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Angular-spectrum optics and controlled paraxial limits

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Uniform phase-multiplier bound for the square-root dispersion law

Statement

Let $k_{0} > 0$ , $0 < δ < 1$ , $z \in R$ , and $s \in [0, δ^{2}]$ . Define

$R (s) : = 1 - \frac{s}{2} - \sqrt{1 - s} .$

Then

$R (s) = \frac{s^{2}}{4 (1 - \frac{s}{2} + \sqrt{1 - s})}, 0 \leq R (s) \leq \frac{s^{2}}{8 (1 - δ^{2})^{3 / 2}},$

and

$∣ e^{i z k_{0} \sqrt{1 - s}} - e^{i z k_{0} (1 - s / 2)} ∣ \leq \frac{∣ z ∣ k_{0} s^{2}}{8 (1 - δ^{2})^{3 / 2}} .$

More generally, for all real $a, b$ , $∣ e^{i a} - e^{i b} ∣ \leq ∣ a - b ∣ .$

Statement references

Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$

The reals are complete

Squaring is monotone on the nonnegatives

$\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$

Parity and the Pythagorean identity for sine and cosine

Real and imaginary parts, complex conjugation, and modulus

Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive

The complex exponential by its power series

$\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential

The derivatives of sine and cosine are cosine and minus sine

A function differentiable at $c$ is continuous at $c$

Limit point, isolated point, adherent point, derived set, and dense subset of $R$

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$

Facts & Assumptions

Given: Reals $k_{0} > 0$ , $0 < δ < 1$ , $z \in R$ , and $s \in [0, δ^{2}]$ .

[F1]

The real numbers form a complete ordered field, and each nonnegative real has a unique nonnegative square root; a positive radicand has a positive root. The reals are complete Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$

[F2]

For real $t$ , $e^{i t} = \cos t + i \sin t$ and $e^{i π} = - 1$ . The parity and Pythagorean identities give $\cos (- t) = \cos t$ , $\sin (- t) = - \sin t$ , and $\cos^{2} t + \sin^{2} t = 1$ ; with the complex-modulus definition this gives $∣ e^{i t} ∣ = 1$ and $∣ \cos t ∣, ∣ \sin t ∣ \leq 1$ . For $w = a + i b$ , conjugation is $\overline{w} = a - i b$ and $∣ w ∣ = \sqrt{a^{2} + b^{2}}$ . $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ Parity and the Pythagorean identity for sine and cosine Real and imaginary parts, complex conjugation, and modulus

[F3]

The complex exponential satisfies $e^{a + b} = e^{a} e^{b}$ for complex $a, b$ . The complex exponential by its power series $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential

[F4]

The functions $\sin$ and $\cos$ are differentiable on $R$ , with derivatives $\cos$ and $- \sin$ , and $\sin 0 = 0$ . Every real $t$ is a limit point of $R$ : for $ε > 0$ , $t + ε / 2$ is a distinct point within distance $ε$ . Thus both functions are continuous on $R$ . The derivatives of sine and cosine are cosine and minus sine A function differentiable at $c$ is continuous at $c$ Limit point, isolated point, adherent point, derived set, and dense subset of $R$ The reals are complete

[F5]

If a real function is continuous on a closed nondegenerate interval and differentiable on its interior, its endpoint difference equals its derivative at some interior point times the interval length. The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$

[F6]

Complex conjugation is an involution, $\overline{z w} = \overline{z} \overline{w}$ , and the complex modulus is multiplicative: $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ . Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive

Proof

1.1givenF1algebra

Rationalizing the definition of $R$ gives $R (s) = ((1 - s / 2)^{2} - (1 - s)) / (1 - s / 2 + \sqrt{1 - s}) = s^{2} / (4 (1 - s / 2 + \sqrt{1 - s}))$ . The denominator is positive because $s < 1$ .

