Aperture Antenna Electric Field Calculator

Aperture Antenna Electric Field Calculator Guide: Rectangular Apertures, Sinc Patterns, Far Field, and Fraunhofer Distance

Aperture antennas radiate because electromagnetic fields distributed across an opening combine in the observation region. In the far field, the angular radiation pattern is closely related to the spatial Fourier transform of the aperture illumination.

For an ideal rectangular aperture with uniform amplitude and phase, the far-field pattern separates into two orthogonal sinc factors—one associated with aperture width and the other with aperture height.

The existing calculator implements a simplified scalar magnitude model for that idealized case. It accepts the peak aperture electric field E₀, aperture width a, aperture height b, wavelength λ, observation distance r, and spherical observation angles θ and φ.

The field magnitude scales with aperture area ab, decreases approximately as 1/r in the far field, and is modulated by the angular factors sinc(X) and sinc(Y).

The calculator defines X = (πa/λ)sinθ cosφ and Y = (πb/λ)sinθ sinφ, with sinc(u) = sin(u)/u and sinc(0) defined by its limiting value 1.

At boresight, θ = 0, both X and Y are zero. The sinc factors therefore equal one, giving the maximum value predicted by this simplified scalar model.

Away from boresight, phase differences across the finite aperture cause partial constructive and destructive interference. This creates a central main lobe, nulls, and sidelobes.

A larger aperture measured in wavelengths produces a narrower main beam. This follows directly from the Fourier-transform relationship: increasing the physical aperture narrows the angular width of its sinc transform.

Wavelength therefore matters in two ways. It appears directly in the amplitude scaling term and also inside the angular sinc arguments. At shorter wavelength, the same physical aperture is electrically larger and generally produces a narrower directional pattern.

The model is valid only in the radiation far field. The current calculator checks the commonly used Fraunhofer criterion r ≥ 2D²/λ, where D is the largest aperture dimension.

That boundary is a practical criterion rather than a discontinuous physical wall. Electromagnetic fields transition gradually from near-field behavior to the asymptotic far-field form, so results near the boundary should be interpreted cautiously.

In the far field, electric- and magnetic-field amplitudes fall approximately as 1/r while average power density falls as 1/r². This is why doubling observation distance approximately halves field amplitude but quarters power density under otherwise identical far-field conditions.

The calculator intentionally reports only field magnitude. It does not preserve phase, vector polarization, aperture phase taper, edge diffraction, reflections, impedance mismatch, or receiving-antenna behavior.

It should therefore be used for electromagnetics education, preliminary antenna-pattern comparison, and sanity checking—not as a replacement for full-wave electromagnetic simulation, calibrated antenna-range measurements, or regulatory field-strength assessment.

How to Calculate the Far-Field Electric Field of a Rectangular Aperture

Enter the aperture field: Provide the assumed uniform peak aperture electric field E₀ in V/m.
Enter rectangular aperture dimensions: Provide width a and height b in meters.
Enter wavelength: Use free-space wavelength in meters. If frequency is known instead, convert with λ = c/f before using this calculator.
Enter observation distance: Provide r in meters from the aperture reference point to the observation point.
Enter θ and φ: Use the angular convention shown by the calculator. Angles entered in degrees should be converted internally to radians before evaluating sine and cosine.
Calculate X and Y: The calculator computes the two normalized spatial-frequency arguments controlling the rectangular-aperture sinc pattern.
Evaluate the field magnitude: Multiply the aperture amplitude scale by the absolute product of sinc(X) and sinc(Y).
Check the far-field boundary: Compare r with approximately 2D²/λ, using D = max(a,b). Treat results inside that boundary as outside the intended model domain.

Formula and variables

The simplified rectangular-aperture far-field model consists of an overall amplitude scale E₀ab/(λr) multiplied by the product of two sinc factors. Each sinc factor represents interference across one aperture dimension. The calculator evaluates the magnitude only and applies sinc(0) = 1 by continuity.

|E| = [E₀ab/(λr)] |sinc(X)sinc(Y)|, X = (πa/λ)sinθ cosφ, Y = (πb/λ)sinθ sinφ, sinc(u) = sin(u)/u

E — Far-field electric-field magnitude: Estimated electric-field amplitude at the observation point, in V/m.
E₀ — Peak aperture electric field: Uniform electric-field magnitude assumed across the ideal aperture.
a — Aperture width: One physical dimension of the rectangular aperture.
b — Aperture height: The orthogonal physical dimension of the rectangular aperture.
λ — Wavelength: Free-space wavelength of the radiated electromagnetic wave.
r — Observation distance: Distance from the aperture reference point to the observation point.
θ — Polar observation angle: Angular direction away from aperture boresight under the calculator’s spherical-coordinate convention.
φ — Azimuthal observation angle: Angular orientation around the aperture axis under the calculator’s spherical-coordinate convention.
D — Maximum aperture dimension: max(a,b), used in the Fraunhofer-distance check.

