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Subsections

INSTRUMENTATION

(Entire section in one PDF file).

Often, astronomers use additional optics between the telescope and their detector. These, in conjunction with a detector, make up an instrument.

Location of optics

Before going into specifics, consider the effect of placing optics at different locations within an optical system, like a telescope.

Optics placed in or near a focal plane will affect images at different field angles differently. Optics in a focal plane will not affect the image quality at any given field angle; however, such optics might be used to control the location of an image of the pupil of the telescope.

Optics placed in or near a pupil plane will affect images at all field angles similarly, and will have an effect on the image quality.


\begin{shaded}
\textit{Understand the implications of putting optics in different locations.}
\end{shaded}

Refractive optics and chromatic aberration

In many instruments, lenses are used rather than mirrors: they can be cheaper and lead to more compact designs. Recall, however, that when lenses are used, chromatic effects will arise, because the index of refraction of glasses changes with wavelength. While they can often be minimized by the use of use of multiple elements to make achromatic combinations, they are not always negligible. In particular, if an instrument is used at multiple wavelengths, some refocussing may be required.

Field Flatteners

As we've discussed, all standard two-mirror telescopes have curved focal planes. It is possible to make a simple lens to correct the field curvature. We know that a plane-parallel plate will shift an image laterally, depending on the thickness of the plate. If we don't want to affect the image quality, only the location, we want the correcting element to be located near the focal plane.

Consequently, we can put a lens right near the telescope focal plane to flatten the field. For a field which curves towards the secondary mirror, one finds that the correct shape to flatten the field is just a plano-concave lens with the curved side towards the secondary. Often, the field flattener is incorporated into a detector dewar as the dewar window.

Focal plane reimagers

A focal reimager is a reimaging system which demagnifies/magnifies the telescope focal plane. In a simple form, it consists of two lenses: a collimator and a camera lens. The collimator lens is placed such that the telescope focal plane is put at the focal length of the collimator, so that it converts the telescope beam into a collimated beam (note that the focal ratio of the collimating lens itself will be larger than that of the telescope so that the beam underfills the lens to allow for off-axis light as well). The camera lens then refocuses the light light with the desired focal ratio. The magnification of the system is given by:

m = $\displaystyle {f_{camera}\over f_{collimator}}$

Consequently, the scale in the image plane of the focal reimager is just the scale in the telescope focal plane multiplied by the ratio of the focal ratio of the camera to that of the telescope.

Note that with a focal plane reimager, one does not necessarily get a new scale ``for free''. The focal reimaging system may introduce additional aberrations giving reduced image quality. In addition, one always loses some light at each additional optical surface from reflection and/or scattering, so the more optics in a system, the lower the total throughput.

Note that it is possible to do focal reduction/expansion without reimaging, i.e., by putting optics in the converging beam.


\begin{shaded}
\textit{Understand the basic design and effect of a focal plane r...
...some rays and to determine if a reimager magnifies or
demagnifies.}
\end{shaded}

Pupil reimagers

Often, an additional lens, called a field lens is placed in or near the telescope focal plane. This does not affect the focal reduction but is used to reimage the telescope pupil somewhere in the reimager. One reason this may be done is to minimize the size that the collimator lens needs to be to get off-axis images. The size of the field lens itself depends on the desired size of the field that one wishes to reimage.

Another use of reimaging the pupil is when one is building a coronagraph, an imaging system designed to observe faint sources nearby to very bright ones. The problem in seeing the faint source is light from the bright one, both from scattered light, from diffraction, and sometimes, from detector effects (e.g., charge bleeding in a CCD). A partial solution is to put an occulting spot in the telescope focal plane which removes most of the light from the bright object. However, the diffraction structure is still a problem. It turns out you can remove this by reimaging the pupil after the occulting spot and putting a mask in around the edges which are the source of the diffraction; this mask is called a Lyot stop. The resulting image in the focal plane of the focal reducer is free of both bright source and diffraction structure.

Note that for really high contrast imaging, you also need to consider other sources of far-field light including light scattered from small-scale features on optical elements, and far-field light from seeing. Minimizing the former required very smooth optics, while minimizing the latter requires high-performance adaptive optics (e.g. ``extreme-AO'').

