Simulation Methods and Models

The two-dimensional multiple multipole program (MMP) (Hafner 1990) has been used to investigate diffraction in the evanescent near field of metallic gratings, suspended in various dielectric media. MMP is a semi-analytic technique, in that the electromagnetic fields are approximated by a set of basis functions that are solutions to Maxwell's equations. The program numerically sets the coefficients to these basis functions to minimise the errors for the boundary conditions at matching points distributed along the boundary. Once the coefficients have been determined a full vector solution can be found at any point. Finite-difference time domain (FDTD) techniques have also been used to study ENFOL and related techniques, and these are in agreement with the MMP results. Simulations are performed for chrome transmission gratings of pitch p and thickness t suspended in a medium with index of refraction n. Fig. 17.11 illustrates a typical geometry that has been simulated and defines the Cartesian coordinate system and illumination polarisation. In general only the TM polarisation is considered as for gratings of this scale very little light TE light is transmitted due to the polarising nature of the grating (McNab and Blaikie 2000). A unit cell consisting of one period of the grating is computed while specialised periodic boundaries take into account the effects of surrounding conductors to obtain a solution for effectively an infinitely long grating. Note that, an evanescent exposure is defined here as an exposure where all the transmitted diffracted orders are evanescent. The zeroth propagating order is also generally present but does not provide contrast for photolithography as it does not contain any modulation in the x direction; it simply appears as an intensity offset. For incident illumination normal to the grating an evanescent exposure occurs when p < '/Jn. where /-/« is the effective wavelength.

TM TE

B, alr H, $ ® —V- -—1- ®

air

— unit cell

chrome

X = 436 nm air

Fig. 17.11. Simulation model illustrating suspended chrome gratings mask to simulate exposure into a substrate held in intimate contact with the mask. The TE and TM polarisation orientations are indicated

Fig. 17.11. Simulation model illustrating suspended chrome gratings mask to simulate exposure into a substrate held in intimate contact with the mask. The TE and TM polarisation orientations are indicated

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