section_path: Compound lenses
section_name: Compound lenses
paragraph_text: Compound lenses
  
  Simple lenses are subject to the optical aberrations discussed above. In many cases these aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations. A compound lens is a collection of simple lenses of different shapes and made of materials of different refractive indices, arranged one after the other with a common axis.
  
  In a multiple-lens system, each lens treats the previous lens's image as an object, and produces a new image of it, so that the imaging is cascaded through the lenses. This is easy to understand when the image from one lens is before the front focus of the next lens, but it remains true even when the intermediate image is located within the focal length of the next lens or even beyond it.
  
  For a two-lens system the object distances of each lens can be denoted as s o 1 {\textstyle s_{o1}} and s o 2 {\textstyle s_{o2}}, and the image distances as and s i 1 {\textstyle s_{i1}} and s i 2 {\textstyle s_{i2}}. If the lenses are thin, each satisfies the thin lens formula
  
  1 f j = 1 s o j + 1 s i j, {\displaystyle {\frac {1}{f_{j}}}={\frac {1}{s_{oj}}}+{\frac {1}{s_{ij}}},}
  
  If the distance between the two lenses is d {\displaystyle d}, then s o 2 = d − s i 1 {\textstyle s_{o2}=d-s_{i1}}. (The 2nd lens images the image of the first lens.)
  
  FFD (Front Focal Distance) is defined as the distance between the front (left) focal point of an optical system and its nearest optical surface vertex. If an object is located at the front focal point of the system, then its image made by the system is located infinitely far way to the right (i.e., light rays from the object is collimated after the system). To do this, the image of the 1st lens is located at the focal point of the 2nd lens, i.e., s i 1 = d − f 2 {\displaystyle s_{i1}=d-f_{2}}. So, the thin lens formula for the 1st lens becomes
  
  1 f 1 = 1 F F D + 1 d − f 2 → F F D = f 1 (d − f 2) d − (f 1 + f 2). {\displaystyle {\frac {1}{f_{1}}}={\frac {1}{FFD}}+{\frac {1}{d-f_{2}}}\rightarrow FFD={\frac {f_{1}(d-f_{2})}{d-(f_{1}+f_{2})}}.}
  
  BFD (Back Focal Distance) is similarly defined as the distance between the back (right) focal point of an optical system and its nearest optical surface vertex. If an object is located infinitely far away from the system (to the left), then its image made by the system is located at the back focal point. In this case, the 1st lens images the object at its focal point. So, the thin lens formula for the 2nd lens becomes
  
  1 f 2 = 1 B F D + 1 d − f 1 → B F D = f 2 (d − f 1) d − (f 1 + f 2). {\displaystyle {\frac {1}{f_{2}}}={\frac {1}{BFD}}+{\frac {1}{d-f_{1}}}\rightarrow BFD={\frac {f_{2}(d-f_{1})}{d-(f_{1}+f_{2})}}.}
  
  A simplest case is where thin lenses are placed in contact (d = 0 {\displaystyle d=0}). Then the combined focal length f of the lenses is given by
  
  1 f = 1 f 1 + 1 f 2. {\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}\,.}
  
  Since 1/ f is the power of a lens with focal length f, it can be seen that the powers of thin lenses in contact are additive. The general case of multiple thin lenses in contact is
  
  1 f = ∑ k = 1 N 1 f k {\displaystyle {\frac {1}{f}}=\sum _{k=1}^{N}{\frac {1}{f_{k}}}}
  
  where N {\textstyle N} is the number of lenses.
  
  If two thin lenses are separated in air by some distance d, then the focal length for the combined system is given by 1 f = 1 f 1 + 1 f 2 − d f 1 f 2. {\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}-{\frac {d}{f_{1}f_{2}}}\,.}
  
  As d tends to zero, the focal length of the system tends to the value of f given for thin lenses in contact. It can be shown that the same formula works for thick lenses if d is taken as the distance between their principal planes.
  
  If the separation distance between two lenses is equal to the sum of their focal lengths (d = f + f), then the FFD and BFD are infinite. This corresponds to a pair of lenses that transforms a parallel (collimated) beam into another collimated beam. This type of system is called an afocal system, since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type of optical telescope. Although the system does not alter the divergence of a collimated beam, it does alter the (transverse) width of the beam. The magnification of such a telescope is given by
  
  M = − f 2 f 1, {\displaystyle M=-{\frac {f_{2}}{f_{1}}}\,,}
  
  which is the ratio of the output beam width to the input beam width. Note the sign convention: a telescope with two convex lenses (f > 0, f > 0) produces a negative magnification, indicating an inverted image. A convex plus a concave lens (f > 0 > f) produces a positive magnification and the image is upright. For further information on simple optical telescopes, see Refracting telescope § Refracting telescope designs.