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I am confused on how to calculate the final image position after the two lenses. I understand that the image position of the first lens can be treated as the object position of the second lens, but in this circumstance (see image) the image position of the first lens is behind the second lens.

I don't understand how you would do it mathematically, and I don't know how you could get an accurate measurement from a ray diagram.

Also when drawing the ray diagram for the second lens, the horizontal ray would be easy to figure because it would go through the focus of the second lens, but would you just have to guess on the other two rays if they don't pass through a focus?

enter image description here

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If the image distance of the first lens is past the second lens, you would just have to plug in a negative object distance for the second lens.

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For a multiple lens optical system, the first order (paraxial approximation) characteristics are often derived using ray transfer matrix analysis. For each surface a 2x2 matrix is used to represent the refractive power (depending on curvature and ratio of refractive indices) of that surface. Another 2x2 matrix represents the distance between surfaces. A sequence of such matrices, ordered in the direction rays pass through the lenses, are multiplied to produce a system matrix from which the characteristic principle and focal points are calculated.

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