---
title: "Cross-dispersed Echelle Spectrograph Interactive Lab | Sandbox Physics"
description: "How can high diffraction orders separate fine colors without piling them onto the same pixels?"
canonical_url: "https://sandboxphysics.com/labs/echelle-spectrograph/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/echelle-spectrograph/"
last_modified: "2026-11-03"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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 |
| --- | --- | --- | --- |

Physics tutorial

## One camera direction is not enough for overlapping orders

**Background** A ruled reflective surface sends light into several bright directions. At a high diffraction order, two nearby wavelengths leave at more distinct angles. Another order can send a different wavelength along that same path.

**Why it matters** A camera needs a second way to tell those coincident colors apart before it can measure a narrow spectral line.

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 misconception** A bright pixel already comes with its wavelength.

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

#### A flat frame corrects detector response

**Typical misconception** Detector flat correction makes the spectrum an absolute photon flux.

**Better mental model** The 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

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Trace stripes from continuum pixels and calibrate their wavelengths with independent known-line exposures.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How an echelle spectrograph gives overlapping colors a second direction**](https://sandboxphysics.com/stories/instruments/echelle-spectrograph/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)