MATLAB Spectrometer Spectrum Generation Tutorial | Engineering Lab

Light dispersion is the phenomenon where different wavelengths refract at different angles when passing through material media. Isaac Newton first explained this, demonstrating that colors come from light decomposition rather than substance properties. Spectroscopy analyzes electromagnetic radiation emitted or absorbed by substances, with each element producing a characteristic spectrum. There are two types: emission spectra (from excited substances releasing characteristic frequencies) and absorption spectra (from substances absorbing specific frequencies). The visible light spectrum consists of seven colors: red, orange, yellow, green, cyan, blue, and violet.

Spectroscopy disperses light into component wavelengths using diffraction gratings, which work by exploiting interference: light passing through closely spaced parallel lines creates path length differences causing constructive and destructive interference at different angles. Longer wavelengths spread more than shorter ones, separating light into a spectrum. Astronomical spectrographs use slit masks to isolate target objects and prevent spectral overlap, dichroic mirrors to separate wavelengths, and detectors to record the dispersed light. The resulting two-dimensional image shows the spectrum along one axis and spatial position along another.

Diffraction gratings separate light into its component wavelengths through constructive interference. Each slit acts as a secondary wave source (Huygens principle), causing different wavelengths to diffract at different angles. The grating equation determines diffraction angles for each wavelength. Higher diffraction orders spread wavelengths more widely, but standard gratings concentrate most energy in the first order. This dispersion enables spectrographs to produce spectra where wavelength is mapped to position.

This section covers the foundational concepts of light dispersion and spectroscopy. It begins with the applications of dispersion phenomena in understanding natural phenomena like rainbows and in spectrometers. The lesson reviews key optical concepts including the optical center, principal axis, focal points, and focal planes of lenses. It then introduces the dispersion of white light through prisms, explaining how white light separates into a continuous spectrum of colors from red to violet. This sets the stage for understanding how spectrometers work by utilizing the dispersion principle to analyze light sources.

Dispersion is the splitting of white light into constituent colors when passing through a prism. Different colors travel at different speeds in the same medium due to varying refractive indices. Red light has maximum speed and minimum refractive index, bending least; violet light has minimum speed and maximum refractive index, bending most. According to Cauchy's formula μ = a + b/λ² + c/λ⁴, refractive index decreases with increasing wavelength. Angular dispersion θ = (μ_v - μ_r) × A measures the angular separation between extreme colors. Mean deviation δ̄ = (μ_y - 1) × A uses the mean refractive index approximately equal to yellow light's refractive index.

Digital images are represented as two-dimensional arrays of pixels with coordinates and intensity values. Three representation methods exist: 3D surface plots showing intensity variations, displayed images on screens, and mathematical matrices where rows and columns correspond to pixel coordinates. Grayscale images use single intensity values (0-255 for 8-bit), while color images use RGB models with three components per pixel. The alpha channel controls transparency. Higher bit depths provide more intensity levels but require more storage.

Digital images are created through digitization, which converts analog images to digital format via sampling (digitizing coordinate values x,y) and quantization (digitizing amplitude values f(x,y)). Digital images can be represented in three ways: three-axis representation with spatial coordinates and intensity values, two-axis representation with coordinates and intensity, and binary representation where 0 represents black and 1 represents white. The number of intensity levels is typically an integer power of 2 (L=2^k), and images are efficiently stored as m×n matrices where each element represents a pixel's intensity value.

An image represents light intensity in a 2D field. Images are analog (continuous signals, 0 to infinity intensity) or digital (discrete signals, bit-dependent intensity). Digital images consist of pixels with coordinates (x,y) and amplitude f(x,y). RGB color model combines red, green, and blue channels (24-bit = 16,777,216 colors). Grayscale conversion uses: Gray = 0.299R + 0.587G + 0.114B. Binary images have 1-bit depth (0=black, 1=white) created through thresholding where values ≥ threshold become 1, < threshold become 0.

Digital images are discrete representations of continuous visual scenes, stored as matrices where each element represents a pixel's intensity. Pixel values range from 0 (black) to 255 (white) in grayscale images, with intermediate values representing different shades of gray. Color images use RGB composition (Red, Green, Blue) where each pixel combines three intensity values. Binary images use only 0 (black) and 1 (white) for thresholded representations. The mathematical representation f(x,y) maps spatial coordinates to intensity values, enabling analytical processing approaches.

Digital images are represented as two-dimensional matrices with rows and columns, where intensity at each point (x,y) is denoted as f(x,y). The horizontal axis represents positive X direction, vertical axis positive Y direction, with origin at top-left. Pixel intensity values use 8-bit unsigned integers, yielding 256 possible levels (0-255). True color (RGB) images have three channels (red, green, blue) represented as [rows × columns × 3]. Gray scale images use single intensity planes with 256 shades from black (0) to white (255). Binary images use logical values (0 for black, 1 for white) created through thresholding operations.

The normal Dimitra workflow involves taking observation images, accessing the data reduction tab, and clicking the Run button for automatic processing. Wavelength calibration converts pixel positions to physical wavelengths through polynomial fitting. The horizontal axis changes from pixels to physical units. The process requires identifying several reference points (typically emission lines) and calculating their matching wavelengths. Once reference points are established, the software can determine wavelengths for all pixels in the spectrum. This conversion is fundamental to all spectroscopic analysis.

Wavelength calibration converts pixel positions to physical wavelengths using arc lamp spectra with known emission lines. The same reduction pipeline applied to scientific data is applied to arc lamp spectra, then spectral lines are identified and matched against laboratory wavelength databases. Statistical criteria select the best-fitting lines, typically requiring 5-6 well-fitted lines for accurate calibration. A polynomial function (cubic or quartic) is fitted to the identified line positions, establishing the wavelength solution. This calibration enables accurate measurement of spectral features like line strengths, velocities, and elemental abundances. The fitted parameters can be saved and reused for consistency across multiple observations.

This segment covers ISIS software operation: (1) ISIS requires configuration of directories containing spectral images, spectrograph type selection (e.g., ALP 600 for DS7), and CCD pixel size entry; (2) Background subtraction is performed by selecting regions adjacent to the spectrum containing only sky emission; (3) The software calculates the tilt of the spectrum and applies corrections for accurate wavelength calibration; (4) Wavelength calibration converts pixel positions to wavelengths using reference sources like neon lamps with known spectral lines; (5) The process involves identifying reference line positions and fitting polynomial functions (typically 3rd or 4th order) to map pixels to wavelengths; (6) Linear fitting is insufficient due to non-linear dispersion; (7) The quality is assessed by RMS error, with good calibration having errors less than 0.1 Å for most amateur applications.

Calibration images are essential for establishing the dispersion law that maps pixel positions to physical wavelengths. A calibration lamp containing argon and neon produces numerous emission lines at precisely known positions, which the software (Démétria) automatically recognizes. The dispersion law defines which pixel corresponds to which wavelength, enabling accurate wavelength identification in object spectra. Exposure times should maximize signal without saturation, and multiple calibration images (2-3) are recommended for redundancy against cosmic ray damage.

To obtain accurate wavelength information from an image sensor, a calibration process maps pixel positions to corresponding wavelengths. First, capture the spectrum of a known light source (like a tungsten lamp) to establish reference pixel-wavelength relationships. Each column of pixels represents a specific wavelength region. By averaging intensity values across multiple pixel columns and plotting intensity versus pixel position, you create a reference calibration curve. This allows conversion of unknown sample spectra from pixel coordinates to actual wavelength values for quantitative analysis.