IN17 · Give overlapping orders another direction

Cross-dispersed Echelle Spectrograph

How can high diffraction orders separate fine colors without piling them onto the same pixels? High orders spread nearby wavelengths farther apart, but different orders can send different colors in the same direction. A second disperser separates them in the perpendicular direction. The camera then records separate spectral stripes, each calibrated with known wavelengths.

Rotate an echelle grating, separate two orders across a real camera plane, and recover wavelength from stored detector pixels and independent lamp frames.

Order 111 largest bin $548.55\,\mathrm{nm}$

Order 110 largest bin $553.56\,\mathrm{nm}$

Minimum trace spacing $41.0\,\mathrm{px}$

Largest withheld-line error $0.000125\,\mathrm{nm}$

Give each order its own stripe

Slit width

w/\mathrm{\mu m}

20

Echelle rotation

\theta/{}^\circ

0

Cross-disperser ruling

n_c/\mathrm{mm}^{-1}

900

Camera pointing

\psi/{}^\circ

0

Source doublet spacing

\Delta\lambda/\mathrm{nm}

0.05

Drag the gold echelle or its mount to rotate it. Changed optics need fresh lamp frames; the saved observations keep their own calibration.

Two raw spectral orders are ready.

Detector, extraction and model

Extraction half-width in pixelsSource brightness multiplier

1

Exposure in seconds

1

Random seed

Vacuum wavelengths; column increases along camera U and row increases along camera V. The two order identities are manually assigned to synthetic known-line standards. A separate line checks each wavelength fit.

Finite slit and Gaussian main-lobe blur precede pixel-area integration and shot/read noise. An independent uniform flat measures detector response. Spectra retain aperture losses and have flat-corrected ADU units; absolute flux is not calibrated.

Saturation, merged orders, missed camera or failed line checks withhold wavelength inference. Order colors identify stripes. Sparse groove markings, 100-times slit opening and slowed light markers are schematic. Polarization, scattered light and detailed lens aberrations are omitted.

Model assumptions

A compact in-plane quasi-Littrow teaching mount uses a 31.6-groove/mm reflective echelle and a perpendicular thin transmission grating. It is a companion to the ordinary grating Lab. All wavelengths are vacuum nanometres. A two-band source illuminates orders 110 and 111 inside a finite camera; other orders and losses are included in photon allocation. Scalar blaze weights, a Gaussian main-lobe PSF, finite slit width, pixel integration, Poisson counts and read noise are modeled. Detector flat illumination is independent of lamp throughput. Continuum traces and manually assigned synthetic known-line standards determine extraction and wavelength fits; independent check lines are withheld. Saturation, order overlap, clipped apertures or stale calibration withhold wavelength inference. Polarization, scattered light, real lens aberrations, atmospheric refraction and absolute radiometric calibration are omitted. Slit openings are magnified 100 times; moving light markers show routes at a schematic speed.

Pixels first, wavelengths after calibration

Raw camera observation

$c/\mathrm{px}\quad;\quad r/\mathrm{px}\quad(\mathrm{ADU})$

Log display above a 50-ADU display pedestal; row zero is at the bottom. Colored outlines show the actual extraction apertures. Drag vertically to change their half-width.

Two extracted spectra

$\lambda_{\mathrm{vac}}/\mathrm{nm}\quad;\quad C_{\mathrm{flat}}/\mathrm{ADU}$

Turquoise is order 111; amber is order 110. Largest-bin labels do not split unresolved source lines. Invalid fits leave only the raw camera visible.

Lamp frames, checks and saved observations

Inspect a calibration exposure

\widehat\lambda(c)=a_0+a_1\xi+a_2\xi^2,\qquad\xi=\frac{c-63.5}{64}

Four line centroids per order determine the quadratic. The fifth wavelength is checked only afterwards. Exports retain all six raw frames, axes, seeds, standard definitions, pixel windows and processing results.

Frozen observations
Observation	Calibration	Aperture	Valid

Start with the essentials

Focus question: How can high diffraction orders separate fine colors without piling them onto the same pixels?
One-sentence intuition: A perpendicular disperser turns coincident orders into separate camera stripes.

Core mathematical model

Directions of a reflective grating

m\lambda=d(\sin\alpha+\sin\beta),\qquad d=(31.6\,\mathrm{mm}^{-1})^{-1}

Angles are measured from the outward grating normal, with incidence and reflection on opposite tangent sides. Wavelength is vacuum wavelength; use the same length unit for pitch and wavelength. Order is an integer. The scene uses a fixed 80-degree incident beam and a small echelle rotation; the two arms have a separate 8-degree narrow-beam mount clearance.

Perpendicular wavevector change

u_{\mathrm{out}}=u_{\mathrm{in}},\qquad v_{\mathrm{out}}=v_{\mathrm{in}}+\lambda n_c,\qquad w_{\mathrm{out}}=\sqrt{1-u_{\mathrm{out}}^2-v_{\mathrm{out}}^2}

The thin transmission grating has grooves along its U direction and adds tangential wavevector along V. Direction cosines share its orthogonal local frame; wavelength in millimetres times rulings per millimetre is dimensionless. The positive root transmits forward. A linked camera mount follows the fixed central reference wavelength as ruling density changes.

Extract from corrected camera pixels

C_m(c)=\sum_{r\in\mathcal A_m(c)}\left[\frac{D(r,c)-B(r,c)}{F(r,c)}-b_m(c)\right]

Raw ADU minus a saved equal-exposure dark is divided by the normalized uniform detector flat. Actual row windows follow quadratic traces fitted from continuum pixels. Median side rows estimate background. The sum has flat-corrected ADU units and retains finite-aperture losses. Four known line centroids per order fit wavelength; a fifth line checks it independently.

Common difficulties

The camera is not a wavelength ruler yet

Typical misconceptionA bright pixel already comes with its wavelength.

Better mental modelPixel coordinates become wavelengths only after known-line calibration and an independent line check.

A flat frame corrects detector response

Typical misconceptionDetector flat correction makes the spectrum an absolute photon flux.

Better mental modelThe independent uniform flat removes relative pixel response. Blaze, aperture throughput and other optical losses remain; the extracted spectrum is not absolutely flux calibrated.

Run the experiment

01
Follow the two arms
Open Separated arms, then drag the gold echelle slowly.
What to observe: The incident and reflected beams are distinct. Camera points shift with the same ray equations; changed optics require fresh lamp frames.
02
Bring the stripes together
Compare Two clean orders, Orders move closer and Orders can no longer be extracted.
What to observe: Reducing cross dispersion decreases camera-row separation. Once the continuum no longer resolves two traces, extraction stops.
03
Blur within an order
Compare A wide slit blurs colors and A pair merges within one order.
What to observe: Separating orders does not remove the slit and diffraction width within each stripe. The processing never inserts hidden source-line identities.
04
Change processing after saving
Keep this observation, then drag vertically on the raw image to change the aperture.
What to observe: The preview refits and extracts the same pixels using a new window. Saved pixels, lamp frames and processing remain unchanged in CSV and JSON.

Cross-dispersed Echelle Spectrograph

Pixels first, wavelengths after calibration

Raw camera observation

Two extracted spectra

One camera direction is not enough for overlapping orders

Core mathematical model

Directions of a reflective grating

Perpendicular wavevector change

Extract from corrected camera pixels

Common difficulties

The camera is not a wavelength ruler yet

A flat frame corrects detector response

Run the experiment

Follow the two arms

Bring the stripes together

Blur within an order

Change processing after saving

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