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Uniform rectangular aperture: the sinc-squared Fraunhofer pattern and its exact zeros and maxima

Physical scope

Exact computation in the adopted scalar aperture-diffraction model for a uniform rectangular aperture: the Fraunhofer amplitude $4 a b A_{0} sinc (k a n_{1}) sinc (k b n_{2})$ and the sinc-squared model intensity pattern with its zeros and central maximum, up to the explicit Fraunhofer error of the parent theorem. Excluded: measured diffraction records, absolute radiometric calibration, polarization and vector effects, screen interaction and aperture back-reaction, apparatus models, sampling laws and every empirical or statistical claim.

Example

Assume Countable Choice ( ${A C}_{ω}$ ) as carried by Controlled Fraunhofer and Fresnel estimates for the scalar aperture model, with intensity-error transfer and by the Riemann-Lebesgue comparison and Fubini suppliers used in the integral reduction below (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Work in the scalar aperture model of The adopted scalar aperture-diffraction model and Controlled Fraunhofer and Fresnel estimates for the scalar aperture model, with intensity-error transfer, with frequency $ω > 0$ , $k = ω / c$ , aperture plane $y_{3} = 0$ , model field $u = G_{k} * α$ and $I (x) = \frac{1}{2} ∣ u (x) ∣^{2}$ . Fix lengths $a > 0$ , $b > 0$ in metres and $A_{0} \in C ∖ {0}$ in $V m^{- 2}$ , so that $α$ has the model unit $V$ ; identify the rectangle $R : = [- a, a] \times [- b, b] \subset R^{2}$ with the planar aperture ${(y_{1}, y_{2}, 0) : (y_{1}, y_{2}) \in R}$ , let $ȷ : R^{2} \to R^{3}$ be $ȷ (y_{1}, y_{2}) = (y_{1}, y_{2}, 0)$ , and define the uniform rectangular aperture distribution as the plane-embedding pushforward

$α (E) : = A_{0} λ_{2} (R \cap ȷ^{- 1} (E)) (E \subseteq R^{3} Borel) .$

This is a finite complex measure on $R^{3}$ , supported on the embedded compact rectangle $S = ȷ (R)$ with $∣ y ∣ \leq a^{'} : = \sqrt{a^{2} + b^{2}}$ and total variation $∥ α ∥_{T V} = 4 a b ∣ A_{0} ∣$ (Axis-parallel rectangles in $R^{m}$ and their volume, The measure with density $f$ relative to $μ$ , Integration against a signed or complex measure, and the class L^1(nu) = L^1(|nu|)). Then: (a) [Fraunhofer amplitude] with $sinc ξ : = \sin ξ / ξ$ for $ξ \neq 0$ and $sinc 0 : = 1$ , the Fraunhofer amplitude factor of Controlled Fraunhofer and Fresnel estimates for the scalar aperture model, with intensity-error transfer(a) is exactly

$\int e^{- i k n \cdot y} d α (y) = 4 a b A_{0} sinc (k a n_{1}) sinc (k b n_{2}) (n = (n_{1}, n_{2}, n_{3}) \in S^{2}),$

an identity for every direction $n$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The complex exponential by its power series, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , The derivatives of sine and cosine are cosine and minus sine, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections, Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable, Riemann–Lebesgue comparison for distribution test integrands, The Lebesgue integral is linear on $L^{1} (μ)$ ). (b) [Fraunhofer intensity] in the Fraunhofer regime of Controlled Fraunhofer and Fresnel estimates for the scalar aperture model, with intensity-error transfer(a) with aperture radius $a^{'} : = \sqrt{a^{2} + b^{2}}$ , i.e. for $r \geq 2 a^{'}$ , the model intensity is

$I_{F} (r n) = \frac{a^{2} b^{2} ∣ A_{0} ∣^{2}}{2 π^{2} r^{2}} {sinc}^{2} (k a n_{1}) {sinc}^{2} (k b n_{2})$

up to the Fraunhofer absolute error of that theorem, transferred to $I$ by its clause (c). The pattern vanishes exactly where $k a n_{1} \in π Z ∖ {0}$ or $k b n_{2} \in π Z ∖ {0}$ , and, as a mathematical pattern on all $S^{2}$ , has its global maximum exactly at the two poles $n = (0, 0, \pm 1)$ . The model observes only $x_{3} > 0$ ; its allowed directions have $n_{3} > 0$ , where the central maximum is only $n = (0, 0, 1)$ . (c) [Scope] this is an exact computation inside the adopted ideal scalar model, not a measured diffraction record: no absolute radiometric calibration, polarization content, screen interaction, apparatus model or sampling law is asserted, and the Fraunhofer/Fresnel error bounds of Controlled Fraunhofer and Fresnel estimates for the scalar aperture model, with intensity-error transfer are inherited rather than re-estimated.

