a | Lateral x-period of photomask
and simulation domain or rate parameter of Mack's
`a'-model. |

a, a_{k} | Constant relating concentration to absorption coefficient. |

a^{pq} | Excitation vector at air/photoresist interface. |

A | Bleachable absorption coefficient of Dill's `ABC'-model. |

A_{FB},
A_{NB} | Bleachable absorption coefficient after full and no prebake. |

A_{ac} | Pattern location factor in Zernike expansion of aberration function. |

A_{k}(x, z) | Slowly varying electric field amplitude in beam propagation model. |

A_{pq}, A_{s} | Amplitude of a spherical wave emerging from a source point. |

A_{0} | Normalizing constant in aerial image calculation. |

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. |

B_{FB},
B_{NB} | Non-bleachable absorption coefficient after full/no prebake. |

_{N} | Numerical factor of truncation error bound. |

(x;t) | Magnetic induction vector. |

, | Boundary matrices at upper and lower interface. |

c | Light velocity. |

c_{0} | Vacuum light velocity. |

C | Exposure rate of Dill's `ABC'-model. |

Set of complex numbers. | |

C(x;t) | Concentration of absorbing photoresist species. |

(z) | Relation matrix of a homogeneous planar layer. |

d_{i} | Image-to-lens distance in projection printing. |

d_{o} | Object-to-lens distance in projection printing. |

d_{g} | Mask-to-wafer distance in proximity printing. |

D | Actual photoresist thickness. |

D_{eff} | Effective photoresist thickness. |

D_{h} | Post-exposure bake diffusivity of generated acid. |

D_{h, 0}, D_{h, 1} | Parameters for the post-exposure bake diffusivity of generated acid. |

D_{pre} | Prebake diffusivity of photoactive compound. |

D_{0}, D_{100} | Characteristic energy densities of photoresist contrast curve. |

DOF | Depth of focus. |

(x;t) | Electric displacement vector. |

^{2} | Set of considered Fourier coefficients. |

(z) | Diagonal oscillation factor of the relation matrix of a homogeneous planar layer. |

e_{i} | Unit vector. |

e_{l} | Vector comprising lateral electric plane wave amplitudes traveling upwards and downwards a homogeneous planar layer. |

e_{x}(z),
e_{y}(z) | Vectors comprising Fourier coefficients of the lateral electric field phasors. |

E(x) | Exposure energy within photoresist. |

E_{w} | Exposure energy generating feature of width w. |

E_{0} | Exposure energy for complete photoresist clearance. |

E_{1}, E_{2}, E_{3} | Rate parameter of Dill's `E'-model. |

E_{pre} | Activation energy for prebake rate constant. |

E_{y, k}(x, z) | Electric field amplitude in beam propagation model and truncation error analysis. |

EL | Exposure latitude. |

(x;t) | Electric field vector. |

E(x) | Complex-valued phasor of time-harmonic electric field. |

(x) | Scaled electric field phasor. |

E_{i}^{pq}(x, y) | Electric field phasor of aerial image due to one source point. |

E_{i, nm}^{pq} | Plane wave amplitudes of incident electric field phasor due to one source point. |

E_{i, 0, nm}^{pq} | Plane wave amplitudes of incident electric field phasor due to one source point in case of blank photomask. |

E_{k}(x) | Electric field phasor at a time-step. |

E_{l}^{+},
E_{l}^{-} | Electric plane wave amplitudes traveling upwards and downwards in a homogeneous planar layer. |

E_{l}(x) | Electric field phasor in a homogeneous planar layer. |

E_{nm}(z) | Fourier coefficient of electric field phasor. |

E_{r, nm}^{pq} | Plane wave amplitudes of reflected electric field phasor due to one source point. |

E_{s, nm}^{pq} | Plane wave amplitudes of outgoing electric field phasor due to one source point. |

