E.1 Linear Approximation

(E.1) | |

(E.2) | |

(E.3) | |

(E.4) | |

(E.5) | |

(E.6) | |

(E.7) | |

(E.8) |

If one defines the transmission function:

(E.9) |

with and and being the aperture area which are transmitting and not transmitting light respectively. The resulting electric field on the projection side after diffraction at an aperture given by the transmission function yields

taking the solution of a spherical wave for diffraction of a planar wave at an infinite small aperture opening

(E.11) |

with A as the Amplitude of the incident planar wave. Equation (E.10) yields

The lengths and are given by

For the far field approximation and are linear functions of and . Therefore we calculate the series expansion of and around and , with and given by

These distances are no functions of and . Using (E.14) in (E.13) yields

assuming that the second and third term is much smaller than the first one, the squareroot in (E.15) can be written as

(E.16) |

and

(E.17) |

Finally R is calculated by the series expansion to the second order in and

This result is only valid if and . This holds especially for the far field where is big in relation to the aperture size

(E.19) |

Therefore the third term in (E.18) is negligible and this gives

(E.20) |

Using the approximation

(E.21) |

which is valid for the far field, and the transformation of the coordinates

(E.22) |

the total optical path can be written as

(E.23) |

substituting this optical path in the phase factor of the diffraction integral in (E.12) yields

(E.24) |

with

(E.25) |

is not a function of and . The coordinates of the source and of the projection point are now included in . Finally the electrical field at the projection point P' emanating from the source point P and diffracted at the aperture with the transmission function can be given as

The integral of (E.26) is exactly the FOURIER transform of the transmission function

This important result implies, that under the given assumptions,

If as shown in Figure E.3 the points , and are on a line, then and (E.26) together with (E.27) reduces to

With (E.28), (E.26) can be normalized to

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