Scenario 1: Rectangular Aperture Observed Off Boresight

A uniformly illuminated aperture is 0.10 m wide and 0.05 m high with E₀ = 10 V/m. The wavelength is 0.03 m, the observation distance is 100 m, θ = 30°, and φ = 0°.

Peak aperture field: 10 V/m
Width a: 0.10 m
Height b: 0.05 m
Wavelength λ: 0.03 m
Distance r: 100 m
θ: 30°
φ: 0°

X = (π × 0.10 / 0.03) × sin(30°) × cos(0°).
X ≈ 5.236.
Y = (π × 0.05 / 0.03) × sin(30°) × sin(0°).
Y = 0.
sinc(Y) = 1.
Evaluate sinc(X) = sin(X)/X.
Apply |E| = E₀ab/(λr) × |sinc(X)sinc(Y)|.

Result: |E| ≈ 2.75665 × 10⁻3 V/m.

The off-boresight sinc factor reduces field magnitude relative to the boresight value. This is the same example and result implemented by the current calculator.

Understanding your results

Far-field magnitude

This is the scalar magnitude predicted by the idealized aperture model.

It does not include field phase or polarization direction.

Boresight value

At θ = 0, X = Y = 0 and both sinc factors equal one.

The simplified model therefore reaches its maximum angular factor at boresight.

Pattern reduction

A value below the boresight field indicates destructive phase addition across part of the aperture.

The sinc factors control that angular reduction.

Far-field warning

If r is smaller than the Fraunhofer estimate, the angular far-field approximation may be unreliable.

Near-field analysis requires a more complete aperture integral or electromagnetic solver.

Null

When either sinc factor is zero, the simplified field magnitude is zero.

These ideal nulls arise from complete cancellation across one aperture dimension.

Assumptions

The aperture is planar and rectangular.
Aperture illumination is uniform in amplitude.
Aperture phase is uniform.
Propagation is through free space.
The observation point lies in the Fraunhofer far-field region.
A scalar field-magnitude model is adequate.
Edge effects are neglected.
Losses are neglected.
The calculator uses the stated spherical angle convention consistently.
Wavelength is positive.
Aperture dimensions and observation distance are positive.

Limitations

The model does not include aperture illumination taper.
It does not include phase errors across the aperture.
It does not calculate polarization components.
It does not retain complex field phase.
It does not include aperture efficiency explicitly.
It does not model impedance mismatch or feed losses.
It does not include reflections, scattering from nearby structures, radomes, ground effects, or multipath.
It does not include mutual coupling or feed-network effects.
It does not model near-field or Fresnel-region field structure.
The Fraunhofer boundary is an approximate model-validity criterion, not an exact discontinuity in electromagnetic behavior.
The simple field scale implemented here is an idealized scalar expression and should not be substituted for a full vector aperture-field derivation when accurate absolute field levels are required.
The calculator does not calculate receiving-antenna power.
Received power requires additional information such as receiving-antenna effective area or gain, polarization alignment, impedance, and system losses.
The calculator is not intended for electromagnetic exposure compliance or regulatory certification.
Full-wave numerical simulation or calibrated antenna-range measurement is required when detailed sidelobes, cross-polarization, phase, near fields, feed geometry, or real-aperture losses matter.

Common mistakes

Entering frequency where the calculator expects wavelength.
Using centimeters or millimeters without converting to meters.
Using degrees directly in programming-language trigonometric functions that expect radians.
Using an observation point inside the far-field boundary without recognizing the approximation failure.
Forgetting the limiting definition sinc(0) = 1.
Using sin(u) instead of sin(u)/u for the sinc function.
Confusing field amplitude pattern with power pattern.
Forgetting that power pattern is proportional to |E|².
Assuming aperture width and height affect the beam in the same angular plane regardless of φ.
Treating the largest aperture dimension as a radius rather than D itself in the Fraunhofer-distance check.
Assuming a field null means real antennas always produce exactly zero measured power.
Interpreting the scalar magnitude as a complete polarization-resolved electromagnetic field.

