a singular tool—it’s more practical when making adjustments or understanding their combined effect.

The total magnification m of the combination is a product of magnification (m₁, m₂) of individual lenses.

The magnification resulting from multiple lenses in contact is the result of multiplying the magnification produced by each lens. This is important in applications such as microscopes and cameras, where the goal is to achieve high magnification through combinations of optics. Thus, total magnification can be determined accurately by understanding how each lens contributes to the final image.

Consider using multiple sets of binoculars; the way they enhance your view is multiplied with each set you add. Each magnifying lens in a telescope or microscope acts like a pair of binoculars, enhancing what you see by combining their effects.

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