Paraxial approximation

The paraxial approximation assumes that every ray stays close to the optical axis and makes a small angle $\theta$ with it, so that the trigonometric functions in the laws of refraction and propagation can be replaced by the first terms of their series:

$$\sin\theta \approx \tan\theta \approx \theta, \qquad \cos\theta \approx 1.$$

With these replacements, Snell's law becomes $n_1\theta_1 = n_2\theta_2$, every spherical surface focuses perfectly, and ray height and angle change linearly from element to element. All of first-order optics rests on this: focal lengths, the thin-lens and mirror equations, principal planes, magnification and the ray-transfer (ABCD) matrices. The result is often called Gaussian optics.

Size of the error

The relative error of $\theta$ as an estimate of $\sin\theta$ grows as $\theta^2/6$:

At 30° incidence on glass of index 1.5, the exact refraction angle is 19.47°, while the paraxial formula gives 20.00°. For design work the approximation is usually taken as adequate below about 10°, a cone of roughly f/3 and slower, and as a first estimate to be refined by exact ray tracing at larger angles.

Relation to aberrations

The next term of the series, $\sin\theta \approx \theta - \theta^3/6$, is where third-order (Seidel) aberrations come from: spherical aberration, coma, astigmatism, field curvature and distortion are the departures from perfect imaging produced by the cubic terms. Paraxial optics defines the ideal image, its position and size, against which these departures are measured. A worked case: a concave spherical mirror of radius 200 mm has a paraxial focus at 100.00 mm. A ray parallel to the axis at height 25 mm (an f/2 aperture) crosses the axis at 99.21 mm, 0.79 mm closer to the mirror; the third-order estimate $h^2/(4R)$ gives 0.78 mm. At 5 mm height the difference is only 0.03 mm. This is the spherical aberration that the paraxial treatment omits. A system corrected for it and satisfying the Abbe sine condition extends good imaging to large apertures, where the relevant quantity is $\sin\theta$, as in the numerical aperture, and $\theta$ itself no longer suffices.

Paraxial wave optics

The same approximation applied to waves gives the paraxial wave equation. Writing a beam as $E = A(x,y,z)\,e^{ikz}$ and assuming the envelope $A$ changes slowly over a wavelength along $z$ (the slowly varying envelope approximation), the second derivative $\partial^2 A/\partial z^2$ is neglected:

$$\nabla_\perp^2 A + 2ik\,\frac{\partial A}{\partial z} = 0.$$

Its solutions include the Gaussian beam and the Hermite-Gaussian and Laguerre-Gaussian modes of laser resonators, and its integral form is the Fresnel diffraction integral, in which the spherical wavelets of Huygens' construction are replaced by parabolic ones. The condition here is that the divergence half-angle $\lambda/(\pi w_0)$ stay below about 0.2 rad, which requires a waist radius larger than about 1.6 wavelengths. Tightly focused beams from high-NA objectives fall outside it and need vector diffraction theory.

Where it matters in practice

Paraxial formulas give the starting point for nearly every optical layout: lens positions from the imaging equation, beam sizes from the Gaussian-beam formulas, resonator stability from the $g_1 g_2$ criterion. They are exact in the limit of small apertures and fields, so they correctly predict image location and magnification even in well-corrected wide-aperture systems; what they do not predict is image quality. Laser diodes are a common case of misuse: a fast-axis divergence of 25–40° FWHM lies outside the paraxial range, and Gaussian-beam formulas then overstate the angle.

Common questions

What does "paraxial" mean?

Literally "near the axis". A paraxial ray is one whose height above the axis and angle to it are both small enough that sines and tangents can be replaced by the angles themselves.

Up to what angle is the paraxial approximation valid?

It depends on the accuracy required. The error in $\sin\theta$ is 0.5% at 10° and 2% at 20°; image positions computed paraxially remain the reference at any aperture, but blur from aberrations grows rapidly beyond about 10° half-angle in uncorrected optics.

Is the paraxial approximation the same as the Fraunhofer approximation?

No. The paraxial (Fresnel) approximation keeps the quadratic phase across an aperture; the Fraunhofer approximation further drops it, which is valid only far from the aperture, as described under Fresnel number.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017); M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999); W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); A. E. Siegman, Lasers (University Science Books, 1986).