1.2F2F3F4F5F6algebra

For real $a, b$ , put $d : = a - b$ and $t : = d / 2$ . By [F3], $e^{i a} - e^{i b} = e^{i b} e^{i t} (e^{i t} - e^{- i t})$ . From the Euler formula and parity in [F2], $e^{- i t} = \overline{e^{i t}}$ , so $e^{i t} - e^{- i t} = 2 i \sin t$ ; its modulus is $2 ∣ \sin t ∣$ . By [F2] and [F6], the prefactor has modulus $∣ e^{i b} e^{i t} ∣ = ∣ e^{i b} ∣ ∣ e^{i t} ∣ = 1$ , and therefore $∣ e^{i a} - e^{i b} ∣ = 2 ∣ \sin t ∣$ . If $d \neq 0$ , apply [F5] to $\sin$ on $[\min (0, t), \max (0, t)]$ ; [F2] gives $∣ \cos ∣ \leq 1$ , so [F4] yields $∣ \sin t ∣ \leq ∣ t ∣$ . The same inequality is immediate for $d = 0$ . Thus $∣ e^{i a} - e^{i b} ∣ \leq ∣ a - b ∣$ .

2.1givenF1step 1.1algebra

Put $t : = \sqrt{1 - δ^{2}}$ , so $0 < t \leq 1$ . Since $1 - s \geq 1 - δ^{2}$ , the nonnegative roots obey $\sqrt{1 - s} \geq t$ : otherwise the difference of their squares, factored as a product of a positive sum and positive difference, would give $1 - s < 1 - δ^{2}$ . Hence $1 - s / 2 + \sqrt{1 - s} \geq 1 - δ^{2} / 2 + t = (1 + t)^{2} / 2 \geq 2 t^{3}$ ; the last inequality follows from $(1 + t)^{2} \geq 4 t$ and $t \geq t^{3}$ . Substitution into step 1.1 proves $0 \leq R (s) \leq s^{2} / (8 t^{3})$ .

3.1givenstep 2.1step 1.2algebra∎

Set $a : = z k_{0} \sqrt{1 - s}$ and $b : = z k_{0} (1 - s / 2)$ in step 1.2. Then $∣ a - b ∣ = ∣ z ∣ k_{0} R (s)$ , so step 2.1 gives the stated multiplier bound.

DefinitionDefinition: AI-adaptedProof: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Forward angular-spectrum beam

Physical scope

Idealized classical scalar optics in a fixed inertial laboratory chart and SI units, for square-integrable plane data whose two-dimensional spatial-frequency support lies in a forward propagating cone strictly inside the vacuum light cone. The compact-aperture postulate supplies scalar-optics context but does not imply that every compact-aperture output belongs to this band-limited finite-energy class. The scalar propagation operators are not asserted to be exact Maxwell fields; exact vector plane-wave modes are treated separately. No empirical claim, measurement, or quantization is made.

Definition

Fix a named positively oriented inertial laboratory chart with coordinates $(x, y, z, t)$ , spatial domain $R^{3}$ , and time domain $R$ . Adopt SI units, a vacuum speed $c > 0$ , an angular frequency $ω > 0$ , and $k_{0} : = ω / c$ (so $[k_{0}] = m^{- 1}$ ). Let $x_{⊥} : = (x, y) \in R^{2}$ . Use the unitary two-dimensional Fourier transform $F_{2}$ whose Schwartz-space convention is $\hat{f} (ξ) = \int_{R^{2}} f (x_{⊥}) e^{- 2 π i x_{⊥} \cdot ξ} d^{2} x_{⊥},$ and whose inverse is its unitary $L^{2}$ extension. The variable $ξ$ is in cycles per metre; its associated angular spatial wavevector is $q : = 2 π ξ$ in $m^{- 1}$ , with $∣ q ∣^{2} : = q_{x}^{2} + q_{y}^{2}$ .

Choose $0 < δ < 1$ and $u_{0} \in L^{2} (R^{2}; C)$ with $F_{2} u_{0}$ supported, up to null sets, in $D_{δ} : = {ξ : ∣ 2 π ξ ∣ \leq δ k_{0}}$ . For $ξ \in D_{δ}$ put $k_{z} (ξ) : = \sqrt{k_{0}^{2} - ∣ 2 π ξ ∣^{2}},$ which is real and strictly positive there. For each $z \in R$ , define the exact multiplier on all of $R^{2}$ by $m_{e x a c t} (z, ξ) : = {\begin{cases} e^{i z k_{z} (ξ)}, & ξ \in D_{δ}, \\ 0, & ξ \notin D_{δ}, \end{cases}$ and define the paraxial multiplier on all of $R^{2}$ by $m_{p a r} (z, ξ) : = e^{i k_{0} z} e^{- i z ∣ 2 π ξ ∣^{2} / (2 k_{0})} .$ The exact scalar and paraxial scalar propagators are the $L^{2}$ equivalence classes $u_{e x a c t} (z) : = F_{2}^{- 1} ⁣ [m_{e x a c t} (z, ξ) F_{2} u_{0} (ξ)], u_{p a r} (z) : = F_{2}^{- 1} ⁣ [m_{p a r} (z, ξ) F_{2} u_{0} (ξ)] .$ The associated real scalar fields are the real parts of these phasors multiplied by $e^{- i ω t}$ . If $u_{0}$ carries $V m^{- 1}$ , both propagated fields carry the same unit. “Exact” here names the exact positive axial dispersion multiplier inside this scalar model; it does not assert exact Maxwell validity. Forward propagation is $z \geq 0$ ; the formulas also define the mathematical continuation to negative $z$ .

