H. Kirchauer and S. Selberherr: Three-Dimensional Photolithography Simulation
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2 Imaging Simulation


In this section we describe our aerial image simulator. Although the basic concept follows the vector-valued extension [7] [8] of the scalar theory of Fourier optics [9], the module is described in greater length for the following two reasons. Firstly, the disrectization procedure of the distributed light source is of crucial importance for a numerically efficient implementation of the differential method. Secondly, a semi-analytical algorithm for the computation of the Fourier coefficients of the mask transmission function is proposed, which has the great advantage of avoiding aliasing errors due to a sampling of the mask transmission function.

To apply the concept of Fourier optics we first reduce the projection printing system to its essential parts as shown in Fig. 3.





Figure 3: The projection printing system is reduced to its essential parts.


The mask-pattern is thereby assumed to be laterally periodic with periods and . The photomask is infinitesimally thin with ideal transitions of the transmission characteristic. The piecewise constant transmission function is real-valued (zero or one) for binary masks , in case of phase-shift masks it is complex-valued with module less or equal than one.

For the simulation of arbitrary illumination forms the distributed light source is discretized into mutually independent coherent point sources . In Fig. 4 we show the discretization for an annular and a quadrupole aperture.




We choose the point source locations in such a way that all the individual contributions to the EM field are periodic. The reason therefore lies in the usage of the differential method for the exposure/bleaching module. The periodicity guarantees that the ordinary differential equation system of (6) and the boundary matrices (10) are equal for all point sources. As a consequence the linear system (12) has to be assembled just once. The additional costs due to the partial coherence lie therefore only in the solution of (12) for multiple right hand sides. Hence, the periodic requirement is essential for a numerically efficient exposure/bleaching simulation.

The spacing between the point sources is chosen to assure that the lateral wavevector components and of the waves incident onto the photomask equal an integer multiple of the sampling frequencies and in the Fourier domain. This requirement is illustrated in the wavevector diagram of Fig. 5.




The resulting image on top of the wafer due to one coherent point source can now be expressed by a superposition of discrete diffraction orders. Because of the periodicity of the EM field this superposition corresponds to a Fourier expansion and writes to



whereby stands vicariously for the two EM field vectors and . The diffraction orders are homogeneous plane waves with wavevectors and amplitudes following from the vector-valued diffraction theory [7],



is the actinic wavelength and the time dependence of the EM field is a time-harmonic one, i.e., and with angular frequency .

In (1) stands for the Fourier coefficients of the mask transmission function. As illustrated in Fig. 6 they are computed by first triangulating the piecewise constant transmission function and then superposing weighted analytical Fourier transforms of the triangular patterns. This semi-analyitcal algorithm has the great advantage of avoiding any errors due to aliasing. Aliasing would occur by conventional sampling of the transmission function and application of the Fast Fourier Transform (FFT) algorithm, because the mask transmittance is not a low pass function [12].





The second term of (1) is the vector-valued counterpart to the pupil-function of the scalar diffraction theory [9] and follows from ray-tracing [7] [8] trough the optical system (cf. Fig. 3). is essentially a low-pass filter (no evanescent waves can travel towards the wafer) and accounts for the polarization state, defocus and higher order aberrations terms. Hence, the vector-valued illumination spectrum (1) is band-limited and the Fourier backward transform can be accomplished using FFT techiques without any aliasing errors [12]. As the Fourier forward transform, i.e., the Fourier coefficients of the mask transmission function, is computed with the above described semi-analyitcal algorithm, the EM field is calculated aliasing free.

The aerial image itself is the light intensity incidenting on top of the wafer and is thus given by the real part of the vertical component of the Poynting vector of the EM field. It is calculated by a weighted incoherent superposition of all mutually independent terms and writes to



whereby the asterisk denotes complex conjugation. For a uniform bright source the weights are determined by the portion of the discretization rectangles within the illumination cone (cf. Fig. 5).

Simulation results of the aerial image module are shown in Fig. 1 and Fig. 11.



next up previous
Next: 3 Exposure/Bleaching Simulation Up: Abstract Previous: 1 Introduction

H. Kirchauer and S. Selberherr: Three-Dimensional Photolithography Simulation