Practical use cases

Scenario 2: Boresight field estimate

Set θ = 0.

Both sinc factors become one, leaving only the overall aperture/distance amplitude scale.

Scenario 3: Compare two aperture widths

Keep wavelength and height fixed while increasing width a.

The radiation pattern becomes narrower in the corresponding angular dimension.

Scenario 4: Compare two wavelengths

Keep the physical aperture fixed while reducing λ.

The aperture becomes electrically larger and the angular sinc pattern narrows.

Scenario 5: Distance scaling

Compare the same angular direction at 100 m and 200 m.

In the far field, the model predicts approximately half the field magnitude at twice the distance.

Scenario 6: Find a pattern null

Choose an angular direction that makes X or Y equal to a nonzero integer multiple of π.

The corresponding sinc term becomes zero in the ideal model.

Planning and decision guide

The far field is the Fourier transform of aperture illumination

Aperture theory expresses the far-field angular distribution as the spatial Fourier transform of the aperture field distribution.

A rectangular uniform aperture therefore produces separable sinc behavior in its two dimensions.

Scenario 7: Why rectangular apertures give sinc patterns

A one-dimensional uniform rectangular function transforms into a sinc function.

A two-dimensional rectangular aperture produces the product of two such transforms.

The power pattern is the squared magnitude of the field pattern

If normalized electric-field amplitude follows sinc(X)sinc(Y), normalized power follows sinc²(X)sinc²(Y).

This distinction matters when comparing field plots with antenna gain or radiation-intensity plots.

Scenario 8: Half field does not mean half power

If field amplitude is reduced to 0.5 of its boresight value, power density scales approximately with 0.5².

That corresponds to one quarter of the normalized power under the same impedance conditions.

sinc must be handled at zero by its limit

sin(u)/u gives 0/0 numerically at u = 0.

Mathematically the limit is 1.

Recommended implementation

If |u| is very small, return a stable near-zero approximation or exactly 1 at u = 0.

Do not allow NaN at boresight.

Scenario 9: Boresight

θ = 0 gives X = 0 and Y = 0.

The implementation must return sinc(0)sinc(0) = 1 rather than NaN.

The amplitude scale increases with aperture area

The implemented model contains ab in the numerator.

Holding E₀, λ, r, and angular factors constant, doubling aperture area doubles the scalar amplitude scale in this simplified relation.

The angular pattern also changes when dimensions change

Increasing a or b changes more than amplitude because those dimensions also appear inside X and Y.

Larger apertures produce narrower angular features.

Aperture size is best understood relative to wavelength

The ratios a/λ and b/λ determine the electrical size of the aperture.

Two physically different antennas can have similar normalized patterns when these ratios are similar.

Scenario 10: Electrically scaled apertures

An aperture twice as large operated at twice the wavelength has the same a/λ and b/λ ratios.

Its normalized angular sinc pattern can therefore remain similar under the idealized model.

Wider aperture gives a narrower beam in that dimension

MIT and other antenna-theory treatments show first nulls scaling roughly with λ divided by aperture dimension.

This is a direct Fourier-transform result.

First null occurs when the sinc argument reaches ±π

For X, nulls occur when X = ±π, ±2π, and so forth.

The same applies independently to Y.

Scenario 11: Principal-plane first null

If φ = 0, Y = 0 and the width dimension controls the sinc variation.

The first X null satisfies approximately a sinθ/λ = 1.

Small-angle approximation gives θ_null ≈ λ/a

When θ is small, sinθ ≈ θ in radians.

The first-null angle therefore scales inversely with aperture width.

The orthogonal plane behaves similarly with b

At φ = 90°, the width-related factor can become constant while the height-related sinc controls the principal-plane pattern.

This produces different beamwidths for a non-square rectangular aperture.

Scenario 12: Wide but short aperture

If a is much larger than b, the beam is narrow in the angular dimension governed by a and broader in the dimension governed by b.

This anisotropy is expected, not a calculation error.

A square aperture produces symmetric principal-plane scaling

If a = b and the illumination is otherwise symmetric, the two principal dimensions have matching normalized beamwidth behavior.

