= Diffraction order
{title2=$m$}
A diffraction order is the <integer> $m$ labeling a principal <constructive interference> direction of a <diffraction grating>: adjacent elements differ in path length by $m$ <wavelengths>. In the normal-incidence transmission geometry described above,
$$
\sin\theta_m=\frac{m\lambda}{d},\qquad \left|\frac{m\lambda}{d}\right|\leq1.
$$
Positive and negative <diffraction orders> lie on opposite sides of the forward normal. The condition expresses the range of the <sine> of a real <angle>, so it also limits the propagating <diffraction orders>.
For $\lambda=600\,\mathrm{nm}$ and $d=2\,\mathrm{\mu m}$, $\sin\theta_m=0.3m$. Taking the <inverse sine> gives
$$
\begin{aligned}
m=0 &: \quad \theta_0=0^\circ,\\
m=\pm1 &: \quad \theta_m=\pm17.458^\circ,\\
m=\pm2 &: \quad \theta_m=\pm36.870^\circ,\\
m=\pm3 &: \quad \theta_m=\pm64.158^\circ.
\end{aligned}
$$
These are seven allowed propagating <diffraction orders>. Order $m=4$ would require $\sin\theta_4=1.2$, and $m=-4$ would require $-1.2$, so neither has a propagating far-field direction. The allowed <diffraction orders> are equally spaced in $\sin\theta$, while their actual <angles> become farther apart toward grazing propagation.
Each principal peak has finite width. For the ideal uniform coherent array, the nearest zeros around order $m$ obey
$$
\sin\theta_{m,\pm}=\frac{m\lambda}{d}\pm\frac{\lambda}{Nd}.
$$
Thus the one-sided distance from the order direction to the first zero is $\Delta(\sin\theta)=\lambda/(Nd)$. Near the first <diffraction order> in our example, $N=10$ puts the zeros at $0.27$ and $0.33$, while $N=40$ puts them at $0.2925$ and $0.3075$. The respective one-sided widths are $0.03$ and $0.0075$; the full zero-to-zero widths are $0.06$ and $0.015$. These are calculated coherent peak widths, with the slit envelope omitted, and do not describe a measured instrument response. https://farside.ph.utexas.edu/teaching/315/Waveshtml/node93.html[Richard Fitzpatrick: Principal maxima and their neighbouring zeros, equations 10.39-10.42]{ref}.
\Image[/finite-grating-interference.png]
{title=<Single-slit diffraction> envelope and calculated first-order peak widths for $N=10$ and $N=40$}
Four times as many illuminated slits therefore give a fourfold narrowing in this coordinate without moving the <diffraction order>. That sharper directional selection helps distinguish nearby <wavelengths> in a <spectrograph>. Actual performance also depends on illumination, <optical coherence>, and source width; the separate <slit-limited resolving power of a grating> describes the contribution of the entrance slit.
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