Installation of a Large Concave Diffraction Grating at the State Optical Institute in Leningrad

S. Frisch and F. Gerasimov, Leningrad

1. As is known, in the eighties of the last century Rowland first introduced concave diffraction gratings into spectroscopic practice.

The rulings of such a grating are made on the surface of a spherical concave mirror at equal distances from one another, measured along the chord. The main advantage of a concave diffraction grating as compared with a plane one is its focusing action.

If the grating and the slit illuminating it are placed on a circle whose radius is half the radius of curvature of the grating, then the diffraction spectra of various orders will be arranged along that same circle, known as the Rowland circle.

The gratings ruled by Rowland still belong among the best. They have one of the following numbers of rulings per unit length: 20,000; 14,438; and 10,000 per 1 inch, or, equivalently: 7874.1; 5684.4; and 3937.1 rulings per 1 cm. The most successful and widespread gratings have 14,438 or 10,000 rulings per 1 inch. Rowland’s gratings are ruled on mirrors made of so-called speculum metal, which is an alloy of copper (about 68%) and tin (about 32%).

The condition determining the position of a line with wavelength \(\lambda\) has the form

\[ b(\sin i + \sin \varphi)=k\lambda, \tag{1} \]

where \(i\) is the angle of incidence of the light on the grating, \(\varphi\) is the angle of diffraction, \(b\) is the grating constant, and \(k\) is the order of the spectrum. The linear dispersion of the grating \(\frac{dl}{d\lambda}\) is equal to

\[ \frac{dl}{d\lambda}=\frac{kR}{b\cdot \cos\varphi}, \tag{2} \]

where \(R\) is the radius of curvature of the grating. Thus only near the normal to the grating, where the angle \(\varphi=0\), can the dispersion be considered constant and the spectrum, consequently, normal.

Finally, let us note that the resolving power of the grating \(\dfrac{\lambda}{\delta\lambda}\) is equal to

\[ \frac{\lambda}{\delta\lambda} = kN, \tag{3} \]

where \(N\) is the total number of grooves of the grating.

The methods of mounting concave gratings are very varied: in the Abney mounting the grating and the cassette are fixed immovably, while the slit is moved along the Rowland circle. In Rowland’s own mounting the slit is fixed immovably, and the grating and cassette are placed at the ends of a movable diameter. The so-called Eagle mounting is analogous to autocollimation mountings; in order that different parts of the spectrum may fall on the cassette, the grating is rotated about a vertical axis. Finally, according to the proposal of Paschen and Runge, both the slit and the grating are fixed immovably, while the photographic plates are placed at different positions on the Rowland circle. The last two types of mountings are the most widespread, with the Eagle mounting being used chiefly for vacuum spectrographs. The Paschen–Runge mounting, although bulky, is convenient in that it is stationary and makes it possible to photograph simultaneously broad spectral regions in different orders. In general it is the most suitable for precise spectral measurements and is used in a number of foreign laboratories possessing large concave diffraction gratings.

1. The installation of a large concave diffraction grating described below was constructed and tested at the State Optical Institute by the authors of the present article. It is carried out according to the Paschen–Runge scheme, is the first and so far the only installation of this type in the Soviet Union, and is intended for various precise spectroscopic work.

The grating itself was made by Rowland at the end of the nineteenth century and is distinguished by good quality. Its data are as follows:

To make use of the full resolving power of the grating, especially for long exposures (reaching 10 hours or more), constancy of the temperature regime of the grating and absence of vibration are required. In view of this, the grating is located in a special room, without windows, on the first floor of an institute annex. A schematic plan of the room is shown in Fig. 1. Its total area is \(80\ \text{m}^2\). In the middle of the room there is, resting directly on the ground and not connected with the rest of the building, a foundation in the form of a triangle built of rubble stone. This foundation carries three massive concrete pillars, on which there freely rests a reinforced-concrete truss, whose horizontal projection is also shown in Fig. 1. This truss, distinguished by great massiveness—

ness (its weight is more than 9 t), carries the slit \(S\), the grating \(G\), and the cassettes, arranged along the Rowland circle \(aa'a''\) of radius \(r=320\) cm. Thus all parts of the apparatus are placed on one and the same reinforced-concrete frame, which reduces their vibrations relative to one another. In addition, provision is made for inserting rubber gaskets between the frame and the tables; for this purpose 6 jacks have been brought under the frame, on which it can be raised.

