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Figure 3: When a parallel beam of light illuminates an object, unique patterns of light are formed by a lens placed one focal length away. At this “Fourier plane,” the resulting spatial distribution of light is proportional to the Fourier transform of the object and is typically referred to as the object’s Fourier spectrum. If the object contains fine details, the resulting diffracted light may not enter and pass through the lens, resulting in lost information. The same diffracted light can be partially transferred when the incident light beam is tilted.
light cone will rotate (but otherwise not change), allowing a new segment of the cone to enter the imaging lens and pass onto the digital image sensor. If the specimen is illuminated at a relatively large angle, then regions of the light cone that would not have originally entered the lens at all will now be able to pass through it and onto the image sensor. Accordingly, Fourier ptychography uses an objective lens,
situated far away from the specimen, that forms images over a large FOV and illuminates the specimen with a planar light wave from a series of angles. For each unique illumination angle, Fourier ptychography captures a low-resolution image of the specimen of interest. Tere are, of course, many possible ways to tilt an incident illumination beam to provide multi- angle illumination. Programmable light emitting diode (LED) arrays, which are inexpensive, readily available, and contain hundreds to thousands of individually addressable light sources in a small package, provide an extremely simple and effective means to achieve multi-angle illumination without any moving parts. Tus, an LED array oſten comprises the only hardware required to turn a standard digital microscope into a Fourier ptychographic microscope.
Fourier Transform Operations within the Microscope Te key to understanding Fourier ptychography, as the
name suggests, is the Fourier transform. From a purely signal- processing perspective, the Fourier transform decomposes an arbitrary function into a sum of sinusoidal waves, where low- frequency waves correspond to slow variations within the function and high- frequency waves correspond to its rapid
2022 May •
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variations. By describing the optical properties of a specimen across space with a function, one can connect its resulting Fou- rier transform’s lower frequencies to large and slowly varying spatial features, while its higher frequencies naturally define the specimen’s fine details. To understand the connection between the Fourier
transform and the microscope, it is illustrative to think of a plane wave interacting with a diffraction grating (Figure 3). Te higher the groove frequency of the grating, the larger the resulting diffraction angle of light, relative to its incident angle. Similarly, as the groove frequency approaches 0, the diffrac- tion angle approaches 0. Tere is thus a one-to-one relationship between the groove frequency of the grating and its resulting diffraction angle. If a microscope lens is positioned above the grating to capture and focus the diffracting light, it will focus light to one or more points at specific locations, whose distance from the optical axis is proportional to the groove frequency. It is also clear that the lens can capture the diffracted light from the grating only if its acceptance angle (that is, the lens numerical aperture [NA]) is large enough. If light is diffracted at a high angle and cannot enter the lens, it is possible to rotate the incident plane wave to an oblique angle to cause some of the resulting diffracted light to pass through. From a Fourier transform perspective, the diffraction
grating can be thought of as a very simple sample that con- tains only one dominant sinusoidal oscillation across space (that is, spatial frequency). A biological sample typically con- tains many low and high spatial frequencies, each of which has a one-to-one relationship with a diffraction angle. An arbitrary sample can thus be modeled as a superposition of
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