List of Symbols

a	Lateral x-period of photomask and simulation domain or rate parameter of Mack's `a'-model.
a, ak	Constant relating concentration to absorption coefficient.
apq	Excitation vector at air/photoresist interface.
A	Bleachable absorption coefficient of Dill's `ABC'-model.
AFB, ANB	Bleachable absorption coefficient after full and no prebake.
Aac	Pattern location factor in Zernike expansion of aberration function.
Ak(x, z)	Slowly varying electric field amplitude in beam propagation model.
Apq, As	Amplitude of a spherical wave emerging from a source point.
A0	Normalizing constant in aerial image calculation.
$\underline{\mathbf{A}}$	Excitation matrix comprising all source point contributions or transformation matrix in analytical calculation of pattern Fourier coefficients.
b	Lateral y-period of photomask and simulation domain.
B	Non-bleachable absorption coefficient of Dill's `ABC'-model.
BFB, BNB	Non-bleachable absorption coefficient after full/no prebake.
$\mathcal {B}$N	Numerical factor of truncation error bound.
$\mathcal {B}$(x;t)	Magnetic induction vector.
$\underline{\mathbf{B}}_{0}^{}$, $\underline{\mathbf{B}}_{h}^{}$	Boundary matrices at upper and lower interface.
c	Light velocity.
c0	Vacuum light velocity.
C	Exposure rate of Dill's `ABC'-model.
$\mathbb{C}$	Set of complex numbers.
C(x;t)	Concentration of absorbing photoresist species.
$\underline{\mathbf{C}}_{l}^{}$(z)	Relation matrix of a homogeneous planar layer.
di	Image-to-lens distance in projection printing.
do	Object-to-lens distance in projection printing.
dg	Mask-to-wafer distance in proximity printing.
D	Actual photoresist thickness.
Deff	Effective photoresist thickness.
Dh	Post-exposure bake diffusivity of generated acid.
Dh, 0, Dh, 1	Parameters for the post-exposure bake diffusivity of generated acid.
Dpre	Prebake diffusivity of photoactive compound.
D0, D100	Characteristic energy densities of photoresist contrast curve.
DOF	Depth of focus.
$\mathcal {D}$(x;t)	Electric displacement vector.
$\mathbb{D}$2	Set of considered Fourier coefficients.
$\underline{\mathbf{D}}_{l}^{}$(z)	Diagonal oscillation factor of the relation matrix of a homogeneous planar layer.
ei	Unit vector.
el	Vector comprising lateral electric plane wave amplitudes traveling upwards and downwards a homogeneous planar layer.
ex(z), ey(z)	Vectors comprising Fourier coefficients of the lateral electric field phasors.
E(x)	Exposure energy within photoresist.
Ew	Exposure energy generating feature of width w.
E0	Exposure energy for complete photoresist clearance.
E1, E2, E3	Rate parameter of Dill's `E'-model.
Epre	Activation energy for prebake rate constant.
Ey, k(x, z)	Electric field amplitude in beam propagation model and truncation error analysis.
EL	Exposure latitude.
$\mathcal {E}$(x;t)	Electric field vector.
E(x)	Complex-valued phasor of time-harmonic electric field.
$\widetilde{\mathbf{E}}$(x)	Scaled electric field phasor.
Eipq(x, y)	Electric field phasor of aerial image due to one source point.
Ei, nmpq	Plane wave amplitudes of incident electric field phasor due to one source point.
Ei, 0, nmpq	Plane wave amplitudes of incident electric field phasor due to one source point in case of blank photomask.
Ek(x)	Electric field phasor at a time-step.
El+, El-	Electric plane wave amplitudes traveling upwards and downwards in a homogeneous planar layer.
El(x)	Electric field phasor in a homogeneous planar layer.
Enm(z)	Fourier coefficient of electric field phasor.
Er, nmpq	Plane wave amplitudes of reflected electric field phasor due to one source point.
Es, nmpq	Plane wave amplitudes of outgoing electric field phasor due to one source point.