Fig. 1.

Fig. 2.

The light source \(Q\) is located in the niche \(A\) (Fig. 1), connected with the adjacent working room and separated from the grating room by heat-insulating walls. The light emitted by the source, projected by the lens \(L\), passes through the window \(O\) and falls on the slit of the grating \(S\).

The requirements imposed on the constancy of temperature when working with a diffraction grating are very high. A change in temperature leads to thermal expansion of the grating itself, which changes its constant \(b\), as a result of which the spectral lines are displaced. A quite insignificant change in temperature is sufficient for two barely resolved lines to merge. Indeed, differentiating relation (1) for given \(k\), \(i\), and \(\varphi\), we obtain

\[ d\lambda=\frac{db}{k}(\sin i+\sin\varphi). \]

Since the displacement of a line must be less than the distance \(\delta\lambda\) between two barely resolved lines, the change in the grating constant \(db\) must satisfy the inequality

\[ db<\frac{k\delta\lambda}{\sin i+\sin\varphi}. \]

Taking into account that, by formula (3),

\[ db=\frac{\lambda}{K\cdot N}, \]

where \(N\) is the total number of rulings of the grating, we obtain

\[ db<\frac{\lambda}{N(\sin i+\sin \varphi)} . \tag{4} \]

In the installation described, the angle of incidence \(i\) is close to \(45^\circ\). Taking the angle of diffraction \(\varphi=30^\circ\) and \(\lambda=5000\) Å, we obtain, for \(N=57000\),

\[ db<\frac{5\cdot 10^{-5}}{57000\cdot 1.2}\simeq 7\cdot 10^{-10}\ \text{cm}. \]

Since the coefficient of linear expansion \(\alpha\) for the speculum metal from which the grating is made is \(\alpha=0.000018\), and the grating constant is \(0.0025\ \text{cm}\), the temperature change must not exceed \(0.02^\circ\).

Fig. 3.

The regulation of the temperature inside the room in which the grating is installed is carried out automatically. Along the walls 8 electric furnaces are installed (\(c_1, c_2, c_3,\ldots\) in Fig. 1), connected in pairs in series and supplied with alternating current at 110 V. The current in the furnaces is 7 A, so that their total power is \(\sim 3500\) W. Inside the room a toluol thermometer is installed, shown in Fig. 2.

The tube \(AA'\), 2.5 cm in diameter and 56 cm long, contains \(440\ \text{cm}^3\) of toluol, which has a large coefficient of thermal expansion, \(a=0.00110\). Upon thermal expansion the toluol moves the mercury column \(B\), which correspondingly closes or breaks the contact between the wire \(b\) and the pin \(C\), the position of which can be regulated.

An electromagnetic relay \(R_1\) (Fig. 3) is included in the circuit \(bb'\) with an electromotive force of 20 V. This relay is located in the adjoining room and accordingly switches on or switches off the secondary

circuit \(cc'\) with an electromotive force of 12 V. A second relay is included in this circuit, consisting of an electromagnet \(M\), which, when switched on, tilts an aluminum board on which 4 mercury switches \(D\) are arranged. On a larger scale one such switch is shown in Fig. 4. The switch consists of a slightly bent glass tube with two leads \(K_1\) and \(K_2\). This tube is partly filled with mercury. The air has been pumped out of the tube, and it is filled with dry hydrogen at a pressure somewhat below atmospheric. When the tube is bent to one side, the mercury flows over and makes contact between the leads \(K_1\) and \(K_2\); when it is bent to the other side, it correspondingly breaks the contact. Each of these switches is inserted in the circuit of a pair of furnaces and operates very reliably.

Fig. 4.

In the room where the grating is located, another 4 fans have been installed under the ceiling for mixing the air; however, investigation of the thermal regime of the room shows that, apparently, one can dispense with them as well.

