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Manufacturers often quote the resolution of a stereomicroscope in terms of the greatest number of 'line pairs per millimeter' (parallel lines) that can be discerned, or sometimes the 'numerical aperture' (NA) of the microscope objective is given. But often manufacturers of generic stereomicroscopes don't include information about resolution in their specification sheets, leaving assessment to the purchaser.

Resolution can be measured by observing a prepared test target with lines etched at varying microscopic spacings, but such targets cost hundreds of dollars. Or one can just look through a microscope at a few selected objects and form a subjective opinion of sharpness. But are there inexpensive ways to get a more objective measure of resolution?

Here are three methods I used to estimate the resolution of a stereomicroscope (these could work with any optical system, not just stereomicroscopes):

(Click on any of them for 'how to' information about that method)

Using an Airy pattern to estimate microscope resolution | Using an FFT (fast fourier transform) to estimate microscope resolution | Using an MTF (spatial frequency response) to estimate microscope resolution |

Not always visible. Works only at magnifications high enough to resolve features at the scale of an Airy pattern, eg., few microns. | Easy. Works with just an image from the microscope but gives only a rough estimate. | Best. Works with any optical system but requires building or finding a straight-edge target. |

As discussed below, all of these methods exploit the fact that even with perfect lenses, a microscope with a finite aperture cannot focus light perfectly because the aperture blocks higher spatial frequency components necessary to make sharp focus. The aperture causes the microscope's resolution, even with the best optics, to be diffration-limited.

If optics are good, the higher the numerical aperture (NA), the sharper the image. A modest stereomicroscope might have an NA of 0.05; Olympus says its SZX16 has an NA of 0.30 and is able to resolve up to 900 cycles (line pairs) per mm.

If the optics have imperfections (aberrations, etc), then a microscope's resolution may not reach the diffraction limit and thus numerical aperture alone might over-estimate a microscope's performance; measures based on MTF and FFT would provide more information because they reflect end-to-end system performance.

'Numerical aperture' (NA) (common in microscopy) and ƒ/stop (common in photography) both relate focal length to aperture; one is approximately the reciprical of twice the other, both are dimensionless ratios:

NA ≅ 1 / 2ƒ

ƒ ≅ 1 / 2NA

For perfect optics limited only by diffraction, a larger aperture (ie., larger NA, or smaller ƒ/stop value) yields a sharper image. However, larger-aperture optics are more difficult to produce without aberration errors, etc, so usually there is a trade-off; the sharpest image from a camera lens is often one or two ƒ/stops from its lowest (widest aperture) value.

A point source of light viewed with diffraction-limited optics appears as an Airy diffraction pattern (because higher frequencies necessary to make it sharper were blocked by the finite aperture). The apparent radial distance on the subject plane to the first minimum (first dark ring) of an Airy diffraction pattern is given by:

r_{Airy} [length] = 0.61 λ / NA

where λ is the light wavelength under monochromatic illumination (eg., green-yellow = 550 nm).

By the Rayleigh criterion, two point sources of light are defined as resolvable by a microscope if they are separated by more than the apparent radial distance on the subject plane to the first minimum (first dark ring) of their Airy diffraction patterns. (More on Rayleigh criterion.)

Resolution is also often expressed in terms of the minimum spacing between pairs of lines that can be resolved. Various measures are related as follows:

1 line pair (LP) = 2 line widths (LW) = 1 spatial wavelength (cycle)

Spatial frequency is the inverse of spatial wavelength. Thus a resolution of 10 μm would be equivalent to a spatial frequency of 100 cycles/mm.

Resolution can also be expressed in terms of angle of view.

Lens imperfections (eg., aberrations, dispersion) and diffraction cause light from the subject to be distorted and diffused, resulting in decreased contrast and resolution. A measure of how well contrast is carried from the subject plane to the image is the modulation transfer function (MTF), which measures the image contrast compared to subject contrast as a function of spatial frequency (here are other MTF references: Nikon MicroscopyU, and IDEX).

Diffraction-limited MTF

The MTF function for a uniformly lit circular aperture with perfect optics, limited only by diffraction, is:

MTF(ν) = 2/π (φ - cos(φ) sin(φ)),

where ν is spatial frequency (cycles/length) and φ = arccos(νλ / 2NA).

