Electron microscope lens defects

We are know about the next lens defects: spherical aberration, chromatic aberration, astigmatism.

Spherical aberration is an inability of a lens to focus all incident beams from a point source to a point. The outer zones of the lens have a greater strength, and light rays or electrons originating from a point are not imaged at a point. The one point the envelope of the imaged rays has a minimum diameter known as the circle or disc of least confusion. This limiting disc has a diameter d_s given by: d_s =1/2*C_s*α³ =>C_s- spherical aberration coefficient, α - semi–angular aperture of the lens.

Chromatic aberration, the other major distortion, is caused by the faster (shorter wavelength) electrons focusing at a different position from the slower ones, a problem that only a perfect electron beam could solve completely. The chromatic aberration coefficient C_s is the parameter which expresses this quality , and the resolution d_c as limited by chromatic effects is given by

Where α is the objective semi-angular aperture.

Astigmatism is a defect of magnetic field asymmetry resulting in differing lens strengths in two directions at right angles. The amount of the astigmatism of the lens is defined as the distance between the focal lines. The astigmatism of an objective lens in a good microscope will be typically 1 μm or less.

Characteristics parameters of an electron microscope lens

Electromagnetic lens contain soft-iron pole-pieces and copper coils. The soft-iron pole-pieces sit in the hole down the middle of the lens are surrounded by copper coils through which the current runs to magnetize the pole pieces. As the result creates a electromagnetic field (or electrostatic), depending on current and number of copper coils: B=mNI/L

The resolution of the electron lens is the minimum resolvable distance in the object. The resolution of electromagnetic lens is customarily defined in terms of the Rayleigh criterion. The finite size of the lens results in diffraction of the rays at the outermost collection angle of the lens, usually defined by a limiting aperture. This diffraction results in a point being imaged as a disk (called the Airy disk). The distance apart of the two incoherent point sources is defined as the theoretical resolution of the lens r_th and is given by the radius of the Airy disk: r_th =0.61*(λ/α).

The focal length of a lens is a measure of its strength, and it is defined as the distance between the lens and the beam cross-over (which is also known as the focal point) when all the beams coming into the lens are parallel to one another. The higher the field region of a condenser-objective lens the shorter its focal length.

The Depth of field, D_ob is a measure of how much of the object that we are looking at remains in focus at the same time; the term depth of focus refers to the distance over which the image can move relative to the object and still remain in focus. D_ob=d_ob/a_ob . When d=1nm, α=5*10^-3radians, =>D_ob=200nm

The Depth of focus, D_im is the extent of the region around the image plane in which the image will appear to be sharp. This depends on magnification, M_T. D_im=(d_ob/a_ob)M_T²

Both depth of field and depth of focus are strongly dependent on changes in aperture (hence the semi angle α) and working distance (d_ob).