At a fixed $ξ \in D_{δ}$ , write $K (ξ) : = (2 π ξ_{x}, 2 π ξ_{y}, k_{z} (ξ))$ . For complex amplitudes $e, b \in C^{3}$ at frequency $ω$ , the real-field encoding assigns the maps $E, B : R^{3} \times R \to R^{3}$ by $E (x, t) : = Re ⁣ (e e^{i (K \cdot x - ω t)}), B (x, t) : = Re ⁣ (b e^{i (K \cdot x - ω t)}) .$ In the SI field interpretation, $e$ has units $V m^{- 1}$ and $b$ has units tesla. The candidate is an exact source-free vacuum plane-wave mode precisely when these real fields are jointly $C^{1}$ and satisfy, pointwise on $R^{3} \times R$ , $div E = 0, div B = 0, curl E = - \partial_{t} B, curl B = c^{- 2} \partial_{t} E,$ the zero-source specialization of the adopted smooth vacuum Maxwell branch. The longitudinal electric coefficient is $e_{z}$ , and $e_{⊥} : = (e_{x}, e_{y})$ has norm $∣ e_{⊥} ∣^{2} : = ∣ e_{x} ∣^{2} + ∣ e_{y} ∣^{2}$ . No scalar/vector polarization factorization is part of this definition.

Definition references

Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields

Framework-local SI units, the vacuum constants and the exact/measured status of their values

Fourier inversion on Schwartz space

Plancherel theorem

The Axiom of Countable Choice ( ${A C}_{ω}$ )

The complex exponential by its power series

Real and imaginary parts, complex conjugation, and modulus

The adopted scalar aperture-diffraction model

Classical transverse Fourier amplitudes

The microscopic vacuum Maxwell equations: smooth and weak branches

Complex plane-wave fields: real parts of complex exponentials

The reals are complete

Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$

Squaring is monotone on the nonnegatives

$\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$

Uniform phase-multiplier bound for the square-root dispersion law

Facts & Assumptions

Given: A fixed inertial chart, $c > 0$ , $ω > 0$ , $0 < δ < 1$ , and $u_{0} \in L^{2} (R^{2}; C)$ with Fourier support in $D_{δ}$ .

[F1]

The Schwartz Fourier convention is the displayed $e^{- 2 π i x \cdot ξ}$ integral, and under countable choice it extends uniquely to a unitary map on complex $L^{2}$ ; its inverse is the unitary inverse. Fourier inversion on Schwartz space Plancherel theorem The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F2]

The laboratory chart has spatial coordinates in metres and time in seconds; $c$ carries $m s^{- 1}$ , $ω$ carries $s^{- 1}$ , and $E$ carries $V m^{- 1}$ . Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields Framework-local SI units, the vacuum constants and the exact/measured status of their values

[A1]

The adopted scalar aperture model takes a finite complex measure supported on a compact planar aperture and assigns an outgoing scalar Helmholtz field in a polarization-insensitive regime; it does not assert exact Maxwell validity or that every aperture output is band-limited or square-integrable on a plane. The adopted scalar aperture-diffraction model

[F3]

The existing transverse Fourier-field definition uses a smooth compactly supported three-dimensional wavevector amplitude supported away from zero and imposes the bilinear condition $k \cdot a (k) = 0$ ; its three-dimensional Schwartz-packet class is distinct from the present two-dimensional fixed-frequency $L^{2}$ data. Classical transverse Fourier amplitudes

[F4]