The indicated furnace system with the regulator ensures constancy of the temperature inside the room with an accuracy up to \(0.2^\circ\). In this case the temperature oscillations occur with a comparatively short period of 10—15 minutes about the mean position, whose constancy is maintained very well over an indefinitely long time.

In order that the grating itself preserve constancy of temperature still better, it is placed in a thick-walled cast-iron box, covered on the outside with a layer of asbestos. The box has openings for the entrance and exit of the beam of light incident on the grating. Owing to the great thermal inertia of the box, the temperature in it remains constant within approximately \(0.02—0.03^\circ\). The grating itself, also possessing a large heat capacity, preserves constancy of temperature still better.

In Fig. 5 the course of the temperature inside the box where the grating is placed is presented over the course of a working day: once in the absence of regulation (solid line), the other time with regulation in operation (dashed line). As is evident, in the second case the oscillations do not exceed \(0.05^\circ\). Measurement of the temperature inside the room and near the grating itself can be carried out with the aid of thermocouples and a mirror galvanometer located in the neighboring room, which makes it possible to monitor the constancy of the temperature also during exposures.

1. In most mountings of concave gratings according to the Paschen-Runge scheme, two steel strips are arranged along the entire Rowland circle. A photographic plate can be pressed against these strips at any point, which makes it possible to photograph simultaneously any portions of the spectrum in different orders. Since the entire installation

if the grating is carried out in a darkened room, then no additional protection of the photographic plates from light is required.

However, focusing under such conditions is difficult, and therefore for the apparatus described here a more complicated, but also more accurate, cassette system was constructed.

Fig. 5.

Along the entire Rowland circle, at distances of 30 cm from one another, along the normals to the circle, lengths of optical rail 20 cm long are arranged. On each such length of rail \(A\) (Fig. 6) there is a rider \(B\), which can be moved by screw \(C\). The position of the rider is determined by means of a scale and vernier \(a\) with an accuracy of up to 0.1 mm.

Each rider carries a platform \(D\) and a post \(E\). The riders are arranged in such a way that the front faces of all the posts exactly coincide with the actual focal line of the grating, which differs somewhat from the theoretical Rowland circle.

Fig. 6.

The cassettes are transferable. They are made of two brass strips fastened together. The rear sides of the strips are machined along a circle of radius equal to the radius of the Rowland circle (320 cm). The length of the cassette is 60 cm. Each cassette can be placed on the platforms \(D\) of any two adjacent riders and pressed by springs \(f\) against the posts \(E\). In this way it is automatically positioned along the focal line of the grating. The photographic plates are pressed against the cassette from behind by means of two special clamps. The general view of a pair of riders with a cassette placed on them is shown in Fig. 7. The distance between the edges of the cassette is 5.5 cm; this distance determines the width

of the photographic plate used. The length of the plate is practically determined by the thickness of the glass: with thick glass, a long plate

Fig. 7.

does not withstand bending in the cassette and breaks. With thin glass one can use plates 20–30 cm long.

Fig. 8.

The rails are set up, in the first approximation, by eye, by observing in a short-focus eyepiece the bright lines of the visible spectrum of mercury or iron. A more precise adjustment is achieved by photographing individual portions of the spectrum.

The grating itself is mounted on a special stand (Fig. 8). The stand permits rotation of the grating about a horizontal axis perpendicular to the plane of the grating, and about a vertical axis. Rotation about the vertical axis makes it possible, when necessary, to change the angle of incidence \(i\). The possibility of rotating the grating about the horizontal axis is necessary for setting the grooves of the grating parallel to the slit. The slit is likewise fixed on a separate stand with three adjusting screws.

