Sensitivity calibration

There are so many variables in the optical path of a spectrograph that sensitivity calibration is necessarily totally empirical. A black body at a known temperature is the only feasible method of absolute calibration, and even then one must be sure that there has been a total suppression in the spectrograph of other orders and of scattered light. Reliable absolute calibration is such an onerous task that one must question the need for it in most circumstances. For almost all practical purposes a comparison with a standard source is sufficient. In the visible, a reasonable approximation – no more – can be obtained by making a spectrogram of a surface coated with freshly deposited magnesium oxide from burning magnesium ribbon which will scatter the light of the midday sun on a clear day. In the tropics and at moderate temperate latitudes, the midday sun is an approximation to a black body at 6000 K. Ordinary white card will not do as a scatterer, neither will a painted white matt surface because both are likely to be fluorescent under the UV solar radiation.

The resulting spectrum, after correction for the CCD pixel-to-pixel variations of sensitivity, may not look much like a textbook black body spectrum and for good reason. Firstly there are the basic variations with wavelength of the CCD overall sensitivity.

The earliest reference to optical spectroscopy that we have in modern times appears to be the phenomenon of colours in Isaac Newton's Opticks, in which he describes his famous experiments with prisms and the shaft of sunlight coming through the hole in his window shutter. There was much philosophical conjecture at the time but scientific silence from then on until William Hyde Wollaston (1766–1828) in 1802, also in Cambridge, used a lens to focus images of a narrow, sunlight-illuminated slit through a prism on to a screen. Wollaston appears to have observed the dark lines across the spectrum transverse to the dispersion direction but ascribed them to the divisions between the colours. He may be forgiven for this, because with a single lens the spectral resolution would have been derisory. At about the same time William Herschel (1738–1822) discovered the infra-red radiation by the rise in temperature of the bulb of a thermometer when he held it beyond the red part of the spectrum in his spectroscope. Joseph von Fraunhofer (1787–1826) saw more dark lines but did not guess or deduce their origin. The currently accepted explanation – the absorption of continuous white light by vapours in the atmosphere of the Sun – was given by Gustav Kirchhoff and Robert Bunsen in the University of Heidelberg who, we may be reasonably certain, passed a collimated beam through their prism before focusing it, and thereby secured a reasonable resolution.

First steps

Once the task for the spectrograph has been defined, a suitable type may be chosen and a catalogue search made for a possible manufactured model. Many factors affect the decision. The first of course is fitness for purpose. Time and cost of manufacture may be factors and the facilities available for local or in-house design and construction are primary considerations. A decision on whether to buy or to design and build a dedicated instrument must rest on such an appreciation.

It is a mistake when doing fundamental or academic research to construct a more elaborate, high-performance or expensive instrument than the immediate task demands, possibly in the hope or expectation that it may prove useful for some other investigation at some later date. In the author's experience this is almost never the case and the chief result is delay and unnecessary expense. In the long term the instrument is a sad relic, cannibalised of its optical components and left to decay in the attic with its ingenious mechanisms and precision micrometers.

Initial layout

Once the spectrograph type has been chosen and its main parameters such as wavelength range, resolution, type of detector, number of resolved elements, étendue etc. have been decided, a sketch can be made, and the traditional back-of-an-envelope is as good a place as any for this.

This is the branch of optics which deals with image-forming instruments, including of course spectrographs and interferometers. Such instruments employ lenses, mirrors and prisms and it is the art of combining these elements to make useful devices which is the subject of the next two chapters.

Rays and wavefronts

Image-forming instruments are intended to project images of real objects on a screen or focal surface – usually though not always plane, and instrumental optics is the study of ways of doing this.

The subject comprises two chief parts: instrument design and lens design.

The former is the design of instruments so that light is conveyed through their optical components from one to another to arrive eventually at the focal surface. The technique is essentially a graphical one, using a drawing board or a computer drafting program to lay out the components at their proper places.

The latter involves the accurate tracing of rays through various optical surfaces of different types of glass, and reflections from mirrors of various shapes to achieve a correction of the optical aberrations and to ensure that light of all wavelengths from a point on an object is focused to a corresponding point on its image.

