1 · Aperture field
Complex aperture distribution
\[E_a(x,y)=A(x,y)e^{j\Phi(x,y)}\]
Amplitude \(A\) is constrained from 0 to 1. Phase \(\Phi\) is edited from −180° to +180°. Together they define the complex excitation across the aperture.
Reference
Equations and conventions
The calculator uses normalized wavelength units, so the numerical model sets \(\lambda=1\) and \(k=2\pi/\lambda=2\pi\). The aperture lies in the XY plane, +Z is boresight, \(\theta\) is measured from +Z, and \(\phi\) is measured from +X toward +Y.
2 · Coordinates
Direction cosines
\[u=\sin\theta\cos\phi,\qquad v=\sin\theta\sin\phi\]
These variables map a far-field direction onto the aperture's spatial-frequency coordinates. They are used in both the continuous integral and discrete summation.
3 · Continuous 2D aperture
Far-field aperture factor
\[F(\theta,\phi)=\int_0^{L_x}\int_0^{L_y} A(x,y)e^{j\Phi(x,y)}e^{jk(xu+yv)}\,dy\,dx\]
This is the spatial Fourier-transform relationship between the complex aperture field and its far-field pattern. The implementation evaluates this integral numerically on a sampled grid. The same expression reduces to a single integral for the 1D aperture.
4 · Discrete array
Array-factor summation
\[F(\theta,\phi)=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1} A_{m,n}e^{j\Phi_{m,n}}e^{jk(x_m u+y_n v)}\]
For discrete mode, element locations are generated from the requested aperture dimensions and spacing \(d_x/\lambda\) and \(d_y/\lambda\). In 1D only \(d/\lambda\) is needed. This tool models the array factor only; no individual element pattern is multiplied into the result.
5 · Beam steering
Linear steering phase
\[\Phi_{steer}(x,y)=-k\left(xu_0+yv_0\right)\]
Here \(u_0=\sin\theta_0\cos\phi_0\) and \(v_0=\sin\theta_0\sin\phi_0\). With the sign convention used above, this phase ramp places the coherent maximum at the requested steering direction.
6 · Normalization
Normalized pattern in dB
\[F_N=\frac{|F|}{\max|F|},\qquad P_{dB}=20\log_{10}(F_N)\]
The displayed pattern is normalized so its peak is 0 dB. A user-selected floor such as −40 dB limits the visual range and avoids displaying numerical values near negative infinity.
7 · Physical aperture limit
Maximum theoretical directivity
\[D_{max}=\frac{4\pi A_{phys}}{\lambda^2},\qquad A_{phys}=L_xL_y\]
For a 10λ × 10λ rectangular aperture, \(A_{phys}=100\lambda^2\), so \(D_{max}\approx1256.6\), or about 31.0 dBi. This is a directivity limit, not gain, because conductor, dielectric, feed, mismatch, and other losses are not modeled.
8 · Directivity estimates
2D aperture and 1D line-array directivity
\[D_{est,2D}=\eta_{ap}D_{max}\]
\[D_{1D}=\frac{4\pi U_{max}}{\int_{4\pi}U(\theta,\phi)d\Omega}\]
For a 2D aperture, the calculator uses the physical-aperture limit multiplied by the sampled illumination/phase efficiency. For a 1D line aperture or linear array, physical area is undefined, so the calculator instead evaluates the full-sphere array-factor directivity under an isotropic-element/line-source assumption. The displayed 1D “max” value is the uniform broadside reference for the same line geometry.
9 · 2D cuts
Signed-angle convention
\[\alpha\in[-90^\circ,90^\circ],\quad u=\sin\alpha\cos\phi_c,\quad v=\sin\alpha\sin\phi_c\]
For Cartesian and polar cuts, a signed angle is used. Positive and negative angles represent opposite sides of boresight in the selected cut plane. XZ corresponds to \(\phi_c=0^\circ\), YZ to \(\phi_c=90^\circ\), and Custom uses the user-entered cut azimuth.
10 · Design synthesis
First-order aperture and spacing rules
\[\mathrm{HPBW}_{deg}\approx\frac{50.8\lambda}{L}\]
\[\frac{d}{\lambda}\leq\frac{1}{1+\sin\theta_{max}}\]
The Design page uses the broadside uniform-aperture HPBW approximation to estimate required electrical length. For a scanning discrete array, the spacing inequality is used as a visible-region grating-lobe limit; Auto spacing adds a user-selectable safety margin in Advanced mode.
11 · Scan loss
Projected planar aperture
\[\Delta G_{scan}\approx-10\log_{10}(\cos\theta)\]
For the 2D planar-aperture design estimate, scan loss includes the first-order projected-area reduction. The actual element pattern can create additional scan loss and is not inferred from S-parameters alone.
12 · Periodic unit cell
Active mismatch from scan-dependent S11
\[\eta_{mismatch}=1-|\Gamma_{active}|^2,\qquad L_m=-10\log_{10}(\eta_{mismatch})\]
When a periodic-unit-cell CSV is imported, the Design page evaluates active S11 over the requested scan region. If several spacing candidates are present, Auto spacing favors a scan-safe candidate with stronger worst-case active return loss.
13 · Finite array network
Active reflection from an S-matrix
\[\mathbf b=\mathbf S\mathbf a,\qquad \Gamma_{active,m}=\frac{b_m}{a_m}\]
\[\eta_{accepted}=1-\frac{\|\mathbf b\|^2}{\|\mathbf a\|^2}\]
For an uploaded finite-array Touchstone matrix, ApertureLab constructs the steering excitation vector \(\mathbf a\), computes the reflected wave vector \(\mathbf b\), and estimates total accepted power. This captures network-level mutual-coupling and mismatch behavior for the supplied finite array, but it still does not provide an embedded element radiation pattern.
14 · EM import workflow
Recommended simulation data
For a periodic unit cell, sweep scan angle and export frequency_ghz, dx_lambda, dy_lambda, theta_deg, phi_deg, s11_db. Include multiple spacing candidates if you want the Design page to compare spacing using EM data. For a finite array, export a full Touchstone S-matrix and specify the port grid. Radiation/embedded-element-pattern data is a separate future refinement needed for a fully realized-gain pattern prediction.
15 · Important scope
What is not included
The Main pattern simulator remains an aperture/array-factor model and does not include element pattern, polarization, conductor/dielectric loss, feed-network loss, platform scattering, or radome effects. The Design page can optionally use periodic active-S11 or a finite-array S-matrix to add mismatch/coupling information, but S-parameters alone do not create a full-wave radiation pattern or realized-gain model.