Diffraction Grating: Grating Equation and Resolving Power

Derivation

Diffraction Grating: Grating Equation and Resolving Power

D-208 Home PU-206 Threads light · waves Depends on Fraunhofer Diffraction as a Fourier Transform

Statement

For a grating of \(N\) identical slits with pitch \(d\), the Fraunhofer amplitude is the Fourier transform of the aperture, which factorises into a single-slit form factor and an array (structure) factor \(A(\theta)=\sum_{n=0}^{N-1}e^{-in\phi}\) with \(\phi=\tfrac{2\pi d}{\lambda}\sin\theta\). This yields principal maxima at \(d\sin\theta=m\lambda\), each of intensity \(\propto N^{2}\) and angular half-width \(\propto 1/N\), and hence a chromatic resolving power \(R=\lambda/\Delta\lambda = mN\).

Why it matters

The grating is the workhorse dispersive element of spectroscopy: unlike a prism, its dispersion follows from a single geometric condition, and its resolving power is set purely by the order \(m\) and the number of illuminated lines \(N\) — not by the material. The same interference algebra governs X-ray diffraction from crystal planes, phased-array antennas, and acousto-optic modulators.

Deriving the grating from the Fourier transform of the aperture makes the two competing scales explicit: the slit shape controls where energy goes (the envelope), while the periodic array controls the sharp lines. Separating them is what lets us reason about missing orders, blazing, and instrumental linewidth in one framework.

Assumptions

Fraunhofer (far-field) regime.If \(N^2 d^2/\lambda \gtrsim z\) the quadratic phase across the aperture cannot be dropped; the pattern is Fresnel, principal maxima blur and the FT relation between aperture and field no longer holds.

Coherent, monochromatic, normally incident plane-wave illumination.If the source has finite bandwidth or angular size, each component makes its own pattern and the maxima wash out — this finite coherence is precisely what caps the achievable \(R\).

Identical, equally spaced, infinitely thin (scalar) slits.Pitch or width errors spread each principal maximum and raise the background between orders; scalar theory drops polarization, so it fails for \(d\sim\lambda\) where vector/resonance effects (Wood's anomalies) appear.

Transmission is a genuine convolution \(t(x)=s(x)*\!\sum_n\delta(x-nd)\).If the slit response depends on position (e.g. shadowing at large angles, or a blaze that tilts each facet) the form and array factors no longer separate and the simple product \(U=S\cdot A\) breaks.

Finite but large \(N\), Rayleigh criterion adopted for "just resolved".Rayleigh is a convention (peak-on-first-zero); a different criterion (e.g. Sparrow, or a fitted-centroid analysis) shifts the numerical resolving power by an \(O(1)\) factor even though \(R\propto mN\) is unchanged.

Derivation

\[ t(x) = s(x) * \sum_{n=0}^{N-1}\delta(x-nd) \]

Model the grating aperture as a single-slit profile \(s(x)\) replicated at \(N\) equally spaced sites; convolution with a shifted delta is a rigid copy. A

\[ U(\theta) \;\propto\; \tilde{t}(k_x) = \int_{-\infty}^{\infty} t(x)\,e^{-i k_x x}\,dx, \qquad k_x = k\sin\theta = \frac{2\pi}{\lambda}\sin\theta \]

Prior result (Fraunhofer diffraction as a Fourier transform): the far-field amplitude is the FT of the aperture transmission evaluated at spatial frequency \(k_x=k\sin\theta\). A

\[ U(\theta) \;\propto\; \underbrace{\tilde{s}(k_x)}_{S(\theta)\ \text{form factor}}\;\underbrace{\sum_{n=0}^{N-1} e^{-i n k_x d}}_{A(\theta)\ \text{array factor}} \]

