4.1.1 Scalar Diffraction Theory

Light is an electromagnetic wave with coupled electric and magnetic fields traveling through space. In a homogeneous isotropic medium, like free space or a lens with constant refractive index, the electric and magnetic field vectors form a right-handed orthogonal triad with the direction of propagation. Disregarding polarization, the field can be described by a scalar function $\mathcal {U}$(x;t) representing either the electric or the magnetic field amplitude. As the light used in microlithography exhibits a strong monochromacity (cf. Section 2.3.1) the time dependence of the field is a harmonic one and can thus be explicitly written as

 

$$\mathcal{U}(\mathbf{x}; t) = \operatorname{Re}\left[U(\mathbf{x})e^{-j\omega t}\right].$$ (4.1)

Here, $\omega$ denotes the angular frequency of the light, and the complex-valued amplitude or phasor  U(x) depends on the spatial coordinates x = (x  y  z)^T only. If both quantities $\mathcal {U}$(x;t) and U(x) represent an optical wave, they must satisfy the Helmholtz equation^a

 

$$\left[ \triangle - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} ... ...qquad{\text{and}}\qquad\quad{}\left[ \triangle + k^2 \right] U(\mathbf{x}) = 0.$$ (4.2)

The Helmholtz equation directly follows from the Maxwell equations under the condition of a homogeneous medium and in absence of sources. The quantity k is termed the wavenumber or propagation constant of the medium and is related to the light velocity c, the angular frequency $\omega$, and the vacuum wavelength $\lambda$ by^b

$$k=\frac{\omega}{c} = \frac{2\pi \nu}{\lambda} {}\quad\qquad{\text{and}}\qquad\quad{}c = \frac{c_0}{\nu},$$ (4.3)

whereby $\nu$ is the refractive index of the medium, e.g., in vacuum $\nu_{0}^{}$ = 1.

The complex disturbance  U(x) at any observation point x_o in space can be computed from the Helmholtz equation (4.2) with the help of Green's theorem.^c If a unit-amplitude spherical wave expanding about the observation point x₀ is chosen as Green's function, i.e.,

 

$$G(\mathbf{x}) = \frac{e^{jk_0r_o}}{r_o}{}\quad\qquad{\text{with}}\quad\qquad{} r_o=\Vert\mathbf{x}_o-\mathbf{x}\Vert,$$ (4.4)

the so-called integral theorem of Helmholtz and Kirchhoff is obtained [11, p. 377]:

 

$$U(\mathbf{x}_o) = -\frac{1}{4\pi} \iint\limits_\mathcal{S}\left[ ... ...athbf{x}) - U(\mathbf{x}) \frac{\partial G(\mathbf{x})}{\partial n} \right] ds.$$ (4.5)

This relation plays a central role in the derivation of the scalar theory of diffraction, for it allows the field U(x) at an observation point x_o to be expressed in terms of the ``boundary values'' on any closed surface $\mathcal {S}$ surrounding x_o.

The integral theorem of (4.5) can readily be used to study diffraction effects occurring at a plane object. Therefore the integration surface $\mathcal {S}$ is segmented into three disjoint parts, i.e., $\mathcal {S}$ = $\mathcal {A}$ $\cup$ $\mathcal {B}$ $\cup$ $\mathcal {C}$. The boundary $\mathcal {A}$ is chosen across the object location in the screen plane, $\mathcal {B}$ is the opaque part of the screen, and $\mathcal {C}$ is a half-sphere containing the observation point x_o. A graphical illustration of this specific choice is given in Figure 4.1. The solution of the diffraction problem is found by specifying Kirchhoff boundary conditions on $\mathcal {A}$ and $\mathcal {B}$, and the Sommerfeld radiation condition on $\mathcal {C}$:

 

$$\begin{aligned}{\text{on } \mathcal{A}:} &\qquad\quad U(\mathbf{x... ...ac{\partial U(\mathbf{x})}{\partial n}-jkU(\mathbf{x})\right) = 0,\end{aligned}$$

whereby U_s(x) is the field incident on the screen. The Kirchhoff boundary conditions are an idealization to the real field distribution on the screen by assuming that the field and its derivative across $\mathcal {A}$ is exactly the same as they would be in the absence of the screen. In the geometrical shadow across $\mathcal {B}$ the field is simply set to zero. Although these assumptions are reasonable, they fail in the immediate neighborhood of the rim of the opening. Thus, the range of validity is related to the wavelength $\lambda$ and the geometrical dimension w of the opening (cf. Figure 4.1). The Sommerfeld radiation condition on $\mathcal {C}$ is satisfied, if the disturbance U(x) vanishes at least as fast as a spherical wave. Since the illuminating light U_s(x) invariably consists of a linear combination of spherical waves, this requirement is always fulfilled.

