Paraxial Approximation¶

In analyzing lens systems, it can be helpful to make the approximation, referred to as the paraxial appoximation [3], that the distance r and angle $\theta$ of rays from the optic axis is small. This can allow approximations such as $\sin \theta \approx \theta$ and $\cos \theta \approx 1$ . In a cylindrically symmetric lens, we may also assume that the radial force is a linear function of r by ignoring $O(r^3)$ terms in the series expansion $E_r(x,r) = \frac{1}{2} r \phi(x,0)'' + O(r^3)$ [1]. Transfer Matrix methods can also be applied. The term occurs in other areas as well, notably light optics [4] [5].

SIMION-specific: SIMION does not impose the paraxial approximation because it can just as well, in the general case, perform a complete calculation of the ray (direct ray tracing), but the paraxial approximation is still a useful concept for describing and reasoning about systems, and it can be observed. See also Lens and SIMION Example: lens_properties.

As an example, load the SIMION Example: einzel (3D cylindrical Einzel lens) and execute these lines in the command bar (using the SIMION 8.1 simion.pas API):

pa = simion.wb.instances[1].pa
vpp = (pa:field_vc(44,0,0) - pa:field_vc(42,0,0))/2
for y=0,18 do local ex,ey,ez = pa:field_vc(43,y,0); print(y,ex,ey,-vpp*y/2) end

The output is

0.00044623609977634  -0                -0
0.00044428927495233  -0.17781293264966 -0.17765444177787
0.00043840684034535  -0.35588612396623 -0.35530888355574
0.00042866422852228  -0.53440720053051 -0.53296332533361
0.00041519012360425  -0.71337103479918 -0.71061776711149
0.00039816206681564  -0.89249053069886 -0.88827220888936
0.0003778052310679   -1.0710844420651  -1.0659266506672
0.00035438893978323  -1.2479573211516  -1.2435810924451
0.00032822043240799  -1.4212758476776  -1.421235534223
0.00029963579692094  -1.5884533017798  -1.5988899760008
0.00026899862027108  -1.7460625920167  -1.7765444177787
0.0002366972330492   -1.8898080997316  -1.9541988595566
0.00020313782007975  -2.0145954910926  -2.1318533013345
0.00016874647809573  -2.1147421766658  -2.3095077431123
0.00013397228273959  -2.1843626535966  -2.4871621848902
9.926759618395e-005  -2.217935562564   -2.6648166266681
6.5083447026382e-005 -2.2110096583242  -2.8424710684459
3.1859044696603e-005 -2.1609408204639  -3.0201255102238
-0                   -0                -3.1977799520017

Notice that at low values of radius (y, also known as r), the second column is fairly constant and the last two columns (for measured and estimated radial field component) match closely.

Related topics: Picht equation [2] [1] and paraxial ray equation.


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Paraxial Approximation¶

See Also¶