E_{l, nm}^{+},
E_{l, nm}^{-} | Plane wave amplitudes of propagating downwards and upwards a homogeneous planar layer. |

f | Focal length of the projection lens. |

f_{x}, f_{y} | 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. |

G_{0} | Initial photoacid generator. |

G(x) | Green's function. |

G(x;t) | Photoacid generator concentration. |

_{xs,} | Geometrical factor of truncation error bound. |

(z) | System sub-matrices of the ordinary differential equation system. |

h(x;t) | Generated acid concentration normalized by initial concentration of photoacid generator. |

h_{exp}(x) | Normalized generated acid concentration of photoacid generator after exposure. |

h_{air}(x) | Normalized generated acid concentration in ambient air. |

h_{exp}(x) | Normalized generated acid concentration after exposure. |

h(x, y;x^{}, y^{}) | Impulse response of the projection lens. |

(x, y) | Shift-invariant impulse response of the projection lens. |

h_{x}(z),
h_{y}(z) | Vectors comprising Fourier coefficients of the lateral magnetic field phasor. |

H(x;t) | Generated acid concentration. |

(x;t) | Magnetic field vector. |

H(x) | Complex-valued phasor of time-harmonic magnetic field phasor. |

(x) | Scaled magnetic field phasor. |

(z) | System matrix of ordinary differential equation system. |

H_{i}^{pq}(x, y) | Magnetic field phasor of aerial image due to one source point. |

H_{i, nm}^{pq} | Plane wave amplitudes of incident magnetic field phasor due to one source point. |

H_{l}^{+},
H_{l}^{-} | Magnetic plane wave amplitudes traveling upwards and downwards in a homogeneous planar layer. |

H_{nm}(z) | Fourier coefficient of magnetic field phasor. |

i,
i_{nm} | Normalized wavevectors in image space. |

_{nm} | Wavevector in image space scaled by numerical aperture. |

I(z;t) | Intensity of a plane wave traveling in vertical direction. |

I_{i}(x, y) | Aerial image intensity. |

I_{i}^{pq}(x, y) | Aerial image intensity due to one source point. |

I_{i, 0}^{pq}(x, y) | Aerial image intensity due to one source point in case of a blank photomask. |

I_{r}(z) | Relative intensity variation within photoresist. |

I_{0} | Exposure intensity of aerial image intensity in case of a blank photomask. |

I_{}(,) | Generating function for Fourier coefficients of rectangular-shaped mask patterns. |

I_{}(,) | Generating function for Fourier coefficients of triangular-shaped mask patterns. |

Identity matrix. | |

k | Wavenumber. |

k_{0} | Vacuum wavenumber. |

k, k_{1}, k_{2} | Characteristic parameters of lithography process. |

k_{evap} | Rate constant for evaporation of generated acid into ambient. |

k_{enh} | Rate constant of enhancement reaction in Mack's `a'-model. |

k_{inh} | Rate constant of inhibition reaction in Mack's `a'-model. |

k_{peb, 1},
k_{peb, 2} | Post-exposure bake rate constants. |

k_{pre}(T) | Prebake rate constant. |

k | Wavevector within resist. |

k_{l}^{+},
k_{l}^{-} | Wavevectors traveling upwards and downwards in a homogeneous planar layer. |

K | Rate constant for chemically amplified photoresist. |

K_{pre} | Arrhenius coefficient for prebake rate constant. |

, | Boundary sub-matrices. |

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. |

L_{nm}(z;t_{k}) | Defocus factor in the transfer matrix method. |

Propagation matrix of a stratified medium. | |

(h) | Propagation matrix of a homogeneous planar layer. |

m_{x}, m_{y} | 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. |

m_{exp}(x) | Photoactive compound after exposure. |

m_{p}(x) | Fundamental solution after diffusion from a unit impulse source. |

m_{th} | Threshold concentration of photoactive compound in Mack's `a'-model. |

M | Magnification of the projection printing system. |

M_{0} | 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. |

N_{ODE} | Rank of ordinary differential equation system. |

N_{p} | Number of shooting points. |

N_{TRI} | Number of triangles. |

N_{x}, N_{y} | Number of considered Fourier coefficients of electromagnetic field. |

N_{x}^{,},
N_{y}^{,} | Number of considered Fourier coefficients of permittivity and its reciprocal. |

