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Astigmatism caused by oblique incidence on a spherical mirror and its correction

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 spherical mirror.

Tangential component of a ray of light incident obliquely on a spherical mirror

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,

BP length calculation

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.

Sagittal component of a ray of light incident obliquely 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.

The angle dφ is assumed to be very small.

Each direction vector is expressed as follows:

Direction vectors of incident and reflected light, and normal vector of the reflecting surface

Substituting ① and ② into ③,

Astigmatism occurring when light is incident obliquely onto a spherical mirror_213.png

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

Astigmatism occurring when light is incident obliquely onto a spherical mirror_214.png

Here, k is a positive real number,

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

Astigmatism occurring when light is incident obliquely onto a spherical mirror_216.png

Substituting ③' and ⑧ into ⑦, we get

Astigmatism occurring when light is incident obliquely onto a spherical mirror_215.png

Compare this with ⑥

Astigmatism occurring when light is incident obliquely onto a spherical mirror_217.png

Therefore, from ⑤,

Astigmatism occurring when light is incident obliquely onto a spherical mirror_218.png

Here, since d𝜑 is an infinitesimal quantity, cos⁡ d𝜑 ≅ 1,

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).

  • Toroidal surface shape for astigmatism correction

So far, we have treated the reflecting surface as a spherical surface, but here we consider a toroidal surface, which has a shape with different radii of curvature in the tangential and sagittal directions.

The toroidal surface can be illustrated as follows:

Diagram of the shape of a toroidal mirror

In this case, the shape of the toroidal surface can be expressed by the following formula [1] [2]:

Formula for the shape of a toroidal mirror

where:

Tangential radius of curvature: R t = r/cosθ

Sagittal radius of curvature: R s = r cosθ

Then, the focal length in both the tangential and sagittal directions is r/2, which is the same, and astigmatism can be eliminated.

However, the aberration that occurs when light is incident obliquely on a spherical mirror is not just astigmatism; coma aberration also becomes a problem. One way to correct this is to use multiple mirrors to cancel out the coma aberration that occurs in each mirror. Czerny-Turner spectrometers are designed based on this concept.

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