Huygens-Fresnel Principle

This is the idea that turned light from a bundle of rays into a wave you can compute. Its claim is disarmingly simple: wherever a wave has already reached, imagine every point of that wavefront as a tiny fresh source, each sending out its own expanding spherical ripple. A moment later the real wave is nothing but the sum of all those little ripples, added together with their phases. Run the picture forward through empty space and the ripples rebuild a smooth advancing wavefront — light just keeps going. Block half of them with an obstacle and the survivors no longer cancel each other's sideways spread, so light bends into the shadow. That leak into the shadow, computed as a sum of secondary wavelets, is diffraction. Christiaan Huygens proposed the bare geometric version — new wavefront equals the envelope of the wavelets — and it already delivers the laws of reflection and refraction. But it is mute about brightness, and taken literally it predicts a backward-travelling wavefront nobody has ever seen. Augustin Fresnel repaired both faults in one stroke in 1818 by insisting the wavelets interfere: give each a definite amplitude, a phase $e^{ikr}/r$ , a $1/\lambda$ scale, a quarter-wave head start, and an obliquity weight that fades to zero backward, and the coherent sum reproduces real diffraction patterns to the decimal. Those extra factors looked like lucky guesses for sixty years until the companion article aq-optics-fresnel-kirchhoff-diffraction-integral shows how Kirchhoff derived every one of them from the wave equation itself. Everything downstream — single slits, the Airy disc, zone plates, the whole of Fourier optics — is this one sum, evaluated in a particular limit.

Key takeaways

Every point of a wavefront is treated as a source of a secondary spherical wavelet $\propto e^{ikr}/r$ ; the field ahead is the coherent (phase-respecting) superposition of all of them.
Huygens' original envelope construction (1690) explains reflection and refraction but gives no intensities and predicts a spurious backward wave.
Fresnel (1818) added amplitude, phase, a $1/\lambda$ scale, and an obliquity factor $K(\chi)=\tfrac12(1+\cos\chi)$ — the coherent sum then predicts full diffraction patterns.
The obliquity factor vanishes backward, $K(\pi)=0$ , which is exactly what removes the unobserved backward wavefront.
On an unobstructed wavefront the wavelet sum splits into half-period Fresnel zones of nearly equal area whose contributions alternate in sign and sum to about half the first zone — so light propagates essentially straight.
Key idea: replace a wave by a fictitious sheet of point emitters on any surface it has reached, each radiating $E(Q)\,e^{ikr}/r$ weighted by an obliquity factor, and the physical field downstream is their coherent sum — diffraction is just the interference of these wavelets when part of the wavefront is blocked.

The core insight

Replace the propagating field by a fictitious sheet of point emitters on any surface the wave has already reached. Each emits a spherical wavelet $E(Q)\,e^{ikr}/r$ with an obliquity weight $K(\chi)$ that is strong forward and zero backward, and the field at any point ahead is their coherent sum. Nothing else is needed: reflection, refraction, and every diffraction pattern are consequences of adding these secondary wavelets as phasors.

Where this lives today

Every time a piece of software "propagates a field forward by a distance $z$ ", it is evaluating a discretized Huygens–Fresnel sum. Beam-propagation methods that design laser cavities and photonic waveguides, the reconstruction step in digital holography, the point-spread-function modelling behind every serious telescope and microscope, and the phase-retrieval algorithms that turn diffraction patterns into images of viruses at a synchrotron — all are the wavelet integral in numerical clothing, usually run as a fast Fourier transform in the far-field limit. The principle also underwrites diffractive optics as a manufacturing technology: Fresnel zone plates focus X-rays and extreme-ultraviolet light where no glass lens can, computer-generated holograms shape laser beams into arbitrary patterns, and the flat "metalenses" now appearing in cameras are engineered arrays of sub-wavelength scatterers whose combined field is designed, wavelet by wavelet, exactly as Fresnel would recognise.