Pupil reimagers are also widely used in IR systems to reduce emission via cold pupil stops. The issue here is that the telescope itself contributes infrared emission which acts as additional background in your observations. There is little you can do about emission from the primary, since you need to see light from the primary to see your object! However, you can block out emission from regions of the pupils which are obscured already, for example, by the secondary and/or secondary support structures. To do this you put a mask in the pupil plane. Obviously, however, the mask needs to be colder than the telescope itself or else the mask would contribute the background, so it is usually placed within the dewar that contains the detector and camera optics.


\begin{shaded}
\textit{Understand why one might want to access a pupil in
an instrument. Know the principles of a coronagraph works.}
\end{shaded}

Filters

Filters are used in optical systems (usually imaging systems) to restrict the observed wavelength range. Using multiple filters thus provides color information on the object being studied. Generally, filters are loosely classified as broad band ( > $ \sim$ 1000Åwide), medium band (100 < $ \sim$ 1000 Å), or narrow band ( 1 < $ \sim$ 100 Å).

Perhaps a better distinction between different filters is by the way that they filter light. Many broad band filters work by using colored glass, which has pigments which absorb certain wavelengths of light and let others pass. Bandpasses can be constructed by using multiple types of colored glass. These are generally the most inexpensive filters.

A separate filter technique uses the principle of interference, giving what are called interference filters. They are made by using two partially reflecting plates separated by a distance d apart. The priciple is fairly simple:

Interference filter diagram When light from the different paths combines constructively, light is transmitted; when it combines destructively, it is not. Simple geometry gives:

m$\displaystyle \lambda$ = 2nd cos$\displaystyle \theta$

It is clear from this expression that the passband of the filter will depend on the angle of incidence. Consequently narrowband filters will have variable bandpasses across the field if they are located in a collimated beam; this can cause great difficulties in interpretation! If the filter is located in a focal plane or a converging beam, however, the mix of incident angles will broaden the filter bandpass. This can be a serious effect in a fast beam. Bandpasses of interference filters can also be affected by the temperature.

Since interference filters will pass light at integer multiples of the wavelength, the extra orders often must be blocked. This can be done fairly easily with colored glass.

The width of the bandpass of a narrowband filter is determined by the amount of reflection at each surface. Both the wavelength center and the width can be tuned by using multiple cavities and/or multiple reflecting layers, and most filters in use in astronomy are of this more complex type.

The same principles by which interference filters are made are used to make antireflection coatings.

Note filters can introduce aberrations, dust spots, reflections, etc; one needs to consider these issues when deciding on the location of filters in an optical system.


\begin{shaded}
\textit{Understand how filters work and the difference between
a ...
...l configuration can modify the bandpass of an
interference filter.}
\end{shaded}

Fabry-Perot Interferometer

A Fabry-Perot system makes use of a tunable interference filter. The filter is tuned in wavelength by adjusting one of

A tunable interference filter is called an etalon. Often, etalons are made to provide very narrow bandpasses, on the order of 1Å.

A picture taken with a Fabry-Perot system covers multiple wavelengths because the etalon is located in the collimated beam between the two elements of the focal reducer. At each etalon setting, one observes an image which has rings of constant wavelength. By tuning the etalon to give different wavelengths at each location, one build up a ``data cube'', through which observations at a constant wavelength carve some surface. Consequently, to extract constant wavelength information from the Fabry-Perot takes some reasonably sophisticated reduction techniques. It is further complicated by the fact that to get accurate quantitative information, one requires that the atmospheric conditions be stable over the entire time when the data cube is being taken.


\begin{shaded}
\textit{Know what a Fabry-Perot system is.}
\end{shaded}

Spectrographs

A spectrograph is an instrument which separates different wavelengths of light so they can be measured independently. Most spectrographs work by using a dispersive element, which directs light of different wavelengths in different directions.

A conventional spectrograph has a collimator, a dispersive element, a camera to refocus the light, and a detector. The performance of a spectrograph is characterized by the dispersion, which gives the amount that different wavelengths are separated, and the resolution, which gives the smallest difference in wavelength that two different monochromatic sources can be separated. There are different sorts of dispersive elements with different characteristics; two common ones are prisms and diffraction gratings, with the latter the most commonly in use in astronomy.