NA | Numerical aperture of a lens. |

NA_{c} | Numerical aperture of the condenser lens. |

NA_{p} | Numerical aperture of the projection lens. |

NILS | Normalize image log-slope. |

o,
o_{nm} | Normalized wavevectors in object space. |

p_{ij}^{nm} | Elements of the characteristic matrix of a stratified medium. |

P(x;t) | Exposure product concentration. |

PAT_{k}(x, y) | Indicator function of a mask pattern. |

_{k, nm} | Fourier coefficients of a mask pattern. |

P(x, y) | Discrete pupil function of the projection lens. |

(x, y) | Physical pupil function of the projection lens. |

P(n, m : p, q) | Vector-valued pupil function of the projection lens. |

Characteristic matrix of a stratified medium. | |

Orthogonal factor of the lower boundary matrix. | |

, | 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. |

r_{min},
r_{min} | Minimal and maximal development rate of photoresist. |

r_{o} | Spatial distance to observation point. |

r_{o}, | Polar coordinates of pattern relative to optical axis. |

r_{resin} | Development rate of photoresist resin. |

r_{s} | Spatial distance to point source. |

r_{x}, r_{y} | Powers of the number of considered Fourier coefficients. |

r_{0} | Development rate for exposure for complete photoresist clearance. |

r_{1}, r_{2} | Curvature radii of a lens. |

R | Resin or universal gas constant. |

R_{i} | Rate parameter of Kim's `R'-model. |

RECT(x, y) | Indicator function of a rectangular-shaped mask pattern. |

Set of real numbers. | |

_{b}^{c}() | Radial polynomials. |

_{nm} | Fourier coefficients of a rectangular-shaped mask pattern. |

(z) | System sub-matrices of the ordinary differential equation system. |

Upper triangular factor of the lower boundary matrix. | |

, | Sub-matrices of the upper triangular factor of the lower boundary matrix. |

s,
s_{nm} | Normalized wavevectors in source space. |

s,
s^{pq} | Shooting parameters due to one source point. |

, | Reduced shooting parameters due to one source point. |

S | Solvents. |

(x;t) | Electric current density. |

SD | Simulation domain. |

Shooting parameter matrix comprising all source points. | |

Reduced shooting parameter matrix comprising all source points contributions. | |

Sub-matrix of the orientation factor of the relation matrix of a homogeneous planar layer. | |

t_{CPU} | Run-time of simulation. |

t_{exp} | Exposure time. |

t_{k} | Transmission of a mask pattern. |

t_{k} | Time-step. |

t_{peb} | Post-exposure time. |

t_{pre} | Prebake time. |

t(x, y) | Mask transmission function. |

T | Temperature. |

T_{nm} | Fourier coefficients of mask transmission function. |

_{nm} | Fourier coefficients of a triangular-shaped mask pattern. |

TCC(p^{}, q^{} : p^{}, q^{}) | 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. |

u_{l} | Vector comprising lateral Fourier coefficients electromagnetic field in a homogeneous planar layer. |

u^{pq}(z) | Solution vector due to one source point. |

U(x) | Scalar complex-valued phasor of time-harmonic electromagnetic field disturbance. |

U_{l}^{1}(x, y), U_{l}^{2}(x, y) | Scalar field phasor at entrance/exit pupil of the projection lens. |

(x;t) | Scalar electromagnetic field disturbance. |

U_{i}(x) | Scalar image field phasor. |

U_{i}^{pq}(x) | Scalar image field phasor due to one source point. |

U_{i, nm}^{pq} | Plane wave amplitudes of image due to one source point. |

U_{i, 0, nm}^{pq} | Plane wave amplitudes of image due to one source point in case of a blank photomask. |

U_{o}(x) | Scalar field phasor at an observation point. |

U_{o}^{pq}(x),
U_{o}^{s}(x) | Scalar field phasor at an observation point due to one source point. |

U_{r}(x;t_{k}) | Scalar field phasor in photoresist at a time-step in the transfer matrix method. |

U_{r, nm}(z;t_{k}) | Plane wave amplitudes in photoresist at a time-step in the transfer matrix method. |

