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Astronomical Spectroscopy Notes from Richard Gray, Appalachian State, and D. J. Schroeder 1974 in “Methods of Experimental Physics, Vol. 12-Part A Optical and Infrared”, p.463. See also Chapter 3 in “Stellar Photospheres” textbook. Elements Resolution Grating Equation Designs.

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elements resolution grating equation designs

Astronomical SpectroscopyNotes from Richard Gray, Appalachian State, andD. J. Schroeder 1974 in “Methods of Experimental Physics, Vol. 12-Part A Optical and Infrared”, p.463.See also Chapter 3 in “Stellar Photospheres” textbook

ElementsResolutionGrating EquationDesigns


Schematic Spectrograph



Detector (CCD)


Converging light

from telescope


(prism or


slit spectrographs
Slit Spectrographs
  • Entrance Aperture: A slit, usually smaller than that of the seeing disk
  • Collimator: converts a diverging beam to a parallel beam
  • Dispersing Element: sends light of different colors into different directions
  • Camera: converts a parallel beam into a converging beam
  • Detector: CCD, IR array, photographic plate, etc.

Why use a slit?

  • A slit fixes the resolution, so that it does
  • not depend on the seeing.
  • A slit helps to exclude other objects in
  • the field of view

A spectrograph should be

designed so that the slit

width is approximately

the same as the average

seeing. Otherwise you

will lose a lot of light.


Design Considerations: Resolution vs Throughput

Without the disperser, the spectrograph optics

would simply reimage the slit on the detector.

With the disperser, monochromatic light passing

through the spectrograph would result in a single

slit image on the detector; its position on the detector

is determined by the wavelength of the light.

This implies a spectrum is made up of overlapping

images of the slit. A wide slit lets in a lot of light,

but results in poor resolution. A narrow slit lets in

limited light, but results in better resolution.


Design Considerations: Projected slit width



Collimator focal length

Camera focal length

Let s = slit width, p = projected slit width (width of slit on detector).

Then, to first order:

Optimally, p should have a width equal to two pixels on the detector.Resolution element Δλ = wavelength span associated with p.



Prisms: disperse light into a spectrum

because the index of refraction is a

function of the wavelength. Usually:

n(blue) > n(red).

Diffraction gratings: work through

the interference of light. Most modern

spectrographs use diffraction gratings.

Most astronomical spectrographs use

reflection gratings instead of transmission


A combination of the two is called a



Diffraction Gratings

Diffraction gratings are made up of very narrow grooves which

have widths comparable to a wavelength of light. For instance,a 1200g/mm grating has spacings in which the groove width is

about 833nm. The wavelength of red light is about 650nm.

Light reflecting off these grooves will interfere. This leads

to dispersion.


The Grating Equation

Light reflecting from grooves A and

B will interfere constructively if the

difference in path length is an

integer number of wavelengths.

The path length difference will

be a + b, where a = d sinα and

b = d sinβ. Thus, the two

reflected rays will interfere

constructively if:



Meaning: Let m = 1. If a ray of light of wavelength λ strikes

a grating of groove spacing d at an angle α with the grating

Normal, it will be diffracted at an angle β from the grating.

If m, d and α are kept constant, λ is clearly a function of β.

Thus, we have dispersion.


m is called the order of the spectrum. Thus, diffraction gratings

produce multiple spectra. If m = 0, we have the zeroth order,

undispersed image of the slit. If m = 1, we have two first order

spectra on either side of the m = 0 image, etc.

Diffraction grating

illustrated is a

transmission grating.

These orders will overlap, which produces problems for grating



Overlapping of Orders

If, for instance, you want to observe at 8000Å in 1st order,

you will have to deal with the 4000Å light in the 2nd order.

This is done either with blocking filters or with cross dispersion.

Massey & Hanson 2011arXiv 1010.5270v2.pdf

Overlap equation:

Meaning that a wavelength of λm in the mth order overlaps with a

wavelength of λm+1 in the m+1th order.


Dispersion & Resolution

Dispersion is the degree to which the spectrum is spread out.

To get high resolution, it is really necessary to use a diffraction

grating that has high dispersion. Dispersion (dβ/dλ) is given by:

Thus, to get high resolution, three strategies are possible:

long camera focal length (f3), high order (m), or small

grating spacing (d). The last has some limitations. The

first two lead to the two basic designs for high-resolution

spectrographs: coudé (long f3) and echelle (high m).

grating spectrographs
Grating Spectrographs
  • Reciprocal dispersion P=(d cosβ)/(mf3) (often quoted in units of Å/mm)
  • Free spectral range m(λ+Δλ)=(m+1)λ  Δλ=λ/mλ difference between two orders at same β
  • Blaze angle with max. intensity whereangle of incidence = angle of reflection
blaze wavelength
Blaze wavelength
  • β – θB = θB – α
  • θB = (α+β)/2δ/2 = (β-α)/2
  • Insert in grating eq.λB=2d sinθB cos(δ/2)
  • Blaze λ in other ordersλm = λB /m
  • Manufacturers giveθB for α=β (Littrow)

Three basic optical designs for spectrographs

Littrow (not commonly used in


Ebert: used in astronomy, but

p = s. Note camera = collimator.

Czerny-Turner: most versatile

design. Most commonly used

in astronomy.


High-resolution spectrographs: Echelle

Echelle grating: coarse grating (big d) used

at high orders (m ~ 100; tan θB = 2).

Kitt Peak 4-m Echelle

Orders are separated by cross

dispersion: using a second

disperser to disperse λ in a direction perpendicular to the

echelle dispersion.