MTF is zero at the 'cut-off' frequency where φ is 0, ie., ν = 2NA / λ; at that point, no contrast is transmitted. MTF50 (the spatial frequency at which contrast is degraded by 50%) is obtained when φ = 1.155, therefore:

1.155 = arccos(ν_{MTF50} λ / 2NA)

cos(1.155) = ν_{MTF50} λ / 2NA

NA = ν_{MTF50} λ / 0.808

Similarly, MTF30:

NA = ν_{MTF30} λ / 1.17

By the Rayleigh Criterion, two point light sources in the subject plane are considered resolvable when their Airy disc centers are separated by at least the radius of an Airy disc. Taking the distance between two such points as one spatial wavelength, the corresponding spatial frequency, the 'Rayleigh Criterion frequency' ν_{Rayleigh}, is:

ν_{Rayleigh} = 1 / r_{Airy} = NA / 0.61 λ

We can find the MTF corresponding to the Rayleigh frequency, as follows:

φ_{Rayleigh} = arccos(ν_{Rayleigh} λ / 2NA) = arccos((NA / 0.61 λ)(λ / 2NA)) = arccos(1 / 1.22)

MTF(ν_{Rayleigh}) = 0.0894 ≅ 9%

We can define ν_{MTF09} (= ν_{Rayleigh}) as the spatial frequency corresponding to the Rayleigh criterion at which the MTF of a diffraction-controlled optical system is at 9%, meaning there is still some contrast left, but it's faint.

By the method above, NA at ν_{MTF09} is:

NA = ν_{MTF09} λ / 1.638

However, if the optics are not perfect, the MTF curve will not reach the diffraction limit, and the MTF09 figure will be at a spatial frequency lower than ν_{Rayleigh}.

A camera for photomicroscopy ought to be able to record all the details (spatial frequency information) transmitted by the microscope. By the Nyquist theorem, it takes at least two pixels to record a minimum-size detail (a single spatial cycle). Thus to record a subject plane feature of length r_{Airy} would require at least two pixels per r_{Airy}. But r_{Airy} is just the Rayleigh Criterion distance for resolution; there are details visible smaller than r_{Airy} (but not separately resolvable). Thus a more reasonable minimum multiple might be 3 or 4 pixels per r_{Airy}.

Given the field of view (FOV) of a microscope and camera (eg., by viewing a millimeter scale through the microscope+camera), then the camera sensor ought to have at least 3 or 4 * FieldOfView / r_{Airy} pixels wide or high, depending on which FOV dimension is measured.

Another way of saying this is: The image that falls on the camera's sensor must be magnified enough so that light from a subject plane object r_{Airy} long falls upon at least two pixels, or better, three or four pixels.

Low magnification configurations are typically most demanding; there is usually more information in the larger field of view.

Below, the image to the left has approximately five pixels per r_{Airy}, and the image to the right has half that, ie., approximately 2.5 pixels per r_{Airy}. Clearly there is benefit in having more than two pixels per r_{Airy}. Note that features smaller than r_{Airy} (eg., the dots in the matrix of dots in the bottom-left) are visible in both; the Rayleigh Criterion just says that if each of those dots were actually composed of multiple dots separated by less than r_{Airy}, we won't be able to see the separation.

Left: Approx five pixels per r_{Airy} . Right: Approx 2.5 pixels per r_{Airy}

See Review: MicroscopeNet V434B Stereomicroscope where I work out actual camera pixel-width requirements for a typical stereomicroscope arrangement.

Nikon's MicroscopyU and Olympus' Microscopy Reference Center have well-written information pages, interactive tutorials.

Imatest.com. Lots of information and background on MTF and sharpness.

QuickMTF.com. Resolution and MTF.

Clarkvision.com, by R.N. Clark. Good technical info and test results, albeit scattered.

Wikipedia Point spread function, good description of Airy pattern.

How imperfect optics (aberrations) affect the Airy diffraction pattern: Diffraction patterns and aberrations, at telescopeOptics.net