Aberration Balancing
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In the section on Aberration Balancing we considered the case of a
two-element laser beam expander where two lenses are separated by a distance which is the sum of their focal lengths, so
that the overall system focal length is infinite. This system does not focus incoming collimated light, but it does change
the beam diameter. By definition, each of the lenses is operating at the same f-number. A 4x Beam Expander The equation for longitudinal spherical aberration shows that for two lenses with the same f-number, aberration varies directly with the focal lengths of the lenses. The sign of the aberration is the same as focal length. Thus, it should be possible to correct the spherical aberration of this Galilean-type beam expander, which consists of a positive focal length objective and a negative diverging lens. If a plano-convex lens of focal length f1 oriented in the normal direction is combined with a plano-concave lens of focal length f2 oriented in its reverse direction, the total spherical aberration of the system is |
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After setting this equal to zero, we obtain |
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To make the magnitude of aberration contributions of the two elements equal so they will cancel out, and thus correct the
system, select the focal length of the positive element to be 3.93 times that of the negative element. The figure below shows a beam-expander system made up of catalog elements, in which the focal length ratio is 4:1. This simple system is corrected to about 1/6 wavelength at 632.8nm, even though the objective is operating at f/4 with a 20-mm aperture diameter. This is remarkably good wavefront correction for such a simple system; one would normally assume that a doublet objective would be needed and a complex diverging lens as well. This analysis does not take into account manufacturing tolerances. |
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Corrected 4x beam expander |
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A Lower-Power Beam Expander A beam expander of lower magnification can also be derived from this information. If a symmetric-convex objective is used together with a reversed plano-concave diverging lens, the aberration coefficients are in the ratio of 1.069/0.403 = 2.65. The figure below shows a system of catalog lenses that provides a magnification of 2.7 (the closest possible given the available focal lengths). The maximum wavefront error in this case is only 1/4 wave, even though the objective is working at f/3.3. The relatively fast speed of these objectives is a great advantage in minimizing the length of these beam expanders. They would be particularly useful with Nd:YAG and argon-ion lasers, which tend to have large output beam diameters. |
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Corrected 2.7x beam expander |
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High Numerical Aperture Lenses These same principles can be utilized to create high numerical aperture objectives that might be used as laser focusing lenses. The figure below shows an objective consisting of an initial negative element, followed by two identical plano-convex positive elements. Again, all of the elements operate at the same f-number, so that their aberration contributions are proportional to their focal lengths. To obtain zero total spherical aberration from this configuration, we must satisfy |
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Therefore, a corrected system should result if the focal length of the negative element is just about half that of each of
the positive lenses. In this case, f1= -25 mm and
f2 = 50 mm yield a total system focal length of about 25 mm and an
f-number of approximately f/2. This objective, corrected to 1/6 wave and shown in the figure below, has the additional
advantage of a very long working distance. |
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Spherically corrected 25-mm EFL f/2.0 objective |
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| Optics Guide Copyright 2002 Melles Griot Inc. |