The physical picture

Freeze an advancing plane wavefront and picture every point on it suddenly emitting a small expanding spherical shell of light. Along the forward direction the shells all crest together, and their common tangent surface is the next wavefront — the wave marches on with no sign that anything unusual happened. The magic only becomes visible when you take some of the emitters away. Cover half the wavefront with a screen and the wavelets that used to interfere away the sideways spreading are gone; what remains spills into the region geometry says should be dark. Diffraction is not light "going around a corner" so much as an interference sum that is suddenly missing half its terms.

Figure

Huygens–Fresnel construction: points on a wavefront emit secondary wavelets whose forward envelope is the next wavefront; the backward envelope (dashed) is never seen

Why the two odd factors Fresnel bolted on? The obliquity factor is the wavelet's way of saying emit strongly ahead, not at all behind: it equals $1$ straight forward and $0$ straight back, which is precisely why the backward wavefront in Huygens' bare picture is never observed. The $-i = e^{-i\pi/2}$ prefactor supplies a quarter-wave phase advance and the $1/\lambda$ fixes the scale, so that summing wavelets over an open wavefront rebuilds exactly the wave you started with. Neither factor is a fudge; both fall out of Kirchhoff's later derivation with no freedom left.

Figure

The obliquity factor K(χ) = ½(1 + cos χ) as a polar curve: full forward (K = 1), half at 90°, and zero backward (K = 0)

Finally, the wavelets must be added as phasors, never as intensities. Two wavelets leaving opposite edges of a slit arrive at a screen point along paths of different length $r$ ; the phase gap $k\,\Delta r$ decides whether they reinforce or cancel. The whole pattern is that competition summed over every open point of the aperture — a continuous interference calculation. Organizing the sum into concentric half-period Fresnel zones, within which the phase $kr$ advances by $\pi$ , turns it into a slowly converging alternating series whose value is about half the first zone. That half-zone result is why an unobstructed spherical wave rebuilds itself with the right amplitude, and why a zone plate that blocks the cancelling zones can focus light.

The core equations

The whole principle lives in one integral. Read it as an instruction: to find the field at $P$ , walk over every point $Q$ of a surface the wave has reached, launch a spherical wavelet weighted by the local field, tilt its strength by the obliquity factor, and add up the phases. The prefactor and the wavelet form are not arbitrary — the spherical wavelet is the outgoing Green's function of the wave equation, and the constant is fixed by demanding the sum reproduce a freely propagating wave.

E(P)=\frac{-i}{\lambda}\iint_{\Sigma} E(Q)\,\frac{e^{ikr}}{r}\,K(\chi)\,\mathrm{d}S \qquad (1)

where: $E(P)$ — complex field at the observation point $P$ ; $E(Q)$ — field on the wavefront at the secondary source $Q$ ; $r$ — distance from $Q$ to $P$ ; $k=2\pi/\lambda$ — wavenumber; $K(\chi)$ — obliquity factor depending on the angle $\chi$ between $QP$ and the forward normal; $-i/\lambda = e^{-i\pi/2}/\lambda$ — prefactor giving a quarter-wave phase advance and the $1/\lambda$ scale; $\mathrm{d}S$ — area element of $\Sigma$ . The integral is a coherent sum of spherical phasors, not a sum of brightnesses. The obliquity factor is the piece that makes the sum physical rather than merely plausible. It must weight forward emission fully and backward emission not at all, and the specific form that Kirchhoff's derivation produces is the following, symmetric between the incoming and outgoing directions in its general version.

K(\chi)=\tfrac12\left(1+\cos\chi\right) \qquad (2)

where: $\chi$ — angle between the forward wavefront normal at $Q$ and the direction $QP$ ; $K(0)=1$ is full forward emission and $K(\pi)=0$ is zero backward emission. Its vanishing at $\chi=\pi$ is exactly what deletes the backward wavefront that Huygens' bare envelope wrongly kept. Before Fresnel, Huygens had only the geometry: the next wavefront is the envelope of the wavelets after they have expanded for a time $\Delta t$ . This already carries the ray-optics content — it yields straight-line propagation, reflection, and Snell's law — but it assigns no amplitude and, read literally, keeps a backward envelope.