A real plane-wave electric field is the componentwise real part of a complex amplitude times $e^{i (K \cdot x - ω t)}$ . Complex plane-wave fields: real parts of complex exponentials

[A2]

The homogeneous system in the Definition is the source-free specialization of the adopted smooth vacuum Maxwell postulate, so in particular $div E = 0$ . This is a physical model assumption, not an empirical observation. The microscopic vacuum Maxwell equations: smooth and weak branches

[F5]

Complex exponential and real-part operations are defined on every complex number; the real-field encoding is componentwise. The complex exponential by its power series Real and imaginary parts, complex conjugation, and modulus

[F6]

The real numbers are a complete ordered field. The reals are complete

[F7]

Every nonnegative real has a unique nonnegative square root, and a positive real has a positive root. Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$

[F8]

Squaring preserves and reflects the order on nonnegative real numbers. Squaring is monotone on the nonnegatives

[F9]

For real $θ$ , $∣ e^{i θ} ∣ = 1$ ; for $w = a + i b$ , $∣ Re w ∣^{2} = a^{2} \leq a^{2} + b^{2} = ∣ w ∣^{2}$ , hence $∣ Re w ∣ \leq ∣ w ∣$ . $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ Real and imaginary parts, complex conjugation, and modulus Squaring is monotone on the nonnegatives

[F10]

For real $a, b$ , $∣ e^{i a} - e^{i b} ∣ \leq ∣ a - b ∣$ . Uniform phase-multiplier bound for the square-root dispersion law

Verification

1.1givenF6F7F8algebra

For $ξ \in D_{δ}$ , put $q : = 2 π ξ$ , $a : = k_{0}^{2} - ∣ q ∣^{2}$ and $b : = k_{0}^{2} (1 - δ^{2})$ . The support condition gives $0 < b \leq a \leq k_{0}^{2}$ . By [F6] and [F7] the roots in the definition exist; uniqueness and $k_{0} > 0$ give $\sqrt{b} = k_{0} \sqrt{1 - δ^{2}}$ and $\sqrt{k_{0}^{2}} = k_{0}$ . Applying [F8] to these nonnegative roots yields $0 < k_{0} \sqrt{1 - δ^{2}} \leq k_{z} (ξ) \leq k_{0}$ .

1.2givenF2algebra

The transverse and axial coordinates are measured in metres, while $q, k_{z}, k_{0}$ are in $m^{- 1}$ ; thus every propagation-phase exponent is dimensionless. The multipliers preserve the unit of $u_{0}$ .

1.3givenA1

[A1] fixes a separate scalar optical model, while the present Definition additionally restricts its plane data by $L^{2}$ and $D_{δ}$ support; hence no existence or band-limit claim for every compact-aperture output is being made.

2.1step 1.1F9F10algebra

For $ξ, η \in D_{δ}$ , rationalizing gives $∣ k_{z} (ξ) - k_{z} (η) ∣ = \frac{∣ ∣ 2 π ξ ∣^{2} - ∣ 2 π η ∣^{2} ∣}{k_{z} (ξ) + k_{z} (η)} .$ The denominator is at least $2 k_{0} \sqrt{1 - δ^{2}} > 0$ by step 1.1, and the numerator is continuous in $(ξ, η)$ ; hence $k_{z}$ is continuous on $D_{δ}$ . By [F10], $∣ e^{i z k_{z} (ξ)} - e^{i z k_{z} (η)} ∣ \leq ∣ z ∣ ∣ k_{z} (ξ) - k_{z} (η) ∣$ , so the exact exponential branch is continuous on $D_{δ}$ ; since $D_{δ}$ is closed, its zero extension $m_{e x a c t}$ is measurable on $R^{2}$ . The paraxial phase is a continuous real polynomial in the spatial frequencies; applying the same [F10] estimate to its real phase values makes $m_{p a r}$ continuous and hence measurable on $R^{2}$ . [F9] gives modulus one for $m_{e x a c t}$ on $D_{δ}$ and for $m_{p a r}$ everywhere; $m_{e x a c t}$ is zero off $D_{δ}$ .

