Astronomical Spectroscopy

Notes 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

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ElementsResolution

Grating EquationDesigns

Schematic Spectrograph

Converging lightfrom telescope

Slit

Collimator

Disperser(prism or grating)

CameraDetector (CCD)

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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.

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Why use a slit?

1) A slit fixes the resolution, so that it does not depend on the seeing.2) A slit helps to exclude other objects in the field of view

A spectrograph should bedesigned so that the slitwidth is approximatelythe same as the averageseeing. Otherwise youwill lose a lot of light.

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Design Considerations: Resolution vs Throughput

Without the disperser, the spectrograph optics would simply reimage the slit on the detector.

With the disperser, monochromatic light passingthrough the spectrograph would result in a singleslit 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 overlappingimages 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.

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Design Considerations: Projected slit width

f2 f3

Let s = slit width, p = projected slit width (width of slit on detector).Then, to first order:

pf

fs

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2

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

Collimator focal length Camera focal length

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Design Considerations: Spectral Resolution vs. Spectral Range

R

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Dispersers

Prisms: disperse light into a spectrumbecause the index of refraction is afunction of the wavelength. Usually:n(blue) > n(red).

Diffraction gratings: work throughthe interference of light. Most modernspectrographs use diffraction gratings.Most astronomical spectrographs usereflection gratings instead of transmissiongratings.

A combination of the two is called aGrism.

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Diffraction Gratings

Diffraction gratings are made up of very narrow grooves whichhave widths comparable to a wavelength of light. For instance,a 1200g/mm grating has spacings in which the groove width isabout 833nm. The wavelength of red light is about 650nm.Light reflecting off these grooves will interfere. This leadsto dispersion. 9

The Grating Equation

d

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 willbe a + b, where a = d sinα andb = d sinβ. Thus, the tworeflected rays will interfereconstructively if:

m d (s in sin )

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m d (s in sin )

Meaning: Let m = 1. If a ray of light of wavelength λ strikesa grating of groove spacing d at an angle α with the gratingNormal, 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.

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m d (s in sin )

m is called the order of the spectrum. Thus, diffraction gratingsproduce multiple spectra. If m = 0, we have the zeroth order,undispersed image of the slit. If m = 1, we have two first orderspectra on either side of the m = 0 image, etc.

Diffraction gratingillustrated is atransmission grating.

These orders will overlap, which produces problems for gratingspectrographs. 12

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.

Overlap equation: m m

m

m 1 1

Meaning that a wavelength of λm in the mth order overlaps with awavelength of λm+1 in the m+1th order.

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Massey & Hanson 2011arXiv 1010.5270v2.pdf

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Dispersion & Resolution

Dispersion is the degree to which the spectrum is spread out.To get high resolution, it is really necessary to use a diffractiongrating that has high dispersion. Dispersion (dβ/dλ) is given by:

d

d

m

d

cos

Thus, to get high resolution, three strategies are possible:long camera focal length (f3), high order (m), or smallgrating spacing (d). The last has some limitations. Thefirst two lead to the two basic designs for high-resolutionspectrographs: coudé (long f3) and echelle (high m).

Rf

p

d

d 3

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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

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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) 17

Blaze function FWHM≈λ/m

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Three basic optical designs for spectrographs

Littrow (not commonly used inastronomy).

Ebert: used in astronomy, butp = s. Note camera = collimator.

Czerny-Turner: most versatiledesign. Most commonly usedin astronomy.

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High-resolution spectrographs: Echelle

Echelle grating: coarse grating (big d) usedat high orders (m ~ 100; tan θB = 2).

Orders are separated by crossdispersion: using a seconddisperser to disperse λ in a direction perpendicular to theechelle dispersion.

Kitt Peak 4-m Echelle

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Hamilton echelle spectrum format:Vogt 1987, PASP, 99, 1214

m λ