\text{new wavefront} = \text{envelope of wavelets of radius } r = v\,\Delta t \qquad (3)

where: $v=c/n$ — phase speed in a medium of index $n$ ; $\Delta t$ — time between successive wavefronts; $r$ — wavelet radius. Equation (3) is all geometry; equations (1)–(2) are what interference adds on top. Finally, the sum has a natural bookkeeping in concentric zones. Divide the wavefront (or aperture) into annuli whose edges differ in path to $P$ by half a wavelength; the last relation counts how many such zones an aperture spans, which is what decides whether the diffraction is near-field or far-field.

r_m=\sqrt{m\,\lambda\,z}\,,\qquad N_F=\frac{a^2}{\lambda z} \qquad (4)

where: $r_m$ — outer radius of the $m$ -th half-period Fresnel zone seen on a plane a distance $z$ from the axial point; $a$ — aperture radius; $N_F$ — Fresnel number, the count of zones the aperture spans. $N_F\gg1$ is the near (Fresnel) field; $N_F\ll1$ is the far (Fraunhofer) field where the pattern becomes the Fourier transform of the aperture.

Derivation AFrom Huygens' envelope to Fresnel's coherent sum

The goal is to see precisely what interference has to add to Huygens' picture, and why the two mysterious factors — the obliquity weight and the $-i/\lambda$ prefactor — are forced rather than chosen. We start from the bare geometric envelope, promote each wavelet to a spherical phasor, insert the minimal weighting that kills the backward wave, and then fix the overall constant by insisting that an unobstructed wavefront rebuild the very wave that launched it. Each ingredient is added only because the previous form fails a specific physical test.

Step 1. Take Huygens' construction: every wavefront point emits a sphere of radius $r=v\,\Delta t$ , and the forward tangent surface is the new wavefront. Why: this reproduces rectilinear propagation and Snell's law, so any repair must keep the envelope's forward behaviour — but it gives no amplitude and, fatally, a backward envelope too.

Step 2. Replace the envelope by a superposition: let each wavelet be a spherical phasor $\mathrm{d}E(P)=C\,E(Q)\,(e^{ikr}/r)\,\mathrm{d}S$ . Why: the $1/r$ conserves energy over the expanding shell and the $e^{ikr}$ carries the phase that makes wavelets interfere — the ingredient that lets the same construction compute a diffraction pattern, not just a wavefront position.

Step 3. Weight each wavelet by an obliquity factor $K(\chi)$ with $K(0)=1$ and $K(\pi)=0$ ; Kirchhoff's theory pins it to $K(\chi)=\tfrac12(1+\cos\chi)$ . Why: without a factor that vanishes backward, the superposition still radiates a wave toward the source — the very defect we are trying to remove — so the weighting is mandatory, not cosmetic.

Step 4. Fix $C$ by requiring the open-wavefront sum to reproduce the original spherical wave. Carrying the Fresnel-zone sum through, the amplitude and quarter-wave phase match only if $C=1/(i\lambda)=-i/\lambda$ . Why: the constant is not free — self-consistency of propagation determines it uniquely, and Kirchhoff's Green's-theorem derivation later returns exactly this value. *Verified: obliquity $K(0)=1,\,K(\pi)=0$ and the prefactor $-i/\lambda=1/(i\lambda)$ reproduced symbolically, matching Hecht §10.1 and Born & Wolf §8.2, in verify_aq-optics-huygens-fresnel-principle.py.*

Go deeper · Derivation B · 1 minHalf-period zones and why light travels nearly straight

Derivation BHalf-period zones and why light travels nearly straight

Here we evaluate the wavelet sum for the simplest case — an unobstructed wavefront observed on axis — and extract two results at once: that the wave rebuilds itself, and that the rebuilt amplitude equals only about half the first zone's contribution. The trick is to slice the wavefront into annular zones whose edges differ in path to the observation point by exactly $\lambda/2$ , so that consecutive zones arrive in antiphase and the sum becomes a slowly converging alternating series.

