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205
3.3. Diffraction
r O can be considered to be the source of a spherical outgoing wave.
The waves emanating from neighboring object points are assumed
to radiate coherently with respect to one another. This can only
happen if all object points radiate monochromatically with a constant phase relationship. It therefore represents an approximation.
The waves from all object points propagate coherently through the
optical system. The integral over r O represents a superposition of
amplitudes over the entire object plane. The complex amplitude
u O in the object plane is convolved with the function h to form
the amplitude u I in the Gaussian image plane. This physical significance of this can be appreciated by considering the important
special case of a point object on axis. In this case the amplitude
in the object plane is given by
u O (r O ) = δ(r O ),
(3.264)
where the right-hand side is the Dirac delta function. From the
property of the delta function, it follows immediately that
−1
kar I
kar I
u I (r I ) ∼
J 1
,
(3.265)
Z 2
Z 2
where the ratio a/Z 2 is the tangent of the semiangle of the cone
of rays at the image plane z I . The square of this functional form,
which represents the intensity, is known as an Airy disk. Physically, this is precisely the diffraction pattern of the aperture. The
kernel h is called the point spread function, since it represents the
blurring of every image point relative to an ideal image.
To this point we have assumed imaging without aberrations. We
now inquire into the effect of spherical aberration. This is depicted
in Figure 3.12, where the spherical aberration gives rise to an additional path length increment d S . The spherical aberration in the
Gaussian image plane was found earlier to be
δr S = C S α
3 ,
(3.266)
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�
�
205
3.3. Diffraction
r O can be considered to be the source of a spherical outgoing wave.
The waves emanating from neighboring object points are assumed
to radiate coherently with respect to one another. This can only
happen if all object points radiate monochromatically with a constant phase relationship. It therefore represents an approximation.
The waves from all object points propagate coherently through the
optical system. The integral over r O represents a superposition of
amplitudes over the entire object plane. The complex amplitude
u O in the object plane is convolved with the function h to form
the amplitude u I in the Gaussian image plane. This physical significance of this can be appreciated by considering the important
special case of a point object on axis. In this case the amplitude
in the object plane is given by
u O (r O ) = δ(r O ),
(3.264)
where the right-hand side is the Dirac delta function. From the
property of the delta function, it follows immediately that
−1
kar I
kar I
u I (r I ) ∼
J 1
,
(3.265)
Z 2
Z 2
where the ratio a/Z 2 is the tangent of the semiangle of the cone
of rays at the image plane z I . The square of this functional form,
which represents the intensity, is known as an Airy disk. Physically, this is precisely the diffraction pattern of the aperture. The
kernel h is called the point spread function, since it represents the
blurring of every image point relative to an ideal image.
To this point we have assumed imaging without aberrations. We
now inquire into the effect of spherical aberration. This is depicted
in Figure 3.12, where the spherical aberration gives rise to an additional path length increment d S . The spherical aberration in the
Gaussian image plane was found earlier to be
δr S = C S α
3 ,
(3.266)