Off-axis azimuthal combinations still follow the two-dimensional product.

Uniform illumination produces characteristic sidelobes

The abrupt field truncation at aperture edges creates sinc sidelobes.

Tapering the illumination can reduce sidelobe levels but generally broadens the main beam.

The current calculator does not model taper

Its sinc pattern specifically corresponds to uniform rectangular illumination.

A tapered aperture requires Fourier transforming the actual illumination distribution.

Scenario 13: Why real antenna patterns differ

A real horn or dish feed may illuminate the aperture more strongly at the center than at the edges.

Its sidelobe pattern therefore differs from the uniform-aperture sinc result.

The calculator reports magnitude rather than signed sinc amplitude

The underlying sinc function changes sign between lobes.

Taking the absolute value removes phase reversal information from the displayed scalar magnitude.

A negative sinc lobe is not negative electric-field magnitude

The sign represents phase in the scalar pattern factor.

Magnitude remains nonnegative.

Complex phase contains additional information

A full aperture field includes phase factors such as exp(−jkr).

The current magnitude calculator intentionally discards those complex phases.

The far-field amplitude falls approximately as 1/r

The implemented expression contains 1/r.

This is consistent with radiating spherical-wave behavior in the far field.

Scenario 14: Double the distance

Holding direction fixed, changing r from 100 m to 200 m halves |E|.

The normalized angular pattern itself does not change in the ideal far field.

Power density falls approximately as 1/r²

Far-field power density is proportional to the squared electric-field magnitude for a fixed medium impedance.

Therefore doubling distance reduces power density by approximately a factor of four.

Field pattern becomes distance-independent after normalization

In the Fraunhofer region, changing r changes overall amplitude but not the normalized angular shape.

That is one defining property of the far field.

Near-field patterns depend on distance

In the Fresnel region, phase curvature across the aperture remains important.

The angular distribution is no longer described fully by the simple asymptotic sinc pattern.

The Fraunhofer criterion uses the largest aperture dimension

The current calculator uses D = max(a,b).

Its check is r ≥ 2D²/λ.

Scenario 15: Far-field distance

a = 0.50 m, b = 0.20 m, λ = 0.03 m.

D = 0.50 m.

r_F ≈ 2 × 0.50² / 0.03 ≈ 16.67 m.

An observation at 5 m would be inside that criterion

The calculator can still evaluate the algebraic formula numerically.

But it should warn that the result is outside the intended Fraunhofer-model domain.

The boundary criterion is approximate

Antenna literature uses the 2D²/λ value as a common far-field or Rayleigh/Fraunhofer distance.

Required measurement distance can depend on desired phase or pattern accuracy.

Do not hide the boundary warning

Model validity matters more than returning a number.

The result should visibly distinguish “far-field criterion satisfied” from “outside recommended far-field region.”

Recommended result badge

Far-field criterion: PASSED.

r = 100 m; estimated Fraunhofer boundary = 0.667 m.

Wavelength can be calculated from frequency

In free space, λ = c/f.

This conversion could be offered as a convenience input mode without changing the aperture-field model.

Scenario 16: 10 GHz

Using c ≈ 299,792,458 m/s, λ is approximately 0.02998 m.

That is close to 3 cm.

Frequency input should never be inserted directly into the wavelength field

Frequency and wavelength have different dimensions.

The current page correctly lists this as a common mistake.

Shorter wavelength increases electrical aperture size

At fixed a and b, reducing λ increases a/λ and b/λ.

The beam therefore becomes more directive in the ideal uniform-aperture model.

This does not automatically mean every real antenna becomes more efficient

Losses, feed design, surface tolerances, and material behavior can change with frequency.

The ideal aperture model contains none of those effects.

Angle convention must remain explicit

The implemented X and Y expressions assume a specific spherical coordinate definition.

Mixing a different θ/φ convention can rotate or distort the interpreted pattern.

The UI should include a small coordinate diagram

Aperture normal should be marked as boresight.

θ should show departure from boresight and φ should show azimuth around the axis.

This would prevent more errors than additional prose

Antenna angle conventions differ among textbooks and software packages.

A visual coordinate definition is therefore valuable.

Boresight field under the implemented equation is simple

|E_bore| = E₀ab/(λr).

That result follows because both sinc factors equal one.