  

Figure 4.1: Range of validity of scalar diffraction theories. Depending on the distance z_o between observation point x_o and screen, several approximations can be employed to describe diffraction. For aerial image simulation the far-field theory of Fraunhofer valid for z_o $\gg$ w²/$\lambda$ is adequate since in M : 1-reduction projection systems the patterns to be imaged are of size w $\cong$ M$\lambda$ and we thus get z_o $\gg$ M²$\lambda$ $\cong$ 0.1 mm for M = 10 and $\lambda$ $\cong$ 400 nm.

Inserting the boundary conditions of (4.6) into the integral theorem (4.5) shows that only the illuminating light U_s(x) across the opening $\mathcal {A}$ contributes to the integral, i.e.,

 

$$U(\mathbf{x}_o) = -\frac{1}{4\pi} \iint\limits_\mathcal{A}\left[ ... ...hbf{x}) - U_s(\mathbf{x}) \frac{\partial G(\mathbf{x})}{\partial n} \right] ds.$$ (4.6)

We further assume that the aperture is illuminated with a single expanding spherical wave arising from a point source located in x_s, a distance r_s away from the screen (cf. Figure 4.1),

 

$$U_s(\mathbf{x}) = A_s\frac{e^{jk_0r_s}}{r_s}{}\quad\qquad{\text{with}}\quad\qquad{} r_s=\Vert\mathbf{x}_s-\mathbf{x}\Vert.$$ (4.7)

Hence, both Green's function in (4.4) and the incident disturbance in (4.8) have the shape of spherical waves. A further simplification is obtained by noting that the distances from the screen to the observation point and the location of the point source, r_o and r_s, respectively, are many optical wavelengths. Therefore the following approximations hold for $\lambda$ $\ll$ r_o, r_s:^d

$$\frac{\partial U_s(\mathbf{x})}{\partial n} \cong jk_0A_s\cos(\ma... ... n} \cong jk_0 \cos(\mathbf{n},\mathbf{x}_o-\mathbf{x})\frac{e^{jk_0r_o}}{r_o}.$$ (4.8)

Insertion of these approximations into (4.7) yields together with (4.4) and (4.8) the Fresnel-Kirchhoff diffraction formula

 

$$U(\mathbf{x}_o) = \frac{jA_s}{\lambda} \iint\limits_\mathcal{A}\f... ...bf{x}_o-\mathbf{x}) - \cos(\mathbf{n},\mathbf{x}_s-\mathbf{x})}{2} \right]\,ds.$$ (4.9)

The term in square brackets is called obliquity or inclination factor and was first introduced by Fresnel based on heuristic arguments, when he combined Huygens' envelope construction with Young's principle of interference. The mathematical sound derivation described above was later developed by Kirchhoff. An interesting property of the diffraction formula (4.10) is its symmetry with respect to the source and observation point, a result that is sometimes referred to as the reciprocity theorem of Helmholtz.

For our purposes we rewrite (4.10) by introducing a general linear relationship between the exciting field U_s(x, y) at the screen and the resulting diffraction pattern U(x_o, y_o)^e

 

$$U(x_o,y_o) = \iint\limits_{x,y} h(x_o,y_o;x,y) U_s(x,y) \, dxdy.$$ (4.10)

Note that we are always interested in field values lying in certain planes parallel to the screen. Therefore, only the dependence on the lateral x- and y-coordinate is explicitly written, whereas the vertical position is indicated by subscripts. The integration kernel h(x, y;x_o, y_o) of the superposition integral in (4.11) is given by (cf. (4.8) and (4.10))

 

$$h(x_o,y_o;x,y) = \frac{j}{\lambda} \left[\frac{\cos(\mathbf{n},\m... ... - \cos(\mathbf{n},\mathbf{x}_s-\mathbf{x})}{2}\right] \frac{e^{jk_0r_o}}{r_o}.$$ (4.11)