The necessity of setting the rulings of the grating strictly parallel to the slit is due to the astigmatism of the concave grating. As the general theory of the concave diffraction grating shows, the image of each illuminated point of the slit is stretched in the focal plane into a line. The length of this line \(L\) is determined by the expression

\[ L=(\sin i \cdot \tg i+\sin \varphi \cdot \tg \varphi)\cdot \cos \varphi \cdot h, \tag{5} \]

where \(i\), as before, is the angle of incidence, \(\varphi\) is the angle of diffraction, and \(h\) is the height of the ruled area of the grating. For the grating described, the height \(h\) is equal to 5 cm. If the slit and the rulings of the grating lie in parallel vertical planes, but form an angle \(\alpha\) with one another, then the line produced by the grating becomes a band of width

\[ \delta l=L\cdot \sin \alpha, \tag{6} \]

where \(L\) is determined by formula (5). Obviously, \(\delta l\) must be smaller than the distance between two barely resolvable lines.

For the installation under consideration this requirement leads to the condition that the angle \(\alpha\) must not exceed \(3.5'\).

The focal line of the grating, as was indicated, differs somewhat from the theoretical Rowland circle. This becomes clearest if the grating is turned over without otherwise changing its mounting. As a result of such a reversal, at the place where, for example, the spectrum of the 1st order situated to the right of the central image had fallen, there will fall that spectrum of the 1st order which had previously been situated to the left of the central image, etc. If all spectra were located strictly on the Rowland circle, then such an “exchange” of spectra should not lead to a change in focusing. In fact, however, the focusing changes, and by so much the more the farther the regions taken are from the central image. Thus, photographs made especially for this purpose showed that the focusing for the blue part (\(\lambda = 4358\) Å) of the spectrum of the 1st order changes by 2 cm, while for the same spectral region of the 5th order it changes by 12.5 cm.

1. The grating is mounted in such a way that the angle of incidence \(i\) is equal to \(43^\circ 40'\). In this case, at the place where the Rowland circle intersects the normal to the grating \(GN\) (Fig. 9), the following spectral regions of different orders fall:

\[ \begin{array}{rcl} \lambda\ 17632\ \text{Å} & \ldots\ldots\ldots\ldots & \text{1st order} \\ \lambda\ \phantom{1}8816\ \text{Å} & \ldots\ldots\ldots\ldots & \text{2nd } \\ \lambda\ \phantom{1}5877\ \text{Å} & \ldots\ldots\ldots\ldots & \text{3rd } \\ \lambda\ \phantom{1}4408\ \text{Å} & \ldots\ldots\ldots\ldots & \text{4th } \\ \lambda\ \phantom{1}3526\ \text{Å} & \ldots\ldots\ldots\ldots & \text{5th } \end{array} \]

Thus, at this place there fall the far infrared region of the spectrum of the 1st order and the visible parts of the spectra of the 3rd and 4th orders.

The linear dispersion near the normal according to formula (2) is equal to

\[ \left(\frac{dl}{d\lambda}\right)_0 = \frac{k\cdot R}{b} = \frac{k\cdot 6400}{2.54\cdot 10^{-4}} \ \mathrm{mm}/\text{\AA} = k\cdot 0.252\ \mathrm{mm}/\text{\AA}. \]

Hence the quantity inverse to the dispersion near the normal is equal to

\[ \left(\frac{d\lambda}{dl}\right)_0 = 3.97\ \text{\AA}/\mathrm{mm}. \]

The boundaries of the visible part of the spectrum of the 1st order \((\lambda = 4000\ \text{\AA}\) and \(\lambda = 7600\ \text{\AA})\) correspond to diffraction angles \(\varphi\) equal to \(\varphi_1=-32^\circ 50'\) and \(\varphi_2=-23^\circ 10'\). The total length of the visible part of the spectrum of the 1st order is equal to 104 cm. In Fig. 10 a portion, reproduced at natural size, of a photograph with the iron spectrum in the 1st order is presented.

Fig. 9.

Since the visible part of the spectrum of the 1st order lies far from the normal, for it, according to formula (2), the dispersion is not constant and differs appreciably from the dispersion near the normal. The course of the dispersion in the spectrum of the 1st order in the region from \(\lambda = 3000\ \text{\AA}\) to \(\lambda = 8000\ \text{\AA}\) is presented in Fig. 11. The values of the dispersion determined experimentally for different spectral regions and in different orders agree well with the theoretical ones. The nonconstancy of the dispersion affects measurements. When determining wavelengths from normals one has to use three-term interpolation formulas and select the constants of the formulas separately for each interval of 100–150 Å. Under these conditions, in the 1st order the accuracy of measurements can be brought to \(\pm 0.005\)—\(0.006\ \text{\AA}\).

