Now one must ask the question if this level of magnification, 30,000:1, is simply beyond the useful range, meaning it is empty magnification. How do we determine a range of useful magnification for digital microscopy when an image is observed on a monitor? First, it is important to understand the resolution or resolving power of the microscope system and the viewing distance from the monitor.

### Microscope system resolution

The system resolution for a digital microscope (or stereo microscope with a digital camera) is influenced by three main factors:

(1) Diffraction-limited light microscope resolution, using Rayleigh’s criterion and the numerical aperture [9]:

where NA is the numerical aperture and λ is the wavelength of light in nm.

(2) Image sensor (camera sensor) resolution [9]:

where *M*
_{
TOT PROJ
} is the magnification from the sample to the sensor (Equation 3), the “sensor bin mode” refers to the binning mode which is 1 for full frame (1 × 1) and 2 for 2 × 2 pixel binning, (see Figure 2a), and “pixel size” refers to the sensor pixel size in μm.

Figure 2 (a) Examples of pixel binning modes for image sensors: no binning (full frame, 1 × 1), double binning (2 × 2), triple binning (3 × 3), and quadruple binning (4 × 4). (b) Image sensor detection of black/white line pairs, used to measure the resolution limit of a microscope, requires a minimum of two pixels (red squares) per line pair (Nyquist rate). However, better image results are obtained if three or more pixels per line pair are used.

(3) Display (monitor) resolution [9]:

where *M*
_{
DIS
} is the total lateral magnification (Equation 4) and the monitor pixel size is in mm.

The basis for the camera sensor and display monitor resolution limit is the Nyquist sampling theorem for digital signal processing (see Figure 2b) [14, 15]. This theorem assumes that at least two pixels are needed to resolve one line pair. One can simply imagine a microscope image of a sample showing line pairs projected onto the camera sensor and then displayed on the monitor. If a single line in the image corresponds to a single line of pixels, as shown in Figure 2b , say for the monitor, then to determine the minimum line pair spacing on the sample resolvable by the monitor, one can just divide the size of 2 pixels by the total lateral display magnification (*M*
_{
DIS
}). If the same exercise is done now for the camera sensor, that is, a single line in the image corresponds to a single line of pixels, then dividing by the total projection magnification from sample to sensor (*M*
_{
TOT PROJ
}) determines the minimum line pair spacing resolvable by the camera. To write these calculations out for more clarity:

The reciprocal of the minimum resolvable line pair spacing gives the resolution limits shown above in Equations 7 and 8. Simply add a conversion (µm to mm) and pixel binning factor for the camera sensor to arrive at the final form of Equation 7.

For this article, as stated above, the scenario of a 1-to-1 correspondence is assumed between the pixels of the sensor and monitor. For this specific case, using Equation 4b and converting the monitor pixel size units from mm to μm, it becomes clear that the resolution limit of the sensor and monitor are identical. Example calculations demonstrating this will be given in a later section of this article.

The resolution limit of the digital microscope system resolution is determined by the *smallest* of the three resolution values above. The diffraction limit of the light microscope (Equation 6) still governs the ultimate level of detail that can be observed and recorded. This last point is important: the best resolution of a microscope generally is measured from a recorded image rather than a viewed image; methods for this fall outside the topics of this article [16].

### Useful viewing distance

The viewing distance is the distance between the observer’s eyes and the displayed image. The range for a useful viewing distance is affected by the system resolution of the microscope and the visual angle of the observer [17, 18]. The minimal angle of resolution depends on the intensity of light emitted or reflected from the observed object and the contrast between its specific features. The angle ranges, on average, from 2.3 to 4.6 minutes of arc (low to high light intensity and contrast) for human eyes [9, 19–21]. This angular range indicates the optimal performance, in terms of visual acuity (resolution) and contrast sensitivity, of the human eye averaged over a large population varying in age from young to old. Thus, on average, an eye is capable of distinguishing details on a monitor which have a separation distance corresponding to an angular difference of 2.3 to 4.6 minutes of arc (0.038 to 0.077 degrees; 0.669 to 1.338 × 10^{-3} radians) for a specific viewing distance. To understand how the range for a useful viewing distance is determined, imagine a person observing an image displayed on a monitor which shows the smallest line pair spacing resolvable by the digital microscope system. It follows that the actual line pair spacing on the sample would then be:

where *M*
_{
DIS
} is the total magnification (Equation 4). To determine the minimal angle of resolution for the observer’s eye necessary in order to see two separate lines in the pair, the minimal line spacing observed on the monitor must be divided by the viewing distance:

The minimal resolvable line pair spacing on the sample is the inverse of the microscope system resolving power or resolution, thus:

As noted above, the minimum angle of resolution for the eye falls between 6.669 × 10^{-4} and 1.338 × 10^{-3} radians, so rearranging the equation above for viewing distance (in mm) and setting it equal to the lower and upper values of the angle of resolution, the *useful viewing distance range* can be expressed as (converted from units of mm to meters):

Again, *M*
_{
DIS
} is the total lateral magnification (Equation 4), and the *system resolution* refers to the light microscope system resolution limit as discussed above (Equations 6–8). For the following calculations, it is assumed that the viewing distance is always within the useful range.

### Range of useful magnification

To understand how to determine the range of useful magnification for digital microscopy—the lowest and highest magnification values for an image displayed on a monitor where the system resolution limit is clearly observed for optimal and non-optimal illumination—it is first necessary to mention briefly the “perceived” magnification from visual observation of an image or object. Everyday experience shows that the perceived size of an object depends on the distance from which it is observed. Using geometrical optics and the relationship between angular and lateral magnification, the following can be derived:

where *M*
_{
DIS
} is the total magnification (Equation 1) and 250 refers to the standard reference for the viewing distance in mm, which is based on the average near point for the human eye, that is, the closest point on which the eye can focus [22]. The Appendix at the end of this article provides a more detailed derivation of Equation 10.