The dispersion depends on the characteristic of the dispersing element. Various elements can be characterized by the angular dispersion, d$ \theta$/d$ \lambda$, or alternatively, the reciprocal angular dispersion, d$ \lambda$/d$ \theta$. In practice, we are often interested in the linear dispersion, dx/d$ \lambda$ = f2d$ \theta$/d$ \lambda$ or the reciprocal linear dispersion, d$ \lambda$/dx = $ {1\over f_2}$d$ \lambda$/d$ \theta$ where the latter is often referred to simply as the dispersion in astronomical contexts, and is usually specified in Å/mm or Å/pixel.

If the source being viewed is extended, it is clear that any light which comes from regions parallel to the dispersion direction will overlap in wavelength with other light, leading to a very confused image to interpret. For this reason, spectrographs are usually used with slits or apertures in the focal plane to restrict the incoming light. Note that one dimension of spatial information can be retained, leading to so-called long-slit spectroscopy. If there is a single dominant point source in the image plane, or if they are spaced far enough (usually in combination with a low dispersion) that spectra will not overlap, spectroscopy can be done in slitless mode. However, note that in slitless mode, one can be significantly impacted by sky emission.

The resolution depends on the width of the slit or on the size of the image in slitless mode, because all a spectrograph does is create an image of the focal plane after dispersing the light. The ``width'' of a spectral line will be given by the width of the slit or the image, whichever is smaller. In reality, the spectral line width is a convolution of the slit/image profile with diffraction. The spatial resolution of the detector may also be important.

Note that throughput may also depend on the slit width, depending on the seeing, so maximizing resolution may come at the expense of throughput.

Given a linear slit or image width, $ \omega$ (or an angular width, $ \phi$ = $ \omega$/f, where f is the focal length of the telescope) and height h (or $ \phi^{\prime}_{}$ = h/f), we get an image of the slit which has width, $ \omega^{\prime}_{}$, and height, h$\scriptstyle \prime$, given by

h$\scriptstyle \prime$ = h$\displaystyle {f_2\over f_1}$

$\displaystyle \omega^{\prime}_{}$ = r$\displaystyle \omega$$\displaystyle {f_2\over f_1}$

where we have allowed that the dispersing element might magnify/demagnify the image in the direction of dispersion by a factor r, which is called the anamorphic magnification.

Using this, we can derive the difference in wavelength between two monochromatic sources which are separable by the system.

$\displaystyle \delta$$\displaystyle \lambda$ = $\displaystyle \omega^{\prime}_{}$$\displaystyle {d\lambda\over dx}$

$\displaystyle \delta$$\displaystyle \lambda$ = r$\displaystyle \omega$$\displaystyle {f_2\over f_1}$$\displaystyle {d\lambda\over dx}$

The bigger the slit, the lower the resolving power.

The resolution is often characterized in dimensionless form by

R $\displaystyle \equiv$ $\displaystyle {\lambda\over \delta\lambda}$ = $\displaystyle {\lambda f_1 \over r \omega f_2 (d\lambda/dx)}$

Note that there is a maximum resolution allowed by diffraction. This resolution is given aproximately by noting that minimum angles which can be separated is given by approximately $ \lambda$/d2, where d2 is the width of the beam at the camera lens, from which the minimum distance which can be separated is:

$\displaystyle \omega_{{min}}^{}$ = f2$\displaystyle {\lambda\over d_2}$

The slit width which corresponds to this limit is given by:

$\displaystyle \omega^{\prime}_{}$ = r$\displaystyle \omega$$\displaystyle {f_2\over f_1}$ = f2$\displaystyle {\lambda\over d_2}$

or

$\displaystyle \omega$ = $\displaystyle {f_1 \over r}$$\displaystyle {\lambda\over d_2}$

and the maximum resolution is

Rmax = $\displaystyle {d_2 \over f_2 (d\lambda/dx)}$ = d2$\displaystyle {d\theta\over d\lambda}$


\begin{shaded}
\textit{Understand how a typical astronomical spectrograph works....
...on, and what
about the spectrograph determines what these will be.}
\end{shaded}

Astronomical spectrographs

Slitless spectographs: generally need to work at low dispersion (or narrow spectral range) to avoid spectrum overlap. Issue with background: since light from all field angles is included, this effectively disperses object light, but not background.

Long slit spectrographs: standard spectrograph as discussed above. Avoids spectrum overlap by limiting spectra to a line in the sky.

Image slicers: preserving resolution and flux.

Fiber spectrographs: multiobject data. Use fibers to select objects, then line up the other ends of fibers into a pseudo-slit.

