Paraxial Formulae for Lenses in Air
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The following formulae are based on the behavior of paraxial
rays, which are always very close and nearly parallel to the
optical axis. In this region, lens surfaces are always very n
early normal to the optical axis, and hence all angles of
incidence and refraction are small. As a result, the sines
of the angles of incidence and refraction are small (as used
in Snell's law) and can be approximated by the angles themselves
(measured in radians). The paraxial formulae do not include effects of spherical aberration experienced by a marginal ray - a ray passing through the lens near its edge or margin. All effective focal length values ( f ) tabulated in this catalog are paraxial values which correspond to the paraxial formulae. The following paraxial formulae are valid for both thick and thin lenses unless otherwise noted. The refractive index of the lens glass, n, is the ratio of the speed of light in vacuum to the speed of light in the lens glass. All other variables can be found in the subsection Definition of Terms. NOTE: For Quick Approximations, much time and effort can be saved by ignoring the differences among f, fb, and ff in these formulae (assume f = f b = ff) by thinking of s as the lens-to-object distance, by thinking of s" as the lens-to-image distance, and by thinking of the sum of conjugate distances s + s" as being the object-to-image distance. This is known as the thin-lens approximation. |
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Focal Length |
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where n is the refractive index,
tc is the center thickness, and
the sign convention previously given for the radii
r1 and r
2 applies. For thin lenses,
tc = 0 (by definition),
and for plano lenses either r1
or r2 is infinite. In either case
the second term of the above equation vanishes, and we are left with the
familiar Lens Maker's formula |
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Surface Sagitta and Radius of Curvature (refer to the figure below) |
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An often useful approximation is to neglect s/2. |
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| Surface sagitta and radius of curvature |
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Symmetric Lens Radii (r2 = -r1) |
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With center thickness contrained, | |
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where, in the first form, the + sign is chosen for the square
root if f is positive, but the
- sign must be used if f is negative.
In the second form, the + sign must be used regardless of the sign
of f. With edge thickness constrained,
the equation for r1 becomes
transcendental: |
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where f is the lens diameter. This equation can be solved
by numerical methods. |
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Plano Lens Radius Since r2 is infinite, |
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Principal-Point Locations (signed distances from vertices) |
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where the above sign convention applies. For symmetric lenses (r2 = r1) |
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If either r1 or
r2 is infinite, l'Hôpital's rule
from calculus must be used. Thus, for plano-convex lenses in the correct
orientation, |
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For flat plates, by letting r1 approach
infinity in a symmetric lens, we obtain A1H = A2
H" = tc/2n.
These results are useful in connection with the following paraxial lens combination
formulae. |
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Hiatus or Interstitium (principal-point separation) |
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which, in the thin-lens approximation (exact for plano lenses), becomes |
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Solid Angle The solid angle subtended by a lens, for an observer situated at an on-axis image point, is |
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where this result is in steradians, and where |
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is the apparent angular radius of the lens clear aperture. For an observer at an
on-axis object point, use s instead of s.
To convert from steradians to the more intuitive sphere units, simply divide
W by 4p If the Abbé
sine condition is known to apply, q may be calculated
using the arc sine function instead of the arc tangent. |
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Back Focal Length |
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where the sign convention presented above applies to A2H" and to the
radii. If r2 is infinite, l'Hôpital's rule
from calculus must be used, whereby |
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Front Focal Length |
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where the sign convention presented above applies to A1H and to the radii.
If r1 is infinite, l'H&ocric;pital's rule from
calculus must be used, whereby |
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Edge-to-Focus Distances |
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for positive lenses, |
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and, |
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where s1 and s
2 are the sagittas of the first and second surfaces. Bevel is neglected.
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Magnification or Conjugate Ratio |
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| Optics Guide Copyright 2002 Melles Griot Inc. |