Facts & Assumptions

Given: Countable Choice; the data $a > 0$ , $b > 0$ , $A_{0} \in C ∖ {0}$ , the rectangle $R = [- a, a] \times [- b, b]$ , the embedding $ȷ (y_{1}, y_{2}) = (y_{1}, y_{2}, 0)$ , the pushforward $α$ of the Example, and the model field $u = G_{k} * α$ , $I = \frac{1}{2} ∣ u ∣^{2}$ of the scalar aperture model.

[F1]

Parent model and theorem: the model field and $I = \frac{1}{2} ∣ u ∣^{2}$ are those of The adopted scalar aperture-diffraction model, and Controlled Fraunhofer and Fresnel estimates for the scalar aperture model, with intensity-error transfer provides, for a finite complex measure supported in $∣ y ∣ \leq a_{0}$ with $r \geq 2 a_{0}$ and all $n \in S^{2}$ , the Fraunhofer estimate $u (r n) = e^{i k r} (4 π r)^{- 1} \int e^{- i k n \cdot y} d α (y) + error$ with $∣ error ∣ \leq ∥ α ∥_{T V} (π r)^{- 1} [a_{0} / r + k a_{0}^{2} / r]$ , uniformly in direction, together with the intensity-error transfer $∣ \frac{1}{2} ∣ u_{a p p r o x} ∣^{2} - I ∣ \leq \frac{1}{2} δ (∣ u ∣ + ∣ u_{a p p r o x} ∣)$ whenever $∣ u_{a p p r o x} - u ∣ \leq δ$ and the period-average identity of Period averages of products of real parts of complex exponentials.

[F2]

Rectangle, area and density measure: $R$ is a nondegenerate closed rectangle in $R^{2}$ with two-dimensional Lebesgue measure $λ_{2} (R) = 4 a b$ , its embedded image $S = ȷ (R)$ is compact with $∣ y ∣ \leq a^{'} = \sqrt{a^{2} + b^{2}}$ and the corners attaining $a^{'}$ , by Axis-parallel rectangles in $R^{m}$ and their volume and A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included. The nonnegative restricted measure $λ_{2} ∣_{R}$ has density $1_{R}$ as in The measure with density $f$ relative to $μ$ ; the complex pushforward itself is defined by the displayed formula in the Example and verified in step 1.1.

[F3]

Measure-theoretic toolbox: $ȷ$ is continuous, hence the inverse image of every Borel set is Borel; a finite complex measure on the Borel sets is countably additive and its total variation is as in A complex measure is a finite-valued countably additive set function and The total variation |nu|(E) from countable measurable partitions; the integral against it is defined by Integration against a signed or complex measure, and the class L^1(nu) = L^1(|nu|), and for any bounded Borel function $h$ on a finite-variation measure space, $h$ is integrable and $∣ \int h d α ∣ \leq \int ∣ h ∣ d ∣ α ∣ \leq ∥ h ∥_{\infty} ∥ α ∥_{T V}$ by Integrals against signed or complex measures are bounded by total variation; this includes the simple-function bound (A continuous map has Borel preimages of Borel sets, Simple integrals are bounded by total variation, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F4]

Integral comparison: for a bounded Borel function on a nondegenerate rectangle that is Riemann integrable, the Lebesgue integral exists and agrees with the Riemann integral (Riemann–Lebesgue comparison for distribution test integrands); a continuous function on a compact rectangle is Riemann integrable (Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable); the Riemann double integral factors through iterated one-dimensional integrals when the sections are integrable (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections); and the Lebesgue integral is complex-linear on $L^{1}$ (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F5]

One-dimensional calculus and the exponential: $e^{i θ} = \cos θ + i \sin θ$ , so $e^{- i k n_{1} y_{1}}$ decomposes into real and imaginary parts, and the complex exponential is the power-series exponential ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The complex exponential by its power series, Real and imaginary parts, complex conjugation, and modulus); the chain and algebra of derivatives, the sine and cosine derivatives and the second fundamental theorem give the antiderivative computations below (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivatives of sine and cosine are cosine and minus sine, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ); on the aperture $n \cdot y = n_{1} y_{1} + n_{2} y_{2}$ for the Euclidean inner product (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F6]

Sinc calculus: the zeros of sine are exactly the integer multiples of $π$ (The zero sets of sine and cosine and the least positive common period 2 pi); $\sin$ is odd, $\sin^{2} + \cos^{2} = 1$ , and cosine is strictly decreasing on $[0, π]$ with $\cos 0 = 1$ (Parity and the Pythagorean identity for sine and cosine, Signs, monotonicity intervals, and ranges of sine and cosine); continuous functions on closed intervals are Riemann integrable, the Riemann integral is monotone, and it is additive over subintervals (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ ).