Step 1. Define the $m$ -th half-period zone as the annulus whose distance to $P$ runs from $r_0+(m-1)\lambda/2$ to $r_0+m\lambda/2$ . Why: a half-wavelength step in path is a $\pi$ step in phase, so successive zones contribute with opposite sign — the structure that makes the sum tractable.

Step 2. Show the zones have nearly equal area. For a wavefront a distance $z$ from the axial point the zone radii are $r_m=\sqrt{m\lambda z}$ , so the area between zones $m-1$ and $m$ is $\pi(r_m^2-r_{m-1}^2)=\pi\lambda z$ , independent of $m$ . Why: equal areas mean the wavelet amplitudes $|E_m|$ change only through the slowly varying obliquity factor, so the series terms shrink gradually and monotonically.

Step 3. Sum the alternating series $E=|E_1|-|E_2|+|E_3|-\cdots$ . For monotonically, slowly decreasing terms this telescopes to $E\approx |E_1|/2$ . Why: pairing each term with the average of its neighbours leaves only half the first term — a standard result for a slowly varying alternating series.

Step 4. Read off the physics. Carrying the phase through, the on-axis field equals the original spherical wave — propagation works — while the amplitude being half the first zone explains why blocking alternate zones (a zone plate) can more than double the on-axis field and focus light. Why: the half-zone amplitude is the quantitative fingerprint of the $-i/\lambda$ prefactor, tying this derivation back to the first. *Verified: zone radii $r_m=\sqrt{m\lambda z}$ , constant zone area $\pi\lambda z$ , and the alternating-series sum $\to |E_1|/2$ checked numerically in verify_aq-optics-huygens-fresnel-principle.py.*

Limiting cases

The one integral contains every standard diffraction regime as a way of approximating the wavelet phase $kr$ .

Case	Result	Meaning
$\lambda\to0$ (ray limit)	wavefront $\to$ envelope of wavelets	stationary-phase contribution dominates; diffraction spreading vanishes and geometry (rays, reflection, refraction) is recovered
near field, $N_F\gtrsim1$	$kr\approx kz+\tfrac{k}{2z}[(x-x')^2+(y-y')^2]$	quadratic phase kept ⟹ the Fresnel diffraction integral, a convolution with a chirp kernel
far field, $N_F\ll1$	$E\propto\iint E(x',y')\,e^{-ik(xx'+yy')/z}\,\mathrm{d}x'\mathrm{d}y'$	pattern is the Fourier transform of the aperture — sinc for a slit, Airy for a disc
backward direction $\chi=\pi$	$K(\pi)=\tfrac12(1+\cos\pi)=0$	the obliquity factor deletes the backward wavefront of the naive envelope
on-axis open wavefront	$E\approx \|E_1\|/2$	half-period zones alternate and sum to half the first zone — free propagation, and the basis of zone-plate focusing

Common trap“Huygens' construction by itself explains diffraction.”

It does not. Huygens' bare envelope is geometry: it locates the next wavefront and so gives reflection and refraction, but it assigns no amplitude, computes no fringe pattern, and — taken at face value — predicts a backward-travelling wavefront that is never seen. Diffraction is the interference of the wavelets, and that requires Fresnel's additions: a definite amplitude and phase $e^{ikr}/r$ , the $-i/\lambda$ prefactor, and above all the obliquity factor $\tfrac12(1+\cos\chi)$ that vanishes backward. Drop the obliquity factor and the backward wave returns; drop the $-i/\lambda$ and an open wavefront no longer reproduces itself. A second, quieter trap: the wavelets are spherical. Approximating them as plane waves throws away the wavefront curvature that is the whole source of diffractive spreading.