3.1step 2.1F1F5F9algebra

Both multipliers are measurable and bounded by one, so their products with $F_{2} u_{0}$ are in $L^{2}$ . Since $F_{2} u_{0}$ vanishes almost everywhere off $D_{δ}$ , the exact multiplier's modulus one on $D_{δ}$ and the paraxial multiplier's modulus one everywhere show that each product has the same $L^{2}$ norm as $F_{2} u_{0}$ . The unitary inverse in [F1] defines the propagated classes. For each fixed $t$ , [F9] gives $∣ Re (e^{- i ω t} u) ∣ \leq ∣ e^{- i ω t} u ∣ = ∣ u ∣$ , so the real scalar fields are in $L^{2}$ and independent of the representative.

4.1givenF3F4A2∎

The fixed-frequency plane-mode clause uses the real fields of [F4] and the zero source pair in [A2]; its Maxwell equations are the homogeneous specialization of the adopted postulate. It is not identified with the distinct three-dimensional Schwartz Fourier-packet construction in [F3].

Remarks

The following normalized cross-section uses $Q = q_{x} / k_{0}$ and $K_{z} = k_{z} / k_{0}$ , sets $q_{y} = 0$ , and sketches the illustrative value $δ = 1 / 2$ . The exact admissible region in the Definition is the disk $∣ q ∣ \leq δ k_{0}$ ; the diagram is only a cross-section and is not used in any proof.

\begin{tikzpicture}[x=2.2cm,y=1.7cm]
  \draw[->] (-1.15,0) -- (1.2,0) node[right] {$Q$};
  \draw[->] (0,0) -- (0,1.2) node[above] {$K_z$};
  \draw[thick,domain=-1:1,samples=80]
    plot (\x,{sqrt(1-\x*\x)});
  \draw[dashed,thick,domain=-0.5:0.5,samples=40]
    plot (\x,{1-\x*\x/2});
  \draw[dotted] (-0.5,0) -- (-0.5,1.08);
  \draw[dotted] (0.5,0) -- (0.5,1.08);
  \node[above right] at (0.56,0.84) {$1-Q^2/2$};
  \node[above left] at (-0.88,0.47) {$\sqrt{1-Q^2}$};
  \node[below] at (-0.5,0) {$-\delta$};
  \node[below] at (0.5,0) {$\delta$};
\end{tikzpicture}

Physics theoremStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Paraxial phase and longitudinal-field bounds

Physical scope

Conditional result for the ideal band-limited scalar angular-spectrum model and smooth source-free monochromatic vacuum plane-wave modes in the same fixed inertial chart and SI convention. It bounds the scalar Helmholtz phase approximation and the longitudinal electric amplitude only; it does not say that finite-aperture data have the required band limit, that the scalar model is exact Maxwell theory, or that a finite sample or measurement obeys the bound. No empirical conclusion or angular-momentum decomposition is asserted.

Statement

Let $k_{0} > 0$ , $0 < δ < 1$ , and $z \in R$ . For every $u_{0} \in L^{2} (R^{2}; C)$ whose Fourier transform is supported, up to null sets, in $D_{δ} : = {ξ : ∣ 2 π ξ ∣ \leq δ k_{0}}$ ,

${∥ u_{e x a c t} (z) - u_{p a r} (z) ∥}_{L^{2} (R^{2})} \leq \frac{∣ z ∣ k_{0} δ^{4}}{8 (1 - δ^{2})^{3 / 2}} {∥ u_{0} ∥}_{L^{2} (R^{2})},$

where the propagated fields are exactly those of the angular-spectrum Definition. For a source-free monochromatic vacuum plane-wave mode with $q : = 2 π ξ$ , $∣ q ∣ \leq δ k_{0}$ , $k_{z} : = \sqrt{k_{0}^{2} - ∣ q ∣^{2}}$ , and complex electric amplitude $e = (e_{⊥}, e_{z})$ , use the complex-bilinear convention $q \cdot e_{⊥} : = q_{x} e_{x} + q_{y} e_{y}$ with no conjugation,

$e_{z} = - \frac{q \cdot e_{⊥}}{k_{z}}, ∣ e_{z} ∣ \leq \frac{δ}{\sqrt{1 - δ^{2}}} ∣ e_{⊥} ∣ .$

Statement references

Forward angular-spectrum beam

Uniform phase-multiplier bound for the square-root dispersion law

Plancherel theorem

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Complex plane-wave fields: real parts of complex exponentials

The microscopic vacuum Maxwell equations: smooth and weak branches

Real and imaginary parts, complex conjugation, and modulus

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$

Squaring is monotone on the nonnegatives

Framework-local SI units, the vacuum constants and the exact/measured status of their values

The reals are complete

Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$

Quarter-turn values and shifts by pi/2 and pi

Facts & Assumptions

Given: $k_{0} > 0$ , $0 < δ < 1$ , $z \in R$ , scalar data $u_{0}$ supported in the angular cone, and an exact source-free monochromatic vacuum plane-wave mode at $∣ q ∣ \leq δ k_{0}$ .

