229
3.3. Diffraction
where we assume the final ray slopes (x I
� , y I
� ) and the magnification
M are identical for the physical system and the equivalent confocal
system.
This completes the correspondence between the aberrations of
the physical system and the Zernike coefficients (a mn , b mn ). The
Zernike coefficients uniquely specify the aberrations. Strictly
speaking, these coefficients are a function of object position O. In
a well-designed system, the coefficients do not vary greatly across
the object plane.
3.3.8 Relationship between diffraction and the
Heisenberg uncertainty principle
We have seen in the foregoing sections that diffraction and interference follow naturally from the Fresnel–Kirchhoff relation (3.244).
In turn, this relation represents a stationary-state solution of the
spatial part of Schr¨ odinger’s equation (3.230) for a free particle
wave function. It is of great interest to consider diffraction and
interference from a closely related point of view, namely, Heisenberg’s uncertainty principle. This is the subject of the present
section.
We begin with diffraction from two parallel slits. This is shown
schematically in Figure 3.18. A plane wave is incident from the
top of the figure, with propagation direction normally incident on
a screen S. The screen has two infinitely long parallel slits, oriented
out of the page. A diffracted ray I emanates from the left slit, and
a diffracted ray II emanates from the right slit. The two rays are
assumed to be parallel to one another, and propagate at an angle
θ relative to the central axis. A thin lens L causes the two rays to
converge at a viewing plane P. The axial spacing between S and L
3.3. Diffraction
where we assume the final ray slopes (x I
� , y I
� ) and the magnification
M are identical for the physical system and the equivalent confocal
system.
This completes the correspondence between the aberrations of
the physical system and the Zernike coefficients (a mn , b mn ). The
Zernike coefficients uniquely specify the aberrations. Strictly
speaking, these coefficients are a function of object position O. In
a well-designed system, the coefficients do not vary greatly across
the object plane.
3.3.8 Relationship between diffraction and the
Heisenberg uncertainty principle
We have seen in the foregoing sections that diffraction and interference follow naturally from the Fresnel–Kirchhoff relation (3.244).
In turn, this relation represents a stationary-state solution of the
spatial part of Schr¨ odinger’s equation (3.230) for a free particle
wave function. It is of great interest to consider diffraction and
interference from a closely related point of view, namely, Heisenberg’s uncertainty principle. This is the subject of the present
section.
We begin with diffraction from two parallel slits. This is shown
schematically in Figure 3.18. A plane wave is incident from the
top of the figure, with propagation direction normally incident on
a screen S. The screen has two infinitely long parallel slits, oriented
out of the page. A diffracted ray I emanates from the left slit, and
a diffracted ray II emanates from the right slit. The two rays are
assumed to be parallel to one another, and propagate at an angle
θ relative to the central axis. A thin lens L causes the two rays to
converge at a viewing plane P. The axial spacing between S and L