E.2 Circular Aperture

(E.30) |

and

(E.31) |

Using polar coordinates for the transformed coordinates and defined in the previous section

(E.32) |

gives

(E.33) |

We are now reviewing the case with the source being on the optical axis of the aperture () which yields

(E.34) |

where

is the radial distance of the projection point from the z-axis in the projection plane. is the angle of the projection point seen from the aperture (refer to Figure E.2). The FOURIER transform of can be calculated from (E.27). The differential area element is in polar coordinates

as depicted by Figure E.4 The exponential factor in the integral of (E.27) in the above defined coordinate system gives

Therefore (E.27) is in polar coordinates

The inner integrand of this solution is the well known BESSEL function of zeroth order and is defined as

(E.37) |

by using this definition with (E.27) the FOURIER transform of a circular aperture gives

which is independent of as a consequence of the rotational symmetry of the aperture. Applying a coordinate transformation to the integral of (E.38) yields

To solve this integral one can use a relation between BESSEL functions of different order

integrating (E.40) with gives

(E.41) |

Using this result in (E.39) gives finally

(E.42) |

or

(E.43) |

In the center of the diffraction pattern with the properties of the BESSEL function give

yielding

(E.44) |

Taking (E.29) the electrical field of the diffraction pattern behind a circular aperture is finally

(E.45) |

The average optical intensity is proportional to the square of the absolute electrical field. Therefore the intensity for diffraction behind a circular aperture is

with

By using the (E.14) and (E.35) in (E.46) one obtains

Substituting from (E.47) into above equation yields

With the substitution

(E.50) |

(which is dimensionless) the intensity is finally

The (for most cases valid) assumption that the aperture radius is much bigger than the wavelength ( ) the intensity is

Which is the well known intensity distribution for a circular aperture given in many textbooks. The variable reduces to

(E.53) |

for the assumption mentioned above.

R. Minixhofer: Integrating Technology Simulation into the Semiconductor Manufacturing Environment