Principles of
solid immersion lens (SIL)
A major motivation for microscope observation is to observe with high resolution. The resolution limit (Abbe's resolution limit) is expressed by the following formula:
・What is a solid immersion lens?
A major motivation for microscope observation is to observe with high resolution. The resolution limit (Abbe's resolution limit) is expressed by the following formula:
Here, λ is the wavelength and NA is the numerical aperture.
Therefore, in order to raise the resolution limit and observe finer objects, it is necessary to increase the numerical aperture. The numerical aperture is expressed by the following formula:
Here, φ is the collection angle of the objective lens, and n is the refractive index of the medium. As can be seen from this, the maximum numerical aperture in air is 1, and the only way to increase the numerical aperture beyond this is to increase the refractive index of the medium. A liquid immersion lens uses a liquid as the medium, while a solid immersion lens uses a solid.
It is known that there are two types of solid immersion lenses when the entrance surface is spherical:
(i) Hemispherical
When the focal point coincides with the spherical center of the incident surface, this focal point is stigmatic. The numerical aperture is n times greater than without the solid immersion lens.
(ii) Hyperhemispherical type (Weierstrass spherical type)
When the focusing position is located r/n below the spherical center of the incident surface, this focusing point will be free of aberration. At this time, the NA with the solid immersion lens is n^2 times that without the solid immersion lens, so the magnification is also n^2 times.
It is obvious that the focal point in (i) is aberration-free, but it is difficult to intuitively understand why the focal point in (ii) is aberration-free, so we will derive it in the next chapter.
・Principle of Weierstrass spherical solid immersion lens
A diagram of a super-hemispherical solid immersion lens and the light rays passing through it is shown below.
Here, r, t, and n are constants that do not depend on φ, and the other parameters are expressed as functions of φ.
The coordinates of point O that is imaged to a single point are defined as follows:
The coordinates of point O that is imaged to a single point are defined as follows:
(i) When nr = t,
From ⑤, ⑥, ⑦, and ⑧,
From ⑧a, it can be seen that since d is a constant independent of φ, the image formed at point P is aberration-free.
Furthermore, if we rewrite ⑥⑦a,
Therefore, if the maximum values of φ, θ are φ MAX, θ MAX, respectively,
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Objective observation numerical aperture:
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Numerical aperture for solid immersion lens observation:
Therefore, the numerical aperture for solid immersion lens observation is n^2 times that of objective lens observation, and the magnification is also n^2 times.