= Single-slit diffraction
{title2=$I(\theta)/I(0)=(\sin u/u)^2,\quad u=\pi a\sin\theta/\lambda$}
<Single-slit diffraction> is the spreading of <light> transmitted by one slit. Consider an ideal rectangular slit of width $a$, illuminated normally by a uniform <plane wave> with a single <wavelength> $\lambda$ in the surrounding medium. This <monochromatic wave> has uniform <wave amplitude> and equal incident <wave phases> across the slit, providing the required <optical coherence>. In the scalar <Fraunhofer diffraction> model, the normalized angular <wave intensity> is
$$
\frac{I(\theta)}{I_0}=\left(\frac{\sin u}{u}\right)^2,\qquad u=\frac{\pi a\sin\theta}{\lambda},\qquad I_0=I(0),\qquad \lim_{u\to0}\frac{\sin u}{u}=1.
$$
Here $\theta$ is measured from the forward normal in the plane across the slit width. The <sinc function> uses the unnormalized convention $\operatorname{sinc}u=\sin u/u$. The <limit> makes the central value one. This ideal envelope neglects additional direction-dependent propagation and detector factors; near the forward direction it describes the normalized screen <irradiance>. https://openstax.org/books/university-physics-volume-3/pages/4-2-intensity-in-single-slit-diffraction[Ling, Sanny and Moebs, University Physics Volume 3, §4.2: single-slit intensity]{ref}.
The noncentral zeros satisfy
$$
a\sin\theta_m=m\lambda,\qquad m\in\mathbb Z\setminus\{0\},\qquad \left|\frac{m\lambda}{a}\right|\leq1.
$$
The restriction follows from the range of the <sine> of a real <angle>. The first minima have $\sin\theta=\pm\lambda/a$ when $\lambda/a\leq1$. If $a<\lambda$, this model has no noncentral zero in the forward hemisphere; equality puts the first zeros at grazing directions, outside the <small-angle approximation> and off any finite perpendicular screen. https://openstax.org/books/university-physics-volume-3/pages/4-1-single-slit-diffraction[Ling, Sanny and Moebs, University Physics Volume 3, §4.1: positions of diffraction minima]{ref}.
For $0<\lambda/a<1$, define the central width as the separation between the two first minima. Its angular width and its width on a perpendicular screen are
$$
\Delta\theta=2\arcsin(\lambda/a)\approx\frac{2\lambda}{a},\qquad y=L\tan\theta,\qquad W=2L\tan\!\bigl(\arcsin(\lambda/a)\bigr)\approx\frac{2L\lambda}{a}.
$$
The angular approximation uses <radians> and requires $\lambda/a\ll1$; the screen approximation uses $y\approx L\theta$. The <tangent> relation is exact straight-line geometry, even though the underlying <Fraunhofer diffraction> field is approximate. This width is distinct from the <full width at half maximum>.
For $\lambda=500\,\mathrm{nm}$, $a=0.1\,\mathrm{mm}$ and $L=1\,\mathrm m$, the positive first minimum is
$$
\theta_1=\arcsin(0.005)\approx0.286480^\circ,\qquad y_1=\frac{L(0.005)}{\sqrt{1-0.005^2}}\approx5.00006\,\mathrm{mm},\qquad W\approx10.0001\,\mathrm{mm}.
$$
The dimensionless far-field parameter is $a^2/(\lambda L)=0.02$. Doubling the slit width to $a=0.2\,\mathrm{mm}$ gives $W\approx5.00002\,\mathrm{mm}$ and parameter $0.08$; both curves below use the <Fraunhofer diffraction> approximation. Choosing $L=2\,\mathrm m$ halves those parameters and doubles both screen widths.
\Image[/single-slit-diffraction-width.png]
{title=Normalized <single-slit diffraction> on a screen: a narrower slit produces a wider central maximum}
Each curve is normalized to its own $I_0$; the arrows measure between first minima. The calculation uses $\theta=\arctan(y/L)$ before evaluating the <sinc function>, rather than replacing $\sin\theta$ by $y/L$. The lower diagram shows the exact screen geometry with the vertical scale enlarged. Normalization removes the change in peak brightness when the slit width changes.
The <wave interference> explanation is especially simple: divide the slit into equal narrow strips and add their <complex amplitudes>, rather than their <wave intensities>. With <wavenumber> $k=2\pi/\lambda$, the continuous slit gives
$$
\frac{A(\theta)}{A(0)}=\frac1a\int_{-a/2}^{a/2}e^{ikx\sin\theta}\,dx=\frac{e^{iu}-e^{-iu}}{2iu}=\frac{\sin u}{u}.
$$
Integrating the <complex exponential> gives the endpoint difference; its continuous value at zero is one. Taking the squared <complex modulus> gives the stated <wave intensity>. This is the <Fourier transform of an interval indicator> expressed in angular coordinates. Feynman's continuous-source argument obtains the same pattern by letting the number of sources tend to infinity. https://www.feynmanlectures.caltech.edu/I_30.html#Ch30-S2[Feynman, Leighton and Sands, I §30–2: continuous distributions of oscillators]{ref}.
At the positive first minimum, the <wave phases> span exactly $2\pi$ across the slit. Every strip in one half has a partner $a/2$ away, with equal <wave amplitude> and <phase difference> $\pi$, so the pairs cancel. At $\theta=0$, all the <wave phases> align instead. https://openstax.org/books/university-physics-volume-3/pages/4-1-single-slit-diffraction[Ling, Sanny and Moebs, §4.1: cancellation between the slit halves]{ref}.
\Image[/single-slit-aperture-phase-cancellation.png]
{title=<Complex amplitudes> across a slit: aligned contributions at broadside and cancellation at the first minimum}
The drawing uses $N=8$ equally weighted midpoint samples $x_j/a=-1/2+(j+1/2)/8$, with $j=0,\ldots,7$. At the first minimum their <complex amplitudes> are a common rotation of the eight <roots of unity>, so their head-to-tail polygon closes exactly. This finite sum illustrates cancellation; the continuous <integral>, rather than eight point sources, defines the slit pattern at general <angles>. The circle shows the continuous range of <wave phases>; coloured arrows show the eight samples.
A narrower <aperture> requires a larger <angle> to accumulate the cancelling <phase difference>, so narrowing the slit widens the beam. This is a useful limit on <geometrical optics>: confining <light> more tightly at an opening increases its subsequent angular spread. At fixed incident <irradiance> and slit height, reducing $a$ also reduces transmitted <power>, exposing a practical balance between <diffraction-limited resolution> and light throughput. In a <diffraction grating>, this <single-slit diffraction> envelope modulates the <wave interference> between different slits.
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