Verification

technique · direct

1.1givenF2F3algebra

[The pushforward is a finite complex measure with the stated support and total variation] For disjoint Borel sets $E_{m} \subseteq R^{3}$ the inverse images $ȷ^{- 1} (E_{m})$ are disjoint and Borel by [F3], and countable additivity of $λ_{2}$ gives $α (⋃_{m} E_{m}) = A_{0} \sum_{m} λ_{2} (R \cap ȷ^{- 1} (E_{m})) = \sum_{m} α (E_{m})$ , while $α (\emptyset) = 0$ ; hence $α$ is a complex measure, and it is finite because $λ_{2} (R) = 4 a b$ by [F2]. If $E \cap S = \emptyset$ then $R \cap ȷ^{- 1} (E) = \emptyset$ , so $α$ is supported in $S$ . For every countable Borel partition ${E_{m}}$ of $R^{3}$ , the defining values sum to $∣ A_{0} ∣ \sum_{m} λ_{2} (R \cap ȷ^{- 1} (E_{m})) = ∣ A_{0} ∣ λ_{2} (R) = 4 a b ∣ A_{0} ∣$ by countable additivity and [F2]; taking the supremum over partitions gives $∥ α ∥_{T V} = 4 a b ∣ A_{0} ∣ < + \infty$ .

2.1step 1.1F2F3algebra

[Pairing identity] For a complex simple Borel function $s$ , its defining sum gives $\int s d α = A_{0} \int_{R} s \circ ȷ d λ_{2}$ , since each level set pulls back to $R \cap ȷ^{- 1} (E)$ , by [F3]. For bounded Borel $f$ , choose $M > 0$ bounding both real coordinates. For each positive integer $N$ , partition $[- M, M]$ at $t_{j} : = - M + 2 M j / N$ , $0 \leq j \leq N$ , into half-open intervals $[t_{j}, t_{j + 1})$ except for the last interval, which includes $M$ . On the Borel preimage of the cell with coordinate indices $j, k$ , set $s_{N} : = t_{j} + i t_{k}$ . This gives a finite-range Borel simple function with $∥ f - s_{N} ∥_{\infty} \leq 2 \sqrt{2} M / N$ . By [F3], the error in the left integral is at most $2 \sqrt{2} M ∥ α ∥_{T V} / N$ , and the error in the right integral is at most $8 a b ∣ A_{0} ∣ \sqrt{2} M / N$ , using $λ_{2} (R) = 4 a b$ . Passing to the limit in the simple-function identities proves $\int f d α = A_{0} \int_{R} f \circ ȷ d λ_{2}$ .

3.1step 2.1F4F5algebra

[Fraunhofer amplitude] Fix $n \in S^{2}$ and put $f (y) : = e^{- i k n \cdot y}$ , which is continuous on the compact support; by step 2.1, $\int e^{- i k n \cdot y} d α (y) = A_{0} \int_{R} e^{- i k (n_{1} y_{1} + n_{2} y_{2})} d λ_{2} (y_{1}, y_{2})$ because $y = ȷ (y_{1}, y_{2})$ and $n \cdot y = n_{1} y_{1} + n_{2} y_{2}$ by [F5]. The integrand is continuous on the compact rectangle $R$ , hence Riemann integrable, so the Lebesgue integral equals the Riemann double integral by [F4], and the Riemann double integral factors as the product of the one-dimensional integrals $\int_{- a}^{a} e^{- i k n_{1} y_{1}} d y_{1}$ and $\int_{- b}^{b} e^{- i k n_{2} y_{2}} d y_{2}$ by [F4]. For $n_{1} \neq 0$ the map $y_{1} \mapsto e^{- i k n_{1} y_{1}} / (- i k n_{1})$ is an antiderivative with derivative $e^{- i k n_{1} y_{1}}$ by the chain, algebra and trigonometric derivative rules of [F5], so the second fundamental theorem gives $\int_{- a}^{a} e^{- i k n_{1} y_{1}} d y_{1} = (e^{- i k n_{1} a} - e^{i k n_{1} a}) / (- i k n_{1}) = 2 \sin (k a n_{1}) / (k n_{1}) = 2 a sinc (k a n_{1})$ ; for $n_{1} = 0$ the integral is $2 a = 2 a sinc 0$ by the separate value of sinc. The second factor is analogous, giving the displayed identity $4 a b A_{0} sinc (k a n_{1}) sinc (k b n_{2})$ for every direction $n$ .

