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.