Lens Selection
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Having discussed the most important factors that affect a lens' or a lens system's performance, we will now address the
practical matter of selecting the optimum catalog components for a particular task. The following useful relationships are important to keep in mind throughout the selection process:
Produce a collimated beam from a quartz halogen bulb having a 1-mm-square filament. Collect the maximum amount of light possible and produce a beam with the lowest possible divergence angle. This problem, illustrated in the figure below, involves the typical tradeoff between light-collection efficiency and resolution (where a beam is being collimated rather than focused, resolution is defined by beam divergence). To collect more light, it is necessary to work at a low f-number, but because of aberrations, higher resolution (lower divergence angle) will be achieved by working at a higher f-number. |
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Collimating an incandescent source |
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In terms of resolution, the first thing to realize is that the minimum divergence angle (in radians) that can be achieved
using any lens system is the source size divided by system focal length. An off-axis ray (from the edge of the source)
entering the first principal point of the system exits the second principal point at the same angle. Therefore, increasing
system focal length improves this limiting divergence because the source appears smaller. An optic that can produce a spot size of 1 mm when focusing a perfectly collimated beam is therefore required. Since source size is inherently limited, it is pointless to strive for better resolution. This level of resolution can be achieved easily with a plano-convex lens. While angular divergence decreases with increasing focal length, spherical aberration of a plano-convex lens increases with increasing focal length. To determine the appropriate focal length, set the spherical aberration formula for a plano-convex lens equal to the source (spot) size: |
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This ensures a lens that meets the minimum performance needed. To select a focal length, make an arbitrary f-number choice. As can be seen from the relationship, as we lower the f-number (increase collection efficiency), we decrease the focal length, which will worsen the resultant divergence angle (minimum divergence = 1mm/f ). In this example, we will accept f/2 collection efficiency, which gives us a focal length of about 120 mm. For f/2 operation we would need a minimum diameter of 60 mm. The 01 LPX 209 fits this specification exactly. Beam divergence would be about 8 mrad. Finally, we need to verify that we are not operating below the theoretical diffraction limit. In this example, the numbers (1-mm spot size) indicate that we are not, since |
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Coupling an Incandescent Source into a Fiber In the Numeric Aperture and Magnification example set we considered a system in which the output of an incandescent bulb with a filament of 1 mm in diameter was to be coupled into an optical fiber with a core diameter of 100 µm and a numerical aperture of 0.25. From the optical invariant and other constraints given in the problem, we determined that system focal length = 9.1 mm, diameter = 5 mm, s = 100 mm, s" = 10 mm, NA" = 0.25, and NA = 0.025 (or f/2 and f/20 respectively). The singlet lenses that match these specifications are the plano-convex 01 LPX 003 or biconvex lenses 01 LDX 003 and 01 LDX 005. The closest achromat would be the 01 LAO 001. We can immediately reject the biconvex lenses because of spherical aberration. We can estimate the performance of the 01 LPX 003 on the focusing side by using our spherical aberration formula: |
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We will ignore, for the moment, that we are not working at the infinite conjugate. This is slightly smaller than the 100-μm spot size we are trying to achieve. However, since we are not working at infinite conjugate, the spot size will be larger than given by our simple calculation. This lens is therefore likely to be marginal in this situation, especially if we consider chromatic aberration. A better choice is the achromat. Although a computer ray trace would be required to determine its exact performance, it is virtually certain to provide adequate performance. Symmetric Fiber-to-Fiber Coupling Couple an optical fiber with an 8-μm core and a 0.15 numerical aperture into another fiber with the same characteristics. Assume a wavelength of 0.5 μm. This problem, illustrated in the figure below, is essentially a 1:1 imaging situation. |
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Symmetric fiber-to-fiber coupling |
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We want to collect and focus at a numerical aperture of 0.15 (or f/3.3), and we need a lens with an 8-μm spot size at this
f-number. Based on the discussion in the Lens Combination section,
our most likely setup is either a pair of identical plano-convex lenses or a pair of achromats, faced front to front.
To determine the necessary focal length for a plano-convex lens, we again use the spherical aberration estimate formula: |
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This formula yields a focal length of 4.3 mm and a minimum diameter of 1.3 mm. The 01 LPX 423 meets these criteria. The biggest
problem with utilizing these tiny, short-focal-length lenses is the practical considerations of handling, mounting, and
positioning them. Since using a pair of longer focal length singlets would result in unacceptable performance, the next step
might be to use a pair of the slightly longer focal length, larger achromats, such as the 01 LAO 001. The performance data,
given in the section on Spot Size, shows that this combination does
provide the required 8-mm spot diameter. Because fairly small spot sizes are being considered here, it is important to make sure that the system is not being asked to work below the diffraction limit: |
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Since this is half the spot size caused by aberrations, it can be safely assumed that diffraction will not play a significant
role here. An entirely different approach to a fiber-coupling task such as this would be to use a pair of spherical ball lenses (e.g., the 06 LMS series), or a gradient-index lens (e.g., the 06 LGT series). Diffraction-Limited Performance Determine at what f-number a plano-convex lens being used at an infinite conjugate ratio with 0.5-μm wavelength light becomes diffraction limited (i.e., the effects of diffraction exceed those caused by aberration). To solve this problem, set the equations for diffraction-limited spot size and third-order spherical aberration equal to each other. The result depends upon focal length, since aberrations scale with focal length, while diffraction is solely dependent upon f-number. Substituting some common focal lengths into this formula, we get f/8.6 at f = 100 mm, f/7.2 at f = 50 mm, and f/4.8 at f = 10 mm. |
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When working with these focal lengths (and under the conditions previously stated), we can assume essentially
diffraction-limited performance above these f-numbers. Keep in mind, however, that this treatment does not take into
account manufacturing tolerances or chromatic aberration, which will be present in polychromatic applications. | |
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| Optics Guide Copyright 2002 Melles Griot Inc. |