Recently D. Gabor in England and the writer have independently reached the same conclusion, namely, that the intensity distribution of images can be described by using positive-definite Hermitian matrices under various conditions of illumination. According to the writer's results, the intensity distribution of an image obtained by a pupil of numerical aperture α can be described as the following quadrature form,
I(
x; α)=(Ø(
x; α),
AØ (
x; α)), (1)
where
A is the intensity matrix having
n,
m-element
Anm=∫∫Γ
12 (
X1,
X12)
E* (
X1)
E(
X2)
u* (
X1-
nπ/
kα)
u (
X2-
mπ/
kα)
dX1dX2, (2)
and ø (
x; α) is the vector,
n-th component of which is given by the
n-th sampling function for the image obtained by the pupil, that is,
ø
n(
x; α)=sin(
kα
x-
nπ)/(
kα
x-
nπ). (3)
It is shown in this paper that the given intensity matrix
A is transformed into another matrix
B when the wave in an image by the pupil is transmitted through another pupil of numerical aperture β, namely,
B=
T'*
AT, (4)
where
T is the transmission matrix concerning the second pupil and
T'* is the matrix,
m,
n-element of which is equal to the complex conjugate of
n,
m-element of the above matrix
T. The equation (4) can be derived as follows; since the transmission function
u of the combined system of the two pupils mentioned above is given by the well-known convolution integral of transmission functions
u and
u' of the first and second pupils respectively, namely,
u(
X-
x)=∫+∞-∞
u(
X-ξ)
u'(ξ-
x)
dξ, (5)
and the first transmission function
u(
X-ξ) can be expressed in series by using the sampling theorem for the image by the pupil of numerical aperture α, the transmission factor
u(
X-
nπ/
kβ) in the matrix element
Bnm is expressed as
u(
X-
nπ/
kβ)=+∞Σ
n=-∞
u(
X-
nπ/
kα)
Tnn', (6)
and
Tnn'=∫
un(ξα)
u'(ξ-
n'π/
kβ)
dξ, (7)
where
un(ξ; α) is the n-th sampling function for the image by the said pupil. Inserting the equation (6) into the
n,
m-element of intensity matrix
B, which is of the form similar to the equation (2), the equation (4) can be obtained, where the transmission matrix T is given by the element
Tnn' obtained above.
In the case of α≥β, the element of transmission matrix can be expressed as
Tnn'=(π/
kα)
u'(
nπ/
kα-
n'π/
kβ),
(8) that is, the
n,
n'-element of the transmission matrix indicates how much complex amplitude of wave can be produced at the
n'-th sampling point in the second image plane by the wave having unit amplitude at the
n-th sampling point in the first image plane.
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