4.1step 3.1F1F2algebra

[Fraunhofer intensity] For $r \geq 2 a^{'}$ the support bound $∣ y ∣ \leq a^{'}$ of [F2] puts the model in the regime of the Fraunhofer estimate of [F1] with aperture radius $a^{'}$ , so $u (r n) = e^{i k r} (4 π r)^{- 1} \cdot 4 a b A_{0} sinc (k a n_{1}) sinc (k b n_{2}) + R (r n)$ with $∣ R (r n) ∣ \leq ∥ α ∥_{T V} (π r)^{- 1} [a^{'} / r + k (a^{'})^{2} / r]$ uniformly in direction by step 3.1. Taking moduli, $∣ u_{F} (r n) ∣ = \frac{a b ∣ A_{0} ∣}{π r} ∣ ⁣ sinc (k a n_{1}) ∣ ∣ ⁣ sinc (k b n_{2}) ∣$ , so $I_{F} (r n) = \frac{1}{2} ∣ u_{F} (r n) ∣^{2} = \frac{a^{2} b^{2} ∣ A_{0} ∣^{2}}{2 π^{2} r^{2}} {sinc}^{2} (k a n_{1}) {sinc}^{2} (k b n_{2})$ , and the intensity-error transfer of [F1] bounds $∣ I (r n) - I_{F} (r n) ∣$ by the Fraunhofer field error.

5.1step 4.1F6algebra

[Zeros of the pattern] By definition of sinc and the zero set of sine in [F6], $sinc ξ = 0$ exactly when $ξ \in π Z ∖ {0}$ ; since $k, a, b > 0$ and the two factors multiply, $I_{F} (r n) = 0$ exactly when $k a n_{1} \in π Z ∖ {0}$ or $k b n_{2} \in π Z ∖ {0}$ .

5.2step 4.1F2F6algebra

[Global maxima] First, $∣ \sin x ∣ \leq ∣ x ∣$ for every real $x$ , with equality exactly at $x = 0$ : for $x > 0$ choose $δ : = \min {x / 3, π / 6} > 0$ ; on $[δ, 2 δ] \subseteq [0, π / 2]$ the cosine is positive and strictly decreasing, so $∣ \cos t ∣ = \cos t \leq \cos δ < 1$ there, while $∣ \cos t ∣ \leq 1$ everywhere; the second fundamental theorem, the integral triangle bound and monotonicity of [F6] give $∣ \sin x ∣ = ∣ \int_{0}^{x} \cos t d t ∣ \leq \int_{0}^{x} ∣ \cos t ∣ d t \leq (x - δ) + δ \cos δ = x - δ (1 - \cos δ) < x$ , and for $x < 0$ oddness of sine in [F6] gives $∣ \sin x ∣ = ∣ \sin (- x) ∣ < ∣ x ∣$ , while $x = 0$ gives equality. Hence $∣ ⁣ sinc ξ ∣ \leq 1$ with equality exactly at $ξ = 0$ ; since $k, a, b > 0$ , the modulus of the product in step 4.1 equals the prefactor exactly when $k a n_{1} = 0$ and $k b n_{2} = 0$ , that is $n_{1} = n_{2} = 0$ , which on $S^{2}$ means $n = (0, 0, \pm 1)$ ; these two poles are therefore exactly the global maxima of the pattern on $S^{2}$ , and the model's half-space $x_{3} > 0$ forces $n_{3} > 0$ , leaving only the positive pole.

6.1step 3.1step 4.1step 5.2F1F3F4given∎

[Scope and choice accounting] The computation is exact inside the adopted scalar model: by [F1] it inherits the Fraunhofer and Fresnel error bounds rather than re-estimating them, it makes no measured-intensity, apparatus, sampling or calibration claim, and no polarization or screen-interaction content is introduced; only model comparisons of $I$ between observation points are read from it, which is clause (c). The only choice principle used is the Countable Choice carried by the parent theorem and by the Riemann-Lebesgue comparison and Fubini interfaces of [F4]; the remaining steps are explicit finite computations, so no further choice is made.

Remarks

The pattern's mathematical maxima on the whole sphere sit at the two poles, but the model field is only defined for $x_{3} > 0$ ; the observable central maximum is the single direction $n = (0, 0, 1)$ , and the lower pole belongs to the extension of the formula, not to the model's observation region.
The zeros $k a n_{1} \in π Z ∖ {0}$ or $k b n_{2} \in π Z ∖ {0}$ are exact zeros of the ideal Fraunhofer factor; a finite measurement could not verify exact darkness, and no such empirical claim is made here.

Uniform rectangular aperture: the sinc-squared Fraunhofer pattern and its exact zeros and maxima

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources