I'll take a shot at this, although the answer is probably more complicated. Let's say we define the resolution in terms of the largest spatial frequency that can be measured by the sensor. This could be intuitively related to say the resolving power (ability to separate two point sources). In Fourier optics, the spatial frequencies associated with a coherent source can be thought of as plane waves propagating at different angles to the normal, with higher frequencies at higher angles. My thinking is, the largest angle measured by the image sensor will tell you the spatial frequency cut off. This is related to the extent of the image sensor at the source. So, for instance, say you have a 5 mm radius sensor at a distance of 2 mm, then the largest angle is atan(2/5) ~ 22 degrees, which corresponds to a numerical aperture of ~ 0.37. Hence, the spatial resolution will be lamba/(2*NA) ~ 680 nm for 500 nm illumination wavelength. On the other hand, if you place your sensor far away (say 1 m from source), this leads to a numerical aperture of only 0.005, giving you a spatial resolution of ~ 50 micron. Realistically, the answer (and ability to separate two close by point sources) will depend on SNR, type of illumination, sample structure and can also likely be enhanced using techniques like having a coded aperture or structured illumination.. I think lensless imaging is an active area of research.
Hey Kamrul,
I would agree with Indrasen's assessment. I would also suggest to think in terms of spatial frequncies. If your detector is placed in the optical far-field, you will directly measure the Fourier transform. The numercial aperture of your detector will then limit the spatial frequency extent of your Fourier transform, i.e. your detector acts as a low pass filter. If you are measuring with a lensless setup in the optical near-field, you can just as much think in terms of the spatial frequency spectrum by using the angular spectrum formalism (ASPW); the ASPW will lead to the same reasoning as Indrasan's argument above.
However, in most lensless microscopes you will have to do some sort of phase retrieval that allows you to refocus the wave field into the specimen plane. The ability to successfully do phase retrieval typically depends on some sort of data redundancy (as in ptychography) or a priori knowledge (as in single shot coherent diffraction imaging or holography [where the reference wave is assumed to be known]). Departure from idealized assumptions (a priori knowledge), mathematical models, systematic experimental errors and noise will lead to a decrease in resolution. For this reason, the so called Fourier ring correlation (FRC*) is used in electron microscopy and x-ray ptychography to assess the reproducibilty of computationally retrieved spatial frequency spectra. Let's say you you are computationally reconstructing a microscopic sample with two different randomized initial guesses (assuming some sort of iterative reconstruction algorithm). Then the two different initial guesses will yield two slightly different reconstructions. From my experience in this field, typically the low frequencies are well reconstructed, the high frequencies are less reproducible. But only reproducible spatial frequencies are resolved spatial frequencies. Therefore I would recommend using the FRC for resolution estimation.
Finally, it is always worth taking a resolution test target (such as spokes targets). While those are typically binary test objects and biological specimen are harder to reconstruct, they give you at least an idea of what the resolution of your microscopes is.
Having said that, take also a look at this ** reference. Be also aware to always distinguish between half pitch and full pitch resolution (***). Simply stated, if you are able to resolve a harmonic grating (sinusoid), then due to the Shannon-Nyquist sampling criterion it takes two samples per spatial period to digitally represent this signal. Then the half period is the half pitch, the full period is the full pitch.
Best wishes,
Lars
* Van Heel, M., & Schatz, M. (2005). Fourier shell correlation threshold criteria. Journal of structural biology, 151(3), 250-262.
** Horstmeyer, R., Heintzmann, R., Popescu, G., Waller, L., & Yang, C. (2016). Standardizing the resolution claims for coherent microscopy. Nature Photonics, 10(2), 68.
*** Zheng, G., Horstmeyer, R., & Yang, C. (2015). Corrigendum: Wide-field, high-resolution Fourier ptychographic microscopy. Nature Photonics, 9(9), 621.
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