POSES - Series 2
“Numerical aperture (NA)” represents the angular value of entrance cone of incident light into the objective lens. It can also be interpreted as the maximum limit of the angle of an objective lens to accept the incident light. Similarly, the f/# (read as “f-number”) is defined as the ratio between the focal length and diameter of the objective lens.?
NA and f/# are calculated by the following formulas:
where: n = refractive index of the medium between objective lens and sample; d = diameter of the lens in objective lens; f = focal length of the objective lens; α = half angle of the maximum cone of incident light that can be accepted by the objective lens.
From both formulas above, the relation between NA and f/# can be derived:
A question arised then is: what is the importance of these two dimensionless values for objective lens? A short answer is NA or f/# will determine the performance of the objective lens in the microscope system.
NA and f/# define optical resolution
The NA or f/# of the objective lens will characterize the “optical resolution” of the microscope system. Optical resolution is defined as the smallest distance between two points on the sample that can be identified by the optical system. It was first introduced by Ernst Abbe, he was also the one introducing NA, in the following formula:
where: λ = the wavelength of the light source.
It was found later that this formula was not applicable in some applications. A more reliable formula known as the “Rayleigh criterion” was proposed then by John William Strut. It is based on resolving capability of the objective lens to resolve two Airy disks of two points on the sample. The “Airy disk” is a concentric pattern of bright and dark rings due to the diffraction occurred on light when it is passing through a circular aperture, in this case is the objective lens’ aperture.
where: θ is the angular resolution of the objective lens.
Using small angle approximation, the angular resolution θ is converted into lateral resolution ΔL:
Hence, the formula is reorganized to reflect that the resolution of the objective lens in the microscope system is governed by NA and f/#:
NA and f/# define optical throughput and image contrast
NA and f/# generally are physically associated to the diameter of aperture in the objective lens. It is very natural then that both are considered as the control parameters of the incident light amount into the objective lens, which in turn will define the optical throughput of the microscope.
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Furthermore, the amount of the incident light into the objective lens and the optical throughput of the microscope will determine the image contrast. The image contrast refers to how distinguish the difference between the brightest and darkest intensity on the image can be identify.
where: Imax = the value of brightest intensity on image; Imin = the value of darkest intensity on image.
NA and f/# define optical depth of field (DOF)
“Depth of field (DOF)” denotes the distance of the closest and farthest plane where the focus of the lens or imaging system can be maintained. DOF implies the resolving power of the lens or imaging system on axial direction. Similar as the resolution discussed earlier, which refers to lateral resolution, DOF obviously depends on the NA and f/#. The larger the NA, the smaller the f/#, the larger the aperture, the smaller the DOF, which can be estimated from the following formula:
To be noted, often DOF can ambiguously stand for “depth of focus”, which has different meaning from “depth of field”. Both terms principally refer to the axial resolution. One way to distinguish their difference is that depth of field is associated to axial resolution on object plane, while depth of focus is associated to one on image plane. Further discussion about depth of field and depth of focus are kept at advanced level and complex microscopy system, where more delicate details of optical design are taken into accounts.
Moving forward, to avoid confusion for basic understanding purpose of microscope system: DOF is assigned to depth of field and it means axial resolution, whereas resolution means lateral resolution.
NA and f/# define working distance
Working distance is the distance measured from the sample to the front edge of the objective lens when it is in focus, which in general is relatively short. Typically, with the increase of magnification, the NA is also increasing while f/# is decreasing as the focal length of lens is getting shorter, causing the working distance to be shorter.
ELWD = extra long working distance; SLWD = super long working distance
The demand to provide objective lens with long working distance is always limited by the challenge to provide high NA for better resolution. Therefore, there should be a trade-off between these two parameters in the design and manufacturing of objective lens. Moreover, with parfocal length as one of boundary conditions, the objective lens with desired magnification will be designed to have optimized NA and its associated working distance, while it can be interchangeable easily with other objective lens with the same parfocal length during the usage.
Summary
NA and f/#, which are inversely proportional, control the performance of objective lens which generally is represented as its resolution, optical throughput and image contrast, DOF, and working distance. Typically, increase of NA and equally decrease of f/# will lead to increase of resolution, increase of optical throughput and image contrast, decrease of DOF, and decrease of working distance.
Additionally, there is practical restriction for dry type objective lens to achieve high NA (typically the NA is less than 1.0) due to the limitation by aberration compensation. In order to go beyond NA of 1.0, the objective lens is designed to work on medium having refractive index larger than 1.0 (medium for dry type objective lens is air with refractive index of 1.0). Some applications use water as medium, of which refractive index is around 1.33; while others use special oil or glycerol as medium, of which refractive index is around 1.5.
Written by: Mr Harsono Cahyadi (Ph.D.) | Application & Research Scientist at Phaos Technology