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We have studied Two lens system, I wanted to know how to formulate when there are Multiple Lenses. Like How the formula for Effective Focal length changes when there are $2$ or more lenses.

For example: Consider a $3$ thin lens system, the effective focal length would be $F=f_1+f_2+f_3-d_1f_1f_2-d_2f_2f_3$ ?

Where $d_1$ is the distance between lens $1$ and $2$, and $d_2$ is the distance between lens $2$ and $3$.

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Take a look at the ray transfer matrix formalism.

This will allow you to model the propagation of rays, in the paraxial approximation, across media characterised by their refractive index, as well as refraction at optical interfaces.

So, this is the tool to model systems consisting of an arbitrary number of thin lenses, as well as thick lenses, prisms, etc. If you want to apply it, all you need is a bit of matrix algebra. Pick the matrices corresponding to the optical elements in your setup, multiply and what you get is a representation of your system from which you can read off its properties.

There is even an extension for Gaussian beams and optical resonators if you want to dig more deeply into the topic.

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