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):

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.

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.

Numerical aperture isn't everything, of course, but if the optics are perfect, diffraction limits resolution. If the optics have imperfects (aberrations, etc), then the microscope will not be diffraction-limited and thus NA alone will not say as much (or is misleading); MTF and FFT will provide more information.

If optics are good, the higher the NA, the sharper the image. Stereomicroscopes typically have NA ranging from 0.05 to 0.30 (eg., Olympus SZX16), able to resolve up to 900 cycles (line pairs) per mm.

General background

'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 diffraction-limited optics, 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 compromise; the sharpest image from a camera lens is often one or two ƒ/stops from its lowest (widest aperture) value. Aperture diffraction is why pinhole cameras cannot produce sharp images.

'Resolution' relating to microscopy is a measure of how well objects in the subject plane can be seen. Resolution is somewhat subjective and depends on eye quality and perception, which vary between people. One 'objective' measure of resolution is established by the Rayleigh criterion which defines the minimum resolvable distance, r_Airy, as the radius to the first minimum (dark ring) of an Airy diffraction pattern when a point source of light is viewed through the microscope.

r_Airy [length/cycle] = 0.61 λ / NA

where λ is the wavelength of the light (eg., green-yellow, at 550 nm).

Resolution has units of distance in the subject plane per cycle. Resolution is also often expressed in terms of the minimum spacing between pairs of lines that can be resolved. These various measures are related as follows:

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

Spatial frequency, ν [cycles/length], = 1 / spatial wavelength. Thus a resolution of 10 μm would be equivalent to a spatial frequency of 100 cycles/mm.

Lens imperfections (eg., aberrations) 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's another MTF reference: Nikon MicroscopyU site).

Diffraction-limited MTF

The MTF function for a uniformly lit circular aperture with perfect optics (diffraction-limited) 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 / 0.404

Similarly, MTF30:

NA = ν_MTF30 λ / 0.585

According to the Rayleigh Criterion, two points in the subject plane are considered resolvable when their Airy disc centers are separated by at least the radius of an Airy disc. Thus the Rayleigh 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%

Thus the Rayleigh criteria corresponds to an MTF of 9%, meaning there is still some contrast left, but it's faint.

The NA given ν_MTF09 (the frequency at which the MTF is 9%, the Rayleigh frequency) is as follows:

NA = ν_MTF09 λ / 0.819

How many pixels does a microscope camera need?

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 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 for a table presenting the camera pixel-width requirements for a typical stereomicroscope arrangement.

Helpful links

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.