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various gratings composed of different spatial frequencies, and light that diffracts from it may be accurately described by the specimen’s spatial Fourier transform at a particular plane (the Fourier plane). A key goal of high-resolution microscope design is to create a large Fourier space that passes a wide range of spatial frequencies. While the traditional method of achieving this is to increase the NA of the microscope objec- tive, Fourier ptychography achieves this via a strategy that relies on tilted illumination and computation.
Putting the “Fourier” in Fourier Ptychography With the above insights in place, we can now develop a
mathematical picture for Fourier ptychography. Following from the information above, if a thin sample is illuminated with a perpendicular plane wave, then the beauty of physics specifies that the resulting spatial pattern of light, formed at a particular plane behind a lens, is described by the sample’s Fourier trans- form. Due to the limited size of the microscope objective lens, only a small segment of this Fourier spectrum can pass through the imaging system and onto the image sensor (see diffracted spectrum in Figure 2). Mathematically, one can thus model the effect of a lens on a specimen’s diffracted spectrum as a low- pass filter, which allows a narrow window of frequencies in the Fourier domain to pass unperturbed and otherwise filters out higher frequencies. To form an image, the filtered Fourier spectrum that
has passed through the microscope objective lens must next propagate to the image plane. Without dwelling on details,
this process may be once again represented by a second Fou- rier transform, which returns the spectrum’s spatial frequen- cies back into a spatial map of the specimen of interest, albeit at reduced resolution. When the specimen is illuminated with an angled plane wave, the specimen’s Fourier spec- trum will shift across the fixed lens aperture (Figure 2b), thus allowing measurement of a new windowed region of Fourier components. The amount of shift is specified by the angle of illumination where a larger angle yields a larger shift. By capturing many uniquely illuminated images at appropriately selected angles, Fourier ptychography records many segments of the Fourier transform that cover a large area (Figure 4). Fourier ptychography’s subsequent compu- tational goal is to combine the captured image data into a large, stitched Fourier spectrum composite, thus resolving higher-resolution features when inverted back into a final image.
The Phase Problem Diffraction is a wave phenomenon, which requires
Fourier ptychography to consider light as an optical field that includes both an amplitude and a phase. Let’s assume for a moment that it is possible for a digital image sensor to measure a complex-valued optical field at each image sensor pixel. In this case, after recording a sequence of uniquely illuminated images, Fourier ptychography’s com- putational goal for high-resolution image formation is rela- tively straightforward. To reconstruct the sample’s enlarged
Figure 4: Fourier ptychography acquires multiple wide-area images, each under illumination from a different angle provided by an LED array. In the workflow above, the left image shows the central LED illuminating the center of the resolution sample. In the top row of the workflow, each low-magnification image is mathematically connected to a windowed component of the specimen’s Fourier spectrum (here, 9 unique windows as shown top right). The central window is further processed in the bottom row of the workflow. The computational goal of Fourier ptychography is to synthesize a much larger Fourier spectrum, which is digitally Fourier transformed into the final high-resolution, large-area result. As the lack of measured phase prevents direct tiling in the Fourier domain, Fourier ptychography applies a phase retrieval algorithm to successfully produce a final high-resolution result as shown in the red boxed region in the bottom left panel of the workflow.
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www.microscopy-today.com • 2022 May
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