Scenario 17: Boresight version of the worked example

E₀ = 10 V/m, a = 0.10 m, b = 0.05 m, λ = 0.03 m, r = 100 m.

The simplified boresight scale is E₀ab/(λr).

Off-axis field can be presented relative to boresight

Normalized amplitude = |sinc(X)sinc(Y)|.

This removes the absolute field scale and isolates the antenna-pattern factor.

Normalized power can also be useful

Normalized power pattern = sinc²(X)sinc²(Y).

A dB pattern can be calculated with 10 log10 of normalized power or equivalently 20 log10 of normalized field magnitude.

Scenario 18: Convert normalized field to dB

If normalized |E| = 0.1, field-pattern level is 20 log10(0.1) = −20 dB.

Normalized power is 0.01, which is also −20 dB using 10 log10.

A future dB-pattern output would improve engineering utility

Antenna patterns are frequently plotted in decibels.

The current linear V/m result should remain primary because that is the implemented calculator intent.

Ideal sinc nulls can produce numerical precision issues

Floating-point trigonometric evaluation may return a tiny residual rather than exactly zero.

Values sufficiently close to theoretical nulls can be normalized carefully for display.

Do not normalize every small field to zero

Very weak but nonzero fields can be physically meaningful.

Null handling should be based on known sinc-zero structure or a documented numerical tolerance.

Received power is a separate calculation

Electric field at the observation point does not by itself specify power delivered to a receiver.

A receiving antenna adds effective aperture, polarization, impedance, and loss considerations.

The existing page correctly rejects received-power interpretation

Its FAQ states that receiving-antenna gain or effective area, polarization, impedance, and losses are additionally required.

That boundary should remain.

Aperture efficiency is not in the implemented formula

Real aperture antennas often have an effective aperture smaller than their physical aperture.

The simplified uniform model should not silently add an assumed efficiency factor.

If aperture efficiency is added later, label it separately

Do not alter the current ideal-field equation without changing the model description.

A new advanced mode could include illumination efficiency or gain relationships.

Polarization matters in real antenna fields

The scalar magnitude gives no Eθ/Eφ decomposition.

Cross-polarized fields therefore cannot be obtained from the current model.

Uniform phase is also a strong assumption

Phase errors across an aperture distort and steer the far-field pattern.

Arrays and phased apertures intentionally manipulate this phase distribution.

Scenario 19: Beam steering

A linear phase gradient across an aperture can shift the main beam away from boresight.

The current uniform-phase calculator cannot model that steering.

The current page is ideal for teaching diffraction-like behavior

Rectangular-aperture radiation and Fraunhofer diffraction are mathematically closely related through Fourier transforms.

That gives the page a strong educational niche distinct from generic antenna link-budget calculators.

Do not turn it into a gain calculator

Gain, effective aperture, power density, and electric-field strength are related but distinct quantities.

This page should retain its field-distribution intent.

The strongest result output should include normalized pattern factor

Far-field magnitude in V/m.

Normalized amplitude |sinc(X)sinc(Y)|.

Fraunhofer boundary.

Far-field pass/warning state.

Recommended result audit trail

sinc(X).

sinc(Y).

Boresight scale E₀ab/(λr).

Final magnitude.

This makes antenna calculations reproducible

A user can identify whether a surprising result comes from aperture scaling, one of the sinc factors, or the far-field validity condition.

That is considerably more useful than returning one unexplained V/m value.

Frequently asked questions

What does this aperture antenna calculator calculate?

It estimates the far-field electric-field magnitude of an ideal uniformly illuminated rectangular aperture antenna.

What formula does the calculator use?

|E| = E₀ab/(λr) × |sinc(X)sinc(Y)|, with X = (πa/λ)sinθ cosφ and Y = (πb/λ)sinθ sinφ.

What is a rectangular aperture antenna?

It is an antenna model in which radiation is associated with an electromagnetic field distributed across a rectangular opening or equivalent aperture.

Why does a rectangular aperture produce a sinc pattern?

The far field is proportional to the Fourier transform of the aperture illumination, and the transform of a uniform rectangular distribution is a sinc function in each dimension.

What is sinc?

In this calculator, sinc(u) = sin(u)/u with sinc(0) defined by its limit as 1.

Why is sinc zero at certain angles?

At those directions, contributions from different parts of the aperture cancel destructively in the ideal model.