Worked examples

1 · Sizing the Fresnel zones of a plane wave (confidence builder)

A collimated beam of green light, $\lambda=550$ nm, illuminates a plane. Find the radii of the first and tenth half-period Fresnel zones referred to an axial point $z=1$ m away, show the zones carry equal area, and say how many zones a $2$ mm-radius aperture spans.

Step 1. Zone radii follow $r_m=\sqrt{m\lambda z}$ . First zone: $r_1=\sqrt{1\cdot550\times10^{-9}\cdot1}=\sqrt{5.50\times10^{-7}}=\mathbf{0.742\ mm}$ . Tenth zone: $r_{10}=\sqrt{10\cdot550\times10^{-9}\cdot1}=\sqrt{5.50\times10^{-6}}=\mathbf{2.35\ mm}$ . The zones crowd inward as $\sqrt{m}$ : the tenth is only $\sqrt{10}\approx3.16$ times the first, not ten times.

Step 2. Equal areas. The area between consecutive zones is $\pi(r_m^2-r_{m-1}^2)=\pi\lambda z=\pi\cdot550\times10^{-9}\cdot1=\mathbf{1.73\ mm^2}$ , the same for every $m$ . Equal areas are why the wavelet amplitudes fall only slowly (through the obliquity factor), which is what makes the alternating zone sum converge to half the first zone.

Step 3. Zone count. A $2$ mm-radius aperture spans $N_F=a^2/(\lambda z)=(2\times10^{-3})^2/(550\times10^{-9}\cdot1)=4\times10^{-6}/5.5\times10^{-7}=\mathbf{7.3}$ zones. Because $N_F\approx7\gg1$ , this geometry sits firmly in the near (Fresnel) field: the shadow of the aperture still has sharp-ish edges dressed with fringes, not a broad Airy pattern.

Step 4. The lever. Move the screen to $z=10$ m and $N_F$ drops tenfold to $0.73<1$ : the same aperture now diffracts into the far field, and the pattern becomes the Fourier transform of the hole. The single dimensionless number $N_F$ tells you which limit of the Huygens–Fresnel integral you are in. $r_1=0.742$ mm, $r_{10}=2.35$ mm, equal zone area $1.73$ mm², $N_F\approx7.3$ .

2 · The obliquity factor and the missing backward wave (exam level)

Evaluate the obliquity factor $K(\chi)=\tfrac12(1+\cos\chi)$ straight forward, at $60°$ , at $90°$ , and straight backward, and use the values to explain why a Huygens–Fresnel wave does not radiate toward its source.

Step 1. Forward, $\chi=0$ : $K=\tfrac12(1+1)=\mathbf{1}$ — full strength, as it must be for the wave to keep advancing.

Step 2. Oblique angles: at $\chi=60°$ , $K=\tfrac12(1+0.5)=\mathbf{0.75}$ ; at $\chi=90°$ (sideways along the wavefront), $K=\tfrac12(1+0)=\mathbf{0.5}$ — a sideways wavelet still contributes at half strength, which is what lets light spread into the shadow.

Step 3. Backward, $\chi=180°$ : $K=\tfrac12(1+\cos180°)=\tfrac12(1-1)=\mathbf{0}$ . Every wavelet's backward emission is switched off exactly, so summing them produces no wave travelling back toward the source — the defect of Huygens' bare envelope is cured term by term.

Step 4. Why this shape and not a step function? A hard forward-only cutoff would violate the smooth way Kirchhoff's boundary integral joins the incoming and outgoing directions; the gentle $\tfrac12(1+\cos\chi)$ is what the wave equation actually delivers, and its half-value at $90°$ is quantitatively necessary to reproduce measured near-field patterns. $K = 1,\,0.75,\,0.5,\,0$ at $\chi = 0°,60°,90°,180°$ .