Slitlets: multiobject data. Break up single long slit into individual slitlets, avoiding overlap by the slitmask design. Note that each slitlit will have it's own wavelength calibration.

Integral field spectrographs. Get spectra information over 2D field. Either use fibers to accomplish, or optical configuration, e.g. with lenslets.


\begin{shaded}
\textit{Understand the different types of astronomical
spectrographs.}
\end{shaded}

Dispersing elements

Prisms

Perhaps the simplest conceptual dispersing element is a prism, which disperses light because the index of refraction of many glasses is a function of wavelength. From Snell's law, one finds that:

$\displaystyle {d\theta\over d\lambda}$ = $\displaystyle {t\over d}$$\displaystyle {dn\over d\lambda}$

where t is the base length, and d is the beamwidth. Note that prisms do not have anamorphic magnification (r = 1). The limiting resolution of a prism, from above is:

Rmax = $\displaystyle {d_2 \over f_2 (d\lambda/dx)}$ = d2$\displaystyle {d\theta\over d\lambda}$

Rmax = t$\displaystyle {dn\over d\lambda}$

One finds that dn/d$ \lambda$ $ \propto$ $ \lambda^{{-3}}_{}$ for many glasses.

So dispersion and resolution are a function of wavelength for a prism. In addition, the resolution offered by a prism is relatively low compared with other dispersive elements (e.g. gratings) of the same size. Typically, prisms have R < 1000. Consequently, prisms are rarely used as the primary dispersive element in astronomical spectrographs. They are occasionally used as cross-dispersing elements.

Gratings

Diffraction gratings work using the principle of multi-slit interference. A diffraction grating is just an optical element with multiple grooves, or slits (not to be confused with the slit in the spectrograph!). Diffraction gratings may be either transmissive or reflective. Bright regions are formed where light of a given wavelength from the different grooves constructively interferes.

The location of bright images is given by the grating equation:

m$\displaystyle \lambda$ = $\displaystyle \sigma$(sin$\displaystyle \theta$ + sin$\displaystyle \alpha$)

for a reflection grating, where $ \sigma$ is the groove spacing, m is the order, and $ \alpha$ and $ \theta$ ($ \beta$ in picture above!) are the angles of incidence and diffraction as measured from the normal to the grating surface.

The dispersion of a grating can then be derived:

$\displaystyle {d\theta\over d\lambda}$ = $\displaystyle {m\over \sigma \cos\theta}$

One can see that the dispersion is larger at higher order, and for a finer ruled grating. The equation can be rewritten as

$\displaystyle {d\theta\over d\lambda}$ = $\displaystyle {\sin\theta + \sin \alpha \over \lambda \cos\theta}$

from which it can be seen that high dispersion can also be achieved by operating at large values of $ \alpha$ and $ \theta$. This is the principle of an echelle grating, which has large $ \sigma$, and operates at high m, $ \alpha$ and $ \theta$, and gives high dispersion and resolution. An advantage of this is that one can get a large fraction of the light over a broad bandpass in a series of adjacent orders.

Typical gratings have groove densities between 300 and 1200 lines/mm. Echelle gratings have groove densities between 30 and 300 lines/mm.

One can derive the anamorphic magnification for a grating by looking at how $ \theta$ changes as $ \alpha$ changes at fixed $ \lambda$. One finds that:

r = $\displaystyle {d\theta\over d\alpha}$ = $\displaystyle {\cos \alpha\over \cos \theta}$ = $\displaystyle {d_1\over d_2}$

where the d's are the beam diameters. Note that higher resolution occurs when r < 1, or $ \theta$ < $ \alpha$.

The limiting resolution can be derived:

Rmax = $\displaystyle {d_2 \over f_2 (d\lambda/dx)}$ = d2$\displaystyle {d\theta\over d\lambda}$

Rmax = $\displaystyle {d_2 m\over \sigma \cos\theta}$ = $\displaystyle {m W\over \sigma}$ = mN

where W is the width of the grating ( = d2/cos$ \theta$), and N is the total number of lines in the grating.

Note that light from different orders can fall at the same location, leading to great confusion! This occurs when

m$\displaystyle \lambda^{\prime}_{}$ = (m + 1)$\displaystyle \lambda$

or

$\displaystyle \lambda^{\prime}_{}$ - $\displaystyle \lambda$ = $\displaystyle {\lambda\over m}$

The order overlap can be avoided using either an order-blocking filter or by using a cross-disperser. The former is more common for small m, the latter for large m.

