Electromagnetic Waves, Antennas, and RF: Huygens Principle and Electromagnetic Field Equivalence - Exercises

Electromagnetic Waves, Antennas, and RF: Huygens Principle and Electromagnetic Field Equivalence - Exercises and Complete Worked Solutions

This companion to EM29 develops calculation skill with Huygens surfaces, electromagnetic field equivalence, equivalent electric and magnetic surface currents, far-field phase, and finite-aperture superposition. All exercises are stated first. Complete worked solutions follow in Part II. The sign conventions are the same as EM29: the phasor convention is e^iωt, outgoing waves contain e^−ikr, and the unit Normal n points from region 1 to region 2 [1, 2, 3, 4].

The principal surface jump relations are

(1)

and

(2)

For Love exterior equivalence, where the replacement field is zero on the interior side,

(3)

Three numerical ideas are used repeatedly in the problems. A far-field phase error is the difference between the exact propagation phase −kR and the phase obtained after replacing R by its far-field approximation. A discrete Huygens-source approximation replaces a continuous surface or line integral by a finite weighted sum. The resulting difference from the continuous-aperture result is the aperture sampling error.

Part I: Exercises

Exercise 1: outgoing Green-function phase and amplitude

At f = 1.00 GHz in free space, compare the outgoing scalar Green function

(4)

at

(5)

Find the magnitude ratio |G(R₂)∕G(R₁)| and the phase change from R₁ to R₂.

Exercise 2: surface currents from general field jumps

A planar interface has

(6)

The tangential fields on the two sides are

H1	= (0.20x + 0.10y) A/m,	(7)
H2	= (−0.10x + 0.40y) A/m,	(8)
E1	= (10x + 5y) V/m,	(9)
E2	= (4x + 11y) V/m.	(10)

Find J_s and M_s.

Exercise 3: Love currents for a normally incident plane wave

A free-space plane wave propagates in the +z direction with

(11)

Take the Huygens surface at z = 0 with n = z. Find the Love-equivalent J_s and M_s that reproduce the exterior wave and give zero interior field. Use η₀ = 376.73 Ω.

Exercise 4: reversing the chosen surface normal

Repeat Exercise 3 with the geometric surface normal reversed:

(12)

Using the corresponding Love formulas with n′, determine how the signed equivalent currents change.

Exercise 5: complementary interior-field equivalence

Let region 1 be the interior of a closed surface and region 2 the exterior, with n pointing outward. Construct a replacement problem that preserves the original fields E,H in region 1 but forces the replacement fields to zero in region 2. Derive the required J_s and M_s.

Exercise 6: oblique TE plane wave on a Huygens plane

A free-space plane wave propagates in the x-z plane at

(13)

from the +z axis. Its electric field is TE-polarized,

(14)

with

(15)

For the plane z = 0 with n = z, find H, J_s, and M_s.

Exercise 7: complex polarization and equivalent currents

At z = 0, a normally propagating +z free-space wave has phasor electric field

(16)

Find H, J_s, and M_s for n = z. Verify the magnitude relation between M_s and J_s.

Exercise 8: two discrete Huygens sources

Two equal in-phase elementary Huygens sources are separated by

(17)

along the x axis. In the x-z plane, the far-field phase difference is

(18)

Ignoring the common element pattern, find the magnitude of the coherent sum at 𝜃 = 0^∘, 30^∘, and 90^∘, normalized so that one source alone has amplitude 1.

Exercise 9: quantify a far-field phase approximation

For an aperture point at

(19)

an observation distance

(20)

and observation angle

(21)

use

(22)

and

(23)

Find the path-length error, the associated phase error in degrees, and the ratio between the exact 1∕R amplitude factor and the common far-field factor 1∕r.

Exercise 10: discrete Huygens approximation to a continuous aperture

A uniform line aperture has width

(24)

Its normalized continuous aperture factor is

(25)

Approximate the aperture by N equally weighted midpoint Huygens sources,

(26)

where the x_n are the subinterval midpoints. At 𝜃 = 10^∘, evaluate |A| and |A _N| for N = 4, 8, 16, 32. Report the percentage magnitude error.

Exercise 11: first null of a uniform aperture

A uniformly illuminated one-dimensional aperture has width

(27)

Find the first null angle measured from broadside.

