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 focusing position coincides with the spherical center of the incident surface, the light rays do not bend before or after entering the medium, so this focusing point is aberration-free. In addition, since the focusing angle does not change before or after entering the medium, the numerical aperture is n times greater than without a solid immersion lens.
(ii) Hyperhemispherical type (Weierstrass spherical type)
It has long been known that when the focusing position is located r/n lower than the center of the spherical plane of incidence, this focusing point is aberration-free [1] . In this case, the NA with a 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 the Weierstrath spherical immersion lens [1]
The super-hemispherical solid immersion lens and the light rays passing through it are illustrated as follows.
Here, r, t, and n are constants independent of φ, while the other parameters are expressed as functions of φ.
From the diagram, the following can be said:
From ①, ②, ③, and ④,
(i) When nr = t,
From ⑤, ⑥, ⑦, ⑧,
From ⑧a, since d is a constant independent of φ, it has been shown that the image formed on point P is aberration-free.
Furthermore, if we rewrite ⑥⑦a,
Therefore, if the maximum values of φ, θ are φMAX, θMAX, respectively,
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Numerical aperture for objective lens observation:
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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.
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References
[1] Weierstrass, K., “Zur Theorie der Abbildung durch Linsen und Spiegel,” Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 1890.
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Update History
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2026-03 Newly released