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I know the formula that when two lenses are separated by a distance $d,$ then the resultant focal length is given by :-

$$\frac 1f = \frac1f_1 + \frac1f_2 - \frac d{f_1\cdot f_2}$$

However , there are two doubts that arise in my mind.

1) My book says that a combination of two thin lenses can only be replaced with a thick lens. What exactly is a thick lens? Also, how do we find out the position of the lens i.e. the position of the pole of this single replaced lens?

I assume it to be the half of the distance between the two lenses($d,$ in the above formula). Am I correct?

2)Also , what if we have more than 2 lenses? In that case do we first replace two lenses by a single lens, and then use that new lens with the other lens/lenses to find its focal length and keep repeating it until we run out of lenses? In this approach, I still need some distance of the pole of the new lens with respect to something OR any distance in order to find out the new lens's position (i.e. the ans to my 1st question).

Or is there a formula to find that in one go Like $1/f_1 + 1/f_2 + \ldots + 1/f_n - d/{f_1f_2\ldots f_n}\;?$

Or is it something more complicated? I was able to prove the formula for 2 lenses but not for any $n$ number of them. I guess that is the case because my book says that the theory of a thick lens is very complicated and far beyond the scope of the book.

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The thin-lens formula $$ \frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}$$ is an approximation which assumes that the distance from the front interface of the lens (the transition from index of refraction outside the lens to inside) to the back interface (transition from inside to outside) is small enough to be ignored compared to radii of curvature of the interfaces. It also approximates that the important light paths are close to the middle/axis of the lens (paraxial rays).

When two thin lenses are separated by a distance, $d$, which is not small enough to ignore compared to the radii of curvature, one essentially has introduced the effect of a lense which is thick, that is, one in which the front interface is separated from the back by a non-negligible distance, and the thin-lens approximation can't account for the focusing ability of the combination. Essentially, the thick lens is described as having 2 planes of focusing power, separated by a finite, fixed distance. For purposes of finding/using lens power values, distance to the object is measured from one plane, and distance to the image is measured from the other plane.

You can't use a single plane to measure the distance to both image and object. Those planes will not necessarily be equal distances from the physical middle of the lens. It depends on the specific radius of curvature of each surface.

More than two lenses is extremely complicated if you are trying to reduce the system to a single set of parameters that resemble a single non-thin lens. It's better to encounter each lens separately and step through them one at a time.

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  1. A thick lens is a more complicated device than a thin lens. Theoretically, it is defined by the curvature radius of each of the two sides of the lens, the thickness and the refractive index of the material (4 variables). Note that your equivalent thick-lens is under-constrained: 4 variables > 3 parameters. By playing with these variables, you can get the thick lens to behave like your two thin lenses or any optical system in air, because the determinant of the ray-transfer matrix is 1 for a thick lens. Therefore, it might be possible to replace the two lenses by just one thick lens half way in between the two lenses.

  2. There is a way to obtain the focal length of an array of lenses as described here. The trick is to keep track of the front and back distances of the equivalent lens. Note that the formula you provided is incorrect because the last term $d/f_1 ... f_n$ does not have units of 1/distance.

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