Exercise 12: half-power beamwidth of the 4λ aperture

For a uniform aperture with

(28)

solve numerically for the positive angle 𝜃_hp at which

(29)

Find the full half-power beamwidth

(30)

Exercise 13: steering a continuous aperture with a phase ramp

A uniform line aperture of width D = 4λ is given the excitation phase

(31)

with

(32)

Show where the main beam points. Find the phase slope dϕ∕dx in rad/λ, the total phase change from x = −D∕2 to x = +D∕2, and the phase step between samples spaced by d = λ∕2.

Exercise 14: conducting-screen aperture equivalence

An aperture in a large conducting screen has tangential electric field

(33)

With n = z directed into the radiating half-space, use the common aperture-equivalence convention

(34)

Find M_s and state its units.

Exercise 15: recover tangential fields from Love currents

For Love exterior equivalence with outward normal n, suppose the equivalent currents are known. Starting from

(35)

derive formulas for the tangential fields E_t and H_t in terms of J_s and M_s.

Exercise 16: synthesis check from surface fields to a sampled aperture pattern

A 4λ uniform aperture is represented by N = 16 equally spaced midpoint Huygens samples. At 𝜃 = 10^∘, compare the sampled magnitude |A ₁₆| with the exact continuous magnitude |A|. Then state what would happen to the sampled pattern if every equivalent current sample were multiplied by the same complex constant Ce^iϕ₀.

Part II: Complete Worked Solutions

Solution 1: outgoing Green-function phase and amplitude

The free-space wavelength is

(36)

Hence

(37)

The magnitude of the Green function is 1∕(4πR), so

(38)

The unwrapped phase change is

Δϕ	= −k(R2 − R1)	(39)
	= −(20.9585)(0.25)	(40)
	= −5.2396 rad	(41)
	= −300.21∘.	(42)

Modulo 360^∘, this is equivalent to

(43)

Thus increasing range reduces amplitude by the geometric 1∕R factor and advances the negative outgoing-wave phase by 300.21^∘.

Solution 2: surface currents from general field jumps

First form the field jumps:

H2 − H1	= (−0.30x + 0.30y) A/m,	(44)
E2 − E1	= (−6x + 6y) V/m.	(45)

Using

(46)

we obtain

Js	= z × (−0.30x + 0.30y)	(47)
	= −0.30x − 0.30y A/m,	(48)

and

Ms	= −z × (−6x + 6y)	(49)
	= 6x + 6y V/m.	(50)

Therefore

(51)

Solution 3: Love currents for a normally incident plane wave

For a +z free-space plane wave,

(52)

Numerically,

(53)

Then

Js	= z × H = −5.31 × 10−3x A/m,	(54)
Ms	= −z × E = −2.00y V/m.	(55)

Hence

(56)

Solution 4: reversing the chosen surface normal

With

(57)

the same algebraic Love formulas give

(58)

and

(59)

Therefore

(60)

Equivalent-current signs are orientation dependent. A surface normal must always be stated together with the current convention.

Solution 5: complementary interior-field equivalence

For the desired replacement problem,

(61)

in the interior, while

(62)

in the exterior. The jump conditions give

Js	= n × (0 − H) = −n × H,	(63)
Ms	= −n × (0 − E) = +n × E.	(64)

Thus

(65)

The signs are the reverse of the usual Love exterior-equivalence currents because the prescribed field jump has been reversed.

Solution 6: oblique TE plane wave on a Huygens plane

For a plane wave,

(66)

Using

(67)

and E = E₀y,

k ×y

= sin 𝜃z − cos 𝜃x.

(68)

Hence

(69)

The Love electric surface current is

Js	= z × H	(70)
	= −y A/m	(71)
	= −6.90 × 10−3y A/m.	(72)

The equivalent magnetic surface current is

Ms	= −z × (3.00y)	(73)
	= +3.00x V/m.	(74)

Therefore

(75)

Only the tangential part of H contributes to J_s on the z = 0 surface.

Solution 7: complex polarization and equivalent currents

For propagation along +z,

H	= z × (2x + iy)	(76)
	= (−ix + 2y) A/m.	(77)

Then

Js	= z × H	(78)
	= −(2x + iy) A/m,	(79)

and

Ms	= −z × E	(80)
	= ix − 2y V/m.	(81)

The electric-field magnitude is

(82)

Therefore

(83)

so

(84)

The result holds for normal-incidence free-space plane waves even when the polarization state is complex.

