Biomedical Engineering Reference
In-Depth Information
to electrical engineers. The treatment given
here is brief and is not intended to be rigorous
but rather to merely provide a few practical
techniques.
light arriving at θ 1 = 0 , it is easy to see that, for
the air-glass interface, R 0. 04 , which means
4% of the light is lost to reflection, which is not
an insignificant amount. This same effect occurs
at every boundary, so the back side of the lens
exhibits reflection, as do any boundaries associ-
ated with additional lenses. This can cause spu-
rious reflections to bounce around inside an
optical system, greatly degrading the image
contrast and causing undesirable noise spots in
the image.
To reduce this effect, antireflective (AR) coa-
tings have been developed that incorporate one
or more thin films (where the thickness of the
coating is typically on the order of λ/ 4 ). Destruc-
tive interference of the reflected light (and con-
structive interference of the transmitted light)
greatly reduces R and allows more light to be
transmitted to the photodetectors. The effective-
ness of an AR coating is dependent on both the
wavelength and the angle of incidence. The term
coated optics is sometimes used to describe AR
optics. Essentially all optical components (lenses,
beam splitters, turning mirrors, etc.) are availa-
ble with AR coatings, and their use is highly
recommended. See Chapter 12 on biomimetic
AR coatings.
1.2.2.1 Point-Spread Function
For the purposes of optical design, the two
domains linked by the Fourier transform F
{}
are the spatial domain (i.e., distance) and the
spatial frequency domain (i.e., cycles per unit
distance), where the distance coordinates are
usually assumed to be measured transverse
to the optical axis, usually at the focal plane.
Recall from the theory of Fourier transforms
that convolution in one domain is equivalent
to multiplication in the other domain; this will
be useful. Every optical component (aperture,
lens, etc.) has a point-spread function (PSF)
defined in the spatial domain at the focal plane,
which describes how an infinitesimally small
(yet sufficiently bright) point of light (the opti-
cal equivalent of a Dirac delta function δ( x o , y o )
at the object plane) is spread (or smeared) by
that component. A perfect component, in the
absence of both aberrations and diffraction,
would pass the point unchanged. The PSF of
an optical component is convolved in the spa-
tial domain with the incoming light. This means
that a perfect component would require a PSF
that was also a delta function δ( x , y ) ; how much
the PSF deviates from δ( x , y ) determines how
much it smears each point of light. Since dif-
fraction is always present, it provides the limit
on how closely a PSF can approach δ( x , y ) ; any
aberrations simply make the PSF deviate even
further from the ideal.
Assume that light enters the sensor system
though an aperture and lens, and that the lens
focuses an image at the focal plane. The ampli-
tude transmittance of the aperture can be
described mathematically by a simple aperture
function A ( x a , y a ), which is an expression of how
light is transmitted through or is blocked by the
aperture at the aperture plane. For example, an
1.2.2 Fourier Optics Approach
A particularly powerful and practical method
of dealing with design considerations such as
apertures, lenses, photodetector size, and spa-
tial sampling is called Fourier optics . The Fou-
rier approach can even be continued (with a
change of domains from space to time) to the
electronics associated with obtaining an image
from a given sensor system. The classic refe-
rence for Fourier optics is the excellent topic
by Goodman [10] , although Wilson [11] is also
very helpful; a succinct treatment can be found
in Hecht [7, Ch. 11] . The method is very simi-
lar to the Fourier approach to the design and
analysis of circuits and systems that is familiar
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