Fig. 10.

The grating also gives well-defined spectra of higher orders up to the 8th, which is very substantial in solving those spectroscopic problems where great resolving power is required.

power (hyperfine structure of lines, the Zeeman effect, rotational structure of molecular spectra, etc.). The instrument permits observations up to \(\lambda = 8000\ \text{Å}\) in the 4th order and up to \(\lambda = 4000\ \text{Å}\) in the 8th order.

The resolving power of the grating, by the usual definition, is equal to \(kN\), where \(k\) is the order of the spectrum and \(N\) is the total number of rulings. Thus the grating described should theoretically resolve, at \(\lambda = 5000\ \text{Å}\), two lines separated from one another by

\[ \delta\lambda = 0.089\ \text{Å} \quad \text{in the 1st order} \]

\[ \delta\lambda = 0.044\ \text{Å} \quad \text{in the 2nd ”} \]

\[ \delta\lambda = 0.030\ \text{Å} \quad \text{in the 3rd ”} \]

\[ \delta\lambda = 0.022\ \text{Å} \quad \text{in the 4th ”} \]

The linear distance between such two barely resolved lines is \(\delta l = 0.32\ \text{mm}\) for the visible part of the spectrum in the 1st order, and \(\delta l = 0.30\ \text{mm}\) for lines near the normal, where the dispersion has its smallest value. It follows from this that in photographs the linear distance between two resolved lines is very small. To use the full resolving power of the grating, one must work with a narrow slit and pay attention to precise focusing, absence of shaking, and constancy of temperature, as was stated above.

Fig. 11.

To determine the practical resolving power, objects with narrow structure were photographed in various orders. When photographing in the 1st order, in the iron spectrum one could easily observe the resolution of two lines \(\lambda = 3830.85\) and \(3830.75\), i.e. separated from one another by \(0.10\ \text{Å}\), which gives a practical resolving power of 38,000. In higher orders (up to the 6th), the blue mercury line \((\lambda = 4358\ \text{Å})\), possessing hyperfine structure, was photographed.

In Fig. 12 an 8-fold enlarged photograph of this line, produced in the 5th order, is reproduced. In addition to the central bright and somewhat blurred component, 4 sharper and weaker components are also visible on the sides. Moreover, observations were made of the Zeeman effect on the line \(\lambda = 5852\) Å. In a magnetic field this line gives a triplet close to the “normal” one. The middle component of this triplet was extinguished by a Nicol prism, and thus a doublet was obtained, the width of which depended on the applied magnetic field. By decreasing the magnetic field, it was possible to narrow this doublet to such an extent that its two components merged. In Fig. 13 microphotometric curves obtained from such a doublet in fields of strength 2840, 1916, 1500, and 0 gauss are presented. The exposures were made in the 4th order. As is seen, at a magnetic-field strength of 1500 gauss, for which the theoretical width of the doublet is equal to \(\delta\lambda = 0.050\) Å, its resolution can still be quite clearly observed.

Fig. 12.

1. Every grating actually realized in practice differs to a greater or lesser degree from the ideal type of grating with strictly periodic reflecting and non-reflecting rulings. As a consequence

Fig. 13.

of this, the amplitude of the individual reflected beams is not quite the same, and the path difference between them is likewise not quite constant. To calculate the resulting oscillations in the focal surface of a real grating, one should sum a series of oscillations with different amplitudes and nonconstant phase difference, i.e., deal with a sum of the form

\[ \sum_{n=1}^{N} A_n \cos [2\pi \nu t + n\Delta + f(n)\cdot \Delta], \]

where \(N\) is the total number of rulings of the grating, and the term \(f(n)\Delta\) may be considered as an error in the phase of the \(n\)-th beam.

Very often this error has a periodic character, connected with errors in the screw of the dividing engine. Periodic errors lead to the appearance of extra maxima—false lines, called “ghosts.” These ghosts, if their position and intensity are not known in advance, may be mistaken for real lines and become a source of misunderstandings.

