Biomedical Engineering Reference
In-Depth Information
phases 0,
π
/2,
π
, and 3
π
/2, and with k g parallel to the x axis. We will call these images
s 1 ,
...
, s 4 , and their Fourier transforms S 1 ,
...
, S 4 . These are described by:
s 1 5 Iðx; yÞ½
1 1 cos
ðk g x 1 φðx; yÞÞ
(16.3)
s 2 5 Iðx; yÞ½
1 1 sin
ðk g x 1 φðx; yÞÞ
(16.4)
s 3 5 Iðx; yÞ½ 1 2 cos ðk g x 1 φðx; yÞÞ
(16.5)
s 4 5 Iðx; yÞ U ½
1 2 sin
ðk g x 1 φðx; yÞÞ
(16.6)
1
1
S 1 5 !
δ ð !
2 Φð !
ð 2 !
2 ! g Þ 1
1 ! g Þ
2 Φ
Þ
Þ 1
; ...
(16.7)
( x , y ) is its phase; represents convolution
where I ( x , y ) expresses the image intensity and
φ
and the vector ! g is parallel to the x axis.
!
Φð !
Þ
and
Þ
are the transforms of
I ( x , y ) and
exp[i
φ
( x , y )], respectively. It then follows that the transform of the complex image
I ( x , y )
exp[i
φ
( x , y )] is given by
ðS 2 2 S 4 Þ δ !
2 ! g
!
Þ 5 ½S 1 2 S 3 1 i
(16.8)
This way, the complete phase structure of the image can be constructed with super-
resolution approaching 2.
We now ask whether 2 k m is the limiting spatial frequency of SIM. If the modulation grating
is projected through the microscope condenser onto the object, then k g 5 k m is indeed the
largest shift which can be projected since it uses the whole NA of the optical system
( Figure 16.4A ). But if the grating can be put in contact with the object, then there is the
possibility of using a subwavelength grating with spatial frequency k g . k m , in which case
the illumination wave propagates evanescently into the object. Then, the interaction of the
evanescent wave with object structure having frequency k obj close enough to k g will give
rise to a propagating wave of frequency
j , k m . This frequency can be imaged by
the microscope aperture. Clearly, such an imaging process would not be easy to carry out,
particularly because to cover a wide range of object frequencies, it would be necessary to
use many subwavelength gratings in contact with the object, but the process is not
forbidden for fundamental reasons. Figure 16.6 shows a possible way in which this might
be implemented to gain a modest degree of super-resolution of a thin air-immersed object;
use of an actual subwavelength grating in contact with the object would be better.
Obviously, this implementation requires that the object be thinner than the evanescent
penetration depth, 2
jk obj 2 k g
q
k obj 2 k 0
π=
:
Work using this approach has been done by Gur et al.
[25] , who used a flowing suspension of nanoparticles in contact with an object to
superimpose on it a series of independent subwavelength illumination patterns with varying
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