Spectrographs disperse light using different kind of dispersing elements, such as prisms, transmission and reflection gratings or grisms. The most common grating in astronomical spectroscopy is the reflection grating and we will use it as an example to establish the basic principles. A reflection grating has a series of closely spaced grooves so that light reflects off the grating surface as though there was a series of narrow parallel mirrors (see Fig A.2). Constructive interference between the reflected beams of light at specific angles for a given wavelength and not for other wavelengths results in the creation of a spectrum. The constructive interference condition is fulfilled when the path difference between adjacent rays is equal to multiples,m, of the wavelength of the illuminating light. This leads to the grating equation:
m·λ
d = sin Θ + sini (A.1)
wherem is called the spectral order, dis the spacing between the repeated element in the grating and Θ andiare the angles of diffraction and incidence, respectively, as measured from the normal to the grating surface.
dΘ
dλ = m
d·cos Θ (A.2)
The beam is dispersed into multiple rays of light. The 0th order is not dispersed (white light) because all the wavelengths are deflected by the same angle. Given an order m ≥ 1, red light is dispersed at a greater angle than blue light (see Fig A.3), thus producing a spectrum.
Figure A.2: A reflection diffraction grating: N is the normal to the plane of the grating,
BLAZE is the normal to the facets of the grating, d is the spacing between the repeated element in the grating,δis the tilting angle of the facets to the grating plane,Θandiare the angles of diffraction and incidence as measured from N while β and α are the same angles but measured from BLAZE, γ is the collimator-camera angle (figure from Oliver 2004).
Also, the dispersion increases at higher orders, which is why high resolution spectrographs work at high m (UVES as a typical order value of 100). The blue end of order m+1 will overlap the red end of order m. Therefore it may be necessary to insert a colored glass filter to isolate the order of interest (order-separating filter) or a further dispersing element who separates them in the direction perpendicular to the dispersion (cross disperser). The useful spectral range (over which orders will not overlap) is approximately:
λred−λblue∼= λcentralm
the higherm, the narrower the spectral rangeλred−λblue.
Figure A.3: Dispersion increases with the order and the light wavelength as it can be see for blue and red light in the left and centre panel. The maximum intensity of the light corresponds to the0th order (no blaze). In the case of a blazed grating a large percentage of
the incident light is casted into a prescribed order (1st) as shown in the right panel (figure
from Oliver 2004).
The light ray with the maximum intensity is the one that follows the reflection law (incidence angle is equal to the refraction angle) and, in not-blazed gratings, it is in coincidence with the 0th order spectrum, in which there is no dispersion. Blazing is a technique of angling the surface
of the grooves in a grating so that it shifts the position of the maximum intensity from the 0th
order spectrum to a higher order spectrum, thus rendering the higher order brighter. The blaze condition is fulfilled when the reflection law is satisfied with respect to the normal to the facets of the grating, i.e. whenα=β (see Fig. A.2). An example can be see in Fig A.3, right panel, where the maximum intensity corresponds to the 1st order. The grating equation at the blaze condition
is
m·λB
d = 2 sinδcosγ/2
since 2δ = i+ Θ and γ = i−Θ, where λB is the wavelength for which the blaze condition is
satisfied,γ is the collimator-camera angle which is fixed by the design of the spectrograph andδ is the tilting angle of the facets to the grating plane.
Grism
A grism is a combination of a diffraction grating and a prism. Here a grating has been glued to the surface of a prism. The equivalent of the grating equation (Eq. A.1) becomes
m·λ
d =n·sin Θ + sini
wheren is the refraction index of the prism glass and the grating material, which are assumed to be equal. Considering the typical case shown in Fig A.4 where the input prism face and the facets are both normal to the optical axis, the most useful configuration is when the light is undeviated (Θ =−i), allowing the camera and the collimator to be in line. The advantage of this configuration
is that the monochromatic image of the target will appear at the same location as a direct image obtained with the grism removed.
Figure A.4: Grism scheme (figure from Oliver 2004).
Echelle
To increase the resolution of a grating one can either decrease the spacing between groove d, or increase the orderm. Combining Eq. A.1 and Eq. A.2, the dispersion equation can be rewritten as:
dΘ
dλ =
sin Θ + sini λ·cos Θ
from which it can be seen that high dispersion can also be achieved by operating at values ofiand Θ close to 90◦. This is the principle of an echelle grating, which has large d, and operates at high
m (typically more than 50),i and Θ, and gives high dispersion. To prevent the overlapping of the orders a prism or a grating is mounted with its dispersion perpendicular to the echelle dispersion (cross-dispersion) so that it separates the various orders.