Convolution theorem: the FT of a convolution is the product of FTs. The FT of the delta comb is the phased sum over sites. This is the key separation of "what a slit does" from "how the array interferes". B

\[ A(\theta) = \sum_{n=0}^{N-1} e^{-in\phi} = \frac{1-e^{-iN\phi}}{1-e^{-i\phi}} = e^{-i(N-1)\phi/2}\,\frac{\sin(N\phi/2)}{\sin(\phi/2)}, \qquad \phi \equiv k_x d = \frac{2\pi d}{\lambda}\sin\theta \]

Finite geometric series summed in closed form, then symmetrised by factoring \(e^{-iN\phi/2}\) from the numerator and \(e^{-i\phi/2}\) from the denominator to expose real sines. B

\[ I(\theta) = |U|^2 \;\propto\; |S(\theta)|^2 \left[\frac{\sin(N\phi/2)}{\sin(\phi/2)}\right]^{2} \]

Intensity is the squared modulus; the pure phase \(e^{-i(N-1)\phi/2}\) drops out. The bracket is the interference (grating) factor; \(|S|^2\) is the diffraction envelope. A

\[ \lim_{\phi\to 2m\pi}\frac{\sin(N\phi/2)}{\sin(\phi/2)} = \pm N \;\Rightarrow\; I_{\text{peak}} \propto N^{2}\,|S|^2 \]

At \(\phi=2m\pi\) both sines vanish; l'Hôpital (or small-angle expansion about the zero) gives the ratio \(N\). Principal maxima therefore scale as \(N^2\): the coherent \(N\)-fold amplitude, squared. B

\[ \phi = 2m\pi \;\Rightarrow\; \frac{2\pi d}{\lambda}\sin\theta = 2m\pi \;\Rightarrow\; \boxed{\,d\sin\theta = m\lambda\,},\quad m\in\mathbb{Z} \]

Symbolic substitution of the principal-maximum condition into the definition of \(\phi\). Numbers enter only later. A

\[ \sin\!\Big(\frac{N\phi}{2}\Big)=0,\ \tfrac{N\phi}{2}=p\pi\ (p\neq Nm) \;\Rightarrow\; \Delta\phi_{\text{half}} = \frac{2\pi}{N},\qquad d\cos\theta\,\Delta\theta = \frac{\lambda}{N} \]

The numerator zeros nearest a principal maximum sit at \(N\phi/2=(Nm\pm1)\pi\), a phase half-width \(2\pi/N\). Differentiating the grating equation \(d\sin\theta=m\lambda\) at fixed \(m\) converts this to angular half-width \(\Delta\theta=\lambda/(Nd\cos\theta)\): the lines sharpen as \(1/N\). B

\[ \text{First zero at } d\sin\theta = \Big(m+\tfrac{1}{N}\Big)\lambda. \quad\text{Rayleigh: peak of }(\lambda+\Delta\lambda)\text{ on this zero:}\quad m(\lambda+\Delta\lambda) = \Big(m+\tfrac{1}{N}\Big)\lambda \]

Two lines are "just resolved" when the order-\(m\) principal maximum of \(\lambda+\Delta\lambda\) falls on the first minimum of \(\lambda\) at the same angle (Rayleigh criterion). Equate the two \(d\sin\theta\) values. B

\[ m\,\Delta\lambda = \frac{\lambda}{N} \;\Rightarrow\; \boxed{\,R \equiv \frac{\lambda}{\Delta\lambda} = mN\,} \]

Cancel \(m\lambda\) from both sides and rearrange. The resolving power depends only on the order and the number of illuminated lines. A

Result

\[ d\sin\theta = m\lambda, \qquad I_{\text{peak}}\propto N^{2}, \qquad \Delta\theta = \frac{\lambda}{Nd\cos\theta}, \qquad R = \frac{\lambda}{\Delta\lambda} = mN \]

Reading. Each spectral order \(m\) is diffracted to the angle where the path difference between adjacent slits, \(d\sin\theta\), is a whole number of wavelengths. Adding slits does not move the maxima — it makes them taller (\(\propto N^2\)) and narrower (\(\propto 1/N\)), which is why resolving power grows linearly with \(N\). Going to higher order \(m\) also improves resolution proportionally, at the cost of overlapping orders and reduced free spectral range.

