A microscope's Numerical Aperture (NA) limits the finest details that can be resolved. A very simplified imaging system is shown in the figure below. A lens system takes light from the specimen and focuses it onto an image plane. At some point in the system, an aperture limits the angles of light that can travel through the lens system to create the final image.
The goal of this example is to provide an analysis script that will post-process simulation monitor data such that it can be compared to experimental images obtained from a microscope with a given numerical aperture. To do this, near field monitor data is decomposed into a series of plane waves, which propagate at different angles. Any plane waves with angles outside of the NA are then discarded. Finally, the remaining light is re-focused onto an image plane. The script microscopy_imaging.lsf does this calculation, and produces an image of the final result. This example does not include a magnification factor for the lens system, but that can be added and is used in some other examples. It does include a defocus setting which allows us to consider the imaged light at a specified distance from the focal plane.
A set of 2D example files are also provided.
Image Gaussian beam propagating through free space
The first example (microscopy_imaging_beam.fsp) images a Gaussian beam propagating in free space. The beam is positioned 3 microns away from the waist, its center is offset in x-direction by 1um, and propagates at a 30 degree angle.
The left figure below shows electric field intensity at the monitor (near field). In this image, the spatial offset of the source is visible but it is impossible to discern the direction of propagation. The right figure shows the far field projection of the light. In this case, the direction of propagation is visible. We can see that the source is a highly focused beam that contains plane waves at many angles, but centered at 30 degrees.
The left figure below shows the far field result from the imaging script with Numerical Aperture (NA) of 1. The far field profile looks very much like the near field because all of the light in the near field image is being collected at the image plane. The right figure shows the image when NA = 0.3. As you can see, much of the detail of the original near field has been lost but the beam spot is still small. If we continue to reduce the NA, for example to 0.05, then the imaged beam will become much larger than the original near field beam.
Image fields above photonic crystal cavity
Looking at a resonant mode of the cavity, we expect the near field to be quite different from the far field because most of the energy is trapped in the cavity. The light tends to be traveling in the plane of the PC, reflecting back and forth. Therefore, when the light finally does leave the cavity, it tends to be at grazing angles to the surface, rather than at normal incidence. This is a very important consideration when using an imaging system with an NA.
More information about this cavity and its modes can be found in the Photonic crystal cavity section. This example uses the mode at 243.4 THz.
The left figure below shows electric field intensity at the monitor (near field) obtained from mode_profiles analysis group, located above the PC cavity. Many sub-wavelength features are visible.The right figure shows the far field projection of the light. The majority of the power is radiating at 45 degrees.
near and far field @ 243.4THz
The left figure below shows the output of the imaging script with NA=1. For this plot we used microscopy_imaging.lsf script. The profile is quite different from the near field because much of the near field energy is bound, rather than propagating. The right figure shows the output of the imaging script with NA=0.5. The smaller aperture causes more loss of detail. However, some key features such as the zero at the center are still preserved.
far field @ 243.4THz, NA=1 and NA=0.5