Spectrometer

An electron spectrometer measures an electron energy distribution. A time of flight design infers kinetic energy from calibrated travel times.

An electron energy analyzer selects electrons by kinetic energy, allowing their count rate to be recorded as an energy scan.

A spectrometer that selects charged particles using the bending of their trajectories in a magnetic field. For a uniform magnetic field of magnitude $B$, the perpendicular momentum component is $p_{\perp }=|q|Br$, where $q$ is the electric charge and $r$ is the curvature radius of the projected trajectory.

Fourier transform spectroscopy measures an interferogram as a function of optical path length difference. Its Fourier transform recovers the spectrum. An imaging implementation obtains an interferogram at every detector pixel, but a scanned cube is not simultaneous at all wavelengths.

Multiplying a finite interferogram by a smooth window reduces sharp-edge artifacts. In the recovered radiation spectrum this convolves the true spectral density with the window's Fourier transform. Smoother windows generally lower sidelobes while broadening the main lobe, so a resolution figure must specify the window and criterion. Apodization does not recover unmeasured long-delay information.^[ref]

For $N$ equally intense spectral lines at ${\omega }_n={\omega }_0+n\Delta \omega$, sum their delayed self-correlations as a geometric series. The result isThe interferometric visibility revives every $T=2\pi /\Delta \omega$. There are $N-1$ zeros between adjacent revivals, at $\tau =jT/N$ for $j=1,\dots ,N-1$. The removable singularity at a revival has unit modulus. Revival spacing fixes line spacing; the internal zero count fixes $N$ under the equal-intensity model. The interferogram alone does not establish a fixed relative phase between the source lines or prove the emission is a coherent pulse train.

A Michelson interferometer mirror displacement $x$ changes round-trip optical path difference by $2x$, giving delay $\tau =2x/c$. An interferogram recorded over a finite delay range is multiplied by a window; the recovered spectrum is the true spectrum convolved with that window's Fourier transform. Its characteristic frequency resolution scales inversely with delay range. Consequently a mirror travel range $L$ gives $\delta \nu \propto c/L$ and $\delta \lambda \simeq ({\lambda }^2/c)\delta \nu \propto {\lambda }^2/L$. Numerical width factors depend on the scan definition, window and resolution criterion; the inverse-travel scaling does not.

A spectrograph records the optical spectrum dispersed by an instrument. Its spectral resolving power describes which nearby wavelengths it separates.

Spectrophotometry measures received light as a function of wavelength, accounting for the instrument's wavelength-dependent response.

A sample holder providing a defined geometric light path through a chemical solution for spectrophotometry, as in the trypsin colorimetric assay.

A spectrophotometer measures radiant power in selected wavelength bands. Comparing a chemical solution in a cuvette with a reference blank gives transmittance and absorbance.

A spectral resolution element is a spectral interval at the adopted instrumental resolution. At spectral resolving power $R$, its width is $\Delta \lambda =\lambda /R$. The number of photodetector detector pixels sampling it is distinct from $R$.

Linear spectral dispersion is the derivative of the photodetector coordinate in the dispersion direction with respect to wavelength. Near the optical axis, $q=f_{\mathrm{cam}}d\beta /d\lambda$.

An integral field spectrograph records a spectrum for each of many spatial elements across a two-dimensional field, producing a spectral data cube. Field reformatters include lenslet arrays, optical fibers and image slicers.

With fixed spectral resolving power, wavelength interval, diffraction grating geometry, photodetector sampling and collimator focal ratio, a beam diameter scales as the spaxel width $\theta$, but the optical camera focal length stays fixed. A simple on-axis beam-envelope model therefore gives collimator and disperser volumes proportional to ${\theta }^3$ and optical camera volume proportional to ${\theta }^2$. With a fixed number of spaxels per photodetector, replication multiplies these volumes by $M\propto {\theta }^{-2}$. A fixed photodetector field adds transverse clearances and changes the small-$\theta$ limit; these exponents are not universal mechanical-volume laws.

A spaxel is one spatial sampling element in a spectral data cube. For a square field of angular side $\beta$ sampled at angular side $\theta$, the number of spaxels is $(\beta /\theta {)}^2$, before edge rounding.

An image slicer divides a two-dimensional image into narrow strips and optically rearranges them into a long pseudo-slit. A spectrograph then disperses each strip while preserving its position along the slit.

A spectral data cube stores intensity $I(x,y,\lambda )$ with two spatial coordinates and one spectral coordinate. The third coordinate is wavelength, not an additional spatial dimension.

Spectral resolving power is $R=\lambda /\Delta \lambda$, where $\Delta \lambda$ is the wavelength separation just resolved under the specified instrumental and source conditions.

A cross-dispersed echelle spectrograph uses an echelle grating for strong dispersion and a cross-disperser in the perpendicular direction to separate overlapping diffraction orders. It records many orders simultaneously in an echellogram.

A cross-disperser is a second dispersing element, often an optical prism or diffraction grating, oriented to separate optical spectra perpendicular to their principal dispersion direction.

An echellogram is the two-dimensional detector pattern of separated diffraction orders from a cross-dispersed echelle spectrograph.

A multi-object spectrograph records optical spectra of many selected objects simultaneously. The main designs are multi-slit spectrographs and fiber-fed spectrographs. The ESO instrumentation overview describes both arrangements.

A multi-slit spectrograph puts separate slits at target positions in the focal plane. Each slit transmits the target plus nearby sky; dispersion places its spatially resolved optical spectrum on the detector.

A fiber-fed spectrograph places optical fibers on selected targets and reformats their output into a common entrance slit. Dedicated sky fibers sample sky brightness. Fiber coupling, transmission, and output beam spreading affect throughput and resolution.