Units check. \(d\sin\theta\) is a length \(=\) \(m\lambda\) a length (\(m\) dimensionless, \(\sin\theta\) dimensionless). \(\Delta\theta = \lambda/(Nd\cos\theta)\) is (length)/(length) \(=\) radians. \(R=\lambda/\Delta\lambda = mN\) is dimensionless — a pure number, as a resolving power must be.

Limiting cases

\(N=1\): array factor \(\to 1\); only the single-slit envelope survives — no sharp orders, \(R\to0\).
\(N=2\) (Young's double slit): \(\big[\cos(\phi/2)\big]^2\) fringes, broad maxima, \(R=2m\) — the small-\(N\) end of the same formula.
\(N\to\infty\): principal maxima become delta-like; \(R\to\infty\) formally, capped in practice by source coherence and aberrations.
\(m=0\): \(d\sin\theta=0\) for all \(\lambda\); the zeroth order is undispersed (white) and carries no spectral information, \(R=0\).
\(d\to\lambda\): only \(m=0,\pm1\) exist (since \(|\sin\theta|\le1\)); at \(d<\lambda\) a single normally incident beam gives only \(m=0\).

Breaks when

Order does not exist: if \(|m\lambda/d|>1\) there is no real \(\theta\); the grating equation has no solution and that order is simply absent.
Missing orders: when a principal maximum coincides with a zero of the single-slit envelope (\(d/a = m/m'\) for integer ratios, \(a\) the slit width), \(|S|^2=0\) suppresses that order despite \(d\sin\theta=m\lambda\) being satisfied.
Fresnel / near field: at insufficient propagation distance the FT relation fails, maxima broaden and shift, and \(R=mN\) no longer applies.
Finite source bandwidth or divergence: the incident light is not truly monochromatic/collimated, so the instrumental linewidth is set by the source, not by \(N\); adding lines stops improving \(R\).
\(d\lesssim\lambda\) (subwavelength): scalar theory breaks; polarization-dependent resonances (Wood's anomalies) and evanescent orders require a full vector (rigorous coupled-wave) treatment.

Failure modes

Confusing \(d\) with slit width \(a\): \(d\) is the period (line spacing); \(a\) sets the envelope. Using "lines/mm" as \(a\) mislocates the maxima.
Reading "lines/mm" as \(d\) directly: \(d = 1/(\text{lines per unit length})\); 600 lines/mm means \(d=1/600\) mm, not \(600\) of anything.
Treating maxima as evenly spaced in \(\theta\): they are evenly spaced in \(\sin\theta\); the angular gaps widen with \(m\).
Using \(R = mN\) with the total ruled lines instead of the illuminated ones: only the \(N\) lines the beam actually covers contribute.
Forgetting the envelope can kill an order: quoting an order that is a missing order because it lands on a single-slit zero.
Small-angle-approximating a grating: grating angles are large; \(\sin\theta\approx\theta\) is usually invalid and \(\cos\theta\neq1\) in \(\Delta\theta\).
Dropping \(m\) in resolving power: assuming \(R=N\); it is \(mN\), so second order doubles the resolution.

Discussion

The physical content of the derivation is the clean factorisation \(U = S(\theta)\,A(\theta)\). The array factor \(A\) is periodic in \(\phi\) with period \(2\pi\) and knows only about the lattice of slits — its zeros and peaks give the sharp orders. The form factor \(S\) is aperiodic and knows only about the shape of one slit — it modulates the heights of those orders. This is exactly the crystallographer's split into a "reciprocal lattice" (where the Bragg spots sit) and a "structure/form factor" (how bright each spot is). The grating equation \(d\sin\theta=m\lambda\) is the one-dimensional Laue condition; \(d\sin\theta = m\lambda\) and \(2d\sin\theta=m\lambda\) (Bragg) are the same interference statement counted from different geometries.