[F1]

The exact and paraxial fields are the $L^{2}$ Fourier multipliers $e^{i z k_{z} (ξ)}$ and $e^{i k_{0} z} e^{- i z ∣ 2 π ξ ∣^{2} / (2 k_{0})}$ on the support disk; the $F_{2}$ convention is $e^{- 2 π i x \cdot ξ}$ . Forward angular-spectrum beam

[F2]

Under countable choice, $F_{2}$ is unitary on complex $L^{2}$ and preserves its norm. Plancherel theorem The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F3]

For $0 \leq s \leq δ^{2}$ , the difference between phases $z k_{0} \sqrt{1 - s}$ and $z k_{0} (1 - s / 2)$ has multiplier modulus at most $∣ z ∣ k_{0} s^{2} / [8 (1 - δ^{2})^{3 / 2}]$ . Uniform phase-multiplier bound for the square-root dispersion law

[F4]

A real plane wave is $E (x, t) = Re (e e^{i (K \cdot x - ω t)})$ ; the exponential is differentiable and its phase derivative follows the chain rule. Complex plane-wave fields: real parts of complex exponentials The complex exponential is entire and its complex derivative is itself The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$

[A1]

In the smooth source-free vacuum branch, the adopted Maxwell equations include $div E = 0$ . The microscopic vacuum Maxwell equations: smooth and weak branches

[F5]

The Euler formula is $e^{i θ} = \cos θ + i \sin θ$ . The quarter-turn theorem gives $\sin (π / 2) = 1$ , $\cos (π / 2) = 0$ , and the shifts $\sin (x + π) = - \sin x$ , $\cos (x + π) = - \cos x$ . Taking $x = - π / 2$ in the shift identities yields $\sin (- π / 2) = - 1$ and $\cos (- π / 2) = 0$ , hence $e^{- i π / 2} = - i$ . $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ Quarter-turn values and shifts by pi/2 and pi

[F6]

Complex modulus is the Euclidean modulus of real and imaginary parts, and the real transverse Euclidean norm is induced by the dot product; extend a real dot product bilinearly when its second argument is complex. Order of squares reflects order for nonnegative real numbers. For real $r > 0$ and $w = a + i b$ , $∣ w / r ∣ = \sqrt{(a / r)^{2} + (b / r)^{2}} = ∣ w ∣ / r$ . Real and imaginary parts, complex conjugation, and modulus The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ Squaring is monotone on the nonnegatives

[F7]

In the fixed SI chart, $z$ is measured in metres, $k_{0} = ω / c$ in $m^{- 1}$ , and $δ$ is dimensionless. Forward angular-spectrum beam Framework-local SI units, the vacuum constants and the exact/measured status of their values

[F8]

The root $k_{z} = \sqrt{k_{0}^{2} - ∣ q ∣^{2}}$ is nonnegative, and $\sqrt{k_{0}^{2} (1 - δ^{2})} = k_{0} \sqrt{1 - δ^{2}}$ is nonnegative. Since $∣ q ∣ \leq δ k_{0}$ , their squares satisfy $k_{z}^{2} \geq k_{0}^{2} (1 - δ^{2})$ ; monotonicity of squaring on nonnegative reals therefore gives $k_{z} \geq k_{0} \sqrt{1 - δ^{2}} > 0$ . Forward angular-spectrum beam The reals are complete Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ Squaring is monotone on the nonnegatives

Proof

1.1givenF1algebra

On the Fourier support put $s : = ∣ 2 π ξ ∣^{2} / k_{0}^{2}$ . Then $0 \leq s \leq δ^{2}$ , $k_{z} = k_{0} \sqrt{1 - s}$ , and the paraxial axial phase is $k_{0} (1 - s / 2)$ .

