Here is how I calculate. Assumptions (which has been modified to be aligned to 550 nm wavelength from your example):
Wavelength of light: 550 nm
Photo tube magnification: 2.5X (Olympus PE 2.5X)
I am using the Reyleigh Criteria: Resolution = 0.61*Wavelength/NA
Example Objective:
40X NA 0.95
1. Maximum resolution at specimen plane: 0.61*550/0.95 = 353.158 nm (this is the resolving power of the objective and the pixel pitch needed if the sensor was just above the specimen)
2. Applying Nyquist theorem meaning that we need to sample at least twice that to achieve good results: 353.158/2 -> We need to resolve/sample 176.579 nm
3. The light is moving on through the objective magnifying 40X. This is the resolution we need to meet at the objective backplane. 176.579 * 40 = 7063.16 nm = 7.063 um.
3. The light is moving on through the photo eyepiece and has a magnification factor of 2.5: 7.063 * 2.5 =17.65 um. This is the resolution we need to meet at the sensor plane.
4. The full frame sensor has a size of 35.6 x 23.8 mm.
5. 17.65 um = 0.01765 mm: Required X-resolution: 35.6/0.01765 = 2017 pixels
6. 17.65 um = 0.01765 mm: Required Y-resolution: 23.8/0.01765 = 1348 pixels
Sensor resolution: 2017 x 1348 = 2.72 MP.
60X NA 1.20
1. Maximum resolution at specimen plane: 0.61*550/1.20 = 279.58 nm (this is the resolving power of the objective)
2. Applying Nyquist theorem meaning that we need to sample at least twice that to achieve good results: 279.58/2 -> We need to resolve/sample 139.79 nm
3. The light is moving on through the objective magnifying 60X. This is the resolution we need to meet at the objective backplane. 139.79 * 60 = 8387.5 nm = 8.387 um.
3. The light is moving on through the photo eyepiece and has a magnification factor of 2.5: 8.387 * 2.5 = 20.969 um. This is the resolution we need to meet at the sensor plane.
4. The full frame sensor has a size of 35.6 x 23,8 mm.
5. 20.969 um = 0.02969 mm: Required X-resolution: 35.6/0.020969 = 1698 pixels
6. 20.969 um = 0.02969 mm: Required Y-resolution: 23,8/0.020969 = 1135 pixels
Sensor resolution: 1698 x 1135 = 1.93 MP.
Using The Nikons webpage and filling in data for the 40X NA0.95 but with a 1-inch sensor [12.8 x 9.6 mm] (biggest selectable) and a 1X tube magnification I get a pixel size of 5.8 um.
Looking at the three formulas at the Nikon page, it appears they are using the first formula: 0.5*wavelength/NA. I am using formula 2: 0.61X*wavelength/NA, but change it below to compare
Link to Nikon's calculations:
https://www.microscopyu.com/tutorials/m ... resolution
Using my formula above applied and calculating It manually I get:
1. Maximum resolution at specimen plane: 0.61*550/0.95 = 353.158 nm (this is the resolving power of the objective and the pixel pitch needed if the sensor was just above the specimen)
2. Applying Nyquist theorem meaning that we need to sample at least twice that to achieve good results: 353.158/2 -> We need to resolve/sample 176.579 nm
3. The light is moving on through the objective magnifying 40X. This is the resolution we need to meet at the objective back plane. 176.579 * 40 = 7063.16 nm = 7.063 um.
3. The light is moving on through the photo eyepiece and has a magnification factor of 1: 7.063 * 1 = 7.063 um. This is the resolution we need to meet at the sensor plane.
4. The 1 inch sensor has a size of 9.6 x 12.8 mm.
5. 7.063 um = 0.007063 mm: Required X-resolution: 12.8/0.007063 = 1812 pixels
6. 7.063 um = 0.007063 mm: Required Y-resolution: 9.6/0.007063 = 1359 pixels
Sensor resolution size: 1812 x 1359 = 2.46 MP.
Using Nikons slightly more conservative formula with 0.5*wavelength/NA we get:
1. Maximum resolution at specimen plane: 0.5*550/0.95 = 289.47 nm (this is the resolving power of the objective and the pixel pitch needed if the sensor was just above the specimen)
2. Applying Nyquist theorem meaning that we need to sample at least twice that to achieve good results: 289.47/2 -> We need to resolve/sample 144.74 nm
3. The light is moving on through the objective magnifying 40X. This is the resolution we need to meet at the objective backplane. 176.579 * 40 = 5789.47 nm = 5.79 um.
3. The light is moving on through the photo eyepiece and has a magnification factor of 1: 5.79 * 1 = 5.79 um. This is the resolution we need to meet at the sensor plane.
4. The 1 inch sensor has a size of 9.6 x 12.8 mm.
5. 5.79 um = 0.00579 mm: Required X-resolution: 12.8/0.00579 = 2211 pixels
6. 5.79 um = 0.00579 mm: Required Y-resolution: 9.6/0.00579 = 1658 pixels
Sensor resolution: 2211 x 1658 = 3.66 MP. (Slightly more than 2.46 due to the more conservative formula of 0.5 instead of 0.61)
If I recalculate my resolution table I previously presented and change 0.61 to 0.5 (same as Nikon) and also increase the Nyquist theorem requirement from 2 to 3 (to be even further on the safe side), I get the following:
Code: Select all
Magnification Megapixel
4X (NA 0.16) 25,81
10X (NA 0.4) 25,81
20X (NA 0.75) 22,69
40X (NA 0.95) 9,10
60X (NA 1.35) 8,17
60X (NA 1.20) 6,45
100X (NA 1.4) 3,16