Editing Lens (section)

== Compound lenses <span class="anchor" id="compound_lens_anchor"></span> ==
{{See also|Photographic lens|Doublet (lens)|Triplet lens|Achromatic lens}}
Simple lenses are subject to the [[#Aberrations|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, if the purpose of the system is to image an object, then the system design can be such that each lens treats the image made by the previous lens as an object, and produces the new image of it, so the imaging is cascaded through the lenses.<ref>{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages=178 |language=English |chapter=Thin-Lens Combinations}}</ref><ref>{{Cite web |last=Vlasenko |first=Alexey |date=2011 |title=Lecture 9 Notes: 07 / 13 - Multiple-lens systems |url=https://courses.physics.ucsd.edu/2011/Summer/session1/physics1c/lecture9.pdf |url-status=live |archive-url=https://web.archive.org/web/20240418224408/https://courses.physics.ucsd.edu/2011/Summer/session1/physics1c/lecture9.pdf |archive-date=18 April 2024 |access-date=2024-04-19 |website=Physics 1C, Summer Session I, 2011 - University of California San Diego}}</ref> As shown [[#Derivation 2|above]], the Gaussian lens equation for a spherical lens is derived such that the 2nd surface of the lens images the image made by the 1st lens surface. For multi-lens imaging, 3rd lens surface (the front surface of the 2nd lens) can image the image made by the 2nd surface, and 4th surface (the back surface of the 2nd lens) can also image the image made by the 3rd surface. This imaging cascade by each lens surface justifies the imaging cascade by each lens.

For a two-lens system the object distances of each lens can be denoted as <math display="inline">s_{o1}</math> and <math display="inline">s_{o2}</math>, and the image distances as and <math display="inline">s_{i1}</math> and <math display="inline">s_{i2}</math>. If the lenses are thin, each satisfies the thin lens formula

<math display="block">\frac{1}{f_j} = \frac{1}{s_{oj}} + \frac{1}{s_{ij}},</math>

If the distance between the two lenses is <math>d</math>, then <math display="inline">s_{o2} = d - s_{i1}</math>. (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.<ref>{{Cite journal |last=Paschotta |first=Dr Rüdiger |title=focal distance |url=https://www.rp-photonics.com/focal_distance.html |access-date=2024-04-29 |website=www.rp-photonics.com |language=en |doi=10.61835/6as |archive-date=29 April 2024 |archive-url=https://web.archive.org/web/20240429224301/https://www.rp-photonics.com/focal_distance.html |url-status=live }}</ref> 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., <math>s_{i1} = d - f_2 </math>. So, the thin lens formula for the 1st lens becomes<ref>{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages=181 |language=English |chapter=Back and Front Focal Lengths}}</ref>

<math display="block">\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)}. </math>

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

<math display="block">\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)}.</math>

A simplest case is where thin lenses are placed in contact (<math>d = 0</math>). Then the combined focal length {{mvar|f}} of the lenses is given by

<math display="block">\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}\,.</math>

Since {{math|1/''f''}} is the power of a lens with focal length {{mvar|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

<math display="block">\frac{1}{f} =\sum_{k = 1}^N \frac{1}{f_k}</math>

where <math display="inline">N</math> is the number of lenses.

If two thin lenses are separated in air by some distance {{mvar|d}}, then the focal length for the combined system is given by
<math display="block">\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}-\frac{d}{f_1 f_2}\,.</math>

As {{mvar|d}} tends to zero, the focal length of the system tends to the value of {{mvar|f}} given for thin lenses in contact. It can be shown that the same formula works for thick lenses if {{mvar|d}} is taken as the distance between their principal planes.<ref name="Hecht-2017" />

If the separation distance between two lenses is equal to the sum of their focal lengths ({{math|1=''d'' = ''f''{{sub|1}} + ''f''{{sub|2}}}}), 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 [[Refracting telescope|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

<math display="block">M = -\frac{f_2}{f_1}\,,</math>

which is the ratio of the output beam width to the input beam width. Note the sign convention: a telescope with two convex lenses ({{math|''f''{{sub|1}} > 0}}, {{math|''f''{{sub|2}} > 0}}) produces a negative magnification, indicating an inverted image. A convex plus a concave lens ({{math|''f''{{sub|1}} > 0 > ''f''{{sub|2}}}}) produces a positive magnification and the image is upright. For further information on simple optical telescopes, see [[Refracting telescope#Refracting telescope designs|Refracting telescope § Refracting telescope designs]].