A monotonic change of phase leads to additional focal properties of the grating, while aperiodic errors lead to the appearance of a continuous background. Thus, from the experimental point of view, periodic errors and the ghosts caused by them are of the greatest interest.

There are two kinds of ghosts: a) the so-called Rowland ghosts (first noted by Rowland), situated symmetrically about the real line at small distances from it; b) Lyman ghosts (first noted by Lyman), also situated symmetrically about the line, but at large distances from it. The intensity of Rowland ghosts increases with the order of the spectrum and may reach several percent of the intensity of the line itself. As theory shows, Rowland ghosts are situated at positions corresponding to wavelengths

\[ \lambda = \lambda_0 \pm \frac{n i_0}{k \cdot q}, \tag{7} \]

where \(\lambda_0\) is the wavelength of the line itself, \(q\) is the period of the error, \(k\) is the order of the spectrum, and \(n\) is an integer equal to 1, 2, 3. As the number \(n\) increases, the intensity of the ghosts rapidly decreases.

Lyman ghosts are situated at positions corresponding to wavelengths

\[ \lambda = \pm \frac{n_1}{n_2}\lambda_0, \tag{8} \]

where \(\frac{n_1}{n_2}\) is a rational fraction, for example \(4/5\) or \(6/5\), etc. The intensity of these ghosts is insignificant (usually not more than \(0.1\%\) of the intensity of the line itself), and they are complex diffraction maxima, whereas Rowland ghosts are distinguished by sharpness and therefore can especially easily be mistaken for real lines.

In the investigation of the grating described, special attention was paid to Rowland ghosts. The mercury line \(\lambda = 4358\) Å was photographed in various orders with exposures several times (up to 12) greater than normal. In Fig. 14, \(a\), a photograph taken in the 1st order is shown; the positions of the ghosts are marked by arrows, while the remaining lines are real, weaker mercury lines, whose ghosts are not visible. In Fig. 14, \(b\), a photograph of the same line in the 5th order is shown; the positions of the ghosts are again marked by arrows. As can be seen, the ghosts here are considerably more intense than in the 1st order. Measurements give good agreement with formula (7). In the 1st order the ghosts

are separated by \(\pm 8.4 n\) Å from the line, and in the 5th order by \(\pm 1.7 n\) Å, which corresponds to an error period of 515 rulings. Microphotometry showed that the intensity of the ghosts in the spectrum of the 1st order does not exceed \(0.1\%\), and in the spectrum of the 4th order, \(1\%\) of the intensity of the line itself.

Lyman ghosts, because of their low intensity, were not investigated.

Apparently, certain features of extra-focal photographs are connected with the phenomenon of ghosts. If the grating were ideal, then in photographs taken out of focus (nearer to or farther from the focus), instead of a line there should have been obtained a uniformly illuminated broadened band. In reality, extra-focal photographs reveal a series of more or less sharp bands. Thus, the grating described, at a distance from the focus of only 4 mm, gives instead of one line two fairly sharp lines of approximately equal intensity.

Fig. 14.

Fig. 15.

In Fig. 15 \(a, b, c,\) four photographs are shown of the blue mercury line \(\lambda = 4358\) Å, made in the 1st order. The first of these photographs corresponds to exact focusing; the other three were made respectively at distances of 7.5, 15, and 50 mm in front of the focus. (The photographs are enlarged 4 times relative to the original.) As can be seen, at a distance of 7.5 mm from the focus the line appears triple, and at a distance of 50 mm from the focus a blurred band is obtained, revealing clear nonuniformities in illumination.

An exhaustive theory of the indicated properties of extrafocal photographs has not yet been given. Wulff connects the appearance of individual bands with a monotonic change in the grating constant in separate regions and with the additional focal properties of individual parts of the grating that arise as a result. Brodersen and Zeising pointed out the connection of these bands with ghosts. By cutting out with a slit, placed in the focal surface of the grating, only the principal line and blocking the ghosts, they obtained uniform illumination within the band formed under these conditions beyond the focus.