The \(N^2\) peak height and \(1/N\) width are two faces of one fact: energy is conserved, so as each principal maximum narrows its peak must rise to keep the integrated intensity \(\propto N\). That the resolving power is \(mN\) and not, say, \(N^2\) is because resolution is set by the ratio of line separation to line width in wavelength, and both the dispersion (\(\propto m\)) and the sharpness (\(\propto N\)) enter linearly. Higher orders resolve better because a given \(\Delta\lambda\) produces a larger angular splitting when the path difference is counted more times.

There is a coherence-theoretic ceiling that the ideal formula hides. Writing \(R=mN\) presumes every illuminated slit radiates in fixed phase relation to every other — i.e. the transverse coherence length of the beam spans all \(N\) slits, and the temporal coherence length \(\ell_c\sim\lambda^2/\Delta\lambda_{\text{source}}\) exceeds the maximum path difference \(Nd\sin\theta = mN\lambda\). The condition \(\ell_c \gtrsim mN\lambda\) rearranges to \(\lambda/\Delta\lambda_{\text{source}} \gtrsim mN\): the source's own coherence must already be at least as good as the grating's ideal \(R\). This is not a coincidence — it is the statement that an interferometer cannot manufacture spectral information finer than the light's own coherence permits, and it is the deep reason real spectrographs are coherence- or aberration-limited long before they are \(N\)-limited.

Common misconceptions. A grating does not "split white light into colours" by refraction the way a prism does — it does so by interference, and its dispersion runs the opposite way (blue deviates least, red most, within an order). More slits do not brighten a given wavelength by moving its maximum; they concentrate the same average power into narrower, taller lines. And the zeroth order is genuinely useless for spectroscopy: all wavelengths pile up at \(\theta=0\).

Worked examples

\[ \textbf{First- and second-order angles for sodium light.} \]

Grating of 600 lines/mm, wavelength \(\lambda = 589\ \text{nm}\), normal incidence. Find \(\theta_1,\theta_2\). A

\[ d = \frac{1}{600\ \text{mm}^{-1}} = 1.667\times10^{-3}\ \text{mm} = 1.667\ \mu\text{m} \]

Pitch is the reciprocal of lines per unit length. A

\[ \sin\theta_m = \frac{m\lambda}{d}:\quad \sin\theta_1 = \frac{589\times10^{-9}}{1.667\times10^{-6}} = 0.3534,\quad \sin\theta_2 = 0.7068 \]

Grating equation solved for \(\sin\theta\), symbol first then numbers. A

\[ \theta_1 = 20.7^\circ,\qquad \theta_2 = 45.0^\circ \]

Reading. The two orders are far from evenly spaced in angle (20.7° then a 24° jump) because they are evenly spaced in \(\sin\theta\), not \(\theta\). A third order would need \(\sin\theta_3=1.06>1\) — it does not exist. Units check. \(m\lambda/d\) is (length)/(length), dimensionless, as \(\sin\theta\) must be.

\[ \textbf{Resolving the sodium doublet.} \]

Lines at \(\lambda = 589.0\) and \(589.6\ \text{nm}\) (\(\Delta\lambda = 0.6\ \text{nm}\)). With the same 600 lines/mm grating, what illuminated width \(w\) is needed in first order? B

\[ R = \frac{\lambda}{\Delta\lambda} = \frac{589.0}{0.6} = 982 \]

Required resolving power from the mean wavelength and the separation. A

\[ R = mN \;\Rightarrow\; N = \frac{R}{m} = \frac{982}{1} = 982\ \text{lines} \]

Invert the resolving-power law at order \(m=1\). A

\[ w = \frac{N}{\text{lines per mm}} = \frac{982}{600\ \text{mm}^{-1}} = 1.64\ \text{mm} \]

Convert number of illuminated lines to physical beam width using the line density. A

\[ w \approx 1.64\ \text{mm}\ \ (m=1); \qquad w \approx 0.82\ \text{mm}\ \ (m=2) \]

Reading. A beam only 1.6 mm wide already resolves the sodium doublet in first order; second order halves the required aperture because \(R=mN\) doubles with \(m\). This assumes the source itself has \(R\gtrsim10^3\) of coherence. Units check. \(N\) (lines) divided by lines/mm gives mm, a length.