Physics of the portrait

The principle is one of the rare ideas named for the geometry rather than a person — and rightly so, because it took two people a century and a half apart to make it work.

Origins · 1 minHuygens' principle and how this was discovered

Huygens' principle

1690 envelope · 1818 coherent sum · wave optics and diffraction theory

Huygens' principle is the founding proposition of wave optics, and its history is a two-act story. In act one, Christiaan Huygens set it out in his Traité de la Lumière (published 1690, presented 1678): light is a disturbance in a medium, every point a wavefront reaches becomes the source of a secondary spherical wavelet, and the new wavefront is the forward envelope of all the wavelets. With nothing more, this construction derives the laws of reflection and refraction and even the double refraction of calcite — a triumph, at a time when Newton's corpuscular theory held the field. Yet it was incomplete: it said nothing about how bright the light was, and its literal envelope included a backward wavefront that does not exist. Act two came in 1818, when Augustin Fresnel — answering a French Academy prize question on diffraction — grafted Thomas Young's principle of interference onto Huygens' geometry. Fresnel insisted the wavelets carry amplitude and phase and be summed coherently, with an obliquity factor to suppress the backward wave. The combined Huygens–Fresnel principle predicted diffraction fringes so accurately that Siméon Poisson, a hostile judge, deduced from it the "absurd" consequence of a bright spot at the centre of a disc's shadow — which François Arago promptly observed, turning a intended refutation into a spectacular confirmation. The heuristic became rigorous only in 1882, when Gustav Kirchhoff derived Fresnel's postulated factors from the wave equation, the subject of the companion article on the Fresnel–Kirchhoff integral.

History

1678

— Huygens presents his wavelet construction to the Paris Academy

1690

— Huygens publishes the Traité de la Lumière with the envelope principle

1801

— Thomas Young revives the wave theory with his interference principle

1818

— Fresnel's prize memoir fuses interference with Huygens' wavelets

1819

— Poisson predicts, and Arago observes, the bright spot in a disc's shadow

1882

— Kirchhoff derives the amplitude, phase and obliquity factors from the wave equation

Going further

Texts. Hecht, Optics, 5th ed., §10.1 (The Huygens–Fresnel Principle) and §10.3 (Fresnel zones) · Born & Wolf, Principles of Optics, 7th ed., §8.2 (Huygens–Fresnel principle) · Goodman, Introduction to Fourier Optics, 4th ed., Ch. 3 (Fresnel and Fraunhofer diffraction) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., §4.1.

Provenance. The wavelet form $e^{ikr}/r$ , the prefactor $-i/\lambda=1/(i\lambda)$ , and the obliquity factor $K(\chi)=\tfrac12(1+\cos\chi)$ with $K(0)=1$ , $K(\pi)=0$ are from Hecht §10.1 and Born & Wolf §8.2; the Fresnel-zone radii $r_m=\sqrt{m\lambda z}$ , the constant zone area $\pi\lambda z$ , and the half-first-zone sum are the standard results of Hecht §10.3; the near/far-field selector $N_F=a^2/\lambda z$ and the paraxial reduction to the Fresnel and Fraunhofer integrals follow Goodman Ch. 3–4. All five are reproduced in verify_aq-optics-huygens-fresnel-principle.py. The rigorous justification of the once-postulated factors is developed in the companion article on the Fresnel–Kirchhoff diffraction integral.

Prerequisites. Scalar wave equation and Helmholtz equation · Complex amplitude and phasor superposition · Interference of coherent waves. Leads to. Fresnel-Kirchhoff Diffraction Integral · Rayleigh-Sommerfeld Diffraction Theory · Fresnel Diffraction Regime · Fraunhofer Diffraction Regime · Fresnel Zones and Zone Plates · Fresnel Number.

Is the sheet of secondary sources real, or just a calculation trick?

