6 M. R. SCHROEDER
directions).
How should one go about designing such an ideal scatterer for sound (or
light or radar waves)? Curiously, one answer comes from a classical branch
of number theory that has exercised the great Gauss for a long time: quadratic
residues. Consider a surface structure whose reflection coefficient rn varies in
equidistant steps along one axis according to
rn =
exp(27iin2/p)
, n = 0 , ±1, ±2, (3)
where p is a prime and n2 may be replaced by (n2)modp, its least
nonnegative residue modulo p.
It is easy to show that the discrete Fourier Transform (DFT) of rn has
constant magnitude. As a physical consequence, the intensities of the
wavelets scattered into different directions from a surface with reflection
coefficients (3) will have equal magnitudes (in the customary Kirchhoff
approximation of diffraction theory [1]). The scattering angles ak are given
by the wavelength X and the step size w (corresponding to An = 1):
sina
k
= ^ , | k | - ^ , (4)
pw A,
yielding 2|_pw/Xj + 1 different angles. (The "Gauss brackets" LJ stand
for rounding down to the nearest integer.) The different reflection coefficients
are realized by * 'wells'' of different depths
dn = —-
(n2)modp
,
2p
as illustrated in Fig. 1 for p = 17. Such wells give a round-trip phase change
of 2dn 2K/X in accordance with the phase requirement of (3). In (4), X is
the longest wavelength to be scattered. For any integral submultiple of that
wavelength, X/m, the reflection coefficients (3) are changed to r™, which for
m # 0(mod p) has the same flat Fourier property as (3). For w one chooses
typically half the smallest wavelength to be scattered over ±n/2. Figure 2
shows the diffraction pattern of the grating in Fig. 1 for one third the longest
wavelength.
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