The sampling theorem for the amplitude of waves by a circular aperture is derived; namely, the complex amplitude
F (ρ, φ) in the image plane is expressed as
F(ρ, φ)=_??_
Fn(λ
ns/κα)
Cns (ρ, φ) (1)
Fn(λ
ns/κα)=1/2π_??_
F(λ
ns/κα, φ)exp(-
inφ)
dφ (2)
Cns(ρ, φ)=exp(
inφ)
Jn(καρ)2λ
ns/
Jn'(λ
ns){(καρ)
2-λ
ns2} (3)
where α is aperture constant, ρ, φ polar coordinates, κ=2π/λ and λ wave length, λ
ns s-th zero of the Bessel function
Jn(χ). The sampling functions {
Cns} satisfy the orthogonal relation
1/2π_??_
Cns(ρ, φ)
Cmt*(ρ, φ)ρ
dρ
dφ=2δ
nmδ
st/{κα
Jn'(λ
ns)}
2 (4)
and
Cns is unity upon a sampling circle of radius λ
ns/κα and is zero upon the other sampling circles of the same order (Fig. 1). The sampling coefficient
Fn(λ
ns/κα) is obtained by the integration with angle φ of the complex amplitude at a sampling circle multiplied by exp (-
inφ). At each sampling circle of order zero a sampling coefficient is obtained and at each sampling circle of non-zero order two sampling coefficients
Fn(λ
ns/κα) and
F-n(λ
ns/κα) are obtained. There is an important relation between the above sampling coefficient and the coefficient of Fourier-Bessel expansion of the pupil function:
Kns=4π
Fn(λ
ns/κα)/(
i)
n{κα
Jn'(λ
ns)}
2 (5)
and the pupil function ƒ (
r, θ) is expressed as
ƒ (γ, θ)=_??__??_
Knsexp(
inθ)
Jn(
rnsγ/α) (6)
The number of sampling coefficients whose sampling circles are included within a circle of area
S of the image, namely, the number of degrees of freedom, is estimated as πα
2S/λ
2 by considering the distribution of zeros of
Jn(χ) and by using known results for rectangular apertures.
That the Fourier transform of intensity distribution of an image by a circular aperture α vanishes outside the region of a circle of radius 2α in various degrees of coherence of illumination, is shown by considering that the intensity is described by a series of products
CnsCmt* and Fourier transforms of
CnsCmt* vanish outside the circle mentioned above as it is clarified by means of convolution integrals or analytically in Appendix 2. Because of the limited spectrum of the intensity distribution mentioned above the sampling theorem for intensity distribution is obtained by putting 2α in place of α into equations (1)_??_(3).
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