Depending on the distance z_o between the observation point x_o and the screen, the kernel h(x_o, y_o;x, y) can be simplified. The denominator r_o is approximated by z_o provided that z_o is much larger than the lateral extent w of the object. For the same reason and by assuming that the source as well as the observation point are located near the optical axis, the inclination factor is approximately given by cos$\delta$ (cf. Appendix A). As shown in Figure 4.1 the angle $\delta$ lies between the line from source to observation point and the normal vector n. The approximation of the exponential factor is more involved, since r_o is multiplied by the very large wavenumber k₀. Hence, a truncation of Taylor series expansions is employed,

$$\begin{aligned}k_0 r_o &= k_0 \sqrt{ z_o^2 + (x-x_o)^2 + (y-y_o)^... ...rac{xx_o+yy_o}{z_o^2} + \frac{x^2+y^2}{2z_o^2} \mp \ldots \right).\end{aligned}$$

• Fresnel approximations are obtained by truncating the Taylor series after the first term, i.e.,
   

  $$h(x_o,y_o;x,y) \cong j\cos\delta\,\frac{e^{jk_0z_o}}{\lambda z_o} e^{j\frac{k_0}{2z_o}\left((x-x_o)^2+(y-y_o)^2\right)}.$$ (4.12)

  The resulting integral relation (4.11) involves Fresnel's integrals.^f Due to the square dependence on the integration variables (x, y), no analytical solution exists. Tabulated values can be found in, e.g., [115, ch. 7]. The approximation (4.14) holds for (2z_o)³ $\gg$ k₀[(x - x_o)² + (y - y_o)²]²_max or simply z_o $\gg$ w.
• Fraunhofer approximations additionally neglect the square dependence on the integration variables (x, y) resulting in
   

  $$h(x_o,y_o;x,y) \cong j\cos\delta\,\frac{e^{jk_0z_o}}{\lambda z_o} e^{j\frac{k_0}{2z_o}\left(x_o^2+y_o^2\right)} e^{j\frac{k_0}{z_o}(xx_o+yy_o)}.$$ (4.13)

  The integral relation (4.11) corresponds now to a Fourier transform, i.e., the dependence on the integration variables is a linear one. Efficient numerical techniques like FFT can be used for evaluation. The range of validity of (4.15) is given by 2z_o $\gg$ k(x² + y²)_max or z $\gg$ w²/$\lambda$.

Footnotes

... ^a

The symbol $\triangle$ denotes the Laplace operator, i.e., $\triangle$ = ${\frac{\partial^2}{\partial x^2}}$ + ${\frac{\partial^2}{\partial y^2}}$ + ${\frac{\partial^2}{\partial z^2}}$.

... by^b

The quantity c₀ is the vacuum light velocity and has a value of c₀ $\approx$ 2.998  10⁸ m/s.

... theorem.^c

Green's theorem states that if a complex-valued function U(x) is continuous and has continuous single-valued first and second partial derivatives inside a volume $\mathcal {V}$ surrounded by a closed surface $\mathcal {S}$, the following relation holds: $$\iiint\limits_{\mathcal{V}} [G(\mathbf{x})\triangle U(\mathb... ...mathbf{x})\frac{\partial G(\mathbf{x})}{\partial n} \right]ds,$$ whereby the so-called Green's function  G(x) is any other function with the same continuity requirements as U(x) and ${\frac{\partial}{\partial n}}$ = n ^. grad signifies a partial derivative in the inward normal direction n at each point on $\mathcal {S}$.

...:^d

For $\lambda$ $\ll$ r and thus 1/r $\ll$ k₀ we get ${\frac{\partial}{\partial n}}$$\left(\vphantom{ {e^{jk_0r}}/{r} }\right.$e^jk₀r/r$\left.\vphantom{ {e^{jk_0r}}/{r} }\right)$ = cos(n, x)$\left(\vphantom{jk_0-{1}/{r}}\right.$jk₀ - 1/r$\left.\vphantom{jk_0-{1}/{r}}\right)$e^jk₀r/r $\cong$ jk₀cos(n, x)e^jk₀r/r.

...)^e

All integrations and sums go from - $\infty$ to $\infty$ unless specified otherwise.

... integrals.^f

Integrals of the type C(w) = $\int_{0}^{w}$cos($\pi$$\tau^{2}_{}$/2) d$\tau$ and S(w) = $\int_{0}^{w}$sin($\pi$$\tau^{2}_{}$/2) d$\tau$ are referred to as Fresnel's integrals [115, ch. 7].