Page 197 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

INTRODUCTION 177

thus simply
>r=oo
2 2 2
P = ( I( x,y, 0)2nr dr with I(x, y, 0) = 7(0, 0, 0) exp(–2r /Wo) (2r-w 0 >
Jr=0
where r is the radial distance and r 2 is given by x 2 + y 2 according to the Pythagoras
theorem and w 0 is defined as the half-width of the Gaussian beam. Because the power of
the laser beam is normally known, we can integrate Equation (7.3) to find that I (0,0,0)
2
is equal to 2P/(w 0 ). Substituting this result back into Equations (7.2) and (7.3) allows
us to define the irradiance at an arbitrary point inside the resin as

2
2
2
I(x, y, z) = (2P/w 0 )exp(–2r /w 0 )exp(– z/d p) (7.4)
To derive the working curve of the SL process, the energy per unit area in the beam
must be determined and is defined as
/=
I(x, y, z) dt where dt = dx/u s (7.5)
/ .. =- =-00
where v s is the scanning speed along the x-axis. The energy along the x-axis is thus
given by
2
2
E(y, z) = J-— exp (-2y /w 0 ) exp (-z/d p) (7.6)
V w 0u S
The maximum curing depth is obtained at the point when y = 0; the exposure at this
point is taken as the critical exposure E c; thus, we obtain the working curve equation for
curing depth C d

C d = d p In (E max/E c) and E max = E(0, 0) = J- — (7.7)
W 0 v s
The maximum cured line-width l w (2y max) is obtained at the point of z = 0, and is
written as
= w 0 2 I n ( E m a x / E c ) (7.8)
l w
It is assumed here that the laser exposure at the point with maximum depth is equal to that
at the point with maximum line- width and is taken as critical exposure E c. It is known
that the exposure level at which the gel point (i.e. polymerisation) is reached should be
slightly higher than the threshold exposure known as the critical exposure.
The relationship between the curing depth and line-width is finally obtained and is
given by

l w = 2w 0 c d/2d p (7.9.)
It is essential that the precise working curves of the curing depth and line-width are known
in an SL process. More sophisticated models that take account of other factors, such as
dark polymerisation, diffusion of the light source, with calibration performed for each SL
system, and photocurable material may be constructed.