# Two-Lens Systems

This is a continuation from the previous tutorial - laser amplification.

Figure 16 illustrates the general imaging problem where an image is formed of an object by two lenses at a specified magnification and object-to-image distance.

Most imaging problems can be solved by using two equivalent lens elements. An equivalent lens can comprise one lens or multiple lenses and may be represented by the principal planes and power of a single thick lens.

All distances are measured from the principal points of each equivalent lens element. For simplicity, the lenses shown in Fig. 16 are thin lenses. If the magnification $$m$$, object-image distance $$\mathscr{S}$$, and lens powers $$\phi_a$$ and $$\phi_b$$ are known, then the equations for $$s_1$$, $$s_2$$, and $$s_3$$ are given by

\tag{34}\begin{align}s_1&=\frac{\phi_b(\mathscr{S}-s_2)-1+m}{m\phi_a+\phi_b}\\s_2&=\frac{\mathscr{S}}{2}\left[1\pm\sqrt{1-\frac{4[\mathscr{S}m(\phi_a+\phi_b)+(m-1)^2]}{\mathscr{S}^2m\phi_a\phi_b}} \right]\\s_3&=\mathscr{S}-s_1-s_2\end{align}

The equation for $$s_2$$ indicates that zero, one, or two solutions may exist.

If the magnification and the distances are known, then the lens powers can be determined by

\tag{35}\begin{align}\phi_a&=\frac{\mathscr{S}+(s_1+s_2)(m-1)}{ms_1s_2}\\\phi_b&=\frac{\mathscr{S}+s_1(m-1)}{s_2(\mathscr{S}-s_1-s_2)}\end{align}

It can be shown that only certain pairs of lens powers can satisfy the magnification and separation requirements. Commonly, only the magnification and object-image distance are specified with the selection of the lens powers and locations to be determined.

By utilizing the preceding equations, a plot of regions of all possible lens power pairs can be generated. Such a plot is shown as the shaded region in Fig. 17 where $$\mathscr{S}=1$$ and $$m=-0.2$$.

Examination of this plot can assist in the selection of lenses that may likely produce better performance by, for example, selecting the minimum power lenses. The potential solution space may be limited by placing various physical constraints on the lens system.

For example, the allowable lens diameters can dictate the maximum powers that are reasonable. Lines of maximum power can then be plotted to show the solution space.

When $$s_1$$ becomes very large compared to the effective focal length $$efl$$ of the lens combination, the optical power of the combination of these lenses is expressed by

$\tag{36}\phi_{ab}=\phi_a+\phi_b-s_2\phi_a\phi_b$

The effective focal length is $$\phi_{ab}^{-1}$$ or

$\tag{37}f_{ab}=\frac{f_af_b}{f_a+f_b-s_2}$

and the back focal length is given by

$\tag{38}bfl=f_{ab}\left(\frac{f_a-s_2}{f_a}\right)$

The separation between lenses is expressed by

$\tag{39}s_2=f_a+f_b-\frac{f_af_b}{f_{ab}}$

Figure 18 illustrates the two-lens configuration when thick lenses are used. The principal points for the lens combination are denoted by $$P_1$$ and $$P_2$$, $$P_{a1}$$ and $$P_{a2}$$ for lens $$a$$, and $$P_{b1}$$ and $$P_{b2}$$ for lens $$b$$.

With the exception of the back focal length, all distances are measured from the principal points of each lens element or the combined lens system as shown in the figure. For example, $$s_2$$ is the distance from $$P_{a2}$$ to $$P_{b1}$$. The $$bfl$$ is measured from the final surface vertex of the lens system to the focal point.

The next tutorial discusses about achromatic doublets