4.3.1 Abbe's Method

Following Abbe the image due to one coherent source point is first calculated. Either the scalar formulation of (4.54) or the vector form of (4.68) can be used. The aerial image is then obtained by an incoherent superposition of all the contributions and formally writes to

 

$$I_i(x,y) = \sum_{p,q} I_i^{pq}(x,y).$$ (4.73)

The point sources (p, q) are located within the illumination aperture. Beside this no further restrictions on the shape of the aperture are made. Examples for annular and quadrupole illumination are shown in Figure 4.7.

  

Figure 4.7: Source discretization of an annular and quadrupole illumination system. To account for arbitrary illumination forms the light source is discretized into mutually independent coherent point sources located inside the illumination aperture.

The spacing between the individual point sources has to satisfy a specific criterion. As mentioned before, it is chosen in such a way that the lateral wavevector components s_{x, p} and s_{y, q} of the waves incident on the photomask equal an integer multiple of the sampling frequencies M$\lambda$/a and M$\lambda$/b (cf. (4.47)). This choice guarantees that the image on the wafer surface is periodic with periods a and b, which becomes extremely important in context with the exposure/bleaching simulation described in Chapter 6. This requirement is illustrated in the wavevector diagram of Figure 4.8 for the case of conventional illumination.

  

Figure 4.8: Source discretization of a conventional partially coherent illumination system. The spacing between the individual point sources is chosen to yield a periodic field incident on the photomask.

What is missing now is the determination of the amplitudes A_pq of the point sources. Clearly, they depend on the total imaging intensity I₀ as well as on the discretization scheme illustrated in Figure 4.8. The intensity of one point source is proportional to the discretization area w_pq. As it is also related to the square of the light amplitude A_pq² we write

 

$$A_{pq}^2 = A_0^2 w_{pq},$$ (4.74)

where A₀ is a simple normalizing constant. By considering a missing or ``perfectly transparent'' mask, we obtain the relationship between the amplitude A_pq and the total imaging intensity I₀ looked for.

The transmission characteristic t(x, y) of a transparent mask equals to unity, and its Fourier spectrum T_nm thus consists of a delta impulse at zero frequency, i.e.,

 

$$t(x,y) \equiv 1 {}\quad\qquad\Longrightarrow\quad\qquad{}T_{nm} = \delta(n)\delta(m).$$ (4.75)

The scalar and vector approach have to be studied separately from now on:

• Scalar theory. With (4.84) and (4.85) the image spectrum (4.52) writes as

  $$U^{pq}_{i,0,nm} = A_0 \sqrt{w_{pq}}\delta(n-p)\delta(m-q)P(n,m),$$ (4.76)

  and the image amplitude and intensity are obtained from (4.51) and (4.54)

  $$U^{pq}_{i,0}(x,y) = A_0 \sqrt{w_{pq}} P(p,q) e^{j2\pi(px/a+qy/b)}... ...row\quadI^{pq}_{i,0}(x,y) = \frac{1}{2\eta_0} A_0^2 w_{pq} \vert P(p,q)\vert^2.$$ (4.77)

  The aerial image intensity I_{i, 0}(x, y) is thus constant and with the definition of the scalar pupil function (4.69) found to be

  $$I_0 = \frac{M^2 A_0^2}{2\eta_0} \sum_{p,q} w_{pq}\vert F(p,q)\vert^2,$$ (4.78)

  whereby we assume that all point sources (p, q) are collected by the projection lens, i.e., the numerical aperture of the condenser lens must be smaller than that of the projection lens or, equivalently, the partial coherence factor $\sigma$ defined in (2.4) is smaller than unity, $\sigma$$\le$1. From that the normalizing constant A₀ can simply be determined

  $$A_0 = \frac{1}{M}\sqrt{\frac{2\eta_0 I_0}{\displaystyle\sum_{p,q} w_{pq}\vert F(p,q)\vert^2}}.$$ (4.79)

• Vector theory. The electric field amplitude (4.57) writes with (4.84) and (4.85) to

  $$\mathbf{E}^{pq}_{i,0,nm} = A_0 \sqrt{w_{pq}}\delta(n-p)\delta(m-q)\mathbf{P}(n,m:p,q).$$ (4.80)

  Hence, in case of blank illumination a single homogeneous plane wave

  $$\mathbf{E}^{pq}_{i,0}(x,y) = A_0 \sqrt{w_{pq}}\, \mathbf{P}(p,q:p,q) e^{jk_0(i_{x,p}x+i_{y,q}y)}$$ (4.81)

  is incident on the wafer. For a homogeneous plane wave the vertical component of the Poynting vector is simply given by

  $$I^{pq}_{i,0}(x,y) = \frac{1}{2\eta_0} i_{z,pq} \Vert\mathbf{E}^{pq}_{i,0}(x,y)\Vert^2,$$ (4.82)

  and with (4.65) to (4.69) the contribution of a single point source is calculated as follows:ⁿ

  $$I^{pq}_{i,0}(x,y) = \frac{1}{2\eta_0} A_0^2 w_{pq} i_{z,pq}\Vert\mathbf{P}(p,q:p,q)\Vert^2$$

  $$= \frac{1}{2\eta_0} A_0^2 w_{pq} i_{z,pq}\frac{M^2\vert F(p,q)\vert^2}{o_{z,pq}i_{z,pq}} \Vert{\boldsymbol{\Psi}}(p,q:p,q)\Vert^2$$ (4.83)

  $$= \frac{1}{2\eta_0} M^2 A_0^2 w_{pq} o_{z,pq} \vert F(p,q)\vert^2.$$

  As expected the contributions I^pq_{i, 0}(x, y) are constant. Thus we obtain for the blank aerial image

  $$I_0 = \frac{M^2A_0^2}{2\eta_0} \sum_{p,q} w_{pq} o_{z,pq} \vert F(p,q)\vert^2,$$ (4.84)
  from which the constant tex2html_wrap_inline$A_0$ can easily be determined,

  $$A_0 = \frac{1}{M} \sqrt{\frac{2\eta_0 I_0}{\displaystyle\sum_{p,q} w_{pq} o_{z,pq} \vert F(p,q)\vert^2}}.$$ (4.85)

Footnotes

... follows:ⁿ

For (n, m) = (p, q) and thus s_pq = o_pq we get from (4.66) and (4.67) |$\Psi$(p, q : p, q)|² = o_{z, pq}².