U_{s}(x) | Scalar field phasor at the screen or the object. |

(x;t) | Scalar electromagnetic field disturbance. |

U(x) | Complex-valued phasor of time-harmonic electromagnetic field. |

(z) | Solution matrix comprising all source point solution vectors. |

v_{n}(x;t) | Speed along surface normal in level set method. |

v_{i}(z) | Fundamental solution vector of the ordinary differential equation system. |

(z) | Linear independent solution vector of the ordinary differential equation system. |

V_{nm} | Fourier spectrum of aerial image. |

V(x, y) | Spatial function of aerial image. |

(z) | Fundamental solution matrix of ordinary differential equation system. |

(z) | Linear independent solution matrix of ordinary differential equation system. |

w | Feature width. |

w_{h} | Parameter for the post-exposure bake diffusivity of generated acid. |

w_{pq} | Discretization area of aperture inside illumination cone. |

W | Lithographic resolution. |

x_{o} | Observation point. |

x_{p} | Integration point within the photoresist. |

x_{s} | Source point location. |

X | Light insensitive prebake product. |

z_{p} | Shooting point. |

z_{0} | Defocus in air in scaled defocus model. |

_{b}^{c}(,) | Zernike polynomials. |

Zero matrix. | |

Collection angle of a lens. | |

, | Characteristic depth of focus parameters of photoresist. |

, , | Auxiliary angles in approximation of inclination factor. |

, , | Transformed spatial frequencies in analytical calculation of pattern Fourier coefficients. |

(z;t) | Absorption coefficient of a dilute solution. |

(x;t) | Absorption coefficient of the photoactive compound. |

(x;t) | Absorption coefficient of the exposure product. |

Absorption coefficient of the resin. | |

Absorption coefficient of the solvents. | |

Absorption coefficient of the light-insensitive prebake products. | |

Focus levels in case of in-lens filtering. | |

(x) | Reciprocal relative permittivity. |

(x) | Scaled reciprocal relative permittivity. |

(z) | Fourier coefficients of reciprocal relative permittivity. |

(z) | Fourier coefficients of scaled reciprocal relative permittivity. |

Angle of the inclination factor or phase difference in case of in-lens filtering. | |

Penetration depth of vertical solution vector. | |

(x, y) | Thickness function of a lens. |

Coefficients of power series expansion of aberration function. | |

Wafer positioning error resulting in defocus. | |

Thickness of a lens on optical axis. | |

z | Net defocus in scaled defocus model. |

(x) | Absolute permittivity. |

, , | Numerical errors. |

Truncation error. | |

Vacuum permittivity. | |

(x) | Relative permittivity. |

(x) | Scaled relative permittivity. |

(z) | Fourier coefficients of relative permittivity. |

(z) | Fourier coefficients of scaled relative permittivity. |

Free-space resistance. | |

Resist contrast. | |

Lumped amplification constant. | |

(x;t) | Light sensitive photoresist compound. |

(t) | Propagating surface in level set method. |

Extinction coefficient. | |

Actinic wavelength. | |

Oscillation period of vertical solution vector. | |

Vacuum permeability. | |

Refractive index. | |

Angular frequency. | |

, | Polarization angles. |

(x, y) | Phase function of a lens. |

(x;t) | Level set function. |

(x;t) | Stationary level set function. |

(m, m) | Aberrations function. |

(n, m : p, q) | Component of polarization vector parallel to meridional plane. |

(n, m : p, q) | Component of polarization vector perpendicular to meridional plane. |

(n, m : p, q) | Polarization vector. |

_{TE}(n, m : p, q) | Transversal-electric polarization vector. |

_{TM}(n, m : p, q) | Transversal-magnetic polarization vector. |

_{ || }(n, m) | Normalized vector parallel to meridional plane. |

_{}(n, m) | Normalized vector perpendicular to meridional plane. |

, | Polar coordinates of wavevector in image space. |

Partial coherence factor. | |

Prebake diffusion length. | |

(x) | Specific conductivity. |

1998-04-17