El, nm+, El, nm-	Plane wave amplitudes of propagating downwards and upwards a homogeneous planar layer.
f	Focal length of the projection lens.
fx, fy	Spatial frequencies.
f (z, m)	Induction multiplication function in Kim's `R'-model.
F(n, m)	In-lens filter function.
g(x;t)	Photoacid generator normalized by its initial concentration.
g(x| y)(x| y)nm, pq(z)	Coefficients of the ordinary differential equation system.
G0	Initial photoacid generator.
G(x)	Green's function.
G(x;t)	Photoacid generator concentration.
$\mathcal {G}$xs,$\Delta$$\varepsilon$	Geometrical factor of truncation error bound.
$\underline{\mathbf{G}}^{(x\vert y)(x\vert y)}_{}$(z)	System sub-matrices of the ordinary differential equation system.
h(x;t)	Generated acid concentration normalized by initial concentration of photoacid generator.
hexp(x)	Normalized generated acid concentration of photoacid generator after exposure.
hair(x)	Normalized generated acid concentration in ambient air.
hexp(x)	Normalized generated acid concentration after exposure.
h(x, y;x$\prime$, y$\prime$)	Impulse response of the projection lens.
$\tilde{h}$(x, y)	Shift-invariant impulse response of the projection lens.
hx(z), hy(z)	Vectors comprising Fourier coefficients of the lateral magnetic field phasor.
H(x;t)	Generated acid concentration.
$\mathcal {H}$(x;t)	Magnetic field vector.
H(x)	Complex-valued phasor of time-harmonic magnetic field phasor.
$\widetilde{\mathbf{H}}$(x)	Scaled magnetic field phasor.
$\underline{\mathbf{H}}$(z)	System matrix of ordinary differential equation system.
Hipq(x, y)	Magnetic field phasor of aerial image due to one source point.
Hi, nmpq	Plane wave amplitudes of incident magnetic field phasor due to one source point.
Hl+, Hl-	Magnetic plane wave amplitudes traveling upwards and downwards in a homogeneous planar layer.
Hnm(z)	Fourier coefficient of magnetic field phasor.
i, inm	Normalized wavevectors in image space.
$\dot{\boldsymbol\iota}$nm	Wavevector in image space scaled by numerical aperture.
I(z;t)	Intensity of a plane wave traveling in vertical direction.
Ii(x, y)	Aerial image intensity.
Iipq(x, y)	Aerial image intensity due to one source point.
Ii, 0pq(x, y)	Aerial image intensity due to one source point in case of a blank photomask.
Ir(z)	Relative intensity variation within photoresist.
I0	Exposure intensity of aerial image intensity in case of a blank photomask.
I$\mathcal {R}$($\alpha_{nm}^{}$,$\beta_{nm}^{}$)	Generating function for Fourier coefficients of rectangular-shaped mask patterns.
I$\mathcal {T}$($\alpha_{nm}^{}$,$\beta_{nm}^{}$)	Generating function for Fourier coefficients of triangular-shaped mask patterns.
$\underline{\mathbf{I}}$	Identity matrix.
k	Wavenumber.
k0	Vacuum wavenumber.
k, k1, k2	Characteristic parameters of lithography process.
kevap	Rate constant for evaporation of generated acid into ambient.
kenh	Rate constant of enhancement reaction in Mack's `a'-model.
kinh	Rate constant of inhibition reaction in Mack's `a'-model.
kpeb, 1, kpeb, 2	Post-exposure bake rate constants.
kpre(T)	Prebake rate constant.
k	Wavevector within resist.
kl+, kl-	Wavevectors traveling upwards and downwards in a homogeneous planar layer.
K	Rate constant for chemically amplified photoresist.
Kpre	Arrhenius coefficient for prebake rate constant.
$\underline{\mathbf{K}}$, $\underline{\mathbf{K}}^{0}_{}$	Boundary sub-matrices.
$\underline{\mathbf{K}}_{l}^{}$	Orientation factor of the relation matrix of a homogeneous planar layer.
l	Selectivity parameter in Mack's `a'-model.
l (x, y)	Phase transfer function of a lens.
Lnm(z;tk)	Defocus factor in the transfer matrix method.