What is E0?

E₀ is the assumed uniform peak electric-field magnitude across the aperture.

What do a and b represent?

They are the width and height of the rectangular aperture.

What is lambda?

λ is the free-space wavelength.

How do I convert frequency to wavelength?

Use λ = c/f in free space.

Can I enter frequency directly?

The existing calculator expects wavelength, so frequency should first be converted to wavelength.

What is r?

r is the observation distance from the aperture reference point.

What are theta and phi?

They are the observation-direction angles used by the calculator’s spherical-coordinate convention.

What is boresight?

Boresight is the direction normal to the aperture plane, corresponding to θ = 0 in this model.

What happens at boresight?

X and Y are zero, so both sinc factors equal one and the angular pattern factor is maximal.

What is the Fraunhofer distance?

A common antenna far-field estimate is r_F ≈ 2D²/λ, where D is the largest antenna dimension.

What value of D should I use?

For this rectangular aperture calculator, use D = max(a,b).

Can I use the formula inside the Fraunhofer distance?

The calculator can evaluate it numerically, but the simplified far-field approximation may be unreliable there.

Does the electric field decrease as 1/r?

In the far field, radiated field amplitude decreases approximately as 1/r.

How does power density decrease with distance?

Far-field power density decreases approximately as 1/r².

What happens if I double the observation distance?

The ideal far-field electric-field magnitude approximately halves, while power density falls to about one quarter.

What happens if I increase aperture width?

The corresponding far-field beam becomes narrower because the aperture is larger relative to wavelength.

What happens if I decrease wavelength?

For a fixed physical aperture, the antenna becomes electrically larger and the ideal angular beam narrows.

What is a radiation-pattern null?

It is an ideal direction where one of the sinc factors becomes zero and the simplified field magnitude vanishes.

What are sidelobes?

They are secondary maxima outside the main beam produced by the finite aperture distribution.

Does this calculate power pattern?

The primary result is field magnitude. Normalized power is proportional to the square of normalized field magnitude.

How do I convert field pattern to dB?

For a normalized field magnitude E_n, use 20 log10(E_n).

Does the calculator include polarization?

No. It reports scalar magnitude only.

Does it include field phase?

No. Complex phase is discarded in the displayed magnitude.

Does it model a tapered aperture?

No. It assumes uniform aperture illumination.

Does it include aperture efficiency?

Not explicitly.

Does it calculate antenna gain?

No. The model calculates a simplified electric-field magnitude and normalized directional behavior.

Does it calculate received power?

No. Received power requires additional receiving-antenna and system parameters.

Can I use this for a dish antenna?

A real circular dish does not have the same rectangular-aperture sinc pattern. Circular apertures produce a different Bessel-function-based pattern.

Can I use this for a horn antenna?

A horn aperture can sometimes be approximated through aperture theory, but its actual amplitude and phase illumination are generally not perfectly uniform, so the simple rectangular model is only an approximation.

Why do real measurements differ from this calculator?

Real antennas include nonuniform illumination, phase errors, losses, polarization, feed structure, reflections, edge effects, and other departures from the ideal model.

Can this calculator model near fields?

No. Near-field and Fresnel-region points require a more complete electromagnetic field calculation.

How accurate is this aperture antenna electric-field calculator?

Its arithmetic is reproducible for the implemented model. Physical accuracy depends on how closely the real antenna matches the uniformly illuminated rectangular aperture and whether the observation point is sufficiently far into the Fraunhofer region.

Sources and review

6.013 Electromagnetics and Applications — Rectangular Aperture Far Field — Massachusetts Institute of Technology OpenCourseWare. Accessed 2026-09-02.
Rectangular Aperture Antennas — National Telecommunications and Information Administration. Accessed 2026-09-02.
Far-Field Antenna Pattern — Uniformly Illuminated Rectangular Aperture — Brigham Young University. Accessed 2026-09-02.
Rectangular Apertures With Uniform Field at the Aperture — Cornell University. Accessed 2026-09-02.
ITU-R Report M.2109 — Far-Field Distance Criterion — International Telecommunication Union. Accessed 2026-09-02.
Efficient Computation of the Far Field Radiated by an Arbitrary Rectangular-Aperture Distribution — National Institute of Standards and Technology. Accessed 2026-09-02.

Reviewed 2026-09-02 by Dr Akawak Ejigu, DBA.