Traditionally this was done by ray tracing, the accurate computation of ray paths using seven-figure logarithms and trigonometrical tables, but now is done chiefly by iterative ray tracing in a small computer, using a program specifically designed for that purpose.

Fore-optics

The correct use of a spectrograph requires that its aperture and its field be filled with light. More than this is not possible. Less than this is inefficient and may record the spectrum incompletely or incorrectly.

For most efficient use, the light source should be imaged on to the entry slit with a condensing lens, at a magnification which allows the whole slit length and width to be illuminated. The aperture of the condensing lens must be enough to allow the diverging light beam, after passing the slit, to fill the grating with light.

When this is done the condensing lens must itself be imaged on to the grating (not the collimator) by a relay lens somewhere near the entry slit, at a magnification sufficient to fill the grating with the image. If the condenser has been chosen properly in accordance with the first paragraph this will happen automatically. The aperture of the collimator must itself be large enough to avoid vignetting the grating, and the length of the entry slit and the distance of the collimator from the grating may well determine whether this is the case. The sort of fore-optical arrangement which meets these criteria is shown in Fig. 13.1.

However, there are perils in imaging a structured source on to the entry slit unless perhaps the source structure itself is being examined.

When parallel rays pass through a centred, image-forming optical system infinitesimally far from the axis, they arrive at the corresponding paraxial image point. Its position is computed using the Gaussian optical equations.

It is an unfortunate fact that in general parallel rays coming through the system at the outer parts of the apertures or pupils do not arrive at the same image point as the paraxial rays. They arrive in the vicinity of the paraxial image point and their separations from it define the aberrations of the system. Rays which pass the edges of the pupils are called the marginal rays and the separation of these rays from the paraxial image point can be calculated using aberration theory.

In instrument design we arrange the various components of the system so that the aberrations are reduced to zero where possible and minimised where not. The whole theory is non-linear; the various causes of aberration interact with each other and the resulting complication is often deplorable. To ease the analysis we simplify things by treating each aberration as though the others were all absent.

Optical designers are sometimes eccentric and the routes to their designs are often highly individual and hard to follow. Conrady, for example, is regarded as the father of modern optical design but his notation is not to be recommended. In any case, for plain spectrograph optics we need only a small subset of the theory and this is laid out below in reasonably familiar language.

This is to show the stages of evolution of a design for a Fabry–Perot spectrograph for night airglow or astronomical applications.

1. Astronomical considerations required that the resolution should be 1.2 Å at 5183.6 Å, the wavelength of the Mg triplet of solar Fraunhofer lines. The free spectral range was to be about 20 Å, set by the FWHM of the interference filter which was to act as order-sorter. The available finesse was 40 and this constrained the étalon gap thickness to be 28 µm. The order at the centre of the ring system is then 56 µm/0.5183.6 µm = 108.

2. The available CCD camera used a 50 mm F/1.4 ‘Takumar’ lens and a CCD with 1152 × 325 pixels each 25 µm square. The field semi-angle was then 16°. A primary requirement was for spatial resolution, so that both the sky and the Fabry–Perot fringes were to be in focus on the CCD. It was also required that the angle subtended by the CCD on the sky should not be greater than ∼5°.

3. The étalon gap thickness determined the angular radii of the Fabry–Perot rings (assuming m0 = 108 at the centre) which are given by cos θn = (m0– n)/m0 with n = 1, 2, 3 … so that cos θ1 = 7.8°. The next rings are at 11.04° and 13.53°.

4. […]

The extent of the electromagnetic spectrum is too well known to require description here. We shall be chiefly concerned with the so-called ‘optical’ region, of which the boundaries are determined partly by the methods of detection and partly by the methods of dispersing and analysing the radiation. What is common to all parts of the region is the type of optical element and materials of construction of spectroscopic instruments. Beyond the region on the long-wave side, coherent detectors, paraboloidal aerials, waveguides and dipole arrays are used, and on the short-wave (X-ray) side, optical elements other than diffracting crystals and grazing incidence reflectors are generally unknown. Radiation of 100 Å wavelength liberates photoelectrons with more than 100 eV of energy and the appropriate detection methods are those of radiography and nuclear physics.