$\underline{\mathbf{L}}_{l}^{}$	Propagation matrix of a stratified medium.
$\underline{\mathbf{L}}_{l}^{}$(h)	Propagation matrix of a homogeneous planar layer.
mx, my	Stratified medium factors for boundary conditions at photoresist/substrate interface.
m, m(x;t)	Photoactive compound normalized by its initial concentration in case of a conventional photoresist or reactive sites normalized by initial concentration of the photoacid generator in case of chemically amplified resists.
mexp(x)	Photoactive compound after exposure.
mp(x)	Fundamental solution after diffusion from a unit impulse source.
mth	Threshold concentration of photoactive compound in Mack's `a'-model.
M	Magnification of the projection printing system.
M0	Initial photoactive compound.
M(x;t)	Photoactive compound or reactive sites.
n	Reaction order for post-exposure bake or sensitivity parameter in Mack's `a'-model.
n(x;t)	Complex-valued refractive index.
n	Normal vector at screen.
NODE	Rank of ordinary differential equation system.
Np	Number of shooting points.
NTRI	Number of triangles.
Nx, Ny	Number of considered Fourier coefficients of electromagnetic field.
Nx$\tilde{\varepsilon}$,$\tilde{\chi}$, Ny$\tilde{\varepsilon}$,$\tilde{\chi}$	Number of considered Fourier coefficients of permittivity and its reciprocal.
NA	Numerical aperture of a lens.
NAc	Numerical aperture of the condenser lens.
NAp	Numerical aperture of the projection lens.
NILS	Normalize image log-slope.
o, onm	Normalized wavevectors in object space.
pijnm	Elements of the characteristic matrix of a stratified medium.
P(x;t)	Exposure product concentration.
PATk(x, y)	Indicator function of a mask pattern.
$\mathcal {P}$k, nm	Fourier coefficients of a mask pattern.
P(x, y)	Discrete pupil function of the projection lens.
$\mathcal {P}$(x, y)	Physical pupil function of the projection lens.
P(n, m : p, q)	Vector-valued pupil function of the projection lens.
$\underline{\mathbf{P}}^{nm}_{}$	Characteristic matrix of a stratified medium.
$\underline{\raisebox{0pt}[1ex][0pt]{$\mathbf{Q}$}}_{h}^{}$	Orthogonal factor of the lower boundary matrix.
$\underline{\raisebox{0pt}[1ex][0pt]{$\mathbf{Q}$}}_{h}^{1}$, $\underline{\raisebox{0pt}[1ex][0pt]{$\mathbf{Q}$}}_{h}^{2}$	Sub-matrices of the orthogonal factor of the lower boundary matrix.
r(m)	Development rate of the photoresist as function of the photoactive compound.
r(x, y, z)	Development rate of the photoresist.
r(x| y)(x| y)nm, pq(z)	Coefficients of the ordinary differential equation system.
rmin, rmin	Minimal and maximal development rate of photoresist.
ro	Spatial distance to observation point.
ro, $\phi_{o}^{}$	Polar coordinates of pattern relative to optical axis.
rresin	Development rate of photoresist resin.
rs	Spatial distance to point source.
rx, ry	Powers of the number of considered Fourier coefficients.
r0	Development rate for exposure for complete photoresist clearance.
r1, r2	Curvature radii of a lens.
R	Resin or universal gas constant.
Ri	Rate parameter of Kim's `R'-model.
RECT(x, y)	Indicator function of a rectangular-shaped mask pattern.
$\mathbb{R}$	Set of real numbers.
$\mathcal {R}$bc($\rho$)	Radial polynomials.
$\mathcal {R}$nm	Fourier coefficients of a rectangular-shaped mask pattern.
$\underline{\mathbf{R}}^{(x\vert y)(x\vert y)}_{}$(z)	System sub-matrices of the ordinary differential equation system.
$\underline{\mathbf{R}}_{h}^{}$	Upper triangular factor of the lower boundary matrix.
$\underline{\mathbf{R}}_{h}^{1}$, $\underline{\mathbf{R}}_{h}^{2}$	Sub-matrices of the upper triangular factor of the lower boundary matrix.
s, snm	Normalized wavevectors in source space.
s, spq	Shooting parameters due to one source point.
