Lens Combination Formulae
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Many optical tasks require several lenses in order to achieve an
acceptable level of performance. One possible approach to lens
combinations is to consider each image formed by each lens as the
object for the next lens and so on. This is a valid approach,
but it is time-consuming and unnecessary. It is much simpler to
calculate the effective (combined) focal length and principal-point
locations and then use these results in any subsequent paraxial calculations
They can even be used in optical invariant calculations.
For reference purposes, use the illustration below. |
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| Generalization from combinations to systems |
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Effective Focal Length The following formulae show how to calculate the effective focal length and principal-point locations for a combination of any two arbitrary lenses. The approach for more than two lenses is very simple: calculate the values for the first two elements,then perform the same calculation for this combination with the next lens. This is continued until all lenses in the system are accounted for. The expression for the combination focal length is the same whether lens separation distance (d), defined as the distance between the secondary principal point H1" of the first (left-hand) lens and the primary principal point H2 of the second (right-hand) lens, is large or small or whether the focal lengths f1 and f2 are positive or negative: |
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This may be more familiar in the form |
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Notice that the formula is symmetric with the lenses (end-for-end rotation of
the combination) at constant d. The next two formulae
are not. |
Combination Focal-Point Location |
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For all cases, the location of focal point of the combined system (s2"), measured from the secondary principal point
of the second lens (H2"), is given by:
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Combination Secondary Principal-Point Location Because the thin-lens approximation is obviously highly invalid for most combinations, the ability to determine the location of the secondary principal point is vital for accurate determination of d when another element is added. The simplest formula for this calculates how far the secondary principal point of the final (second) element is moved by being part of the combination: |
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It is possible for a lens combination or system to exhibit principal planes that
are far removed from the system. When such systems are themselves combined,
negative values of d may occur. Probably the simplest
example of a negative d-value situation is shown in the
figure below. Meniscus lenses with steep surfaces have external principal
planes. When two of these lenses are brought into contact, a negative value of
d can occur. Click on the link below to see other
examples. | |
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| Extreme meniscus-form lenses with external principal planes (not to scale) |
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| Optics Guide Copyright 2002 Melles Griot Inc. |