Solution 8: two discrete Huygens sources

With d = λ∕2,

(85)

Two equal phasors separated symmetrically in phase have total magnitude

(86)

At broadside,

(87)

At 30^∘,

(88)

At endfire,

(89)

Thus

(90)

Solution 9: quantify a far-field phase approximation

Normalize lengths by λ. With r = 100λ, x = 2λ, and sin 30^∘ = 0.5,

Rexact	= λ	(91)
	= 99.01515λ,	(92)

whereas

(93)

The path error is

(94)

The phase error magnitude is

|Δϕ|	= kΔR	(95)
	= 2π(0.0151504)	(96)
	= 0.09519 rad	(97)
	= 5.45∘.	(98)

For amplitude,

(99)

Therefore

(100)

The approximation is already good in amplitude, while the phase error is the more sensitive quantity.

Solution 10: discrete Huygens approximation to a continuous aperture

At 𝜃 = 10^∘ and D = 4λ,

(101)

Therefore

(102)

Using midpoint samples gives

Thus the discrete Huygens sum converges rapidly toward the continuous aperture integral:

(103)

The numerical trend also shows why a sampled array and a continuous aperture become closely related when the spatial sampling is sufficiently fine.

Solution 11: first null of a uniform aperture

For a uniform line aperture,

(104)

With D = 6λ,

(105)

Therefore

(106)

Increasing aperture width narrows the main lobe.

Solution 12: half-power beamwidth of the 4λ aperture

The half-power condition is

(107)

The positive root inside the main lobe is

(108)

The pattern is symmetric about broadside, so

HPBW	= 2𝜃hp	(109)
	≈ 12.7156∘.	(110)

Thus

(111)

Solution 13: steering a continuous aperture with a phase ramp

The aperture integrand contains the product of the imposed excitation phase and the observation phase:

(112)

All aperture points are in phase when

(113)

For the main beam near broadside,

(114)

The phase slope is

(115)

Expressing x in wavelengths,

(116)

Across D = 4λ,

ΔϕD	= −kD sin 25∘	(117)
	= −8π sin 25∘	(118)
	= −10.622 rad	(119)
	= −608.6∘.	(120)

For d = λ∕2 sampling,

Δϕd	= −kd sin 25∘	(121)
	= −π sin 25∘	(122)
	= −1.328 rad	(123)
	= −76.1∘.	(124)

The phase ramp steers because it precompensates the free-space propagation phase in the selected direction.

Solution 14: conducting-screen aperture equivalence

Using

(125)

with n = z and E_a = 4x V/m,

Ms	= −2z × (4x)	(126)
	= −8y V/m.	(127)

Therefore

(128)

Magnetic surface-current density has units V/m in this equivalence convention.

Solution 15: recover tangential fields from Love currents

Start with

(129)

Cross both sides with n:

(130)

Because H_t is tangential,

(131)

and the vector triple-product identity gives

(132)

Hence

(133)

Similarly,

(134)

Crossing with n gives

(135)

Therefore

(136)

This inversion is useful when a numerical or measured equivalent-current distribution is used as the surface data.

Solution 16: synthesis check from surface fields to a sampled aperture pattern

From Solution 10, the continuous 4λ aperture at 10^∘ has

(137)

For N = 16 midpoint Huygens samples,

(138)

The magnitude difference is

(139)

corresponding to

(140)

relative magnitude error.

Now multiply every equivalent source by the same complex constant Ce^iϕ₀. The whole discrete sum is multiplied by that same factor:

(141)

Therefore the normalized angular pattern is unchanged. The absolute field magnitude is scaled by |C|, the common phase is shifted by ϕ₀, and the power scale changes by |C|². Beam direction, null angles, and normalized sidelobe structure remain unchanged.

Summary

The exercises demonstrate four central habits for equivalence-principle calculations:

1. state the surface normal and region ordering before evaluating cross products;
2. add complex field contributions coherently before taking magnitudes or powers;
3. treat far-field phase accuracy as more restrictive than amplitude accuracy when large apertures are involved;
4. recognize a sampled Huygens surface and a discrete antenna array as closely related spatial-superposition models.

EM30 can now use these skills to derive radiation from continuous apertures without treating the surface integral as a purely formal expression.

References

References