Endomicroscopy - Biomedical Optical Imaging Technologies: Design and Applications - page 270

Fig. 8.10 GRIN lens
operation showing on-axis
( solid lines ) and off-axis
propagation ( dashed lines )

minimum value near the edge of the rod. Following description in [ 49 ], the function
describing refractive index distribution is:

n 2 .r/ D n 0 .1 ˛ 2 r 2 /;

(8.7)

where n 0 is a refractive index on axis, ˛ is a distribution constant, and r is a radial
coordinate. For small ˛2r 2 refractive index distribution is often approximated with
parabolic function and takes following form [ 36 , 49 ]:
n.r / D n 0 1

:

˛ 2 r 2
2

(8.8)

The schematic drawing demonstrating GRIN lens operation is shown in Fig. 8.10 .
The figure demonstrates on-axis (solid lines) and off-axis (dashed lines) propagation
of optical rays. At distances of Z L and Z L /2, a GRIN lens creates an upright image
and inverted image, respectively. Z L /4 corresponds to collimated beam. Therefore,
the length of the component will determine its operating properties and lengths of
multiples of pitch; half and quarter pitch can be selected. A focal length of the GRIN
lens depends also on the length Z of the component [ 49 ]:

1
n 0 ˛ sin .˛Z/ :

f GRIN D (8.9)

NA of the GRIN lens [ 49 ] will depend on the lens diameter D and is:

n 0 D˛
2

NA D : (8.10)

Note that the original application of GRIN lenses was telecommunication applica-
tions so they are not inherently suitable for high NA and large FOV imaging, but
their obvious advantage is both in simplicity and small size. They are available,
for example, through NSG in Japan or GRINTECH in Germany. Several groups
have used GRIN lenses to relay an image of the tissue site to a region outside the
body, where conventional macroscopic optical elements and detectors record the
image. Jung et al. demonstrated full-frame epifluorescence imaging within the brain
of living mice and rats with GRIN relay [ 50 ]. Kim et al. [ 22 ] implemented confocal
fluorescence imaging with the source, detector, and scanning system located
proximal to a triple GRIN lens, 1 mm in diameter. Images were 250 250 m