Math 2.0

Optics, electromagnetic theory Definition

Paraxial approximation

Builds on Snell's lawTaylor series

Definition

Let an optical system have an optical axis. Describe a ray at a plane perpendicular to the axis by its height yy above the axis and its angle θ\theta with the axis. The paraxial approximation keeps only the terms of first order in yy and θ\theta in the laws of ray propagation. In particular it uses sin⁡θ≈θ,tan⁡θ≈θ,cos⁡θ≈1.\sin\theta \approx \theta, \qquad \tan\theta \approx \theta, \qquad \cos\theta \approx 1. A ray is called paraxial when these replacements are accurate. This means that ∣θ∣≪1\abs{\theta} \ll 1 and that yy is small compared with the radii of curvature and the apertures.

Remarks

The replacements are the first terms of the Taylor series: sin⁡θ=θ−θ3/6+⋯\sin\theta = \theta - \theta^3/6 + \cdots, tan⁡θ=θ+θ3/3+⋯\tan\theta = \theta + \theta^3/3 + \cdots, cos⁡θ=1−θ2/2+⋯\cos\theta = 1 - \theta^2/2 + \cdots. Under the approximation, Snell's law becomes n1θ1=n2θ2n_1 \theta_1 = n_2 \theta_2. The third-order terms give the Seidel aberrations.

Example: θ=0.1\theta = 0.1 rad. Check: sin⁡0.1≈0.09983\sin 0.1 \approx 0.09983, so the relative error of sin⁡θ≈θ\sin\theta \approx \theta is about 0.17%0.17\%.

Non-example: θ=1\theta = 1 rad, about 57∘57^\circ. Here sin⁡1≈0.841\sin 1 \approx 0.841, so replacing it by 11 gives an error near 19%19\%, and tan⁡1≈1.557\tan 1 \approx 1.557; the ray is not paraxial.

Reading path

To understand this page from the ground up, read these 24 pages in order. The foundations come first.

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Calculus of variations, concept, no evidence levelCalculus of variationsconceptEuler–Lagrange equation, concept, no evidence levelEuler–Lagrange equati…conceptGeodesic, concept, no evidence levelGeodesicconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionNormal vector, concept, no evidence levelNormal vectorconceptRefractive index, concept, no evidence levelRefractive indexconceptSequence, definition, no evidence levelSequencedefinitionFermat's principle, statement, no evidence levelFermat's principlestatementLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionSeries, definition, no evidence levelSeriesdefinitionSnell's law, statement, no evidence levelSnell's lawstatementContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionPower series, definition, no evidence levelPower seriesdefinitionDerivative function, definition, no evidence levelDerivative functiondefinitionHigher derivative, definition, no evidence levelHigher derivativedefinitionTaylor polynomial, definition, no evidence levelTaylor polynomialdefinitionContinuously differentiable function, definition, no evidence levelContinuously differen…definitionSmooth function, definition, no evidence levelSmooth functiondefinitionTaylor series, definition, no evidence levelTaylor seriesdefinitionParaxial approximation, definition, no evidence levelParaxial approximationdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Calculus of variations concept
  2. Euler–Lagrange equation concept
  3. Geodesic concept
  4. Limit point of a set definition
  5. Normal vector concept
  6. Refractive index concept
  7. Sequence definition
  8. Fermat's principle statement
  9. Limit of a function definition
  10. Real sequence definition
  11. Continuity at a point definition
  12. Derivative definition
  13. Series definition
  14. Snell's law statement
  15. Continuous function definition
  16. Differentiable function definition
  17. Power series definition
  18. Derivative function definition
  19. Higher derivative definition
  20. Taylor polynomial definition
  21. Continuously differentiable function definition
  22. Smooth function definition
  23. Taylor series definition
  24. Paraxial approximation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Paraxial_approximation
About this page
TagBQH7VFR6XH
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Kinddefinition
WritingW0 stub
MSC78A05
DefinesParaxial approximation
UsesSnell's law, Taylor series
Changed3 November 2026

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