Broadly we can identify six wavelength divisions appropriate to optical design techniques:

• 50–15 µm. The far infra-red (FIR), where bolometric, superconductor and semiconductor detectors are the chief methods of detection and measurement, and only specialised materials such as selenium, thallium bromide and various polymer resins such as sulphones have the necessary transparency to make useful refracting components. Optical elements may well be polymer plastics of high dielectric constant. Reflecting elements are coated with gold. New FIR-transparent materials with desirable optical properties are appearing all the time.

The method described here is suitable for the assembly and adjustment of a Čzerny–Turner type of spectrograph. Essentially similar methods are indicated for other types and mountings.

Alignment is best carried out systematically after the instrument has been installed in its place of work or, if it is newly made, close to the workshop which made it.

The optical alignment

The optical table has a number of holes drilled at the vertices of the mirrors and the grating. Into these holes will normally fit the vertical axle about which the angular adjustments of the optical components are made. If a precision milling machine is available, similar holes can be drilled along the optic path so that alignments can be checked at any time.

An alignment tool (Fig. 17.1) should be made, preferably but not necessarily of metal, which will stand on the optical table with a base extension which will fit these holes. It should carry a vertical disc with a 1mm hole at the height of the optic axis.

Stage 1. A laser beam, adjusted to the height of the optic axis, is made to shine through the centre of the entry slit so that it is parallel to the optical table surface. ‘Parallel’ here means that the light from the laser would be able to pass through the hole in the alignment tool no matter where the latter is placed on the table.

In spectrography, where spatial resolution is required, there is a clear subdivision in detection technology between photochemical and photoelectric methods. So far as chemical methods are concerned they possess few advantages over photoelectric methods except possibly for observations at remote sites where a reliable supply of electricity is uncertain or unavailable. They comprise the reduction of silver halide salts to colloidal silver in a solid emulsion coated on to glass or on to a transparent flexible substrate. The detective quantum efficiency (DQE) of silver halide crystals is in the region of 10–4–10–3, the dynamic range is in the region of 100:1 and as photodetectors they are severely non-linear in response, especially where long recording times – greater than 100 s – are involved. ISO ratings, usually quoted in the range of 12 for high-resolution fine-grain emulsions, to 2000 at the higher end of the range for severely coarse high-γ high-sensitivity emulsions, fall rapidly at exposures longer than 10–30 s. The effect, known as ‘reciprocity failure’, is the notion that the blackening produced depends solely on the product of intensity and time and is a necessary assumption if spectrophotometry is to be done.

This sensitivity range may be compared with a liquid-nitrogen cooled CCD detector with a DQE in the region of 0.5 and strict linearity over a dynamic range of 25 000:1, a sensitivity broadly equivalent to an ISO of 3 × 106. Liquid-nitrogen cooled CCD detectors require far more careful screening from light than silver halide emulsions.

Mechanical motions, automatic adjustments and electro-mechanical monitoring and position sensors are so diverse, and the range of transducers and electric motors for remote focusing, wavelength shifting and measuring so vast that there is no point here in listing them or criticising their relative merits. So much depends on the specification of the instrument and the capability of the available workshops that it must be left to the builder to decide how the instrument shall be controlled. The simplest possible mechanical construction is shown here – handles to turn to adjust grating turntables, capstan-bar mirror adjustments and so on, and the improvement of these by other means is clearly up to the designer and the size of his purse. This chapter therefore is confined to those matters which are peculiar to spectrograph construction.

The optical layout

The positions of the optical components will have been decided by the final version of the design coming from the ray-tracing program. The position tolerance for the mirrors is ±0.5mm on both x- and z-axes, the small errors being taken up when focusing.

It remains to dress them in appropriate mountings and assemble the mountings either on to an optical table or into a tube or a space-frame.

The art of constructing a scientific instrument is different from that of engineering mechanisms. The loadings are generally lighter and the precision as high as that of a machine tool or an internal combustion engine, and different criteria apply.