Vergence Tracing in Thick Lens

Thick lenses can be analyzed in terms of vergence. The vergence can be traced by noting its change at each surface.

A thick lens can be characterized by its radii of curvature, its index of refraction, and its thickness. For glass with index nlens = 对于索引为n的透镜， 中文译文中的待填/计算数值依次对应：1：n。实际数值以上方原输入框为准。 the front surface power for lens radius R1 = m is P1 = m-1 透镜半径 R₁ 的前表面功率为 m 时，P₁ = m - 1 中文译文中的待填/计算数值依次对应：1：r1 2：p1。实际数值以上方原输入框为准。 and the back surface power for lens radius R2 = m is P2 = m-1 以及透镜半径R₂ = m时的后表面功率P₂ = m - 1 中文译文中的待填/计算数值依次对应：1：r2 2：p2。实际数值以上方原输入框为准。 厚透镜可以由其曲率半径、折射率和厚度来表征。

Lens thickness d = cm = m

(Note that for a double convex lens, the front surface radius R₁ is positive and the back surface radius R₂ is negative according to the Cartesian sign convention.)

The vergences can be calculated from the lens parameters and the object distance.The change in vergence when the light encounters a refracting surface is equal to the power of the surface:

Using the Cartesian sign convention, the object distance is typically a negative number since it points opposite to the direction of light travel.

Object distance o = m.

The vergences can then be calculated.

V₁ = n₀/o = m^-1

V₂ = V₁ + P₁ = n/i₁ =m^-1

V₃ = n/(i₁ - d) = m^-1

V₄ = V₃ + P₂ = n₀/i =m^-1

The value i₁ calculated as an intermediate value above would be the image distance inside the glass with just the front surface power acting. This calculation presumes that this image distance is greater than the thickness of the lens. It is also presumed that the medium is air so that n₀ = 1.

The exit vergence from the final surface determines the image distance with respect to that surface.

Image distance i = n₀/V₄ = m.

This can be compared with the image distance for a thin lens with the same surface powers:

Image distance i_thin = m.

The power of a thin lens is just the sum of the surface powers:

P_thin = P₁ + P₂ = m^-1.

The equivalent power for the thick lens can be calculated from Gullstrand's equation:

P_thick = P₁ + P₂ - P₁P₂d/n = m^-1.

Note that the calculation does not take into account the change in lens thickness with the angle of the incoming ray. It is typical to do the calculation only for the paraxial rays where the departure from full thickness is negligible.