Finally, the range of useful magnification can be defined by combining Equations 9 and 10. From Equation 10, after converting the near point of the eye from units of mm to meters:

Then, substituting the expression just above for the viewing distance into Equation 9 and dividing all sides by 0.25 and *M*
_{
DIS
}:

Now, by taking the reciprocal of the inequality, remembering to interchange the greater than and lesser than signs, the *range of useful magnification* is determined:

Thus, the *range of useful magnification* is between 1/6 and 1/3 of the microscope system resolution.

What does it mean exactly, the range of useful magnification? It means the optimal magnification range when observing an object via a microscope (or an optical instrument that can magnify) where the finest resolvable features can be seen. Remember that range is based on the average performance of the human eye with respect to visual acuity (resolution) and contrast sensitivity along with the optical system’s resolution limit, as mentioned above. A significant number of individuals will have eyes who perform above or below this average.

*
***Low magnification**
. When the magnification from sample to camera sensor is low, generally 1× or even less, then the camera sensor or monitor are the limiting resolution factors of the microscope system. As an example, a digital microscope using a 1.6× objective (*M*
_{
O
} = 1.6×) with a numerical aperture of 0.05 (*NA* = 0.05), a total tube factor of 1× (*q* = 1×), a 0.32× photographic projection lens (*M*
_{
PHOT
} = 0.32×), a camera sensor with a 3 µm pixel size and binning mode of 1 × 1, a monitor pixel size of 0.3 mm (pixel ratio of 100:1), a 1-to-1 pixel correspondence between the sensor and monitor, and white light (mixture of all visible light wavelengths [400–700 nm] with an average in the green at λ = 550 nm) illumination, then from Equations 6, 7, and 8:

This example shows that, at such low magnification, the resolution limit of camera sensors with pixels sizes larger than 2 μm (and monitors with pixel sizes larger than 0.2 mm) will start to be inferior to the light resolution. Therefore, at low magnification, approximately 1× or less, the sensor or monitor will likely be the limiting factor for the microscope system resolution.

*
***High magnification**
. When the magnification from sample to camera sensor is high, generally 50× or greater, then light diffraction is the limiting resolution factor of the microscope system. Again, to illustrate it with an example, a digital microscope using a 160× objective (*M*
_{
O
} = 160×) with a numerical aperture of 1.4 (*NA* = 1.4), a total tube factor of one (*q* = 1×), a 1× photographic projection lens (*M*
_{
PHOT
} = 1×), a camera sensor with a 6 µm pixel size and binning mode of 1 × 1, a monitor pixel size of 0.3 mm (pixel ratio of 50:1), a 1-to-1 pixel correspondence between the sensor and monitor, and green light (λ = 550 nm) illumination, then from Equations 6, 7, and 8:

Here at high sample-to-sensor magnification, the system resolution of a microscope using a modern camera sensor with a pixel size in the 1–6 μm range (and a monitor pixels size below 0.6 mm) is limited by the light resolution.

For a best-case scenario, the greatest light resolution possible with the smallest wavelength of visible light, 400 nm, and a very high numerical aperture, 1.4, is approximately 5,740 line pairs/mm. From the example above, it is clearly seen that the resolution limit of a camera sensor with a pixel size below 6 μm easily exceeds this value.

For this description of digital microscopy, it is assumed that the image on the monitor is always observed within the range of useful viewing distance. As discussed above, the range of useful magnification is derived based on the average performance of the eye. Whenever a sample is observed with a light microscope, sample features that correspond to the microscope’s resolution limit become resolvable by the eye under optimal illumination conditions at the lower magnification value of the useful range (Equation 11). For non-optimal illumination, then the higher magnification value may be necessary for the eye to resolve the features. Beyond the higher magnification value of the range, no finer details of the sample can be resolved, so it is empty magnification.

The example above for high magnification demonstrated that the best possible resolution attained with a light microscope (objective with a 1.4 numerical aperture and illumination with 400 nm light) is about 5,740 line pairs/mm. Now, the range of useful magnification for the best resolution case can be calculated using Equation 11:

So the highest useful magnification for a light microscope is about 1,900×–2,000×. Magnification values exceeding 2,000× fall under empty magnification. When below the lower end of the range, 957×, again it means that the average eye can no longer distinguish details on a sample with a spatial frequency equivalent to 5,740 line pairs/mm. However, most samples are not uniform with features of the same exact dimensions and spatial frequency everywhere on the surface. Normally, as magnification is decreased, other larger features with lower spatial frequencies (below the resolution limit) become resolvable.

The example for low magnification above, where the image sensor limits the resolution of the microscope (objective with a 0.05 numerical aperture and sensor with 3 µm pixel size), shows a limit of about 85 line pairs/mm. Now, the range of useful magnification for this low magnification case also can be calculated:

So the range of useful magnification for this case, where the sensor limits the resolution of the microscope, is about 14× – 28×. In order to resolve features on a sample with a spatial frequency equivalent to 85 line pairs/mm, the total magnification (*M*
_{
DIS
}) of the observed image should fall between 14× and 28×. Beyond 28× it would be empty magnification. To make sure that sample features with a spatial frequency corresponding to the resolution limit of 85 line pairs/mm can be resolved, then the choice of a monitor with appropriate dimensions and pixel size becomes important.