$\hat{\mathbf{s}}$, $\hat{\mathbf{s}}^{pq}_{}$	Reduced shooting parameters due to one source point.
S	Solvents.
$\mathcal {S}$(x;t)	Electric current density.
SD	Simulation domain.
$\underline{\mathbf{S}}$	Shooting parameter matrix comprising all source points.
$\widehat{\underline{\mathbf{S}}}$	Reduced shooting parameter matrix comprising all source points contributions.
$\underline{\mathbf{S}}_{l}^{}$	Sub-matrix of the orientation factor of the relation matrix of a homogeneous planar layer.
tCPU	Run-time of simulation.
texp	Exposure time.
tk	Transmission of a mask pattern.
tk	Time-step.
tpeb	Post-exposure time.
tpre	Prebake time.
t(x, y)	Mask transmission function.
T	Temperature.
Tnm	Fourier coefficients of mask transmission function.
$\mathcal {T}$nm	Fourier coefficients of a triangular-shaped mask pattern.
TCC(p$\prime$, q$\prime$ : p$\prime{^\prime}$, q$\prime{^\prime}$)	Transmission cross coefficients.
TRI(x, y)	Indicator function of a triangular-shaped mask pattern.
u(z)	Solution vector comprising all Fourier coefficients of the electromagnetic field components.
ul	Vector comprising lateral Fourier coefficients electromagnetic field in a homogeneous planar layer.
upq(z)	Solution vector due to one source point.
U(x)	Scalar complex-valued phasor of time-harmonic electromagnetic field disturbance.
Ul1(x, y), Ul2(x, y)	Scalar field phasor at entrance/exit pupil of the projection lens.
$\mathcal {U}$(x;t)	Scalar electromagnetic field disturbance.
Ui(x)	Scalar image field phasor.
Uipq(x)	Scalar image field phasor due to one source point.
Ui, nmpq	Plane wave amplitudes of image due to one source point.
Ui, 0, nmpq	Plane wave amplitudes of image due to one source point in case of a blank photomask.
Uo(x)	Scalar field phasor at an observation point.
Uopq(x), Uos(x)	Scalar field phasor at an observation point due to one source point.
Ur(x;tk)	Scalar field phasor in photoresist at a time-step in the transfer matrix method.
Ur, nm(z;tk)	Plane wave amplitudes in photoresist at a time-step in the transfer matrix method.
Us(x)	Scalar field phasor at the screen or the object.
$\mathcal {U}$(x;t)	Scalar electromagnetic field disturbance.
U(x)	Complex-valued phasor of time-harmonic electromagnetic field.
$\underline{\mathbf{U}}$(z)	Solution matrix comprising all source point solution vectors.
vn(x;t)	Speed along surface normal in level set method.
vi(z)	Fundamental solution vector of the ordinary differential equation system.
$\hat{\mathbf{v}}_{i}^{}$(z)	Linear independent solution vector of the ordinary differential equation system.
Vnm	Fourier spectrum of aerial image.
V(x, y)	Spatial function of aerial image.
$\underline{\mathbf{V}}$(z)	Fundamental solution matrix of ordinary differential equation system.
$\widehat{\underline{\mathbf{V}}}$(z)	Linear independent solution matrix of ordinary differential equation system.
w	Feature width.
wh	Parameter for the post-exposure bake diffusivity of generated acid.
wpq	Discretization area of aperture inside illumination cone.
W	Lithographic resolution.
xo	Observation point.
xp	Integration point within the photoresist.
xs	Source point location.
X	Light insensitive prebake product.
zp	Shooting point.
z0	Defocus in air in scaled defocus model.
$\mathcal {Z}$bc($\rho$,$\phi$)	Zernike polynomials.
$\underline{\mathbf{0}}$	Zero matrix.
$\alpha$	Collection angle of a lens.
$\alpha$, $\beta$	Characteristic depth of focus parameters of photoresist.
$\alpha$, $\beta$, $\delta$	Auxiliary angles in approximation of inclination factor.