It is a calculation trick — but an exact one, in the appropriate limit. There is no physical layer of little emitters on a wavefront; what is true is that the linear wave equation lets you replace the field on any closed surface by an equivalent distribution of sources on that surface and get the correct field everywhere beyond it. Kirchhoff's derivation makes this precise: Green's theorem turns the field's own values and normal derivatives on the surface into exactly the Huygens–Fresnel integral. So the wavelets are a bookkeeping device for the mathematics of linear superposition, not a claim about hidden machinery in the vacuum — which is why the same integral works whether the "medium" is air, glass, or empty space.

Why do we never see the backward wave that Huygens' envelope seems to predict?

Because the obliquity factor is exactly zero in the backward direction. Huygens' bare construction draws two envelopes, forward and backward, and offers no reason to prefer one; that is a genuine flaw in the geometric picture. Fresnel's coherent version weights each wavelet by $K(\chi)=\tfrac12(1+\cos\chi)$ , which equals $1$ straight ahead and falls smoothly to $0$ straight back. When you sum wavelets over a whole wavefront, the backward contributions are switched off term by term, so no backward-travelling wave is ever built. Kirchhoff's derivation shows this factor is not an add-on but a consequence of the wave equation and the outgoing boundary condition.

What is the difference between the Huygens–Fresnel principle and the Fresnel–Kirchhoff integral?

They are the same integral, arrived at two ways. The Huygens–Fresnel principle is a physical postulate: assume the wavelet form, the prefactor, and the obliquity factor, justified because the answers come out right. The Fresnel–Kirchhoff integral, developed in the companion article, derives that same expression from the scalar Helmholtz equation using Green's theorem, so the amplitude, the $-i/\lambda$ phase factor, and the obliquity factor all emerge with no free parameters. In practice they give identical predictions when the aperture is much larger than a wavelength; the Kirchhoff route additionally exposes a small internal inconsistency in the boundary conditions, later cleaned up by Rayleigh–Sommerfeld theory.

Why must the wavelets be added as amplitudes (phasors) rather than intensities?

Because interference lives entirely in the phases, and squaring too early destroys it. Each wavelet reaches the observation point with its own phase $kr$ set by its path length; whether two wavelets reinforce or cancel depends on the difference of those phases. If you added intensities ( $|E|^2$ ) you would throw the phase away before the interference had a chance to happen, and every diffraction pattern would collapse into a featureless blur. Only after the full complex sum $E(P)$ is assembled do you take $|E(P)|^2$ to get the measurable intensity. This is the same phasor discipline as in two-beam interference, now carried out over a continuous distribution of sources.

How does this one integral give both Fresnel and Fraunhofer diffraction?

By how carefully you approximate the wavelet phase $kr$ . Expand $r=\sqrt{z^2+(x-x')^2+(y-y')^2}$ for a screen a distance $z$ away. Keep the quadratic term in the phase and you get the Fresnel (near-field) integral, a convolution of the aperture field with a quadratic-phase kernel. Push further to $N_F=a^2/\lambda z\ll1$ , where the quadratic aperture-plane phase becomes negligible, and the integral collapses to a plain Fourier transform of the aperture field — the Fraunhofer (far-field) pattern, the Airy disc for a round hole and the sinc for a slit. Same integral, two limits, selected by the Fresnel number.

If the zone sum is only half the first zone, how can a zone plate focus light so strongly?

That is precisely the point of a zone plate. On an open wavefront the half-period zones alternate in sign and mostly cancel, leaving only about half the first zone on axis. A zone plate blocks (or phase-shifts) every other zone, removing the negative contributions, so the survivors add up in phase instead of cancelling. The on-axis amplitude then jumps from "half the first zone" to the sum of many same-sign zones — a bright focus at $f=r_1^2/\lambda$ , produced purely by interference of secondary wavelets rather than by refraction. It is the clearest demonstration that the wavelet sum is real physics: engineer which terms survive and you engineer the focus.