1.2givenF4F5A1algebra

Write $ϕ (x, t) : = K \cdot x - ω t$ . For each spatial coordinate $x_{j}$ , the chain rule and $\exp^{'} = \exp$ in [F4] give $\partial_{x_{j}} E_{j} = Re (i K_{j} e_{j} e^{i ϕ})$ : indeed $\partial_{x_{j}} ϕ = K_{j}$ , and the real-part map commutes with this real derivative. Summing over $j = 1, 2, 3$ and using additivity of real parts yields $div E = Re (i (K \cdot e) e^{i ϕ})$ . By [A1] this is zero for every $(x, t)$ . Put $S : = K \cdot e$ . At $x = 0, t = 0$ , the expression is $Re (i S) = - Im S$ , so $Im S = 0$ . At $x = 0, t = π / (2 ω)$ , [F5] gives $e^{i ϕ} = - i$ and the expression is $Re (S)$ , so $Re S = 0$ . Thus $S = 0$ , that is, $q \cdot e_{⊥} + k_{z} e_{z} = 0$ .

2.1step 1.1F1F2F3algebra

By [F3], the difference of the two Fourier multipliers has modulus at most $C : = ∣ z ∣ k_{0} δ^{4} / [8 (1 - δ^{2})^{3 / 2}]$ throughout the support. Unitarity [F2] therefore gives $∥ u_{e x a c t} (z) - u_{p a r} (z) ∥_{2}^{2} = \int ∣ {\hat{u}}_{0} (ξ) ∣^{2} ∣ e^{i z k_{z}} - e^{i k_{0} z - i z ∣ 2 π ξ ∣^{2} / (2 k_{0})} ∣^{2} d^{2} ξ \leq C^{2} ∥ u_{0} ∥_{2}^{2}$ ; taking nonnegative square roots proves the scalar bound.

2.2step 1.2F6F8algebra

By [F8] and step 1.2, $∣ e_{z} ∣ = ∣ q \cdot e_{⊥} ∣ / k_{z}$ . The identity $(∣ q ∣^{2} ∣ e_{⊥} ∣^{2} - ∣ q \cdot e_{⊥} ∣^{2}) = ∣ q_{x} e_{y} - q_{y} e_{x} ∣^{2} \geq 0$ implies $∣ q \cdot e_{⊥} ∣ \leq ∣ q ∣ ∣ e_{⊥} ∣$ ; using $∣ q ∣ \leq δ k_{0}$ and [F8] yields the displayed longitudinal bound.

3.1F7algebra∎

The scalar error prefactor $∣ z ∣ k_{0} δ^{4} / [8 (1 - δ^{2})^{3 / 2}]$ and the longitudinal prefactor $δ / \sqrt{1 - δ^{2}}$ are dimensionless by [F7]. For a scalar field $u$ measured in $V m^{- 1}$ , the norm $∥ u ∥_{2} = (\int_{R^{2}} ∣ u (x_{⊥}) ∣^{2} d^{2} x_{⊥})^{1 / 2}$ has units $((V m^{- 1})^{2} m^{2})^{1 / 2} = V$ , since transverse area is measured in $m^{2}$ . Thus both sides of the scalar bound have units $V$ , while both sides of the longitudinal amplitude bound have units $V m^{- 1}$ .

Remarks

The scalar bound is conditional on the stated spectral support and its longitudinal scale $∣ z ∣ k_{0} δ^{4}$ . It does not claim a bound for arbitrary aperture data or for exact vector Maxwell errors.

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Paraxial orbital winding in its exact scalar domain

Physical scope

Conditional identity for a smooth, finite-transverse-norm scalar paraxial field in the specified angular-spectrum model, at a fixed propagation plane. The dimensionless scalar rotation generator is not identified with torque, total electromagnetic angular momentum, a photon angular momentum, or a general exact-vector spin/orbital decomposition. No empirical claim is made.

