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In some cases, the second-order approximation is also called "paraxial". The approximations above for sine and tangent do not change for the "second-order" paraxial approximation (the second term in their
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The second-order approximation is accurate within 0.5% for angles under about 10°, but its inaccuracy grows significantly for larger angles.
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of the system, and lies close to the axis throughout the system. Generally, this allows three important approximations (for
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Use of the small angle approximations replaces dimensionless trigonometric functions with angles in radians. When doing
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on optics equations it is important to remember that radians are dimensionless and therefore can be ignored.
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The error associated with the paraxial approximation. In this plot the cosine is approximated by
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expansion is zero), while for cosine the second order approximation is
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For larger angles it is often necessary to distinguish between
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401:Paraxial Approximation and the Mirror
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367:"Paraxial approximation error plot"
405:The Wolfram Demonstrations Project
306:. SPIE Field Guides. Vol. 1.
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304:Field Guide to Geometrical Optics
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302:Greivenkamp, John E. (2004).
182:Ray transfer matrix analysis
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336:"Paraxial Approximation"
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48:paraxial approximation
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276:dimensional analysis
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421:Geometrical optics
403:by David Schurig,
332:Weisstein, Eric W.
310:. pp. 19–20.
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178:first-order
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351:15 January
282:References
382:26 August
224:θ
218:−
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312:ISBN
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176:and
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