Project Cyclops, A Design... - Department of Earth and Planetary ...

APPENDIXO
THE OPTICAL SPECTRUM ANALYZER
If a transparency having the complex amplitude and
transmittance g(x,y) is placed in plane PI of Figure 11-5
and illuminated by a unit amplitude monochromatic
plane wave of wavelength k, then the distribution of
complex amplitude in plane P2 is approximately (ref. 1)
in Chapter 11. Following Thomas, the nth line of
recording has an amplitude transmittance
Sn(X ) = s V" rect , 1 _n_N
1 foo foo F_i2n(ux+vyTdxdy
-- LI'---- 1
and the y variation
Sn(Y ) = 6 { y- [ h-(2n-l)c]}*rect(y)
2
where f is the focal length of the lens. Thomas (ref. 2)
and Markevitch (ref. 3) have shown that the onedimensional
data handling capability of this basically
two-dimensional operation can be greatly increased by
converting the one-dimensional signal to be analyzed,
s(t), into a raster-type, two-dimensional format as shown
in Figure 11-4. The following parameters are defined:
where
rect ( _ ) = O, 1,1_1 elsewhere < 1/2
b = width of spectrum analyzer input window
h = length of spectrum analyzer input window
c = scan line spacing
N = scan lines within input window = hie
and , denotes convolution. The overall transmittance is
g(x,y) = Z S I rect
n=l V ,
a = width of scan line
B0 = maximum signal frequency
k = maximum spatial frequency of recorded signals
V = recording scan velocity = Bo/k
p = frequency resolution in spectrum
T = time window temporal duration of signal within
input window of analyzer
The signal to be analyzed s(t), is suitably limited,
added to a bias to make it non-negative, and then
recorded by some sort of scanned recorder as explained
Y- [ 2 ] , rect
and, neglecting constants, the output plane complex
amplitude is,
G (u,v) =f_ y_ g (x,y) exp -/2n l --'if'- (ux + vy)] dx dy
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