Statement

Fix a propagation plane $z$ and let $u_{z} \in C^{\infty} (R^{2}; C) \cap L^{2} (R^{2}; C)$ be a scalar field in the paraxial model. Assume $u_{z}$ is nonzero as an $L^{2}$ equivalence class. For $r > 0$ write $u_{z} (r \cos ϕ, r \sin ϕ) = f (r) e^{i ℓ ϕ}$ , where $ℓ \in Z$ and $ϕ$ is read modulo $2 π$ . Require a smooth axis extension; explicitly, assume there are $ϵ > 0$ and a smooth complex-valued $g : (- ϵ, ϵ) \to C$ such that $f (r) = r^{∣ ℓ ∣} g (r^{2})$ for $0 < r < \sqrt{ϵ}$ . Define $L_{z}^{s c} : = - i (- y \partial_{x} + x \partial_{y})$ and use the first-variable-linear $L^{2}$ pairing $⟨ v, w ⟩_{2} : = \int_{R^{2}} v (x, y) \overline{w (x, y)} d^{2} x .$ Then the normalized scalar angular-generator expectation is $\frac{⟨ L_{z}^{s c} u_{z}, u_{z} ⟩_{2}}{⟨ u_{z}, u_{z} ⟩_{2}} = ℓ .$

Statement references

Forward angular-spectrum beam

The complex exponential by its power series

Real and imaginary parts, complex conjugation, and modulus

Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$

The complex exponential is entire and its complex derivative is itself

Facts & Assumptions

Given: A fixed plane $z$ , the stated smooth $L^{2}$ scalar field $u_{z}$ with nonzero $L^{2}$ class and the stated integer-winding form.

[F1]

The paraxial scalar field and its SI field interpretation are those of the angular-spectrum Definition. Forward angular-spectrum beam Framework-local SI units, the vacuum constants and the exact/measured status of their values

[F2]

The real partial derivatives are those on the fixed laboratory spatial slice, and complex exponentiation satisfies $\exp^{'} = \exp$ . Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ The complex exponential is entire and its complex derivative is itself

[F3]

The field uses the complex exponential and conjugation/modulus operations defined on all complex numbers. The complex exponential by its power series Real and imaginary parts, complex conjugation, and modulus

[F4]

On each local polar-coordinate chart with $r > 0$ , the smooth map $(r, ϕ) \mapsto (x, y) = (r \cos ϕ, r \sin ϕ)$ has $\partial_{ϕ} x = - y$ and $\partial_{ϕ} y = x$ . The chain rule therefore gives $\partial_{ϕ} u_{z} = - y \partial_{x} u_{z} + x \partial_{y} u_{z}$ there. The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$

Proof

1.1givenF1F2F3F4algebra

For $r > 0$ , [F4] identifies the angular vector field $- y \partial_{x} + x \partial_{y}$ with $\partial_{ϕ}$ . By [F2]–[F3], differentiating the integer winding factor gives $L_{z}^{s c} u_{z} = - i \partial_{ϕ} (f e^{i ℓ ϕ}) = ℓ f e^{i ℓ ϕ} = ℓ u_{z}$ . The required axis form makes $u_{z}$ smooth at the origin: for $ℓ \geq 0$ its angular factor is $(x + i y)^{ℓ}$ , and for $ℓ < 0$ it is $(x - i y)^{∣ ℓ ∣}$ , each multiplied by the smooth radial factor $g (x^{2} + y^{2})$ . The rotation vector field vanishes at the origin, so the same identity holds there as well.

2.1step 1.1givenalgebra∎

Step 1.1 gives $L_{z}^{s c} u_{z} = ℓ u_{z}$ everywhere, hence $L_{z}^{s c} u_{z} \in L^{2}$ because $u_{z} \in L^{2} (R^{2}; C)$ . Therefore $⟨ L_{z}^{s c} u_{z}, u_{z} ⟩_{2} = \int_{R^{2}} ℓ ∣ u_{z} ∣^{2} d^{2} x = ℓ ⟨ u_{z}, u_{z} ⟩_{2}$ ; the denominator is positive since $u_{z}$ is nonzero. Division proves the claimed identity.

Remarks

The axis hypothesis is stated as a smooth extension condition; the bound $f (r) = O (r^{∣ ℓ ∣})$ alone would not guarantee smoothness. No radial integration by parts is used. The scalar generator is dimensionless and does not by itself determine a torque or a photon angular momentum.

5 · Examples, experiments and counterexamples

None yet.

Angular-spectrum optics and controlled paraxial limits

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Uniform phase-multiplier bound for the square-root dispersion law

Statement

Statement references

Facts & Assumptions

Proof

Forward angular-spectrum beam

Definition

Definition references

Facts & Assumptions

Verification

Remarks

Paraxial phase and longitudinal-field bounds

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Paraxial orbital winding in its exact scalar domain

Statement

Statement references

Facts & Assumptions

Proof

Remarks

5 · Examples, experiments and counterexamples

Sources