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Astigmatism occurs when light is obliquely incident on a spherical mirror

Consider a system in which light is reflected obliquely by a spherical mirror as shown below. In this case, the focal length in the sagittal direction (red line) is r/(2cosθ), and the focal length in the tangential direction (blue line) is (r cosθ)/2 (in the paraxial region), resulting in large astigmatism.

This shows the astigmatism that occurs when light is obliquely incident on a spherical mirror.

Below, we will consider the reasons for this.

  • Derivation of the focal length of the tangential component

First, as shown below, consider only the tangential component in a system where light is reflected obliquely by a mirror.

Astigmatism occurring when light is obliquely incident on a spherical mirror_101a.png

Here, dθ and dψ are small quantities.

The conditions shown in the figure can be expressed as follows:

Astigmatism caused by oblique incidence on a spherical mirror_102.png

From Law of Sines,

Therefore, it was shown that the focal length in the tangential direction is (r cos θ)/2.

  • Derivation of the focal length of the sagittal component

Next, as shown below, consider only the sagittal component in a system where light is reflected obliquely by a mirror.

This shows the sagittal component of astigmatism that occurs when light is obliquely incident on a spherical mirror.

Here, vector i is the direction vector of the incident light, vector j is the direction vector of the reflected light, and vector n is the normal vector of the reflecting surface.

Each direction vector is expressed as follows:

Substituting ① and ② into ③,

Moreover, the vectors between each point are expressed as follows:

Here, k is a positive real number,

If we substitute ③' and ⑥ into this and compare the z components, we get

Substituting ③' and ⑧ into ⑦, we get

Compare this with ⑥

Therefore, from ⑤,

Here, if 𝜑 ≅ 0, then cos⁡𝜑 ≅ 1, so

This is the result of calculating the focal length in the sagittal direction.

Therefore, it was shown that the focal length in sagittal direction is r/(2cosθ).

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