$\alpha_{nm}^{}$, $\beta_{nm}^{}$, $\gamma_{nm}^{}$	Transformed spatial frequencies in analytical calculation of pattern Fourier coefficients.
$\alpha$(z;t)	Absorption coefficient of a dilute solution.
$\alpha_{M}^{}$(x;t)	Absorption coefficient of the photoactive compound.
$\alpha_{P}^{}$(x;t)	Absorption coefficient of the exposure product.
$\alpha_{R}^{}$	Absorption coefficient of the resin.
$\alpha_{S}^{}$	Absorption coefficient of the solvents.
$\alpha_{X}^{}$	Absorption coefficient of the light-insensitive prebake products.
$\beta$	Focus levels in case of in-lens filtering.
$\chi$(x)	Reciprocal relative permittivity.
$\tilde{\chi}$(x)	Scaled reciprocal relative permittivity.
$\chi_{nm}^{}$(z)	Fourier coefficients of reciprocal relative permittivity.
$\tilde{\chi}_{nm}^{}$(z)	Fourier coefficients of scaled reciprocal relative permittivity.
$\delta$	Angle of the inclination factor or phase difference in case of in-lens filtering.
$\delta_{z,nm}^{}$	Penetration depth of vertical solution vector.
$\Delta$(x, y)	Thickness function of a lens.
$\Delta_{abc}^{}$	Coefficients of power series expansion of aberration function.
$\Delta_{\mathrm{Def}}^{}$	Wafer positioning error resulting in defocus.
$\Delta_{0}^{}$	Thickness of a lens on optical axis.
$\Delta$z	Net defocus in scaled defocus model.
$\epsilon$(x)	Absolute permittivity.
$\varepsilon$, $\varepsilon_{1}^{}$, $\varepsilon_{2}^{}$	Numerical errors.
$\varepsilon_{\mathrm{trunc}}^{}$	Truncation error.
$\varepsilon_{0}^{}$	Vacuum permittivity.
$\varepsilon$(x)	Relative permittivity.
$\tilde{\varepsilon}$(x)	Scaled relative permittivity.
$\varepsilon_{nm}^{}$(z)	Fourier coefficients of relative permittivity.
$\tilde{\varepsilon}_{nm}^{}$(z)	Fourier coefficients of scaled relative permittivity.
$\eta_{0}^{}$	Free-space resistance.
$\gamma$	Resist contrast.
$\gamma_{\mathrm{amp}}^{}$	Lumped amplification constant.
$\gamma$(x;t)	Light sensitive photoresist compound.
$\Gamma$(t)	Propagating surface in level set method.
$\kappa$	Extinction coefficient.
$\lambda$	Actinic wavelength.
$\lambda_{z,nm}^{}$	Oscillation period of vertical solution vector.
$\mu_{0}^{}$	Vacuum permeability.
$\nu$	Refractive index.
$\omega$	Angular frequency.
$\phi$, $\psi$	Polarization angles.
$\phi$(x, y)	Phase function of a lens.
$\phi$(x;t)	Level set function.
$\phi_{s}^{}$(x;t)	Stationary level set function.
$\Phi$(m, m)	Aberrations function.
$\Psi_{\parallel}^{}$(n, m : p, q)	Component of polarization vector parallel to meridional plane.
$\Psi_{\perp}^{}$(n, m : p, q)	Component of polarization vector perpendicular to meridional plane.
$\Psi$(n, m : p, q)	Polarization vector.
$\Psi$TE(n, m : p, q)	Transversal-electric polarization vector.
$\Psi$TM(n, m : p, q)	Transversal-magnetic polarization vector.
$\Psi$ || (n, m)	Normalized vector parallel to meridional plane.
$\Psi$$\perp$(n, m)	Normalized vector perpendicular to meridional plane.
$\rho_{nm}^{}$, $\phi_{nm}^{}$	Polar coordinates of wavevector in image space.
$\sigma$	Partial coherence factor.
$\sigma_{\mathrm{pre}}^{}$	Prebake diffusion length.
$\sigma$(x)	Specific conductivity.