Problems

A grating has 300 lines/mm. For \(\lambda = 500\ \text{nm}\) at normal incidence, how many diffraction orders (including \(m=0\)) are observable?
Solution
\(d = 1/300\ \text{mm} = 3.333\ \mu\text{m}\). Require \(|m\lambda/d|\le1\Rightarrow |m|\le d/\lambda = 3.333\times10^{-6}/500\times10^{-9} = 6.67\), so \(m=0,\pm1,\dots,\pm6\). That is \(13\) orders total (one \(m=0\) plus six on each side). Order 7 would need \(\sin\theta=1.05>1\).
A grating of 1200 lines/mm is illuminated over a width of 2.0 cm. What is its resolving power and minimum resolvable \(\Delta\lambda\) at \(\lambda = 500\ \text{nm}\) in second order?
Solution
Illuminated lines \(N = 1200\ \text{mm}^{-1}\times 20\ \text{mm} = 2.4\times10^4\). \(R = mN = 2\times2.4\times10^4 = 4.8\times10^4\). \(\Delta\lambda = \lambda/R = 500\ \text{nm}/4.8\times10^4 = 1.0\times10^{-2}\ \text{nm} = 10\ \text{pm}\).
Show that if the single-slit width is \(a=d/3\), the third-order principal maximum is missing.
Solution
The envelope \(|S|^2\propto\big[\sin(\beta)/\beta\big]^2\) with \(\beta=(\pi a/\lambda)\sin\theta\) has zeros when \(a\sin\theta = m'\lambda\). Principal maxima obey \(d\sin\theta=m\lambda\), i.e. \(\sin\theta=m\lambda/d\). At \(m=3\): \(a\sin\theta = (d/3)(3\lambda/d)=\lambda\), which is a single-slit zero (\(m'=1\)). Since \(|S|^2=0\) there, the \(m=3\) order is suppressed regardless of the interference factor. In general orders \(m = (d/a)m'\) are missing; here \(d/a=3\), so \(m=3,6,9,\dots\) vanish.
Light of \(\lambda=633\ \text{nm}\) hits a 500 lines/mm grating at \(20^\circ\) incidence. Find the first-order (\(m=1\)) diffraction angle. (Use \(d(\sin\theta_i + \sin\theta_m)=m\lambda\) for oblique incidence.)
Solution
\(d = 1/500\ \text{mm} = 2.00\ \mu\text{m}\). \(\sin\theta_1 = m\lambda/d - \sin\theta_i = \dfrac{633\times10^{-9}}{2.00\times10^{-6}} - \sin20^\circ = 0.3165 - 0.3420 = -0.0255\). So \(\theta_1 = -1.46^\circ\) (on the opposite side of the normal from a naive normal-incidence estimate). The oblique term shifts the order noticeably.
A grating spectrograph must resolve \(\Delta\lambda = 0.010\ \text{nm}\) at \(\lambda = 600\ \text{nm}\) working in first order. If the grating has 1800 lines/mm, what minimum illuminated width is required, and what is the smallest \(N\) achievable if instead you use third order?
Solution
Required \(R = \lambda/\Delta\lambda = 600/0.010 = 6.0\times10^4\). First order: \(N = R/m = 6.0\times10^4\); width \(w = N/1800\ \text{mm}^{-1} = 33.3\ \text{mm} = 3.33\ \text{cm}\). Third order: \(N = R/m = 6.0\times10^4/3 = 2.0\times10^4\); width \(w = 2.0\times10^4/1800 = 11.1\ \text{mm} = 1.11\ \text{cm}\). Higher order cuts the needed aperture threefold (at the cost of order overlap / reduced free spectral range).