allington-smith02 - Dispersive astronomical spectroscopy...

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Jeremy Allington-Smith Page 1 of 29 13/09/02 Dispersive astronomical spectroscopy Jeremy Allington-Smith, University of Durham, 29 July 2002 (Copyright Jeremy Allington-Smith, 2002) I will start by considering the most usual type of dispersing element encountered in astronomical spectroscopy: the ruled plane reflection grating used in low order. Having used this to establish the basic principles, I will consider other types of dispersing element. 1. Reflection gratings 1.1 Interference condition As shown in Figure 1, the path difference between interfering rays AB and A’B’ is ) sin (sin β α + a where a is the spacing between the repeated element in the grating from which reflection (or refraction) occurs. The interference condition is fulfilled when the path difference is equal to multiples, m , of the wavelength of the illuminating light. This gives rise to the grating equation : ρλ sin sin + = m where a / 1 = ρ is the ruling density, m is the spectral order and λ is the wavelength of light. 1.2 Dispersion By differentiating with respect to the output angle we obtain the angular dispersion m d d cos = α β AB A’ B’ a Figure 1: Interference of light by a diffraction grating. Although considered here as a transmission grating, the same principle applies to a reflection grating with suitable care taken with the sign of the angles. D T f 1 f 2 α β ψ D 1 D 2 Grating Slit Detector Telescope α W s χ f T Figure 2: Illustration of the principle of a generic grating spectrograph showing the definition of quantities used in the text. To be consistent with the grating equation, α and β have the same sign if they are on the same side of the grating normal.
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Jeremy Allington-Smith Page 2 of 29 13/09/02 The linear dispersion is then 2 cos f m dx d d d dx d ρ β λ = = since dx d f = 2 where 2 f is the focal length of the camera (see Figure 2). 1.3 Resolving power In standard textbooks, the resolving power , δλ R (where is the resolution in wavelength), is usually described as being given by the total number of lines in the grating multiplied by the spectral order, hence W m R = * (sometimes this is called the spectral resolution which can lead to confusion with ). But in practice, the resolving power is determined by the width of the image of the slit, s , projected on the detector, s’ . Before going further, it is useful to consider the invariance of Etendue in optical systems (note that fibres and some other optical systems do not conserve Etendue but systems made from normal optics – mirrors and lenses - do). This is normally stated as constant = A n where is the solid angle of radiation incident at a surface of area A in a medium with refractive index n . For our purposes, we may set 1 = n (since we only consider optics in air or vacuum) and consider a one-dimensional analogue: constant ' ' = = a a ω where and a are the opening angle of the beam and the aperture dimension respectively.
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This note was uploaded on 07/17/2008 for the course ASTRO 890 taught by Professor Martini during the Spring '08 term at Ohio State.

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allington-smith02 - Dispersive astronomical spectroscopy...

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