One can compare grating operating in low order, those operating in high order, and prisms, and one finds that higher resolution is available from gratings, and that echelles offer higher resolution than typical low order gratings.

We can also discuss grating efficiency, the fraction of incident light which is directed into a given diffracted order. One finds that for a simple grating, less light is diffracted into higher orders. However, one can construct a grating which can maximize the light put into any desired order by blazing the grating, which involves tilting each facet of the grating by some blaze angle. The blaze angle is chosen to maximize the efficiency at some particular wavelength in some particular order; it is set so that the angle of diffraction for this order and wavelength is equal to the angle of reflection from the grating surface. The blaze function gives the efficiency as a function of wavelength.

A special case of high efficiency is when the angle of incidence equals the angle of diffraction, i.e. the diffracted light at the desired wavelength comes back to the same direction of in the incoming light. This is called the Littrow configuration; high efficiency spectrographs often try to work close to this configuration.

Typical peak efficiencies of reflective diffraction gratings are of order 50-80%. Recently, a new technology for making diffraction gratings, volume phase holographic (VPH) gratings, as been developed, and these are attractive because they offer the possibility of very high efficiencies (> 90% peak efficiency).


\begin{shaded}
\textit{
Understand the principle by which gratings work. Underst...
...ired order. Understand how the groove density affects
dispersion. }
\end{shaded}

Grisms

A grism is a combination of a prism and a diffraction grating. These are combined such that light is dispersed, but light at a chosen central wavelength passed through the grism with direction unchanged. This feature allows grisms to be placed in an imaging system (e.g., in a filter wheel) to provide a spectroscopic (usually low resolution) capability.

Operational items: using a spectrograph

Choice of dispersion: wavelength coverage vs. dispersion/resolution, available gratings, etc. Using grating tilt to select wavelength range.

Choice of slit width (science, seeing).

How to put object in slit. Imaging the slit. Slit viewing cameras.

(DEFER FOLLOWING TO SECTION ON DATA REDUCTION???)

Spectrograph calibration (not including basic detector calibration, to be discussed soon).

Wavelength calibration: correspondance between pixel position (in wavelength dimension) and wavelength. Arc lamps, wavelength solutions. Subtleties: extrapolation, line curvature, flexure (using skylines to calibrate).

Flux calibration: relative fluxes at different wavelengths. Spectrophotometric standards. Subtleties: differential refraction

Spectral extraction: object extraction and sky subtraction. Subtleties: S-distortion, differential refraction: spectral traces. Issues: variation of focus along slit and implications for sky line subtraction, scattered light.

Relative fluxes along slit: slit width variations.

Examples of typical spectra: line lamps, flat fields, stellar spectra, galaxy spectra. Night sky emission.

Non-dispersive spectroscopy

It is also possible to use interference effects to measure spectral energy distributions instead of a dispersing element. The Fabry-Perot is an example of such a type of instrument, although it does not record all wavelengths simultaneously.

Another instrument which uses interference to infer spectroscopy information is the Fourier Transform Spectrometer (FTS), which is basically a scanning Michaelson interferometer. The light from the source is split into two parts using a beamsplitter. One part of light is reflected off a fixed flat mirror and the other is reflected off a mirror which can be moved laterally. The two images are combined to form fringes. The fringe pattern changes as the path length of the second beam is changed. The intensity modulation for a given wavelength ($ \lambda$) or wavenumber ( k = 2$ \pi$/$ \lambda$) is given by:

T(k,$\displaystyle \Delta$x) = $\displaystyle {T_{max}\over 2}$[a + cos(2k$\displaystyle \Delta$x)]

and the flux after integrating over all wavelengths is:

F($\displaystyle \Delta$x) = C$\displaystyle \int$I(k)T(k,$\displaystyle \Delta$x)dk = C$\displaystyle \int$I(k)cos(2k$\displaystyle \Delta$x)dk

where I(k) is the input spectrum. Consequently it is possible to recover the input spectrum by taking the Fourier cosine transform of the recorded intensity. In practice, a discrete Fourier transform is used.

The FTS requires scanning in path spacing. But unlike the Fabry-Perot, it yields information on intensity at all wavelengths simultaneously.


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Next: Detectors Up: AY535 class notes Previous: Telescopes
Jon Holtzman 2016-05-07