Optics 059 · Diffraction, Fourier optics, and computational imaging

Diffraction-Grating Spectrometer

A reflection-grating spectrometer with a real front-side optical route. Resolve sodium lines, change finite groove count and entrance slit, inspect blaze facets, calibrate detector pixels and diagnose overlapping orders with a filter.

Central angle / ideal resolving power

\beta_m,R_{\rm ideal}

$0.50$

Rayleigh limit / slit width in wavelength

\delta\lambda_R,\delta\lambda_{\rm slit}

$50\%$

Illuminated grating length / sensor pitch

L,p

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Diffraction-Grating Spectrometer

Groove density

\rho_g

600\,\mathrm{mm^{-1}}

Illuminated grooves

N_g

3000

Incidence from outward normal

\alpha

40^{\circ}

Actual groove blaze angle

\varphi

11.60^{\circ}

Uniform incoherent entrance slit width

w

10\,\mathrm{\mu m}

Ideal camera focal length

f

200\,\mathrm{mm}

Synthetic source long line (sodium fixed)

\lambda_0

589.30\,\mathrm{nm}

Tracked and calibrated order

m

1

Source: sodium / single / half-wavelength pair

s

0

Incident long-pass cutoff (zero = off)

\lambda_{\rm cut}

0\,\mathrm{nm}

Display gain exponent, raw data unchanged

g

1

Start with the sodium doublet. Widen the entrance slit until two peaks merge, then increase the groove count and compare the raw central detail. Move blaze: the groove facets change, the envelope changes, but ideal spectral positions do not. In overlap, first and second order land on the same pixel; insert the long-pass filter to remove the short wavelength before diffraction. Inspect the optical plane, uniformly magnified groove profile and actual sensor front. Camera tracks the chosen central order. Gain changes the coloured display only; Play marks the incoming route, not photons.

Scalar finite-groove reflection model and Casini–Nelson facet response, uniform incoherent slit average, ideal angular camera, 4096 finite-pixel records. Relative response, not absolute grating efficiency. NIST sodium air wavelengths; illustrative source weights, synthetic other sources, UV/IR false colours. Explicit gain; ideal references separate from acquired pixels. No coating, polarization, aberrations, noise or Maxwell solver. Microscopic groove detail is a separate uniform magnification, not a floating detector. Near-coincident two-arm geometry is obstructed; no Littrow beam splitter.

CONTINUE EXPLORING

Aperture, propagation, measurement

Change one physical assumption at a time: slit count, coherence, aperture shape or propagation distance.

Start with the essentials

Focus question: How do groove count, blaze, slit width, and order overlap set spectral resolution?
One-sentence intuition: An angle is not a unique wavelength: $m_1\lambda_1=m_2\lambda_2$ . Different wavelengths in different orders can share a pixel. A known order or an order-sorting filter is needed before assigning wavelength.

Core mathematical model

Signed reflection geometry

\begin{aligned}\sin\beta_m&=m\lambda/d-\sin\alpha\\\beta_0&=-\alpha\\|m\lambda/d-\sin\alpha|&\le1\end{aligned}

Normal is the positive laboratory horizontal axis, grooves vertical. Source direction and outgoing direction have positive normal components; the incoming propagation direction points toward the grating. Zero order obeys ordinary reflection. Orders outside the sine bound have no propagating ray.

Exact finite array, not invented peaks

\begin{aligned}t&=\frac d\lambda(\sin\alpha+\sin\beta)\\A_N(t)&=\frac1N\sum_{j=0}^{N-1}e^{2\pi ijt}\\J_\lambda&=E_\lambda\,|A_N|^2\end{aligned}

Complete uniformly illuminated grooves add complex amplitudes. Integer phase is a principal maximum; the first null is one over groove count away. Every order, its side lobes and its width come from this one array, multiplied by a scalar facet envelope. Normalization is fixed, not set by the current brightest pixel.

Blaze is facet reflection

\begin{aligned}\alpha+\beta_b&=2\varphi\\b&=d\frac{\cos\alpha}{\cos(\alpha-\varphi)}\\E_\lambda&=k\,\operatorname{sinc}^2\!\left(\frac{\pi b}{\lambda}S\right)\\S&=\sin(\alpha-\varphi)+\sin(\beta-\varphi)\end{aligned}

The uniformly magnified sawtooth has a rectangular apex and facet normal set by blaze. Casini–Nelson scalar facet integration uses the illuminated facet length, not its projected width. A shadow factor reduces outgoing directions hidden by neighbouring facets. This relative scalar response is not a calibrated metallic-grating efficiency or a polarization solution.

Ideal resolution is only one limit

\begin{aligned}R_{\rm ideal}&=|m|N\\\delta\lambda_R&=\lambda/(|m|N)\\\delta\lambda_{\rm slit}&\simeq\frac{d\cos\alpha}{|m|}\frac w{f_c}\end{aligned}

Rayleigh spacing applies to an ideal narrow line and zero slit. Uniform incoherent slit positions produce incidence angles through the ideal collimator, then intensities are averaged. The displayed slit wavelength width is the small-angle geometric full width, not a quadrature addition to Rayleigh spacing. The recorded profile also includes finite pixel averaging.

Calibrate the actual receiving plane

\begin{aligned}u&=f\tan(\beta-\beta_c)\\\lambda_m(u)&=\frac d m\,[\sin\alpha+\sin\beta(u)]\\\beta(u)&=\beta_c+\arctan(u/f)\end{aligned}

The camera arm tracks the central wavelength in the selected order. Its actual 12 mm sensor normal points back toward the camera pupil. The data row and the physical face use the same 4096 pixel records. Enlarged curves remain outside the apparatus; colour bands repeat a one-dimensional record vertically for visibility.

Overlap, then sort before diffraction

\begin{aligned}\lambda_2&=\lambda_1/2\\1\lambda_1&=2\lambda_2\\J_{\rm record}&=\sum_\ell w_\ell T(\lambda_\ell)\,\langle J_\ell\rangle\end{aligned}

The half-wavelength pair is an explicitly synthetic source, not a sodium emission spectrum. Its long first-order line and short second-order line share an angle. A perfect long-pass filter removes the short source contribution before it reaches the grating; surviving channel data stays unchanged. Infrared and ultraviolet are false-coloured, not visible light.

Common difficulties

More grooves do not fix every blur

Typical misconceptionThe ideal groove-count resolving power guarantees two lines will be separated, even with a wide entrance slit.

Better mental modelInspect acquired pixels. Slit averaging, sensor pitch and the chosen angular window can dominate. A calculated ideal resolving power is not a measured instrument resolution.

Run the experiment

01
Scene 1: Resolve sodium doublet
Use sodium. Compare a zero-width slit with a wide slit; vary groove count and inspect the central raw profile.
What to observe: More grooves narrow the finite-array peaks; a wide slit can still merge the sodium doublet.
02
Scene 2: Blaze without moving lines
Use blaze. Move the facet angle and inspect the uniformly magnified groove profile; compare the fixed ideal line tick.
What to observe: Blaze changes the facet normal and relative envelope without changing the grating-equation line positions.
03
Scene 3: Overlapping orders and filter
Use overlap. Read the candidate orders at the central pixel, insert the 500 nm long-pass filter, and compare both source channels.
What to observe: The filter removes a source channel, not a diffraction order by number; overlap is an order ambiguity at one pixel.